Numerical Modeling and Simulation of a Double Tube Heat Exchanger Adopting a Black Box Approach

Agniprobho Mazumder. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 2) April 2016, pp.35-41
www.ijera.com 35 | P a g e
Numerical Modeling and Simulation of a Double Tube Heat
Exchanger Adopting a Black Box Approach
Agniprobho Mazumder*, Dr. Bijan Kumar Mandal**
*(Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur,
Howrah, West Bengal 711103, India,
** (Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur,
Howrah, West Bengal 711103, India,
ABSTRACT
The double tube heat exchangers are commonly used in industry due to their simplicity in design and also their
operation at high temperatures and pressures. As the inlet parameters like temperatures and mass flow rates
change during operation, the outlet temperatures will also change. In the present paper, a simple approximate
linear model has been proposed to predict the outlet temperatures of a double tube heat exchanger, considering it
as a black box. The simulation of the heat exchanger has been carried out first using the commercial CFD
software FLUENT. Next the linear model of the double tube heat exchanger based on lumped parameters has
been developed using the basic governing equations, considering it as a black box. Results have been generated
for outlet temperatures for different inlet temperatures and mass flow rates of the cold and hot fluids. The results
obtained using the above two methods have then been discussed and compared with the numerical results
available in the literature to justify the basis for the assumption of a linear approximation. Comparisons of the
predicted results from the present model show a good agreement with the experimental results published in the
literature. The assumptions of linear variation of outlet temperatures with the inlet temperature of one fluid
(keeping other inlet parameters fixed) is very well justified and hence the model can be employed for the
analysis of double tube heat exchangers.
Keywords – Black Box, Heat Exchanger, Linear Model, Numerical Modeling
NOMENCLATURE
T Temperature of fluid (o
C)
ṁ Mass flow rate of fluid (kg/s)
Cp Specific heat capacity of fluid (J/kg-o
C)
A Area (m2
)
U Overall heat transfer coefficient (W/m2
-o
C)
h Convection coefficient (W/m2
-o
C)
k Thermal conductivity of tube material
(W/m-o
C)
d Diameter (m)
L Length (m)
Re Reynolds number (-)
Pr Prandtl number (-)
Nu Nusselt number (-)
ρ Density of fluid (kg/m3
)
μ Dynamic viscosity of fluid (kg/m-s)
λ Thermal conductivity of fluid (W/m-o
C)
ε Effectiveness of heat transfer (-)
Subscripts
1 Inlet, interior
2 Outlet, exterior
h Hot fluid
c Cold fluid
in Internal tube
an Annular space
I. Introduction
Heat exchanger is a device which controls
the temperature of a system or a substance by adding
or removing thermal energy. Although there are
different types of heat exchangers with varying sizes
[1, 2], they have a basic similarity. All of them use a
thermally conducting element usually in the form of
a plate or a tube to separate the two fluids, such that
one can transfer thermal energy to the other even
without being mixed. The double tube heat
exchanger involves two concentric tubes, where
generally, a hot fluid flows through the interior tube
and a cold fluid flows through the annular space.
This type of heat exchangers are widely used in food
and oil refinery industries and are an important
element of various types of installations like steam
power labs, heating and air conditioning systems.
The most widely used types of double tube heat
exchangers are counter flow heat exchangers
because of their high effectiveness. While there are
many advantages of double tube heat exchangers
like simple structures, operation in parallel and
counter flow, operation at relatively low flow rates,
low costs, etc., their disadvantages are related to low
values of the overall heat transfer coefficients that
leads to large heat transfer areas [3].
The mathematical modeling of heat
exchangers has been treated extensively in literature.
RESEARCH ARTICLE OPEN ACCESS

Heat exchanger models to predict the outlet
temperatures of the fluids [4] and a heat transfer
coefficient calculation program based on the EES
software for the double tube heat exchangers [5]
have been developed. The solution of the
mathematical models of the heat exchangers using
numerical algorithms has been performed [6].
Simulators based on these models have been
developed which allow the determination of the
outlet temperatures of the two fluids between which
the heat transfer is realized, by using data about heat
exchanger geometry and the values of inlet
temperatures and mass flow rates of the two fluids
[7, 8].
In the present paper the authors proposed
another approach for evaluating the outlet
temperatures of the double tube heat exchangers in a
simplified approximate linear form. In this work, the
heat exchanger has been considered as a „black box‟
to examine how the input variables affect the output.
1.1 Geometry and arrangement of the heat
exchanger
Heat exchangers are typically classified
according to flow arrangement and type of
construction. The simplest heat exchanger is one for
which the hot and cold fluids move in the same or
opposite directions in a concentric tube (or double-
tube) construction. The schematic diagrams of
double tube heat exchanger have been shown in Fig.
1-a and Fig. 1-b for the counter flow and parallel
flow arrangements of the two participating fluids
respectively. In the parallel flow arrangement, the
hot and cold fluids enter at the same end, flow in the
same direction, and leave at the same end. In the
counter-flow arrangement, the fluids enter at
opposite ends, flow in opposite directions, and leave
at opposite ends.
Figure 1-a: Flow arrangement of counter-flow
double tube heat
exchanger
Figure 1-b: Flow arrangement of parallel flow
double tube heat exchanger
The geometrical construction of the double
tube heat exchanger is as shown in Fig. 2. The hot
fluid is assumed to flow through the inner tube and
the cold fluid is assumed to flow through the annular
space.
Figure 2: Geometrical dimensions of the double
tube heat exchanger
II. Modeling And Simulation Using Ansys
(Fluent)
The modeling and simulation has been
carried out using a double tube counter-flow heat
exchanger using water as the hot as well as the cold
fluid. The geometrical dimensions and inlet
parameters of the heat exchanger to be investigated
are presented in Table 1 and Table 2 respectively.
The tubes of the heat exchanger being examined are
made of copper.
Table 1: Geometrical dimensions
Geometrical characteristic Variable Value
Interior diameter of the
interior tube [m]
di1 0.027
Exterior diameter of the
interior tube [m]
de1 0.030
exterior tube [m]
di2 0.040
exterior tube [m]
de2 0.043
Length of the tube [m] L 1
Table 2: Inlet parameters
No. mc [kg/s] mh [kg/s] Tc1 [o
C] Th1 [o
C]
1 0.03 0.1 17 52
2 0.03 0.1 18 52
3 0.03 0.1 19 52
4 0.03 0.1 20 52
5 0.03 0.1 21 52
6 0.03 0.1 17 51
7 0.03 0.1 17 50
8 0.03 0.1 17 49
9 0.03 0.1 17 48
10 0.04 0.1 17 52
11 0.05 0.1 17 52
12 0.06 0.1 17 52
13 0.07 0.1 17 52
The heat exchanger has been modeled using
Fluent in such a way that the interior of the heat
exchanger was visible for the ease of defining the
surfaces, symmetries, cell zone and boundary

conditions. In modeling the heat exchanger, the
outer surface was considered to be perfectly
insulated. The model after meshing has been
presented in Fig. 3.
Figure 3: Meshed model
The temperature distributions for the third
set of input as shown in Table 2 in the vicinity of the
cold fluid entry side (and simultaneously hot fluid
exit) and cold fluid exit side (and simultaneously hot
fluid entry) have been presented in Fig. 4-a and Fig.
4-b respectively.
Figure 4-a: Contour of static temperature (cold fluid
entry side)
Figure 4-b: Contour of static temperature (hot fluid
entry side)
The results of this simulation have been
discussed in detail and also compared with the
experimental results and that obtained using the
proposed linear model in section 4.
III. The Mathematical Model
To mathematical model is based on certain
assumptions. The flow is single phase and in steady
state regime. The heat transfer to the surrounding
environment is neglected, i.e., the outer surface of
the heat exchanger is perfectly insulated. The heat
exchanger is considered a system with lumped
parameters. Considering the heat balance between
the hot and the cold fluid and the heat transfer, the
following equations can be written:
phh
Cm (Th1 – Th2) = pcc
Cm (Tc2 – Tc1) (1)
phh
Cm (Th1 – Th2) = UATlm (2)
The logarithmic mean temperature difference, Tlm,
for counter-flow heat exchangers is given as:
and for parallel flow heat exchangers as:
The system of equations given by (1) and (2)
represents a system of transcendental equations with
two variables, having the form:
f1(Th2,Tc2) = phh
Cm (Th1 – Th2)– pcc
Cm (Tc2 – Tc1)
= 0 (5)
f2(Th2,Tc2) = phh
Cm (Th1 – Th2) – UATlm = 0 (6)
The solution of the system of transcendental
equations is rather complex. Therefore, in this paper
an attempt has been made to simplify the set of
equations, using certain assumptions to obtain the
outlet temperatures of the hot and cold fluids as
linear functions of the inlet temperatures.
3.1 Linear model using the black box approach
A black box model is a system which can
be viewed in terms of its inputs and outputs (or
transfer characteristics). The heat exchanger system
as a black box model is shown in Fig. 5. The inputs
are the inlet temperatures and mass flow rates of the
hot and cold fluids and the outputs are the outlet
temperatures of the two fluids. The black box
approach of heat exchanger analysis has been
attempted before [9]. It was specifically modeled for
counter-flow heat exchangers having five constant
coefficients. In the present work, an attempt has
been made to put forward a general purpose linear
relationship between the outlet temperatures, the
inlet temperatures and heat capacity rate ratios using
the black box model. Steady state, single phase flow,
constant physical property values of the fluid and
insulated outer surface conditions are assumed for
the analysis. The logarithmic mean temperature
difference has been approximated by arithmetic

mean temperature difference (within about 1.4%
error).
Figure 5: Inlet and outlet parameters of a double
tube heat exchanger
The basic mathematical model remains the
same, i.e., here the linear model is developed by
simplifying the equations (1) and (2). The equation
(1) remains the same and Tlm in equation (2) is
replaced by the arithmetic mean temperature
difference, Tam, which can be approximated for the
analysis of counter-flow heat exchangers and is
given as:
Now, using equation (1), Th2 may be written in terms
of Tc2 as:
Here it is assumed that the thermodynamic
properties remain constant. Now, after some
simplification, equation (2) may be written as:
The overall heat transfer coefficient is based on the
cooler side and the heat transfer surface area, A, is
given by the relation:
A = Πde1L (10)
Here, two non-dimensional parameters, γ and C, are
introduced for the heat capacity rate ratios:
Now, using equations (7) to (12), the cold fluid
outlet temperature, Tc2 has been obtained as:
Similarly, Th2 has been obtained using equations (8)
and (13) and is given as:
The effectiveness, ε, of the heat exchanger may be
given as:
With the above said assumptions, the set of
transcendental equations has been converted to a set
of linear equations involving two unknowns. The
overall heat transfer coefficient has been treated as a
constant value for a particular set of data.
3.2 Estimation of overall heat transfer coefficient
The overall heat transfer coefficient, U, has
a known expression and can be written as:
The calculation of the convection coefficients have
been carried out by using the Reynolds, Prandtl and
Nusselt similitude criteria relation. The Reynolds
similitude criteria for the fluid flowing through the
interior tube and the annular space are obtained from
the following two equations respectively:
vh and vc are the velocities of the hot and cold fluids
respectively, which are given as:
The cross sectional areas in the interior tube and the
annular space are given as:
Ain = 0.25Πdi1
2 (21)
Aan = 0.25Π(di2
2 – de1
2)
The term deh in equation (18) is known as the
equivalent hydraulic diameter and is given as:
deh = di2 – de1 (22)
The Prandtl similitude criteria for the interior fluid
and that flowing through the annular space are given
by the following two equations respectively:
The Nusselt numbers in the circular section of the
interior tube and the annular space has been
calculated for the laminar, intermediate and
turbulent flowing regime [10] and its expressions are
given in equations (25) and (26) respectively.
In the circular section of the interior tube:
Rein<2300 (laminar):
Nuin = 1.86[Rein Prin (di1/L)]0.33
2300≤Rein<104 (intermediate):
Nuin =
0.023 Rein
0.8Prin
0.4[1-(6e5/Rein
1.8)]
Rein≥104 (turbulent):
Nuin = 0.023 Rein
0.8Prin
0.4
(25)

In the annular space:
Rean<2300 (laminar):
Nuan = 4.05 Rean
0.17Pran
0.33
2300≤Rean<104 (intermediate):
Nuan =
1.86[ReanPran(deh/L)]0.33[1-(6e5/Rean
1.8)]
Rean≥104 (turbulent):
Nuan = 0.023 Rean
0.8Pran
0.4
The convection coefficients hi and he are determined
respectively from the following equations:
Some representative values of the physical
properties of some common fluids and thermal
conductivities of some common tube materials have
been incorporated into the simulation program in
order to calculate the convection coefficients and
hence the overall heat transfer coefficient using
equation (16).
IV. Results And Discussion
The primary focus here was to justify and
validate the assumption of a linear model by
comparing the results with those obtained from
ANSYS (Fluent) simulation and also the
experimentally obtained data. The basis of the
assumption of a linear model was that the Fluent
simulation produced a similar linear plot (Fig. 6-9),
by varying one inlet parameter (temperature of one
fluid) and keeping other inlet parameters fixed. The
simulations and linear model were primarily based
on the counter-flow heat exchangers using water as
the hot as well as the cold fluid. This is due to the
fact that because of their high effectiveness, counter-
flow heat exchangers are most widely used in
practical applications. The results obtained from the
existing simulator model [8, 9] (sim), Fluent
simulation (ANS) and those obtained from the
proposed linear model (sim1) were then compared to
validate the proposed model. The double tube heat
exchanger under consideration is made of copper,
whose geometrical dimensions have been presented
in Table 1. The inlet parameters have been presented
in Table 2. The comparative results of the outlet
temperatures of the heat exchanger are presented in
Table 3.
Table 3: The comparison between sim, ANS and sim1 data
No. Th2[o
C] Tc2[o
C] ε
sim ANS sim1 sim ANS sim1 sim ANS sim1
1 49.06 48.50 48.22 26.78 31.00 29.61 0.279 0.400 0.360
2 49.14 48.60 48.26 27.54 31.60 30.46 0.281 0.400 0.366
3 49.21 48.70 48.30 28.29 32.20 31.32 0.282 0.400 0.373
4 49.28 48.80 48.35 29.04 32.80 32.17 0.283 0.400 0.380
5 49.36 48.90 48.39 29.79 33.40 33.03 0.284 0.400 0.388
6 48.16 47.60 47.30 26.47 30.60 29.32 0.279 0.400 0.362
7 47.25 46.70 46.39 26.16 30.20 29.03 0.278 0.400 0.364
8 46.34 45.80 45.48 25.85 29.80 28.74 0.277 0.400 0.367
9 45.43 44.90 44.57 25.55 29.40 28.45 0.276 0.400 0.369
10 48.61 48.50 48.02 25.47 29.25 26.95 0.242 0.350 0.284
11 48.25 48.50 47.88 24.49 27.50 25.24 0.214 0.300 0.235
12 47.95 46.75 47.78 23.74 26.63 24.04 0.193 0.275 0.201
13 47.71 46.75 47.69 23.13 25.75 23.15 0.175 0.250 0.176
Table 4: Statistical parameters of the double tube
heat exchanger (linear model)
Using Table 3, the statistical parameters
associated to output variables of the proposed linear
model have been presented in Table 4.
The adoption of a linear model for the
double tube heat exchanger has been validated by
comparing the variation of outlet temperatures with
inlet temperature of one fluid (hot or cold), keeping
other inlet parameters like mass flow rates and the
inlet temperature of the other fluid constant, and
hence plotting the results obtained from our relation
with that from Fluent and sim data and comparing
them (Fig. 6-9). As can be seen from these figures,
the outlet temperatures obtained from the existing
simulator model as well as those obtained from
Fluent simulation too vary approximately linearly
with the inlet temperatures of both the fluids,
provided other inlet parameters are kept constant.
The slight differences in the values of the outlet
temperatures arise due to the assumptions
incorporated in developing the linear model.
Statistical
parameters
Output Variables
against ANS against sim
Th2 Tc2 Th2 Tc2
Max. absolute
deviation [o
C] 1.03 2.6 0.97 3.24
Max. relative
deviation [%] 2.20 10.09 1.97 10.87
(26)

Figure 6: Tc2 vs Tc1 (other parameters constant)
Figure 7: Th2 vs Tc1 (other parameters constant)
Figure 8: Tc2 vs Th1 (other parameters constant)
Figure 9: Th2 vs Th1 (other parameters constant)
The effectiveness of the heat exchanger as
obtained from Fluent simulation and the proposed
linear model have been presented in Table 3 and as
seen from Fig. 10, the effectiveness values of the
two models are linear and parallel to the line (y = x)
(but not along the line due to certain errors as
already discussed) which indicates that the proposed
linear model is fairly accurate.
Figure 10: Comparison of Effectiveness
To further validate the results, the data from
the linear model was compared with the
experimental data [8] along with Fluent and the
existing model data for a different heat exchanger
whose geometrical dimensions and inlet parameters
are provided in Table 5 and Table 6 respectively.
Table 7 shows the comparative analysis which
indicates that the results obtained for the linear
model hold good and are somewhat better than the
data predicted by the existing simulator model for
this case.
Table 5: Geometrical dimensions
Geometrical characteristic Variable Value
interior tube [m]
di1 0.012
interior tube [m]
de1 0.026
exterior tube [m]
di2 0.014
exterior tube [m]
de2 0.028
Length of the tube [m] L 0.935
Table 6: Inlet parameters
No. mc [kg/s] mh [kg/s] Tc1 [o
C] Th1 [o
C]
1 0.0256 0.0528 11.8 55.3
2 0.0256 0.0583 11.8 55.3
3 0.0256 0.0597 11.7 55.3
4 0.0256 0.0639 11.7 55.3
5 0.0278 0.0528 11.6 55.3
6 0.0358 0.0611 11.5 55.3
7 0.0511 0.0639 11.0 55.3
8 0.0611 0.0667 10.9 55.3
9 0.0486 0.0681 10.7 55.3

Table 7: Comparison of sim, ANS and sim1 data against experimental results
No. Th2[o
C] Tc2[o
C]
exp ANS sim1 sim exp ANS sim1 sim
1 49.2 48.77 49.72 50.7 24.5 27.02 23.32 21.5
2 49.5 48.78 50.06 51.0 25.1 27.03 23.75 21.8
3 49.6 48.76 50.14 51.1 25.2 26.96 23.76 21.8
4 49.9 50.94 50.37 51.3 25.4 26.96 24.03 21.9
5 49.4 48.75 49.62 50.6 23.3 25.80 22.39 20.7
6 49.2 48.73 49.85 50.8 22.0 23.54 20.78 19.2
7 48.5 48.65 49.64 50.6 19.7 20.96 18.07 16.9
8 48.2 48.64 49.63 50.6 18.6 19.78 17.09 16.1
9 47.5 48.61 49.89 50.8 17.7 20.73 18.26 17.0
The slight differences compared with
experimental results arise due to flow rate
inconsistencies, heat losses in the experimental
setup, small scale heat transfer apparatus and
measurement inaccuracies. Also steady state flow is
not perfectly realized in the laboratory setup.
V. Conclusion
In the present paper, a simpler approximate
linear model was proposed for the determination of
the outlet temperatures in a double tube heat
exchanger. Water was used as heat transfer fluids. In
the development of the linear model the heat
exchanger has been treated as a „black box‟. The
linear model has been validated by comparing the
outlet temperatures of the two fluids, calculated with
the proposed model, with the results from the
existing simulator model, FLUENT simulation and
finally with the experimentally obtained data for the
same operating conditions. It is observed that the
assumption of linear variation of outlet temperatures
with the inlet temperature of one fluid (keeping other
inlet parameters fixed) is very well justified. Even
the slopes of the temperature plots are almost
accurate. Taking into account the assumptions in the
calculation of the output variables compared to
experimentally determined values and those
obtained from simulation models (ANS and sim), the
deviation is seen to be small. Hence, taking all these
factors into account, it may be considered that the
proposed linear model can be employed for the
analysis of double pipe heat exchanger.
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Numerical Modeling and Simulation of a Double Tube Heat Exchanger Adopting a Black Box Approach