Analysis and Comparison of the Performance of Concurrent and
Countercurrent Flow Double Pipe 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
Double Pipe Heat Exchangers ...................................................................................... i
LIST OF TABLES ............................................................................................................ vi
LIST OF FIGURES ......................................................................................................... vii
LIST OF SYMBOLS ........................................................................................................ ix
ACKNOWLEDGMENT .................................................................................................. xi
ABSTRACT .................................................................................................................... xii
1. Introduction and Background ...................................................................................... 1
1.1
Heat Exchanger Analysis Theory....................................................................... 1
1.1.1
Log Mean Temperature Difference ........................................................ 3
1.1.2
Heat Exchanger Effectiveness (ε) ......................................................... 3
1.1.3
NTU Method .......................................................................................... 3
1.1.4
Thermal Entrance in a Tube or Pipe ...................................................... 4
1.2
Description and History of Previous Graetz Problem Solutions........................ 6
1.3
Finite Element Analysis Theory ........................................................................ 7
2. Problem Description and Methodology ....................................................................... 9
2.1
Defining Material Properties .............................................................................. 9
2.2
Methodology and Approach ............................................................................... 9
2.2.1
Finite Element Analysis Modeling ........................................................ 9
2.2.2
Defining Variable Temperature and Velocity ...................................... 10
3. Results and Discussion .............................................................................................. 11
3.1
The Graetz Problem Results............................................................................. 11
3.1.1
The Graetz Problem COMSOL Model ................................................ 11
3.1.2
The Graetz Problem COMSOL Mesh .................................................. 13
3.1.3
The Graetz Problem Study Results ...................................................... 17
3.1.4
The Graetz Problem Calculations ........................................................ 18
iii
3.1.5
3.2
3.3
3.4
3.5
Turbulent Flow with Constant Wall Temperature ............................... 20
Flow in a Pipe with Axial Conduction ............................................................. 24
3.2.1
Laminar Flow with a Pipe Wall COMSOL Model .............................. 24
3.2.2
Laminar Flow with a Pipe Wall Problem Calculations ........................ 25
3.2.3
Turbulent Flow in a Pipe with Axial Conduction ................................ 27
Flow in a Concurrent Flow Heat Exchanger .................................................... 29
3.3.1
Laminar Flow in a Concurrent Heat Exchanger COMSOL Model ..... 29
3.3.2
Turbulent Flow in a Concurrent Heat Exchanger ................................ 32
3.3.3
Laminar Flow in a Concurrent Heat Exchanger Problem Calculations 34
3.3.4
Laminar Flow in a Counter-current Heat Exchanger COMSOL Model
.............................................................................................................. 42
3.3.5
Turbulent Flow in a Counter-Current Heat Exchanger ........................ 44
3.3.6
Laminar Flow in a Counter-current Heat Exchanger Problem
Calculations .......................................................................................... 47
Flow in a Concurrent Flow Heat Exchanger with Fouling .............................. 49
3.4.1
Laminar Flow in a Concurrent Heat Exchanger with Fouling COMSOL
Model ................................................................................................... 49
3.4.2
Turbulent Flow in a Concurrent Heat Exchanger with Fouling ........... 49
3.4.3
Laminar Flow in a Concurrent Heat Exchanger with Fouling Problem
Calculations .......................................................................................... 49
Flow in a Counter-Current Flow Heat Exchanger with Fouling ...................... 50
3.5.1
Laminar Flow in a Counter-current Heat Exchanger with Fouling
COMSOL Model .................................................................................. 50
3.5.2
Turbulent Flow in a Counter-Current Heat Exchanger with Fouling .. 50
3.5.3
Laminar Flow in a Counter-current Heat Exchanger with Fouling
Problem Calculations ........................................................................... 50
4. Conclusion ................................................................................................................. 51
5. References.................................................................................................................. 52
6. APPENDIX................................................................................................................ 53
6.1
Laminar Flow Concurrent Heat Exchanger Data ............................................. 53
iv
6.2
Laminar Flow Counter-Current Heat Exchanger Data .................................... 55
v
LIST OF TABLES
Table 1: COMSOL Model Initial Conditions ................................................................. 11
Table 2: Mesh Effectiveness ........................................................................................... 15
Table 3: Graetz Problem Comparison ............................................................................ 19
Table 4: Results from Mesh Refinement ........................................................................ 20
Table 5: Change in Temperature Along the Channel of Water ...................................... 26
Table 6: Fluid Properties ................................................................................................ 35
vi
LIST OF FIGURES
Figure 1: Basic Heat Exchanger Design ........................................................................... 3
Figure 2: Graetz Problem Temperature Profile ................................................................ 4
Figure 3: Nusselt Number for Various Pr Numbers ......................................................... 5
Figure 4: Graetz Problem Geometry............................................................................... 12
Figure 5: Graetz Velocity Profile ................................................................................... 13
Figure 6: Graetz Temperature Profile ............................................................................. 13
Figure 7: User Defined Mesh ......................................................................................... 15
Figure 8: Centerline Temp vs Mesh Element Number ................................................... 16
Figure 9: Initial Value Variance ..................................................................................... 17
Figure 10: Graetz Problem Centerline Temperature ...................................................... 18
Figure 11: Laminar Flow Velocity Profile ..................................................................... 20
Figure 12: Turbulent Flow Velocity Profile ................................................................... 21
Figure 13: Turbulent Flow Centerline Temperature ....................................................... 22
Figure 14: Turbulent Model for the Graetz Problem ...................................................... 23
Figure 15: Velocity Profile for Flow Through a Pipe ..................................................... 24
Figure 16: Temperature Profile of Flow Through a Pipe ............................................... 25
Figure 17: Outflow Temperature Distribution for the Graetz Problem .......................... 27
Figure 18: Outflow Temperature Distribution for the Graetz Problem with a Pipe Wall
......................................................................................................................................... 27
Figure 19: Velocity Profile for Laminar Flow in a Pipe ................................................. 28
Figure 20: Velocity Profile for Turbulent Flow in a Pipe .............................................. 28
Figure 21: Temperature Profile of Turbulent Flow Through a Pipe ............................. 29
Figure 22: Velocity Profile for Concurrent Heat Exchanger .......................................... 30
Figure 23: Temperature Profile for Concurrent Heat Exchanger ................................... 30
Figure 24: Concurrent Flow Heat Exchanger Temperature Change .............................. 31
Figure 25: Temperature Change Across the Outlet Flow ............................................... 32
Figure 26: Laminar Flow Developing Velocity Profile .................................................. 32
Figure 27: Velocity Profile for a Turbulent Concurrent Flow Heat Exchanger ............. 33
Figure 28: Temperature Profile for a Turbulent Concurrent Flow Heat Exchanger ...... 34
Figure 29: Turbulent Concurrent Flow Heat Exchanger Temperature Change ............. 34
vii
Figure 30: Hot Fluid Outlet Temperature vs Cooling Water Flow Rate ........................ 39
Figure 31: Change in Hot Fluid Temperature vs Cooling Water Flow Rate .................. 40
Figure 32: Temperature Change in the Fluids vs the Difference in Inlet Temperatures 41
Figure 33: Velocity Profile for Countercurrent Heat Exchanger .................................... 42
Figure 34: Temperature Profile for Countercurrent Heat Exchanger ............................. 43
Figure 35: Outlet of the Inner Pipe, Inlet of the Outer Pipe ........................................... 43
Figure 36: Inlet of the Inner Pipe, Outlet of the Outer Pipe ........................................... 44
Figure 37: Counter-current Flow Heat Exchanger Temperature Change ....................... 44
Figure 38: Turbulent Flow Arrow Velocity Profile ........................................................ 45
Figure 39: Velocity Profile for a Turbulent Counter-current Flow Heat Exchanger ..... 45
Figure 40: Temperature Profile for a Turbulent Counter-current Flow Heat Exchanger 46
Figure 41: Turbulent Counter-current Flow Heat Exchanger Temperature Change ...... 46
Figure 42: Hot Fluid Outlet Temperature vs Cooling Water Flow Rate for CounterCurrent Flow .................................................................................................................... 47
Figure 43: Change in Hot Fluid Temperature vs Cooling Water Flow Rate for CounterCurrent Flow .................................................................................................................... 48
Figure 44: Temperature Change in the Fluids vs the Difference in Inlet Temperatures
for Counter-Current Flow ................................................................................................ 49
viii
LIST OF SYMBOLS
A= Area (m^2)
𝐶𝐶 = Heat capacity rate for the cold fluid (𝑚̇𝑐 × 𝐶𝑝,𝑐 )
𝐶ℎ = Heat capacity rate for the hot fluid (𝑚̇ℎ × 𝐶𝑝,ℎ )
𝐶𝑝 = Specific Heat at Constant Pressure (J/ kg K)
𝐶𝑟 = Heat capacity ratio (
𝐶𝑚𝑖𝑛
⁄𝐶
)
𝑚𝑎𝑥
𝐶𝑚𝑖𝑛 = Minimum of 𝐶𝐶 and 𝐶ℎ
𝐶𝑚𝑎𝑥 = Maximum of 𝐶𝐶 and 𝐶ℎ
D= Diamater of a circular tube (m)
h= Heat transfer coefficient (W/m^2 K)
k= Thermal conductivity (W/m K)
L= Flow length of a tube (m)
𝐿
𝐿∗ = Dimensionless length 𝐷𝑅𝑒𝑃𝑟
𝑚̇= Mass flow rate (Kg/s)
Nu= Nusselt number (hD/k)
P= Pressure (N/m^2)
Pe= Peclet Number (Re Pr)
Pr= Prandtl Number (µ c/ k)
q= Heat energy (J)
𝑞̇ = Heat transfer rate (W or J/s)
r= Radial distance of a circular tube (m)
Re= Reynolds Number (unitless)
𝑅𝑓 = Fouling factor ( m^2 k/w)
T= Temperature (K)
𝑇𝑚∗ (𝐿)=Dimensionless Temperature
𝑇𝑚 (𝐿)= Outlet Temperature (C, K)
𝑇𝑤 = Wall Temperature (C)
𝑇𝑜 = Initial temperature of fluid flow (C)
U= Overall heat transfer coefficient (W/m^2 k)
ix
V= Velocty (m/s)
µ= Dynamic viscosity (Pa s)
ε= Heat exchanger effectiveness
Subscripts c and h denote cold and hot fluid flow
Subscripts i and o denote inlet and outlet fluid flow, or inner and outer pipe
x
ACKNOWLEDGMENT
I’d like to thank my family (Ken, Marj, and Dan Onarheim), friends, girlfriend (Jessica
Baker), and advisor (Professor Ernesto Gutierrez-Miravete) for supporting me during
work on this project and my master’s degree.
xi
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.
xii
1. Introduction and Background
There are many uses for heat exchangers from car radiators, to air conditioners,
to large condensers in power plants. Just in submarines alone, heat exchangers are used
for: hydraulic cooling, air conditioning and ventilation, electrical device cooling, cooling
of different types of coolant systems, in purification means, and in the nuclear reactor
and steam generators themselves to provide the means of propulsion. 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.
Basically, one fluid flows through a pipe while the second fluid flows through the
annulus between the inner pipe and outer pipe hence making the pipe walls of the inner
tube the heat transfer surfaces. 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.
Examining the marketplace for applications for concentric tube heat exchangers
or double pipe heat exchangers, one finds that they are used in areas where extreme
temperature crosses are needed, there are high pressure and temperature demands, and
there are low to medium surface area requirements for the job.
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:
1
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 
[3]
The effectiveness of the types of heat exchangers is as follows:

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:
1
Do
UA 
[6]
R f ,i ln( D i ) 1
R f ,o
1



hi Ai
Ai
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.
2
Figure 1: Basic Heat Exchanger Design
1.1.1
Log Mean Temperature Difference
In order to determine the amount of heat to be transferred in a heat exchanger or
the force at which the heat from fluid flow will be transferred, the log mean temperature
difference is calculated. As the name suggests, it is the logarithmic average of the hot
and cold fluid channels of a heat exchanger at both the inlet and outlets. The log mean
temperature difference is defined in terms of ΔT1 and ΔT2 which are defined depending
on whether flow is concurrent or counter current. The larger the temperature difference,
the larger the value of heat that is transferred. The basic equation is:
∆𝑇𝐿𝑀 =
∆𝑇2 −∆𝑇1
∆𝑇
𝐿𝑁( 2⁄∆𝑇 )
1
[7]
For concurrent flow: ∆𝑇2 = 𝑇ℎ,𝑜 − 𝑇𝑐,𝑜 𝑎𝑛𝑑 ∆𝑇1 = 𝑇ℎ,𝑖 − 𝑇𝑐,𝑖
For counter-current flow: ∆𝑇2 = 𝑇ℎ,𝑜 − 𝑇𝑐,𝑖 𝑎𝑛𝑑 ∆𝑇1 = 𝑇ℎ,𝑖 − 𝑇𝑐,𝑜
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
[8]
The effectiveness equation is usually defined by the type of heat exchanger. The
equations for effectiveness include the value of NTU (number of transfer units) and Cr
(ratio of heat capacities). These values are arranged into different equations depending
upon the type of heat exchanger.
1.1.3
NTU Method
This is another method in determining the heat transfer rate and is based on the
“number of transfer units.” For any heat exchanger, the effectiveness can be found to be
a function of the NTU and ratio of heat capacity rates. By definition NTU is:
𝑈𝐴
𝑁𝑇𝑈 = 𝐶
𝑚𝑖𝑛
3
[9]
As shown above, the effectiveness of a double pipe heat exchanger, whether it be
concurrent or countercurrent, can be solved based on the NTU number and the ratio of
heat capacity rates of the fluids, 𝐶𝑟 . This method is typically used when some of the inlet
or outlet temperature data is not available or needs to be solved for. Using this method,
the amount of heat transferred can be determined by the following equation:
𝑞 = 𝜀 × 𝐶𝑚𝑖𝑛 × (𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 )
[10]
Therefore the outlet temperatures for the hot and cold fluids can be calculated as
follows:
𝑞
𝑇𝑐,𝑜 = 𝑇𝑐,𝑖 + ⁄𝐶
𝑚𝑎𝑥
𝑞
𝑇ℎ,𝑜 = 𝑇ℎ,𝑖 − ⁄𝐶
𝑚𝑖𝑛
[11]
[12]
To determine the heat capacity rate for each fluid, the mass flow rate for the fluid is
multiplied by the specific heat of the fluid. The smaller value of these is labeled Cmin
while Cmax is denoted as the larger value.
1.1.4
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 [11]. For incompressible Newtonian fluid flow, the equation of energy
becomes:
𝐷𝑇
𝜌𝑐𝑝 𝐷𝑡 = 𝑘∇2 𝑇 + 𝜑
[13]
Neglecting dissipation and any conduction axially, equation 8 reduces to the following:
4
𝜕𝑇
𝑢 𝜕𝑥 =
𝛼 𝜕
𝜕𝑇
(𝑟 𝜕𝑟 )
𝑟 𝜕𝑟
[14]
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. [11]
Figure 3: Nusselt Number for Various Pr Numbers
5
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 +
.075⁄ ∗
𝐿
1+.05⁄ 2
[15]
𝐿∗3
1.2 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
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
6
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.
1.3 Finite Element Analysis Theory
By definition, finite element analysis is the term applied to the numerical
technique which is used to solve partial differential equations and/ or integral equations.
The system takes the equations and approximates them into a system of ordinary
differential equations and then uses standard numerical solving techniques to solve the
problem. The COMSOL computer program used in this project is a finite element
program. A typical finite element program consists of:
a pre-processor, a mesh
generator, a processor or solver, and a post-processor. The pre-processor part of the
program consists of building a model of the item to be analyzed and the application of
boundary conditions. The boundary conditions consists of any constraints or loads being
applied in the statics/ dynamics region or any velocity or temperature conditions for the
fluid dynamics and heat transfer aspects. In additions to boundary condition definition,
the properties of the materials involved are also defined, and many programs have a
library in which the properties of common materials are stored and able to be used for
definition.
The mesh generator breaks up the model into elements which are geometric bodies
which produce the stiffness or material properties of part of the structure. The element
geometry is defined by nodal locations or conductivity. These elements can be modified
to be smaller or larger or coarser or more refined. The mesh created from the model
applies the geometric and boundary conditions as well as the material properties to the
entire structure. The processor portion of the finite element program has the equations of
heat transfer, fluid flow, as well as solid property equations in order to solve the defined
model. In the COMSOL program there are 3 different types of non-linear solver which
can be used for this purpose. How the solver develops a solution can also be modified by
increasing or decreasing the tolerance of convergence that is required for a solution to be
obtained, or by changing the order in which the solver solves the equations. The post-
7
processor portion of the program allows examination of the results in the form of 1D,
2D, and 3D plots of velocity and temperature profiles as well as arrow, surface, and
contour plots. It is this portion of the program that allows the finite element analysis to
be used whatever fashion is needed.
8
2. Problem Description and Methodology
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 Modeling
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
9
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
In the COMSOL computer application, temperature, velocity, and various fluid
parameters are easily defined and changed by the lefthand tab. For the Graetz problem,
non-isothermal flow was used to define the fluid flow parameters and temperature
distribution, but in the later models, conjugate heat transfer equations were added. This
allowed for laminar flow parameters as well as heat transfer equations to be added. For
the fluid flowing both an inlet and outlet point was chosen. Under these the velocity field
incoming is defined as well as if there is any viscous stress at the outlet. Now that the
velocity is defined, the heat transfer in solids is added when conjugate heat transfer is
used for models with pipe walls, or heat transfer in non-isothermal flow is used. Under
this tab (right clicking on the flow tab) these are many applications that can be defined
from heat flux, heat conduction, cooling, insulation, to temperature definition and
outflow. For the purposes of the models in this paper, temperature is defined in this
method both for incoming fluid as well as the constant wall temperature as defined in the
beginning models. Now that temperature and velocity of the fluid and/ or tubing or pipe
wall is defined and the models can be meshed and solved. The parameters are easily
changed and many iterations with various values can be performed.
10
3. Results and Discussion
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:
Table 1: COMSOL Model Initial Conditions
Flow Parameters
L=
1.0 m
D=
.1 m
k=
0.64
0.000547 Pa s
µ=
988 kg/m^3
ρ=
Cp=
4181 J/Kg k
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.
11
Figure 4: Graetz Problem Geometry
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.
12
Figure 5: Graetz Velocity Profile
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
13
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
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.
14
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
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 2: Mesh Effectiveness
Mesh Effectiveness
Number of Mesh Elements
2150
Centerline Outlet Temp (F)
306.0347
15
2279
2408
2948
3388
3696
4095
4500
5875
8183
10, 376
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.
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
16
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
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.
17
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
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
[16]
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
[17]
The dimensionless length value is defined as
𝐿
(1)
𝐿∗ = 𝐷𝑅𝑒𝑃𝑟 = (.1)(18.062)(3.57) = .15508
18
[18]
The outlet temperature is defined as
𝑇𝑚 (𝐿) = 𝑇𝑤 − (𝑇𝑤 − 𝑇𝑜 ) ∗ 𝑇𝑚∗ (𝐿)
[19]
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
[20]
[21]
𝑇𝑚 (𝐿) = 𝑇𝑤 − (𝑇𝑤 − 𝑇𝑜 ) ∗ 𝑇𝑚∗ (𝐿) = 30 − (30 − 50) ∗ (. 07997)
𝑇𝑚 (𝐿) = 31.5994 C = 304.7494 K
[22]
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
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 3: 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.
19
Table 4: 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
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.
20
% Error
0.388015
1.91009
1.703853
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
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.
21
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
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.
22
Figure 14: Turbulent Model for the Graetz Problem
23
3.2 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.
3.2.1
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
24
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.
3.2.2
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
[23]
𝑄𝑤𝑎𝑡𝑒𝑟 = 𝜌𝐶𝑝 ∆𝑇∆𝑉 = 𝜌𝐶𝑝 ∆𝑇𝜋𝑅 2 ∆𝐿
[24]
𝑞̇ 2𝜋𝑅∆𝐿
∆𝑇 = 𝜌𝐶
𝑝 𝜋𝑅
2 ∆𝐿
2𝑞̇
= 𝜌𝐶
𝑝𝑅
[25]
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.
25
Table 5: 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.
26
Figure 17: Outflow Temperature Distribution for the Graetz Problem
Figure 18: Outflow Temperature Distribution for the Graetz Problem with a Pipe Wall
3.2.3
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. In order to demonstrate that the
laminar flow had a developing velocity profile while the turbulent model had velocity
which was already developed with very little change, the velocity field z component was
plotted for both models, and the velocity graph of the centerline velocity across the
27
length of the pipe was shown. This velocity relationship is shown in figures 19 and 20
for both models below.
Figure 19: Velocity Profile for Laminar Flow in a Pipe
Figure 20: Velocity Profile for Turbulent Flow in a Pipe
This shows that for laminar flow, the velocity was changing from approximately
10 X 10^-5 m/s to approximately 20 X 10^-5 m/s, while for turbulent flow the step
change between the entrance and exit was only from 10 to 10.2.
28
The centerline temperatures 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 problem. Figure 19 shows
the temperature profile for the turbulent model with velocity at .0001 m/s.
Figure 21: 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.
3.3 Flow in a Concurrent Flow Heat Exchanger
3.3.1
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
29
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.
Figure 22: Velocity Profile for Concurrent Heat Exchanger
Figure 23: Temperature Profile for Concurrent Heat Exchanger
30
For concurrent flow heat exchangers the hotter fluid will lower in temperature as
it loses heat to the cooler fluid which will then rise in temperature due to the heat
transfer. A 1D plot was made to determine this temperature development. First a line
graph of the temperature distribution along the centerline (the hotter fluid) was made.
Then a second curve was created of the temperature along the length of the pipe in the
middle of the flow in the outer tube. Figure 24 below shows this gradual temperature
change in both flow paths. This is the correct cure form already proven for concurrent
flow heat exchangers.
Figure 24: Concurrent Flow Heat Exchanger Temperature Change
Looking at the end of the 1 m heat exchanger, the flow closest to the centerline
was the hottest for the inner fluid and the flow closest to the outside of the inner pipe
was the hottest for the outer fluid. This is due to the flow closest to the inside wall of the
inside pipe experiences more of the heat transfer to the colder fluid of the outer pipe. The
flow closest to the outside wall of the inner pipe receives more of the heat energy and
therefore has a higher temperature nearest the inner pipe for the colder outer flow. This
leads to a downwards sloping curve from the 0.0 m to the .05m mark for the inner flow
as well as a downward slope from .07 m to 1.2 m when temperature is graphed along the
radius at the outlet of the heat exchanger. In addition to this figure 25 also shows the
slight heat conduction in the steel pipe. It’s very slight, but does show that a portion of
the heat energy is transferred to the pipe wall and not the flow parallel to one another.
31
Figure 25: Temperature Change Across the Outlet Flow
3.3.2
Turbulent Flow in a Concurrent Heat Exchanger
In the laminar flow model, an arrow surface plot of the flow shows the
developing velocity profile of the inner and outer flows, and the typical parabolic shape
of the velocity is shown in figure 26.
Figure 26: Laminar Flow Developing Velocity Profile
32
The velocity profile for the turbulent model of the same concurrent flow heat
exchanger shows that there is very little change in the velocity of either fluid since the
velocity profile as previously discussed is already developed. The temperature profile
and resultant graphs of the centerline of both fluid flows shows very little change in
either the inner or outer fluid’s temperature. In the laminar case, the concurrent flow heat
exchanger yielded a gradually lowering hot fluid temperature with a similar gradually
increase in the cold fluid temperature, but with turbulence applied to the model, the
temperature of both fluids with the .0001 m/s velocity shows little to no change in either
fluid. Figures 27-29 shows the effect the turbulent flow has on a concurrent flow heat
exchanger.
Figure 27: Velocity Profile for a Turbulent Concurrent Flow Heat Exchanger
33
Figure 28: Temperature Profile for a Turbulent Concurrent Flow Heat Exchanger
Figure 29: Turbulent Concurrent Flow Heat Exchanger Temperature Change
3.3.3
Laminar Flow in a Concurrent Heat Exchanger Problem Calculations
In order to analyze the concurrent flow heat exchanger better, an example heat
exchanger was designed in COMSOL using the existing model and an excel spreadsheet
made to document the hand calculated results. In the cases studied, engine oil was
assumed to be flowing through the inner pipe which was made of copper and cooled by
the outer concentric pipe in which water was flowing. Material properties such as
34
dynamic viscosity, density, prandtl number, and thermal conductivity were obtained
from reference [5]. It was noted at this time that in the mesh that was previously used, no
boundary layer elements were added to the outside of the inner pipe where the cooling
water of the outer pipe was flowing across. For the oil and water heat exchanger design,
an additional boundary layer mesh was added to this surface. Comparing results for the
first case (.0001 m/s oil velocity with varying watermK velocity) with and without this
boundary layer showed only a small change in the outlet temperatures. The largest
difference was approximately .5K.
For comparison to the COMSOL model results, the outlet temperatures for the oil
and water were determined using a NTU-effectiveness method. An excel spreadsheet
was used so that during the differing cases which changed the fluid velocities and
temperatures, only these parameters had to be changed in the spreadsheet and the hand
calculated version of the outlet temperatures would automatically update. An example of
these calculations is as follows below for the first case analyzed, oil velocity at .0001
m/s and water velocity .0001 m/s. The hot inner fluid (oil) is flowing through 1 copper
pipe 1 meter in length.
𝐷𝑜𝑢𝑡𝑒𝑟 𝑡𝑢𝑏𝑒 = 𝐷𝑜 = .14𝑚,
𝐴𝑜𝑢𝑡𝑒𝑟 𝑡𝑢𝑏𝑒 = 𝐴𝑜 = 𝜋𝐷𝐿 = 𝜋(. 14𝑚)(1𝑚) = .4398𝑚2
𝐷𝑖𝑛𝑛𝑒𝑟 𝑡𝑢𝑏𝑒 = 𝐷𝑖 = .10𝑚,
𝐴𝑖𝑛𝑛𝑒𝑟 𝑡𝑢𝑏𝑒 = 𝐴𝑖 = 𝜋𝐷𝐿 = 𝜋(. 10𝑚)(1𝑚) = .3142𝑚2
The cross sectional area of each fluid flow is:
𝐴𝑜𝑖𝑙 = 𝜋𝑟 2 = 𝜋(. 05𝑚2 ) = .007854𝑚2
𝐴𝑤𝑎𝑡𝑒𝑟 = 𝜋(𝑟𝑜 2 − 𝑟𝑖 2 ) = 𝜋((.12𝑚)2 − (.07𝑚)2 ) = .0298454𝑚2
The inlet temperature of each fluid and its corresponding properties due to that
temperature is as follows:
Table 6: Fluid Properties
Fluid Parameters for Oil
T=
125 C
T=
398.15 K
k=
0.134 w/mK
0.00915 Pa s
µ=
ρ=
826 kg/m^3
Pr=
159
Cp=
2328 J/Kg K
Fluid Parameters for Water
T=
20 C
T=
293.15 K
k=
0.600 w/mK
0.001003 Pa s
µ=
ρ=
998.2 kg/m^3
Pr=
6.99
Cp=
4182 J/Kg K
35
The mass flow rates are then calculated and used to determine the heat capacity rates.
826𝑘𝑔
𝑚
2)
(.
𝑚̇𝑜𝑖𝑙 = 𝜌𝐴𝑣 = (
)
007854𝑚
(.
0001
) = .0006487 𝑘𝑔/𝑠
𝑚3
𝑠
998.2𝑘𝑔
𝑚
𝑚̇𝑤𝑎𝑡𝑒𝑟 = 𝜌𝐴𝑣 = (
) (. 029845𝑚2 ) (. 0001 ) = .002979 𝑘𝑔/𝑠
3
𝑚
𝑠
𝐽
. 0006487𝐾𝑔
𝐶𝑜𝑖𝑙 = 𝐶𝑝,𝑜𝑖𝑙 × 𝑚̇𝑜𝑖𝑙 = (2328
)(
) = 1.5102 𝑊/𝐾
𝐾𝑔𝐾
𝑠
𝐶𝑤𝑎𝑡𝑒𝑟 = 𝐶𝑝,𝑤𝑎𝑡𝑒𝑟 × 𝑚̇𝑤𝑎𝑡𝑒𝑟 = (4182
𝐽
. 002979𝐾𝑔
)(
) = 12.4588 𝑊/𝐾
𝐾𝑔𝐾
𝑠
From this it can be defined for the analysis purposes that Cmin is Coil and Cmax is
Cwater. This yields our ratio of heat capacity rates to be:
𝐶𝑟 =
𝐶𝑚𝑖𝑛
𝐶𝑜𝑖𝑙
1.5102
=
=
= .12122
𝐶𝑚𝑎𝑥 𝐶𝑤𝑎𝑡𝑒𝑟 12.4588
The Reynolds number for the oil flow and then the Nusselt number for the heat transfer
from the oil to the water are as follows:
𝑅𝑒 =
𝑁𝑢 = 3.66 +
𝜌𝑣𝐷
4𝑚̇𝑜𝑖𝑙
4 × .0006487𝑘𝑔/𝑠
=
=
= .9027
𝜇
𝜋𝐷𝑖 𝜇𝑜𝑖𝑙 𝜋 × .10𝑚 × .00915 𝑃𝑎 𝑠
. 0668𝐴
𝐷
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝑅𝑒𝑃𝑟 ( 𝑖⁄𝐿) = (. 9027)(159)(. 10) = 14.35
1 + .04𝐴.667
𝑁𝑢 = 4.435
The heat transfer coefficient of the inner pipe wall is expressed as follows:
𝑊
𝑘𝑜𝑖𝑙 × 𝑁𝑢 . 134 𝑚𝐾 × 4.435
ℎ𝑖 =
=
= 5.9435 𝑊⁄𝑚2 𝐾
𝐷𝑖
. 10𝑚
The overall heat transfer coefficient is expressed in terms of UA. For this overall
coefficient, the heat transfer coefficient of the outer wall of the inner pipe is required.
For the purposes of the analysis it is assumed to be approximately half of the value for
the heat transfer coefficient of the inner wall. This overall coefficient is defined as
follows with the thermal conductivity of copper being 393.11 W/mK:
36
𝑈𝐴 =
=
1
𝐷
ln( 𝑜⁄𝐷 )
1
1
+
+ 2𝜋𝐿𝐾 𝑖
ℎ𝑖 𝐴𝑖 ℎ𝑜 𝐴𝑜
1
ln(. 14𝑚⁄. 10𝑚)
1
1
+
+
(5.9435𝑊/𝑚2 𝐾)(.3142𝑚2 ) (. 5 ∗ 5.9435𝑊/𝑚2 𝐾)(.4398𝑚2 ) 2𝜋(1𝑚)(393.11𝑊/𝑚𝐾)
𝑈𝐴 = 0.768769 𝑊/𝐾
The value for the number of heat transfer units is:
𝑁𝑇𝑈 =
𝑈𝐴
. 768769 𝑊/𝐾
=
= .50903
𝐶𝑚𝑖𝑛
1.5102 𝑊/𝐾
Now that the heat capacity ratio and NTU values are determined for this concurrent
concentric tube heat exchanger the effectiveness value is calculated as follows:
1 − 𝑒 [−𝑁𝑇𝑈(1+𝐶𝑟 )] 1 − 𝑒 [−.50903(1+12122)]
𝜀=
=
= .387872
1 + 𝐶𝑟
1 + .12122
The equation for heat transferred in the NTU-effectiveness method is in terms of this
effectiveness value as well as the minimum heat capacity.
1.5102𝑊
𝑞 = 𝜀 × 𝐶𝑚𝑖𝑛 × (𝑇ℎ,𝑖 − 𝑇𝑐,𝑖 ) = (. 387872) (
) (398.15𝐾 − 293.15𝐾)
𝐾
= 61.50782𝑊
From equations 11 and 12 we know the overall energy balance gives the outlet
temperatures of the fluids by subtracting or adding the value of the heat transferred
divided by the heat capacity of the fluid to the inlet temperature of that fluid. For this
case:
𝑞
𝑇𝑐,𝑜 = 𝑇𝑐,𝑖 + ⁄𝐶
= 293.15𝐾 + 61.50782𝑊⁄12.4588𝑊/𝐾 = 298.0869
𝑚𝑎𝑥
𝑞
𝑇ℎ,𝑜 = 𝑇ℎ,𝑖 − ⁄𝐶
= 398.15𝐾 − 61.50782𝑊⁄1.5102𝑊/𝐾 = 357.4235𝐾
𝑚𝑖𝑛
As a double check for this calculation, the log mean temperature difference was
determined using the outlet temperatures calculated and then compared to the log mean
temperature difference determined by equation 2.
𝑞 = 𝑈𝐴∆𝑇𝐿𝑀 ∴ ∆𝑇𝐿𝑀 =
𝑞
61.50782𝑊
=
= 80.017𝐾
𝑈𝐴 . 768789𝑊/𝐾
37
∆𝑇𝐿𝑀 =
(357.4235𝐾 − 298.0869𝐾) − (398.15𝐾 − 293.15𝐾)
∆𝑇2 − ∆𝑇1
=
=
∆𝑇2
(357.4235𝐾
−
298.0869𝐾)
𝐿𝑁 ( ⁄∆𝑇 ) 𝐿𝑁
⁄(398.15𝐾 − 293.15𝐾)
1
∆𝑇𝐿𝑀 = 80.017𝐾 ∴ 𝑇ℎ𝑒 𝑐ℎ𝑒𝑐𝑘 𝑖𝑠 𝑆𝐴𝑇
After completing the model generation in COMSOL, a the study of the heat
exchanger consisted of running the model with the same oil velocity of .0001 m/s but the
cooling flow velocity was increased from .0001 m/s to .001 m/s and then .01 m/s.
Maintaining the same fluid velocities for both, the inlet temperature of the cooling flow
was increased thus lowering the temperature difference between the fluids. Figure 30
below shows that as the cooling water flow increases the outlet temperature of the oil
lowers. For each increase of velocity (each increment was ten times the previous), the
outlet temperature of the hot fluid lowered by approximately 2K. So therefore as the
velocity increases for the colder fluid, the heat capacity rate for the cooling fluid will
increase which will decrease the ratio between the capacity rates and therefore change
the effectiveness of the heat exchanger. In the case of this concurrent flow heat
exchanger, the effectiveness increases which therefore increases the amount of heat
transferred, allowing the temperature of the oil to drop more and the temperature of the
water to raise more.
38
370
Oil Flow Oulet Temperature (K)
369
368
367.05901
367
366
365.1556
365
363.7862
364
Th,o (oil)
363
362
361
360
0
0.002
0.004
0.006
0.008
0.01
0.012
Cooling Water Velocity (m/s)
Figure 30: Hot Fluid Outlet Temperature vs Cooling Water Flow Rate
Figure 31 shows that as the cooling water flow increases the temperature change
of the hotter fluid increases. This is due to the fact that oil temperature is the lowest for
the larger the cooling flow.
39
35
34.3638
34.5
Change in Oil Temperature (K)
34
33.5
32.9944
33
32.5
Δ in Oil Temp
32
31.5
31.09099
31
30.5
0
0.002
0.004
0.006
0.008
0.01
Cooling Water Velocity (m/s)
Figure 31: Change in Hot Fluid Temperature vs Cooling Water Flow Rate
As mentioned above, the velocity of the oil and water was held constant at .0001
m/s and the inlet temperature of the water was increased from 293.15 K to 303.15 K and
to 313.15K. Figure 32 depicts the temperature changes in both the hot and cold fluids as
a the temperature drop between the fluids increase. For the smaller difference between
the inlet temperature of the oil and water, 85F, the change between the inlet and outlet
for both the cold and hot fluids is the smallest. But as the temperature difference increase
to 95F and 105F, the temperature change between both the cold and hot fluids increases
linearly.
40
Change in Fluid Temperature Between Inlet and Outlet (K)
35
31.09099
28.83715
30
26.30021
25
20
16.62035
15.05942
Th,o-Th,i (Oil)
13.50046
15
Tc,o-Tc,i (Water)
10
5
0
75
80
85
90
95
100
105
110
Difference in Inlet Temperatures Between Fluids (K)
Figure 32: Temperature Change in the Fluids vs the Difference in Inlet Temperatures
For each case the results were compared to the COMSOL values and the percent
difference calculated. Most of the results were in the range of 2-3% different. These
results are part of the results spreadsheet located in the results section. There are a
couple possible reasons for the difference between the actual (COMSOL) and calculated
values. First, the heat transfer coefficient for the outer portion of the inner pipe was
estimated in the hand calculations, and the COMSOL model used the previously
determined value from the material library. For better results, if this coefficient could be
user defined in the finite element program or the value the program uses recorded for use
in the hand calculations, a more accurate solution might have been obtained. This
affected the overall heat transfer coefficient and therefore the NTU value and the
effectiveness of the heat exchanger. Secondly, the material property values used in the
calculations were based on the inlet temperatures of the oil and water. To create a better
representation of the actual case, these should have been based off the average
temperature of the fluids. If a more in depth study could have been performed, the outlet
41
temperature should have initially been guessed and several iterations of the calculations
performed until the value of the outlet temperature settles out to a near constant value. In
this method the specific heat values, prandtl numbers, thermal conductivity numbers,
viscosity, and densities would be based off the average temperature of the fluids (inlet
temperature plus outlet temperature divided by 2).
3.3.4
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.
Figure 33: Velocity Profile for Countercurrent Heat Exchanger
42
Figure 34: Temperature Profile for Countercurrent Heat Exchanger
Figure 35: Outlet of the Inner Pipe, Inlet of the Outer Pipe
43
Figure 36: Inlet of the Inner Pipe, Outlet of the Outer Pipe
Looking at the difference in temperature profile of the inner fluid vs the temperature
profile of the outer fluid, the proven results of a counter-current heat exchanger are
obtained.
Figure 37: Counter-current Flow Heat Exchanger Temperature Change
3.3.5
Turbulent Flow in a Counter-Current Heat Exchanger
For turbulent flow in the counter-current heat exchanger, the velocity profile was
almost exactly that of turbulent flow in a singular pipe. The extent of the velocity
44
distribution was between 9.8-10.2 X 10^-5 m/s which as discussed is due to the
developed flow already entering both pipes due to the turbulence being applied to the
model. Figure 35 shows that both velocity profiles are developed prior to heat transfer,
and figure 36 shows the extent of velocity distribution throughout the model.
Figure 38: Turbulent Flow Arrow Velocity Profile
Figure 39: Velocity Profile for a Turbulent Counter-current Flow Heat Exchanger
Although the velocity profiles were different between the concurrent and counter
current flow heat exchangers with turbulence applied, the temperature profiles between
45
the 2 types of heat exchangers were almost identical, and very little change is seen
between the hot and cold fluid along the length of the center of each fluid. Figures 37
and 38 show the turbulent temperature profiles in the counter-current type heat
exchanger.
Figure 40: Temperature Profile for a Turbulent Counter-current Flow Heat Exchanger
Figure 41: Turbulent Counter-current Flow Heat Exchanger Temperature Change
46
3.3.6
Laminar Flow in a Counter-current Heat Exchanger Problem Calculations
375.54683
376
Oil Flow Oulet Temperature (K)
375
374
373
372
371.25821
371
Th,o (oil)
370
369
368
367.44359
367
0
0.002
0.004
0.006
0.008
0.01
Cooling Water Velocity (m/s)
Figure 42: Hot Fluid Outlet Temperature vs Cooling Water Flow Rate for Counter-Current Flow
47
32
30.70641
Change in Oil Temperature (K)
30
28
26.89179
26
Δ in Oil Temp
24
22.60317
22
20
0
0.002
0.004
0.006
0.008
0.01
Cooling Water Velocity (m/s)
Figure 43: Change in Hot Fluid Temperature vs Cooling Water Flow Rate for Counter-Current
Flow
48
Change in Fluid Temperature Between Inlet and Outlet (K)
35
31.33479
28.62676
30
25.842385
25
22.60317
20.96748
19.16949
20
Tc,o-Tc,I (Water)
15
Th,i-Th,o (Oil)
10
5
0
75
80
85
90
95
100
105
110
Difference in Inlet Temperatures Between Fluids (K)
Figure 44: Temperature Change in the Fluids vs the Difference in Inlet Temperatures for CounterCurrent Flow
3.4
Flow in a Concurrent Flow Heat Exchanger with Fouling
3.4.1
Laminar Flow in a Concurrent Heat Exchanger with Fouling COMSOL
Model
3.4.2
Turbulent Flow in a Concurrent Heat Exchanger with Fouling
3.4.3
Laminar Flow in a Concurrent Heat Exchanger with Fouling Problem
Calculations
49
3.5 Flow in a Counter-Current Flow Heat Exchanger with Fouling
3.5.1
Laminar Flow in a Counter-current Heat Exchanger with Fouling
COMSOL Model
3.5.2
Turbulent Flow in a Counter-Current Heat Exchanger with Fouling
3.5.3
Laminar Flow in a Counter-current Heat Exchanger with Fouling Problem
Calculations
50
4. Conclusion
-Talk about findings of velocity and inlet temp on outlet temp and the difference
between concurrent and countercurrent HX
-Talk about whether laminar or turbulent flow is better and the difference between the 2
-How did fouling affect the heat transfer?
-Talk about findings during the process of the project that the initial values and mesh
refinement can change the ending results unless a proper mesh is produced and verified
to give consistent results. This may involves changing mesh conditions (boundary
layers), changing the tolerance of solution convergence, or changing the type of nonlinear solver. If not done, the results could be inaccurate analysis and results that in the
industrial and business application could lead to developing and marketing the wrong or
improperly designed heat exchanger that not only could cause damage but could be a
personnel hazard in the industrial workplace.
51
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.
52
6. APPENDIX
6.1 Laminar Flow Concurrent Heat Exchanger Data
CASE 1:
Iteration No.
Velocity of oil= .0001 m/s
1
2
3
Ao (m^2)
Ai (m^2)
Tc,I (Celsius)
Vc, I (m/s)
Th,I (Celsius)
Vh, I (m/s)
A oil flow
A water flow
ρ Oil (kg/m^3)
ρ Water (kg/m^3)
Mc (kg/s)
Mh (kg/s)
Cpc (j/kg*k)
Cph (j/kg*k)
0.439823
0.314159
20
0.0001
125
0.0001
0.007854
0.029845
826
998.2
0.002979
0.000649
4182
2328
0.439823
0.314159
20
0.001
125
0.0001
0.007854
0.029845
826
998.2
0.029791
0.000649
4182
2328
0.439823
0.314159
20
0.01
125
0.0001
0.007854
0.029845
826
998.2
0.297914
0.000649
4182
2328
Cc (w/k)
Ch (w/k)
Cmin/Cmax
12.45877
1.510264
0.121221
124.5877
1.510264
0.012122
1245.877
1.510264
0.001212
μ (Pa s)
Pr
Re
k oil (w/m*k)
Nusselt Number
hi (w/m^2*k)
k Copper (w/m*k)
UA (w/k)
NTU
ε
q (w)
Tc,o (Celsius)
Tc,o (Kelvin)
Tc,o (COMSOL)
Tc,o Percent Diff
Th,o (Celsius)
Th,o (Kelvin)
Th,o (COMSOL)
Th,o Percent Diff
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.387872
61.50782
24.93691
298.0869
309.7703
3.771614
84.27347
357.4235
367.059
2.625067
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.397797
63.08173
20.50632
293.5063
295.9225
0.816473
83.23133
356.2313
365.1556
2.443964
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.398809
63.2422
20.05076
293.0508
293.1495
0.033675
83.12507
356.1251
363.7862
2.105942
53
CASE 2:
Iteration No.
Velocity of water and oil= .0001 m/s
1
2
3
4
Ao (m^2)
Ai (m^2)
Tc,I (Celsius)
Vc, I (m/s)
Th,I (Celsius)
Vh, I (m/s)
A oil flow
A water flow
ρ Oil (kg/m^3)
ρ Water (kg/m^3)
Mc (kg/s)
Mh (kg/s)
Cpc (j/kg*k)
Cph (j/kg*k)
0.43982297
0.31415927
20
0.0001
125
0.0001
0.00785398
0.02984513
826
998.2
0.00297914
0.00064874
4182
2328
0.439822972
0.314159265
30
0.0001
125
0.0001
0.007853982
0.02984513
826
995.6
0.002971381
0.000648739
4179
2328
0.439823
0.314159
40
0.0001
125
0.0001
0.007854
0.029845
826
992.2
0.002961
0.000649
4179
2328
0.439822972
0.314159265
20
0.0001
150
0.0001
0.007853982
0.02984513
811
998.2
0.002979141
0.000636958
4182
2440
Cc (w/k)
Ch (w/k)
Cmin/Cmax
12.4587672
1.51026412
0.12122099
12.41740188
1.51026412
0.121624808
12.375
1.510264
0.122042
12.45876723
1.554177302
0.124745673
μ (Pa s)
Pr
Re
k oil (w/m*k)
Nusselt Number
hi (w/m^2*k)
k Copper (w/m*k)
UA (w/k)
NTU
ε
q (w)
Tc,o (Celsius)
Tc,o (Kelvin)
Tc,o (COMSOL)
Tc,o Percent Diff
Th,o (Celsius)
Th,o (Kelvin)
Th,o (COMSOL)
Th,o Percent Diff
0.00915
159
0.90273224
0.134
4.43545035
5.94350347
393.111
0.76876929
0.5090297
0.38787175
61.5078227
24.9369108
297.936911
309.77025
3.82003733
84.2734662
357.273466
367.05901
2.66593206
0.00915
159
0.90273224
0.134
4.435450351
5.94350347
393.111
0.768769293
0.509029701
0.38783566
55.64475678
34.48119158
307.4811916
318.20942
3.371436464
88.15561228
361.1556123
369.31285
2.20876087
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.387798
49.78263
44.02284
317.0228
326.6505
2.947377
92.03713
365.0371
371.8498
1.832099
0.00564
104
1.437943262
0.132
4.46366455
5.892037207
391.3795
0.762112671
0.490364047
0.376916352
76.15332906
26.11242891
299.1124289
313.80913
4.683324888
101.0008742
374.0008742
386.5914
3.25680441
54
6.2 Laminar Flow Counter-Current Heat Exchanger Data
CASE 1:
Iteration No.
Velocity of oil= .0001 m/s
1
2
3
Ao (m^2)
Ai (m^2)
Tc,I (Celsius)
Vc, I (m/s)
Th,I (Celsius)
Vh, I (m/s)
A oil flow
A water flow
ρ Oil (kg/m^3)
ρ Water (kg/m^3)
Mc (kg/s)
Mh (kg/s)
Cpc (j/kg*k)
Cph (j/kg*k)
0.439823
0.314159
20
0.0001
125
0.0001
0.007854
0.029845
826
998.2
0.002979
0.000649
4182
2328
0.439823
0.314159
20
0.001
125
0.0001
0.007854
0.029845
826
998.2
0.029791
0.000649
4182
2328
0.439823
0.314159
20
0.01
125
0.0001
0.007854
0.029845
826
998.2
0.297914
0.000649
4182
2328
Cc (w/k)
Ch (w/k)
Cmin/Cmax
12.45877
1.510264
0.121221
124.5877
1.510264
0.012122
1245.877
1.510264
0.001212
μ (Pa s)
Pr
Re
k oil (w/m*k)
Nusselt Number
hi (w/m^2*k)
k Copper (w/m*k)
UA (w/k)
NTU
ε
q (w)
Tc,o (Celsius)
Tc,o (Kelvin)
Tc,o (COMSOL)
Tc,o Percent Diff
Th,o (Celsius)
Th,o (Kelvin)
Th,o (COMSOL)
Th,o Percent Diff
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.390964
61.99814
24.97627
298.1263
324.4848
8.123192
83.94881
357.0988
375.5468
4.912309
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.39812
63.13294
20.50674
293.5067
294.0418
0.181983
83.19742
356.1974
371.2582
4.05669
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.398841
63.24734
20.05077
293.0508
293.15
0.033855
83.12167
356.1217
367.4436
3.081268
55
CASE 2:
Iteration No.
Velocity of water and oil= .0001 m/s
1
2
3
4
Ao (m^2)
Ai (m^2)
Tc,I (Celsius)
Vc, I (m/s)
Th,I (Celsius)
Vh, I (m/s)
A oil flow
A water flow
ρ Oil (kg/m^3)
ρ Water (kg/m^3)
Mc (kg/s)
Mh (kg/s)
Cpc (j/kg*k)
Cph (j/kg*k)
0.43982297
0.31415927
20
0.0001
125
0.0001
0.00785398
0.02984513
826
998.2
0.00297914
0.00064874
4182
2328
0.439822972
0.314159265
30
0.0001
125
0.0001
0.007853982
0.02984513
826
995.6
0.002971381
0.000648739
4179
2328
0.439823
0.314159
40
0.0001
125
0.0001
0.007854
0.029845
826
992.2
0.002961
0.000649
4179
2328
0.439822972
0.314159265
20
0.0001
150
0.0001
0.007853982
0.02984513
811
998.2
0.002979141
0.000636958
4182
2440
Cc (w/k)
Ch (w/k)
Cmin/Cmax
12.4587672
1.51026412
0.12122099
12.41740188
1.51026412
0.121624808
12.375
1.510264
0.122042
12.45876723
1.554177302
0.124745673
μ (Pa s)
Pr
Re
k oil (w/m*k)
Nusselt Number
hi (w/m^2*k)
k Copper (w/m*k)
UA (w/k)
NTU
ε
q (w)
Tc,o (Celsius)
Tc,o (Kelvin)
Tc,o (COMSOL)
Tc,o Percent Diff
Th,o (Celsius)
Th,o (Kelvin)
Th,o (COMSOL)
Th,o Percent Diff
0.00915
159
0.90273224
0.134
4.43545035
5.94350347
393.111
0.76876929
0.5090297
0.39096374
61.9981432
24.9762663
297.976266
324.48479
8.1694195
83.9488074
356.948807
375.54683
4.95225125
0.00915
159
0.90273224
0.134
4.435450351
5.94350347
393.111
0.768769293
0.509029701
0.390937449
56.08978617
34.51703075
307.5170308
331.77676
7.312064066
87.86094237
360.8609424
377.18252
4.327235957
0.00915
159
0.902732
0.134
4.43545
5.943503
393.111
0.768769
0.50903
0.39091
50.18212
44.05512
317.0551
338.9924
6.471313
91.77262
364.7726
378.9805
3.748976
0.00564
104
1.437943262
0.132
4.46366455
5.892037207
391.3795
0.762112671
0.490364047
0.379811816
76.7383374
26.15938447
299.1593845
332.1002
9.918938781
100.6244639
373.6244639
396.93667
5.87302908
56