Progress in Energy and Combustion Science 31 (2005) 371–421 www.elsevier.com/locate/pecs Heat transfer in ash deposits: A modelling tool-box Ana Zbogar, Flemming J. Frandsen *, Peter Arendt Jensen, Peter Glarborg CHEC Research Group, Department of Chemical Engineering, Technical University of Denmark, Building 229, Lyngby DK-2800, Denmark Received 8 March 2004; accepted 7 August 2005 Available online 11 November 2005 Abstract The objective of this paper is to review the present state-of-the-art knowledge on heat transfer to the surface of and inside ash deposits formed in solid fuel-fired utility boilers, and-based on the review-to propose models for calculation of heat transfer, e.g. in deposition models. Heat transfer will control the surface temperature of the deposit, thereby influencing the physical conditions at the deposit surface, e.g. if the surface is molten. The deposit surface conditions will affect the deposit build-up rate as well as the removal/shedding of deposits: molten deposit may lead to a more efficient particle capturing, but may also flow down the heat transfer surfaces. The heat transfer parameters of prime interest are the convective heat transfer coefficient h, the effective thermal conductivity of the deposit keff, and the surface emissivity 3 of the deposit. The convective heat transfer coefficient is a function of flow characteristics, and can be calculated using different correlation equations, while the other two parameters depend on the deposit properties, and can be calculated using different structure-based models. The thermal conductivity of porous ash deposits can be modelled using different models for packed beds. These models can be divided into two major groups, depending on the way they treat the radiation heat transfer, i.e. the unit cell models and the pseudo homogeneous models. Which model will be suitable for a particular application depends primarily on the deposit structure, i.e. whether deposit is particulate, partly sintered or completely fused. Simple calculations of heat transfer resistances for deposits have been performed, showing that major resistances are in the heat transfer to the deposit (by convection), and the heat transfer through the deposit (by conduction). Very few experimental data on the thermal conductivity of ash deposits, especially at high temperatures where radiation is important, are found in the literature. Although the structure of the deposit is essential for its thermal conductivity, most of the measurements were done on crushed samples. The results obtained using different models were compared with the experimental data published in Rezaei et al. [Rezaei, Gupta, Bryant, Hart, Liu, Bailey, et al. Thermal conductivity of coal ash and slags and models used. Fuel 2000;79:1697–1710.], measured on crushed coal ash samples. Although errors of the predictions were very high in most cases, two models were proposed as suitable for heat conductivity calculations, i.e. the Yagi and Kunii model for particulate deposits, and the Hadley model for sintered and fused deposits. This literature study showed the need for a wide range of experimental data, which would help in evaluating and improving the existing thermal conductivity models. Also, it is necessary to formulate a more accurate model for the thermal conductivity of solid mixtures, in which potentially important sources of errors can be identified. q 2005 Published by Elsevier Ltd. Keywords: Heat transfer; Radiation; Conductivity; Convection; Deposits; Combustion; Models * Corresponding author. Tel.: C45 4525 2883; fax: C45 4588 2258. E-mail address: ff@kt.dtu.dk (F.J. Frandsen). 0360-1285/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.pecs.2005.08.002 372 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Contents 1. 2. 3. 4. 5. 6. 7. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Fundamental description of an ash deposit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Heat transfer through ash deposits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Thermal properties of a deposit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Heat convection in a boiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Heat convection to the tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Heat convection to the fluid inside the tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models for the effective thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Theoretical boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Unit cell models for packed beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Additive effective thermal conductivity models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Resistance network models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Pseudohomogeneous models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Dedicated models developed for ash deposits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of thermal conductivity of ash deposits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Estimation of the general parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Structural parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Radiation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Thermal conductivity of gas and solid mixtures gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Solid mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Influence of various factors on emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Models for emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Experimental data on emissivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental validation of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal conductivity of ash deposits obtained by different models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. The thermal conductivity of solid material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Simple models for the thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Complex structure models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Radiative heat transfer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. The exchange factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2. Additive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3. Resistance network models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 374 374 374 375 377 377 378 379 379 380 381 394 397 398 399 399 399 401 403 403 404 404 405 406 408 411 411 412 412 413 413 414 414 416 420 Nomenclature cp d D h k keff kr n p R heat capacity diameter diffusivity coefficient convective heat transfer coefficient thermal conductivity effective thermal conductivity radiative thermal conductivity refractive index porosity, i.e. volume fraction of gas phase heat resistance r T u ratio between the thermal conductivities of discrete and continuous phase temperature velocity Greek letters a surface absorptivity 3 surface emissivity k ratio between the thermal conductivities of solid and gas phase m dynamic viscosity A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 r s f c density Stefan–Boltzmann constant (5.67!10K8 W/m2 K4) volume fraction radiative exchange factor Subscripts c continuous phase 1. Introduction In solid fuel combustion, non-combustible material in the fuel forms an ash fraction, which may be deposited inside the boiler. Ash deposits, formed on a furnace walls and a convective pass tubes during fuel thermal conversion, see Fig. 1, may seriously inhibit heat transfer to the working fluid and hence reduce the overall process efficiency [11–13]. Heat is transferred between a hot flue gas and the steam inside the heat transfer tubes by convection, conduction, and radiation. Convective heat transfer to the deposit surface and to the working fluid is not influenced by the deposit characteristics, but mainly by the flow characteristics of the fluids and by the geometry of the system. On the other hand, the formation of porous, low-conductivity deposit layers on the heat transfer surfaces affect the conductive and radiative heat transfer. The physical properties of a deposit, which are thus the most important for the heat transfer, are the effective thermal conductivity, keff, and the surface emissivity, 3. These properties will be discussed in details throughout this paper, both qualitatively and quantitatively. Ash deposit acts as a heat transfer resistance between the flame/hot flue gases, and the working fluid, e.g. by changing the radiation characteristics of the heat transfer surfaces, Fig. 1. Deposit build-up on superheaters after 1 week of co-firing (coal and straw), Amager Power Station, Denmark. d eff g m p rad s 373 discontinuous phase effective gas phase mixture particle radiation solid phase i.e. the colour and the surface smoothness, properties that directly influence the surface emissivity. The effective thermal conductivity and the emissivity depend on the deposit structure. As shown in Fig. 2, an ash deposit consists of several layers, having different compositions, porosities, and microstructures, and, thus, different thermal conductivities. As the result of the thermal resistance, the deposit surface temperature increases. This may lead to deposit softening and melting, and, conditions favourizing accelerated deposit formation, may be introduced. In Section 2, the basic theory on heat transfer within a utility boiler is presented, i.e. the overall heat transfer equations, correlations for convective heat transfer for different geometries, and the various deposit properties influencing the heat transfer characteristics, are outlined. The thermal conductivity is highly dependent upon the deposit structure, i.e. whether the deposit is particulate, partly sintered or completely fused. Thus, deposit structures are also explained in Section 2. In Section 3, different models for predicting the thermal conductivity within a porous structure are presented. Due to its porous structure, a deposit layer can be modelled as packed bed. Depending on the actual theoretical handling of the radiation effects, two groups of models can be distinguished, i.e. the unit cell models, which treat radiation as a local effect taking place between adjacent particle surfaces, and, void boundary surfaces in the unit cell, and pseudo homogeneous models, which considers the packed bed to be a continuum for radiation. In Section 4, we show how to estimate the different parameters, needed for the calculation of the effective thermal conductivity of an ash deposit, i.e. the thermal conductivities of the gas and the solid phase, and the emissivity. These parameters are later used to calculate the thermal conductivity of ash. In Section 5, experimental data on thermal conductivities of solids, available in the literature, are presented. Different experiments were done on the synthetic oxide-mixture powders and crushed coal 374 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Fig. 2. Layered structure of ash deposits. ashes, with only one published work dealing with the in situ measurements of thermal conductivity of ash. In Section 6, a comparison between model predictions and a set of experimental results from Rezaei et al. [1], performed on crushed coal ash samples, is presented. A discussion, considering the applicability of the mentioned models, is presented. 2. Theoretical background 2.1. Fundamental description of an ash deposit Ash deposits are porous materials, whose physical and chemical characteristics change with time, during the process of initiation, growth and maturation, due to retention of different ashy material and deposit consolidation. Deposit consolidation includes sintering, where the contact between ash particles is increased. Un-sintered deposits consist of distinct particles that appear isolated from their neighbours. When the deposit is sintered, solid phases are interconnected and forms large plate-like features near the outside surface of the deposit, where the temperature is highest. Scanning Electron Microscopy (SEM) analyses indicate that ash deposits have a layered structure, with a relatively un-sintered innermost layer. The initial layer, adjacent to the heat transfer surface, usually has a particulate structure, consisting of a continuous gas phase and discrete solid particles. Due to its high porosity, the thermal conductivity of this layer is very low, causing decreased heat transfer and increase of the deposit outer layer temperature. According to Robinson et al. [2], this layer largely determines the overall effective thermal conductivity of the deposit. On the other hand, due to higher surface temperatures, the outermost layer can be completely fused with voids embedded in a solid continuous phase. This layer has high thermal conductivity. 2.2. Heat transfer through ash deposits In utility boilers, heat needs to be transferred from hot flue gases (Tg) to the steam (Ts). As shown in Fig. 3, heat is first transferred from the flame/hot flue gas to the deposit surface (Td) by radiation and convection. Consider a composite cylindrical wall, consisting of ash deposit and a tube wall. Heat is assumed to be conducted through the deposit, and through the tube wall (Tt). Finally, heat is transferred by convection to the steam. The deposit influences the heat transfer by forming an insulating layer on the tube surface, and by changing the radiation properties of the surface. As shown in Fig. 3, the energy transferred to the deposit surface, should be equal the energy conducted through the Steam Tube Deposit Flue gas Tg Ts Q = Qconvection + Qradiation Qconduction Qreflected = Qradiation x ρ Qemitted = Qradiation x ε Fig. 3. Heat transfer to and through ash deposits. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 composite wall: Qconduction Z Qconvection C Qradiation KQreflected KQemitted (1) If heat conduction through the tube wall and steam film is neglected, and the flame emissivity is equal to unity (for large boilers), Eq. (1) can be rewritten for a cylindrical geometry: Qcond Z Td KTt Zhconv ðTg KTd ÞCa$s$Tg4 K3$s$Td4 dd dd ln 2$p$kd dto 375 dependent on the temperature and the chemical composition of the respective phases. Since the thermal conductivity of a solid phase is two to three orders of magnitude greater than the one of a gas phase, heat conduction through a deposit, will primarily occur through the solid phase [2]. The emissivity is affected by: (1) the deposit surface conditions (i.e. particulate, partly fused or completely molten), (2) the size distribution of particles in the surface layer, and (3) the chemical composition (i.e. colour of the deposit). (2) where subscripts d and t refers to deposit, respectively, to the inner tube inner diameter of the tube. A more detail discussion about heat transfer mechanisms together with an example of a heat transfer resistance calculation is provided in Appendix A. A calculation showed that heat convection from the flues gases to the deposit, and conduction through the deposit are the major thermal resistances in the overall heat transfer. Ash deposits are often characterized by an effective thermal conductivity, keff, which is used to represent the combined conductive and radiative heat transfer that may occur both in parallel and in series. The effective thermal conductivity is defined such that the following form of the heat transfer equation for one-dimensional steady-state conduction is satisfied:
 dQ d dT ZK k ðTÞ Z0 (3) dy dy eff dy An overall effective thermal conductivity, representing the entire ash deposit thickness, can be calculated by integrating the local value with the respect to the location (y) and dividing it by the deposit thickness. 2.2.1. Thermal properties of a deposit The deposit thermal properties, i.e. the thermal conductivity and the surface emissivity, are of great importance for an accurate heat transfer model. The thermal properties are strongly influenced by the deposit physical structure, i.e. the particle size distribution, the porosity and the degree of sintering. The radiative properties depend solely on the surface conditions, while the conductive properties depend on physical data throughout the deposit. The effective thermal conductivity of a two-phase gas–solid system depends on: (1) the thermal conductivity of solid phase ks, (2) the thermal conductivity of gas phase kg, (3) the porosity, (4) the size distribution of pores or particles, and (5) the deposit sintering state. The properties ks and kg are 2.2.1.1. The influence of the temperature. The thermal conductivity of deposits generally increases when the temperature increases. Fig. 4 shows experimental data on the temperature dependency of the deposits thermal conductivity, as obtained by Rezaei et al. [1]. It can be seen that thermal conductivity is slightly influenced by the temperature change up to the sintering point. In this area, the thermal conductivity changes primarily due to increase of the gas phase thermal conductivity. Authors defined the initial sintering temperature, as temperature where the deposit thermal conductivity starts to be significantly influenced by the increasing temperature. For the ash sample shown in Fig. 4, the initial sintering temperature is identified to be equal 650 8C. According to Anderson et al. [3], two effects appear at high temperatures: (1) the radiation becomes very important (for glasses at temperatures above 500 8C), causing a reversible thermal conductivity change, and, (2) sintering occurs, causing an irreversible morphology, and thereby a change in the thermal conductivity. As stated by Gupta et al. [4], sintering for coal ashes occurs in the temperature range 900–1200 8C. Fig. 4. Effect of temperature on thermal conductivity of ash (the arrows indicating heating and cooling) [1]. 376 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Fig. 4 shows that the deposit has a higher thermal conductivity after fusion. This is due to the irreversible changes in the physical structure and porosity. Experiments showed that the change of porosity in the two pellets, before and after heating, was zero for the pellet with porosity 0.48, while there was less then 5% of porosity reduction for the pellet with porosity 0.3. This, together with the data in Fig. 4, indicate that the existence of irreversibility in thermal conductivity measurements for heating and cooling cycles can be attributed to the better connections between particles, more than to a reduction in porosity. The emissivity is affected by the temperature increase due to surface structural changes, i.e. through the particle size increase due to sintering. Influence of particle size on emissivity is explained in Section 2.2.1.3. 2.2.1.2. The influence of the chemical composition. The chemical composition determines indirectly the structure of the deposit at a given temperature. The silica ratio1 is known to affect the fusion characteristics of a coal ash, i.e. the sintering temperature. According to Raask [5], iron-rich slags have been observed to have a lower sintering temperature above, which their density increases rapidly when compared to low-iron slags. Hence, iron-rich slags may have better conductivity properties. The thermal conductivity of solid oxide mixtures is approximately equal to the weighted mass portions of the oxides. This correlation is not apparent for ashes, probably because the ash is not composed of the oxides, even though an ash analysis obtained by the analysis of elements, is generally reported in terms of equivalent oxides [6]. According to Wall et al. [7], the chemical composition of a deposit affects the emissivity due to the presence of colouring agents (e.g. iron [8]), which increase the absorptivity of the deposit, and together with the particle size distribution, will influence the temperature of the onset of sintering, and the course of fusion of the deposit. approximately the same for all particle diameters of the same packing material. As the temperature increases, the difference in effective thermal conductivity increases. Since it is shown that the effect of particle size is more significant at high temperatures (in [9], above 800 K), it indicates that the particle size distribution influence the radiative properties of the deposit. The initial deposit consisting mainly of sub micronsize particles of sulfates and chlorides constitutes a low-emittance surface. Thus, even a thin initial deposit layer can significantly reduce the radiative heat transfer to boiler tubes. As Wall et al. [6] showed, below sintering temperature the emissivity of coal ash decreases when the temperature increase, and then increases sharply at higher temperatures, as sintering and fusion of the ash occur. As the deposit melts to form a slag, the absorbance increases to values approaching 0.9, as shown in Fig. 5. The particle size distribution strongly affects the emissivity of unsintered samples. With an increase in the mean particle size (diameter), the emissivity increases, because small particles reflect a significant portion of the incident radiation, while the large particles absorb most of it. Fig. 6 shows the effect of particle size, on the emissivity. Also, it is show that a marked increase in the surface emissivity results from a formation of a sintered or fused matrix. 2.2.1.3. The effect of the particle size. According to Gupta et al. [4], experiments in [9,10] showed that an increase in the mean particle size leads to an increase in the effective thermal conductivity. For the alumina–air packed beds, used by Nasr et al. [9], the effective thermal conductivity at low temperatures (350 K) is 1 Silica ratioZ[(SiO2)/(SiO2CFe2O3CCaOCMgO)]. Fig. 5. Expected trends in deposit properties during their growth, according to Wall et al. [6]. 377 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Re Z r$u$d m (5b) Pr Z m$cp k (5c) where d is a characteristic dimension. Gas properties were calculated at the film temperature, i.e. at an average between the flue gas temperature, and the deposit surface temperature. Correlations, which are used for heat transfer coefficient calculations, are different in case of heat transfer from the flue gas (heat transfer onto the cylinder) and from the tube wall to the steam (heat transfer inside the tube). Fig. 6. The effect of particle size and temperature on emittance for particulate ash and the change when a slag layer (sms) is formed on heating. The transition path from a particulate layer to slag on heating is also indicated [6]. 2.2.1.4. The effect of the porosity. A decrease in porosity leads to increase in the effective thermal conductivity of the deposit. Porosity is a function of sintering conditions, and time. According to Gupta et al. [4], even a half percent increase in porosity can decrease the effective thermal conductivity by an order of magnitude. The effective thermal conductivity varies significantly with the deposit type. The slag structure thermal conductivity is considerably higher than the particulate structure thermal conductivity, especially in the porosity range 0.2–0.8 [1]. The initial stages of sintering are accompanied by an increase in the deposit thermal conductivity. Subsequent sintering continues to densify the deposit, but has little effect on the deposit thermal conductivity. 2.3. Heat convection in a boiler As stated above, in a utility boiler, the heat is transferred partly by convection from the flame/hot flue gases to the deposit surface and from the tube wall to the stream inside the tubes. Correlations, which are used for the calculation of the heat transfer coefficient h (W/m2 K), can be as: h Z f ðNu; Re; PrÞ 2.3.1. Heat convection to the tubes First, correlations for the convective heat transfer to a cylindrical tube as shown in Fig. 7 will be presented. As stated in Incropera and DeWitt [14], calculation of the Nusselt number can be done using the empirical correlation of Hilpert, based on experimental values of the average heat transfer coefficient from a heated cylinder: Nu Z C$Rem $Pr 1=3 (6) Values for parameters C and m in the above equation are given in Table 1. Re, Nu and Pr are calculated for the characteristic dimension d, equal to the cylinder diameter, dc. All the properties are evaluated at the film temperature. An approach of Gnielinski is presented in [15], where the characteristic dimension d equals dcp/2. Correlations, which are valid over the ranges 10!Re!107 and 0.6! Pr!1000, are shown qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7) Nul;0 Z 0:3 C Nu2l;lam C Nu2l;turb where pffiffiffiffiffiffi Nul;lam Z 0:664 Re$Pr 1=3 (8a) Flue gas Tube (4) Steam where Nu is the Nusselt number, Re is the Reynolds number and Pr is the Prandtl number of the actual system. These dimensionless parameters are defined as: Nu Z h$d k (5a) Fig. 7. Heat transfer to cylinder tubes. 378 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Table 1 Constants of Eq. (6) for cylindrical cylinder in cross-flow [14] Re C M 0.4–4 4–40 40–4000 4000–40,000 40,000–400,000 0.989 0.911 0.683 0.193 0.0266 0.330 0.385 0.466 0.618 0.805 fA Z 1 C 0:7 ðb=aK0:3Þ j1:5 ðb=a C 0:7Þ2 (11) where Transverse pitch ratio: aZ SQ D (12a) Longitudinal pitch ratio 0:8 Nul;turb Z 0:037$Re $Pr 1 C 2:443$ReK0:1 ðPr 2=3 K1Þ (8b) Whitaker [16] recommends the following correlation for the mean Nusselt number:  1=4 1=2 2=3 0:4 mN Nu Z ð0:4$Re C 0:06$Re ÞPr (9) ms in the range 1!Re!105, 0.67!Pr!300, and 0.25! mN/m0!5.2. Subscript N stands for the viscosity at the bulk gas temperature, and s for the viscosity at the surface temperature. The values of the viscosity and the thermal conductivity in Re and Pr are estimated at those at the approaching stream temperature. In the case of a tube bundle, heat transfer coefficient can be calculated as follows: (10) Nul0;bundle Z fA $Nul;0 where fA is a function of the tube bundle spatial geometry. In the case of an in-line arrangement, as shown in Fig. 8, the parameter fA may be calculated from: SQ bZ SL D (12b) with parameters shown in Fig. 9. Void ratio: j Z 1K p ; bR 1 4a (12c) j Z 1K p ; b! 1 4ab (12d) Nul,o is calculated using Eq. (7), with RejZ(r.ud)/ jm. In the case of a staggered arrangement, as shown in Fig. 9, the parameter fA may be calculated from: fA Z 1 C 2 3$b (13) According to [14], each correlation is reasonable over a certain range of conditions, but for the most engineering calculations one should not expect accuracy to be much better than 25%. 2.3.2. Heat convection to the fluid inside the tubes If we assume turbulent flow of steam inside the heat transfer tubes (ReO10,000), an average value of the heat transfer coefficient can be calculated using SL SL D SQ ugas ugas Fig. 8. In-line arrangement in a tube bundle. Fig. 9. Staggered arrangement in a tube bundle. 379 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Table 2 Values of 3l parameter Re 4 10 2!104 5!104 105 106 l/d 1 2 5 10 15 20 30 40 50 1.65 1.51 1.34 1.28 1.14 1.50 1.40 1.27 1.22 1.11 1.34 1.27 1.18 1.15 1.08 1.23 1.18 1.13 1.10 1.05 1.17 1.13 1.10 1.08 1.04 1.13 1.10 1.08 1.06 1.03 1.07 1.05 1.04 1.03 1.02 1.03 1.02 1.02 1.02 1.01 1 1 1 1 1 Miheiev correlation [17] Nu Z 0:021$Re 0:80 $Pr 0:43 3. Models for the effective thermal conductivity  Pr Prw 0:25 (14) where Prw is the value of the Prandtl number at the tube wall temperature. Dimensionless numbers are calculated for the cylinder inner diameter. This equation can be used for ReO104 and PrZ0.7–2500, and ratio tube length to tube diameter ratio l/dO50. If l/d!50, then calculated heat transfer coefficient should be multiplied by 3l, which values are given in Table 2. Since the temperature at the tube inner diameter and steam can be considered very close, the ratio of viscosities or Prandtl numbers can be neglected (set equal to 1). One such correlation is given by DittusBoelter [14]: From a modelling point-of-view, ash deposits are porous structures that can be approximated as packed beds. Thus, the thermal conductivity of deposits may be estimated using well-documented models for the thermal conductivity of packed beds. Un-sintered (particulate) deposits should be considered as systems with continuous gas phase and discontinuous solid phase. Fused deposits (slags) should be considered as systems with discontinuous gas phase (voids), and continuous solid phase. Sintered deposits can be treated as complex systems, where both phases are continuous. Thermal conductivity models can be divided into two main groups depending upon the handling of radiative effects in the system: unit cell models and pseudo homogeneous models. 3.1. Theoretical boundaries 4=5 Nu Z 0:023$Re $Pr 0:4 (15) This equation has been confirmed experimentally for the range of conditions: 0.7!Pr!160, ReO10000, l/dO10. This expression should be used only for small to moderate temperature differences between fluid and wall [14], with all the properties evaluated at the bulk fluid temperature. Theoretically, the most general approach to calculate the upper and the lower boundaries value of the thermal conductivity correspond to the cases schematically shown in Fig. 10. Assuming that the gas and the solid phase conduct heat independently, these boundaries can be formulated as: (1) The upper boundary value, where heat is transferred in a direction parallel to the layers as shown gas flow Maximum K Minimum K Rayleigh CPS CSP Fig. 10. Models for thermal conductivity estimations (maximum and minimum k—for theoretical boundaries, Eqs. (16a), (16b), (17a) and (17b)), according to [4]. 380 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 in Fig. 10a. Here, the heat resistance is calculated as a parallel connection of thermal resistances: keff Z ks C (16a) Since the thermal resistance is equal RZ1/ (thermal conductivity, k), the maximum effective thermal conductivity can be calculated from: keff;max k Z p C ð1KpÞ s kg kg (16b) (2) The lower boundary value, where heat is transferred in a direction normal to the layers as shown in Fig. 10b. Here, the heat transfer resistance is calculated as a serial connection of thermal resistances: Rtot Z p$Rg C ð1KpÞRs (17a) As in the previous case, the minimum effective thermal conductivity is calculated as: keff;min 1  Z p kg p C ðk1K s =kg Þ (17b) These are not rigorous theoretical bounds, but are made in intention to provide a rational basis for comparison. For example, radiation can result in an effective thermal conductivity greater than the upper limit provided in Eq. (16a) and (16b) and deposits with small sized pores could have thermal conductivity less than the lower limit in Eq. (17a) and(17b). Measurements presented in Robinson et al. [2], showed that the overall trend in the thermal conductivity of particulate deposits follows the lower theoretical limit. This is most probably due to the weak solid–solid contact of deposited particles, which is a limiting factor, while the deposit structure is particulate. According to Bird et al. [16], the first major contribution to the estimation of the conductivity of heterogeneous solids was made by Maxwell (1873), who’s equation represent a material consisting of noninteracting spheres suspended in a fluid matrix. As stated in Nimick and Leith [18], these equations were shown to be the most stringent upper and lower boundaries for homogeneous, isotropic, two-phase mixtures. The upper-bond equation, describing low- 1 kgKks p p C 1K 3ks (18) while, the lower-bond equation, where solid spheres are suspended in a fluid matrix is: keff Z kg C 1Kp C 3kpg (19) 1 ksKkg These equations can be derived using the steadystate temperature distribution in a sphere (T1) and in the surrounding medium (T0). The volume fraction p of embedded gas spheres (porosity), is taken to be sufficiently small that the spheres do not ‘interact’ thermally; that is, one needs to consider only the thermal condition in a large medium containing only one embedded sphere. Due to the non-interacting spheres assumption, these equations are limited to porosities close to 0, i.e. Eq. (18), or 1, i.e. Eq. (19). This is why the model cannot be used for estimating the thermal conductivity of a granular porous media, but only the limiting cases. These two approaches to the theoretical boundary limits are shown in Fig. 11, where the effective thermal conductivity is plotted versus porosity. 3.2. Unit cell models for packed beds The unit cell models treat radiation as a local effect taking place between adjacent particle surfaces, and void boundary surfaces in the unit cell, i.e. long range effects of radiation are neglected. Vortmeyer [19] lists five assumptions under which these models are developed: Effective thermal conductivity, kef 1 p 1Kp Z C Rtot Rg Rs conductivity (fluid) spheres in a high-conductivity (solid) matrix, is 1 0.8 0.6 0.4 0.2 0 0 0.2 equ. 16b 0.4 0.6 Porosity equ. 17b equ. 18 0.8 1 equ. 19 Fig. 11. Theoretical boundary limits for the effective thermal conductivity. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 † † † † † 381 Particle diameter[wavelength Grey emitting surfaces Opaque material Absence of free convection DT/T/1 across a particle layer Unit cell models treat effective thermal conductivity in two ways: (1) By adding the radiative conductivity to the conduction component, i.e. additive models: keff Z kgKs C krad (20) (2) By including the radiative conductivity as a resistance in the network of resistances representing the unit cell, i.e. resistance network models. 3.2.1. Additive effective thermal conductivity models 3.2.1.1. Simple models. The simple models are restricted to low temperatures, since they neglect radiative heat transfer through the packed bed. According to [3,4], radiation will affect the thermal conductivity in the temperature range: 900–1250 8C. Neglecting radiative transfer is reasonable if it can be assumed that the conductive component is independent of the radiative component. The models have a simplified approach to the system structure, and thus can be applied only when one phase is continuous and the other is discontinuous, i.e. for particulate or fused deposits. Using these models, a heat conductivity component, kgKs, can be obtained. The usual assumption is that temperature isotherms are normal to the direction of heat transfer, i.e. due to an infinite lateral conductivity assumption the temperature gradient perpendicular to the heat flow does not exist. The larger the ratio ks/kg, the greater the errors introduced by this assumption, which is only totally valid when ksZkg [20], will be. Lord Rayleigh [4,10] derived an equation for estimating the thermal conductivity of a two-phase system, schematically shown in Fig. 10c. The model system consists of a cubic array of uniform spheres (discrete phase), being embedded in a continuous phase: keff ð2 C rÞ=ð1KrÞK2p Z ð2 C rÞ=ð1KrÞ C p kc Fig. 12. Comparison between experimental data and estimations based on the Rayleigh model for the thermal conductivity of sintered ash, at a porosity of 0.3 [1]. spheres, the Rayleigh model is limited to a discrete phase volume fraction of up to p/6, which is explained in Appendix B. Predictions based on the Rayleigh model has been compared to experimental results presented by Rezaei et al. [1]. Fig. 12 indicates that the thermal conductivities of unsintered ash samples, during the heating cycle, are closer to the thermal conductivity of particulate structure. The thermal conductivity of samples during the cooling cycle is closer to predictions for the slag type structure, due to partial sintering. Fig. 13 shows that experimental results for sintered ash samples are closer to predictions for fused material. Although an exact mathematical calculation is being provided, this model is rigid and artificial, permitting the spheres to be located only in the center of their unit cubical species, which radically departs from the powders encountered in practice. Russel [4] derived a thermal conductivity model, schematically shown in Fig. 14, for an array of cubes in a cubical structure: (21) where p refers to the volume fraction of the discrete phase. Like most of the other models that consider Fig. 13. Comparison between the experimental data and estimations based on the Rayleigh model for thermal conductivity of unsintered ash samples, at a porosity of 0.3 [1]. 382 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Fig. 14. Structure used in the Russel model [4]. keff rp2=3 C ð1Kp2=3 Þ Z 2=3 kc rðp KpÞ C ð1Kp2=3 C pÞ (22) where p refers to the volume fraction of the discrete phase. When r is large, Eq. (22) can be simplified to: keff 1 Z kc 1Kp1=3 (23) The Russell model can be applied over the complete range of porosity. Since the mathematical treatment is unrigorous, the model permits an arbitrary, irregular location of obstacles in their unit cubical species, and hence is more realistic than the Rayleigh model [10]. On the other hand, pores of cubical shape are not likely to be found within ash deposits. Woodside [20] considers a cubical array of noninteracting spherical particles in a continuous medium, shown in Fig. 15. Snow, with the density range 0.10– 0.48 g/cm3, was used in the experiments. The thermal resistance of the model cube against the heat conduction, in the direction shown in Fig. 15, equals the thermal resistivity of the composite material. The total thermal resistance in the cube is: ðR 1 1KR dx Z C 2 k kg ks $p$r =4 C kg ð1Kp$r 2 =4Þ (24) 0 The first term in Eq. (24) represents thermal resistance of the shaded gas layer, and the second term is the thermal resistance of the composite gas– solid layer of thickness dx. Since r2ZR2Kx2, the resulting equation, derived by integration of Eq. (24), is  1=3
 2      kg 6S a K1 a C1 1K ln Z 1K p 2a aK1 keff (25) where Fig. 15. Representative unit cell of a material consisting of uniform solid spheres, distributed in a cubic lattice in a gas, used for estimation of the thermal conductivity by Woodside [20]. " 4 a Z 1C pðks =kg K1Þð6S=pÞ2=3 S Z 1Kp Z p$R3 =6 Z #1=2 rKrg ; rS Krg (26a) (26b) 0% S% p=6 ðsimple cubic cellÞ If the system density approaches the solid phase density, then S approaches 1, and k approaches the solid thermal conductivity. On the other hand, if the system density approaches the gas density, then S approaches 0, and k approaches the gas thermal conductivity. If the subscripts s and g are interchanged in Eqs. (25), (26a) and (26b), Eq. (25) can be rearranged to become the equation for the conductivity of a material consisting of uniform spherical pores distributed in a cubic lattice solid. Krupiczka [21] developed a model for calculating the effective thermal conductivity of granular material of spherical and cylindrical grains embedded in a continuous fluid phase, schematically shown in Fig. 16. The simplifying assumption is that the total heat flux through the unit cell consists of the independent heat fluxes, Q1 and Q2. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 B ZK0:057 Q2 φ α log k Z 0:48$log dp K1:75 Fig. 16. Cubic element of the model of spheres under consideration [21]. The quantity of heat conducted only by the gas is:  p Q1 Z kg $A$DT Z kg 1K (27) 4 where A is the area normal to heat transfer direction, when a cubic cell has unity diameter, DT is the temperature gradient. The quantity of heat that proceeds from the gas through surface of the sphere to the solid is ð   vt2 Q 2 Z kg dF (28) vr rZ1 n;rZ1 F dFn;rZ1 Z ðr cos 4Þr d4 da Z cos 4$d4$da (31b) According to the experiments, Eq. (19) can predict the effective thermal conductivity of particulate beds (powders in different gases) with an accuracy of G 30%. Boow and Goard [5,22] investigated the thermal conductivity and the total emittance of coal ash deposits and synthetic slags. Their experiments showed that these properties are influenced by the particle size and the chemical composition of the material. Empirical correlations for the thermal conductivity of ashes, in the particle size range of 20–300 mm and at TZ700 8C, were obtained: for colorless synthetic slag: Q1 ρ 383 (29) (32) for iron-containing slag: log k Z 0:56$log dp K1:63 (33) where k [Z] W/m K, and dp [Z] mm. The underlying experimental results are provided in Fig. 17. Applying laboratory-prepared ashes, Boow and Goard obtained exceptionally low values of the thermal conductivity, from 0.02 to 0.06 W/m K, in the temperature range of 500–1000 K (less than for air). In order to reduce the thermal conductivity of a porous body below that of air or boiler flue gas, the pore size must be less than 0.5 mm. Leach [23] developed models for foam like materials (cellular material, plastic foams, other lightweight materials), where air is considered to be the discrete phase. Model unit cells are shown in Fig. 10d and e. where dFn is the area of the sphere slice shown in Fig. 16, and t2 is the temperature distribution with respect to the y-axis, in the external zone. When Eqs. (27) and (28) are combined, the effective thermal conductivity of the system can be estimated. Since the expression for the temperature distribution is quite complicated, an attempt was made to approximate the final equation for thermal conductivity by a simple function. The final model equation is given as:  ACB logðks=kgÞ keff k Z s (30) kg kg where A and B are parameters derived based on experiments. For the model based on spheres, the parameters A and B can be estimated from: A Z 0:280K0757$log p (31a) Fig. 17. The influence of mean particle size on the thermal conductivity at 975 K. , furnace deposits, ! precipitator and cyclone dusts, C laboratory ashes, 6 primary superheater, C furnace floor deposits (pfc) and cyclone dust, $, :, & synthetic slag [22]. 384 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 The Cubic Parallel Series (CPS) model considers a composite material to consist of vertical side cell walls and the air inside the cell. This composite material is put in series with the horizontal solid layers. The model is analog to the Russell model, with gas as the discrete phase. It predicts a higher thermal conductivity than other models. The Cubic Series Parallel (CSP) model considers horizontal cell walls and the air inside the cell as a composite material. This composite material is put in parallel with the vertical cell walls. This model is equivalent to the model of Diesser and Eian. The main difference between CSP and CPS models is the manner in which heat flow at the cell corner is treated: CSP assumes zero horizontal conductivity and CPS assumes infinite horizontal conductivity. The model equations are: CSP yCSP C 1 Z M CPS yCPS C 1 Z f2KM C ðMK1Þ½x C ð1KxÞ1=3 g 1 C ðMK1Þð1KxÞ1=3 (34) MKðM K1Þð1KxÞ2=3 1Kð1K1=MÞ½ð1KxÞ2=3 Kð1KxÞ (35) k Z y C1 kg (36) ks ZM kg (37) x Z 1Kp (38) These models are valid for fused deposits; for particulate deposits discrete and continuous phase have to be interchanged. For low density foams or slags (x/1 and M[1), CPS, CSP, and spherical models, can be reduced to 1 1 y z ð2MK1Þx C q M$x2 3 9 (39) where qZ1, 2, 3 (CSP, spherical models, CPS). Rezaei et al. [1] showed (Fig. 18) that all these models (CPS, CSP, Rayleigh) lie within the two extreme limits for a given ratio of the thermal conductivity of gas to the thermal conductivity of solid. The Rayleigh model (spheres) gives values that are between the CPS and CSP models (cubes). 3.2.1.2. Thermal conductivity of complex structures. In models considering complex structures, both fluid and gas phase are continuous, as is in reality the case with a Fig. 18. Three simple models for thermal conductivity of particulate (P) and slag (S) type structures (radiation neglected) [1]. sintered deposit. Radiation is not discussed here, so the influence of the particle size will be neglected. Parameters, which would be required for a complete analysis, are the fraction of each continuous phase, and the porosity of each phase. But with the present knowledge it is not possible to provide such information, so simplified models will be used. Brailsford and Major [24] derived expressions for the thermal conductivity of two-phase media for various types of structures, based on the Maxwell theory. The unit cell, which is shown in Fig. 19, consists of particles, i.e. phase 1, surrounded by phase 2, which is in turn surrounded by a material having an average conductivity equal to that which we wish to calculate. The temperature distribution in each region satisfies the Laplace equation: V2 T Z 0 (40) The general solution in each of the regions, in spherical coordinates, with the origin at the centre of the suspended particle, is 385 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 equation is: 2 n 0 0 k0 p0 C k1 p1 ð2k3k C k2 p2 ð2k3k 0Ck1 Þ 0Ck2 Þ o kZ n 0 0 p0 C p1 ð2k3k C p2 ð2k3k 0Ck1 Þ 0Ck2 Þ 1 r ? Fig. 19. Unit cell for Brailsford and Major model.  vT vz  d cos q r2 (41a)  c r1 ! r! r2 T Z br C 2 cos q r (41b) r! r1 T Z ar cos q (41c) rO r2 T Z zC ex o (45) Brailsford and Major also developed a model for complex systems, which can be interpreted as a random two phase assembly, containing regions of a gascontinuous-phase, and a solid-continuous-phase embedded in a random mixture of the same two phases. They regard both phases as continuous and assume the porosities in these two phases to be similar. The material used in the experiments was sandstone. Using Eq. (45) with k0Zk the following equation was where zZr cos q, and a, b, c, d are constants. Conditions of a uniform external temperature gradient requires that dZ0. This gives the equation n    o r3 r3 2 1K r13 C kk12 1K2 r13 k 2   2 o (42) Z n r13 r13 k1 k2 1K 3 2C 3 C r2 k2 r2 where r1 is the radius of spherical particles of phase 1, surrounded by phase 2 out to radius r2. If phase 1 (gas) is the continuous phase, then n  o k2 k2 k k1 3 k1 C 2$p1 1K k1 n  o Z (43) k2 k2 3Kp1 1K kk21 where p1 is the volume fraction of phase 1 ðp1 Z r13 =r23 Þ. This equation is presented in Fig. 20, as curve 4. In case phase 2 (solid) is the continuous phase, then:   1Kk1 =k2 1K2p 1 2Ck1 =k2 k  Z (44) k2 1 C p 1Kk1 =k2 1 2Ck1 =k2 This equation is equivalent to Maxwell’s result, and is illustrated in Fig. 20 as curve 3. For three (0 (continuous), 1, 2) or more phase systems, the model Fig. 20. (1) Minimum conductivity, Eq. (17a) and (17b); (2) maximum conductivity, Eq. (16a) and (16b); (3) phase 2 continuous, Eq. (43); (4) phase 1 continuous, Eq. (42); (5) random mixture, Eq. (45), [24]. 386 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 obtained: k Z fð3p1 K1Þk1 C ð3p2 K1Þk2 C ½fð3p1 K1Þk1 C ð3p2 K1Þk2 g2 C 8k1 k2 1=2 g=4 (46) where k1, k2 are thermal conductivities of fluidcontinuous phase and gas-continuous phase and p2Z 1Kp1. Eq. (46) is shown in Fig. 20, as curve 5. If it is assumed that sandstone can be subdivided into small regions, each of which may be regarded as a fluidcontinuous or solid-continuous region, the appropriate conductivities within these regions are given in (43) or (44). The resultant, overall thermal conductivity can be obtained by using Eq. (46). The volume fraction of the fluid-continuous phase pf is related to the volume fraction p of fluid phase through a simple oneparameter equation: pf Z pfa C ð1KaÞpg (47) where a is a factor relating to the structure of the deposit, Eq. (47) reflects the physically reasonable assumption that the phase having the major portion of continuous phase is likely to predominate; thus pf/1 when p/1, and pf/0 when p/0. The result of fitting experimental data for sandstone, shown in Fig. 21, gives the value aZ0.8. Rezaei et al. [1] compared results obtained using the Brailsford and Major model, with the measurements of sintered ash samples. According to Figs. 22 and 23, the agreement is satisfactory. The model describing a random distribution of mixed phases, gives results, which are within 20% of the measured values for a partially sintered sample. Nozad et al. [25] used the method of volume averaging for estimating the thermal conductivity of two- and three-phase systems. The volume averaging method provides a tool for establishing macroscopic governing equations from microscopic equations and boundary conditions. It considers the volume to be Fig. 22. Comparison between the experiments and prediction results based on mixed structure (Brailsford and Major) model for thermal conductivity of sintered ash samples [1]. Fig. 21. Comparison of the experimental thermal conductivities of dry sandstone at 0 8C with the theoretical predictions assuming either phase continuous or a random distribution [24]. Fig. 23. Comparison between the experiments and prediction results based on mixed structure (Brailsford and Major) model for thermal conductivity of sintered ash samples [1]. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 huge, compared to the small-scale deviations in volume fraction of each component, but small enough so that the temperature, vary only slightly over the volume. The experimental measurements included three fluids (air, glycerol, and water) and five solids (glass, stainless steel, bronze, urea-formaldehyde, and aluminium) with particle diameters ranging from 2.5 to 4.0 mm. The measurements were made on a fixed bed of unconsolidated spherical particles, at average temperature of 45 8C. In order to illustrate this method, the governing differential equation and boundary conditions for transient heat conduction, in the system, are given in the form ðrcp Þb vTb Z Vðkb VTb Þ; in Vb vt (48) where subscripts b, s represents the continuous, respectively, discontinuous phases, Tb, Ts are the local temperatures in the b- and s-phases, respectively, Abs is the interfacial area between two phases, Abe, Ase are the bounding surfaces of the entrances and exits for phases, and nbs is the unit outwardly directed normal for the b-phase over the area Abs(nbsZKnsb). B:C:1 Tb Z Ts ; at Abs (49a) B:C:2 nbs $kb VTb Z nbs $ks VTs ; at Abs (49b) ðrcp Þs vTs Z Vðks VTs Þ; at Vs vt (49c) B:C:3a Tb Z g1 ðtÞ; at Abe (49d) B:C:3b Ts Z g2 ðtÞ; at Ase (49e) resistance analogy for cubical particles and square contacts keff 1  (51) Z rð1Ka2 Þ C  1 ks C 2 1Ka 2 a$r r$a Cð1KrÞc p Z a3 C 3$c2 ð1KaÞ Abs ð1KrÞ rV ð 1 ðn f 2 bs 1 Abs C f1 nbs ÞdA (50) where keff is the effective thermal conductivity tensor (kcal/ms K), ps, pb are the volume fraction of phases s and b, respectively, I is the unit tensor, V is the averaging volume (m3), f is the vector that relates PhTi to Tb (m) and is represented like f Z f0 C p$f1 C p2 f2 C/. Finally, a model in which both phases are considered to be continuous is derived using the electrical (52) where r is the ratio of the thermal conductivity of the discontinuous phase to the thermal conductivity of the continuous phase and a and c are the ratios of particle size, and neck size to the unit cell size as given in Fig. 24. Such structure may form when the particulate phase starts to sinter, establishing contacts between the particles, thereby resulting in a solid phase continuous. At the same time, the continuity of the fluid phase is not destroyed. Experimental validation of the model is illustrated in Fig. 25, where the experimental point having the highest value of r represents the author’s result for the air-aluminum system. Hadley [26] developed a theoretical model for predicting the thermal conductivity of consolidate, based on the volume averaging method. Consolidate mixture of two metal powders were used in the experiments: brass powder (highly angular particles) and stainless steel powder (approximately spherical particles). Consolidation of the particles was done by cold pressing. A multiphase mixture containing n phases was considered, and an averaging volume of arbitrary shape The solution to the model is ð keff ð1KrÞ 1 ðn f Z ðpb C ps rÞI C V 2 bs 0 kb C f0 nbs ÞdA C 387 Fig. 24. Model for particle–particle contact [25]. 388 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 constant for a continuous solid, and, kZks/kg. For granular systems, a is small and the first term in Eq. (57) is dominant for moderate values of k (for pZ0.4, the first term will be dominant for kz100). For consolidated materials, a is moderate (for porosities less than 0.1) and the second term may be dominant over most of the range of k. The values for f o and a are experimentally determined. If vacuum is used as a pore fluid, then k/N. If this is used in Eq. (57), the thermal conductivity at the low gas pressure (close to vacuum) can be predicted as: Fig. 25. Comparison between the theoretical predictions for the effective thermal conductivity, obtained using the Nozad model, and different experimental measurements [25]. was constructed. Spatial derivates of volume-averaged quantities may be defined as the change in that quantity per unit distance movement of the averaging center (point which represent averaging volume), with the shape and orientation of the averaging volume fixed. When volume averaging the temperature, respectively, the heat flux, we obtain: hTi Z n X hFi Z n X iZ1 iZ1 hTi i (53) n X k¼i hVTi i hFii ZK k 1Kp Z 2$a$ ks 2 Cp (58) Thus, a measurement of the conductivity, made with pore pressures low enough to eliminate the influence of the pore fluid, will provide a value for a. Fig. 26 shows values of a obtained from Eq. (58) plotted vs. d (percent theoretical density), i.e. dZ1Kporosity. The figure shows that a depends primarily upon the material porosity, and that it is approximately independent of the other parameters, such as a particle shape or the brass volume fraction (due to its angular fraction, brass will (54) iZ1 After introducing intrinsic averages in the above equation, the final working equations are: VhTi Z n X iZ1 ¼ kVhTi Z pi hVTi ii n X iZ1 pi k¼i hVTi ii (55) (56) Combination of the equations for conduction through a continuous media (the Maxwell upper formula, Eq. (18)), with the equation for suspension of particles (obtained from reformulating Eqs. (55) and (56)), gives a model for the thermal conductivity of a two-phase system k p$f0 C ð1Kp$f0 Þk Z ð1KaÞ kf 1Kpð1Kf0 Þ C pð1Kf0 Þk Ca 2ð1KpÞk2 C ð1 C 2$pÞk ð2 C pÞk C 1Kp (57) where a is a mixing parameter (function of degree of consolidation), f0 is a parameter which is approximately Fig. 26. Values of consolidation parameter a determined from experimental measurements of evacuated samples. The curve represents an approximate fit [26]. PTD (d): percent theoretical density (dZ1Kp). A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 389 cause the more efficient packing). This dependence can be presented as: p Z 0:3K0:572 a Z 1:5266ð1KpÞ8:7381 (59) p Z 0:0975K0:3 a Z 0:7079ð1KpÞ6:3051 (60) In general, a is the most sensitive parameter in this theory, as it scales the matrix conduction component. The parameter f0 should be determined from measurements using a higher conductivity pore fluid such as water (in that way the second term in Eq. (42) is reduced, so that maximum sensitivity to f0 is obtained). Even though it is not understood why the angular brass particles should require a higher value of fo than the more spherical steel particles, a linear formula for determination of fo was obtained: fo Z 0:8 C 0:1pb (61) where pb is a volume fraction of brass. The value of f0 is between 0.8 for the stainless steel, and, 0.9, for brass samples. In order to compare predictions by the Hadley model with previous two-phase measurements, the author used the large number of two-phase experiments compiled and catalogued by Crane and Vachon. The parallel lines in Fig. 27 represent 20% deviation from unity. The comparison with the Crane and Vachon list gives an average error of 22%. Nozard et al. [25] compared Hadley’s model to their own numerical calculations, for a two-phase granular material, which is shown in Fig. 28. Since the parameter a, defined in the work of Hadley, represents the fraction of heat conducted through matrix contact only, the expected relationship between a and the Fig. 28. Comparison of the Hadley model with the numerical calculations of Nozard et al., and previous two-phase heat conductivity data [26]. relative ‘bridge’ area fraction, defined by Nozad, is evident. Nimick and Leith [18] developed a semi-empirical model for estimating the effective thermal conductivity of a granular porous media, in which convection and radiation are neglected. The authors assumed that granular porous medium comprises regions of solidcontinuous material and fluid-continuous material. A porous medium composed of fused silica powder and air was used in the experiments. Using Eq. (18), with the substitutions: pZxfc, ksZksc, kgZkgc, the authors obtained the following equations
 3$xfc ð1KAÞ keff Z ksc 1K (62) 2 C A C xfc ð1KAÞ where ksc Z ks C kfc Z kf C AZ 1 kfKks psc psc C 1K 3$ks 1Kpfc pfc C 3$k f 1 ksKkf kfc ksc (63a) (63b) (63c) psc Z pKpm ðmO 1Þ pfc Z pn ð0! n! 1Þ Fig. 27. Thermal conductivities determined from the Hadley model vs. measured values for a large sampling of previously published twoand three-phase data [26]. (63d, e) xfc C xsc Z 1 (63f) p Z psc ð1Kxfc Þ C pfc xfc (63g) where p is the bulk porosity in the packed bed, psc is the solid continuous region (psc(p), pfc is the fluid continuous region (pfcOp), and xfc is the volume fraction of the fluid-continuous region. In order to 390 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 estimate the parameters in Eq. (63a,b), i.e. the thermal conductivity of the solid-continuous phase, and the gascontinuous phase, Eqs. (18) and (19) should be used, respectively, with the appropriate use of porosity. Parameters m and n, which are the greatest drawback to the use of this model, can be determined using leastsquare curve-fitting techniques (in this case: mZ2.127, nZ0.929, with a correlation coefficient 0.999). No assumptions have been made about geometric characteristics, so the values of m and n derived for fused silica should be applicable for any powder that is approximately similar to the fused silica powder (silicarich glasses, anhydrous silicate minerals). Expected values for granular media, with a large kf/ks-ratio: m2[1.9; 2.3], and n2[0.88; 0.98]. Fig. 29 shows the data obtained with fused-silica, together with the boundaries given by Eqs. (18) and (19), and the predictions obtained using Eq. (62), with mZ2.127 and nZ0.929. The semi-empirical model defined by Eqs. (62) and (63) gives an excellent representation for the fused silica. Fig. 30 shows that Hadley’s model underpredicts the Nimick and Leith least-square-based fit of fused silica-air measurements by 3–8%. 3.2.1.3. Additive models that include radiation effects. Models, which consider radiation as a mechanism contributing to the overall heat transfer, will be discussed below. The effective thermal conductivity is affected by radiation in two ways: 1. Radiation across solid transparent medium 2. Radiation across voids (important here) Fig. 29. Comparison of model calculations based on Eq. (62) with experimental data for fused silica powder (Nimick, 1990). Fig. 30. Comparison of the prediction of Hadley [26] with experimental data, bounding estimates and Eq. (62) [18]. The contribution of the first mechanism (photon conductivity) is usually implemented into the thermal conductivity of the solid medium with zero porosity, and can be determined by solving radiation and conduction simultaneously. The contribution by the second mechanism, which is important in this case, is dependent on the porosity and the pore size distribution in the case of slags, or particle size distribution in the case of particulate materials. Laubitz [10] measured the effective thermal conductivity of several powders (MgO, Al2O3, ZrO2) and developed the model equivalent to Eq. (20). Powders with particle sizes dpZ0.011, 0.040, 0.140 cm were used (several powders of uniform dp, and one of graded particle size), in the temperature range TZ100– 1000 (8C). The Laubitz model is based on the onedimensional radiant heat transfer through a void, containing solid cubical obstacles. In order to include radiation in the models outlined previously, some structural changes have to be made. To obtain a finite radiation path, particles must be randomly positioned in their respective unit cells, which is possible only in the models by Woodside and Russell. The Laubitz model for the effective thermal conductivity of powders, which includes the contributions of a two-phase gas–solid conductivity (first term) and an equivalent radiation thermal conductivity (second term), is d keff Z 2$kðRÞ C 4$s$T 3 $3 ð1Kp2=3 C p4=3 Þ p (64) 391 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 z t1 t2 d r x –x Fig. 31. Thermal conductivity for Al2O3 and ZrO2 (porous bubblesceramic shell surrounding air) [10]. CCC experimental points for Al2O3; !!! experimental points for ZrO2. where k(R) is the thermal conductivity calculated according to the Russell model (W/m K). A detailed derivation of the radiative contribution is provided in Appendix C. Reasons for doubling the value of the Russel model for the thermal conductivity are given in Appendix D. Experimental results obtained by Laubitz are shown in Figs. 31 and 32. Laubitz’s experiments showed that this model cannot be used for calculating the thermal conductivity of powders with graded particle size. Botterill [27,28] concluded that the Laubitz model gives arbitrary predictions, which are good for alumina but not for silica. Though this model predicts higher values of the thermal conductivity compared to the other models investigated in [27,28], it predicts more accurate by the temperature dependence of the conductive component of the effective thermal conductivity. Schotte [29] developed a model in order to determine the radiation contribution to the thermal conductivity of packed beds. When considering the radiation from a plane located on one side of a particle, to a plane located on the far x Fig. 33. Model for derivation of the radiation contribution to the thermal conductivity of a packed bed [29]. side of the particle, as shown in Fig. 33, two mechanisms may be included into the analysis. First, there is radiation from the particle, in series with the conduction through the particle, i.e. the first term in Eq. (65), and radiation across the void space past the particle, i.e. the second term of Eq. (65)      p 2 ks hdp dt p 2 p dt d Kh d q ZK dp ks C hdp 4 dx 4 p 1Kp p dx (65) where q is the rate of heat transfer. In order to determine the thermal conductivity correction caused by the radiation heat transfer, the rate of heat transfer across the total area can be written as:   p 2 1 dt dp (66) q ZKkr 4 1Kp dx The radiation contribution is now found by equating the right-hand sides of Eqs. (65) and (66) kr Z 1Kp C p$kr0 C k10 1 ks (67) r where kr0 Z 4$s$3$dp $T 3 (68) According to Vortmeyer [19], the final equation of this model can be written as keff ðk Þ 1Kp Z eff c C 1 C p$Nuw ð2 1 kg kg C X Nu ð2K0:2643Þ w K0:264$3ÞX Fig. 32. Thermal conductivity for Al2O3 (graded particle size): CCC experimental points [10]. (69) where (keff)c is the effective thermal conductivity without radiation, XZks/kg, and: 392 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Nuw Z 4$s$dp $T 3 1 2=3K0:264 ks (70) Godbee and Ziegler [30] developed a model for the thermal conductivity of powders, applying MgO, Al2O3, and ZrO2 powders (dpZ211–1023 mm) in their experiments. They assumed one dimensional heat transfer, with a unit cell given in Fig. 34, isotherms being parallel planes perpendicular to the direction of heat transfer, and with the pressure sufficiently high to have no influence on the gas conductivity. Godbee and Ziegler expressed the effective thermal conductivity as the sum of three components (71) keff Z ksc C kgsc C kr where ksc is the conductivity by solid only, kgsc is the conductivity by solid and gas, and kr is the conductivity by radiation. The influence of gas conductivity on the thermal conductivity of powders is greater than the influence of the solid conductivity, so the authors assumed ksc to be negligible. Thus, the simplified expression for kgsc is kgsc Z kc Ds =X ð1KS=X 2 ÞCðkd =kc ÞðS=X 2 Þ   C 1K DXs (72) where Ds is the solid length parallel to the heat flow (considered to be equal to the mean particle size), S is the solid area perpendicular to heat flow, and X is the length of representative cell. This equation was used in the calculations done by Botterill [27,28]. Some investigations, e.g. Woodside [20] have shown that the thermal conductivity of a powder–gas system decreases with decreasing pressure, at a much faster rate than can be explained by the decrease in conductivity of the pure gas by reduced pressure. In order to correct for this, the following equation is derived kgsc Z h kg Ds =X ðkg =kg Þð1KS=XK2 ÞCðkd =kg ÞðS=X 2 Þ where i   C 1K DXs kg Z kg ð1KS=X 2 Þ C kg0 ðS=X 2 Þ (73) (74a) kg kg0 Z h  2Ka  g  T i 1 CZ a 1Cg Pdf2 N (74b) pr and a is the thermal accommodation coefficient for the gas–solid surface, g is the ratio of the gas heat capacity at constant pressure, respectively, at constant volume, d is the distance between close parallel plates as shown in Fig. 34, i.e. dZXKDs, T is the absolute temperature, P is the absolute pressure, f is the molecular diameter of gas as determined from viscosity (using the kinetic theory for gases), NprZhCpg/kg is the Prandtl number, h is the viscosity of gas, Cpg is the gas heat capacity at constant pressure, and ZZ1.26!10K19 for cm g K units (1.26!1024 m kg K) (a constant comprised of fundamental constants and conservation factors). The parameters describing the bed in this model were all evaluated from a log probability plot of the sieve size against the accumulative weight fraction, according to Woodside [20] Ds =X Z ðVd =aÞ1=3 (75) where a h S=D2s (76) and Vd is the volume fraction of the discrete phase, and a is a shape factor [20]. The quantity a is a property of particles, and can be related to the heterogeneous body through a volume balance of two phases. For a, the authors used the following equation aZ D ðb Da 2 eKu =2 du ð2pÞ1=2 (77) where u Z ð1=Sln ÞlnðD=D50% Þ Fig. 34. Unit cell model for model of Godbee and Ziegler [30]. (78) and Da, Db are the lower and upper points of truncation (mm) (or smallest and largest particle diameter), D is the mean particle diameter, D50% is the median particle diameter, Sln is the logarithmic (to the base e) standard deviation. Results for some of the powders used in these experiments are given in Table 3. A value for S/X2 is obtained with Eq. (75) and the fact that SDsZVdX3: A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 393 Table 3 Parameters of magnesia, alumina and zirconia powders obtained from screen analysis [30] expression, covering a number of radiative models, is derived Powder Median particle size, D50% (mm) Shape factor, a kr Z 4$s$c$d$T 3 MgO MgO Al2O3 ZrO2 ZrO2 180 235 166 198 333 0.739 0.735 0.737 0.853 0.851 where c is the radiative exchange factor. Radiative exchange factors for different models are provided in Table 4. This parameter is further discussed in Section 6. The radiative exchange factor, c, is a function of the size and the emissivity of the pores or particles. Botterill’s study showed that: S Z ða$Vd2 Þ1=3 X2 (79) The equation for radiative conductivity is given as kr Z 4n2r s3ð1=Vd K1ÞDs T 3 (80) where nr is the refractive index of the media between surfaces. According to Botterill [27,28], the Godbee and Ziegler model gives a conductive component that is consistent with the experimental results, and a radiative component that is generally larger than those of the other unit cell models. Fig. 35 shows how this model predicts the temperature dependence of thermal conductivity of silica sand. As the emissivity increases, the temperature dependence of the model curve increases but even with the maximum emissivity value of unity, the temperature dependence never attains that experimentally observed value. Botterill [27,28] tested a number of published models, using experimental data obtained on alumina and sand, in the temperature range TZ400–950 (C. An (81) † Models, which characterize packing using the voidage, predict similar values of thermal conductivity. † Models, which take into account the particle size distribution, give very similar estimates (Godbee and Ziegler; Bauer, Schlunder; Zehner). † According to the experiments, most of the models reported cannot deal with packing substantially different to the originally tested. † Good agreement with the model predictions is obtained, when experimental measurements at a ambient temperature are used (when radiation is negligible). † None of the tested models could predict the temperature dependence of the effective thermal conductivity at high temperatures. Possible reason for this could be that particles are partially transparent to radiation and not opaque as the unit cell and pseudohomogeneous models assume (Al, Si transmit certain frequencies). † According to Botterill, the models of Godbee and Ziegler, and Kunii and Smith give reasonable predictions at low temperatures. The radiative models of Zehner, and Kunii and Smith give good estimates for the thermal conductivity for packed bed of particles at high temperatures with some modifications (given in Nasr [9]). Table 4 Radiation exchange factor for various unit cell models [9,28] Fig. 35. Comparison between the effective thermal conductivity for a bed of 410 mm silica sand, of different emissivities, and predictions by the Godbee and Ziegler model; (1) exp. values; (2) 3/0; (3) 3/0.37; (4) 3/1; (5) 3/0 and silica conductivity perpendicular to c-axis. [28]. Authors c Godbee and Ziegler Laubitz Schotte Zehner; Bauer and Schlunder Kunii and Smith 3/(1Kp) ðð1Kð1KpÞ2=3 C ð1KpÞ4=3 Þ=ð1KpÞÞ3 3 3/(2K3) p: porosity; 3: emissivity. 1=ð1C ðp=1KpÞð1K3=23ÞÞ 394 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Mills and Rhine [31] measured the thermal properties of gasifier coal slags in the temperature range 298–1800 K, and developed a model for the radiant component of the heat conduction. The radiant conductivity increases with the optical thickness, i.e. a$d (where a is the absorption coefficient and d is the slag thickness), until a constant value is achieved. At this point a$d is greater than 3.5, and the slag is said to be ‘optically thick’. The radiative thermal conductivity of optically thick slag can be determined as 16$s$n2 $T 3 kr Z 3$a 2 ∆l 3 4 6 1 Fig. 36. A model of heat transfer in a packed bed (adapted from [34]). (82) where n is the refractive index. Since it is very difficult to measure the absorption coefficient of molten slags, measurements made at 298 K, have been assumed to give an approximate value for a at high temperature. Since the absorption coefficient is highly dependent on the concentration of Fe2C in the slag, Miles and Rhine suggested that it could be determined, using the following empirical rule (particle size range 2–4.5 mm): a Z 11ðwt% FeOÞ 5 (83) 6. Heat transfer by convection, solid–fluid–solid 7. Heat transfer by lateral mixing When the Reynolds number is small, the boundary layers around the solid packing are thick, and therefore mechanisms 1, 3, 4 and 5 above, are predominant. In case of gas-filled voids, the model equation is dp $hrv keff bð1KpÞ C p$b Z k  g 1 kg kg g ks C ð1=4ÞCðdp $hrs =kg Þ (84) According to Mills, the absorption coefficient predicted using this rule, are in reasonable agreement with the experimental data reported by Finn et al. b Z lp =dp (85a) g Z ls =dp (85b) 3.2.2. Resistance network models The following group of models includes the radiative conductivity as a resistance in the network of resistances, representing the unit cell. In these models, it is assumed that there is dependence between the conductive and the radiative component. Yagi and Kunii [33] developed a semi-empirical model for estimating the thermal conductivity of packed beds. The model is divided into two terms, one term being independent of the fluid flow, and the other one dependent on the lateral mixing of the fluid in the packed beds. Heat transfer mechanisms, considered are (Fig. 36): 4 Z lv =dp (85c) hrs Z 0:1952ð3=ð2K3ÞÞððt C 273Þ=100Þ3 (85d) 1. Thermal conduction through the solid phase 2. Thermal conduction through the contact surface between neighboring particles 3. Radiant heat transfer between adjacent solid surfaces 4. Radiant heat transfer between neighboring void spaces 5. Thermal conduction through the gas film near the contact surfaces between adjacent particles 2 hrv Z 4  0:1952 p 1K3 1 C 2ð1K pÞ 3 3  5ððt C 273Þ=100Þ3 (85e) where hrs is the effective radiation heat transfer coefficient at the contact surface, hrv is the effective radiation heat transfer coefficient of the voids, t is the temperature (8C), lp is the average length between the centers of two neighboring solids in the direction of heat flow, ls is the equivalent thickness a layer of solid should have to represent the same thermal resistance as the sphere, i.e. lsZ2dp/3, and lv is the equivalent thickness a layer of fluid should have to represent the same thermal resistance as the fluid film. Petersen (as stated in [34]) formulated a following expression for the calculation of parameter lv  0:3716 kg p1:7304 (86) lv Z 0:15912$b kair 395 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 where kair is the thermal conductivity of air, calculated using the following correlation [34] kair ðW=mKÞ Z2:286 !10K11 $T 3 K7:022 K8 2 K4 !10 $T C 1:209 !10 $T K5:321 !10K3 ð87Þ where T [Z] K. In practical packed beds, b takes a limited value between 0.82 and 1, for a wide range of packing characteristics. A possible way to estimate the parameter lp, which is used to determine b, is given in Appendix E. Parameters g and f are to be determined experimentally, and according to existing data they can be approximated as g z1, respectively, f z0.04. Kunii and Smith [35] developed a model for the effective thermal conductivity of beds of unconsolidated particles containing a stagnant fluid, which is then extended to the beds of consolidate particles (materials like sandstone or porous metals). The authors assumed 1-dimensional heat flow with constant linear heat flux, through the gas and the solid, and solid–solid contact conduction to be negligible for atmospheric pressure conditions. This model is semi-empirical, involving parameters that were determined using an extensive comparison with the available experimental data [28]. Similar to the Yagi and Kunii model, heat transfer is assumed to occur in the vertical direction accounting the following mechanisms: 1. Heat transfer through the fluid in the void space by conduction, and radiation between adjacent voids (when the voids are assumed to contain a nonabsorbing gas) 2. Heat transfer through a solid phase a. Heat transfer through contact surfaces, between the solid particles. b. Conduction through the stagnant fluid near the contact surface. c. Radiation between surfaces of solid (when voids are assumed to contain non-absorbing gas). d. Conduction through a solid phase. Mechanisms 1 and 2 are in parallel to each other. Mechanism d is in series with the combined result of mechanisms a, b and c. Mechanism a is only important at low pressures, and when it is neglected, the model is simplified to
 b$hrv $dp keff bð1KpÞ 3 Zp 1C (88) C2 kg kg 6 1  g7 C k5 4 1 dp hrs dC kg where gZ2/3, and b is in the range 0.9 (for close packing) to 1.0 (loose packing) d Z d2 C ðd1 Kd2 Þ pK0:26 ; 0:216 0:26! p! 0:476 d Z d1 ; pO 0:476 d Z d2 ; p! 0:260 (89a) d1 Z   1 2 0:352 kK k  1  2 ln½kK0:545ðkK1ÞK0:455 kK K 3k k  1 2 0:072 kK k  1  2 d2 Z  ln½kK0:925ðkK1ÞK0:075 kK K 3k k 4$s$T 3 i hrv Z h p ð1K3Þ 1 C ð1K pÞ 23 hrs Z 4$s$T 3  3  2K3 (89b) (89c) (89d) (89e) The radiant heat transfer coefficients were calculated from expressions derived by Yagi and Kunii. The data reduction procedure involved a temperature-dependent fluid thermal conductivity, a temperature-independent solid thermal conductivity, and the solid-surface emissivity. According to the model, the temperature dependence of the heat conductivity of solids can be neglected (25% decrease of ks leads to 5% decrease of effective thermal conductivity). The model of Kunii and Smith was evaluated using the experimental data in Nasr et al. [9]. It can be seen that the accuracy of model predictions depends on the particle composition and diameter. Good results were obtained for alumina–air packed bed (dpz6.6 mm), at temperatures below 900 K. For smaller particle size (dpz2.8 mm), better results were obtained for glass–air beds, at temperatures below 600 K. According to Botterill [27,28], the model gives reasonable predictions of the conductive component for the silica beds, at lower temperatures, when radiant transfer is negligible (up to around 900 8C). As mentioned, this model was extended to predict the effective thermal conductivity of consolidate porous media. The consolidation might occur by partially clogging with a cementing substance or by sintering. The idea of clogging of the original bed of the unconsolidated particles, with a cementing substance or by sintering, is shown in Fig. 37. 396 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Heat flow z Dp Contact area dr Fig. 37. Kunii and Smith model of consolidated porous media [35]. α r Now, heat is flowing in the vertical direction through the void space, p 0 and through the solid fraction, 1Kp 0 . In this latter part, the flow is in series through a solid of thickness l 0 , and a fluid of thickness lv. The resulting equation for the effective thermal conductivity is     ð1Kp 0 Þ 1 C p 0 d 0 kg dp $hrv p keff p Z p0 C p 1C d C ðp 0 =pÞ p ks ks ks   1 C kg dp ðhpChrs Þ 1 d ks C ks (90) where p 0 is the void fraction of the bed of consolidated porous media and hp is the heat transfer coefficient representing the heat transfer rate through the contact surface between solid particles in a bed of unconsolidated particles or between clogged particles in a consolidated bed. The heat transfer through the contact surface between consolidated particles, as represented by the dimensionless group hpdp/ks is difficult to quantify. It is regarded as a consolidation parameter and determined by comparing experimental data with the form of Eq.(90), when radiation was neglected. This mechanism is influenced by the characteristics of solid material and the type of consolidation, but does not depend upon the fluid in the pores. The experiments showed that when a value of 0.075 is assumed for the dimensionless group, agreement with the experimental data is satisfactory. Bauer and Schlunder [36] developed a model for random packing of uniform spheres, which takes into account the radiation and pressure effects. The same data reduction procedure, as in Kunii and Smith [35], was applied. The model assumes heat transfer by conduction, through a bed of non-conductive particles surrounded by a conducting gas. Analogy with the diffusion in a packed bed was considered. Fig. 38 shows the unit cell for the model. Heat flow is divided into three parallel paths: † Conduction and radiation through the gas-filled pffiffiffiffiffiffiffiffiffiffi voids within the area fraction 1K 1Kp 1-α 1-√(1-p) √(1-p) Fig. 38. Unit cell for model of Bauer and Schlunder [36]. † Conduction through the solid and the gas phase, with radiation between solid surfaces; within the pffiffiffiffiffiffiffiffiffiffi area fraction ð1KaÞ 1Kp † Solid–solid conduction, within the area fraction pffiffiffiffiffiffiffiffiffiffi a 1Kp where p is the bed voidage, and a is the area fraction of the solid-solid contact conduction path. According to Nasr et al. [9], the final model equations are:   pffiffiffiffiffiffiffiffiffiffi keff k Z ð1K 1KpÞ 1 C p r kg kg
 pffiffiffiffiffiffiffiffiffiffi ks kso C 1Kp a C ð1KaÞ (91) kg kg  1 2  0 kg ks kr ks kr C C K1 B kg kg kg kg ks kso 2 A K BK1 Z 4 ln@ N N B kg N2    k B C1 C r KB 2$B kg    kg kr N Z 1C KB kg ks
 1 1 C Biox   ks Z ks Biox Z Sox kox ks dp  (92a) (92b) (92c) (92d) A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421  3  x $T 3 kr Z 4$s 2K3 R (92e) xR Z Rform $dp (92f) B Z Cform aZ  1Kp p 10=9 23$r2k 1 C 22$rk4=3 (92g) (92h) where Cform is the particle shape factor, Rform is the determinative influence coefficient for the transport mechanism resulting from thermal radiation (for further details, see [9]), xR is the radiation effect factor, r2k is the contact conduction factor, and (Sox/kox) is the oxidation effect factor ([9]) equal to the ratio between the oxide layer thickness, and its thermal conductivity. Parameters Cform, Rform, r2k , (Sox/kox) must be determined experimentally. For monodisperse spherical packing CsphereZ1,25 when RsphereZ1. The experiments done by Nasr et al. [9] showed that this model overpredicts the measured values of the effective thermal conductivity at low temperatures (around 20%, for temperatures below 700 K), but gives reasonably good results at higher temperatures (where radiation is important). 3.3. Pseudohomogeneous models This group of models considers packed beds to be a continuum for radiation. The bed is considered to be a pseudo homogeneous medium, through which radiation can penetrate freely, so equations describing radiation through an absorbing, emitting, and scattering medium are used: dI ZKða C bÞI C bK C asT 4 dz (93) dK K ZKða C bÞK C bI C asT 4 dz (94) where a is the absorption cross-section per unit volume of bed (mK1), b is the backscattering crosssection per unit volume of bed (mK1), I is the forward component of the radiative flux (W/m2), and K is the backward component of the radiative flux (W/m2), (both fluxes are parallel to the z-axis). According to [37], Eq. (93) states that in traversing 397 the distance dz, the radiant flux density in the forward direction is decreased by bIdz due to back scattering and aIdz due to absorption; it is increased by bKdz due to backscattering of the backward radiant flux density and asT4dz due to radiation. Eq. (94) is the analogous balance for the backward radiant flux density. Eqs. (93) and (94) imply that the spectral distribution of the intercepted and reemitted radiation is the same, which is a reasonable assumption except for the situation with extreme temperature gradients. The net radiative flux is: qr Z I KK (95) It was assumed that the medium is grey, the scattering is isotropic, and that the Schuster–Schwarzchild approximation (two flux model) is valid, i.e. radiation can be scattered only in two directions, backward and forward. The framework for pseudo homogeneous models, requires radiative heat transfer through a packed bed to be governed by the equation of heat transfer. Thus, the radiative transport (i.e. the radiative heat flux) may be accurately evaluated even for a comparatively thin packed bed, by numerical solution of the heat transfer equation. This is not possible for the cell models, because they do not take into account the long-range effects of radiative transfer through the voids of a packed bed. It is required that the radiative properties, which appear in the equation of heat transfer, are known. It is essential for the pseudo homogeneous models, to describe these quantities in terms of the void fraction, the particle size and the surface reflectivity of the particles. According to Kamiuto et al. [38], reflectivity cannot be determined by independent scattering theory such as Mie theory,2 since it is not appropriate for this purpose. Vortmeyer [19] had compared the unit cell and pseudo homogeneous models in handling radiation in packed beds. His conclusions are: (1) In the cell models, the opacity of the packed bed is large, and the particles are much larger than the wavelength of radiation. (2) In the pseudo homogeneous models, the dispersed phase is considered to be a continuum for radiation. The effective thermal conductivity is given in terms 2 Only the interactions of a single particle with light of arbitrary wavelength are considered. 398 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 of the effective scattering and an extinction coefficient.3 T1f θ T1s 3.4. Dedicated models developed for ash deposits T2s r Using three characteristic values of the solid volume fraction, mean particle size, and tortuosity, Baxter [39] developed a model connecting the average thermal conductivity to the structural properties of a boiler ash deposit. The void space was assumed to be nonconducting. The equation for 1D, transient heat transfer through a cylindrical surface, may be written as:
   vT 1 v vT 1 v2 T Z keff r r$Cp (96) C 2 vt r vr vr r vq2 The model system for heat transfer through a cylindrical wall is shown in Fig. 39. When the transient term is ignored, the model for temperature field through the cylindrical body is   q$r r TðrÞ Z T1 K 1 ln (97) r1 k where r1, r2 are radial directions of the deposit surface. Eq. (97) reduces to a linear dependence of the deposit temperature on distance, i.e. plate geometry, in the limit of a small deposit thickness relative to the radius of curvature. The relationship between the effective thermal conductivity, the thermal conductivity, the porosity, and the tortuosity, is given by: keff Z kð1KpÞ t (98) The tortuosity represents the shortest average path length between two points, divided by the straight-line distance between the same points. Thus, as the solid phase becomes less connected, the tortuosity increases. The change in tortuosity as a function of the degree of sintering is illustrated in Fig. 40. Two values of the tortuosity were used this study, 1 and 2. Deposits with a solid fraction lower than 0.5, and tortuosities higher than 2, are common in many systems. According to Fig. 41, the temperature range scales linearly on the tortuosity and inversely on the porosity, meaning that a change in 3 The fraction of light lost to scattering and absorption per unit distance in a participating medium. Normally, it is given in standard units as a fraction per meter. It equals the sum of the absorption coefficient and the scattering coefficient. Heat flow 1 – hot fluid (flue gas) 2 – coldfluid (steam) f – fluid s – surface Fig. 39. Heat transfer through a cylindrical wall. either property, changes the difference between the deposit surface temperature and the tube surface by the same factor. The extent of curvature is determined by the system geometry, not by the real deposit physical properties. Robinson et al. [2] made in situ measurements of the thermal conductivity of deposits formed during the combustion of a mixture of coal and wheat straw (65/35 wt%). This is the first time, the deposit conductivity has been measured on original, noncrushed deposit samples. The model describes the thermal conductivity of a layered structure, since the analysis of ash deposits showed that sintering creates a layered deposit structure with a relatively unsintered innermost layer. Assuming steady state, two-dimensional heat transfer through the deposit and a uniform deposit thermal conductivity, the temperature distribution within the deposit can be described using Eq. (130). The average radial temperature gradient at the inside edge of the deposit is 1 dT Z 2p 2p ð 0 vT j ðqÞdq vr in (99) where ((T/(r)(in(q) is the radial temperature gradient as a function of q, determined from a numerical solution of Eq. (40). The effective thermal conductivity of the deposit is keff Z Q 2$p$rin $L$dT (100) where rin is the radius of the inside surface of the deposit, and L is the length of the tube. If the deposit is treated as a two-layer material (the simplest case), its overall conductivity is keff Z L kun 1 LÞ C ð1K ksi (101) A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 399 Fig. 40. Conceptual illustration of the changes in contacting efficiency and tortuosity with sintering/melting [39]. where L is the thickness of the unsintered layer, normalized by the total deposit thickness (this value was kept constant, Lz0,4), kun is the thermal conductivity of the unsintered (inner) layer of the deposit and ksi is the thermal conductivity of sintered (outer) layer of the deposit. 2. Estimation of the conduction part of the thermal conductivity of the zero-density material at the given temperature. 3. Estimation of the effective thermal conductivity of the slag or particulate material or partially sintered deposit. 4. Taking into account the effects of radiation. 4. Estimation of thermal conductivity of ash deposits According to Gupta et al. [4], practical estimation of the thermal conductivity of ash deposits should consist of: 1. Estimation of the thermal conductivity of the zerodensity material at room temperature. 4.1. Estimation of the general parameters 4.1.1. Structural parameters The mean particle size, dp, which characterizes the bed packing, can be calculated as " #K1 N X Dmi dp Z (102) dpi iZ1 where Dmi is the weight fraction of the ith sieve fraction, and dpi is the mean sieve spacing for the ith sieve fraction (m). The structural parameter c, as defined in Robinson et al. [2] is: cðfÞ Z Fig. 41. The deposit surface temperature as a function of the porosity and emissivity, assuming no intra-deposit radiative heat transfer and a non-conducting gas phase. Tortuosity is assumed to be unity [39]. kmeas ðfÞKklow ðfÞ ; 0% cðfÞ% 1 kup ðfÞKklow ðfÞ (103) where kmeas is the measured value of deposit thermal conductivity, kup is the upper limit of the thermal conductivity of deposits, defined by Eq. (18), klow is the lower limit of the thermal conductivity of deposits, defined by Eq. (19), and f is the volume fraction of solid in the deposit. This parameter provides a useful measure of the microstructure of the deposit. Using 400 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Eqs. (18) and (19) as boundary values of the thermal conductivity, the structural parameter can have values between 0 and 1. Thus, as shown in Fig. 9, values close to 0 will indicate a deposit with a more layered structure, while values close to 1 indicate a more columnar structure. 4.1.1.1. Porosity change with time. The temperature profile through a deposit layer increases with time, which causes densification of the deposit. This affects the porosity of the deposit, and thereby the thermal conductivity, which will again affect the rate of heat transfer by conduction, and thereby the temperature profile in the deposit. According to Rezaei et al. [40], during the process of deposit densification, the porosity will decrease while the extent of shrinkage DV/V will increase. Two groups of models, which describe the porosity change during shrinkage, can be distinguished [40]. The first group is based on geometrical assumptions that relate the porous system to an idealized system, i.e. assembles of spherical, viscous particles whose dimensions and physical properties (surface tension) and viscosity remain constant during processing. The second group of models, which are based on a phenomenological approach to densification process. Frenkel (in [40]) was first to propose a model for the sintering of viscous material from geometric assumptions. He modelled the first stage of vitrification, by equating the energy variation due to a decrease in the surface area, to the energy dissipated by viscous flow DV=V ZK9$s$t=ð4$h$ro Þ (104) where s is the surface tension (according to Senior [41] swT, can be neglected), t is time of sintering, h is the ash viscosity (hwexp T [41]), and r0 is the initial pore radius (developed on the basis of Scherer’s structural model for densification of a cubic lattice; the pore radius was set equal to half the average particle diameter). According to this model, the volume shrinkage is proportional to the duration of the thermal treatment. The shrinkage level or degree of sintering, is defined as the ratio of the neck radius, x, to the radius of particles, r, as shown in Fig. 42. Using the Pythagoras Theorem, we can obtain the shrinkage level as x=r Z ððr2=3 Kr0 Þ=ðr2=3 ÞÞ1=2 (105) where r0 is the density of the initial powder, and r is the density of the sintered product. The derivation of Eq. (105) is shown in Appendix F. The powder density at r0 x r Fig. 42. Schematic diagram of the sintering of two spheres [40]. time t, can be expressed as a function of the porosity p rðtÞ Z rs ð1KpðtÞÞ (106) where rs is the density of solid phase. For the initial density r0Zr (tZ0), i.e. p (tZ0)Zp0. Thus, the porosity of sintered product at time t can be obtained from Eqs. (105) and (106) as: p Z 1K 1Kp0 ð1Kx2 =r 2 Þ2=3 (107) This model is limited to a shrinkage level of 0.3 [40]. Assuming that during the time period t, the original radius r0 of the particle does not change much, i.e. rtzr0, the ratio of the pore volume at time t to the initial pore volume is: Vpt =Vpo Z 1K3=2ðx2 =r02 Þ (108) Comparing this equation to Eq. (104), x2 equals 3str/2m, thus for the ratio of pore volumes we obtain: Vpt =Vpo Z 1K9$s$t=ð4$m$ro Þ (109) Ivsen (in [40]) connected the pore volume change to the density change during sintering, using phenomenological assumptions, as: Vpt r ðrKrt Þ Z 0 rt ðrKr0 Þ Vpo (110) According to Rezaei et al. [40], the porosity at time t can be obtained using Eqs. (106) and (110), as:   9st 1K 4mr po  p  pZ (111) 9st 1Kpo C 1K 4mr p o p Mackenzie and Shuttleworth (1949), suggested the following band between the actual porosity p at time t, and the initial porosity po (source: Senior [41]):
 p 3$s$t exp K (112) p0 2$rp $m This model can be used for sintering from app. 50–0% porosity. 401 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 In the above models, an estimate of the surface tension and the viscosity in each deposit sub-layer is needed. The surface tension of an amorphous phase can be estimated from its chemical composition. As stated in Rezaei et al. [40], Tillotson and Oppen showed that the surface tension of a material could be expressed as a linear function of its oxide composition: N X sZ ðxi $si =100Þ (113) iZ1 where si is the surface tension corresponding to each oxide i, and xi is its molar percentage. The surface tension of each oxide is calculated from   0:004 si Z Ai C DT ½N=m (114) 100 where Ai (N/m) is a component-specific constant, provided in Table 5 for some oxides present in ash deposits, DT is the temperature difference to reference temperature TrefZ1300 8C (negative value, if the actual temperature is less than Tref). Concerning the viscosity of a multicomponent oxide mixture, Vargas et al. [42] provided an outline of the existing experimental viscosity data, as well as models for estimation of the viscosity as a function of the chemical composition and the temperature. In this work, the models of Hoy et al. (1965), Watt and Fereday (1969), Urbain et al. (1981), Kalmanovitch and Frank (1988), and Senior and Srinivasachar (1995), among others, are discussed. Senior [41] experimentally determined a change in deposit porosity with time, for coal-derived ash deposit, as shown in Fig. 43. 4.1.2. Radiation parameters Radiation parameters such as the refractive index and other optical parameters, can be used in different models in order to estimate the emissivity of a deposit. Thus, some theoretical and semi-empirical methods for estimating these parameters are outlined below. Table 5 Pure component surface tensions, valid at 1300 8C Component Ai (N/m) Component Ai (N/m) K2O PbO TiO2 SiO2 Na2O 0.01 0.12 0.25 0.29 0.295 Li2O CaO MgO Al2O3 0.45 0.51 0.52 0.58 The coefficients are to be used in Eq. (107), for estimation of pure oxides surface tensions [40]. Fig. 43. Densification of Wyodak-Rochelle deposit [41]. When radiation incident onto a particle, it is absorbed (and then emitted), or scattered. The particle properties defining these interactions are: † Particle size parameter [43] x Z pd=l (115) where d is the particle diameter and l is the wavelength of the incident radiation. † Complex refractive index m Z nKiki (116) where n is the real refractive index. The parameter ki is the absorption index, which is a measure of the attenuation caused by the absorption of energy per unit of distance that occurs in an electromagnetic wave of a given wavelength propagating in a material medium of a given refractive index. Both indices depend on the wavelength of the light, and the temperature. Complex refractive index is a function of material, particle size, and temperature. Values of m valid in room temperature, for slags with composition similar to ash are provided by Goodwin and Mitchner [44]. Due to the lack of reliable and sufficient experimental data, most studies have assumed that the optical constants n and ki for ash deposits are independent of the wavelength, with n taken as 1.5 (because constituent oxides have a real index of about 1.5 up to the wavelength of 5 mm), and ki ranging from 0.005 to 0.05 [45]. Based on a limited amount of experimental data, the complex refractive index of coal ash is known to have a spectral character, with n varying from 1.5 to 2, and k varying from very low values of 0.001 to about 1. Bhattacharya [45] showed that for the coarse size fraction of the fly ash (big particles, in the range 1.563– 212.65 mm), the effect of the real index is insignificant. For the fine size fraction, where the effect of the real index is more pronounced, the difference in total emittance over 402 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 the entire temperature range 800–1800 K, is around 0.05. Calculations with a real index of 1.5, very closely match the calculations using the spectral values. These calculations suggest that the choice of real index is not critical for the estimation of the hemispherical emittance of an opaque ash deposit, unless the deposit is composed of very fine particles. On the other hand, for both size fractions, a significant effect of absorption index (iron content) is observed. For a smooth deposit, which is analogous to molten a slag, no significant variation with the real as well as the absorption index was observed. It was also observed that the real index, does not depend on the chemical composition. Goodwin and Mitchner [44] investigated the influence of the chemical composition of coal ash slags on their optical properties n and k. A mixture rule was developed to allow calculation of the refractive index n, as a function of its composition in the wavelength range 1–8 mm. If the composition is expressed on a mass basis, the value of n for the slag may be written as simplified dispersion equations of the form X  Xm;i  n2i K1  n2 K1 Z r ri n2 C 2 n2i C 2 i Fi ðlÞ Z ðai $l2 Kbi Þ=ðci $l2 Kdi Þ (117) where r is the sample density, Xm,i is the mass fraction of component i in the slag, and ri and ni are the density and refractive index, respectively, for component i in a pure reference’ state, for which refractive index data are available. The slag components are taken to be the oxides: SiO2, Al2O3, CaO, Fe2O3, TiO2, and MgO. The refractive index of pure substances that can be found in combustion systems is given in Table 6. The reference states of these oxides are assumed to be fused silica, saphire (Al2O3), crystalline CaO, hematite (Fe2O3), rutile (TiO2), and crystalline MgO, respectively. The refractive index values for these minerals were taken from the literature and were fit to a Table 6 Refractive index of pure substances that can be found in combustion systems Component Silicon dioxide Aluminum oxide Iron (III) oxide Titanium oxide Calcium oxide Magnesium oxide Potassium oxide Sodium oxide SiO2 Al2O3 Fe2O3 TiO2 CaO MgO K2O Na2O Transmitted color Refractive index Colorless White Red or black Colorless Colorless Colorless White White 1487 1768 3042 2615 1873 17,364 – – n2i K1 Z Ci C ðBi l2 Þ=ðl2 Kl20;i Þ (118) where Bi, Ci, and li are three parameters, the values of which are provided in Table 7 for each oxide component, along with their density in the reference state. Owing to the insufficient experimental data, the value of n for Fe2O3 is taken to be independent of wavelength and equal to its value at visible wavelengths. The sample density (g/cm3) is given as r Z 2:54 C 0:00978ðFe2 O3 Þ (119) where (Fe2O3) denotes the weight percent Fe2O3. But since the above equation slightly underpredicts the values of n, the authors tried to improve it by using two empirical adjustments, obtaining a final mixture rule X n2 K1 Zr Xm;i $Fi ðlÞ 2 n C2 i (120) where (121) Parameters ai, bi, ci and di are given in Table 8. In order to develop a correlation for the absorption index, ki, it was assumed that at infrared wavelengths ki could be written as the sum of two terms describing the infrared tail of the charge transfer band, and the absorption due to Fe2C, respectively: ki ðlÞ Z a1 expðb=lÞ C a2 gðlÞ (122) The exponent b and the function g(l) are taken to be independent of composition. The final equation is obtained as ki ðlÞ Z 3:61 !10K7 $r2 $rð1 KrÞðFe2 O3 Þ2 expð1:75=lÞ C r$rðFe2 O3 Þ !½0:0963 C 0:0011r$rðFe2 O3 ÞgðlÞ (123) where r is the sample density (Fe2O3) is the Fe2O3 weight percent, and r is the ferrous ratio: Table 7 Dispersion equation parameters for pure oxides [44] Oxide Density (g/cm3) Ci Bi lo,i (mm) SiO2 Al2O3 CaO Fe2O3 TiO2 MgO 2.20 3.97 3.31 5.24 4.86 3.58 1.104 2.082 2.31 8.364 5.031 1.962 0.8975 5.281 11.32 0.0 7.764 2.470 9.896 17.93 33.90 0.0 15.60 15.56 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Table 8 Parameters defining the function Fi(l) in the mixture rule for n [44] Oxide ai bi ci di SiO2 Al2O3 CaO Fe2O3 TiO2 MgO 0.9389 1.914 4.250 1.647 2.720 1.278 53.00 174.0 827.7 0.00 260.0 136.9 5.001 10.36 16.63 11.36 15.80 7.433 420.0 1634 6102 0.00 1954 1201 rZ ½Fe2C ½Fe2C C ½Fe3C (124) In the case of ash deposits, in the compositional analysis all the iron is presented as FeC3, thus parameter r can be taken to be 0. The values of the function g(l) are shown in Fig. 44. 4.2. Thermal conductivity of gas and solid mixtures gas mixtures According to Botterill [27,28], the gas mixture in a porous structure can be approximated with air. In the present study, the thermal conductivity of air (W/m K) has been calculated as a function of temperature, using data from Touloukian [47] kg Z 6 !10K12 T 3 K3 !10K8 T 2 C 9 !10K5 T C 0:0007 (125) where T [Z] K. 403 4.2.1. Solid mixtures Wall et al. [6] indicated that the thermal conductivity of a deposit solid-phase cannot be approximated using a weighted average, based on the mass fraction of the oxides, although it presumably could be approximated as the weighted average of the species actually present in the deposit. The particulate ash is seen to have a thermal conductivity below that of the oxides. According to Wall et al. [48], the thermal conductivity of an actual furnace deposit is many times higher than that of the particulates and is closer to that of the sintered or fused ash, which in turn is close to the data for refractory bricks. Robinson et al. [2] used a constant value for the thermal conductivity of solid, ksZ3 W/m K, which is comparable to the thermal conductivity of silicacontaining materials, at high temperatures. Rezaei et al. [1] developed an empirical expression for the thermal conductivity of the solid phase, assuming it to be dependent only on temperature ks Z 0:0015 !T 1:1 (126) where ks is the thermal conductivity of the solid phase (W/m K), and T is the temperature (K). The equation is obtained using experimental data for sintered ash samples. In the present study, the thermal conductivity of the solid phase was calculated as the sum of weighted average of the oxides present in the ash deposit. The thermal conductivity of the oxides present in the ash, has been derived using data from Touloukian [47] and Raznjevic [49]. 4.2.1.1. Models for the thermal conductivity of solid mixtures. Ratcliffe [50] developed an additive formula based on derived oxide factors, which are claimed to enable the thermal conductivities of most glasses within a precision of 5% or better. In the measurements, the electrically heated disc method was used. An additive formula, for computing the thermal conductivities at three temperatures (K100, 0, 100 8C), from the percentage weight compositions of oxides, were derived by solving 22 equations for the unknown oxide factors 105 k Z x1 a C x2 b C x3 c C/C x12 m Fig. 44. The two functions appearing in the correlation for ki (Eq. (122)). The function e1.75/l characterize the tail of the Fe2C–Fe3C charge transfer band, and the function g(l) characterized the Fe2C absorption spectrum [44]. (127) where k (cal/cm s C) is the thermal conductivity of glass at a particular temperature, {x1,x2,.} are the percentage weight compositions of component oxides, and {a,b,c,.} are the factors for the particular oxides, given in Table 9. 404 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 The most important drawback of this approach is that the experimental points used for the correlation, are completely out of the range of interest for ash deposits. As some results showed, extrapolation to higher temperatures will not give good results. If the oxides of concern are present in low concentrations, or not distributed quite uniformly throughout the glass, the factors could be considerably in error. 4.3. Emissivity The emittance, 3, is a surface radiative property, which is defined as the ratio of a body’s emission rate to that from a black body 3ðTÞ Z EðTÞ Eb ðTÞ (128) between total and spectral emittance is 3t ðTs Þ Z 1 EB ðTs Þ N ð 3v ðTs ÞeB ðl; Ts Þdl (129) vZ0 where EB (Ts) is the emissive power, eB (l, Ts) is the spectral black body flux density, and Ts-surface temperature. According to Wall [52], if 3l does not vary with wavelength, meaning that 3t do not vary with temperature. According to the energy conservation law (the first law of thermodynamics), all radiation incident on a surface must be reflected, absorbed or emitted a Cr Ct Z 1 (130) where a is the absorbance, r is the reflecttance, and t is the transmittance. In the absence of non-linear effects (i.e. Raman effect, etc.), Eq. (131) can also be applied to the spectral properties (this can be applied for deposits). Kirchoff analysis, outlined in Appendix G, showed that emissivity is equal to absorptivity: where E is the energy emitted by a body, and Eb is the energy emitted by a blackbody. Since the spectral radiation emitted by a real surface differs from the blackbody distribution, the emissivity can assume different values according to whether one is interested in the emission at a given wavelength or in a given direction, or in integrated averages over wavelength and direction. Spectral, directional emissivity 3l,q of a surface at the temperature T is defined as the ratio of the intensity of the radiation emitted at the wavelength l in direction of q to the intensity of the radiation emitted by a blackbody at the same value of T and l. Total, directional emissivity 3q represents a spectral average of 3l,q. The spectral hemispherical emissivity, 3l represents the body’s emission in all directions. Total, hemispherical emissivity, 3, represents an average over all possible directions and wavelengths. The connection 3l Z 1Krl Table 9 Factors for calculating thermal conductivity at three temperatures from wt% composition, derived from author’s measurements assuming linear relation [50] † The physical state (morphology) of the deposit surface (fused, sintered or packed particles). † The chemical composition (minimal dependence). Oxide The deposit morphology influences the scattering from surfaces: SiO2 K2O Na2O PbO Sb2O3 B2O3 Al2O3 ZnO CaO BaO Fe2O3 MgO Temperature K100 8C 0 8C 100 8C 2.44 0.54 K1.24 0.60 K5.11 1.09 3.23 1.95 2.82 0.39 1.61 6.37 3.07 0.58 K1.29 0.76 K4.16 1.59 3.72 2.02 3.17 0.46 1.90 5.92 3.44 0.39 K0.67 0.96 1.12 2.49 2.14 1.64 2.39 0.75 1.73 4.53 al Z 3l (131) For opaque slabs, transmissivity is equal to zero (tlZ0), so emissivity is: (132) According to Wall et al. [7], deposit must be thick enough (around 1 mm) in order to be considered as opaque. 4.3.1. Influence of various factors on emissivity According to Markham et al. [53], the emittance depends on: 1. Fused deposits solidify from the molten state to become glassy in nature, which lead to high emittance values. According to the experiments obtained by Markham et al. [53], in the temperature range TZ300–1630 K, the measured values for the emittance of ash was 3R0,9. In the molten or liquid state, the spectral emittance is only slightly higher than in the fused state. 2. Sintered or powdery deposits consist of individual ash particles that are weakly held together. The bed A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 of particles of the powdery and sintered deposits increases the scattering efficiency by increasing the number of boundary surfaces available for scattering (reflecting) of the penetrating wave. Due to their particle-like morphology sintered deposits exhibited a wave-length-dependent emittance [53]. The deposit morphology, and thus its emissivity, depends on the surface temperature of the deposit. Curve A in Fig. 45 shows that the emittance of a particulate ash, is decreasing with temperature increase up to the sintering point. Also, sintered ash (B) and fused ash (C) have significantly higher emittance. The marked increase in emittance of sintered and fused ash is caused partially by reduction in the reflective surface of particulate ash and partly by the change from ferric to ferrous state of iron at high temperatures. A similar emittance dependence on the temperature and the degree of sintering have been found by Boow and Goard [22], and Wall et al. [7]. Bhattacharya [8] showed that the chemical composition of the deposit material affects the emittance and the thermal conductivity due to the presence of coloring agents that increase the absorptivity of the deposit, and together with the particle size influence the temperature of the onset of sintering and the fusion of the deposit material. Emittance of smooth deposits, which are analogous to molten deposit showed very little dependence on the chemical composition. Iron in the ferrous valency state (Fe2C), gives a black color to wustite (FeO), hematit (Fe2O3) and silicate particles when dissolved in concentrations above 3% by weight. According to Raask [5], the visual appearance of ash (for coal) depends largely on the amount and the mode distribution of iron in ash. Fig. 46 shows the influence of iron on the emissivity of coal ash. Fig. 45. Emittance characteristics of boiler deposit: (A) particulate ash; (B) sintered deposit; (C) slag. Arrows indicate direction of heat flow [54]. 405 Experiments done by Wall et al. [52] showed that synthetic mixtures of the oxides Al2O3, Fe2O3 and SiO2 give the same trends of emissivity, as those for ashes of the same Fe2O3-content. It was showed that the total emissivity of coal ash with high iron-content, might increase before the sintering temperature for ashes is reached. The trends reported by Boow and Goard [22] were: † A reduction of the total emittance of particulates with temperature until initial sintering, and fusion, occurs, consistent with the spectral effects † A systematic increase in the total emittance with increasing particle size, prior to the point of sintering † An increase in the total emittance with the ironcontent (Fe2O3) and the carbon-content (due to the presence of unburnt coal in the ash) prior to the point of sintering † The values of 3t for a given Ts varied only slightly for the laboratory ashes, despite the wide range of silica ratios examined. On the other hand, samples that had similar composition but different thermal history have quite different emissivtiy values. 4.3.2. Models for emissivity Estimating the emissivity requires certain assumptions: (1) black body distribution, (2) a transmittance zero for wavelengths where data are available, and (3) a normal spectral emittance independent of temperature. Boow and Goard [22] measured the emittance of the following materials: glass powder, laboratory-prepared and boiler ashes of different characteristic sizes. Fig. 46. Total emissivity of opaque particulate layer, for two different iron contents [8]. 406 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 The equations are applicable for the particle diameter range dpZ7–420 mm at TZ775 K. For colored glass powder: curvature of the tubes, so the effective emittance, 3e, is related to the flat surface emittance, 3r, obtained from the experiments: 3r Z 0:25 log d C 0:13 3e Z (133) For typical silicate ash deposit on boiler tubes before a sintered matrix develop 3r Z 0:30 log d C 0:16 (134) where 3r is the total surface emittance (dimensionless ratio), and d is the particle diameter (mm). Eqs. (134) and (135) are not applicable for calculating the initial emittance of a clean metal target. The effective thermal conductivity of unsintered ash depended on the emittance 3r, and the relationship at TZ775 K is: 3r log k Z 0:753r K1:55 p3r 1 C ðpK1Þ3r Wall and Becker [52] showed a method to obtain total emissivity values from spectral data. A (Planckweighted) total emissivity can be found from the spectral band emissivities of a finite number of adjacent bands as 3t ðT0 Þ Z n P jZ1 3lj ðT0 ÞDEb;lj ðT0 Þ n P jZ1 (135) The correlation coefficient is 0.94, for a variety of ashes and powder glasses. The results are shown in Fig. 47. Wall et al. [7] claimed that the results for emissivity 3Z0.55–0.70 as obtained by Boow and Goard [22], Mulcahy et al. [55] and Goodridge and Morgan (1971), may be too low, and that the data should be reexamined and compared with the results of reflective measurements. They suggested that the data obtained from the experiments are not necessarily applicable to real systems, because experiments are usually done on flat surface deposits while real deposits are rarely flat. A thin layer of deposit on boiler walls will follow the (136) (137) DEb;lj ðT0 Þ where T0 is the ash temperature, and; ljC1 3lj ðT0 Þ Z Ð lZlj 3l eb ðl; T0 Þdl DEb;lj ðT0 Þ (138a) where 3lj (T0) is the (Planck-weighted) band emissivity 3l,j, at the ash temperature T0, 3l is the emissivity at a certain wavelength, eb(l,T0) is the spectral blackbody flux density (W/m2) (given by Planck’s law), and DEb;lj ðTÞ Z Eb ðljC1 ; TÞKEb ðlj ; TÞ (138b) where DEb,lj(l,T) is the incremental band emissive power (W/m2) and Eb ðTÞ Z N ð eb ðl; TÞdl (138c) lZ0 where Eb(T) is the total blackbody flux density (W/m2). This integral has been evaluated as Eb ðTÞ Z sT 4 (138d) lj is the nominal wavelength position of the band given by lj Z ll C lu 2 (138e) where subscripts l and u refer to the lower and upper wavelength limits of a band, respectively. Fig. 47. Influence of particle size and heating on total emittance of particles prepared by crushing a synthetic slag of 5% Fe2O3 [22]. 4.3.3. Experimental data on emissivities Markham et al. [53] and Wall et al. [7] showed that the chemistry of a deposit actually has a weak influence on its emissivity. Fig. 46 shows how the emissivity depends on the iron-content. Comparing the emissivity A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 for particulate layers, with 1 and 5% of iron, showed that the ratio is almost constant, and has a value of 3(5% Fe2O3)/3(1%Fe2O3)Z1.12, which will be used to determine the influence of iron on the emissivity of particulate deposit in this study. According to Wall et al. [6], the iron-content does not have any influence on the emissivity values for slags. Boow and Goard [22], Bhattacharya [8] and Wall et al. [6,48] showed that, normal emittance of slags is approximately constant and that it has values greater than or equal to 0.9. Wall et al. [48] reported values of 0.955 for normal emittance and 0.875 for hemispherical emittance, both values measured at 1000 K, for a 5% Fe2O3 slag. The connection between normal (3n) and hemispherical (3h) emittance is determined by Wall [6] to be: 3h Z 0:923n (139) Mills and Rhine [31] have done experiments on gasification slags, and obtained results for the total normal emittance, in the temperature range TZ1070– 1800 K. A constant value of 3Z0.83 was obtained for the entire temperature range. 4.3.3.1. Estimation of the emissivity. Now we will see how empirical equations for emissivities of particles of different sizes, obtained from the Boow and Goard data [22], can be used to predict the emissivity of an ash. These predictions will be compared to total emissivities experimentally determined by Wall et al. [7], and Bhattacharya [8]. The iron-content will be included, knowing the following experimental result: 3(5% Fe2O3)/3(1%Fe2O3)Z1.12. Ash deposits in all experiments have particle sizes less than 44 mm (Bhattacharya [8]: 3.06–11.43 mm; Wall and Becker [52]: Morwell 22 mm, Tarong 9 mm). In order to predict the emissivity, correlation equations were used. Analysis of the results shows: † Correlation equations for smaller particles underestimate experimental results, except in the case of the Morwell coal ash data, in which the iron content is very high (around 10%). Its average error is around 30%. † Correlation equations for larger particles are generally able to predict the emissivity values, except in the case of the Morwell coal ash data, when it seriously overpredicts them. The average error is around 5%. Since different studies [5,22,53] showed that particle size has a big influence on the value of the emissivity of 407 a deposit, experimental results obtained by Boow and Goard [22] given in Fig. 6, can be used for the correlation. They used synthetic slag samples with the same chemical composition (5% Fe2O3), but different particle size. The procedure, which is used in this study, in order to estimate the emissivity vs. particle size and ironcontent dependency, is: † At temperatures below the sintering temperature, the emissivity is determined using the correlating equations based on the Boow and Gorad [22] data. † At temperatures above the fusion temperature, i.e. for fused and molten deposits, a constant value of 0.955 for the normal emissivity can be used. For estimating the hemispherical emissivity, Eq. (139) will be used. † Between the sintering and the fusion temperatures, the linear correlation between 3sint and 3fused developed using the Boow and Goard [22] data, can be used. † The concentration of iron will be taken into account by using the empirical value of 1.12 for ratio 3(5% Fe2O3)/3(1%Fe2O3). 4.3.3.2. Investigation of the chemistry influence on the emissivity. An attempt was made to calculate the emissivity based on the chemical composition. Emissivity data, provided by Wall et al. [6] and Touloukinan [47], for pure oxides, were used for the predictions, and the results were compared with the experimental data. The emissivity of ash deposits is determined as the emissivity of a mixture (the sum of the weighted average of the compounds) of pure oxides, present in the ash. A big drawback of this approach is that it does not give the dependence of the emissivity on the particle size. Total emissivities of different coal ashes, determined by Wall et al. [7], and ashes with different iron content, determined by Bhattacharya [8], will be used for comparison. The results are shown in Figs. 48 and 49 and from these it can be concluded: † The results obtained using the Touloukian [47] emissivity data, largely overestimate the experimental data. † The results obtained using the Wall et al. [6] emissivity data, underestimate the experimental data. † In the data obtained from Wall’s results (average error around 20%), a better agreement is obtained on the estimate for ash with lower iron content. 408 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Emissivity Emissivity, data: Bhattacharya, 1997 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 700 900 1100 1300 T, K 1500 1700 Exp (Bhattacharya, 1%Fe) Touloukian, 1%Fe Wall, 1%Fe Exp(Bhattacharya, 5%Fe) Touloukian, 5%Fe Wall, 5%Fe Fig. 48. Data obtained from Bhattacharya [8] compared to emissivity obtained by weighted average based on data of Touloukian [47] and Wall et al. [6]. † This method cannot correctly predict the dependence of the emissivity on the iron-content. the models of Rayleigh, i.e. Eq. (21), Russel (22), Woodside (25), and Diesler and Eian. Figs. 59 and 60 show that the Rayleigh (LR) and Russel (R) models give similar results. The Woodside (W) model and the model by Deisler and Eian’s (DC E), deviate considerably from (LR), particularly for large p and ko/km. Anderson et al. [3] have performed experiments on fly ash, slags and particulate (fouling) deposits (13.5– 1015 mm), in the temperature range: 300–1420 8C. The equation for estimating the overall thermal conductivity, which represents an average conductivity for the entire ash sample, is ð ð keo h kðTÞdT= dT ZKqH=DT (140) where keo (W/m K) is the overall effective thermal conductivity, q (W/m2) is the heat flux, H (m) is the ash sample thickness, and DT (K) is the difference between extrapolated surface temperatures. The fly ash samples were prepared by loading the asreceived fly ash into the test cell tray. Samples were not 5. Experimental validation of models Laubitz [10] measured the effective thermal conductivity of several powders in the temperature range of 100–1000 8C (see Section 4.1.2.). Figs. 50 and 51 show some of the data on k/km, where km is the conductivity of the solid phase, and k is the effective thermal conductivity of the two-phase medium, obtained from Emissivity: Wall, 1984 1.2 emissivity 1 0.8 0.6 0.4 0.2 0 700 900 1100 T, K 1300 1500 Exp (Wall, Morwell) Touloukian, Morwell Wall,Morwell Exp (Wall, Tarong) Touloukian, Tarong Wall,Tarong Fig. 49. Experimental data reported by Wall et al. [52], compared to estimated values obtained by weighted average based on data of [47] and [6]. Fig. 50. Variation of the effective thermal conductivity with the discrete phase volume fraction, for a fixed value of solid-to-gas conductivities, ko/kmZ100 [10]. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 409 Fig. 52. Overall effective thermal conductivities vs. T, for fly ash [3]. values: ksp Asp kp Ap kh Afc Z C H H H Fig. 51. Variation of the effective thermal conductivity with the ratio of solid-to-gas conductivity, for fixed discrete phase volume fraction, pZp/6 [10]. compacted during the loading process. Fig. 52 shows that the magnitude of keo is quite small, indicating that fly ash will create a large thermal resistance if it is allowed to accumulate on heat transfer surfaces. Results for sample A and B are the same, up to a mean ash temperature of approximately 650 8C despite the disparity in particle size, porosity and silica ratio. For crushed slags the influence of radiation can be neglected for particles up to 100 mm. From Fig. 53, it can be concluded that for smaller particles, the thermal radiation becomes significant at higher temperatures. When Figs. 53 and 54 are compared, it can be seen that the thermal conductivities are approximately twice as large for the larger particles, which is due to the effect of radiative transfer. Particle size is not an important parameter for particles up to 715 mm, but above this particle size, the thermal conductivity an increases with increase in the particle size. Effective thermal conductivities of fouling (superheater) deposits (2/3 solid-porous, 1/3 particulate) were measured. The thermal conductivity is the sum of two (141) where A [m2] is the area, and subscripts represent sp solid-porous, h hybrid and p particulate sample. The results are shown in Fig. 55. From Figs. 60–62 we can conclude that the effective thermal conductivities of all samples were less than 0.5 W/m K and that they were relatively independent of temperature. Results from the crushed slag experiments are compared with the data of Mulcahy and Singer, and presented in Fig. 56. This comparison showed good agreement between experiments and results obtained from the models at temperatures below 1000 8C. Above 1000 8C, a large discrepancy is noticeable, which can be explained by a slightly different silica-content (Anderson: 46%, Mulcahy: 41%) and a difference in the particle size applied (Anderson: 420,715 mm, Mulcahy: 200 mm, smaller particles can be expect to sinter and melt sooner). Fig. 53. Local thermal conductivty for crushed-wall slag deposits with Dp%715 mm [3]. 410 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Fig. 54. Local thermal conductivities for slag deposits with DpZ 1015 mm [3]. Robinson et al. [2] performed experiments under thermal conditions that closely replicate those found in the convective pass of a commercial boiler (see Section 4.1.1). They described a layered structure of the deposit. The outer layer actually goes through a change during sintering, while the inner layer mostly stays unchanged. For kun, a constant value of 0.15 W/m K (the thermal conductivity of deposit at the start of the sintering stage of the experiment) is used, while ksi varies between kun and 3 W/m K (the thermal conductivity of deposit solid phase). With a temperature increase, ksi increases (due to sintering in the outer layer), while kun remains constant. According to the results, once ksi is roughly 6 times greater than kun, the overall thermal conductivity of the deposit is insensitive to changes of ksi. A sensitivity analysis reveals that if L is grater than 0.2, and the ratio of ksi-to-kun is greater than about 3, and then the critical parameter is kun. The authors use the constant value of 0.15 W/m K for kun, which is the Fig. 55. Local thermal conductivities for hybrid and particulate fouling deposits [3]. Fig. 56. Comparison of present data with those of previous experiments [3]. measured deposit thermal conductivity at the beginning of the sintering. Since, in practice, the value of the overall deposit thermal conductivity is relatively insensitive to the assumed functional form of ksi, it may be stated that ksi varies linearly with the solid fraction of the sintered layer. Now, the effective thermal conductivity is largely determined by the thermal conductivity of the unsintered inner layer. The relatively low temperatures (!600 8C—the temperature of a heat transfer surface in a typical boiler) will most likely prevent substantial sintering of the innermost layer. During this experiment, the average deposit temperature was increased by approximately 200 8C. The averaged measured thermal conductivity was w0.14 W/m K. Since these deposits were unsintered and loose, it can be expected that this value represent the lower extreme of the range of deposits that might form in real boilers. According to Fig. 57, the success of the model, supports the conclusions about the importance of a layered deposit structure in determining the effects of sintering on the deposit thermal conductivity, and confirms that an unsintered layer with a low value of thermal conductivity, will limit the overall deposit thermal conductivity. Fig. 57e also compares the measurements to theoretical limits the deposit thermal conductivity. The thick vertical line indicates the transition between the growth and the sintering phases of the experiment. Vertical bars in (e) indicate estimate of experimental uncertainty, G20%. Fig. 58 shows a comparison of the measurements of the deposit thermal conductivity shown in Fig. 57e to predictions of the two-layered structure model as a function of time. Fig. 59 shows measurements of the deposit solid fraction and the deposit effective thermal conductivity as a function of time. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 411 Fig. 58. Comparison of the measurements of deposit thermal conductivity shown in Fig. 57e, to predictions of the two-layered structure model, as a function of time. The dashed lines show the changes in the thermal conductivity of the unsintered, kun, and sintered, ksi, layer of the deposit. The elapsed time corresponds to the sintering stage of the experiment. Vertical bars indicate an estimate of the experimental uncertainty on selected data points [2]. in the literature. Most of the experiments were done on crushed deposits whose original microstructure, which is very important for both the thermal and the radiative properties, is destroyed. Robinson et al. [2] developed a technique for measuring thermal conductivity of ash deposits in situ on the original microstructure. Experimental data from the work of Rezaei et al. [1] is used in order to evaluate the results obtained from the different models. In this study, coal ash deposits were used, whose content is given in Table 10. 6.1. The thermal conductivity of solid material Fig. 57. Measurements of (a) average deposit thickness, (b) average probe and deposit surface temperature, and (c) deposit solid fraction, (d) deposit structural parameter, and (e) deposit effective thermal conductivity as a function of time [2]. 6. Thermal conductivity of ash deposits obtained by different models Very few experimental data of the (effective) thermal conductivity of ash deposits, especially at high temperatures where radiation is important, are available Three different approaches for determining the thermal conductivity of the solid phase of an ash deposit will be examined: (1) the weighted-average of component oxides, (2) the Kobayashi model, and, (3) the Rezaei empirical model. The model predictions are compared to experimental data derived on a highly sintered ash sample A, with porosity pZ0.15, as shown in Fig. 60. The analysis shows that a fused ash sample, with a low porosity will have a thermal conductivity, which is constantly increasing with increasing temperature, and that the experimental data are lower than any of the model predictions. The experimental data is best described by the Rezaei model. Even though the weighted-averaged model gave poor results, it will be used in the predictions instead of the empirical model. 412 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 All the models underpredict the thermal conductivity of the unsintered deposit. For the sample with pZ 0.30, the difference is around 45% up to the sintering point, above which difference rapidly increases. The thermal conductivity of the sintered ashes is over predicted. The Russel model predicts the highest values for thermal conductivity, the CSP model predicts the lowest values, and the Rayleigh model lies in between. The experimental data show that the thermal conductivity decreases with the porosity, but with much less sensitivity to this parameter than the models predict. By comparing the curve of the thermal conductivity of the solid with the curve of the thermal conductivity of porous material, we can see the great similarity between them. This indicates that the models used here predict great influence of the thermal conductivity of the solid phase on the effective thermal conductivity of fused ash deposits. 6.3. Complex structure models Fig. 59. Measurements of (a) the deposit solid fraction, and (b) the deposit effective thermal conductivity, as a function of time. These measurements were conducted under the same conditions, but terminated after different periods of sintering (0, 1, 4, or 12 h) in order to generate samples for SEM analysis of the deposit microstructure. The vertical line at 0 h indicates the start of sintering portion of the experiment. The theoretical limits are calculated from the solid fraction of the deposit sintered for 12 h. The elapsed time for some of the thermal conductivity measurements has been slightly shifted for visual clarity. Vertical bars indicate experimental uncertainty for selected data points [1]. 6.2. Simple models for the thermal conductivity Predictions by the Rayleigh, Russell and CSP models (Eqs. (21), (22) and (34)) will now be compared to experimental data. Fig. 61 shows the comparison for two porosities of unsintered deposit B, while Fig. 62 shows predictions for two porosities of sintered deposit B. Complex structures, which assume the continuity of both the solid and the gas phase, would be expected to give better estimations of the thermal conductivity of sintered deposits. The Brailsford and Major, the Hadley and Nimick, and the Leith models (Eqs. (46), (57) and (62)) will now be compared to the experimental data for sintered deposits. Fig. 63 shows the comparison for two porosities of the sintered deposit A. Based on the comparison shown in Fig. 63, the following can be concluded: † The Nimick and Leith model gives the highest predictions of the thermal conductivity. † Both the Nimick and Leith, and the Brailsford and Major models are greatly influenced by the thermal conductivity of the solid phase, which is evident from the shape of the resulting curves. Also, the models are greatly influenced by the porosity, which is not detected in the experimental data. † Predictions based on the Hadley model give fair agreement with experimental results. This model shows a big sensitivity to porosity, so the agreement Table 10 Ash content from the study of Rezaei et al. [1] Sample name SiO2 Al2O3 Fe2O3 CaO MgO K2O Na2O TiO2 dp (mm) Sintering temp, 8C Fusion temp, 8C A B 34.37 48.49 23.75 22.95 4.85 7.07 13.41 4.77 4.32 1.36 0.95 0.66 5.04 0.88 1.22 1.01 8 20 600 670 1200 1440 413 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Thermal conductivity of solid Sample A; sintered; p=0.30 and p=0.15 10.0 7 6 8.0 k, W/mK k, W/mK 5 6.0 4.0 4 3 2 2.0 1 0 0.0 0 300 600 Rezaei 900 T, C WT 1200 Kobayashi 1500 with the experimental data actually depends on porosity. 6.4. Radiative heat transfer models 6.4.1. The exchange factor The exchange factor, as given in Boterill [27,28], is determined using the Laubitz, the Godbee and Ziegler, the Zehner, the Bauer and Schlunder, and the Kunii and Smith models. These predictions are shown in Fig. 64. We can see that the Goodbee and Ziegler model and the Laubitz model (additive models) predict an increase in the exchange factor with increasing porosity. Resistance network models like the Zehner, and the Bauer and Schlunder model, predicts that exchange the factor does not depend on the porosity, and the Kunii Sample B; unsintered; p=0.30 and p=0.50 1.2 300 900 Rayleigh (1) Rayleigh (2) 1500 T, C 1800 Russell (1) Russell (2) CSP (1) CSP (2) 1200 Fig. 62. Experimental results (sintered ash) compared to models predictions. and Smith model predict that the exchange factor will decrease with a porosity increase. Fig. 65 shows the predictions for the radiative conductivity of a porous material. At lower temperatures, the radiative conductivity slowly increases with the temperature, and then at temperatures above app. 600 8C, the radiative conductivity starts to increase more rapidly. The variation of the radiative thermal conductivity, with porosity, for the different models is the same as the variation of the exchange factor. Again the Kunii and Smith model predict that radiative conductivity will decrease when the porosity increases, but the reason for that can be that the exchange factor predicted by their model is more sensitive to the value of the emissivity. We can also see that absolute values of the radiative conductivity are very small, so according to these models, this value will not affect the effective thermal conductivity much, not even at high temperatures. On-set of sintering 1 600 exp,p=0.3 (1) exp, p=0.15 (2) exp Fig. 60. Thermal conductivity of solid obtained by different models. Sample A; sintered; p=0.30 and p=0.15 6.0 0.8 5.0 0.6 4.0 k, W/mK k, W/mK 0 1800 0.4 0.2 3.0 2.0 1.0 0 0 300 600 900 1200 1500 1800 exp, p=0.30 (1) exp, p=0.50 (2) Rayleigh (1) Rayleigh (2) T, C 0.0 0 T, C Russell (1) Russell (2) CSP (1) CSP (2) Fig. 61. Experimental results (unsintered ash) compared to models predictions. 300 600 900 1200 1500 1800 exp, p=0.30 (1) B&M (1) N&L (1) Hadley (1) exp, p=0.15 (2) B&M (2) N&L (2) Hadley (2) Fig. 63. Experimental results (sintered ash) compared to models predictions. 414 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Sample A; particulate; p=0.29 and p=0.58 2 1.2 1.6 0.8 1.2 k, W/mK Exchange factor Exchange fator; p=0.2 and p=0.5 1.6 0.4 0.4 0.0 0 300 600 G&Z; p=0.2 900 T, C KS; p=0.2 L; p=0.2 1200 1500 1800 0 L; p=0.5 G&Z; p=0.5 KS; p=0.5 6.4.2. Additive models The Laubitz, and the Godbee and Ziegler models, i.e. Eqs. (64) and (71), will be compared to experimental results, below. Since the Laubitz model uses twice the value of the Russell’s thermal conductivity, which overpredicts the values of the thermal conductivity of sintered ash, the Laubitz model will be used only for particulate deposits. Fig. 66 shows the Laubitz predictions for particulate ash sample A. If the values of the Russell model predictions for the thermal conductivity are compared to the radiative conductivity, it can be seen that the value of radiative conductivity is almost insignificant in comparison to twice the value of the Russell’s conductivity. Fig. 67 shows the Godbee and Ziegler predictions for sintered ash sample A. The Godbee and Ziegler model overpredicts the experimental results, for both the sintered and the fused Radiative conductivity, p=0,2 and p=0,5 0.025 0.020 0.015 0.010 0.005 0.000 0 300 600 900 0 300 600 ZBS; p=0.5 Fig. 64. Exchange factor, determined using different models, for porosities pZ0.2 and 0.5. kr, W/mK 0.8 1200 1500 1800 T, C GZ, p=0.2 KS, p=0.2 L, p=0.5 KS, p=0.5 L, p=0.2 GZ, p=0.5 ZBS, p=0.5 Fig. 65. Radiative conductivity determined for two different porosities, when different model for exchange factor are used. exp, p=0.29 (1) 900 T, C L (1) 1200 1500 exp, p=0.58 (2) 1800 L (2) Fig. 66. Predictions based on Laubitz model compared to experimental data. deposits. Furthermore, the model shows great sensitivity to porosity. 6.4.3. Resistance network models The following resistance network models will be tested in action: the Yagi and Kunii model (particulate deposits, Eq. (100)), the Kunii and Smith model (particulate deposit, Eq. (104) and fused deposits, Eq. (106)), and the Zehner, Bauer and Schlunder model (particulate deposit, Eq. (107)). These models use parameters that should be determined experimentally. In the present work, constant values for these parameters will be assumed, which are chosen as averaged values used in the original studies, or have already proposed by the authors. Most of these models are developed for porous media where the continuous phase is a gas, which may explain why the influence of the thermal conductivity of the solid phase is less important. The results obtained using the Yagi and Kunii model are shown in Figs. 68 and 69. The Yagi and Kunii model is due to its semiempirical character applied only to unsintered ash sample, where gas is the continuous phase. The following can be concluded: † The Yagii and Kuni model gives far better agreement with the experimental results than the models that were previously used. † This model gives results that are influenced more by porosity than the experiments showed to be the real case, but less than the models previously tested. 415 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Sample A; sintered; p=0.30 and p=0.15 Sample B; particulate; p=0.30 and p=0.50 1.6 7 6 1.2 k, W/mK k, W/mK 5 4 3 0.8 0.4 2 1 0 0 0 0 300 600 exp, p=0.30 (1) 900 T, C GZ (1) 1200 1500 exp, p=0.15 (2) 1800 GZ (2) Fig. 67. Predictions based on the Godbee and Ziegler model compared to experimental data, corresponding to two different porosities. † The difference between the results obtained using bZ1 and 0.90 are about 10%. † The results for the higher porosities are in better agreement with the experiments when bZ1, and for lower porosities when bZ0.9. Kunii and Smith developed models, which can be applied for unsintered (Eq. (104)) as well as sintered deposits (Eq. (106)). These models are also semiempirical with a parameter b, which is to be determined experimentally. Since the authors suggested that this parameter should take values between 0.9 and 1.0, the influence of this parameter will be investigated further, below. Fig. 70 shows predictions of this model for particulate ash deposits, and Fig. 71 for sintered deposits. Use of different values of b leads to a difference in thermal conductivity value of approximately 5% (between bZ0.9 and 1). This model is again very sensitive to a change in the porosity, for lower porosities the thermal conductivity of the ash is 300 600 exp, p=0.30 (1) exp,p=0.50 (2) 900 B=1 (1) B=1 (2) 1200 1500 B=0.95 (1) B=0.95 (2) T, C 1800 B=0.9 (1) B=0.9 (2) Fig. 69. Predictions based on the Yagi and Kunii model, compared to exp. data. overpredicted, and for higher porosities, the thermal conductivity was underpredicted. The predictions for sintered deposits, based on the Kunii and Smith model, show a great sensibility to the thermal conductivity of the solid phase. This model includes two porosity parameters, f and f 0 (before and after sintering). Three different cases were shown considering these porosities: KS, 1: the two porosities are the same; KS, 0.9: a reduction in porosity of 10% (f 0 /fZ0.9); KS, 0.8: a reduction in porosity of 20% (f 0 /fZ0,8). The difference between the case with no reduction in porosity and 20% reduction is about 20%. Finally the Zehner, Bauer and Schlunder model will be presented. Instead of the experimentally determined parameters in this model, their values are approximated as constants: rkZ0 (contact conduction is neglected), CsphereZ1.25, RsphereZ1 (Sox/kox)Z0 (an oxidation factor which is assumed to be zero). Results are shown in Fig. 72. The trend of the results obtained using the Zehner, Bauer and Schlunder model, is quite similar to the trend Sample A; particulate; p=0.29 and p=0.58 2 Sample A; particulate; p=0.29 and p=0.58 1.6 1.6 k, W/mK k, W/mK 1.2 0.8 0.4 1.2 0.8 0.4 0 0 300 600 exp, p=0.29 (1) exp,p=0.58 (2) 900 B=1 (1) B=1 (2) 1200 1500 B=0.95 (1) B=0.95 (2) 1800 T, C B=0.9 (1) B=0.9 (2) Fig. 68. Predictions based on the Yagi and Kunii model, compared to experimental data. 0 0 300 600 exp, p=0.29 (1) exp, p=0.58 (2) 900 B=0.90 (1) B=0.90 (2) 1200 1500 B=0.95 (1) B=0.95 (2) T, C 1800 B=1 (1) B=1 (2) Fig. 70. Predictions based on the Kunii and Smith model, for a gascontinuous porous material, compared to experimental data. 416 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Sample A; sintered; p=0.15 and p=0.30 3 k, W/mK 2.5 2 1.5 1 0.5 0 300 600 exp, p=0.15 (1) exp, p=0.30 (2) 900 KS, 1 (1) KS, 1 (2) 1200 1500 KS, 0.9 (1) KS, 0.9 (2) T, C 1800 KS, 0.8 (1) KS, 0.8 (2) Fig. 71. Predictions based on the Kunii and Smith model for solidcontinuous porous material, compared to experimental data. of experimental results. The absolute values of the thermal conductivity are much closer at higher, than at lower temperatures. This model also shows great sensitivity to the porosity, so for low porosities it overpredicts thermal conductivity of ash, while it underpredicts the thermal conductivity at higher porosities. 7. Conclusion The aim of this work has been to review the present knowledge on heat transfer in porous media, which can be applied to estimate the thermal conductivities of ash deposits. Besides the different approaches to heat conduction modeling, determination of the thermal conductivity of solid mixtures and the ash deposit surface emissivity has been presented. Although the weighted-average method did not give satisfactory results, it was anyway used in the discussion, due to the lack of other available models. Thus, it is highly Sample A; particulate; p=0.29 and p=0.58 1.6 Appendix A. Calculation of heat transfer resistances The total heat transfer between a flue gas, and the steam inside a superheater tube consists of convective and radiative heat transfer from the hot flue gases (g) to deposit surface (d), heat conduction through deposit and tube wall, and convection from the inner tube wall to the steam (s). The heat transferred (W/m), can be expressed in terms of heat resistances: qL Z KL ðTg KTs Þ Z 1.2 k, W/mK recommended that the thermal conductivity of solid phase, for the particular ash sample, is measured, and an empirical correlation is derived. A method for obtaining surface emissivities was proposed, based on the particle size and iron content of the deposit. It was found that the major factor influencing the thermal conductivity of an ash deposit is the deposit structure, i.e. if the deposit is particulate, sintered or fused. Different models were tested using experimental results for two coal ash samples, obtained by Rezaei et al. [1]. It was found that simple structure models do not give satisfactory results for both nonsintered and sintered samples. Particulate deposits can be modeled using the Yagi and Kuni semiempirical model, while sintered deposits should be modeled using the Hadley complex structure model. Experimental values for fused deposits were not available, but the suggestion is that they can also be modeled as sintered deposits, using the Hadley model. Finally, it can be concluded that a wide range of thermal conductivity models exist, but the need exists for a wide range of experimental data, which would help evaluate and improve these models. Also, it is necessary to formulate a more accurate model for the thermal conductivity of solid mixtures, which is the property that can be identified as a potential important source of errors. 0.8 Z Tg KTs rc Tg KTs rconvCrad C rcond;dep C rcond;tube C rconv;tubesteam (A1) 0.4 T, C 0 0 300 exp, p=0.29 (1) 600 900 ZBS (1) 1200 1500 exp, p=0.58 (2) 1800 ZBS (2) Fig. 72. Predictions based on the Zehner, Bauer and Schlunder model for thermal conductivity, compared to experimental data. where rc [m K/W] is the thermal resistance to heat transfer through the cylindrical wall and KL (W/m K) is heat transfer coefficient. Heat transfer resistances originating from different mechanisms equal one another over normalized heat transfer coefficients, as will be shown. 417 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 The net heat transfer between a flue gas with temperature Tg, and a deposit surface, with temperature Td is The heat transfer characteristics were calculated: qL (W/m) hconvCrad (W/m2 K) kd (W/m K) 1238.47 126.74 0.6328 qL Z qconv C qrad Z hconvCrad ðTg KTd Þpdd Tg KTd Z 1 hconvCrad pdd Z rconvCrad ðTg KTd Þ where dd equals sum of the tube diameter and the deposit thickness. The heat transfer coefficient in this case, i.e. hconvCrad [14,17], equals the summation of hconv, i.e. heat transfer coefficient calculated from the criterial equations and hrad, calculated as: hrad Z 3d sðTg4 KTd4 Þ ðTg KTd Þ (A3) The heat conducted through the deposit to the outside tube wall of temperature Tto is qL Z Td KTto Z rcond;dep ðTd KTto Þ dd 1 2pkd ln dto (A4) where kd is the heat conductivity of the deposit layer, dto is the outer tube diameter, and rcond,depZ1/Kcond,dep is a heat transfer resistance to conduction through the deposit. The heat conducted through the tube wall is qL Z Tto KTti Z rcond;tube ðTto KTti Þ dto 1 2pkt ln dti (A5) where Tti is the temperature of tube inner wall, dti is the tube inner diameter, kt is the heat conductivity of the tube, and rcond,tubeZ1/Kcond,tube is the heat transfer resistance to conduction through the tube. The heat transfer between tube wall and steam can be written as qL Z hs ðTti KTs Þpdti Z ðTti KTs Þ 1 hs pdti Z rconv;tubesteam ðTti KTs Þ The obtained temperature profile is: (A2) Tg (C) Td (C) Tto (C) Tti (C) Ts (C) 900 642.86 570.09 567.69 550 The heat transfer resistances are: rconvCrad (%) rcond,dep (%) rcond,tube (%) rconv,tube-steam (%) 41.10 46.16 1.53 11.21 It is obvious that the major resistances to heat transfer in this system, is the convective heat transfer from flue gases to deposit surface, and heat conduction through the deposit. Appendix B. Maximum discrete phase fraction in thermal conductivity models based on spheres [55] Application of thermal conductivity models, which use spheres to present the discrete phase, is limited to a volume fraction of the discrete phase of p/6. The spheres in these models are arranged in simple cubic cell as shown in Fig. B1. In this case, the overall structure is shown in Fig. B2. This lattice consists of layers of square packed spheres. The layers are stacked so that each sphere is directly above the one in the layer beneath. Since the spheres in the lattice touches each other, the edge length of the cell is two times the sphere radius, r. In this 3D model the packing efficiency, which is equal to the discrete volume fraction, is given by: PE Z ðvolume of spheresÞ=ðvolume of cellÞ (A6) where rconv,tube-steamZ1/Kconv,tube-steam is the heat transfer resistance to convection. Heat transfer coefficients for convection can be calculated using correlations from Section 2. The heat transfer resistances, according to the above equations, were calculated using an iteration method, with a convergence criteria for Tto to be 20 8C higher than the steam temperature. This value is empirical. The heat transfer coefficient for convection is calculated using Eq. (6). Operational conditions, deposit, and tube characteristics are listed in Table B1. For a simple cubic lattice, this is:  1  4 3  8 3r p PE Z 8 Z p=6 Z 52:35% ð2rÞ3 (B1) Appendix C. Calculation of the contribution of radiation to the effective conductivity of powders In the Russell model (1935), the obstacles are assumed to be cubes of length a, located in cubical volumes of the medium of length d. It is then assumed that the obstacles are randomly scattered in the cubical volumes, but in such a manner that the faces of the two 418 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Table B1 A1 Input data for calculation Tgas (C) Tsteam (C) dt (m) u (m/s) kt (W/m K) dep.type p dthick (m) 900 550 0.038 8 25 Fused 0.3 0.005 cubes remain parallel. Such a scattering does not affect the thermal conductivity of a two-phase medium. If the probability that radiation will pass a cube without hitting an obstacle is r, then: LZ a p (C4) The rate of the heat transfer by radiation is a2 r Z 1K 2 d (C1) If L is the average length that radiation travels without hitting an obstacle in the first cube, then: L Z d C rd C r 2 d C/Z d d3 Z 2 1Kr a (C2) If p is the particle volume fraction, then pZ and a3 d3 (C3) Q Z 4sT 3 3ðd 2 Ka2 ÞDT   a2 DTd2 Z 4sT 3L 1K 2 L d 3 (C5) where T is the absolute temperature, 3 is the emissivity of the obstacles, and DT/L is the gradient in the twophase medium, assumed small compared to T. Therefore, the contribution of that radiation effective conductivity is:   a2 a 3 4sT 3L 1K 2 Z 4sT 3 3 ð1Kp2=3 Þ (C6) p d If it is assumed that the radiation that does hit an obstacle in the first cube travels on the average a distance of (dKa), then the contribution by the radiation that does hit an obstacle in the first cube of the medium can be calculated: The rate of heat transfer by this process is Fig. B1. Simple cubic unit cell. QZ a2 dKr K0 DT aKr C LK0 d (C7) where Kr Z 4sT 3 3L (C8) and its contribution to the effective conductivity is a2 Kr K0 p1=3 Kr K0 Z dðaKr C LK0 Þ Kr C K0 =p (C9) Since K0/p[Kr, then Eq. (C9) becomes p4=3 Kr Z p4=3 4sT 3 3 a p (C10) and the total effective conductivity a K Z KðRÞ C 4sT 3 3 ð1Kp2=3 C p4=3 Þ p Fig. B2. Overall structure in simple cubic cell. (C11) where K(R) is the two-phase conductivity calculated by Russell. A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 Appendix D. Qualitative explanation for using doubled value of k(R) in the Laubitz model In actual powders, there are densely packed regions where particles touch each other (contribute to heat transfer chiefly by conduction), and empty air pockets (contribute by radiation). The air pockets have roughly a length of a/p0 (shown in Appendix C), where a is the linear dimension of the particles, and p0 the average particle volume fraction from the aggregate. But when the p-value for the densely packed region is not p0 but p01=3 , then the equivalent conductivity of the powder becomes KðR; p01=3 Þp02=3 , where KðR; p01=3 Þ is the Russell’s conductivity calculated for pZ p01=3 . For the powders used in this experiments KðR; p1=3 Þp2=3 z2KðR; pÞ which gives a rough reason for the factor of two used in Eq. (64). 419 The distance between the centres of the particles can be related to the volume as follows:  3   lp V Z (E3) lpo V0  This gives  1Kp 1=3 lp0 Z 1Kp0 lp (E4) or lp Z lp0  1Kp0 1Kp 1=3 (E5) The last equation relates the distance between particles when the porosity is varied. The initial value of p0 is set equal to the particle diameter. The minimum value of the porosity is calculated for bed of cylinders. Appendix F. Determination of the shrinkage level, based on geometry approach Appendix E. Estimation of the parameter lp in the Yagi and Kunii Model for the effective thermal conductivity The following method of parameter estimation is developed by Fjellerup et al. [34]. Fig. E1 indicates the expansion of a volume partly filled with particles from V0 to V. The amount of particles is constant and it is assumed that the particles will spread homogeneously to volume V. The porosity and the distance between particles will change, and a relation between these two parameters will be derived. The porosity is the porosity between particles (external porosity), and po and p is the porosity before and after expansion, respectively. Parameters lpo and lp are the distance between particle centres before and after expansion, respectively. The initial solid volume equals the final solid volume and the following equation can be derived Vð1KpÞ Z V0 ð1Kp0 Þ (E1) which gives x2 C r 2 Z r02  x 2 r C1 Z (F1)  r 2 0 (F2) r V0 Z ð4=3Þr03 p Z m=r0 (F3) V Z ð4=3Þr 3 p Z m=r (F4)  x 2 r 0 3m 4r0 p B Z@ 3m 4rp 2=3 1 (F5) C 2=3 A K1  2=3 1=2 x r Kr2=3 0 Z r r02=3 (F6) A powder density can be expressed as a function of porosity p and density of solid: r0 Z rs ð1Kp0 Þ 1Kp V Z 1Kp0 V0 (F7) (E2) ro x x – radius of the neck r0 – initial sphere radius r – sphere radius at time t r Fig. E1. Expansion of a porous bed. Fig. F1. Part of Fig. 46. x: radius of the neck; r0: initial sphere radius; r: sphere radius at time t. 420 A. Zbogar et al. / Progress in Energy and Combustion Science 31 (2005) 371–421 r Z rs ð1KpÞ (F8) Using Eqs. (F6)–(F8) porosity at time t can be calculated as:   x 2 3=2 p Z 1Kð1Kp0 Þ 1 C (F9) r Appendix G. Kirchhoff’s Law Kirchhoff’s law can be written as EðTs Þ Z Eb ðTs Þ a (G1) where Ts is the surface temperature, E(T) is the emission from a body at Ts, a is the absorptivity and Eb(T) is the emission from a blackbody, at the temperature Ts. According to Eq. (G1), ratio of emitted energy by a body at the surface temperature and its absorption ability does not depend on physical characteristics of a body; it is the same for all gray bodies and equals to the emission of the blackbody at the surface temperature. Equation for the total, hemispherical emissivity is: 3ðTÞ Z EðTÞ Eb ðTÞ (G2) Combining Eqs. 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