Scope: Chapters 1, 2, 3, 5, 6, 7, 8, 9, 11. Excluded: Pool boiling. Purpose: Dense exam-use formula sheet. No explanations or worked examples.

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Chapter 1 — Introduction

Heat Rate / Heat Flux

$$q''=\frac{\dot{q}}{A},\qquad \dot{q}=q''A$$

Conduction

$$q_x''=-k\frac{dT}{dx}$$ $$q''=k\frac{T_1-T_2}{L}$$ $$\dot{q}=kA\frac{T_1-T_2}{L}$$

Convection

$$q''=h(T_s-T_\infty)$$ $$\dot{q}=hA(T_s-T_\infty)$$

Radiation

$$E_b=\sigma T_s^4$$ $$E=\varepsilon\sigma T_s^4$$ $$\dot{q}_{\text{rad}}=\varepsilon\sigma A(T_s^4-T_{\text{sur}}^4)$$ $$\dot{q}_{\text{abs}}=\alpha GA$$ $$\sigma=5.67\times10^{-8}\ \text{W/(m}^2\cdot\text{K}^4)$$

Energy Balance

$$\dot{E}_{\text{in}}-\dot{E}_{\text{out}}+\dot{E}_g=\dot{E}_{st}$$

Steady, no generation:

$$\dot{E}_{\text{in}}=\dot{E}_{\text{out}}$$

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Chapter 2 — Introduction to Conduction

Thermal Diffusivity

$$\alpha=\frac{k}{\rho c_p}$$

Cartesian Heat Equation

$$\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}+\frac{\dot{q}'''}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

Cylindrical Heat Equation

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) +\frac{1}{r^2}\frac{\partial^2 T}{\partial \phi^2} +\frac{\partial^2 T}{\partial z^2} +\frac{\dot{q}'''}{k} =\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

Spherical Radial Heat Equation

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dT}{dr}\right)+\frac{\dot{q}'''}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

Boundary Conditions

Specified temperature:

$$T_s=T_0$$

Specified heat flux:

• k\frac{\partial T}{\partial n}=q_s”

Adiabatic/symmetry:

$$\frac{\partial T}{\partial n}=0$$

Convection:

• k\frac{\partial T}{\partial n}=h(T_s-T_\infty)

Radiation:

• k\frac{\partial T}{\partial n}=\varepsilon\sigma(T_s^4-T_{\text{sur}}^4)

Interface:

$$T_A=T_B$$

• k_A\frac{\partial T_A}{\partial n}=-k_B\frac{\partial T_B}{\partial n}

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Chapter 3 — One-Dimensional Steady Conduction

Thermal Resistance

$$\dot{q}=\frac{\Delta T}{R_{\text{total}}}$$

Plane wall:

$$R_{\text{cond}}=\frac{L}{kA}$$

Cylinder:

$$R_{\text{cond}}=\frac{\ln(r_2/r_1)}{2\pi kL}$$

Sphere:

$$R_{\text{cond}}=\frac{1}{4\pi k}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$$

Convection:

$$R_{\text{conv}}=\frac{1}{hA}$$

Contact:

$$R_{t,c}=\frac{R_{t,c}''}{A}$$

Composite Plane Wall

$$\dot{q}=\frac{T_{\infty,1}-T_{\infty,2}}{\frac{1}{h_1A}+\sum\frac{L_i}{k_iA}+\frac{1}{h_2A}}$$

Tube Wall with Convection

$$\dot{q}=\frac{T_{\infty,i}-T_{\infty,o}}{\frac{1}{h_iA_i}+\frac{\ln(r_o/r_i)}{2\pi kL}+\frac{1}{h_oA_o}}$$ $$A_i=2\pi r_iL,\qquad A_o=2\pi r_oL$$

Overall Heat Transfer Coefficient

$$\dot{q}=UA\Delta T$$ $$\frac{1}{UA}=R_{\text{total}}$$ $$\frac{1}{U_iA_i}=\frac{1}{h_iA_i}+\frac{\ln(r_o/r_i)}{2\pi kL}+\frac{1}{h_oA_o}$$

Uniform Heat Generation

Plane wall, half-thickness $L$:

$$T(x)=T_s+\frac{\dot{q}}{2k}(L^2-x^2)$$ $$T_{\text{max}}=T_s+\frac{\dot{q}L^2}{2k}$$ $$q_s''=\dot{q}L$$

Solid cylinder:

$$T(r)=T_s+\frac{\dot{q}}{4k}(r_o^2-r^2)$$ $$T_{\text{max}}=T_s+\frac{\dot{q}r_o^2}{4k}$$ $$q'=\dot{q}\pi r_o^2$$

Fins

$$\theta=T-T_\infty,\qquad \theta_b=T_b-T_\infty$$ $$m=\sqrt{\frac{hP}{kA_c}}$$ $$\frac{d^2\theta}{dx^2}-m^2\theta=0$$

Adiabatic tip:

$$q_f=\sqrt{hPkA_c}\,\theta_b\tanh(mL)$$

Convective tip:

$$q_f=M\frac{\sinh(mL)+\frac{h}{mk}\cosh(mL)}{\cosh(mL)+\frac{h}{mk}\sinh(mL)}$$ $$M=\sqrt{hPkA_c}\,\theta_b$$

Infinite fin:

$$q_f=\sqrt{hPkA_c}\,\theta_b$$

Corrected length:

$$L_c=L+\frac{A_c}{P}$$

Fin efficiency:

$$\eta_f=\frac{q_f}{hA_f\theta_b}$$

Overall surface efficiency:

$$\eta_o=1-\frac{A_f}{A_t}(1-\eta_f)$$ $$q_t=\eta_o hA_t\theta_b$$

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Chapter 5 — Transient Conduction

Biot Number

$$Bi=\frac{hL_c}{k}$$

Lumped criterion:

$$Bi<0.1$$

Lumped length:

$$L_c=\frac{V}{A_s}$$

Lumped Capacitance

$$\frac{T(t)-T_\infty}{T_i-T_\infty}=\exp\left(-\frac{hA_s}{\rho Vc_p}t\right)$$ $$\frac{\theta}{\theta_i}=e^{-t/\tau}$$ $$\tau=\frac{\rho Vc_p}{hA_s}$$ $$\dot{q}(t)=hA_s[T(t)-T_\infty]$$ $$Q=\rho Vc_p[T_i-T(t)]$$ $$Q_0=\rho Vc_p(T_i-T_\infty)$$ $$\frac{Q}{Q_0}=1-\frac{\theta}{\theta_i}$$

Fourier Number

$$Fo=\frac{\alpha t}{L^2}\quad\text{plane wall}$$ $$Fo=\frac{\alpha t}{r_o^2}\quad\text{cylinder/sphere}$$

One-Term Approximation

Plane wall:

$$\frac{\theta}{\theta_i}=C_1e^{-\zeta_1^2Fo}\cos\left(\zeta_1\frac{x}{L}\right)$$

Cylinder:

$$\frac{\theta}{\theta_i}=C_1e^{-\zeta_1^2Fo}J_0\left(\zeta_1\frac{r}{r_o}\right)$$

Sphere:

$$\frac{\theta}{\theta_i}=C_1e^{-\zeta_1^2Fo}\frac{\sin(\zeta_1r/r_o)}{\zeta_1r/r_o}$$

Center temperature, all three:

$$\frac{\theta_0}{\theta_i}=C_1e^{-\zeta_1^2Fo}$$

Multidimensional product:

$$\left(\frac{\theta}{\theta_i}\right)_{multi-D}=\prod\left(\frac{\theta}{\theta_i}\right)_{1D}$$

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Chapter 6 — Introduction to Convection

$$q''=h(T_s-T_\infty)$$ $$\dot{q}=hA(T_s-T_\infty)$$ $$Nu=\frac{hL_c}{k_f}$$ $$h=\frac{Nu\,k_f}{L_c}$$ $$Re=\frac{VL_c}{\nu}=\frac{\rho VL_c}{\mu}$$ $$Pr=\frac{\nu}{\alpha}=\frac{\mu c_p}{k_f}$$ $$Gr=\frac{g\beta\Delta TL_c^3}{\nu^2}$$ $$Ra=GrPr$$ $$T_f=\frac{T_s+T_\infty}{2}$$ $$\bar h=\frac{1}{L}\int_0^Lh_x\,dx$$

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Chapter 7 — External Flow

Flat Plate

$$Re_x=\frac{U_\infty x}{\nu},\qquad Re_L=\frac{U_\infty L}{\nu}$$ $$Re_{x,c}\approx5\times10^5$$ $$x_c=\frac{Re_{x,c}\nu}{U_\infty}$$

Laminar local:

$$Nu_x=0.332Re_x^{1/2}Pr^{1/3}$$

Laminar average:

$$\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}$$

Turbulent local:

$$Nu_x=0.0296Re_x^{4/5}Pr^{1/3}$$

Turbulent average from leading edge:

$$\overline{Nu}_L=0.037Re_L^{4/5}Pr^{1/3}$$

Mixed average:

$$\overline{Nu}_L=(0.037Re_L^{4/5}-871)Pr^{1/3}$$ $$h=\frac{Nu\,k_f}{L_c}$$ $$\dot{q}=\bar hA(T_s-T_\infty)$$

Cylinder

$$Re_D=\frac{VD}{\nu}$$

Hilpert:

$$\overline{Nu}_D=CRe_D^mPr^{1/3}$$ $$\bar h=\frac{\overline{Nu}_Dk_f}{D}$$ $$\dot{q}=\bar h\pi DL(T_s-T_\infty)$$ $$q'=\bar h\pi D(T_s-T_\infty)$$

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Chapter 8 — Internal Flow

$$A_c=\frac{\pi D^2}{4},\qquad A_s=\pi DL$$ $$u_m=\frac{\dot{m}}{\rho A_c}$$ $$Re_D=\frac{\rho u_mD}{\mu}=\frac{u_mD}{\nu}=\frac{4\dot{m}}{\pi D\mu}$$ $$D_h=\frac{4A_c}{P}$$

Laminar/turbulent cutoff:

$$Re_D\approx2300$$

Entrance lengths:

$$\frac{x_{fd,h}}{D}\approx0.05Re_D\quad\text{laminar}$$ $$\frac{x_{fd,t}}{D}\approx0.05Re_DPr\quad\text{laminar}$$ $$\frac{x_{fd}}{D}\approx10\quad\text{turbulent estimate}$$

Energy balance:

$$\dot{q}=\dot{m} c_p(T_{m,o}-T_{m,i})$$

Constant heat flux:

$$\dot{q}=q_s''\pi DL$$ $$T_m(x)=T_{m,i}+\frac{q_s''\pi Dx}{\dot{m} c_p}$$ $$T_s-T_m=\frac{q_s''}{h}$$

Constant wall temperature:

$$\frac{T_s-T_{m,o}}{T_s-T_{m,i}}=\exp\left(-\frac{\bar hA_s}{\dot{m} c_p}\right)$$ $$\dot{q}=\bar hA_s\Delta T_{\text{lm}}$$ $$\Delta T_{\text{lm}}=\frac{(T_s-T_{m,i})-(T_s-T_{m,o})}{\ln[(T_s-T_{m,i})/(T_s-T_{m,o})]}$$

Laminar fully developed:

$$Nu_D=3.66\quad\text{constant }T_s$$ $$Nu_D=4.36\quad\text{constant }q_s''$$

Laminar developing:

$$\overline{Nu}_D=3.66+\frac{0.0668(D/L)Re_DPr}{1+0.04[(D/L)Re_DPr]^{2/3}}$$

Turbulent Dittus–Boelter:

$$Nu_D=0.023Re_D^{0.8}Pr^n$$ $$n=0.4\ \text{heating fluid},\qquad n=0.3\ \text{cooling fluid}$$

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Chapter 9 — Free Convection

$$T_f=\frac{T_s+T_\infty}{2}$$

Ideal gas:

$$\beta=\frac{1}{T_f}$$ $$Gr_L=\frac{g\beta|T_s-T_\infty|L^3}{\nu^2}$$ $$Ra_L=Gr_LPr$$ $$Nu=\frac{\bar hL_c}{k_f}$$

Vertical plate:

$$\overline{Nu}_L=\left[0.825+\frac{0.387Ra_L^{1/6}}{[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2$$

Vertical cylinder as vertical plate if:

$$\frac{D}{L}\gtrsim\frac{35}{Gr_L^{1/4}}$$

Horizontal cylinder:

$$\overline{Nu}_D=\left[0.60+\frac{0.387Ra_D^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2$$

Horizontal plate characteristic length:

$$L_c=\frac{A_s}{P}$$

Hot up / cold down:

$$\overline{Nu}_L=0.54Ra_L^{1/4}\quad(10^4\lesssim Ra_L\lesssim10^7)$$ $$\overline{Nu}_L=0.15Ra_L^{1/3}\quad(10^7\lesssim Ra_L\lesssim10^{11})$$

Hot down / cold up:

$$\overline{Nu}_L=0.27Ra_L^{1/4}$$

Convection:

$$\dot{q}_{\text{conv}}=\bar hA_s(T_s-T_\infty)$$

Radiation:

$$\dot{q}_{\text{rad}}=\varepsilon\sigma A_s(T_s^4-T_{\text{sur}}^4)$$

Combined:

$$\dot{q}_{\text{total}}=\dot{q}_{\text{conv}}+\dot{q}_{\text{rad}}$$

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Chapter 11 — Heat Exchangers

$$C_h=\dot{m}_h c_{p,h},\qquad C_c=\dot{m}_c c_{p,c}$$ $$\dot{q}=C_h(T_{h,i}-T_{h,o})=C_c(T_{c,o}-T_{c,i})$$ $$\frac{1}{UA}=\frac{1}{h_iA_i}+R_{f,i}+\frac{\ln(r_o/r_i)}{2\pi kL}+R_{f,o}+\frac{1}{h_oA_o}$$ $$\Delta T_{\text{lm}}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$ $$\dot{q}=UAF\Delta T_{\text{lm}}$$

Parallel:

$$\Delta T_1=T_{h,i}-T_{c,i}$$ $$\Delta T_2=T_{h,o}-T_{c,o}$$

Counterflow:

$$\Delta T_1=T_{h,i}-T_{c,o}$$ $$\Delta T_2=T_{h,o}-T_{c,i}$$ $$C_{\text{min}}=\min(C_h,C_c),\qquad C_{\text{max}}=\max(C_h,C_c)$$ $$C_r=\frac{C_{\text{min}}}{C_{\text{max}}}$$ $$\dot{q}_{\text{max}}=C_{\text{min}}(T_{h,i}-T_{c,i})$$ $$\varepsilon=\frac{\dot{q}}{\dot{q}_{\text{max}}}$$ $$NTU=\frac{UA}{C_{\text{min}}}$$ $$\dot{q}=\varepsilon C_{\text{min}}(T_{h,i}-T_{c,i})$$

Parallel effectiveness:

$$\varepsilon=\frac{1-e^{-NTU(1+C_r)}}{1+C_r}$$

Counterflow, $C_r<1$:

$$\varepsilon=\frac{1-e^{-NTU(1-C_r)}}{1-C_re^{-NTU(1-C_r)}}$$

Counterflow, $C_r=1$:

$$\varepsilon=\frac{NTU}{1+NTU}$$

Phase change, $C_r=0$:

$$\varepsilon=1-e^{-NTU}$$