HEAT TRANSFER EQUATION SHEET - UTRGV …

1 1 heat TRANSFER EQUATION SHEET heat Conduction Rate equations (Fourier's Law) heat Flux : = 2 k : Thermal Conductivity heat Rate : = Ac : Cross-Sectional Area heat Convection Rate equations (Newton's Law of Cooling) heat Flux: = ( ) 2 h : Convection heat TRANSFER Coefficient 2 heat Rate: = ( ) As : Surface Area 2 heat Radiation emitted ideally by a blackbody surface has a surface emissive power: = 4 2 heat Flux emitted : = 4 2 where is the emissivity with range of 0 1 and = 10 8 2 4 is the Stefan-Boltzmann constant Irradiation: = but we assume small body in a large enclosure with = so that = 4 Net Radiation heat flux from surface: = = ( ) = ( 4 4) Net radiation heat exchange rate: = ( 4 4) where for a real surface 0 1 This can ALSO be expressed as: = ( ) depending on the application where is the radiation heat TRANSFER coefficient which is: = ( + )( 2+ 2) 2 TOTAL heat TRANSFER from a surface: = + = ( )+ ( 4 4) Conservation of Energy (Energy Balance) + = (Control Volume Balance).

2 = 0 (Control Surface Balance) where is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and =0 for steady-state conditions. If not steady-state ( , transient) then = heat EQUATION (used to find the temperature distribution) heat EQUATION (Cartesian): + + + = If is constant then the above simplifies to: 2 2+ 2 2+ 2 2+ =1 where = is the thermal diffusivity heat EQUATION (Cylindrical): 1 +1 2 + + = heat Eqn. (Spherical): 1 2 2 +1 2sin 2 +1 2sin sin + = Thermal Circuits Plane Wall: , = Cylinder: , =ln 2 1 2 Sphere: , =(1r1 1r2)4 2 , =1 , =1 _____ General Lumped Capacitance Analysis , + [ ( )+ ( 4 4)] ( , )= Radiation Only EQUATION = 4 , 3 ln + ln + +2 tan 1 tan 1 heat Flux, Energy Generation, Convection, and No Radiation EQUATION =exp( ) ; where = , and = , + Convection Only EQUATION = =exp = 1 ( )=.

3 = 1 exp ; = = If there is an additional resistance either in series or in parallel, then replace with in all the above lumped capacitance equations , where = 1 2 ; = overall heat TRANSFER coefficient, = total resistance, = surface area. Convection heat TRANSFER = = [Reynolds Number] ; = [Average Nusselt Number] where is the density, is the velocity, is the characteristic length, is the dynamic viscosity, is the kinematic viscosity, is the mass flow rate, is the average convection coefficient, and is the fluid thermal conductivity.

4 3 Internal Flow = 4 [For Internal Flow in a Pipe of Diameter D] For Constant heat Flux [ = ]: = ( ) ; where P = Perimeter, L = Length ( )= , + For Constant Surface Temperature [ = ]: If there is only convection between the surface temperature, , and the mean fluid temperature, , use ( ) , = If there are multiple resistances between the outermost temperature, , and the mean fluid temperature, , use ( ) , = = 1 Total heat TRANSFER rate over the entire tube length: = , , = ; = Log mean temperature difference: = ln ; = , ; = , Free Convection heat TRANSFER = ( ) 3 2 [Grashof Number] = ( ) 3 [Rayleigh Number] Vertical Plates: = + 1/6 1+ 9/16 8/27 2; [Entire range of RaL; properties evaluated at Tf] - For better accuracy for Laminar Flow: = + 1/4 1+ 9/16 4/9 ; 109 [Properties evaluated at Tf] Inclined Plates: for the top and bottom surfaces of cooled and heated inclined plates, respectively, the equations of the vertical plate can be used by replacing (g) with ( cos ) in RaL for 0 60.

5 Horizontal Plates: use the following correlations with = where As = Surface Area and P = Perimeter - Upper surface of Hot Plate or Lower Surface of Cold Plate: = 1/4 (104 107) ; = 1/3 (107 1011) - Lower Surface of Hot Plate or Upper Surface of Cold Plate: = 1/4 (105 1010) 4 Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion is met: 35 1/4 Long Horizontal Cylinders: = + 1/6 1+ 9/16 8/27 2 ; 1012 [Properties evaluated at Tf] Spheres: =2 + 1/4 1+ 9/16 4/9 ; 1011 ; [Properties evaluated at Tf] heat Exchangers heat Gain/Loss equations : = ( )= ; where is the overall heat TRANSFER coefficient Log-Mean Temperature Difference: , = , , , , ln , , , , [Parallel-Flow heat Exchanger] Log-Mean Temperature Difference: , = , , , , ln , , , , [Counter-Flow heat Exchanger] For Cross-Flow and Shell-and-Tube heat Exchangers: = , ; where is a correction factor Number of TRANSFER Units (NTU): =.

6 Where is the minimum heat capacity rate in [W/K] heat Capacity Rates: = , [Cold Fluid] ; = , [Hot Fluid] ; = [ heat Capacity Ratio] Note: The condensation or evaporation side of the heat exchanger is associated with = 5 If Pr 10 n = If Pr 10 n = 6 7 8 9 10