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1HEAT TRANSFER EQUATION SHEETHeat Conduction Rate Equations (Fourier's Law)π‘‘π‘‘π‘‘π‘‘π‘Šπ‘Šπ‘Šπ‘Š Heat Flux: π‘žπ‘žπ‘₯π‘₯β€²β€² π‘˜π‘˜π‘‘π‘‘π‘‘π‘‘π‘šπ‘š2β€²β€² Heat Rate: π‘žπ‘žπ‘₯π‘₯ π‘žπ‘žπ‘₯π‘₯ π΄π΄π‘π‘π‘Šπ‘ŠHeat Convection Rate Equations (Newton's Law of Cooling)π‘Šπ‘Š Heat Flux: π‘žπ‘žβ€²β€² β„Ž(𝑇𝑇𝑠𝑠 𝑇𝑇 )π‘šπ‘š2 Heat Rate: π‘žπ‘ž β„Žπ΄π΄π‘ π‘  (𝑇𝑇𝑠𝑠 𝑇𝑇 ) π‘Šπ‘Šk : Thermal Conductivity π‘šπ‘š π‘˜π‘˜Ac : Cross-Sectional Areaπ‘Šπ‘Šh : Convection Heat Transfer Coefficient π‘šπ‘š2 𝐾𝐾As : Surface Area π‘šπ‘š2Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: 𝐸𝐸𝑏𝑏 𝜎𝜎 𝑇𝑇𝑠𝑠4 Heat Flux emitted: 𝐸𝐸 πœ€πœ€πœ€πœ€π‘‡π‘‡π‘ π‘ 4π‘Šπ‘Šπ‘šπ‘š2 8π‘Šπ‘Šπ‘šπ‘š2where Ξ΅ is the emissivity with range of 0 πœ€πœ€ 1π‘Šπ‘Šand 𝜎𝜎 5.67 10is the Stefan-Boltzmann constantπ‘šπ‘š2 𝐾𝐾44 Irradiation: πΊπΊπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝛼𝛼𝛼𝛼 but we assume small body in a large enclosure with πœ€πœ€ 𝛼𝛼 so that 𝐺𝐺 πœ€πœ€ 𝜎𝜎 π‘‡π‘‡π‘ π‘ π‘ π‘ π‘ π‘ π‘žπ‘žβ€²β€²4 ) Net Radiation heat flux from surface: π‘žπ‘žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ πœ€πœ€πΈπΈπ‘π‘ (𝑇𝑇𝑠𝑠 ) 𝛼𝛼𝛼𝛼 πœ€πœ€πœ€πœ€(𝑇𝑇𝑠𝑠4 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝐴𝐴4 Net radiation heat exchange rate: π‘žπ‘žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ πœ€πœ€πœ€πœ€π΄π΄π‘ π‘  (𝑇𝑇𝑠𝑠4 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠) where for a real surface 0 πœ€πœ€ 1This can ALSO be expressed as: π‘žπ‘žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ β„Žπ‘Ÿπ‘Ÿ 𝐴𝐴(𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 ) depending on the applicationπ‘Šπ‘Š2 )where β„Žπ‘Ÿπ‘Ÿ is the radiation heat transfer coefficient which is: β„Žπ‘Ÿπ‘Ÿ πœ€πœ€πœ€πœ€(𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 )(𝑇𝑇𝑠𝑠2 π‘‡π‘‡π‘ π‘ π‘ π‘ π‘ π‘ π‘šπ‘š2 𝐾𝐾44 TOTAL heat transfer from a surface: π‘žπ‘ž π‘žπ‘žπ‘π‘π‘π‘π‘π‘π‘π‘ π‘žπ‘žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ β„Žπ΄π΄π‘ π‘  (𝑇𝑇𝑠𝑠 𝑇𝑇 ) πœ€πœ€πœ€πœ€π΄π΄π‘ π‘  (𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 ) π‘Šπ‘ŠConservation of Energy (Energy Balance)𝐸𝐸̇𝑖𝑖𝑖𝑖 𝐸𝐸𝑔𝑔̇ πΈπΈΜ‡π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ 𝐸𝐸̇𝑠𝑠𝑠𝑠 (Control Volume Balance) ; 𝐸𝐸̇𝑖𝑖𝑖𝑖 πΈπΈΜ‡π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ 0 (Control Surface Balance)where 𝐸𝐸𝑔𝑔̇ is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and𝑑𝑑𝑑𝑑𝐸𝐸̇𝑠𝑠𝑠𝑠 0 for steady-state conditions. If not steady-state (i.e., transient) then 𝐸𝐸̇𝑠𝑠𝑠𝑠 πœŒπœŒπœŒπœŒπ‘π‘π‘π‘Heat Equation (used to find the temperature distribution) Heat Equation (Cartesian): π‘˜π‘˜ If π‘˜π‘˜ is constant then the above simplifies to:Heat Equation (Cylindrical):Heat Eqn. (Spherical):1 π‘Ÿπ‘Ÿ 1 π‘Ÿπ‘Ÿ 2 π‘˜π‘˜π‘˜π‘˜ π‘˜π‘˜π‘Ÿπ‘Ÿ 2Plane Wall: 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐿𝐿 π‘˜π‘˜π‘˜π‘˜ π‘˜π‘˜ 2 𝑇𝑇 π‘₯π‘₯ 2 2 𝑇𝑇 𝑦𝑦1 π‘Ÿπ‘Ÿ 2 1 2 π‘˜π‘˜π‘Ÿπ‘Ÿ 2 sin πœƒπœƒ 2 π‘˜π‘˜ 2 𝑇𝑇 𝑧𝑧 2 π‘žπ‘žΜ‡1 𝛼𝛼 π‘˜π‘˜ Thermal CircuitsCylinder: 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 π‘žπ‘žΜ‡ πœŒπœŒπ‘π‘π‘π‘ π‘˜π‘˜ π‘Ÿπ‘Ÿ π‘˜π‘˜12πœ‹πœ‹πœ‹πœ‹πœ‹πœ‹ where 𝛼𝛼 π‘˜π‘˜πœŒπœŒπ‘π‘π‘π‘ π‘žπ‘žΜ‡ πœŒπœŒπ‘π‘π‘π‘π‘Ÿπ‘Ÿ 2 sin πœƒπœƒln 2 π‘Ÿπ‘Ÿ1 𝑑𝑑𝑑𝑑 π‘˜π‘˜ sin πœƒπœƒis the thermal diffusivity π‘žπ‘žΜ‡ πœŒπœŒπ‘π‘π‘π‘Sphere: 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 1 1r1 r2( )4πœ‹πœ‹πœ‹πœ‹

𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 1𝑅𝑅𝑑𝑑,π‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ β„Žπ΄π΄21β„Žπ‘Ÿπ‘Ÿ 𝐴𝐴General Lumped Capacitance Analysis4 )]π‘žπ‘žπ‘ π‘ β€²β€² 𝐴𝐴𝑠𝑠,β„Ž 𝐸𝐸𝑔𝑔̇ [β„Ž(𝑇𝑇 𝑇𝑇 ) πœ€πœ€πœ€πœ€(𝑇𝑇 4 ‘π‘,π‘Ÿπ‘Ÿ) 𝜌𝜌𝜌𝜌𝜌𝜌Radiation Only Equation𝑑𝑑 𝜌𝜌𝜌𝜌𝜌𝜌4 πœ€πœ€ 𝐴𝐴𝑠𝑠,π‘Ÿπ‘Ÿ 𝜎𝜎3𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 ln 𝑏𝑏𝑏𝑏𝑇𝑇𝑖𝑖 𝑇𝑇 ‘ π‘ π‘  𝑇𝑇 ln 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇𝑖𝑖𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇𝑖𝑖𝑇𝑇 2 tan 1 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 tan 1 exp( π‘Žπ‘Žπ‘Žπ‘Ž) ; where π‘Žπ‘Ž œŒConvection Only Equation and 𝑏𝑏 𝑇𝑇 𝑇𝑇 β„Žπ΄π΄π‘ π‘ πœƒπœƒ exp 𝑑𝑑 πœƒπœƒπ‘–π‘– 𝑇𝑇𝑖𝑖 𝑇𝑇 𝜌𝜌𝜌𝜌𝜌𝜌 (𝜌𝜌𝜌𝜌𝜌𝜌) 𝑅𝑅𝑑𝑑 𝐢𝐢𝑑𝑑;𝑑𝑑𝑄𝑄 𝜌𝜌𝜌𝜌𝜌𝜌 πœƒπœƒπ‘–π‘– 1 exp 𝐡𝐡𝐡𝐡 ‘–𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠Heat Flux, Energy Generation, Convection, and No Radiation Equation𝑇𝑇 𝑇𝑇 π‘Žπ‘Žπœπœπ‘‘π‘‘ 𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 π‘žπ‘žπ‘ π‘ β€²β€² 𝐴𝐴𝑠𝑠,β„Ž π‘šπ‘šπ‘šπ‘šπ‘šπ‘š 𝜌𝜌𝜌𝜌𝜌𝜌 πœƒπœƒπ‘–π‘–π‘˜π‘˜If there is an additional resistance either in series or in parallel, then replace β„Ž with π‘ˆπ‘ˆ in all the above lumped capacitanceequations, whereπ‘ˆπ‘ˆ 1𝑅𝑅𝑑𝑑 𝐴𝐴𝑠𝑠 𝑅𝑅𝑅𝑅 π‘Šπ‘Šπ‘šπ‘š2 𝐾𝐾 πœŒπœŒπœŒπœŒπΏπΏπ‘π‘πœ‡πœ‡; π‘ˆπ‘ˆ overall heat transfer coefficient, 𝑅𝑅𝑑𝑑 total resistance, 𝐴𝐴𝑠𝑠 surface area.Convection Heat Transfer π‘‰π‘‰πΏπΏπ‘π‘πœˆπœˆ[Reynolds Number]; β„ŽπΏπΏ 𝑁𝑁𝑁𝑁 π‘π‘π‘˜π‘˜π‘“π‘“[Average Nusselt Number]where 𝜌𝜌 is the density, 𝑉𝑉 is the velocity, 𝐿𝐿𝑐𝑐 is the characteristic length, πœ‡πœ‡ is the dynamic viscosity, 𝜈𝜈 is the kinematic viscosity, π‘šπ‘šΜ‡ is the mass flowrate, β„Ž is the average convection coefficient, and π‘˜π‘˜π‘“π‘“ is the fluid thermal conductivity.

𝑅𝑅𝑅𝑅 3Internal Flow4 π‘šπ‘šΜ‡[For Internal Flow in a Pipe of Diameter D]πœ‹πœ‹πœ‹πœ‹πœ‹πœ‹For Constant Heat Flux [π‘žπ‘žπ‘ π‘ ΚΊ ‘π‘ π‘žπ‘žπ‘ π‘ ΚΊ (𝑃𝑃 𝐿𝐿) ; where P Perimeter, L Lengthπ‘žπ‘žπ‘ π‘ ΚΊ Β· 𝑃𝑃π‘₯π‘₯π‘‡π‘‡π‘šπ‘š (π‘₯π‘₯) π‘‡π‘‡π‘šπ‘š,𝑖𝑖 π‘šπ‘šΜ‡ 𝑐𝑐𝑝𝑝For Constant Surface Temperature [𝑇𝑇𝑠𝑠 ‘π‘π‘π‘]:If there is only convection between the surface temperature, 𝑇𝑇𝑠𝑠 , and the mean fluid temperature, π‘‡π‘‡π‘šπ‘š , use𝑇𝑇𝑠𝑠 π‘‡π‘‡π‘šπ‘š (π‘₯π‘₯) 𝑒𝑒𝑒𝑒𝑒𝑒 𝑇𝑇𝑠𝑠 π‘‡π‘‡π‘šπ‘š,𝑖𝑖𝑃𝑃 π‘₯π‘₯π‘šπ‘šΜ‡ π‘π‘π‘π‘β„Ž If there are multiple resistances between the outermost temperature, 𝑇𝑇 , and the mean fluid temperature, π‘‡π‘‡π‘šπ‘š , use𝑇𝑇 π‘‡π‘‡π‘šπ‘š (π‘₯π‘₯)𝑃𝑃 π‘₯π‘₯1 𝑒𝑒𝑒𝑒𝑒𝑒 π‘ˆπ‘ˆ 𝑒𝑒𝑒𝑒𝑒𝑒 𝑇𝑇 π‘‡π‘‡π‘šπ‘š,π‘–π‘–π‘šπ‘šΜ‡ π‘π‘π‘π‘π‘šπ‘šΜ‡ 𝑐𝑐𝑝𝑝 𝑅𝑅𝑑𝑑Total heat transfer rate over the entire tube length:π‘žπ‘žπ‘‘π‘‘ π‘šπ‘šΜ‡ 𝑐𝑐𝑝𝑝 π‘‡π‘‡π‘šπ‘š,π‘œπ‘œ π‘‡π‘‡π‘šπ‘š,𝑖𝑖 β„Ž 𝐴𝐴𝑠𝑠 𝑇𝑇𝑙𝑙𝑙𝑙 π‘œπ‘œπ‘œπ‘œ π‘ˆπ‘ˆ 𝐴𝐴𝑠𝑠 𝑇𝑇𝑙𝑙𝑙𝑙Log mean temperature difference: 𝑇𝑇𝑙𝑙𝑙𝑙 π‘‡π‘‡π‘œπ‘œ 𝑇𝑇𝑖𝑖; 𝑇𝑇𝑠𝑠 ‘π‘π‘π‘; π‘‡π‘‡π‘œπ‘œ 𝑇𝑇𝑠𝑠 π‘‡π‘‡π‘šπ‘š,π‘œπ‘œ ; 𝑇𝑇𝑖𝑖 𝑇𝑇𝑠𝑠 π‘‡π‘‡π‘šπ‘š,𝑖𝑖 π‘‡π‘‡π‘œπ‘œ 𝑇𝑇𝑖𝑖ln Free Convection Heat Transfer𝐺𝐺𝐺𝐺𝐿𝐿 𝑅𝑅𝑅𝑅𝐿𝐿 Vertical Plates: 𝑁𝑁𝑁𝑁𝐿𝐿 0.825 𝑔𝑔𝑔𝑔(𝑇𝑇𝑠𝑠 𝑇𝑇 )𝐿𝐿3𝑐𝑐[Grashof Number]𝑔𝑔𝑔𝑔(𝑇𝑇𝑠𝑠 𝑇𝑇 )𝐿𝐿3𝑐𝑐[Rayleigh Number]𝜈𝜈2𝜈𝜈𝜈𝜈0.387 𝑅𝑅𝑅𝑅𝐿𝐿 1 1/60.492 9/16 𝑃𝑃𝑃𝑃8/272 ; [Entire range of RaL; properties evaluated at Tf ]- For better accuracy for Laminar Flow: 𝑁𝑁𝑁𝑁𝐿𝐿 0.68 1/40.670 𝑅𝑅𝑅𝑅𝐿𝐿0.492 9/16 1 𝑃𝑃𝑃𝑃4/9; 𝑅𝑅𝑅𝑅𝐿𝐿 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 (𝑔𝑔 cos πœƒπœƒ) in RaL for 0 πœƒπœƒ 60 .Horizontal Plates: use the following correlations with 𝐿𝐿 𝐴𝐴𝑠𝑠𝑃𝑃where As Surface Area and P Perimeter- Upper surface of Hot Plate or Lower Surface of Cold Plate:1/41/3 𝑁𝑁𝑁𝑁𝐿𝐿 0.54 𝑅𝑅𝑅𝑅𝐿𝐿 (104 𝑅𝑅𝑅𝑅𝐿𝐿 107 , 𝑃𝑃𝑃𝑃 0.7) ; 𝑁𝑁𝑁𝑁𝐿𝐿 0.15 𝑅𝑅𝑅𝑅𝐿𝐿 (107 𝑅𝑅𝑅𝑅𝐿𝐿 1011 , π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑃𝑃𝑃𝑃)- Lower Surface of Hot Plate or Upper Surface of Cold Plate: 𝐿𝐿 0.52 𝑅𝑅𝑅𝑅1/5𝑁𝑁𝑁𝑁𝐿𝐿(104 𝑅𝑅𝑅𝑅𝐿𝐿 109 , 𝑃𝑃𝑃𝑃 0.7)

Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion ismet:𝐷𝐷𝐿𝐿 351/4𝐺𝐺𝐺𝐺𝐿𝐿Long Horizontal Cylinders: 𝑁𝑁𝑁𝑁𝐷𝐷 0.60 Spheres: 𝑁𝑁𝑁𝑁𝐷𝐷 2 1/40.589 𝑅𝑅𝑅𝑅𝐷𝐷4/90.469 9/16 1 𝑃𝑃𝑃𝑃 1/60.387 𝑅𝑅𝑅𝑅𝐷𝐷8/270.559 9/16 1 𝑃𝑃𝑃𝑃 2; 𝑅𝑅𝑅𝑅𝐷𝐷 1012 [Properties evaluated at Tf ]; 𝑅𝑅𝑅𝑅𝐷𝐷 1011 ; 𝑃𝑃𝑃𝑃 0.7 [Properties evaluated at Tf ]Heat ExchangersHeat Gain/Loss Equations:π‘žπ‘ž π‘šπ‘šΜ‡ 𝑐𝑐𝑝𝑝 (π‘‡π‘‡π‘œπ‘œ 𝑇𝑇𝑖𝑖 ) π‘ˆπ‘ˆπ΄π΄π‘ π‘  𝑇𝑇𝑙𝑙𝑙𝑙 ; where π‘ˆπ‘ˆ is the overall heat transfercoefficient and As is the total heat exchanger surface areaLog-Mean Temperature Difference: 𝑇𝑇𝑙𝑙𝑙𝑙,𝑃𝑃𝑃𝑃 π‘‡π‘‡β„Ž,𝑖𝑖 𝑇𝑇𝑐𝑐,𝑖𝑖 π‘‡π‘‡β„Ž,π‘œπ‘œ 𝑇𝑇𝑐𝑐,π‘œπ‘œ [Parallel-Flow Heat Exchanger]Log-Mean Temperature Difference: 𝑇𝑇𝑙𝑙𝑙𝑙,𝐢𝐢𝐢𝐢 π‘‡π‘‡β„Ž,𝑖𝑖 𝑇𝑇𝑐𝑐,π‘œπ‘œ π‘‡π‘‡β„Ž,π‘œπ‘œ 𝑇𝑇𝑐𝑐,𝑖𝑖 [Counter-Flow Heat Exchanger]ln For Cross-Flow and Shell-and-Tube Heat Exchangers:obtained from the figures by calculating P & R valuesEffectiveness – NTU Method (Ξ΅ – NTU):Number of Transfer Units (NTU): 𝑁𝑁𝑁𝑁𝑁𝑁 π‘ˆπ‘ˆπ‘ˆπ‘ˆπΆπΆπ‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘‡π‘‡β„Ž,𝑖𝑖 𝑇𝑇𝑐𝑐,𝑖𝑖 π‘‡π‘‡β„Ž,π‘œπ‘œ 𝑇𝑇𝑐𝑐,π‘œπ‘œ π‘‡π‘‡β„Ž,𝑖𝑖 𝑇𝑇𝑐𝑐,π‘œπ‘œ ln π‘‡π‘‡β„Ž,π‘œπ‘œ 𝑇𝑇𝑐𝑐,𝑖𝑖 𝑇𝑇𝑙𝑙𝑙𝑙 𝐹𝐹 𝑇𝑇𝑙𝑙𝑙𝑙,𝐢𝐢𝐢𝐢 ; where 𝐹𝐹 is a correction factor; where πΆπΆπ‘šπ‘šπ‘šπ‘šπ‘šπ‘š is the minimum heat capacity rate in [W/K]Heat Capacity Rates: 𝐢𝐢𝑐𝑐 π‘šπ‘šΜ‡π‘π‘ 𝑐𝑐𝑝𝑝,𝑐𝑐 [Cold Fluid] ; πΆπΆβ„Ž π‘šπ‘šΜ‡β„Ž 𝑐𝑐𝑝𝑝,β„Ž [Hot Fluid]πΆπΆπ‘Ÿπ‘Ÿ ‘šπ‘šπ‘šπ‘š[Heat Capacity Ratio]Note: The condensation or evaporation side of the heat exchanger is associated with πΆπΆπ‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘žπ‘ž π‘šπ‘šΜ‡π‘π‘ 𝐢𝐢𝑝𝑝,𝑐𝑐 𝑇𝑇𝑐𝑐,π‘œπ‘œ 𝑇𝑇𝑐𝑐,𝑖𝑖 π‘šπ‘šΜ‡β„Ž 𝐢𝐢𝑝𝑝,β„Ž π‘‡π‘‡β„Ž,𝑖𝑖 π‘‡π‘‡β„Ž,π‘œπ‘œ π‘ˆπ‘ˆπ΄π΄π‘ π‘  ‘šπ‘š πΆπΆπ‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘‡π‘‡β„Ž,𝑖𝑖 𝑇𝑇𝑐𝑐,𝑖𝑖 whereπœ€πœ€ π‘žπ‘žπ‘žπ‘žπ‘šπ‘šπ‘šπ‘šπ‘šπ‘šUse: πœ€πœ€ 𝑓𝑓(𝑁𝑁𝑁𝑁𝑁𝑁, πΆπΆπ‘Ÿπ‘Ÿ ) relations or 𝑁𝑁𝑁𝑁𝑁𝑁 𝑓𝑓(πœ€πœ€, πΆπΆπ‘Ÿπ‘Ÿ ) relations as appropriate4

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Page 726Chapter 11䊏Heat 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 01.000.750.500.25Ρ0.60.50axmC0.8min /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 72711.4䊏Th,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 Effectiveness–NTU 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 [1–3], 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 – ToR to – ti00.10.20.30.40.50.60.70.80.91.0to – tiP Ti – 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 – ToR to – ti00.10.20.30.40.50.60.70.80.91.0to – tiP Ti – 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 – To0.6 R t – toi0.500.10.20.30.40.50.60.7to – tiP Ti – 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 – To0.6 R to – ti0.500.10.20.30.40.50.60.70.80.91.0to – tiP Ti – 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 0.50 T h,o or T c,o T c,i or T h,i T c,o or T h,o T h,i .