Analysis and Comparison of the Performance of Concurrent and
Countercurrent Flow Heat Exchangers
by
David Onarheim
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
APRIL 2012
(For Graduation April 2012)
.
© Copyright 2012
by
David Onarheim
All Rights Reserved
ii
CONTENTS
Analysis and Comparison of the Performance of Concurrent and Countercurrent Flow
Heat Exchangers ........................................................................................................... i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. Introduction and Background ...................................................................................... 1
1.1
Heat Exchanger Analysis Theory....................................................................... 1
1.1.1
Log Mean Temperature Difference ........................................................ 2
1.1.2
Heat Exchanger Effectiveness (ε) ......................................................... 2
1.1.3
Thermal Entrance in a Tube or Pipe ...................................................... 3
1.1.4
Description and History of Previous Graetz Problem Solutions............ 4
2. Problem Description .................................................................................................... 6
2.1
Defining Material Properties .............................................................................. 6
2.2
Methodology and Approach ............................................................................... 6
2.2.1
Finite Element Analysis ......................................................................... 6
2.2.2
Defining Variable Temperature and Velocity ........................................ 7
3. Theory of Solution ....................................................................................................... 8
3.1
The Graetz Problem Results............................................................................... 8
3.1.1
The Graetz Problem COMSOL Model .................................................. 8
3.1.2
The Graetz Problem COMSOL Mesh .................................................. 10
3.1.3
The Graetz Problem Study Results ...................................................... 14
3.1.4
The Graetz Problem Calculations ........................................................ 15
3.1.5
Turbulent Flow with Constant Wall Temperature ............................... 16
4. Results........................................................................................................................ 20
4.1
Flow in a Pipe with Axial Conduction ............................................................. 20
iii
4.2
4.3
4.4
4.5
4.1.1
Laminar Flow with a Pipe Wall COMSOL Model .............................. 20
4.1.2
Laminar Flow with a Pipe Wall Problem Calculations ........................ 21
4.1.3
Turbulent Flow in a Pipe with Axial Conduction ................................ 23
Flow in a Concurrent Flow Heat Exchanger .................................................... 24
4.2.1
Laminar Flow in a Concurrent Heat Exchanger COMSOL Model .... 24
4.2.2
Laminar Flow in a Concurrent Heat Exchanger Problem Calculations 26
4.2.3
Turbulent Flow in a Concurrent Heat Exchanger ................................ 26
Flow in a Counter-Current Flow Heat Exchanger ........................................... 26
4.3.1
Laminar Flow in a Counter-current Heat Exchanger COMSOL Model
.............................................................................................................. 26
4.3.2
Laminar Flow in a Counter-current Heat Exchanger Problem
Calculations .......................................................................................... 29
4.3.3
Turbulent Flow in a Counter-Current Heat Exchanger ........................ 29
Flow in a Concurrent Flow Heat Exchanger with Fouling .............................. 29
4.4.1
Laminar Flow in a Concurrent Heat Exchanger with Fouling
COMSOL Model .................................................................................. 29
4.4.2
Laminar Flow in a Concurrent Heat Exchanger with Fouling Problem
Calculations .......................................................................................... 29
4.4.3
Turbulent Flow in a Concurrent Heat Exchanger with Fouling ........... 29
Flow in a Counter-Current Flow HX with Fouling.......................................... 29
4.5.1
Laminar Flow in a Counter-current Heat Exchanger with Fouling
COMSOL Model .................................................................................. 29
4.5.2
Laminar Flow in a Counter-current Heat Exchanger with Fouling
Problem Calculations ........................................................................... 29
4.5.3
Turbulent Flow in a Counter-Current Heat Exchanger with Fouling .. 29
5. References.................................................................................................................. 30
iv
LIST OF TABLES
Table 1: Mesh Effectiveness ........................................................................................... 12
Table 2: Graetz Problem Comparison ............................................................................ 16
Table 3: Results from Mesh Refinement ........................................................................ 16
Table 4: Change in Temperature Along the Channel of Water ...................................... 22
v
LIST OF FIGURES
Figure 1: Basic Heat Exchanger Design ........................................................................... 2
Figure 2: Graetz Problem Temperature Profile ................................................................ 3
Figure 3: Nusselt Number for Various Pr Numbers ......................................................... 4
Figure 4: Graetz Problem Geometry................................................................................. 8
Figure 5: Graetz Velocity Profile ..................................................................................... 9
Figure 6: Graetz Temperature Profile ............................................................................. 10
Figure 7: User Defined Mesh ......................................................................................... 11
Figure 8: Centerline Temp vs Mesh Element Number ................................................... 13
Figure 9: Initial Value Variance ..................................................................................... 13
Figure 10: Graetz Problem Centerline Temperature ...................................................... 14
Figure 11: Laminar Flow Velocity Profile ..................................................................... 17
Figure 12: Turbulent Flow Velocity Profile ................................................................... 17
Figure 13: Turbulent Flow Centerline Temperature ....................................................... 18
Figure 14: Turbulent Model for the Graetz Problem ...................................................... 19
Figure 15: Velocity Profile for Flow Through a Pipe ..................................................... 20
Figure 16: Temperature Profile of Flow Through a Pipe ............................................... 21
Figure 17: Outflow Temperature Distribution for the Graetz Problem .......................... 23
Figure 18: Outflow Temperature Distribution for the Graetz Problem with a Pipe Wall
......................................................................................................................................... 23
Figure 19: Temperature Profile of Turbulent Flow Through a Pipe ............................. 24
Figure 20: Velocity Profile for Concurrent Heat Exchanger .......................................... 25
Figure 21: Temperature Profile for Concurrent Heat Exchanger ................................... 25
Figure 22: Temperature Change Across the Outlet Flow ............................................... 26
Figure 23: Velocity Profile for Countercurrent Heat Exchanger .................................... 27
Figure 24: Temperature Profile for Countercurrent Heat Exchanger ............................. 27
Figure 25: Outlet of the Inner Pipe, Inlet of the Outer Pipe ........................................... 28
Figure 26: Inlet of the Inner Pipe, Outlet of the Outer Pipe ........................................... 28
vi
ACKNOWLEDGMENT
I’d like to thank family and friends for supporting me during work on this project and
my master’s degree.
vii
ABSTRACT
Concentric tube heat exchangers utilize forced convection to lower the
temperature of a working fluid while raising the temperature of the cooling medium. The
purpose of this project was to use a finite element analysis program and hand
calculations to analyze the temperature drops as a function of both inlet velocity and
inlet temperature and how each varies with the other. These results were compared
between concurrent and countercurrent flow and between concurrent and countercurrent
flow with fouled piping. To determine the best heat transfer rate, both laminar and
turbulent flow was analyzed.
viii
1. Introduction and Background
There are many uses for heat exchangers from car radiators, to air conditioners,
to large condensers in power plants. But for all applications the effectiveness of these
heat exchangers are dependent on many factors. Not only does the viscosity and density
of the fluids affect the heat transfer due to being a factor of the Reynolds number and
therefore Nusselt number, but the inlet velocity (mass flow rate) and temperatures of the
fluids are proportional to the heat transfer rate.
  c  Th  Tc 
q  m
[1]
This project looks at the heat exchange between fluids in concentric tube heat
exchangers. In this type of heat exchanger, forced convection is caused by fluid flow of
different temperatures passing parallel to each other separated by a boundary, pipe wall.
Several assumptions will have to be made to make it easier to focus on the inlet velocity
and temperature dependence on heat exchanger temperature drop. Not only will the
viscosity and density remain constant for the calculations, but specific heat and overall
heat transfer coefficients will be assumed constant. Any effects from potential and
kinetic energy are assumed negligible.
1.1 Heat Exchanger Analysis Theory
Two types of analysis for parallel flow heat exchangers to determine temperature
drops are the log mean temperature difference and the effectiveness-NTU method. Both
methods will be attempted to be used for the project. The equation for heat transfer using
the log mean temperature difference becomes:
q  UATlm  UA 
T2  T1

ln  T2


T
1

[2]
where the only change for parallel and countercurrent flow is how the delta-T’s are
defined. The NTU (number of transfer units) method uses the effectiveness number of
the type of heat exchanger to determine the amount of heat transfer.
q    c min  Th ,i  Tc ,i 
The effectiveness of the types of heat exchangers is as follows:
1
[3]

Parallel Flow:
Counter Flow:  
1  exp[  NTU (1  C r )]
1  Cr
[4]
1  exp[  NTU (1  Cr )]
forCr  1
1  Cr exp[  NTU (1  Cr )
[5]
In general the heat flux is comprised of three factors: the temperature difference,
the characteristic area, and an overall heat transfer coefficient. An approximate value for
the transfer coefficient U (W/m^2 k) is 110-350 for water to oil. In the case where
fouling is present on the heat exchanger tubes, the following can be used in the case of
tubular heat exchangers:
UA 
R f ,i
1


hi Ai
Ai
ln(
1
Do
[6]
)
R f ,o
Di
1

2kL ho Ao
Ao
Rf is defined as the fouling factor with units of m^2 k/w. An approximate value of .0009
is used for fuel oil, while .0001 - .0002 is used for seawater and treated boiler feedwater.
Figure 1: Basic Heat Exchanger Design
1.1.1
Log Mean Temperature Difference
1.1.2
Heat Exchanger Effectiveness (ε)
The effectiveness ε is the ratio of the actual heat transfer rate to the maximum
possible heat transfer rate:

qactual
,0    1
q max
2
[7]
1.1.3
Thermal Entrance in a Tube or Pipe
Figure 2: Graetz Problem Temperature Profile
The development of fluid flow and temperature profile of a fluid after undergoing a
sudden change in wall temperature is dependent on the fluid properties as well as the
temperature of the wall. This thermal entrance problem is well known as the Graetz
Problem. For incompressible Newtonian fluid flow, the equation of energy becomes:
𝐷𝑇
𝜌𝑐𝑝 𝐷𝑡 = 𝑘∇2 𝑇 + 𝜑
[8]
Neglecting dissipation and any conduction axially, equation 8 reduces to the following:
𝜕𝑇
𝑢 𝜕𝑥 =
𝛼 𝜕
𝑟 𝜕𝑟
𝜕𝑇
(𝑟 𝜕𝑟 )
[9]
The velocity distribution is assumed to be known when using this equation and can be
several different types of flow. For low prandtl number materials such as liquid metals
the temperature profile (T) will develop faster than the velocity profile (u) and u will be
constant. For high prandtl number materials such as oils or when the thermal entrance
(sudden change in wall temperature) is fairly far down the entrance of the duct/ tubing
𝑟2
𝑢 = 2𝑢̅(1 − 𝑟 2 ). The velocity profile can also be developing and can be used for any
0
prandtl number material assuming the velocity and temperature profiles are starting at
the same point.
There have been numerous analytical solutions developed for the Graetz problem
with different types of flow. For laminar flow with a developing velocity profile, the
mean nusselt number can be approximated based on the relationship illustrated below
between the log mean nusselt number and the graetz number for various prandtl
numbers.
3
Figure 3: Nusselt Number for Various Pr Numbers
An approximation for the mean nusselt number was given by Hausen (1943) for fluid
with their prandtl number >1 (especially for use with oils). This is given by equation 10
below.
𝑁𝑢𝑚 = 3.66 +
1.1.4
.075⁄ ∗
𝐿
1+.05⁄ 2
𝐿∗3
[10]
Description and History of Previous Graetz Problem Solutions
The classic Graetz problem which continues to provide background for the
developmemt and understanding of compact heat exchangers has been refined and
expanded upon since initially introduced in 1883. The original problem has a fluid with a
fully developed velocity profile and uniform temperature enter a tubing or duct that is
maintained at a constant temperature. This could be heating or cooling the flowing fluid
just as long as it was different from the initial value of the fluid flow. This classic
problem neglected any viscous dissipation, axial heat conduction, or and heat generation
by the fluid. The purpose of the solution to this problem was to determine the
temperature distribution and any connection between the wall temperature and the heat
flux to the fluid. Using a separation of variables technique, Graetz found a solution in the
4
form of an infinite series in which the eigenvalues and functions satisfied the sturmLouiville system. While Graetz himself only determined the first two terms, Sellars,
Tribus, and Klein were able to extend the problem and determine the first ten
eigenvalues in 1956. Even though this further developed the original solution, at the
entrance of the tubing the series solution had extremely poor convergence where up to
121 terms would not make the series converge.
Schmidt and Zeldin in 1970 extended the Graetz problem to include axial heat
conduction and found that for very high Peclet numbers (Reynolds number multiplies by
the prandtl number) the problem solution is essentially the original Graetz problem.
Similar to the original problem which showed poor convergence near the ducting
entrance, they discovered up to a 25% deviation in the local nusselt number which made
the results in this region questionable.
The purpose of this paper is to not redo the various numerical solutions presented by
multiple groups over the past century as there doesn’t appear to be a definitive solution
that has proven convergence everywhere. The Graetz problem will be introduced in a
finite element program with certain dimensions, fluid properties, and tubing temperature
in order to analyze the velocity and temperature changes as a building block to
eventually analyzing a compact heat exchanger for the same.
5
2. Problem Description
For this project, fully developed laminar and turbulent incompressible fluid flow
was analyzed in three heat exchanger cases: parallel flow, countercurrent flow, and flow
in a fouled heat exchanger. The resulting temperature difference was compared and
determined as a function of the inlet velocity and inlet temperatures. The overall
objective for this project was to determine the max temperature difference in these cases
for both laminar and turbulent flow for a variety of flow rates and inlet temperatures. To
simplify the number of variables, water and oil were chosen as the fluids to maintain
viscosities and densities of the fluids constant. The type of heat exchanger used was of
concentric tube design. Water was the cooling medium and oil the working fluid.
2.1 Defining Material Properties
Water was used as the base fluid flowing through tube or pipe. Its material
properties were derived from tables based on the temperature which was being used in
the model. The material was defined in COMSOL using its material browser, but certain
properties were defined by the user prior to computing the model results. These
properties were: thermal conductivity, density, heat capacity at constant pressure, ratio
of specific heats, and dynamic viscosity.
2.2 Methodology and Approach
2.2.1
Finite Element Analysis
A finite element analysis was done using COMSOL for the fluid flow and
convective heat transfer. A 2D axisymmetric model was chosen to depict the tubing the
fluid was flowing through. The type of physics to be applied was then added. For the
baseline model (the graetz problem) the physics used was laminar fluid flow and then
non-isothermal flow was chosen. This allowed for definition of not only the fluid
parameters but also the heat transfer of the constant wall temperature to the fluid. The
third model introduced a second pipe concentric to the first and was analyzed for fluid
flow in the same direction. The fourth model reversed the fluid flow for the cooling
medium, which was chosen as water. The material library was used for definition of
properties for oil and water. The fifth and sixth models added on to the second and third
6
models a layer of fouling for both types of flow and determined the effect on not only
the flow but the resultant temperature differences. These models were repeated using
turbulent flow which added complexity to each model. Post-processing plots developed
in COMSOL were used for analysis. In addition to this, the COMSOL information was
exported to excel to better compare and analyze the data. Hand calculations for the
temperature differences were also done to verify results.
2.2.2
Defining Variable Temperature and Velocity
7
3. Theory of Solution
3.1 The Graetz Problem Results
To begin the COMSOL analysis of temperature difference in fluid flow the base
condition must first be analyzed. The first condition is that of fluid passing through a
tube with a constant wall temperature, as described before this is known as the Graetz
problem. A base model was run in COMSOL and the analysis was compared to hand
calculations to verify. The initial conditions of the problem were as follows:
L=1.0m, D=0.1m, k=.64, µ=.000547 Pa.s, ρ=988 kg/m^3, Cp=4181 J/KgK
3.1.1
The Graetz Problem COMSOL Model
As previously described, the physics used for modeling was non-isothermal
laminar flow. The water was selected to be flowing through a tube or pipe of length 1m
with a diameter of .1m. The inlet flow of the water was set initially at .0001 m/s and
varied for 2 other cases: .01 m/s and .001 m/s. The temperature of the water flowing
into the tubing was set at 50 C or 323.15 K while the wall temperature remained constant
at 30 C or 303.15 K. This temperature difference was also varied for 2 other cases. The
figure below shows the geometry of the model in COMSOL.
Figure 4: Graetz Problem Geometry
8
The material properties of the fluid were then defined. Water at 50 C was used
and the properties used for temperature determination were user defined. The physics
used was non-isothermal flow and laminar flow and heat transfer nodes were applied to
define the fluid flow as well as the heat transferred from the constant wall temperature to
the water. For fluid flow the inlet and outlet points of flow were defined with the water
velocity defined at the inlet point. For heat transfer, the temperature of the water flowing
at the inlet was defined as well as the temperature of the wall. The outlet of fluid flow
was also defined as outflow in terms of the heat transfer physics. After initializing a
mesh of the model, results were obtained for not only the velocity profile but also the
temperature profile.
Figure 5: Graetz Velocity Profile
9
Figure 6: Graetz Temperature Profile
3.1.2
The Graetz Problem COMSOL Mesh
Initially the physics controlled mesh was used in COMSOL but looking at the
study results it was discovered that the results were dependent upon the refinement of
the mesh and the initial values tab of the COMSOL model. The initial values are defined
to only be an initial guess for the final solution derived by the non-linear solver in
COMSOL. However, it was found that varying the temperature in this initial values tab
would vary the centerline outlet temperature even though the temperature of inlet flow
and surface temperature were previously defined. It was also discovered that the initial
tolerance of 10^-3 as defined by COMSOL allowed for a very large variance in the
outlet temperature just by changing the refinement of the model. Ideally refining the
model should change the value slightly as the model becomes more refined since more
elements are added to the mesh, the temperature being solved for should become closer
and closer to the desired value. However by starting at the extremely coarse and going to
the fine mesh, the outlet temperature changed by almost 10 degrees and the change was
not linear.
To streamline the results and take out the uncertainty that was being created by
changing the mesh refinement, the tolerance of the solver was changed to 10^-4 and a
10
different type of mesh was created. Instead of using the triangular type elemental mesh
which COMSOL automatically defines when the physics controlled mesh is selected, the
user controlled mesh option was used and a free quad mesh was defined. This allowed
for more of a rectangle shape to the mesh elements along the length of the tubing toward
the middle of the flow. Along the wall of the tubing boundary layer meshing was added
which refined the mesh elements and added extra elements along the wall where the
temperature and velocity profiles are developing and there is more change to the flow at
this point. This allows for COMSOL to have the solver focus more on the boundary that
has complicated change to it than on the steady flow in the middle of the tubing. Figure
7 shows an example of this mesh with the additional layers applied around the wall of
the tubing.
Figure 7: User Defined Mesh
It took several iterations of attempting to find the best mesh to yield the best
result. Ultimately as the number of elements increases the outlet temperature on the
centerline should level out and gradually approach a certain value instead of varying
higher and lower around several values. By changing the number and thickness of the
boundary layers a more accurate mesh was able to be obtained. The maximum size of
the elements in the mesh were changed while the number of boundary layers kept
11
constant to increase and decrease the number of elements in the model (lowering the
maximum elements size increased the total number of mesh elements in the model).
Table 1 below shows the results from increasing the mesh elements on centerline
temperature for the case of V=.0001m/s. The variance in centerline temperature was
from 306.0347F to 305.2428F for a difference of .7919F instead of 10F. The number of
boundary layers was 40 with the stretching factor at 1.2 and the thickness adjustment
factor at 15.
Table 1: Mesh Effectiveness
Mesh Effectiveness
Number of Mesh Elements
2150
2279
2408
2948
3388
3696
4095
4500
5875
8183
10, 376
Centerline Outlet Temp (F)
306.0347
305.9992
306.0265
305.9083
305.8629
305.8664
305.6118
305.6001
305.2428
305.7506
305.6016
Plotting these numbers on a scatter plot shows that as the element size increases
the outlet temperature gradually gets closer to a constant centerline temperature. Figure 8
shows this relationship. An exponential trendline was added to illustrate the
temperatures gradual approach to a constant value.
12
306.4
306.2
Temperature (F)
306
305.8
Centerline Outlet Temp (F)
305.6
Power (Centerline Outlet
Temp (F))
305.4
305.2
305
0
2
4
6
8
10
12
Number of Elements in the Mesh
Figure 8: Centerline Temp vs Mesh Element Number
Since the initial value for the temperature of the graetz problem was causing an
unexpected variance in the results, its effect on this new mesh was also documented.
Using the most refined mesh (element number of 10, 376) the initial value of
temperature was varied from 283.15 to 323.15 and the resulting centerline temperature
Temperature (F)
was fairly constant as shown in figure 9.
310
309
308
307
306
305
304
303
302
301
300
Centerline Outlet Temp (F)
280
290
300
310
320
330
Initial Value Temp (F)
Figure 9: Initial Value Variance
13
This study proved the change in initial values and mesh refinement only effected
the results by a fraction of a percent vice several percent when boundary layer elements
were used in the mesh. To further refine the mesh and provide more accurate results, the
element size near the center of the fluid flow was enlarged and made more rectangular
by changing the size of the quad elements. This mesh was then proven accurate like the
previous study by verifying that changing the number of elements and initial values
didn’t vary the outcome by more than a percent of a fraction. This type of element array
now proven was applied to the following models which added on to this original Graetz
problem model.
3.1.3
The Graetz Problem Study Results
Using COMSOL’s post-processing capabilities, a 1D line graph was plotted
along the center of the tubing to track the temperature as it changes along the center of
the tubing. Figure 10 shows the temperature trend as the fluid cools from its inlet
temperature to near the constant wall temperature.
Figure 10: Graetz Problem Centerline Temperature
To determine the outlet temperature of the center of flow a point evaluation was
done under the derived values tab of the post-processor results of the model. This
yielded 305.3221 K. In order to verify the results, the velocity was changed at the inlet
14
of the tube and compared to hand calculations for both .001 m/s and .01 m/s inlet
velocity.
3.1.4
The Graetz Problem Calculations
The outlet temperature of the fluid is determined by using the mean nusselt
number of the fluid flow. The nusselt number approximation initially used was eq X
from White’s Viscous Fluid Flow and proposed by Hasusen (1943) for PR>1. First the
Reynolds number is calculated for the initial conditions. For the purpose of analysis the
flow is considered incompressible Newtonian flow.
𝑅𝑒 =
𝜌𝑉𝐷
𝜇
=
(988)(.0001)(.1)
(5.47𝑥10−4 )
= 18.062
[11]
The prandtl number is calculated using the material properties of water at the inlet
temperature.
𝑃𝑟 =
𝐶𝑝 𝜇
𝑘
=
(4181)(5.47𝑥10−4 )
.64
= 3.57
[12]
The dimensionless length value is defined as
𝐿
(1)
𝐿∗ = 𝐷𝑅𝑒𝑃𝑟 = (.1)(18.062)(3.57) = .15508
[13]
The outlet temperature is defined as
𝑇𝑚 (𝐿) = 𝑇𝑤 − (𝑇𝑤 − 𝑇𝑜 ) ∗ 𝑇𝑚∗ (𝐿)
[14]
Since there is a relationship between 𝑇𝑚∗ (𝐿) and the mean nusselt number, if the
nusselt number is obtained from the approximation equation, the outlet temperature can
then be determined. Using equation 10, the nusselt number is calculated.
𝑁𝑢𝑚 = 3.66 +
.075⁄ ∗
𝐿
1+.05⁄ 2
= 3.66 +
𝐿∗3
−1
.075⁄
.15508
1+.05⁄
.155082/3
= 4.0722
∗
𝑁𝑢𝑚 = 4𝐿∗ 𝑙𝑛𝑇𝑚∗ (𝐿) ∴ 𝑇𝑚∗ (𝐿) = 𝑒 (−4𝐿 𝑁𝑢𝑚) = .07997
[15]
[16]
𝑇𝑚 (𝐿) = 𝑇𝑤 − (𝑇𝑤 − 𝑇𝑜 ) ∗ 𝑇𝑚∗ (𝐿) = 30 − (30 − 50) ∗ (. 07997)
𝑇𝑚 (𝐿) = 31.5994 C = 304.7494 K
[17]
This was then compared to the centerline temperature of the fluid at the end of
the tubing (at z-=1.0m) and a percent error was calculated between the expected and
actual (COMSOL value). Table 2 shows this particular case as well as 2 other cases. The
inlet velocity was varied to .001 and .01 m/s and the centerline temperature obtained
15
both by hand and by COMSOL. Overall the derived values of the outlet temperature are
all near the values of the COMSOL model with less than 2% error. The Hausen equation
is noted to have an approximation error of 5%.
Table 2: Graetz Problem Comparison
Inlet V
(m/s)
0.0001
0.001
0.01
Inlet Temp
(C)
50
50
50
Wall Temp
(C)
30
30
30
Expected Value Calc (K)
304.7494
316.646
317.644
COMSOL Value (K)
305.3221
321.9023
323.0481
% Error
0.187572
1.632887
1.672847
Several other numerical methods were attempted in order to create the best
numerical solution for the Graetz problem underneath these initial conditions. As
described previously the mesh was changed in the original model to a mesh which
showed little to no variation in centerline temperature when the number of elements or
initial values changed. Table 3 below shows the comparison to the previous hand
calculations and how they compare to the previous models results.
Table 3: Results from Mesh Refinement
Inlet V
(m/s)
0.0001
0.001
0.01
Exp. Value Calc (K)
304.7494
316.646
317.644
COMSOL Value (K)
305.3221
321.9023
323.0481
% Error
0.187572403
1.632886749
1.672846861
COMSOL Refined (K)
305.93648
322.812
323.15
While the percent error is slightly higher with the refined mesh which included
boundary layer mesh elements, the consistency of the results were far superior to the
original mesh. Originally the results were highly dependent on the initial temperature
value the non-linear solver was using even though they should be mutually exclusive as
well as dependent on element size and amount as described.
3.1.5
Turbulent Flow with Constant Wall Temperature
Originally the Graetz COMSOL models which were modeled using laminar flow.
To analyze and determine the difference flow types have on the velocity and temperature
16
% Error
0.388015
1.91009
1.703853
profiles, turbulence was added to the model. The figure below shows the developing
velocity profile of laminar flow.
Figure 11: Laminar Flow Velocity Profile
The figure below shows the velocity profile for turbulent flow.
Figure 12: Turbulent Flow Velocity Profile
As opposed to the laminar flow, turbulent flow already has a fully developed
flow as it enters, flows, and then exits the tubing. Under the non-isothermal tab, the
RANS turbulence model was turned on essentially talking the same laminar flow graetz
17
problem model but changing the flow from laminar to turbulent. The k-e turbulence type
model was used with Kays-Crawford heat transport. The same quad element mesh with
boundary layer elements added was used with a accuracy tolerance of 10^-4. The
centerline temperature along the length of the tubing had more of a linear relationship
while the laminar flow was more gradual in lowering towards the outlet temperature as
described previously. Figure 13 shows the centerline temperature for turbulent flow. The
main difference between the laminar and turbulent flows was that for both .0001 m/s and
.001 m/s the outlet centerline temperature was approximately 322F whereas in laminar
flow the lowest velocity flow actually lowered the centerline temperature down to
approximately 305F due to the fluid flowing slower and having more time to transport
heat from the wall. In the turbulent flow, the fluid is mixed and temperature more evenly
distributed that for the slowest of velocities the centerline temperature didn’t lower
nearly as much.
Figure 13: Turbulent Flow Centerline Temperature
For a velocity of .0001 m/s the centerline temperature was 322.25503F and for a
velocity of .001 m/s the centerline temperature was 322.86295F. For the largest of the
velocities that have been used (.01 m/s) and the same geometry, the type of solver being
used had to be modified. The same mesh and boundary layer elements were used as
previously described, but with this velocity, geometry, and turbulent model defined the
18
stationary solver would not converge and determine a solution. Due to this, the solver
was changed from a fully coupled to segregated in order for the non-linear solver to
divide up the solution process into substeps. Also the type of solver was changed from
MUMPS to SPOOLES. Once these changes were made, the same mesh and parameters
were solved and a solution obtained. For a velocity of .01 m/s the centerline temperature
was 323.14911F. The figure below shows not only the type of solver used, but the
temperature profile for the turbulent model used for a velocity of .01 m/s.
Figure 14: Turbulent Model for the Graetz Problem
19
4. Results
4.1 Flow in a Pipe with Axial Conduction
To add to the original graetz problem a pipe wall was added and the heat exchange
to the fluid flow from the pipe wall was analyzed. In addition to this the heat conduction
through the pipe wall was taken into account. A pipe wall of .02m was added to the
original model and the same 3 velocity profiles were analyzed. An accuracy tolerance of
10^-4 was used as before as well as the previously defined quad element mesh with
boundary layers applied.
4.1.1
Laminar Flow with a Pipe Wall COMSOL Model
The figures below shows flow through a steel pipe and the resultant velocity and
temperature profiles.
Figure 15: Velocity Profile for Flow Through a Pipe
20
Figure 16: Temperature Profile of Flow Through a Pipe
For a velocity of .0001 m/s the centerline temperature was 305.93998F. For a velocity of
.001 m/s the centerline temperature was 322.83099F. For a velocity of .01 m/s the
centerline temperature was 323.14999F.
4.1.2
Laminar Flow with a Pipe Wall Problem Calculations
To analyze the flow a lumped parameter model was used and the temperature
change determined at various points along the length of the pipe. Heat transferred from
the wall will be equal to the heat transferred to the water.
̇
𝑄𝑤𝑎𝑙𝑙 𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑞̇ × ∆𝐴 = 𝑞2𝜋𝑅∆𝐿
[18]
𝑄𝑤𝑎𝑡𝑒𝑟 = 𝜌𝐶𝑝 ∆𝑇∆𝑉 = 𝜌𝐶𝑝 ∆𝑇𝜋𝑅 2 ∆𝐿
[19]
𝑞̇ 2𝜋𝑅∆𝐿
∆𝑇 = 𝜌𝐶
𝑝 𝜋𝑅
2 ∆𝐿
2𝑞̇
= 𝜌𝐶
𝑝𝑅
[20]
To determine a basic change in temperature and therefore outlet temperature, the
heat flux along the length of the flow was graphed using the original laminar flow graetz
problem model and determined at various points along the flow path. Using an excel
spreadsheet and the above equations, the outlet temperature was determined and could
be used as comparison to the COMSOL value of laminar flow with a pipe wall. The
table below shows the outlet temperature based on the heat flux along the length of the
fluid channel.
21
Table 4: Change in Temperature Along the Channel of Water
Length (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Heat Flux at the Wall (W/m^2)
45.71
58.05
73.29
92.19
115.87
145.86
185.05
240.52
304.94
32844.5
ΔT (F)
-0.00044
-0.00056
-0.00071
-0.00089
-0.00112
-0.00141
-0.00179
-0.00233
-0.00295
-0.31804
Outlet Temp (F)
323.1495574
323.1494379
323.1492903
323.1491073
323.148878
323.1485876
323.1482081
323.147671
323.1470472
322.8319572
While the resultant temperature at the outlet of the pipe has a very close
temperature to that of the .001 m/s velocity, these heat flux values were from the Graetz
problem model of velocity .0001 m/s so there does appear to be a small error in the
calculations. Taking a line average of the outlet of the flow from the centerline to the
inner wall in the model with axial heat condution and from the centerline to the tubing in
the Graetz problem verified that the averages were very similar. The Graetz problem
resulted in an average of 304.78102F while the Graetz problem with a pipe wall resulted
in an average of 304.32395F. Also when a line graph was created of the temperature at
the outlet of the flow for each model the curves shapes were identical with the exception
of the model with the pipe wall which had a small slanted horizontal line to the right
where a small amount of temperature rise was seen across the pipe wall going to outside
to inside. Figures 16 and 17 show the similarity in outlet temperature distribution.
22
Figure 17: Outflow Temperature Distribution for the Graetz Problem
Figure 18: Outflow Temperature Distribution for the Graetz Problem with a Pipe Wall
4.1.3
Turbulent Flow in a Pipe with Axial Conduction
The turbulence model was added to flow in the pipe with a wall and the same
dimensions, velocities, and temperatures were used. The centerline temperature from this
model were extremely similar to the turbulent flow through the tubing with no pipe wall
(original Graetz problem) and the difference between the laminar and turbulent flow in a
pipe wall was similar to the differences between laminar and turbulent flow in the Graetz
23
problem. Figure 19 shows the temperature profile for the turbulent model with velocity
at .0001 m/s.
Figure 19: Temperature Profile of Turbulent Flow Through a Pipe
For a velocity of .0001 m/s the centerline temperature was 322.25321F. For a
velocity of .001 m/s the centerline temperature was 322.83265F. For a velocity of .01
m/s the centerline temperature was 323.1494F. Just as in the Graetz problem when flow
was changed from laminar to turbulent, the centerline temperature does not drop as
much at the lowest velocity due to the better mixing and more evenly distributed flow.
The approximate 322F was similar between both turbulent models as expected. This was
also the main difference between the laminar and turbulent model for flow through a
pipe wall with axial condution.
4.2 Flow in a Concurrent Flow Heat Exchanger
4.2.1
Laminar Flow in a Concurrent Heat Exchanger COMSOL Model
Adding onto the COMSOL model of flow through a pipe with a pipe wall, a
second pipe and pipe wall were added. Flow was defined to be flowing in the same
direction with the outer flow at a lower temperature cooling the inner fluid. For the
purposes of simplifying the model for development, the same type of pipe was used as in
the previous model, the same fluid, water, was used for both sides of the fluid flows, and
24
the same dimensions and temperatures were used. Once the model was made and
analyzed the velocity, temperatures, and materials could be changed for further
investigation.
Figure 20: Velocity Profile for Concurrent Heat Exchanger
Figure 21: Temperature Profile for Concurrent Heat Exchanger
25
Figure 22: Temperature Change Across the Outlet Flow
4.2.2
Laminar Flow in a Concurrent Heat Exchanger Problem Calculations
4.2.3
Turbulent Flow in a Concurrent Heat Exchanger
4.3 Flow in a Counter-Current Flow Heat Exchanger
4.3.1
Laminar Flow in a Counter-current Heat Exchanger COMSOL Model
Adding onto the COMSOL model of flow through a pipe with a pipe wall, a
second pipe and pipe wall were added. Flow was defined to be flowing in the opposite
direction with the outer flow at a lower temperature cooling the inner fluid. For the
purposes of simplifying the model for development, the same type of pipe was used as in
the previous model, the same fluid, water, was used for both sides of the fluid flows, and
the same dimensions and temperatures were used. Once the model was made and
analyzed the velocity, temperatures, and materials could be changed for further
investigation.
26
Figure 23: Velocity Profile for Countercurrent Heat Exchanger
Figure 24: Temperature Profile for Countercurrent Heat Exchanger
27
Figure 25: Outlet of the Inner Pipe, Inlet of the Outer Pipe
Figure 26: Inlet of the Inner Pipe, Outlet of the Outer Pipe
28
4.3.2
Laminar Flow in a Counter-current Heat Exchanger Problem Calculations
4.3.3
Turbulent Flow in a Counter-Current Heat Exchanger
4.4
Flow in a Concurrent Flow Heat Exchanger with Fouling
4.4.1
Laminar Flow in a Concurrent Heat Exchanger with Fouling COMSOL
Model
4.4.2
Laminar Flow in a Concurrent Heat Exchanger with Fouling Problem
Calculations
4.4.3
Turbulent Flow in a Concurrent Heat Exchanger with Fouling
4.5 Flow in a Counter-Current Flow HX with Fouling
4.5.1
Laminar Flow in a Counter-current Heat Exchanger with Fouling
COMSOL Model
4.5.2
Laminar Flow in a Counter-current Heat Exchanger with Fouling Problem
Calculations
4.5.3
Turbulent Flow in a Counter-Current Heat Exchanger with Fouling
29
5. References
[1] Beek, W.J., K.M.K. Muttzall, and J.W. van Heuven. Transport Phenomena. 2nd ed.
New York: John Wiley & Sons, Ltd., 1999.
[2] Bird, Byron R., Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena.
Revised 2nd ed. New York: John Wiley & Sons, Inc., 2007.
[3] Blackwell, B.F. “Numerical Results for the Solution of the Graetz Problem for a
Bingham Plastic in Laminar Tube Flow with Constant Wall Temperature.”
Sandia Report. Aug. 1984.
[4] Conley, Nancy, Adeniyi Lawal, and Arun B. Mujumdar. “An Assessment of the
Accuracy of Numerical Solutions to the Graetz Problem.” Int. Comm. Heat Mass
Transfer. Vol.12. Pergamon Press Ltd. 1985.
[5] Kays, William, Michael Crawford, and Bernhard Weigand. Convective Heat and
Mass Transfer. 4th ed. New York: The McGraw-Hill Companies, Inc.,
2005.
[6] Lemcoff, Norberto. “Heat Exchanger Design.” Groton. 10 July 2008.
[7] Lemcoff, Norberto. “Project: Heat Exchanger Design.” Groton. 17 July 2008.
[8] Sellars J., M. Tribus, and J. Klein. “Heat Transfer to Laminar Flow in a Round Tube
or Flat Conduit—The Graetz Problem Extended.” The American Society of
Mechanical Engineers. New York. 1955.
[9] Subramanian, Shankar R. “The Graetz Problem.”
[10]Valko, Peter P. “Solution of the Graetz-Brinkman Problem with the Laplace
Transform Galerkin Method.” International Journal of Heat and Mass Transfer
48. 2005.
[11]White, Frank. Viscous Fluid Flow. 3rd ed. New York:
Companies, Inc., 2006.
The McGraw-Hill
[13] W.M Kays and H.C. Perkins, in W.M. Rohsenow and J.P Harnett, Eds., Handbook
of
Heat Transfer, Chap. 7, McGraw-Hill, New York, 1972.
[14] TBD
30