Important Heat Transfer Parameters CBE 150A Midterm #3 Review Sheet General Parameters: q or or Heat transfer rate Heat flux (per unit area) Cp Specific heat capacity k Thermal conductivity h Convective heat transfer coefficient Thermal diffusivity R Uo, Ui Thermal resistance Overall heat transfer coefficient (based on outside or inside area respectively) Lc ft, m Characteristic length (r for cylinder or sphere; for slab) Heat exchanger parameters: F, K F, K True mean temperature difference Countercurrent: FG=1 Shell&Tube: FG=f(Z,ηH) 1 use plots for #shell_passes-#tube_passes exchanger Log mean temperature difference (ΔT1=Tfluid1,in Tfluid2,out ; ΔT2=Tfluid1,out Tfluid2,in) Ratio of ΔTshell to ΔTtubes Subscripts: h shell; c tube; a in; b out Heating effectiveness, ratio of ΔTactual to ΔTmax Subscripts: h shell; c tube; a in; b out 1
Important dimensionless numbers: Nusselt number CBE 150A Midterm #3 Review Sheet (Convective transfer / conductive transfer) Prandtl number (momentum diffusivity / thermal diffusivity) Biot number (convection out / conduction through body) Bi<<1 fast conduction, uniform T in body Bi>>1 fast convection, constant T at surface Fourier number (conduction rate / storage rate) Dimensionless temperature (similarly for dimensional average temperature ( )) Tb = bulk fluid; To = initial in the solid Viscosity ratio (accounts for μ change with T) Note: may need to iterate to find Twall,ave 2
Calculating Heat Transfer Rates CBE 150A Midterm #3 Review Sheet General expressions: Internal temperature change (ΔT of stream) General heat transfer from point a to point z (ΔT driving force) There may be one or multiple resistances between a and z Heat transfer in heat exchanger (ΔTTM is true mean driving force averaged between the pipe start and end) Overall resistance to heat transfer Resistances in series (one term per resistance in system) Remember is constant through all resistance layers in series! Resistances in parallel (one term per resistance in system) Forms of resistance terms combine as needed for each of the system resistances to get Rtotal: Slab conduction resistance Cylindrical conduction resistance (using log-mean area) Spherical conduction resistance (using geometric-mean area) Convection resistance Fouling resistance, given in 3
CBE 150A Midterm #3 Review Sheet Example - overall heat transfer coefficient 3 resistances (2 convective, 1 conductive) If you have a hot fluid inside a cylindrical metal pipe with a cold fluid outside, then you can expect to have 3 resistances (2 convective in the fluids and 1 conductive in the solid). The overall heat transfer coefficient can then be written as follows: Multiplying through by or (reference area) we get: and Simplifying with the area definitions (A=πDL, ): and Any of these 3 overall equations can be used to find : And the temperature at any intermediate point can be found from that : Note on temperature subscripts: b = bulk, w = wall, o = outside, i = inside ** Be careful and think through all the resistances you have in your system! Double check that you are using the correct ALM (especially if you have multiple layers of conduction, each with its own ALM), Ai, and Ao, etc. 4
Non-Steady State Problems CBE 150A Midterm #3 Review Sheet In steady state, temperature still changes with location, just not with time! For non-steady state problems, the temperature changes with both time and location. You should only use the Biot number and the Fourier number for non-ss calculations. From Non-Steady State Conduction lecture slides (copied at the end of this review sheet): Fig. 10.5 average temperature (without external thermal resistance) of a slab, cylinder, or sphere with constant surface temperature (denoted Ts or Tb); variables: Fo and Fig. 10.7 average temperature of a slab with external thermal resistance (bulk temperature beyond film denoted Tf ); variables: Fo, Bi, and Fig. 10.8 average temperature of a sphere with external thermal resistance (bulk temperature beyond film denoted Tf ); variables: Fo, Bi, and From class handout with plots: Fig. 11.1.2 transient radial temperature profiles (at position r) in a sphere with constant surface temperature (Bi>>1, little/no external resistance); variables: r/r, Fo, and Fig. 11.1.3 (same as 10.6 above) average temperature of a slab, cylinder, or sphere with constant surface temperature (Bi>>1, little/no external resistance); variables: Fo and Fig. b) transient mid-plane temperature (position at center of thickness) for a slab with external thermal resistance (convective h); variables: Fo, Bi, Fig. d) transient axial temperature (at center-line r=0 position) for a long cylinder with external thermal resistance (convective h); variables: Fo, Bi, Fig. f) transient center temperature (at r=0 position) for a sphere with external thermal resistance (convective h); variables: Fo, Bi, Also figure (a) for equation coefficients, and figures (c), (e), and (g) for the ratio of surface to at the center of the solid slab, cylinder, and sphere, respectively. at the 5
Heat Transfer Correlations CBE 150A Midterm #3 Review Sheet For heat transfer by forced convection, Nu = f(re, Pr, ) and empirical correlations exist for different geometries to calculate the convective heat transfer coefficient (h) in the fluid film. For natural convection (density-driven circulation) the notes give a correlation for Nu = f(gr, Pr), where Gr is the Grashof number. Flow past a single sphere (external forced convection) Flow normal to tubes (external forced convection) Sieder-Tate equation for flow in pipes; use Note: The Colburn analogy in the notes is a rearranged Sieder-Tate equation that gives the j-factor for heat transfer (jh) in terms of the Fanning friction factor. Donohue Correlation for shell side flow; use Ge is average flow rate [=] ; Do = OD of 1 tube Calculating Re in Heat Exchangers Use these to calculate the Reynolds number used in the Sieder-Tate and Donohue Correlations in order to find the outside heat transfer coefficients (ho values) Countercurrent: Shell-and-Tube: Definition of hydraulic radius S = cross sectional area, Lp = wetted perimeter Definition of equivalent diameter Geometric average of horizontal and vertical flow rates Cross-sectional area available for shell horizontal flow Ds = shell ID, Do = OD of 1 tube, N = # of tubes/shell pass fb = fraction of shell cross-section not covered by baffle Cross-sectional area available for shell vertical flow Ds = shell ID, Do = OD of 1 tube, p = tube pitch, P=baffle spacing 6
Figure!10.5:!Averagetemperaturesduringunsteady1stateheatingorcoolingofalargeslab,and infinitelylongcylinder,orasphere.!
Figure!10.7:Changewithtimeoftheaveragetemperatureofaslabwithexternalconvectiveresistance. Figure!10.8:!Changewithtimeoftheaveragetemperatureofaspherewithexternalconvective resistance.