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1HEAT TRANSFER EQUATION SHEETHeat Conduction Rate Equations (Fourier's Law)ππππππππ Heat Flux: πππ₯π₯β²β² ππππππππ2β²β² Heat Rate: πππ₯π₯ πππ₯π₯ π΄π΄ππππHeat Convection Rate Equations (Newton's Law of Cooling)ππ Heat Flux: ππβ²β² β(πππ π ππ )ππ2 Heat Rate: ππ βπ΄π΄π π (πππ π ππ ) ππk : Thermal Conductivity ππ ππAc : Cross-Sectional Areaππh : Convection Heat Transfer Coefficient ππ2 πΎπΎAs : Surface Area ππ2Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: πΈπΈππ ππ πππ π 4 Heat Flux emitted: πΈπΈ πππππππ π 4ππππ2 8ππππ2where Ξ΅ is the emissivity with range of 0 ππ 1ππand ππ 5.67 10is the Stefan-Boltzmann constantππ2 πΎπΎ44 Irradiation: πΊπΊππππππ πΌπΌπΌπΌ but we assume small body in a large enclosure with ππ πΌπΌ so that πΊπΊ ππ ππ πππ π π π π π ππβ²β²4 ) Net Radiation heat flux from surface: ππππππππ πππΈπΈππ (πππ π ) πΌπΌπΌπΌ ππππ(πππ π 4 πππ π π π π π π΄π΄4 Net radiation heat exchange rate: ππππππππ πππππ΄π΄π π (πππ π 4 πππ π π π π π ) where for a real surface 0 ππ 1This can ALSO be expressed as: ππππππππ βππ π΄π΄(πππ π πππ π π π π π ) depending on the applicationππ2 )where βππ is the radiation heat transfer coefficient which is: βππ ππππ(πππ π πππ π π π π π )(πππ π 2 πππ π π π π π ππ2 πΎπΎ44 TOTAL heat transfer from a surface: ππ ππππππππππ ππππππππ βπ΄π΄π π (πππ π ππ ) πππππ΄π΄π π (πππ π πππ π π π π π ) ππConservation of Energy (Energy Balance)πΈπΈΜππππ πΈπΈππΜ πΈπΈΜππππππ πΈπΈΜπ π π π (Control Volume Balance) ; πΈπΈΜππππ πΈπΈΜππππππ 0 (Control Surface Balance)where πΈπΈππΜ is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, andπππππΈπΈΜπ π π π 0 for steady-state conditions. If not steady-state (i.e., transient) then πΈπΈΜπ π π π ππππππππHeat Equation (used to find the temperature distribution) Heat Equation (Cartesian): ππ If ππ is constant then the above simplifies to:Heat Equation (Cylindrical):Heat Eqn. (Spherical):1 ππ 1 ππ 2 ππππ ππππ 2Plane Wall: π π π‘π‘,ππππππππ πΏπΏ ππππ ππ 2 ππ π₯π₯ 2 2 ππ π¦π¦1 ππ 2 1 2 ππππ 2 sin ππ 2 ππ 2 ππ π§π§ 2 ππΜ1 πΌπΌ ππ Thermal CircuitsCylinder: π π π‘π‘,ππππππππ ππΜ ππππππ ππ ππ ππ12ππππππ where πΌπΌ ππππππππ ππΜ ππππππππ 2 sin ππln 2 ππ1 ππππ ππ sin ππis the thermal diffusivity ππΜ ππππππSphere: π π π‘π‘,ππππππππ 1 1r1 r2( )4ππππ
π π π‘π‘,ππππππππ 1π π π‘π‘,ππππππ βπ΄π΄21βππ π΄π΄General Lumped Capacitance Analysis4 )]πππ π β²β² π΄π΄π π ,β πΈπΈππΜ [β(ππ ππ ) ππππ(ππ 4 οΏ½οΏ½οΏ½π,ππ) ππππππRadiation Only Equationπ‘π‘ ππππππ4 ππ π΄π΄π π ,ππ ππ3πππ π π π π π ln ππππππππ ππ οΏ½οΏ½π π ππ ln πππ π π π π π πππππππ π π π π π ππππππ 2 tan 1 πππ π π π π π tan 1 exp( ππππ) ; where ππ οΏ½οΏ½Convection Only Equation and ππ ππ ππ βπ΄π΄π π ππ exp π‘π‘ ππππ ππππ ππ ππππππ (ππππππ) π π π‘π‘ πΆπΆπ‘π‘;π‘π‘ππ ππππππ ππππ 1 exp π΅π΅π΅π΅ οΏ½οΏ½πππ π π π π π Heat Flux, Energy Generation, Convection, and No Radiation Equationππ ππ πππππ‘π‘ πππ π π π π π ππππππππππ πππ π β²β² π΄π΄π π ,β ππππππ ππππππ ππππππIf there is an additional resistance either in series or in parallel, then replace β with ππ in all the above lumped capacitanceequations, whereππ 1π π π‘π‘ π΄π΄π π π π π π ππππ2 πΎπΎ πππππΏπΏππππ; ππ overall heat transfer coefficient, π π π‘π‘ total resistance, π΄π΄π π surface area.Convection Heat Transfer πππΏπΏππππ[Reynolds Number]; βπΏπΏ ππππ ππππππ[Average Nusselt Number]where ππ is the density, ππ is the velocity, πΏπΏππ is the characteristic length, ππ is the dynamic viscosity, ππ is the kinematic viscosity, ππΜ is the mass flowrate, β is the average convection coefficient, and ππππ is the fluid thermal conductivity.
π π π π 3Internal Flow4 ππΜ[For Internal Flow in a Pipe of Diameter D]ππππππFor Constant Heat Flux [πππ π ΚΊ οΏ½οΏ½π πππ π ΚΊ (ππ πΏπΏ) ; where P Perimeter, L Lengthπππ π ΚΊ Β· πππ₯π₯ππππ (π₯π₯) ππππ,ππ ππΜ ππππFor Constant Surface Temperature [πππ π οΏ½οΏ½πππ]:If there is only convection between the surface temperature, πππ π , and the mean fluid temperature, ππππ , useπππ π ππππ (π₯π₯) ππππππ πππ π ππππ,ππππ π₯π₯ππΜ ππππβ If there are multiple resistances between the outermost temperature, ππ , and the mean fluid temperature, ππππ , useππ ππππ (π₯π₯)ππ π₯π₯1 ππππππ ππ ππππππ ππ ππππ,ππππΜ ππππππΜ ππππ π π π‘π‘Total heat transfer rate over the entire tube length:πππ‘π‘ ππΜ ππππ ππππ,ππ ππππ,ππ β π΄π΄π π ππππππ ππππ ππ π΄π΄π π ππππππLog mean temperature difference: ππππππ ππππ ππππ; πππ π οΏ½οΏ½πππ; ππππ πππ π ππππ,ππ ; ππππ πππ π ππππ,ππ ππππ ππππln Free Convection Heat TransferπΊπΊπΊπΊπΏπΏ π π π π πΏπΏ Vertical Plates: πππππΏπΏ 0.825 ππππ(πππ π ππ )πΏπΏ3ππ[Grashof Number]ππππ(πππ π ππ )πΏπΏ3ππ[Rayleigh Number]ππ2ππππ0.387 π π π π πΏπΏ 1 1/60.492 9/16 ππππ8/272 ; [Entire range of RaL; properties evaluated at Tf ]- For better accuracy for Laminar Flow: πππππΏπΏ 0.68 1/40.670 π π π π πΏπΏ0.492 9/16 1 ππππ4/9; π π π π πΏπΏ 109 [Properties evaluated at Tf ]Inclined Plates: for the top and bottom surfaces of cooled and heated inclined plates, respectively, the equations of the verticalplate can be used by replacing (g) with (ππ cos ππ) in RaL for 0 ππ 60 .Horizontal Plates: use the following correlations with πΏπΏ π΄π΄π π ππwhere As Surface Area and P Perimeter- Upper surface of Hot Plate or Lower Surface of Cold Plate:1/41/3 πππππΏπΏ 0.54 π π π π πΏπΏ (104 π π π π πΏπΏ 107 , ππππ 0.7) ; πππππΏπΏ 0.15 π π π π πΏπΏ (107 π π π π πΏπΏ 1011 , ππππππ ππππ)- Lower Surface of Hot Plate or Upper Surface of Cold Plate: πΏπΏ 0.52 π π π π 1/5πππππΏπΏ(104 π π π π πΏπΏ 109 , ππππ 0.7)
Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion ismet:π·π·πΏπΏ 351/4πΊπΊπΊπΊπΏπΏLong Horizontal Cylinders: πππππ·π· 0.60 Spheres: πππππ·π· 2 1/40.589 π π π π π·π·4/90.469 9/16 1 ππππ 1/60.387 π π π π π·π·8/270.559 9/16 1 ππππ 2; π π π π π·π· 1012 [Properties evaluated at Tf ]; π π π π π·π· 1011 ; ππππ 0.7 [Properties evaluated at Tf ]Heat ExchangersHeat Gain/Loss Equations:ππ ππΜ ππππ (ππππ ππππ ) πππ΄π΄π π ππππππ ; where ππ is the overall heat transfercoefficient and As is the total heat exchanger surface areaLog-Mean Temperature Difference: ππππππ,ππππ ππβ,ππ ππππ,ππ ππβ,ππ ππππ,ππ [Parallel-Flow Heat Exchanger]Log-Mean Temperature Difference: ππππππ,πΆπΆπΆπΆ ππβ,ππ ππππ,ππ ππβ,ππ ππππ,ππ [Counter-Flow Heat Exchanger]ln For Cross-Flow and Shell-and-Tube Heat Exchangers:obtained from the figures by calculating P & R valuesEffectiveness β NTU Method (Ξ΅ β NTU):Number of Transfer Units (NTU): ππππππ πππππΆπΆππππππ ππβ,ππ ππππ,ππ ππβ,ππ ππππ,ππ ππβ,ππ ππππ,ππ ln ππβ,ππ ππππ,ππ ππππππ πΉπΉ ππππππ,πΆπΆπΆπΆ ; where πΉπΉ is a correction factor; where πΆπΆππππππ is the minimum heat capacity rate in [W/K]Heat Capacity Rates: πΆπΆππ ππΜππ ππππ,ππ [Cold Fluid] ; πΆπΆβ ππΜβ ππππ,β [Hot Fluid]πΆπΆππ οΏ½οΏ½πππ[Heat Capacity Ratio]Note: The condensation or evaporation side of the heat exchanger is associated with πΆπΆππππππ ππ ππΜππ πΆπΆππ,ππ ππππ,ππ ππππ,ππ ππΜβ πΆπΆππ,β ππβ,ππ ππβ,ππ πππ΄π΄π π οΏ½οΏ½π πΆπΆππππππ ππβ,ππ ππππ,ππ whereππ ππππππππππUse: ππ ππ(ππππππ, πΆπΆππ ) relations or ππππππ ππ(ππ, πΆπΆππ ) relations as appropriate4
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Page 726Chapter 11δHeat Exchangers1.01.0mi /n Cmax .500.751.000min /Cmax 00.83NTU405FIGURE 11.10 Effectiveness of a parallelflow heat exchanger (Equation 11.28).0123NTU45FIGURE 11.11 Effectiveness of acounterflow heat exchanger (Equation 11.29).Th,i or Tc,iTc,o or Th,oTh,i or Tc,iTc,o or Th,oTc,i or Th,iTc,i or Th,iTh,o or Tc,oTh,o or Tc,o1.01.00.80.751.000.6 01.000.750.500.25Ξ΅0.60.50axmC0.8min /C0.25min /Cmax 0C7265:21 E 11.12 Effectiveness of a shell-andtube heat exchanger with one shell and anymultiple of two tube passes (two, four, etc.tube passes) (Equation 11.30).0123NTU45FIGURE 11.13 Effectiveness of a shell-andtube heat exchanger with two shell passes andany multiple of four tube passes (four, eight,etc. tube passes) (Equation 11.31 with n 2).
Page 72711.4δTh,i or Tc,iTc,i or Th,iTh,i or Tc,iTc,o or Th,oTc,i or Th,iTc,o or Th,oTh,o or Tc,oTh,o or Tc,o1.01.0axm 0mixed /Cunm0.80.80.40.20123NTU45FIGURE 11.14 Effectiveness of a singlepass, cross-flow heat exchanger with bothfluids unmixed (Equation 11.32). 0, 7Heat Exchanger Analysis: The EffectivenessβNTU Methodmin /C5:21 NTU45FIGURE 11.15 Effectiveness of a singlepass, cross-flow heat exchanger with one fluidmixed and the other unmixed (Equations11.33, 11.34).
1/19/064:57 PMPage W-37Chapter 11 Supplemental Material11S.1Log Mean Temperature DifferenceMethod for Multipass and Cross-FlowHeat ExchangersAlthough flow conditions are more complicated in multipass and cross-flow heatexchangers, Equations 11.6, 11.7, 11.14, and 11.15 may still be used if the following modification is made to the log mean temperature difference [1]: Tlm F Tlm,CF(11S.1)That is, the appropriate form of Tlm is obtained by applying a correction factor tothe value of Tlm that would be computed under the assumption of counterflow conditions. Hence from Equation 11.17, T1 Th,i Tc,o and T2 Th,o Tc,i.Algebraic expressions for the correction factor F have been developed for various shell-and-tube and cross-flow heat exchanger configurations [1β3], and theresults may be represented graphically. Selected results are shown in Figures 11S.1through 11S.4 for common heat exchanger configurations. The notation (T, t) is usedto specify the fluid temperatures, with the variable t always assigned to the tube-sideTitotiTo1.00.90.8Fc11 supl.qxd0.70.60.56.0 4.0 3.02.0 1.51.0 0.8 0.60.40.2Ti β ToR to β ti00.10.20.30.40.50.60.70.80.91.0to β tiP Ti β tiFIGURE 11S.1 Correction factor for a shell-and-tube heat exchangerwith one shell and any multiple of two tube passes (two, four, etc. tubepasses).
W-384:57 PMPage W-3811S.1 Log Mean Temperature Difference MethodTitotiTo1.00.96.0 4.0 3.02.01.0 0.8 0.61.50.40.20.8F1/19/060.70.60.5Ti β ToR to β ti00.10.20.30.40.50.60.70.80.91.0to β tiP Ti β tiFIGURE 11S.2 Correction factor for a shell-and-tube heat exchanger with twoshell passes and any multiple of four tube passes (four, eight, etc. tube passes).TititoTo1.00.94.0 3.00.82.01.51.0 0.80.60.40.80.90.2Fc11 supl.qxd0.7Ti β To0.6 R t β toi0.500.10.20.30.40.50.60.7to β tiP Ti β tiFIGURE 11S.3 Correction factor for a single-pass, cross-flow heatexchanger with both fluids unmixed.1.0
1/19/064:57 PMPage W-3911S.1 W-39Log Mean Temperature Difference MethodTititoTo1.00.90.8Fc11 supl.qxd0.74.0 3.02.0 1.51.0 0.8 0.60.40.2Ti β To0.6 R to β ti0.500.10.20.30.40.50.60.70.80.91.0to β tiP Ti β tiFIGURE 11S.4 Correction factor for a single-pass, cross-flow heatexchanger with one fluid mixed and the other unmixed.fluid. With this convention it does not matter whether the hot fluid or the cold fluidflows through the shell or the tubes. An important implication of Figures 11S.1through 11S.4 is that, if the temperature change of one fluid is negligible, either P orR is zero and F is 1. Hence heat exchanger behavior is independent of the specificconfiguration. Such would be the case if one of the fluids underwent a phase change.
726 Chapter 11 Heat Exchangers 01 2 3 4 5 NTU Ξ΅ 1.0 0.8 0.6 0.4 0.2 0 1.00 C m in / C m a x a 0.25 0 0.75 0.
