Heat Exchanger Design for use in an Under Hood Shower

Transcription

1 Heat Exchanger Design for use in an Under Hood Shower September 27, 2005 The purpose of this project was to design a heat exchanger that utilizes waste heat produced by an automotive engine to heat river water for showering while camping or 4wheeling. There was a strong emphasis placed on the use of heat exchanger designs that could be constructed by the shade tree mechanic. The engine coolant was assumed to be at 100 C, the inlet water at 4.44 C and the engine coolant mass flow rate was defined as a function of engine RPM. There was a maximum heat exchanger length of 20 in, due to test vehicle installation constraints. To solve the design problem, the situation was modeled using EES. The model was then used to discover the effects of the design variables on the shower water outlet temperature. This led to the construction of a 20 shell and tube heat exchanger with one shell pass and two tube passes. When installed in the test vehicle, this configuration resulted in a hot shower at 1.6 GPM with an engine RPM of 2000, as was predicted by the model. 1

2 Introduction: The purpose of this project was to design a heat exchanger for use in an under hood shower for commercially available vehicles. The heat exchanger was to utilize waste heat produced by the engine to heat river water for showering while camping or 4wheeling. There was a strong emphasis placed on the use of heat exchanger designs which could be easily constructed by the shade tree mechanic without special fabrication tools or skills. Materials were limited to those readily available in local hardware stores and the heat exchanger was developed with cost in mind. The design goal was to maximize the shower water outlet temperature. The test vehicle was a 1990 Toyota 4Runner with a 3.0L V6, although the heat exchanger design was required to be easily adaptable to other applications. This test vehicle required that the heat exchanger fit a size requirement, being no more than 20 L X 2 D, although other applications may alter the design accordingly. The heat exchanger is required to use the radiator fluid flowing through the heater core as its heat source as is depicted in Figure 1. The radiator fluid is assumed to be a 50/50 ethylene glycol and water mixture. The heat exchanger configuration, shower water mass flow rate, and shower water outlet temperature were to be specified through the analysis. The designer was also to specify the design trends which would result in a successful heat exchanger so that future readers could modify the design for their application. In order to tackle this design challenge, a computer model was created using Engineering Equation Solver (EES). This program allows the user to easily determine the effects of manipulating design variables even when there are a high number of equations involved. Unlike many computer codes, it does not matter what order you enter the equations in EES as they are all solved simultaneously. In general, the problem was approached by determining the heat 2

3 transfer within the heat exchanger. This involved creating a program which allowed the heat exchanger geometry to be easily modified such that design solutions could be quickly evaluated. Figure 1: Normal automotive cooling system and the modified version which includes an under hood shower. Figure was modified from a figure available through [3]. 3

4 Methods: The EES code modeling the design problem was developed so that a working prototype could be produced. This code took into account the effects of the primary fluid heat transfer, secondary fluid heat transfer, and ignored the pressure drops in the interest of simplicity. To avoid placing increased demands on the automotive water pump, it was required that the radiator fluid have an equivalent area to flow through as the factory radiator hoses provided. Appendix A contains the engine coolant fluid properties used for this analysis. Copper stood out as the material of choice for two compelling reasons. First, it has the highest thermal conductivity of the commercially available materials and secondly, constructing leak proof connections is made easy through soldering. The water source was assumed to be at a temperature of C, since most water sources in the region are fed by snow run off. The inlet engine coolant temperature was assumed to be C and the pressure in the engine cooling system was assumed to be 150 kpa. The mass flow rate of engine coolant through the heater core is a function of engine RPM and data regarding this is included in Appendix B. The outer surface of the heat exchanger was assumed to be well insulated, an assumption that was validated by insulating it during construction. Finally, it was assumed that the effects of any baffles installed in the heat exchanger were negligible. This assumption was made because the contact area between the baffle and the inner tube is expected to be small and inconsistent considering the construction techniques, thus eliminating any fin effects. However, the baffles will certainly have an effect on the shell side flow, and if a higher degree of accuracy is desired, this assumption should be lifted. The radiator fluid was chosen to be the tube side fluid while the shower water was chosen for the shell fluid. The engine coolant was used in the tubes because no insulation can be perfect 4

5 in practice. Since this fluid is at a significantly higher temperature than the shower water, using it as the shell fluid would result in more losses by conduction and convection through the shell. Equations 1, 2 and 3 present the key heat transfer relationships where the mass flow rates, enthalpies, and inlet temperatures were all known values. q = ε Cmin ( T h,i T c,i ) q = m water Cp water ( T c,o T c,i ) (1) (2) q = m coolant Cp coolant ( T h,i T h,o ) (3) Equation 1 uses the epsilon - NTU method to determine the heat transfer rate and requires knowledge of C min which is the lesser quantity when comparing the product of the mass flow rate and the specific heat capacity of each fluid. Equations 2 and 3 relate the heat transfer rate to the inlet and outlet temperatures of the corresponding fluid. By determining epsilon (efficiency) in equation 1, the outlet temperatures of each fluid would be known. Incopera and DeWitt offer several expressions for determining the efficiency of a heat exchanger based on the configuration. At this point, it was decided that two primary designs would be chosen for comparison. The efficiency relation for a Counter Flow heat exchanger is given by equation 4 while equation 5 is for a Shell and Tube with one shell pass and two tube passes. These heat exchanger configurations are depicted in Figure2. 5

6 Figure2: Counter Flow (left) and Shell and Tube (right) heat exchangers The Counter Flow heat exchanger was chosen because of its simplistic design, ease of construction, and for its reputation of having high efficiencies. The Shell and Tube heat exchanger was chosen because it is also easily constructed, its inlet/outlet configuration was highly desirable for the 4Runner application, and it was also known for high efficiencies. These types of heat exchangers were also geometrically similar enough that switching the epsilon relations and altering a few area definitions, allowed comparison of the heat exchangers through the same code base. ε = 1 exp ( NTU ( 1 C r ) ) 1 C r exp ( NTU ( 1 C r ) ) (4) ε = C r C r exp NTU 1 + C r 2 1 C r exp NTU 1 + C r 2 1 (5) equation 6. Equations 4 and 5 make use of C r to define the efficiencies, which is simply defined by C r = Cmin Cmax (6) More importantly, equations 4 and 5 employ the variable NTU which stands for Number of Transfer Units. Equation 7 defines NTU. NTU = U A inner Cmin (7) In equation 7, U is the overall heat transfer coefficient and A inner is defined as the inside surface area of the inner tube. It actually does not matter if you use A outer (outside surface area of the inner tube) in place of A inner because in that case U will be different. Either way, the product 6

7 UA remains constant. The only unknown is equation 7 is U, which can be found through the use of the resistance network, equation 8. U A inner = R 1 + R f A inner + R 2 + R f A outer + R 3 1 (8) R f is defined as the fouling factor and a value of (m^2 K/W) was assumed for our application based on recommendations by Incopera and Dewitt for applications of river water below 50 0 C. The fouling factor takes into account the loss in heat transfer efficiency which will be incurred over time due to corrosion. R i is the resistance to heat transfer and is defined by equations 9,10, and 11. R 1 = ( h 1 A inner ) 1 (9) R 2 = ln D 2 D 1 2 π k cu L (10) R 3 = ( h 2 A outer ) 1 (11) Equation 9 describes the resistance to convection in the inner tube. h 1 is the local heat transfer coefficient on the inside of the inner tube. Equation 10 is the resistance to conduction through the inner tube wall, where k is the conductivity of copper, L is the length of the inner tube and D 1 and D 2 are the inner and outer diameters of the inner tube respectively. Equation 11 describes the resistance to convection on the outside surface of the inner tube. Inspection of equations 9 to 11 reveals that the only unknowns are the local heat transfer coefficients h 1 and h 2. The local heat transfer coefficient is defined using the definition of the dimensionless parameter, the Nusselt number. Nusselt coolant = h 1 D 1 k coolant (12) 7

8 Nusselt water = h 2 D h k water (13) It is important to use the hydraulic diameter whenever you are considering flow through a non circular cross section, as is the case for the shell fluid in Equation 13. The hydraulic diameter is defined by equation 14 where A c is the cross sectional area and P is the wetted perimeter. D h = 4 A c P (14) At this point the local heat transfer coefficients are still unknown because equations 12 and 13 don t help us determine a value for the Nusselt number. The Nusselt number can be determined only through empirical correlations. However, the correct Nusselt number correlation can only be used once the flow conditions have been characterized. The dimensionless parameter Reynolds number, equation 15, allows us to determine if the flow is laminar or turbulent. Reynolds water := ρ water V water D h µ water (15) For internal pipe flow, Reynolds numbers less than 2300 are considered to be laminar flow while Reynolds numbers greater than 2300 are considered turbulent. Equation 15 requires only knowledge of the fluid properties, which are calculated at the arithmetic mean of the tube inlet and outlet temperatures, and the fluid velocity which can be found by applying conservation of mass through equation 16. m water = ρ water V water A c,water (16) The shower water mass flow rate is a design variable and so it can be considered as a known value. While it may seem that we have gotten side tracked, we now have enough information to determine which empirical Nusselt correlation to employ. These correlations are given by equations 17 and 18 for laminar and turbulent flow respectively. 8

9 nusselt water := 1.86 Re water Pr water L water ( 1 / 3 ) µ water µ s,water 0.14 D h (17) f large nusselt water := ( Re water 1000 ) Pr water f large 8 ( 1 / 2 ) ( Pr water ( 2 / 3 ) 1 ) (18) Relations 17 and 18 were provided by Incopera and Dewitt. Unfortunately, they introduce 2 more unknowns; the dimensionless Prandtl number P r and the friction factor f. The Prandtl number is defined solely by fluid properties, as is made evident by equation 19. Prandltwater := Cp water µ water k water (19) Finally, the friction f is given by the Colebrook equation (eqn 20) where e is the surface roughness of the pipe, taken to be mm [2]. 1 f large 0.5 = 2 log e D h Re water f large (20) At this point, the code allowed for the calculation of the heat transfer rate and it was used to optimize the geometry of the heat exchanger to give the highest possible shower outlet temperature. This code is fairly complicated and pulls together many aspects of fluid dynamics and heat transfer. Writing such a program cannot be tackled all at once. At first, the code was written in small modules which were tested independently before being assembled into the main program. This allowed the programmer to troubleshoot the code as it was developed and monitor the validity of the model. The code is reproduced in its entirety in Appendix C along with an embedded version of the program. 9

10 As stated by Incopera and Dewitt, the results of these heat transfer calculations cannot be expect to be more than +- 10% accurate. This means that there is a range of outlet conditions predicted by this EES code. However, the code still allows the designer to discover the important design trends and to confirm whether or not values lie in the correct range. 10

11 Results: The purpose of the code was to predict what types of heat exchanger configurations and design trends would lead to a successful design. To start, the code was to help predict if a counter flow heat exchanger or a shell and tube heat exchanger with one shell pass and two tube passes would be better suited for this application. Figure 3 graphically depicts the results of this analysis which shows that the choice of heat exchanger configuration depends on the application. Figure 3 was created by holding Cr to a constant , which is representative of a 3.5L engine running at 2000 RPM with a 50/50 ethylene glycol water mixture and a shower water mass flow rate of 1.6 GPM. The effectiveness of each type of heat exchanger was then plotted vs a varying NTU. The graph shows that for NTU s below 1.2, the shell and tube heat exchanger is a better design choice while for values above 1.2 the Counter Flow configuration is more effective. The number of transfer units is dependent on a host of variables but the most notable ones are surface area and the fluid flow conditions. For automotive applications, one can expect NTU s in the range of 0.02 to This means that the shell and tube heat exchanger is a significantly better design decision for this application. After this discovery was made, all other analysis was conducted to discover the important design trends for a shell and tube heat exchanger. 11

12 Figure 3: Heat exchanger effectiveness comparison Next, the effects of the heat exchanger length, and therefore the surface areas available for convection were investigated. Engineering tuition tells us that a longer heat exchanger, with more available surface area, should result in a higher NTU, higher efficiency and therefore a higher shower outlet temperature. Figure 4 shows that this is exactly what the code predicts. 12

13 Figure 4: Shower outlet temperature as a function of heat exchanger length However, Figure 5 is more useful in that it predictss the outlet temperature as a function of lengths that would be practical for an automotive application. Figure 5 reveals the important discovery that small changes in length do not greatly affect the shower outlet temperature. This is expected because small changes in length do not greatly affect the convection surface area. It also shows that a longer heat exchanger is better and so the conclusion to be made from this data is that it is best to construct the largest heat exchanger to fit a given application. 13

14 Figure 5: Shower outlet temperature as a function of applicable lengths Engineering tuition would also predict that the shower outlet temperature should rise as the shower water inlet temperature rises. Figure 6 shows that the model predicts this correlation as well. Figure 4 also shows that reasonable outlet temperatures can be expected for the range of likely inlet temperatures. It also shows that if the user is unsatisfied with the outlet temperature, they could recirculate the water in a bucket for a few moments, thereby raising the inlet temperature and resulting in a higher outlet temperature. While the error bars of +- 10% may seem to predict a wide range of temperatures, this is the best accuracy that modern heat transfer correlations have to offer and the data still allows for visualization of trends. 14

15 Figure 6: Effects of water source temperature on shower outlet temperature The shower water outlet temperature is also heavily dependent on the shower water mass flow rate as is shown by Figure 7. The graph shows that a standard shower head flow rate of 2.5 GPM will not result in a sufficiently warm shower and that a flow rate of 1.6 GPM is much more desirable. This flow rate can be achieved through a water saver style shower head or through a restrictor plate in a standard shower head. Finally, Figure 7 shows that installing a valve in the shower head supply line to control the mass flow rate, is also an effective means of controlling the shower outlet temperature. When the valve is opened, the shower will be colder and when the valve is closed, the water will be warmer. 15

16 Figure 7: Shower water outlet temperature as a function of shower water mass flow rate Finally, the shower water outlet temperature should be heavily dependent on engine RPM as this directly effects the engine coolant mass flow rate. Figure 8 shows that this is in fact the case. Figure 8 also reveals several important phenomena. Two interesting things happen in the circled region of Figure 8. First, this is the engine RPM that first corresponds to an engine coolant flow rate of greater than 1.6 GPM, meaning that there is more coolant flowing through the exchanger than there is shower water. Secondly, this flow rate also happens to correspond to the transition between laminar and turbulent engine coolant flow. Turbulent flow is well known for being more efficient at inducing heat transfer. This result means that you cannot expect to take a hot 16

17 shower at a standard flow rate with the engine only idling. It also means that once turbulent flow is established, changing the engine RPM greatly affects the shower water outlet temperature. Figure 8: Shower water outlet temperature as a function of Engine RPM 17

18 Conclusions: The computer model created using EES was very helpful in identifying the desired heat exchanger configuration and in demonstrating the important design trends. The data shows that for an automotive application, one should build a shell and tube heat exchanger with one shell pass and two tube passes. This heat exchanger should be as long in length as the application allows for, but deviations on the order of inches will be insignificant. Nevertheless, I don t recommend building a heat exchanger less than 12 inches in length, although this statement is based solely on engineering tuition. As is expected, the model demonstrated that the shower outlet temperature is strongly related to the inlet temperature. Furthermore, the data shows that the shower water mass flow rate is extremely important in determining the shower water outlet temperature. Because of this, it is recommended that a valve be used to control the mass flow rate and in turn the outlet temperature. In addition, it is recommended that a shower head dispensing 1.6 GPM be used instead of the household standard of 2.5 GPM to ensure a hot shower. Finally, the model stressed the importance of raising the engine rpm to increase the engine coolant mass flow rate and ensure turbulent flow. The uncertainty involved in these calculations means that the magnitude of the shower water outlet temperature may vary significantly. However these calculations also show that the outlet temperature is in the correct ball park and they are important in discovering design trends. If more time and resources allow, it would be very interesting to compare the results of the EES model to the results of a computational fluid dynamics program such as CFDesign or COSMOSfloWorks. A prototype has been constructed and installed in the test vehicle at a cost of $124, and it has been found that the operation of the shower follows the trends predicted by the EES code. In addition, while temperature readings have not been quantitatively confirmed, qualitative analysis of the system proves it is more than adequate in providing a clean hot shower on the trail. 18

19 Works Consulted [1] Fox, McDonald, and Pritchard. Introduction to Fluid Mechanics. John Wiley & Sons Inc: New York, [2] Incropera and DeWitt. Fundamentals of Heat and Mass Transfer. John Wiley & Sons Inc: New York, [3] [4] 19

20 Appendix A: Engine Coolant Fluid Properties Graphs Produced from information at: 50% Ethlyene Glycol Viscosity (kg/m-s) Temperature (C) 50% Ethylene Glycol Cp (kj/kg C) Temperature(C) 20

21 50% Ethylene Glycol Density (kg/m^3) Temperature (C) 21

22 Appendix B: Heater Core Mass Flow Rate Data Engine 3.5 L Test Temp 42 C Pulley ratio 1.2 (it s a guess) Fluid 50/50 Ethylene Glycol Fluid Temp 100 C Density kg/m^ lb/ft^3 Engine RPM Open Thermostat Closed Thermostat GPM GPM RPM Kg/s Kg/s

23 L Engine/50% Ethylene Glycol y = -3E-16x 4 + 3E-12x 3-5E-09x x R 2 = Heater Core Mass Flow Rate (Kg/s) Engine RPM This data was obtained through one of the Big 3 American car makers. However, the provider of this information asked that the source not be identified. This data is presented without citation in honor of this request. 23

24 Appendix C: EES Code "Heat exchanger for under hood shower, shell and tube with one shell pass and two tube passes" FUNCTION smallinnerdiameter(d_small) IF D_small=(1/4) THEN smallinnerdiameter:=4.82 [mm]*convert(mm,m) IF D_small=(1/2) THEN smallinnerdiameter:=10.92 [mm]*convert(mm,m) IF D_small=(3/4) THEN smallinnerdiameter:=16.92 [mm]*convert(mm,m) IF D_small=(1) THEN smallinnerdiameter:=22.4 [mm]*convert(mm,m) IF D_small=(1+1/4) THEN smallinnerdiameter:=29.21 [mm]*convert(mm,m) IF D_small=(1+1/2) THEN smallinnerdiameter:=35.3 [mm]*convert(mm,m) IF D_small=(2) THEN smallinnerdiameter:=47.24 [mm]*convert(mm,m) FUNCTION smallouterdiameter(d_small) IF D_small=(1/4) THEN smallouterdiameter:=6.35 [mm]*convert(mm,m) IF D_small=(1/2) THEN smallouterdiameter:=12.70 [mm]*convert(mm,m) IF D_small=(3/4) THEN smallouterdiameter:=19.05 [mm]*convert(mm,m) IF D_small=(1) THEN smallouterdiameter:=25.4 [mm]*convert(mm,m) IF D_small=(1+1/4) THEN smallouterdiameter:=31.75 [mm]*convert(mm,m) IF D_small=(1+1/2) THEN smallouterdiameter:=38.1 [mm]*convert(mm,m) IF D_small=(2) THEN smallouterdiameter:=50.8 [mm]*convert(mm,m) FUNCTION largeinnerdiameter(d_large) IF D_large=(1/4) THEN largeinnerdiameter:=4.82 [mm]*convert(mm,m) IF D_large=(1/2) THEN largeinnerdiameter:=10.92 [mm]*convert(mm,m) IF D_large=(3/4) THEN largeinnerdiameter:=16.92 [mm]*convert(mm,m) IF D_large=(1) THEN largeinnerdiameter:=22.4 [mm]*convert(mm,m) IF D_large=(1+1/4) THEN largeinnerdiameter:=29.21 [mm]*convert(mm,m) IF D_large=(1+1/2) THEN largeinnerdiameter:=35.3 [mm]*convert(mm,m) IF D_large=(2) THEN largeinnerdiameter:=47.24 [mm]*convert(mm,m) FUNCTION largeouterdiameter(d_large) IF D_large=(1/4) THEN largeouterdiameter:=6.35 [mm]*convert(mm,m) IF D_large=(1/2) THEN largeouterdiameter:=12.70 [mm]*convert(mm,m) IF D_large=(3/4) THEN largeouterdiameter:=19.05 [mm]*convert(mm,m) IF D_large=(1) THEN largeouterdiameter:=25.4 [mm]*convert(mm,m) IF D_large=(1+1/4) THEN largeouterdiameter:=31.75 [mm]*convert(mm,m) IF D_large=(1+1/2) THEN largeouterdiameter:=38.1 [mm]*convert(mm,m) IF D_large=(2) THEN largeouterdiameter:=50.8 [mm]*convert(mm,m) MODULE friction1(e,d_1,re_coolant:f_small) (1/f_small^0.5)=-2.0*Log10((e/D_1)/ /(Re_coolant*f_small^0.5)) MODULE friction2(e,d_h,re_water:f_large) (1/f_large^0.5)=-2.0*Log10((e/D_h)/ /(Re_water*f_large^0.5)) FUNCTION Reynolds_coolant(rho_coolant,V_coolant,D_1,mu_coolant) Reynolds_coolant:=(rho_coolant*V_coolant*D_1)/mu_coolant 24

25 FUNCTION Reynolds_water(rho_water,V_water,D_h,mu_water) Reynolds_water:=(rho_water*V_water*(D_h))/mu_water FUNCTION nusselt_coolant(re_coolant,pr_coolant,l,d_1,mu_coolant,mu_s_coolant,f_small) IF Re_coolant<=2300 THEN nusselt_coolant:=1.86*((re_coolant*pr_coolant)/(l/d_1))^(1/3)*(mu_coolant/mu_s_coolant)^0.14 "valid for Pr>0.5" IF Re_coolant>2300 THEN nusselt_coolant:=(f_small/8)*(re_coolant- 1000)*Pr_coolant/(1+12.7*(f_small/8)^(1/2)*(Pr_coolant^(2/3)-1)) FUNCTION nusselt_water(re_water,pr_water,l_water,d_h,mu_water,mu_s_water,f_large) IF Re_water<=2300 THEN nusselt_water:=1.86*((re_water*pr_water)/(l_water/d_h))^(1/3)*(mu_water/mu_s_water)^0.14 IF Re_water>2300 THEN nusselt_water:=(f_large/8)*(re_water- 1000)*Pr_water/(1+12.7*(f_large/8)^(1/2)*(Pr_water^(2/3)-1)) FUNCTION Prandltcoolant(k_coolant,Cp_coolant,mu_coolant) Prandltcoolant:=(Cp_coolant*mu_coolant)/(k_coolant*1/1000[W/kW]) FUNCTION Prandltwater(k_water,Cp_water,mu_water) Prandltwater:=(Cp_water*mu_water)/(k_water*1/1000[W/kW]) FUNCTION C_min(m_dot_coolant, Cp_coolant,m_dot_water,Cp_water) IF m_dot_coolant*cp_coolant<=m_dot_water*cp_water THEN C_min:=m_dot_coolant*Cp_coolant IF m_dot_coolant*cp_coolant>m_dot_water*cp_water THEN C_min:=m_dot_water*Cp_water FUNCTION C_max(m_dot_coolant, Cp_coolant,m_dot_water,Cp_water) IF m_dot_coolant*cp_coolant>m_dot_water*cp_water THEN C_max:=m_dot_coolant*Cp_coolant IF m_dot_coolant*cp_coolant<=m_dot_water*cp_water THEN C_max:=m_dot_water*Cp_water "Design Variables" L=42 [in]*convert(in,m) L_water=20 [in]*convert(in,m) D_small=1/2 D_large=2 D_1=smallinnerdiameter(D_small) D_2=smallouterdiameter(D_small) D_3=largeinnerdiameter(D_large) D_4=largeouterdiameter(D_large) T_h_i=ConvertTemp(F,C,212) T_c_i=ConvertTemp(F,C,40) Engine_RPM=2000 m_dot_coolant= -3E-16*Engine_RPM^4 + 3E-12*Engine_RPM^3-5E-09*Engine_RPM^ *Engine_RPM m_dot_water=.157 [kg/s]*.64 25

26 "Material Properties" "Assumes a 50% ethylene glycol and water mixture as coolant" P=150.3 R_f=.0002 e=0.046 [mm]*convert(mm,m) k_coolant=conductivity(water,t=t_1,p=p) k_water=conductivity(water,t=t_2,p=p) k_cu=401[w/m-k] T_1=(T_h_i+T_h_o)/2 T_2=(T_c_i+T_c_o)/2 rho_coolant= *t_ rho_water=density(water,p=p,t=t_2) mu_coolant=interpolate('viscosity','column1','column2',column1=mu_coolant) mu_coolant=mu_s_coolant mu_water=viscosity(water,t=t_2,p=p) mu_water=mu_s_water Cp_coolant= (3.2675*T_ )/1000 Cp_water=CP(Water,T=T_2,P=P) "Conservation of Mass" m_dot_coolant=rho_coolant*v_coolant*pi*(d_1/2)^2 m_dot_water=rho_water*v_water*((pi*(d_3/2)^2)-(2*pi*(d_2/2)^2)) "Energy Equations" CALL friction1(e,d_1,re_coolant:f_small) CALL friction2(e,d_h,re_water:f_large) Cmin=C_min(m_dot_coolant, Cp_coolant,m_dot_water,Cp_water) Cmax=C_max(m_dot_coolant, Cp_coolant,m_dot_water,Cp_water) C_r=Cmin/Cmax q=m_dot_coolant*cp_coolant*(t_h_i-t_h_o) q=m_dot_water*cp_water*(t_c_o-t_c_i) q=epsilon*cmin*(t_h_i-t_c_i) epsilon=2*(1+c_r+sqrt(1+c_r^2)*((1+exp(-ntu*sqrt(1+c_r^2)))/(1-c_r*exp(- NTU*SQRT(1+C_r^2)))))^(-1) A_inner=Pi*D_1*L A_outer=Pi*D_2*L NTU=U*A_inner/(Cmin*1000[J/KJ]) U*A_inner=(R_1+(R_f/A_inner)+R_2+(R_f/A_outer)+R_3)^(-1) R_1=(h_1*A_inner)^(-1) R_2=(LN(D_2/D_1)/(2*Pi*k_cu*L)) R_3=(h_2*A_outer)^(-1) Re_coolant=Reynolds_coolant(rho_coolant,V_coolant,D_1,mu_coolant) Re_water=Reynolds_water(rho_water,V_water,D_h,mu_water) 26

27 Pr_coolant=Prandltcoolant(k_coolant,Cp_coolant,mu_coolant) Pr_water=Prandltwater(k_water,Cp_water,mu_water) Nu_coolant=nusselt_coolant(Re_coolant,Pr_coolant,L,D_1,mu_coolant,mu_s_coolant,f_small) Nu_water=nusselt_water(Re_water,Pr_water,L_water,D_h,mu_water,mu_s_water,f_large) Nu_coolant=h_1*D_1/k_coolant Nu_water=h_2*D_h/k_water D_h=4*(Pi*(D_3/2)^2-2*Pi*(D_2/2)^2)/(Pi*D_3+2*Pi*D_2) "4*Cross sectional area over wetted perimeter" For users of electronic copies of this report, the EES program is embedded below: Under_hood_shower.EES 27