Equation Sheets
Equation Sheets

CHEM E 340 Transport Process II

  
Contents
Description Equations
Heat rate q=q′′A\mathbf{q} = \mathbf{q''}A
Fourier’s law (conduction) q′′=−k∇T\mathbf{q''} = -k \nabla T
Fourier’s law (conduction) qx′′=−kdTdxq_x'' = -k \dfrac{dT}{dx}
Description Equations
Newton’s law of cooling (convection) q′′=h(Ts−T∞)q'' = h (T_s - T_\infty)
Description Equations
Stefan-Boltzman’s law (radiation) Eb=εσTs4E_b = \varepsilon\sigma T_s^4
Gray body ε=α\varepsilon = \alpha
Blackbody ε=α=1\varepsilon = \alpha = 1
Irradiation from isothermal surrounding G=σTsurr4G = \sigma T_{\text{surr}}^4
Absorbed irradiation Gabs=αGG_{\text{abs}} = \alpha G
Net heat flux from radiation qrad′′=εσTs4−αGq_{\text{rad}}'' = \varepsilon\sigma T_s^4 - \alpha G
Net heat flux from radiation
★ gray surface, isothermal surrounding TsurrT_{\text{surr}}		 qrad′′=εσ(Ts4−Tsurr4)q_{\text{rad}}'' = \varepsilon\sigma (T_s^4 - T_{\text{surr}}^4)
Net heat flux from radiation in rate law form qrad′′=hrad(Ts−Tsurr)hrad=εσ(Ts+Tsurr)(Ts2+Tsurr2)q_{\text{rad}}'' = h_{\text{rad}} (T_s - T_{\text{surr}}) \newline h_{\text{rad}} = \varepsilon\sigma (T_s + T_{\text{surr}})(T_s^2 + T_{\text{surr}}^2)
Description Equations
Total energy balance In - Out + Generation = Accumulation\text{In - Out + Generation = Accumulation}
Total energy balance Ein−Eout+Egen=ΔEaccE_{\text{in}} - E_{\text{out}} + E_{\text{gen}} = \Delta E_{\text{acc}}
Total energy balance E˙in−E˙out+E˙gen=E˙acc=dEaccdt\dot{E}_{\text{in}} - \dot{E}_{\text{out}} + \dot{E}_{\text{gen}} = \dot{E}_{\text{acc}} = \dfrac{d E_{\text{acc}}}{dt}
Surface energy balance
★ E˙gen=E˙acc=0\dot{E}_{\text{gen}} = \dot{E}_{\text{acc}} = 0		 E˙in=E˙out\dot{E}_{\text{in}} = \dot{E}_{\text{out}}
Description Equations
Heat Diffusion Equation ∇2T+q˙k=1α∂T∂t\nabla^2 T + \dfrac{\dot{q}}{k} = \dfrac{1}{\alpha}\dfrac{\partial T}{\partial t}
Heat Diffusion Equation ∇⋅q′′=q˙−ρcP∂T∂t\nabla \cdot q'' = \dot{q} - \rho c_P\dfrac{\partial T}{\partial t}
Heat Diffusion Equation ∂T∂t=α∇2T+q˙ρcP\dfrac{\partial T}{\partial t} = \alpha\nabla^2 T + \dfrac{\dot{q}}{\rho c_P}
Volumetric heat generation q˙\dot{q}
Volumetric heat capacity (storage) ρcP\rho c_P
Thermal diffusivity α=kρcP\alpha = \dfrac{k}{\rho c_P}
Direction Heat flux Differential Area
x∈(−∞,∞)x \in (-\infty, \infty) qx′′=−k∂T∂xq_x'' = -k \frac{\partial T}{\partial x} dA=dydzdA = dydz
y∈(−∞,∞)y \in (-\infty, \infty) qy′′=−k∂T∂yq_y'' = -k \frac{\partial T}{\partial y} dA=dxdzdA = dxdz
z∈(−∞,∞)z \in (-\infty, \infty) qz′′=−k∂T∂zq_z'' = -k \frac{\partial T}{\partial z} dA=dxdydA = dxdy

∂∂x(k∂T∂x)+∂∂y(k∂T∂y)+∂∂z(k∂T∂z)+q˙=ρcP∂T∂t \dfrac{\partial}{\partial x} \left( k \dfrac{\partial T}{\partial x} \right) + \dfrac{\partial}{\partial y} \left( k \dfrac{\partial T}{\partial y} \right) + \dfrac{\partial}{\partial z} \left( k \dfrac{\partial T}{\partial z} \right) + \dot{q} = \rho c_P \dfrac{\partial T}{\partial t}

Direction Heat flux Differential Area
r∈[0,∞)r \in [0, \infty) qr′′=−k∂T∂rq_r'' = -k \frac{\partial T}{\partial r} dA=r dϕdzdA = r \ d\phi dz
ϕ∈[0,2π]\phi \in [0, 2\pi] qϕ′′=−k1r∂T∂ϕq_\phi'' = -k \frac{1}{r}\frac{\partial T}{\partial \phi} dA=drdzdA = drdz
z∈[−∞,∞)z \in [-\infty, \infty) qz′′=−k∂T∂zq_z'' = -k \frac{\partial T}{\partial z} dA=r drdϕdA = r \ drd\phi

1r∂∂r(kr∂T∂r)+1r2∂∂ϕ(k∂T∂ϕ)+∂∂z(k∂T∂z)+q˙=ρcP∂T∂t \dfrac{1}{r}\dfrac{\partial}{\partial r} \left( k r\dfrac{\partial T}{\partial r} \right) + \dfrac{1}{r^2}\dfrac{\partial}{\partial \phi} \left( k \dfrac{\partial T}{\partial \phi} \right) + \dfrac{\partial}{\partial z} \left( k \dfrac{\partial T}{\partial z} \right) + \dot{q} = \rho c_P \dfrac{\partial T}{\partial t}

Direction Heat flux Differential Area
r∈[0,∞)r \in [0, \infty) qr′′=−k∂T∂rq_r'' = -k \frac{\partial T}{\partial r} dA=r2sin⁡θ dθdϕdA = r^2 \sin\theta \ d\theta d\phi
θ∈[0,2π]\theta \in [0, 2\pi] qθ′′=−k1r∂T∂θq_\theta'' = -k \frac{1}{r}\frac{\partial T}{\partial \theta} dA=rsin⁡θdrdϕdA = r\sin\theta drd\phi
ϕ∈[0,π]\phi \in [0, \pi] qϕ′′=−k1rsin⁡θ∂T∂ϕq_\phi'' = -k \frac{1}{r \sin\theta}\frac{\partial T}{\partial \phi} dA=r drdθdA = r \ drd\theta

1r2∂∂r(kr2∂T∂r)+1r2sin⁡θ∂∂θ(ksin⁡θ∂T∂θ)+1r2sin⁡2θ∂∂ϕ(k∂T∂ϕ)+q˙=ρcP∂T∂t \dfrac{1}{r^2}\dfrac{\partial}{\partial r} \left( k r^2 \dfrac{\partial T}{\partial r} \right) + \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta} \left( k \sin\theta \dfrac{\partial T}{\partial \theta} \right) + \dfrac{1}{r^2 \sin^2\theta}\dfrac{\partial}{\partial \phi} \left( k \dfrac{\partial T}{\partial \phi} \right) + \dot{q} = \rho c_P \dfrac{\partial T}{\partial t}

Updated on 2022-01-16
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