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 of Previous Graetz Problem Solutions... 4 2. Problem Description... 5 2.1 Defining Material Properties... 5 2.2 Methodology and Approach... 5 2.2.1 Finite Element Analysis... 5 2.2.2 Defining Variable Temperature and Velocity... 6 3. Theory of Solution... 7 3.1 The Graetz Problem Results... 7 3.1.1 The Graetz Problem COMSOL Model... 7 3.1.2 The Graetz Problem COMSOL Mesh... 9 3.1.3 The Graetz Problem Study Results... 13 3.1.4 The Graetz Problem Calculations... 14 3.1.5 Turbulent Flow with Constant Wall Temperature... 15 4. Results... 18 4.1 Flow in a Pipe with Axial Conduction... 18 iii
4.1.1 Laminar Flow with a Pipe Wall COMSOL Model... 18 4.1.2 Laminar Flow with a Pipe Wall Problem Calculations... 19 4.1.3 Turbulent Flow in a Pipe with Axial Conduction... 20 4.2 Flow in a Concurrent Flow Heat Exchanger... 21 4.2.1 Laminar Flow in a Concurrent Heat Exchanger COMSOL Model... 21 4.2.2 Laminar Flow in a Concurrent Heat Exchanger Problem Calculations 21 4.2.3 Turbulent Flow in a Concurrent Heat Exchanger... 21 4.3 Flow in a Counter-Current Flow Heat Exchanger... 21 4.3.1 Laminar Flow in a Counter-current Heat Exchanger COMSOL Model... 21 4.3.2 Laminar Flow in a Counter-current Heat Exchanger Problem Calculations... 21 4.3.3 Turbulent Flow in a Counter-Current Heat Exchanger... 21 4.4 Laminar Flow in a Concurrent Flow Heat Exchanger with Fouling... 21 4.5 Laminar Flow in a Counter-Current Flow HX with Fouling... 21 4.6 Turbulent Flow in a Concurrent Flow Heat Exchanger with Fouling... 21 4.7 Turbulent Flow in a Counter-Current Flow HX with Fouling... 21 5. References... 22 iv
LIST OF TABLES Table 1: Mesh Effectiveness... 11 Table 2: Graetz Problem Comparison... 15 Table 3: Results from Mesh Refinement... 15 Table 4: Change in Temperature Along the Channel of Water... 20 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... 7 Figure 5: Graetz Velocity Profile... 8 Figure 6: Graetz Temperature Profile... 9 Figure 7: User Defined Mesh... 10 Figure 8: Centerline Temp vs Mesh Element Number... 12 Figure 9: Initial Value Variance... 12 Figure 10: Graetz Problem Centerline Temperature... 13 Figure 11: Laminar Flow Velocity Profile... 16 Figure 12: Turbulent Flow Velocity Profile... 16 Figure 13: Turbulent Flow Centerline Temperature... 17 Figure 14: Velocity Profile for Flow Through a Pipe... 18 Figure 15: Temperature Profile of Flow Through a Pipe... 19 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. q m c T h T c [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 UAT lm T2 T UA ln T2 T 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 cmin T h, i Tc, i The effectiveness of the types of heat exchangers is as follows: 1 1 [2] [3] 1
1 exp[ NTU (1 Cr )] Parallel Flow: 1 Cr [4] 1 exp[ NTU (1 Cr )] Counter Flow: forcr 1 [5] 1 C exp[ NTU (1 C ) r r 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 1 h A i i R A o f, i D i 1 i 1 D ln( ) 2kL h o A o R A 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. f, o o [6] 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: q actual,0 1 [7] q max 2
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: [8] Neglecting dissipation and any conduction axially, equation 8 reduces to the following: [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 21. The velocity profile can also be developing and can be used for any 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. 3.66.. [10] 1.1.4 Description of Previous Graetz Problem Solutions 4
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 5
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 6
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 7
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 8
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 9
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 10
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 Number of Mesh Elements Mesh Effectiveness Centerline Outlet Temp (F) 2150 306.0347 2279 305.9992 2408 306.0265 2948 305.9083 3388 305.8629 3696 305.8664 4095 305.6118 4500 305.6001 5875 305.2428 8183 305.7506 10, 376 305.6016 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. 11
Mesh Effectiveness 306.4 306.2 Temperature (F) 306 305.8 305.6 305.4 Centerline Outlet Temp (F) Power (Centerline Outlet Temp (F)) 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 was fairly constant as shown in figure 9. 310 Centerline Outlet Temp (F) Temperature (F) 308 306 304 302 300 280 290 300 310 320 330 Initial Value Temp (F) Centerline Outlet Temp (F) Figure 9: Initial Value Variance 12
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 13
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... 18.062 [11]. The prandtl number is calculated using the material properties of water at the inlet temperature. The dimensionless length value is defined as The outlet temperature is defined as..... 3.57 [12].15508 [13] [14] Since there is a relationship between 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. 3.66.. 3.66.... / 4.0722 [15].07997 [16] 3030 50. 07997 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 14
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%. Inlet V (m/s) Inlet Temp (C) Table 2: Graetz Problem Comparison Wall Temp (C) Expected Value Calc (K) COMSOL Value (K) % Error 0.0001 50 30 304.7494 305.3221 0.187572 0.001 50 30 316.646 321.9023 1.632887 0.01 50 30 317.644 323.0481 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) Exp. Value Calc (K) COMSOL Value (K) % Error COMSOL Refined (K) % Error 0.0001 304.7494 305.3221 0.187572403 305.93648 0.388015 0.001 316.646 321.9023 1.632886749 322.812 1.91009 0.01 317.644 323.0481 1.672846861 323.15 1.703853 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 15
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 16
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 322.25506F and for a velocity of.001 m/s the centerline temperature was 322.86295F. 17
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 14: Velocity Profile for Flow Through a Pipe 18
Figure 15: 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. 2 [18] [19] [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 19
table below shows the outlet temperature based on the heat flux along the length of the fluid channel. Table 4: Change in Temperature Along the Channel of Water Length (m) Heat Flux at the Wall (W/m^2) ΔT (F) Outlet Temp (F) 0.1 45.71 0.00044 323.1495574 0.2 58.05 0.00056 323.1494379 0.3 73.29 0.00071 323.1492903 0.4 92.19 0.00089 323.1491073 0.5 115.87 0.00112 323.148878 0.6 145.86 0.00141 323.1485876 0.7 185.05 0.00179 323.1482081 0.8 240.52 0.00233 323.147671 0.9 304.94 0.00295 323.1470472 1 32844.5 0.31804 322.8319572 4.1.3 Turbulent Flow in a Pipe with Axial Conduction 20
4.2 Flow in a Concurrent Flow Heat Exchanger 4.2.1 Laminar Flow in a Concurrent Heat Exchanger COMSOL Model 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 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 Laminar Flow in a Concurrent Flow Heat Exchanger with Fouling 4.5 Laminar Flow in a Counter-Current Flow HX with Fouling 4.6 Turbulent Flow in a Concurrent Flow Heat Exchanger with Fouling 4.7 Turbulent Flow in a Counter-Current Flow HX with Fouling 21
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. 3 rd Companies, Inc., 2006. ed. New York: 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 22