Analysis and Comparison of the Performance of Concurrent and Countercurrent Flow Double

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)

2 CONTENTS

3 LIST OF TABLES

4 LIST OF FIGURES

5 LIST OF SYMBOLS

A= Area (m^2) = Heat capacity rate for the cold fluid () = Heat capacity rate for the hot fluid () = Specific Heat at Constant Pressure (J/ kg K) = Heat capacity ratio () = Minimum of and = Maximum of and 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) = Dimensionless length = 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) = Heat transfer rate (W or J/s) r= Radial distance of a circular tube (m) Re= Reynolds Number (unitless) = Fouling factor ( m^2 k/w) T= Temperature (K) =Dimensionless Temperature = Outlet Temperature (C, K) = Wall Temperature (C) = Initial temperature of fluid flow (C) U= Overall heat transfer coefficient (W/m^2 k) V= Velocty (m/s) 6 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

7 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.

8 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.

9 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. [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.

10 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: [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. [3] The effectiveness of the types of heat exchangers is as follows: Parallel Flow: [4] Counter Flow: [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: [6] 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.

11 Figure 1: Basic Heat Exchanger Design

12 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: [7] For concurrent flow: For counter-current flow:

13 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: [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.

14 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: [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, 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: [10] Therefore the outlet temperatures for the hot and cold fluids can be calculated as follows: [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.

15 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: [13] Neglecting dissipation and any conduction axially, equation 8 reduces to the following: [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 . 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. [11]

16 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. [15]

17 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 sturm- Louiville system. While Graetz himself only determined the first two terms, Sellars, Tribus, and Klein were able to extend the problem and determine the first ten eigenvalues in 1956. Even though this further developed the original solution, at the entrance of the tubing the series solution had extremely poor convergence where up to 121 terms would not make the series converge. Schmidt and Zeldin in 1970 extended the Graetz problem to include axial heat conduction and found that for very high Peclet numbers (Reynolds number multiplies by the prandtl number) the problem solution is essentially the original Graetz problem. Similar to the original problem which showed poor convergence near the ducting entrance, they discovered up to a 25% deviation in the local nusselt number which made the results in this region questionable. The purpose of this paper is to not redo the various numerical solutions presented by multiple groups over the past century as there doesn’t appear to be a definitive solution that has proven convergence everywhere. The Graetz problem will be introduced in a finite element program with certain dimensions, fluid properties, and tubing temperature in order to analyze the velocity and temperature changes as a building block to eventually analyzing a compact heat exchanger for the same. 18 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- 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. 19 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.

20 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.

21 2.2 Methodology and Approach

22 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 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.

23 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.

24 3. Results and Discussion

25 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 µ= 0.000547 Pa s ρ= 988 kg/m^3 Cp = 4181 J/Kg k

26 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

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

27 mesh of the model, results were obtained for not only the velocity profile but also the temperature profile.

Figure 5: Graetz Velocity Profile

Figure 6: Graetz Temperature Profile

28 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 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.

29 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 Centerline Outlet Temp (F) 2150 306.0347 2279 305.9992 30 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.

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.

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.

31 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 of the tube and compared to hand calculations for both .001 m/s and .01 m/s inlet velocity.

32 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. [16] The prandtl number is calculated using the material properties of water at the inlet temperature. [17] The dimensionless length value is defined as [18] The outlet temperature is defined as [19] 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. [20] [21]

[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%.

33 Table 3: Graetz Problem Comparison

Inlet V Inlet Temp Wall Temp (m/s) (C) (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 4: 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.

34 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.

35 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.

36 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.

37 Figure 14: Turbulent Model for the Graetz Problem

38 39 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.

40 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

Figure 16: Temperature Profile of Flow Through a Pipe

41 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.

42 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. [23] [24] [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.

Table 5: 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

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

43 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.

Figure 17: Outflow Temperature Distribution for the Graetz Problem

Figure 18: Outflow Temperature Distribution for the Graetz Problem with a Pipe Wall 44 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 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

45 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. 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.

46 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.

47 3.3 Flow in a Concurrent Flow Heat Exchanger

48 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 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

49 Figure 23: Temperature Profile for Concurrent Heat Exchanger

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.

50 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.

51 Figure 25: Temperature Change Across the Outlet Flow

52 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

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.

53 Figure 27: Velocity Profile for a Turbulent Concurrent Flow Heat Exchanger

Figure 28: Temperature Profile for a Turbulent Concurrent Flow Heat Exchanger

54 Figure 29: Turbulent Concurrent Flow Heat Exchanger Temperature Change

55 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 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.

The cross sectional area of each fluid flow is:

The inlet temperature of each fluid and its corresponding properties due to that temperature is as follows:

56 Table 6: Fluid Properties

Fluid Parameters for Oil Fluid Parameters for Water T= 125 C T= 20 C T= 398.15 K T= 293.15 K k= 0.134 w/mK k= 0.600 w/mK µ= 0.00915 Pa s µ= 0.001003 Pa s ρ= 826 kg/m^3 ρ= 998.2 kg/m^3 Pr= 159 Pr= 6.99 Cp= 2328 J/Kg K Cp= 4182 J/Kg K The mass flow rates are then calculated and used to determine the heat capacity rates.

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:

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:

The heat transfer coefficient of the inner pipe wall is expressed as follows:

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:

The value for the number of heat transfer units is:

57 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:

The equation for heat transferred in the NTU-effectiveness method is in terms of this effectiveness value as well as the minimum heat capacity.

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:

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.

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

58 transferred, allowing the temperature of the oil to drop more and the temperature of the water to raise more.

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.

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.

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 59 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 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).

60 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

61 Figure 34: Temperature Profile for Countercurrent Heat Exchanger

Figure 35: Outlet of the Inner Pipe, Inlet of the Outer Pipe

62 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

63 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 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

64 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 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

65 Figure 41: Turbulent Counter-current Flow Heat Exchanger Temperature Change

66 3.3.6 Laminar Flow in a Counter-current Heat Exchanger Problem Calculations

Figure 42: Hot Fluid Outlet Temperature vs Cooling Water Flow Rate for Counter-Current Flow

Figure 43: Change in Hot Fluid Temperature vs Cooling Water Flow Rate for Counter-Current Flow

Figure 44: Temperature Change in the Fluids vs the Difference in Inlet Temperatures for Counter- Current Flow

67 3.4 Flow in a Concurrent Flow Heat Exchanger with Fouling

68 3.4.1 Laminar Flow in a Concurrent Heat Exchanger with Fouling COMSOL Model

69 3.4.2 Turbulent Flow in a Concurrent Heat Exchanger with Fouling

70 3.4.3 Laminar Flow in a Concurrent Heat Exchanger with Fouling Problem Calculations

71 3.5 Flow in a Counter-Current Flow Heat Exchanger with Fouling

72 3.5.1 Laminar Flow in a Counter-current Heat Exchanger with Fouling COMSOL Model

73 3.5.2 Turbulent Flow in a Counter-Current Heat Exchanger with Fouling

74 3.5.3 Laminar Flow in a Counter-current Heat Exchanger with Fouling Problem Calculations

75 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 non- linear 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.

76 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: The McGraw-Hill Companies, Inc., 2006.

[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.

77 78 6. APPENDIX

79 6.1 Laminar Flow Concurrent Heat Exchanger Data

1: Velocity of oil= .0001 m/s

Iteration No. 1 2 3

Ao (m^2) 0.439823 0.439823 0.439823 Ai (m^2) 0.314159 0.314159 0.314159 Tc,I (Celsius) 20 20 20 Vc, I (m/s) 0.0001 0.001 0.01 Th,I (Celsius) 125 125 125 Vh, I (m/s) 0.0001 0.0001 0.0001 A oil flow 0.007854 0.007854 0.007854 A water flow 0.029845 0.029845 0.029845 ρ Oil (kg/m^3) 826 826 826 ρ Water (kg/m^3) 998.2 998.2 998.2 Mc (kg/s) 0.002979 0.029791 0.297914 Mh (kg/s) 0.000649 0.000649 0.000649 Cpc (j/kg*k) 4182 4182 4182 Cph (j/kg*k) 2328 2328 2328

Cc (w/k) 12.45877 124.5877 1245.877 Ch (w/k) 1.510264 1.510264 1.510264 Cmin/Cmax 0.121221 0.012122 0.001212

μ (Pa s) 0.00915 0.00915 0.00915 Pr 159 159 159 Re 0.902732 0.902732 0.902732 k oil (w/m*k) 0.134 0.134 0.134 Nusselt Number 4.43545 4.43545 4.43545 hi (w/m^2*k) 5.943503 5.943503 5.943503 k Copper (w/m*k) 393.111 393.111 393.111 UA (w/k) 0.768769 0.768769 0.768769 NTU 0.50903 0.50903 0.50903 ε 0.387872 0.397797 0.398809 q (w) 61.50782 63.08173 63.2422 Tc,o (Celsius) 24.93691 20.50632 20.05076 Tc,o 298.0869 293.5063 293.0508

80 (Kelvin) Tc,o (COMSOL) 309.7703 295.9225 293.1495 Tc,o Percent Diff 3.771614 0.816473 0.033675 Th,o (Celsius) 84.27347 83.23133 83.12507 Th,o (Kelvin) 357.4235 356.2313 356.1251 Th,o (COMSOL) 367.059 365.1556 363.7862 Th,o Percent Diff 2.625067 2.443964 2.105942

CASE 2: Velocity of water and oil= .0001 m/s

Iteration No. 1 2 3 4

0.43982 Ao (m^2) 0.43982297 0.439822972 0.439823 2972 0.31415 Ai (m^2) 0.31415927 0.314159265 0.314159 9265 Tc,I (Celsius) 20 30 40 20 Vc, I (m/s) 0.0001 0.0001 0.0001 0.0001 Th,I (Celsius) 125 125 125 150 Vh, I (m/s) 0.0001 0.0001 0.0001 0.0001 0.00785 A oil flow 0.00785398 0.007853982 0.007854 3982 0.02984 A water flow 0.02984513 0.02984513 0.029845 513 ρ Oil (kg/m^3) 826 826 826 811 ρ Water (kg/m^3) 998.2 995.6 992.2 998.2 0.00297 Mc (kg/s) 0.00297914 0.002971381 0.002961 9141 0.00063 Mh (kg/s) 0.00064874 0.000648739 0.000649 6958 Cpc (j/kg*k) 4182 4179 4179 4182 Cph (j/kg*k) 2328 2328 2328 2440

12.4587 Cc (w/k) 12.4587672 12.41740188 12.375 6723 1.55417 Ch (w/k) 1.51026412 1.51026412 1.510264 7302 0.12474 Cmin/Cmax 0.12122099 0.121624808 0.122042 5673

μ (Pa s) 0.00915 0.00915 0.00915 0.00564 Pr 159 159 159 104 1.43794 Re 0.90273224 0.90273224 0.902732 3262 81 k oil (w/m*k) 0.134 0.134 0.134 0.132 4.46366 Nusselt Number 4.43545035 4.435450351 4.43545 455 5.89203 hi (w/m^2*k) 5.94350347 5.94350347 5.943503 7207 391.379 k Copper (w/m*k) 393.111 393.111 393.111 5 0.76211 UA (w/k) 0.76876929 0.768769293 0.768769 2671 0.49036 NTU 0.5090297 0.509029701 0.50903 4047 0.37691 ε 0.38787175 0.38783566 0.387798 6352 76.1533 q (w) 61.5078227 55.64475678 49.78263 2906 26.1124 Tc,o (Celsius) 24.9369108 34.48119158 44.02284 2891 299.112 Tc,o (Kelvin) 297.936911 307.4811916 317.0228 4289 313.809 Tc,o (COMSOL) 309.77025 318.20942 326.6505 13 4.68332 Tc,o Percent Diff 3.82003733 3.371436464 2.947377 4888 101.000 Th,o (Celsius) 84.2734662 88.15561228 92.03713 8742 374.000 Th,o (Kelvin) 357.273466 361.1556123 365.0371 8742 386.591 Th,o (COMSOL) 367.05901 369.31285 371.8498 4 3.25680 Th,o Percent Diff 2.66593206 2.20876087 1.832099 441

82 6.2 Laminar Flow Counter-Current Heat Exchanger Data

CASE 1: Velocity of oil= .0001 m/s

Iteration No. 1 2 3

Ao (m^2) 0.439823 0.439823 0.439823 Ai (m^2) 0.314159 0.314159 0.314159 Tc,I (Celsius) 20 20 20 Vc, I (m/s) 0.0001 0.001 0.01 Th,I (Celsius) 125 125 125 Vh, I (m/s) 0.0001 0.0001 0.0001 A oil flow 0.007854 0.007854 0.007854 A water flow 0.029845 0.029845 0.029845 ρ Oil (kg/m^3) 826 826 826 ρ Water (kg/m^3) 998.2 998.2 998.2 Mc (kg/s) 0.002979 0.029791 0.297914 Mh (kg/s) 0.000649 0.000649 0.000649 Cpc (j/kg*k) 4182 4182 4182 Cph (j/kg*k) 2328 2328 2328

Cc (w/k) 12.45877 124.5877 1245.877 Ch (w/k) 1.510264 1.510264 1.510264 Cmin/Cmax 0.121221 0.012122 0.001212

μ (Pa s) 0.00915 0.00915 0.00915 Pr 159 159 159 Re 0.902732 0.902732 0.902732 k oil (w/m*k) 0.134 0.134 0.134 Nusselt Number 4.43545 4.43545 4.43545 hi (w/m^2*k) 5.943503 5.943503 5.943503 k Copper (w/m*k) 393.111 393.111 393.111 UA (w/k) 0.768769 0.768769 0.768769 NTU 0.50903 0.50903 0.50903 ε 0.390964 0.39812 0.398841 q (w) 61.99814 63.13294 63.24734 Tc,o (Celsius) 24.97627 20.50674 20.05077 Tc,o (Kelvin) 298.1263 293.5067 293.0508 Tc,o (COMSOL) 324.4848 294.0418 293.15 Tc,o Percent Diff 8.123192 0.181983 0.033855 Th,o (Celsius) 83.94881 83.19742 83.12167 Th,o (Kelvin) 357.0988 356.1974 356.1217 Th,o (COMSOL) 375.5468 371.2582 367.4436 Th,o Percent Diff 4.912309 4.05669 3.081268

83 CASE 2: Velocity of water and oil= .0001 m/s

Iteration No. 1 2 3 4

0.43982 Ao (m^2) 0.43982297 0.439822972 0.439823 2972 0.31415 Ai (m^2) 0.31415927 0.314159265 0.314159 9265 Tc,I (Celsius) 20 30 40 20 Vc, I (m/s) 0.0001 0.0001 0.0001 0.0001 Th,I (Celsius) 125 125 125 150 Vh, I (m/s) 0.0001 0.0001 0.0001 0.0001 0.00785 A oil flow 0.00785398 0.007853982 0.007854 3982 0.02984 A water flow 0.02984513 0.02984513 0.029845 513 ρ Oil (kg/m^3) 826 826 826 811 ρ Water (kg/m^3) 998.2 995.6 992.2 998.2 0.00297 Mc (kg/s) 0.00297914 0.002971381 0.002961 9141 0.00063 Mh (kg/s) 0.00064874 0.000648739 0.000649 6958 Cpc (j/kg*k) 4182 4179 4179 4182 Cph (j/kg*k) 2328 2328 2328 2440

12.4587 Cc (w/k) 12.4587672 12.41740188 12.375 6723 1.55417 Ch (w/k) 1.51026412 1.51026412 1.510264 7302 0.12474 Cmin/Cmax 0.12122099 0.121624808 0.122042 5673

μ (Pa s) 0.00915 0.00915 0.00915 0.00564 Pr 159 159 159 104 1.43794 Re 0.90273224 0.90273224 0.902732 3262 k oil (w/m*k) 0.134 0.134 0.134 0.132 4.46366 Nusselt Number 4.43545035 4.435450351 4.43545 455 5.89203 hi (w/m^2*k) 5.94350347 5.94350347 5.943503 7207 391.379 k Copper (w/m*k) 393.111 393.111 393.111 5 0.76211 UA (w/k) 0.76876929 0.768769293 0.768769 2671 0.49036 NTU 0.5090297 0.509029701 0.50903 4047 0.37981 ε 0.39096374 0.390937449 0.39091 1816 84 76.7383 q (w) 61.9981432 56.08978617 50.18212 374 26.1593 Tc,o (Celsius) 24.9762663 34.51703075 44.05512 8447 299.159 Tc,o (Kelvin) 297.976266 307.5170308 317.0551 3845 332.100 Tc,o (COMSOL) 324.48479 331.77676 338.9924 2 9.91893 Tc,o Percent Diff 8.1694195 7.312064066 6.471313 8781 100.624 Th,o (Celsius) 83.9488074 87.86094237 91.77262 4639 373.624 Th,o (Kelvin) 356.948807 360.8609424 364.7726 4639 396.936 Th,o (COMSOL) 375.54683 377.18252 378.9805 67 5.87302 Th,o Percent Diff 4.95225125 4.327235957 3.748976 908