ME.415 Energy Systems Design Course Overview

This course offers an application-intensive approach covering flow in pipes, pumps, heat exchangers, and thermal system simulation. The ME.415 Add-In for Excel provides unique user-defined functions for solving complex problems efficiently. Learn how to apply these functions directly in Excel spreadsheets using different methods such as direct cell call and user ribbon call.

• Energy Systems
• Design
• Thermal
• Excel
• Simulation

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1. ME 415 Energy Systems Design Tutorial

2. COURSE MATERIAL This is a classic thermal systems design course. It is application intensive and covers flow in pipes and piping systems, pumps and pumping systems, heat exchangers and heat exchanger design, and thermal system simulation.

3. APPLICATION THROUGH EXCEL The ME 415 Add-In offers several unique user- defined functions for application of course material in the Excel environment. As a result, students are able to solve complex problems through elimination of cumbersome hand calculations or reading of charts and graphs.

4. APPLICATION THROUGH EXCEL These user-defined functions are utilized in the Excel Spreadsheet. The functions can be invoked by several methods. Call directly from the cell Requires known function name and argument constraints (specific units, range sizes, etc.) Call from the user ribbon (Excel 2007) Provides function descriptions and input boxes

5. DIRECT CELL CALL METHOD This method requires the user to highlight desired cell(s) for output and type = Function Name(Arg1,Arg2, ) For example, we desire to know the Nusselt number for turbulent flow in a tube. This method requires knowing what arguments the function needs to compute the desired output. Direct cell call method becomes useful when user has gained experience with a specific function or group of functions.

6. USER RIBBON CALL METHOD This method uses the user ribbon and the Insert Function button located at the top of the Excel 2007 window. Advantages of this method are function lists and descriptions that provide details on each argument s requirements . Highlight cell(s) for desired output and select the formulas tab on the user-ribbon as seen on following slide. Then click either Insert-Function button

7. USER RIBBON CALL METHOD From the Insert-Function pop up window displayed on the right, select User Defined from the function category list. Either Insert- Function button will work

8. USER RIBBON CALL METHOD User can select the correct function by scrolling through each function and its description. Once function is selected, spaces are provided for each argument. Some arguments can be optional such as the Quiet argument on the NuDTurbTube function.

9. ME 415 ADD-IN FEATURES The ME 415 Add The ME 415 Add- -In provides tools for several In provides tools for several special design calculations. special design calculations. Heat Transfer Fin Efficiency Heat Exchanger Effectiveness-Number of Transfer Units (NTU) Method Pump Performance Correction for Viscous Fluids Hardy-Cross Flow and Hazen-Williams Head Loss Analysis Friction Factor Calculator (Swamee-Jain and Churchill) Nusselt Number

10. FIN EFFICIENCY The function fin_eff l, ri , and ro. m = SQRT(h/k ) l is total fin length ri is inner radius (circular fins) ro is outer radius (circular fins) The function call from the Excel spreadsheet is =fin_eff(Index,m,l,r =fin_eff(Index,m,l,ri i,r ,ro o) ) or =fin_eff_fintype(m,l,r which will provide an equivalent result for an Index corresponding to the same fin type. fin_eff uses known fin parameters m, =fin_eff_fintype(m,l,ri i,r ,ro o) )

11. FIN EFFICIENCY Study of finned surfaces in heat exchanger design requires analysis of fin efficiency Calculation of fin efficiency fin efficiency can become cumbersome with complex fin geometries. With known fin dimensions, the user-defined function fin_eff fin_eff readily calculates fin efficiency. From calculation of fin efficiency fin efficiency, further analysis of finned surface properties such as total surface effectiveness total surface effectiveness can be easily determined. fin efficiency.

12. FIN EFFICIENCY When using the =fin_eff_fintype names should be used for each specific fin geometry. Straight Rectangular Fins Straight Rectangular Fins =fin_eff_rect(m, l) =fin_eff_rect(m, l) Straight Triangular Fins Straight Triangular Fins =fin_eff_tri(m, l) =fin_eff_tri(m, l) Circular Rectangular Fins Circular Rectangular Fins =fin_eff_rect_c(m, l, r =fin_eff_rect_c(m, l, ri i, r , ro o) ) ri and ro are required arguments here Rectangular Spines (Circular cross Rectangular Spines (Circular cross- -section) Round Pin Fin =fin_eff_pin_R(m, l) =fin_eff_pin_R(m, l) Rectangular Spines (Square cross Rectangular Spines (Square cross- -section) Square Pin Fin m = Sqrt(2*h/k/ ) =fin_eff_pin_S(m,l) =fin_eff_pin_S(m,l) Triangular Spines (Circular Cone cross Triangular Spines (Circular Cone cross- -section) Cone Pin Fin =fin_eff_pin_C(m,l) =fin_eff_pin_C(m,l) =fin_eff_fintype function call, the following function section) section) section)

13. FIN EFFICIENCY An Index value of 1,2,3,4,5, or 6 should be supplied for the appropriate fin geometry. 1 1 Straight Rectangular Fins Straight Rectangular Fins =fin_eff(1,m, l) =fin_eff(1,m, l) 2 2 Straight Triangular Fins Straight Triangular Fins =fin_eff2,m, l) =fin_eff2,m, l) 3 3 Circular Rectangular Fins Circular Rectangular Fins =fin_eff(3,m, l, r =fin_eff(3,m, l, ri i, r , ro o) ) ri and ro are required arguments 4 4 Rectangular Spines (circular cross Rectangular Spines (circular cross- -section) Round Pin Fin =fin_eff(4,m, l) =fin_eff(4,m, l) section) 5 5 - - Rectangular Spines (square cross Rectangular Spines (square cross- -section) Square Pin Fin m = Sqrt(2*h/k/ ) =fin_eff(5,m,l) =fin_eff(5,m,l) section) 6 6 Triangular Spines (Circular Cone cross Triangular Spines (Circular Cone cross- -section) Cone Pin Fin fin_eff(6,m,l) fin_eff(6,m,l) section)

14. FIN EFFICIENCY Function arguments ri and ro are provided as optional. When Index 3 3 is used for a circular circular rectangular fin rectangular fin, ri and ro are required. Otherwise, they should not be supplied.

15. NTU METHOD The function calls from the Excel spreadsheet are =Hx_eff( Index,NTU,C =Hx_eff( Index,NTU,Cmin =Hx_NTU( Index,eff, C =Hx_NTU( Index,eff, Cmin The function Hx_eff Hx_eff uses known parameters NTU, Cmin, Cmax, and No. of Passes to calculate heat exchanger effectiveness effectiveness. NTU=UA/Cmin. Cmin is the smaller of the two capacities Ch and Cc. Cmax is the larger of the two capacities Ch and Cc. Passes is an optional argument (specific to certain heat exchanger types). The function Hx_NTU Hx_NTU uses known parameters effectiveness, Cmin, Cmax, and No. of Passes to calculate NTU NTU. Where Cmin, Cmax, and Passes are same as above. min ,C ,Cmax min ,C ,Cmax max ,Passes) ,Passes) and max ,Passes) ,Passes).

16. NTU METHOD Heat exchanger analysis where only inlet conditions are known uses the Number of Transfer Units (NTU) Method to determine heat exchanger effectiveness effectiveness. Effectiveness Effectiveness NTU NTU relations for some heat exchanger types require iterative calculation which is simplified by user-defined functions Hx_eff Hx_eff and Hx_NTU Hx_NTU.

17. NTU METHOD An Index value of 1-8 should be supplied for the appropriate heat exchanger type. 1 1 Parallel flow: single pass Parallel flow: single pass 2 2 Counterflow: single pass Counterflow: single pass 3 3 Shell and tube (one shell pass; 2,4,6, etc., tube passes) Shell and tube (one shell pass; 2,4,6, etc., tube passes) 4 4 Shell and tube (n shell passes; 2n, 4n, 6n, etc., tube Shell and tube (n shell passes; 2n, 4n, 6n, etc., tube passes) passes) - - - - Passes argument required 5 5 Cross flow (both streams unmixed) Cross flow (both streams unmixed) 6 6 Cross flow (both streams mixed) Cross flow (both streams mixed) 7 7 Cross flow (stream C Cross flow (stream Cmin 8 8 Cross flow (stream C Cross flow (stream Cmax Index 4 requires input of the No. of passes. All other indexes should not have No. of passes supplied. min unmixed) unmixed) max unmixed) unmixed)

18. NTU METHOD The direct cell call method uses =Hx_eff( Index,NTU,C =Hx_eff( Index,NTU,Cmin =Hx_NTU( Index,eff, C =Hx_NTU( Index,eff, Cmin The user-ribbon call method is shown in the figure below. min ,C ,Cmax min ,C ,Cmax max ,Passes) ,Passes) and max ,Passes) ,Passes).

19. VISCOUS PUMP The function calls from the Excel spreadsheet are =Vis_pump_QHE(QHE_Matrix,Vis) =Vis_pump_QHE(QHE_Matrix,Vis) and =Vis_pump_CF(Q Vis_pump_CF(QBE , HBE , Vis). The function Vis_pump_QHE Vis_pump_QHE uses a pre-calculated QHE matrix and viscosity of the pumping fluid to provide corresponding flow and head values for the high viscosity fluid. The user can then generate (plot) a new pump curve with the supplied output. The function Vis_pump_CF Vis_pump_CF uses known best efficiency point (BEP) flow and head values along with the viscosity of the pumping fluid to provide correction factors that correct the pump curve data. The user can multiply these correction factors with original pump data to find corresponding flow and head values for the high viscosity fluid. BE, H BE, Vis).

20. VISCOUS PUMP Because pump performance is greatly affected by highly viscous fluids, a correction method must be used to estimate performance when manufacturer s data is not available. These pump corrections can be found from charts but is simplified through user-defined functions Vis_pump_QHE and Vis_pump_CF Vis_pump_QHE and Vis_pump_CF. With the known best efficiency point (BEP) of a specific pump, the correction factors correction factors for efficiency, flow, Head0.6Q, Head0.8Q, Head1.0Q, and Head1.2Q can be found. Both user-defined functions use a BEP to calculate and output the new data for a high viscosity pumping fluid.

21. VISCOUS PUMP The function Vis_pump_QHE Vis_pump_QHE uses a pre-calculated QHE Matrix QHE Matrix and known viscosity viscosity. The QHE matrix QHE matrix is a 4 x 3 matrix that the user must generate for input into the Vis_pump_QHE function. From a given pump curve (water), determine the BEP (highest efficiency). From this point, the user determines the flow and head at the pump s best efficiency. The 4 x 3 matrix is then generated as follows. Q H E (efficiency) Q H E (efficiency) 0.6*QBE Head@0.6QBE Efficiency@0.6QBE 0.8*QBE Head@0.8QBE Effficiency@0.8QBE 1.0*Q 1.0*QBE Head Head@1.0QBE @1.0QBE 1..2*QBE Head@1.2QBE Efficiency@1.2QBE Viscosity Viscosity (SSU Saybolt Seconds Universal) BE Efficiency@1.0QBE

22. VISCOUS PUMP Vis_pump_QHE Vis_pump_QHE outputs a matrix of cells. To execute the function, the user must highlight the expected output of cells. The output is the same size as the input QHE matrix (4 x 3). Highlight any open cells in a 4 x 3 matrix. Call =Vis_pump_QHE( QHE_Mat(4 x 3),Vis). =Vis_pump_QHE( QHE_Mat(4 x 3),Vis). Once all arguments are entered, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter (Do NOT NOT press OK) must be used to obtain the desired corrected pump curve data. (4 x 3).

23. VISCOUS PUMP The output of the cells is corrected pump curve data for the high viscosity fluid. Ctrl+Shift+Enter (Do not click OK)

24. VISCOUS PUMP The function Vis_pump_CF arguments flow (QBE), head (HBE), and viscosity. Flow (GPM) Head ( ft ) Viscosity (SSU Saybolt Seconds Universal) Vis_pump_CF Vis_pump_CF outputs an array of cells. To execute the function, the user must highlight the expected array of six six cells in any column and call =Vis_pump_CF( Flow,Head,Vis). =Vis_pump_CF( Flow,Head,Vis). Once all arguments are entered, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter (Do NOT NOT press OK) must be used to obtain the desired correction factors. Vis_pump_CF uses known BEP

25. VISCOUS PUMP Output array: C CQ CH (0.6 x QNW) CH (0.8 x QNW) CH (1.0 x QNW) CH (1.2 x QNW) Ctrl+Shift+Enter (Do NOT click OK)

26. VISCOUS PUMP For flows equal to or less than 100 GPM, correction factors for 0.6, 0.8, 1.0, and 1.2 flow rates will be equal. Otherwise, correction factors will vary. Final array output

27. HARDY-CROSS ANALYSIS The function calls from the Excel spreadsheet are =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis) =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis) =Darcy(RngL,RngD,RngQ,RngE,rho,vis) =Darcy(RngL,RngD,RngQ,RngE,rho,vis) and =Hardy_Hazen(RngL,RngD,RngQ,RngN,RngC,tol,k1) =Hardy_Hazen(RngL,RngD,RngQ,RngN,RngC,tol,k1) =HazenWill(RngL,RngD,RngQ,k1,RngC) =HazenWill(RngL,RngD,RngQ,k1,RngC) Darcy-Weisbach and Hazen-Williams are two methods for calculating head loss through pipes. They use unique parameters to determine friction and head-loss through piping systems. With appropriate input arguments, these two methods will provide approximately the same solution.

28. HARDY-CROSS ANALYSIS Hardy Hardy- -Cross Cross formulation is an iterative method for obtaining the steady-state solution for any generalized series-parallel flow network. It can be systematically applied to any fluid flow network. While Hardy Hardy- -Cross Cross flow values can be obtained using solver in Excel, an alternative method that employs user-defined functions Hardy_Darcy Hardy_Darcy and Hardy_Hazen Hardy_Hazen supplies the same solution.

29. HARDY-CROSS AND DARCY-WEISBACH The function Hardy _Darcy guesses for line flow rates, loop-node analysis, pipe roughness, density, and dynamic viscosity to determine flow through the system. Corresponding to the number of pipes in the system, the user should supply a range of lengths (RngL) (RngD) (RngD), initial flow guesses (RngQ) initial flow guesses (RngQ), and epsilon coefficients (RngE) coefficients (RngE). The user also supplies a n n- - connection matrix (RngN) connection matrix (RngN), a density viscosity viscosity. The function call from the Excel spreadsheet is =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis). =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis). Hardy _Darcy uses system geometry, initial lengths (RngL), diameters epsilon values diameters density, and a dynamic dynamic

30. HARDY-CROSS AND DARCY-WEISBACH The user must input a rho viscosity). Typical units for each are lbm/ft3 and ft2/sec, respectively, when units of Q are ft3/sec. Hardy_Darcy Hardy_Darcy outputs an array of cells. To execute the function, the user must highlight the expected array of cells (No. of pipes in system) in any column and call =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis). =Hardy_Darcy(RngL,RngD,RngQ,RngN,RngE,rho,vis). Once all arguments are entered, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter (Do NOT be used to obtain the desired Hardy rho (density) and vis vis (dynamic NOT press OK) must Hardy flow values.

31. HARDY-CROSS AND DARCY-WEISBACH Ctrl+Shift+Enter (Do not click OK)

32. HARDY-CROSS AND DARCY-WEISBACH Final array output The final array output Hardy are shown above. Darcy values values through each pipe can then be found with these known flow rates. Hardy _Darcy Darcy- -Weisbach Weisbach head loss Darcy flow values head loss

33. HARDY-CROSS AND DARCY-WEISBACH The user-defined function Darcy same system geometry and the calculated Hardy_Darcy Hardy_Darcy flow values to find the head loss through each pipe. The function call from the Excel spreadsheet is =Darcy(RngL,RngD,RngQ,RngE,rho,vis). =Darcy(RngL,RngD,RngQ,RngE,rho,vis). Since the Darcy Darcy function also uses range inputs, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter must again be used to obtain the expected array Darcy head loss Darcy head loss values Darcy uses the values.

34. HARDY-CROSS AND DARCY-WEISBACH RngQ RngQ uses new Hardy_Darcy Hardy_Darcy flow values

35. HARDY-CROSS AND HAZEN-WILLIAMS The function Hardy _Hazen initial guesses for line flow rates, and loop-node analysis to determine flow through the system. Corresponding to the number of pipes in the system, the user should supply a range of lengths (RngL) (RngD) (RngD), initial flow guesses (RngQ) initial flow guesses (RngQ), and Hazen coefficients (RngC) coefficients (RngC). The user also supplies a n n- - connection matrix (RngN) connection matrix (RngN), a tolerance K1 va K1 value (k1) (k1). The function call from the Excel spreadsheet is =Hardy_Hazen(RngL,RngD,RngQ,RngN,RngC,tol,k1). =Hardy_Hazen(RngL,RngD,RngQ,RngN,RngC,tol,k1). Hardy _Hazen uses system geometry, lengths (RngL), diameters Hazen- -Williams diameters Williams tolerance value (tol) (tol), and a

36. HARDY-CROSS AND HAZEN-WILLIAMS Typical values for tol respectively when units of Q are ft3/sec. Hardy_Hazen Hardy_Hazen outputs an array of cells. To execute the function, the user must highlight the expected array of cells (No. of pipes in system) in any column and call =Hardy(RngL,RngD,RngQ,RngN,RngC,tol,k1). =Hardy(RngL,RngD,RngQ,RngN,RngC,tol,k1). Once all arguments are entered, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter (Do NOT must be used to obtain the desired Hardy values. tol and k1 k1 are .0001 .0001 and 4.727 4.727 NOT press OK) Hardy flow

37. HARDY-CROSS AND HAZEN-WILLIAMS Ctrl+Shift+Enter (Do NOT click OK)

38. HARDY-CROSS AND HAZEN-WILLIAMS Final array output The final array output Hardy values are shown above. Hazen head loss head loss values values through each pipe can then be found with these known flow rates. Hardy _Hazen _Hazen flow Hazen- -Williams Williams

39. HARDY-CROSS AND HAZEN-WILLIAMS The user-defined function HazenWill same system geometry and the calculated Hardy_Hazen Hardy_Hazen flow values to find the head loss through each pipe. The function call from the Excel spreadsheet is =HazenWill(RngL,RngD,RngQ,k1,RngC). =HazenWill(RngL,RngD,RngQ,k1,RngC). k1 is 4.727 when units for Q are ft3/sec Since HazenWill HazenWill also uses range inputs, the keystroke command Ctrl+Shift+Enter Ctrl+Shift+Enter must again be used to obtain the expected array Hazen Williams head loss Williams head loss values. HazenWill uses the Hazen- -

40. HARDY-CROSS AND HAZEN-WILLIAMS RngQ RngQ uses new Hardy_Hazen Hardy_Hazen flow values

41. FRICTION FACTOR The function calls from the Excel spreadsheet are =fric_Swamee(Eps_Dia, ReD) =fric_Swamee(Eps_Dia, ReD) =fric_Churchill(Eps_Dia, ReD) =fric_Churchill(Eps_Dia, ReD) Eps_Dia is the relative roughness = /D ReD is the Reynolds number = *V*D/ Swamee Swamee- -Jain Jain and Churchill Churchill are two methods for calculating friction factors friction factors, a value necessary for calculating head loss through piping. Each function must be used with caution, as they each represent friction factors for different flow regions.

42. FRICTION FACTOR The Swamee Swamee- -Jain appropriate for use only in a region of turbulent flow. For piping flows Turbulent region ReD > 2300 Darcy-Weisbach is used for ReD<2300 f = 64.0/ReD The Churchill Churchill friction factor calculation is appropriate for use in any region of flow Useful in Laminar Transition Turbulent Jain friction factor calculation is

43. FRICTION FACTOR Since Reynolds number 4000, either fric_Churchill fric_Churchill or fric_Swamee fric_Swamee can be used

44. NUSSELT NUMBERS Optional Inputs in italics Rexc, , Quiet Quiet) ) Rexc, , Quiet Quiet) ) NuxPlate(Re, Pr, NuxPlate(Re, Pr, Rexc NuBarPlate(Re, Pr, NuBarPlate(Re, Pr, Rexc NuDBarCyl(Re, Pr, NuDBarCyl(Re, Pr, Quiet NuDBarSphere(Re, Pr, mu_mus, NuDBarSphere(Re, Pr, mu_mus, Quiet NuDBarTubes(Re, Pr, St_D, Sl_D, NuDBarTubes(Re, Pr, St_D, Sl_D, Aligned NuDBarZTubes(Re, Pr, Prs, St_Sl, NuDBarZTubes(Re, Pr, Prs, St_Sl, Aligned NuDBarLamTube(Re, Pr, D_L, NuDBarLamTube(Re, Pr, D_L, Thermal Quiet Quiet) ) NuDTurbTube(Re, Pr, NuDTurbTube(Re, Pr, Quiet NuDLiqMetals (Re, Pr, NuDLiqMetals (Re, Pr, UniformT Quiet) ) Quiet) ) Aligned, , Nl Nl, , Quiet Aligned, , Nl Nl, , Quiet Thermal, , mu_mus mu_mus, , Quiet) ) Quiet) ) Quiet) ) UniformT, , Quiet Quiet) )

45. NUSSELT NUMBERS Functions return the local (Nu) Nusselt number local (Nu) or average (NuBar) average (NuBar) h L h L = = = NuBar Nu Nu k k The functions are reliable only over certain ranges. An answer will be returned, but it is up to the user to decide if it is adequate. A warning will appear for values outside the reliable range for the function. Quiet Quiet - Each function has an optional Quiet input. True or 1 will turn off the warnings. False if omitted.

46. NUSSELT: FLAT PLATE, LOCAL NuxPlate(Re, Pr, Rexc Rexc, , Quiet Returns the local local Nusselt number at x Inputs based on the film temperature, Tf = (Ts+T )/2 Re - Reynolds number, Rex = V x / Pr - Prandtl number, Pr = Cp / k = / Rexc - Critical Reynolds number. Reynolds number at transition point from laminar to turbulent. If Re < Rexc, then laminar calculation. Otherwise, the calculation is for turbulent flow. If omitted, Recx = 5 X 105 Ranges For laminar, Pr 0.6 For turbulent, Rex 108, 0.6 Pr 60 NuxPlate(Re, Pr, Quiet) ) V, T Turbulent Laminar x Ts

47. NUSSELT: FLAT PLATE, MEAN NuBarPlate(Re, Pr, NuBarPlate(Re, Pr, Rexc Returns the average average Nusselt number from 0 to x Inputs based on the film temperature, Tf = (Ts+T )/2 Re - Reynolds number, Rex = V x / Pr - Prandtl number, Pr = Cp / k = / Rexc Critical Reynolds number. Reynolds number at transition point from laminar to turbulent. If Re < Rexc, then laminar calculation. Otherwise, the calculation is for a mix of laminar and turbulent. If omitted, Recx = 5 X 105 Ranges For laminar, Pr 0.6 For mixed, ReL 108, 0.6 Pr 60 V, T Rex, c Rexc, , Quiet Quiet) ) Turbulent Laminar x Ts

48. NUSSELT: CYLINDER IN CROSSFLOW NuDBarCyl(Re, Pr, NuDBarCyl(Re, Pr, Quiet Returns the average over a cylinder Inputs based on the film temperature, Tf = (Ts+T )/2 Re - Reynolds number, ReD = V D / Pr - Prandtl number, Pr = Cp / k = / Range ReDPr 0.2 Quiet) ) average Nusselt number for crossflow

49. NUSSELT: SPHERE NuDBarSphere(Re, Pr, mu_mus, NuDBarSphere(Re, Pr, mu_mus, Quiet Returns the average average Nusselt number for flow over a sphere Inputs based on the ambient fluid temperature, T , except s Re - Reynolds number, ReD = V D / Pr - Prandtl number, Pr = Cp / k = / mu_mus - / s; viscosity ratio calculated from T and Ts at the surface Range 0.71 Pr 380 3.5 ReD 7.6 X 104 Quiet) )

50. NUSSELT: BANK OF TUBES NuDBarTubes(Re, Pr, St_D, Sl_D, NuDBarTubes(Re, Pr, St_D, Sl_D, Aligned Returns the average average Nusselt number for crossflow over a bank of tubes Inputs based on the film temperature, Tf = (Ts+T )/2 Re - Reynolds number, ReD, max = Vmax D / Pr - Prandtl number, Pr = Cp / k = / St_D - Transverse spacing / Diameter, St / D Sl_D - Longitudinal spacing / Diameter, Sl / D Aligned - True or 1 for Aligned tubes, False or 0 for Staggered tubes. Aligned if omitted. Nl - Number of rows, if less than 10. Allows for correction factor if there are less than 10 rows. If omitted, Nl 10 Vmax Aligned - Vmax = St V / (St-D) Staggered if 2 SD > St +D, same as aligned else Vmax = V St / (SD-D) Ranges Pr 0.7 2000 ReD, max 40,000 Aligned, , Nl Nl, , Quiet Quiet) ) Staggered Aligned SD St St Sl Sl Rows Rows

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