CHEM E 330 Transport Processes I

2021-12-08

Contents

-★- TRANSPORT PHENOMENA

Rate Laws for Diffusive Transport

Description	Equations
General form	flux = -(coefficient)(driving force)
Fourier’s law Heat conduction	$q = -k\dfrac{dT}{dy}$
Fick’s law Species diffusion	$J_A^* = -D_{AB} \dfrac{dc_a}{dy}$
Newton’s law of viscosity Momentum transfer	$\tau_{yx} = -\mu \dfrac{dv_x}{dy}$

Rate laws as concentration gradients

Description	Equations
Fourier’s law	$q_{y} = -\alpha \dfrac{dc_{H}}{dy}$
Fick’s law	$J_A^* = -D_{AB} \dfrac{dc_a}{dy}$
Newton’s law of viscosity	$\tau_{yx} = -\nu \dfrac{dc_{p_x}}{dy}$
Kinematic viscosity	$\nu = \dfrac{\mu}{\rho}$
Thermal diffusivity	$\alpha = \dfrac{k}{\rho \hat{c_P}}$
Diffusivity of A in B	$D_{AB}$
Prandtl number	$\mathrm{Pr} = \dfrac{\nu}{\alpha} = \dfrac{\hat{C}_p \mu}{k}$
Schmidt number	$\mathrm{Sc} = \dfrac{\nu}{D_{AB}} = \dfrac{\mu}{\rho D_{AB}}$

Heat transfer

Description	Equations
Heat flow	$\dot{Q} = \dfrac{Q}{t}$
Heat flux	$q = \dfrac{\dot{Q}}{A}$

Mass transfer

Description	Equations
Mass (species) transport	$N_A = x_A (\sum N_i) + J_A^*$
Diffusion of A through a stagnant layer of B	$N_A = -\dfrac{cD_{AB}}{1 - x_A}\dfrac{dx_A}{dy}$ $N_A = -\dfrac{cD_{AB}}{L}\ln(1-x_A^s)$
Equimolar counter diffusion	$N_A = -cD_{AB}\dfrac{dx_A}{dy}$
Reaction at catalytic surface $\ce{A = 2B} \implies N_B = 2N_A$	$N_A = -x_A N_A - cD_{AB}\dfrac{dx_A}{dy}$

Momentum transfer

Description	Equations
Interpretation of $\tau_{yx}$	1. viscous shear stress exerted on a $y$ -plane in the $+x$ -direction by the fluid of lesser $y$ on that of greater $y$ 2. flux of $x$ -momentum across a $y$ -plane in the $+y$ -direction
Shear strain rate	$\dot{\gamma} = \dfrac{dv_x}{dy}$
Hooke’s law ★ Hookean solid	$\tau_{yx} = -G \dfrac{dx}{dy} = -G \gamma$
Newton’s law of viscosity ★ Newtonian fluid	$\tau_{yx} = -\mu \dfrac{dv_x}{dy} = -\mu\dot{\gamma}$
General Newton’s law of viscosity	$\tau_{yx} = -\eta(\dot{\gamma})\dot{\gamma}$
Viscosity function of power law fluid	$\eta = m\dot{\gamma}^{n-1}$
Newton’s law of viscosity ★ Power law fluid	$\tau_{yx} = -m\dot{\gamma}^n$
Carreau equation ★ Slurry	$\dfrac{\eta - \eta_\infty}{\eta_o - \eta_\infty} = [1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}$

Transport Coefficients of Fluids

Ideal gas: Simple kinetic theory

Description	Equations
Average velocity	$\bar{u} = \sqrt{\dfrac{8 k_B T}{\pi m}}$
Mean free path	$\lambda = \dfrac{1}{\sqrt{2}\pi d^2 n}$
Number density	$n = \dfrac{N}{V}$
Molecular flux in the y-direction	$z = \frac{1}{4}n\bar{u}$
Average distance of molecules from ref plane when they initiate their jump	$\bar{a} = \frac{2}{3}\lambda$
Viscosity of ideal gas	$\mu = \dfrac{1}{3}\rho\bar{u}\lambda = \dfrac{2}{3\pi^{3/2}}\dfrac{\sqrt{mk_BT}}{d^2}$
Thermal conductivity of ideal gas	$k = \dfrac{1}{3}\rho\hat{c_v}\bar{u}\lambda = \dfrac{2\hat{c_v}}{3\pi^{3/2}}\dfrac{\sqrt{mk_BT}}{d^2}$
Diffusivity of ideal gas A in B	$D_{AB} = \dfrac{1}{3}\bar{u}_{AB}\lambda_{AB} = \dfrac{2}{\pi^{3/2}}\dfrac{\sqrt{{k_BT}^3/m_{AB}}}{d^2_{AB} P}$
Mean mass for diffusivity	$m_{AB} = \dfrac{2m_A m_B}{m_A+m_B}$
Mean distance for diffusivity	$d_{AB} = \frac{1}{2}(d_A + d_B)$
Prandtl number of monoatomic ideal gas	$\mathrm{Pr}_{\text{mono}} = 1$
Schmidt number of general ideal gas	$\mathrm{Sc} = 1$

Real gas: Chapman-Enskog equations

★ Moderate pressure

Description	Equations
Lenard-Jones potential	$\varphi(r) = 4\varepsilon \left[\left(\dfrac{\sigma}{r}\right)^{12} - \left(\dfrac{\sigma}{r}\right)^6\right]$
Attractive force	$F_{\text{attr}} = \dfrac{24\varepsilon}{r} \left[\left(\dfrac{\sigma}{r}\right)^{6} - 2\left(\dfrac{\sigma}{r}\right)^{12}\right]$
Viscosity of real gas (analytic)	$\mu = \dfrac{5}{16\pi}\dfrac{\sqrt{\pi m k_BT}}{\sigma^2 \Omega_\mu}$
Thermal conductivity of real gas (analytic)	$k = \dfrac{25}{32\pi} \dfrac{\sqrt{\pi m k_BT}}{\sigma^2 \Omega_k}\hat{c_v} = \dfrac{5}{2}\hat{c_v}\mu$
Viscosity of real gas	$\mu\left(\mathrm{\frac{g}{cm \cdot s}}\right) = 2.6692 \times 10^{-5} \dfrac{\sqrt{\mathcal{M}T}}{\sigma^2 \Omega_\mu}$
Thermal conductivity of monoatomic real gas	$k_{\text{mono}}\left(\mathrm{\frac{cal}{cm \cdot s \cdot K}}\right) = 1.989 \times 10^{-4} \dfrac{\sqrt{T / \mathcal{M}}}{\sigma^2 \Omega_k}$
Thermal conductivity of polyatomic real gas Euken factor	$k_{\text{poly}}\left(\mathrm{\frac{cal}{cm \cdot s \cdot K}}\right) = \left[ \hat{c_p} + \dfrac{5}{4}\dfrac{R}{\mathcal{M}} \right] \mu$
Diffusivity of real gas $T [=] \mathrm{K} \newline P [=] \mathrm{atm} \newline \sigma_{AB} [=]$ Å	$D_{AB}\left(\mathrm{\frac{cm^2}{s}}\right) = 2.63 \times 10^{-3} \dfrac{\sqrt{T^3 / \mathcal{M}_{AB}}}{P\sigma^2_{AB}\Omega_D}$
Mean molar mass for diffusivity	$\mathcal{M}_{AB} = \dfrac{2\mathcal{M}_A \mathcal{M}_B}{\mathcal{M}_A + \mathcal{M}_B}$
Mean distance for diffusivity	$\omega_{AB} = \frac{1}{2}(\omega_A + \omega_B)$
Viscosity at different temperatures	$\mu(T_2) = \mu(T_1)\sqrt{\dfrac{T_2}{T_1}}\dfrac{\Omega_{\mu_1}}{\Omega_{\mu_2}}$
Diffusivity at different temperatures	$D_{AB}(T_2) = D_{AB}(T_1)\left(\dfrac{T_2}{T_1}\right)^{3/2}\dfrac{\Omega_{D_1}}{\Omega_{D_2}}$
$T$ and $P$ dependence of transport coefficients of gases at moderate pressure	$\begin{aligned}\mu &\propto\sqrt{T} \\ k_{\text{mono}} &\propto\sqrt{T} \\ k_{\text{poly}} &= f(T, \hat{c_p}(T)) \\ D_{AB} &\propto T^{3/2}P^{-1} \\ D_{AB} &= D_{BA}\end{aligned}$

Ideal gas mixtures

Description	Equations
Wilke equation Viscosity of gas mixture	$\mu_{\text{mix}} = \displaystyle\sum_{i=1}^{N}\dfrac{x_i \mu_i}{\sum_{j=1}^N x_j \Phi_{ij}}$
Wilke equation Thermal conductivity of gas mixture	$k_{\text{mix}} = \displaystyle\sum_{i=1}^{N}\dfrac{x_i k_i}{\sum_{j=1}^N x_j \Phi_{ij}}$
Wilke equation parameter	$\Phi_{ij} = \frac{1}{\sqrt{8}} \left[ 1 + \frac{\mathcal{M}_i}{\mathcal{M}_j} \right]^{-1/2} \left[ 1 + \left[\frac{\mu_i}{\mu_j}\right]^{1/2} + \left[\frac{\mathcal{M}_i}{\mathcal{M}_j}\right]^{-1/4} \right]^2$
Blanc’s equation Diffusivity of gas mixture	$D_{i, \text{mix}} = \left[ \displaystyle\sum_{j \not= 1}^{N} \dfrac{x_j}{D_{ij}} \right]^{-1}$

Liquids

Description	Equations
Eyring model Viscosity of liquid	$\mu = \dfrac{N_A h}{\tilde{V}}\exp\left[ 0.408 \dfrac{\Delta U_{\text{vap}}}{RT} \right]$
Bridgeman equation Thermal conductivity of liquid	$k = 2.8 \left( \dfrac{N_A}{\tilde{V}}^{2/3} k_B v_s \right)$
Einstein equation	$D_{AB} \approx \dfrac{k_BT}{f}$
Hydrodynamic friction factor	$f = \begin{cases} 6\pi\mu_BR_A & \text{no slip} \\ 4\pi\mu_BR_A & \text{free slip} \end{cases}$
Stoke-Einstein Equation Diffusivity of dilute liquid A	$D_{AB} = \dfrac{k_BT}{4\pi\mu_BR_A}$
Wilke-Chang correlation Diffusivity of dilute liquid A $\tilde{V} [=] \mathrm{cm^3/mol} \newline \mu_B [=] \mathrm{cP} \newline T [=] \mathrm{K}$	$D_{AB}\left(\mathrm{\frac{cm^2}{s}}\right) = 7.4 \times 10^{-8} \dfrac{(\psi_B \mathcal{M}_B)^{1/2} T}{\mu \tilde{V}_A^{0.6}}$
Vigne’s equation Diffusivity of liquid mixture	$D_{AB} = (D_{AB}^0)^{x_B} (D_{BA}^0)^{x_A}$
$T$ dependence of transport coefficients of liquids (no $P$ dependence)	$\begin{aligned} \mu &= Ae^{B/T} \\ D_{AB}\mu_B &\propto T \\ D_{AB} &\not= D_{BA} \end{aligned}$

Shell Balance (Bottom-Up)

Boundary conditions and shell volume

Description	Equations
Rectilinear shell volume	$\Delta V = LW\Delta y$
Cylindrical shell volume	$\Delta V = 2\pi r L \Delta r$
Spherical shell volume	$\Delta V = 4 \pi r^2 \Delta r$
Newton’s law of cooling	$q = h(T_{\text{solid}} - T_{\text{fluid}})$
Relationship between $N_A$ and $c_A$ at boundary	$N_A = k_m (c_{A, \text{solid}} - c_{A, \text{fluid}})$
Reynolds number	$\mathrm{Re} = \dfrac{L_{\text{char}}v_{\text{char}}\rho}{\mu}$
No slip condition	$v_1 = v_2$
Free slip condition	$-\mu_1\left(\dfrac{dv_x}{dy}\right)_1 = 0$
Continuity of stress	$\begin{aligned}\tau_{y, 1} &= \tau_{y, 2} \\ -\mu_1\left(\dfrac{dv_x}{dy}\right)_1 &= -\mu_2\left(\dfrac{dv_x}{dy}\right)_2 \end{aligned}$

Shell balance method

Sketch the system with coordinate system
Sketch the shell that is thin in the direction of transport (change)
Write shell volume $\Delta V$
Write shell balance OIGA of transported quantity
- $\mathrm{out - in = generation - accumulation}$
Take limit as shell thickness approach 0
- Differential equation of flux distribution
Separate variable and integrate
- Flux distribution, $c_1$
Substitute rate law
Separate variable and integrate
- Profile, $c_1, c_2$
Evaluate $c_1, c_2$ using boundary conditions

Axial transport in rectilinear systems

Rectilinear coordinates
No generation
No driving force
Steady state

Description	Equations
Differential equation of flux distribution	$\dfrac{dq}{dy} = 0$
Temperature profile (linear)	$T(y) = T_1 - \dfrac{q}{k} y$
Flux distribution (inverse)	$q(y) = \dfrac{k(T_1 - T)}{y}$
Flux across the whole layer	$q = \dfrac{k(T_1 - T_2)}{H}$

Radial transport in cylindrical systems

Cylindrical coordinates
No generation
No driving force
Steady state

Description	Equations
Differential equation of flux distribution	$\dfrac{d(rq)}{dr} = 0$
Flux distribution (inverse)	$q(r) = \dfrac{k(T_i - T_0)}{r \ln(\frac{R_0}{R_i})}$
Temperature profile (logarithmic)	$T(r) = T_i - \dfrac{T_i - T_0}{\ln(\frac{R_0}{R_i})} \ln\left(\dfrac{r}{R_i}\right)$

Radial transport in spherical systems

Spherical coordinates
No generation
No driving force
Steady state

Description	Equations
Differential equation of flux distribution	$\dfrac{d(r^2 q)}{dr} = 0$
Flux distribution (inverse squared)	$q(r) = \dfrac{k(T_i - T_0)}{r^2 (\frac{1}{R_i} - \frac{1}{R_0})}$
Temperature profile (inverse)	$T(r) = T_i - \dfrac{T_i - T_0}{(\frac{1}{R_i} - \frac{1}{R_0})} \left(\dfrac{1}{r} - \dfrac{1}{R_i}\right)$

Axial transport in rectilinear systems (with generation)

Rectilinear coordinates
With generation
No driving force
Steady state

Description	Equations
Differential equation of flux distribution	$\dfrac{dq}{dy} = S$
Flux distribution (linear)	$q(y) = Sy + \dfrac{k}{H}(T_2 - T_1) - \dfrac{SH}{2}$
Temperature profile (quadratic)	$T(y) = T_1 - \dfrac{S}{2k} y^2 + \left[ \dfrac{SH}{2k} - \dfrac{T_2 - T_1}{H} \right] y$

Flow down inclined plane (falling film)

Rectilinear coordinates
Gravity driving force, but no pressure gradient
Steady state

Description	Equations
Differential equation of flux distribution	$\dfrac{d\tau_{yx}}{dy} = \rho g \cos\beta$
Flux distribution (linear)	$\tau_{yx}(y) = -\rho g \cos\beta (\delta - y)$
Velocity profile (quadratic)	$v_x(y) = \dfrac{g \cos\beta}{2\nu}(2\delta y - y^2)$
★ No entry length effect	$L \gg \delta$
★ No edge effect	$W \gg \delta$
★ Incompressible Newtonian fluid	$\Delta\mu = 0, \Delta\rho = 0$
★ No end effect, no ripple	$\mathrm{Re}_{\text{rippling}} \lesssim 20$
Reynolds number for falling film	$\mathrm{Re} = \dfrac{4\delta \langle v_x \rangle\rho}{\mu}$

Flow descriptors

Description	Equations
Skin friction	$\tau^0 = \rho g \cos(\beta)\delta$
Free surface velocity	$v_x^{\text{surf}} = \dfrac{g \cos\beta}{2\nu}\delta^2$
Volumetric flow rate	$Q = \int v_\perp \ dA$
Volumetric flow rate per unit area	$\dfrac{Q}{W} = \dfrac{g \cos(\beta) \delta^3}{3\nu}$
Average velocity	$\langle v_x \rangle = \dfrac{g\cos(\beta)\delta^2}{3\nu}$
Mass flow rate	$\dot{m} = \rho Q$
Mass flow rate per unit width	$\Gamma = \dfrac{\rho Q}{W} = \dfrac{\rho g \cos(\beta) \delta^3}{3\nu}$
Film thickness given $\Gamma$	$\delta = \sqrt[3]{\dfrac{3\nu \Gamma}{\rho g \cos\beta}}$

Flow in round tube (Hagen-Poiseuille flow)

Cylindrical coordinates
Pressure-gravity driving force
Steady state
No tube bents, constant cross section
Negligible P dependence with r

Description	Equations
Modified pressure	$\mathcal{P} = P + \rho gh$
Pressure-gravity driving force	$-\dfrac{dP}{dz} + \rho g \cos\beta = \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L}$
Differential equation of flux distribution	$\dfrac{d (r\tau_{rz})}{dr} = \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right) r$
Flux distribution (linear)	$\tau_{rz}(r) = \dfrac{1}{2} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right) r$
Velocity profile (quadratic)	$v_z(r) = \dfrac{R^2}{4\mu} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right) \left[ 1 - \left( \dfrac{r}{R} \right)^2 \right]$
★ Incompressible Newtonian fluid	$\Delta\mu = 0, \Delta\rho = 0$
★ Laminar flow	$\mathrm{Re}_{\text{laminar}} \le 2100$
★ Fully developed flow (no entry length effect)	$L_e \approxeq 0.035 D \mathrm{Re}$
Reynolds number for pipe flow	$\mathrm{Re}_{\text{pipe}} = \dfrac{D \langle v_z \rangle\rho}{\mu}$

Flow descriptors

Description	Equations
Skin friction	$\tau_{rz}^0 = \dfrac{1}{2} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right) R$
Volumetric flow	$Q = \dfrac{R^4 \pi}{8 \mu} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right)$
Average velocity	$\langle v_z \rangle = \dfrac{R^2}{8\mu} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right)$
Mass flow rate	$\dot{m} = \dfrac{R^4 \pi\rho}{8\mu} \left( \dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L} \right)$

Laminar flow through porous media

Description	Equations
Darcy’s law - average velocity $\kappa$ - bed permeability	$\langle v \rangle = \dfrac{\kappa}{\mu L}(\mathcal{P}_1 - \mathcal{P}_2)$
Darcy’s law - volumetric flow rate $A$ - empty bed cross section $\varepsilon$ - porosity, void fraction	$Q = \dfrac{\kappa A \varepsilon}{\mu L}(\mathcal{P}_1 - \mathcal{P}_2)$
Blake-Kozeny model Bed permeability	$\kappa = \dfrac{D_p^2}{150} \left( \dfrac{\varepsilon}{1 - \varepsilon} \right)^2$
Effective packing particle diameter	$D_p = \dfrac{6}{a_v} = \dfrac{6 V}{A} \newline D_{p, \text{spheres}} = D$
Bed Reynolds number	$\mathrm{Re}_{\text{bed}} = \dfrac{D_p Q \rho}{\mu A (1-\varepsilon)}$
★ Laminar flow	$\mathrm{Re}_{\text{laminar}} < 10$

Fluid pressure, hydrostatic, manometer

Description	Equations
Equation of hydrostatic	$P_1 - P_2 = \rho g(h_2 - h_1)$
Manometer equation	$P_1 - P_2 = (\rho_m - \rho) gH + \rho g(h_2 - h_1)$
Manometer equation	$\mathcal{P}_1 - \mathcal{P}_2 = (\rho_m - \rho) gH$

Unsteady state transport

Description	Equations
Unsteady state conduction in rectilinear system	$\left(\dfrac{\partial T}{\partial t}\right)_y = \alpha \dfrac{\partial^2 T}{\partial y^2} + \dfrac{S}{\rho \hat{c_p}}$
Unsteady state diffusion in rectilinear system	$\left(\dfrac{\partial c_A}{\partial t}\right)_y = D_{AB} \dfrac{\partial^2 c_A}{\partial y^2} + R_A$
Unsteady state Couette flow (1D rectilinear shear flow)	$\left(\dfrac{\partial v_x}{\partial t}\right)_y = \nu \left(\dfrac{\partial^2 v_x}{\partial y^2}\right)_t$
Unsteady state flow in cylindrical system	$\left(\dfrac{\partial v_z}{\partial t}\right)_r = \nu \left[ \dfrac{\partial^2 v_z}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial v_z}{\partial r} \right] + \dfrac{1}{\rho} \left[\dfrac{\mathcal{P}_1 - \mathcal{P}_2}{L}\right]$

Rate Laws in 3D

Description	Equations
Fourier’s law in 3D	$\utilde{q} = -k \nabla T$
Fick’s law in 3D	$\utilde{J}_A^* = -D_{AB} \nabla c_A$
Newton’s law of viscosity in 3D	$\underset{\approx}{\tau} = -\mu (\underset{\approx}{\Delta} + \underset{\approx}{\Delta}^{\dagger})$
Viscous stress tensor	$\underset{\approx}{\tau} = \begin{bmatrix} \tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz} \end{bmatrix}$
Rate of strain tensor	$\underset{\approx}{\Delta} = \begin{bmatrix} \dfrac{\partial v_x}{\partial x} & \dfrac{\partial v_x}{\partial y} & \dfrac{\partial v_x}{\partial z} \\ \\ \dfrac{\partial v_y}{\partial x} & \dfrac{\partial v_y}{\partial y} & \dfrac{\partial v_y}{\partial z} \\ \\ \dfrac{\partial v_z}{\partial x} & \dfrac{\partial v_z}{\partial y} & \dfrac{\partial v_z}{\partial z} \end{bmatrix}$

Conservation Laws in 3D

Description	Equations
Conservation of thermal energy	$\nabla\cdot\utilde{q} = S - \rho \hat{c_p} \dfrac{\partial T}{\partial t}$
Conduction equation ★ No convection	$\dfrac{\partial T}{\partial t} = \alpha \nabla^2 T + \dfrac{S}{\rho \hat{c_p}}$
Molecular diffusion equation ★ No convection	$\dfrac{\partial c_A}{\partial t} = D_{AB} \nabla^2 c_A + R_A$

-★- FLUID MECHANICS

Navier-Stokes Equation

Description	Equations
Continuity equation	$\dfrac{\partial \rho}{\partial t} + \nabla\cdot(\rho\utilde{v}) = 0$
Continuity equation of incompressible liquid ★ Constant $\rho$	$\nabla\cdot\utilde{v} = 0$
Equation of motion ( $v$ -form)	$\rho\dfrac{D\utilde{v}}{Dt} = -\nabla p + \mu\nabla^2\utilde{v} + \rho g$
Equation of motion ( $\tau$ -form)	$\rho\dfrac{D\utilde{v}}{Dt} = -\nabla p - \nabla\cdot\underset{\approx}{\tau} + \rho g$
Equation of motion ( $x$ -component)	$\begin{aligned} &\rho \left[ \dfrac{\partial v_x}{\partial t} + \utilde{v}\cdot\nabla v_x \right] \\ =& -\dfrac{\partial p}{\partial x} - \left[ \dfrac{\partial \tau_{xx}}{\partial x} + \dfrac{\partial \tau_{yx}}{\partial y} + \dfrac{\partial \tau_{zx}}{\partial z} \right] + \rho g_x \end{aligned}$

Operators

Description	Equations
Gradient operator $\nabla$	Operates on scalar to give a vector, whose magnitude is the maximum rate of change of the scalar with position, and whose direction points in the direction of that change
Divergence operator $(\nabla\cdot)$	Operates on a vector to give a scalar
Divergence of a flux vector $(\nabla\cdot\utilde{f})$	Rate of efflux (outflow) of the transported quantity per unit volume
Laplacian operator	$\nabla^2 = \nabla\cdot\nabla$
Substantial derivative operator	$\dfrac{D}{Dt} = \dfrac{\partial}{\partial t} + \utilde{v}\cdot\nabla$

Generalization to convection

Description	Equations
Thermal energy equation	$\dfrac{DT}{Dt} = \alpha \nabla^2 T + \dfrac{S}{\rho \hat{c_p}}$
Convective diffusion equation	$\dfrac{D c_A}{Dt} = D_{AB} \nabla^2 c_A + R_A$

Flow in conduit

Description	Equations
Mach number	$\mathrm{Ma} = \dfrac{v_{\text{char}}}{v_{\text{sound}}}$
Conduit flow	$\begin{aligned} \dot{m}_1 &= \dot{m}_2 \\ \rho_1 Q_1 &= \rho_2 Q_2 \end{aligned}$
Incompressible conduit flow ★ Constant $\rho$	$\begin{aligned} Q_1 &= Q_2 \\ A_1 \langle v \rangle_1 &= A_2 \langle v \rangle_2 \end{aligned}$

Apply N-S Equations (Top-Down)

Flow between parallel plates

Assumptions	Equations
Rectilinear coordinates	$f(x, y, z)$
Constant $\rho, \mu$	$\frac{\partial \rho}{\partial t} = 0, \frac{\partial \mu}{\partial t} = 0$
Laminar flow	$\mathrm{Re} < \mathrm{Re}_{\text{cr}}$
Steady state	$\frac{\partial}{\partial t} = 0$
$v_x$ component only	$v_y = v_z = 0$
No edge effect	$\frac{\partial}{\partial z} = 0$
No end effect	$\frac{\partial v_x}{\partial x} = 0$
No hydrostatic pressure diff between plates	$b \ll W, L \implies -\frac{\partial p}{\partial y} + \rho g_y = 0$

Description	Equations
$x$ -momentum equation	$\dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} + \mu\dfrac{\partial^2 v_x}{\partial y^2} = 0$
Velocity profile (quadratic)	$v_x(y) = \dfrac{1}{2\mu}\left( \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right)(-y^2 + by)$
Average velocity	$\langle v_x \rangle = \dfrac{b^2}{12\mu}\left( \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right)$
Skin friction at bottom plate	$\tau^0 = \dfrac{b}{2} \left( \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right)$

Couette flow between concentric rotating cylinders

Assumptions	Equations
Cylindrical coordinates	$f(r, \theta, z)$
Constant $\rho, \mu$	$\frac{\partial \rho}{\partial t} = 0, \frac{\partial \mu}{\partial t} = 0$
Laminar flow	$\mathrm{Re} < \mathrm{Re}_{\text{cr}}$
Steady state	$\frac{\partial}{\partial t} = 0$
$v_\theta$ component only	$v_r = v_z = 0$
Axial symmetry	$\frac{\partial}{\partial \theta} = 0$
No end effect	$\frac{\partial v_\theta}{\partial z} = 0$
Vertical orientation	$g_z = -g, g_\theta = g_r = 0$

Description	Equations
$r$ -momentum equation	$-\rho\dfrac{v_\theta^2}{r} = -\dfrac{\partial p}{\partial r}$
$\theta$ -momentum equation	$\mu\dfrac{\partial}{\partial r} \left( \dfrac{1}{r}\dfrac{\partial}{\partial r} (rv_\theta) \right) = 0$
$z$ -momentum equation	$-\dfrac{\partial p}{\partial z} - \rho g = 0$
Velocity profile (general form)	$v_\theta(r) = c_1\dfrac{r}{2} + \dfrac{c_2}{r}$
Velocity profile	$v_\theta(r) = \dfrac{\Omega_0}{1 - \kappa^2}\left[r - \dfrac{(\kappa R)^2}{r}\right]$
Pressure profile	$P - P_{\kappa R} = \dfrac{1}{2}\rho \left(\dfrac{\Omega_0\kappa R}{1-\kappa^2}\right)^2 \left[\left(\dfrac{r}{\kappa R}\right)^2 - \left(\dfrac{\kappa R}{r}\right)^2 - 4\ln\left(\dfrac{r}{\kappa R}\right) \right]$
Shear stress distribution	$\tau_{r\theta} = -2\mu\kappa^2\left(\dfrac{\Omega_0}{1-\kappa^2}\right)\left(\dfrac{R}{r}\right)^2$
Torque	$\mathcal{T} = 4\pi\mu L \Omega_0 R^2\dfrac{\kappa^2}{1 - \kappa^2}$
Couette viscometer	$\mu = \dfrac{\mathcal{T}}{4\pi L \Omega_0 R^2}\dfrac{1 - \kappa^2}{\kappa^2}$

Stoke’s law: Flow around a sphere

Assumptions	Equations
Spherical coordinates	$f(r, \theta, \phi)$
Constant $\rho, \mu$	$\frac{\partial \rho}{\partial t} = 0, \frac{\partial \mu}{\partial t} = 0$
Laminar flow	$\mathrm{Re} < \mathrm{Re}_{\text{cr}}$
Steady state	$\frac{\partial}{\partial t} = 0$
Axial symmetry	$\frac{\partial}{\partial \phi} = 0$
No spinning	$v_\phi = 0$
Vertical orientation	$g_r = -g \cos\theta, g_\theta = g \sin\theta, g_\phi = 0$
$v_\theta$ component only	$v_r = v_z = 0$

Description	Equations
$r$ velocity profile	$v_r = v_\infty \left[ 1 - \dfrac{3}{2}\left(\dfrac{R}{r}\right) + \dfrac{1}{2}\left(\dfrac{R}{r}\right)^2 \right] \cos\theta$
$\theta$ velocity profile	$v_\theta = -v_\infty \left[ 1 - \dfrac{3}{4}\left(\dfrac{R}{r}\right) - \dfrac{1}{4}\left(\dfrac{R}{r}\right)^3 \right] \sin\theta$
Pressure profile	$p = p_0 - \rho gz - \dfrac{3}{2}\dfrac{\mu v_\infty}{R}\left(\dfrac{R}{r}\right)^2 \cos\theta$
Viscous drag	$4\pi\mu v_\infty R$
Pressure force (buoyancy + form frag)	$\frac{4}{3}\pi R^3 \rho g + 2\pi R \mu v_\infty$
Stoke’s law	$v_\infty = \dfrac{2R^2 (\rho_s - \rho)g}{9\mu}$
Falling ball viscometer	$\mu = \dfrac{2R^2 (\rho_s - \rho)g}{9 v_\infty}$

Centrifuge viscometer

Description	Equations
Terminal velocity	$v_\infty = \dfrac{2R^2 (\rho_s - \rho) \omega ^2r}{9\mu}$
Centrifuge viscometer	$\mu = \dfrac{2R^2 (\rho_s - \rho)\omega^2}{9 \ln\left(\frac{R_2}{R_1}\right)} \Delta t$

Turbulence

Transition to turbulence

Geometry	Reynolds Number	Critical Reynolds Number
Circular tube flow	$\mathrm{Re} = \dfrac{D \langle v \rangle \rho}{\mu}$	$\mathrm{Re_c} \approx 2100$
Falling film	$\mathrm{Re} = \dfrac{4 \delta \langle v \rangle \rho}{\mu}$	$\mathrm{Re_c} \approx 1500$
Flow between parallel plates	$\mathrm{Re} = \dfrac{2b \langle v \rangle \rho}{\mu}$	$\mathrm{Re_c} \approx 1780$
Tangential flow in an annulus (Couette flow between rotating cylinders)	$\mathrm{Re} = \dfrac{\Omega_0 R^2 \langle v \rangle \rho}{\mu}$	$\mathrm{Re_c} \approx 50000$

Laminar vs. turbulent

Property	Laminar Flow $(\mathrm{Re} < 2100)$	Turbulent Flow $(\mathrm{Re} \in [10^4, 10^5])$
Velocity profile	$\dfrac{v_z}{v_{z, \max}} = 1 - \left(\dfrac{r}{R}\right)^2$	$\dfrac{v_z}{v_{z, \max}} \approx \left(1 - \dfrac{r}{R}\right)^{1/7}$
Average velocity	$\langle v_z \rangle = \frac{1}{2}v_{z, \max}$	$\langle v_z \rangle \approx \frac{4}{5}\bar{v}_{z, \max}$
Volumetric flow rate	$Q = \dfrac{\pi R^4}{8\mu} \left(\dfrac{\mathcal{P}_0 - \mathcal{P}_1}{L}\right)$	$Q \propto \left(\dfrac{\mathcal{P}_0 - \mathcal{P}_1}{L}\right)^{4/7}$
Entry length	$L_e = 0.035 D \mathrm{Re}$	$L_e \approx 40D$
Derivation	From theory	From experiment

Description	Equations
Velocity decomposition	$v_z = \bar{v}_z + v_z'$
Velocity profile in turbulent flow	$\bar{v}_z = \bar{v}_{z, \max}\left(1 - \dfrac{r}{R}\right)^{1/n}$ $n = \begin{cases} 6 & \mathrm{Re} \in [2\times 10^3, 10^4] \\ 7 & \mathrm{Re} \in [10^4, 10^5] \\ 8 & \mathrm{Re} \in [10^5, 10^6] \end{cases}$

Time-smoothed N-S equation

Description	Equations
Time-smoothed continuity equation	$\nabla\cdot\utilde{\bar{v}} = 0 \newline \nabla\cdot\utilde{v}' = 0$
Time-smoothed equation of motion ( $\tau$ -form)	$\rho\dfrac{D\utilde{\bar{v}}}{Dt} = -\nabla \bar{p} - \nabla\cdot\underset{\approx}{\bar{\tau}}^{\text{total}} + \rho g$
Time-smoothed equation of motion ( $x$ -component)	$\begin{aligned} &\rho \left[ \dfrac{\partial \bar{v}_x}{\partial t} + \utilde{\bar{v}}\cdot\nabla \bar{v}_x \right] \\ =& -\dfrac{\partial \bar{p}}{\partial x} - \left[ \dfrac{\partial \bar{\tau}^{\text{total}}_{xx}}{\partial x} + \dfrac{\partial \bar{\tau}^{\text{total}}_{yx}}{\partial y} + \dfrac{\partial \bar{\tau}^{\text{total}}_{zx}}{\partial z} \right] + \rho g_x \end{aligned}$
Total shear stress (viscous + turbulent)	$\begin{aligned} \bar{\tau}^{\text{total}}_{yx} &= \bar{\tau}_{yx}^{(v)} + \bar{\tau}_{yx}^{(t)} \\ &= \bar{\tau}_{yx} + \rho \overline{v_y' v_x'} \end{aligned}$

Shear stress distribution

Description	Equations
Shear stress distribution in round tube	$\tau_{r\theta} = \dfrac{1}{2}\left[\dfrac{\mathcal{P}_0 - \mathcal{P}_1}{L}\right]r$
Shear stress distribution in general conduit	$\tau_{r\theta} = \left[\dfrac{\mathcal{P}_0 - \mathcal{P}_1}{L}\right] R_H$
Hydraulic radius	$R_H = \mathrm{\dfrac{cross \ sectional \ area}{wetted \ perimeter}}$
Characteristic length	$l_{\text{char}} = 4R_H$
Characteristic velocity	$v_{\text{char}} = \langle v_z \rangle$

Universal velocity profile

Layer	Normalized velocity	Normalized length range
Laminar sublayer	$v^+ = y^+$	$y^+ \in (0, 5)$
Buffer layer	$v^+ = 5 \ln(y^+ + 0.205) - 3.27$	$y^+ \in (5, 30)$
Turbulent core	$v^+ = 2.5 \ln(y^+) + 5.5$	$y^+ \in (30, \infty)$

Description	Equations
Characteristic length	$y_* = \dfrac{\mu}{v_* \rho}$
Characteristic velocity	$v_* = \sqrt{\dfrac{\tau^0}{\rho}}$
Normalized length	$y^+ = \dfrac{y}{y_*}$
Normalized velocity	$v^+ = \dfrac{v}{v_*}$
Eddie viscosity	$\mu^{(t)} = - \dfrac{\bar{\tau}_{yz}^{\text{total}}}{\left(\frac{dv_z}{dy}\right)} - \mu = - \dfrac{\left[\frac{\mathcal{P}_0 - \mathcal{P}_1}{L}\right] \frac{r}{2}}{\left(\frac{dv_z}{dy}\right)} - \mu$

Dynamic Similarity and Dimensional Analysis

Flow around a sphere outside of Stoke’s law

Description	Equations
★ Non-Stoke’s law condition	$\mathrm{Re} \ge 0.1$
Nondimensionalized continuity equation	$\breve{\nabla}\cdot\utilde{\breve{v}} = 0$
x-component of momentum equation	$\dfrac{D\breve{v}_x}{D\breve{t}} = -\dfrac{\partial\breve{p}}{\partial\breve{x}} + \dfrac{1}{\mathrm{Re}}\breve{\nabla}^2 \breve{v}_x + \dfrac{1}{\mathrm{Fr}}\breve{g}_x$
Drag coefficient Friction factor	$c_D = f = \dfrac{F_D}{\frac{1}{2}\rho v_\infty^2 A_{\text{approach}}}$
Drag coefficient in Stoke’s law region	$c_D = \dfrac{24}{\mathrm{Re}}$
Drag coefficient in non-Stoke’s law region	$c_D = \left(\sqrt{\dfrac{24}{\mathrm{Re}}} + 0.5407\right)^2$

Dimensionless groups

Description	Equations
Reynolds number	$\mathrm{Re} = \dfrac{l_0 v_0 \rho}{\mu} = \mathrm{\dfrac{inertial \ forces}{viscous \ forces}}$
Froude number	$\mathrm{Fr} = \dfrac{v_0^2}{gl_0} = \mathrm{\dfrac{inertial \ forces}{gravitational \ forces}}$
Capillary number	$\mathrm{Ca} = \dfrac{\mu v_0}{\sigma} = \mathrm{\dfrac{viscous \ forces}{surface \ tension \ forces}}$
Weber number	$\mathrm{Fr} = \dfrac{l_0 \rho v_0^2}{\sigma} = \mathrm{\dfrac{inertial \ forces}{surface \ tension \ forces}}$
Euler’s number	$\mathrm{Eu} = \dfrac{(\Delta p)D^4}{\rho Q^2}$

Dimensional analysis

Buckingham $\pi$ theorem - A function $f(X_1, X_2, \dots, X_k)$ with dimensional variables $X_i$ can be rewritten in a function $\Phi(\Pi_1, \Pi_2, \dots, \Pi_{k-n})$ with dimensionless variables $\Pi_j$ by enforcing dimensional consistency using $n$ fundamental dimensions.
- Define fundamental dimensions
- Choose stand-in variables for fundamental dimensions
- Rewrite other variables in terms of stand-in variables to get dimensionless groups

Bernoulli Analysis and Applications

N-S equation for steady flow in stream tubes

Assumptions	Equations
Constant density fluid	$\Delta \rho = 0$
1D flow in $z$ direction	$v_r = v_\theta = 0$
Plug flow - uniform velocity across cross section	$\langle v \rangle = v = \mathrm{constant} \newline v_z = v_z(z)$
Inviscid flow	$\mu \approx 0, \mathrm{Re} \ge 10000$
No sharp bends	Straight stream lines

Description	Equations
Continuity equation	$\begin{aligned} Q_1 &= Q_2 \\ A_1 \langle v \rangle_1 &= A_2 \langle v \rangle_2 \end{aligned}$
Equation of motion	$\rho v \dfrac{dv}{dz} = -\dfrac{dp}{dz} - \rho g \dfrac{dh}{dz}$

Bernoulli equation

Description	Equations
Bernoulli equation (energy form)	$p_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = p_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$
Bernoulli equation (head form)	$\dfrac{v_1^2}{2g} + \dfrac{p_1}{\rho g} + h_1 = \dfrac{v_2^2}{2g} + \dfrac{p_2}{\rho g} + h_2$
Bernoulli head	$\mathcal{B} = \dfrac{v^2}{2g} + \dfrac{p}{\rho g} + h = \mathrm{constant}$
Drag coefficient	$c_D = \dfrac{F_D}{\frac{1}{2}\rho v_\infty^2 A_{\text{approach}}}$
Lift coefficient	$c_L = \dfrac{F_L}{\frac{1}{2}\rho v_\infty^2 A_{\text{planform}}}$
Pressure change in contracting conduit $\Delta p \equiv p_1 - p_2$	$\Delta p = \dfrac{8\rho Q^2}{\pi^2 D_1^4}\left[\left(\dfrac{D_1}{D_2}\right)^4 - 1\right] + \rho g (h_2 - h_1)$
Torricelli’s law	$\langle v \rangle = \sqrt{2g\Delta h}$
Pressure at stagnation point	$\begin{aligned} p &= p_{\text{static}} + p_{\text{dynamic}} \\ &= p_{\text{static}} + \textstyle\frac{1}{2}\rho v_\infty^2 \end{aligned}$

Flow-metering devices

Description	Equations
Manometer equation	$\Delta p = (\rho_\mathrm{m} - \rho)gH$
Local velocity Pitot tube	$v = \sqrt{\dfrac{2\Delta p}{\rho}}$
Volumetric flow rate Venturi meter $c_0 \in [0.96, 0.98]$ Orfice meter $c_0 \in [0.40, 0.80]$ Nozzle meter $c_0 \in [0.96, 0.98]$	$Q = c_0\pi D_0^2 \sqrt{\dfrac{\Delta p}{8\rho [1 - (\frac{D_0}{D})^4]}}$
Rotameter	Calibrated specifically to the fluid with falling sphere

Full Bernoulli analysis

Description	Equations
Full Bernoulli equation	$\dfrac{v_1^2}{2g} + \dfrac{p_1}{\rho g} + h_1 = \dfrac{v_2^2}{2g} + \dfrac{p_2}{\rho g} + h_2 + H_{L12}$
Head loss	$H_{L12} = H_{L12f} + H_{L12c}$
Skin friction loss $H_{L12f}$	Viscous work done per unit weight by fluid on walls of conduit in moving from 1 to 2
Skin friction loss (general)	$H_{L12f} = \dfrac{\tau^0 L}{\rho g R_H}$
Skin friction loss for circular tube	$H_{L12f} = \dfrac{4\tau^0 L}{\rho g D}$
Fanning friction factor	$f = \dfrac{\tau^0}{\frac{1}{2}\rho \langle v \rangle^2}$
Skin friction loss for circular tube	$H_{L12f} = \dfrac{2\langle v \rangle^2 L}{g D}f = \dfrac{32Q^2 L}{\pi^2 D^5 g}f$
Skin friction loss for non-circular tube	$H_{L12f} = \dfrac{\langle v \rangle^2 L}{2 g R_H}f = \dfrac{Q^2 L}{2g A_c^2 R_H}f$
Reynolds number for noncircular pipes	$\mathrm{Re} = \dfrac{4R_H \langle v \rangle \rho}{\mu}$
Configurational loss of one fitting in circular tube	$H_{Lc} = e_v\dfrac{\langle v \rangle^2_{\text{downstream}}}{2g}$
Configurational loss of all fittings in circular tube	$H_{L12c} = \dfrac{\langle v \rangle^2_{\text{down}}}{2g} (\sum\limits_i e_{v, i}) = \dfrac{8Q^2}{\pi^2 D^4 g} (\sum\limits_i e_{v, i})$
Total head loss for circular tube	$H_{L12} = \begin{cases} \dfrac{2 \langle v \rangle^2}{Dg} [(\sum\limits_i L_i)f + \frac{D}{4} (\sum\limits_i e_{v, i})] \\ \dfrac{32 Q^2}{\pi^2 D^5 g} [(\sum\limits_i L_i)f + \frac{D}{4} (\sum\limits_i e_{v, i})] \end{cases}$
Kinetic head correction factor	$\alpha = \dfrac{\langle v^3 \rangle}{\langle v \rangle^3}$
Brake horse power	$\mathrm{bhp} = \dfrac{P}{\eta} = \dfrac{H_p \rho g Q}{\eta}$

Fanning friction factor correlations

Description	Equations	Conditions
Hydraulically smooth pipes (Blasius)	$f = \dfrac{0.0791}{\mathrm{Re}^1/4}$	$\mathrm{Re} \in [2100, 10^5]$
Hydraulically smooth pipes (Koo)	$f = 0.0014 + \dfrac{0.125}{\mathrm{Re}^{0.32}}$	$\mathrm{Re} \in [10^4, 10^7]$
Pipes of general roughness (Haaland)	$\dfrac{1}{\sqrt{f}} = -3.6\log_{10} \left[\dfrac{6.9}{\mathrm{Re}} + \left(\dfrac{k/D}{3.7}\right)^{10/9}\right]$	$\mathrm{Re} \in [4\times 10^4, 10^7] \newline k/D < 0.05$
Commercial standard piping (Drew)	$f = 0.0014 + \dfrac{0.090}{\mathrm{Re}^{0.27}}$	$\mathrm{Re} \in [10^4, 10^7] \newline k/D \approx 0.00015$
Full rough conduit	$\dfrac{1}{\sqrt{f}} = 2.28 - 4.0 \log_{10} \left(\dfrac{k}{D}\right)$	$\mathrm{Re} > 10^4 \newline k/D > 0.01$

Kinetic head correction factor

$\mathrm{Re}$	$n$	$\alpha$
$2 \times 10^3 \sim 10^4$	$6$	$1.08$
$10^4 \sim 10^5$	$7$	$1.06$
$10^5 \sim 10^7$	$8$	$1.05$

Flow through packed bed

Description	Equations
Specific area of packing element	$a_v = \dfrac{\text{area of packing element}}{\text{volume of packing element}}$
Effective diameter of packing element (particle)	$D_p = \dfrac{6}{a_v}$
Darcy’s law ★ $\mathrm{Re_{bed}} \lesssim 10$	$\langle v \rangle = \dfrac{\kappa}{\mu} \left[ \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right]$
Volumetric flow rate	$Q = \langle v \rangle \varepsilon A = v_0 A$
Superficial velocity	$v_0 = \langle v \rangle \varepsilon$
Bed Reynolds number	$\begin{aligned}\mathrm{Re_{bed}} &= \dfrac{D_p v_0 \rho}{\mu}\dfrac{1}{1 - \varepsilon} \\ &= \dfrac{D_p \langle v \rangle \rho}{\mu}\dfrac{\varepsilon}{1 - \varepsilon} \\ &= \dfrac{D_p Q \rho}{\mu A}\dfrac{1}{1 - \varepsilon}\end{aligned}$
Tube Reynolds number	$\mathrm{Re_{tube}} = \dfrac{2}{3}\mathrm{Re_{bed}}$
Hydrolic radius	$R_H = \dfrac{D_p\varepsilon}{6(1-\varepsilon)}$
Friction factor of tube ★ $\mathrm{Re_{bed}} \le 10$	$f_{\text{tube}} = \dfrac{24(1-\varepsilon)\mu}{D_p v_0 \rho}$
Friction factor of tube ★ $\mathrm{Re_{bed}} > 1000$	$f_{\text{tube}} = \dfrac{7}{12}$
Bed permeability	$\kappa = \dfrac{D_p^2}{150} \left(\dfrac{\varepsilon}{1-\varepsilon}\right)^2$
Blake-Kozeny equation ★ $\mathrm{Re_{bed}} \le 10$	$\left[ \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right] = 150 \dfrac{\mu v_0}{D_p^2}\dfrac{(1-\varepsilon)^2}{\varepsilon^3}$
Burke-Plummer equation ★ $\mathrm{Re_{bed}} > 1000$	$\left[ \dfrac{\mathcal{P}_0 - \mathcal{P}_L}{L} \right] = \dfrac{7}{4}\dfrac{\rho v_0^2}{D_p}\dfrac{1-\varepsilon}{\varepsilon^2}$
Superficial mass flux	$G_0 = \rho v_0 = \dfrac{\dot{m}}{A}$
Ergun equation ★ $\mathrm{Re_{bed}} \in [10, 1000]$	$\left[ \dfrac{(\mathcal{P}_0 - \mathcal{P}_L)\rho}{G_0^2} \right] \dfrac{D_p}{L}\dfrac{\varepsilon^3}{1-\varepsilon} = 150 \left[ \dfrac{1-\varepsilon}{\frac{D_p G_0}{\mu}} \right] + \dfrac{7}{4} \newline \left[ \dfrac{(\mathcal{P}_0 - \mathcal{P}_L)\rho}{G_0^2} \right] \dfrac{D_p}{L}\dfrac{\varepsilon^3}{1-\varepsilon} = 150 \dfrac{1}{\mathrm{Re_{bed}}} + \dfrac{7}{4}$

Cavitation and vortex motion

Description	Equations
Cavitation number	$\sigma = \dfrac{p_A - p_C}{\frac{1}{2}\rho v_\infty^2}$

Forced vortex flow in rotating cylinder

Description	Equations
Velocity profile	$v_\theta = r\Omega$
Pressure difference ★ 1 defined arbitrarily, 2 defined at center	$p_2 - p_1 = \dfrac{1}{2}\rho\Omega^2 (r_2^2 - r_1^2) + \rho g (z_1 - z_2)$
Height	$h = \dfrac{\Omega^2}{2g} r^2$

Free vortex flow during drainage

Description	Equations
Pressure difference ★ 1 defined arbitrarily, 2 defined at $r \to\infty$	$p_2 - p_1 = \dfrac{1}{2}\rho C^2 \left(\dfrac{1}{r_1^2} - \dfrac{1}{r_2^2}\right) + \rho g (z_1 - z_2)$
Depth	$h = \dfrac{C^2}{2g} \dfrac{1}{r^2}$

Microfluidics*

Validity of continuum description

Description	Equations
Mean free path	$\lambda = \dfrac{1}{\sqrt{2}\pi d^2 n} \newline \lambda(\mathrm{\mu m}) \approx 3.1\times 10^{-3} \dfrac{T(\mathrm{K})}{\sigma^2(\mathrm{\mathring{A}^2}) p(\mathrm{atm})}$
Knudsen number	$\mathrm{Kn} = \dfrac{\lambda}{L_c}$

Characteristics	Range
Molecular flow	$\mathrm{Kn} \in (10, \infty)$
Transition flow	$\mathrm{Kn} \in (0.1, 10)$
N-S equations hold, but no-slip condition fails	$\mathrm{Kn} \in (0.001, 0.1)$
N-S equations hold, and no-slip condition holds	$\mathrm{Kn} \in (0, 0.001)$

Forces in microfluidic flows

Viscous force dominate over inertial forces and gravity forces
- Driving force
  - Pressure
  - Capillary (surface tension) forces
  - Electro-kinetic forces
  - Magnetic forces
- Resisting forces: viscous force, dominated by wall effects

Description	Equations
Reynolds number ★ Creeping flow	$\mathrm{Re} = \dfrac{\text{inertial forces}}{\text{viscous forces}} = \dfrac{Lv\rho}{\mu} \to 0$
Froude number	$\mathrm{Fr} = \dfrac{\text{inertial forces}}{\text{gravity forces}} = \dfrac{v^2}{gL}$
Viscous force dominates gravity force	$\mathrm{\dfrac{Re}{Fr}} = \dfrac{\text{gravity forces}}{\text{viscous forces}} = \dfrac{gL^2}{\mu v} \to 0$

Generalized Hagen-Poiseuille flow

Description	Equations
Differential equation of generalized H-P flow	$0 = \dfrac{\Delta p}{L} + \mu \left(\dfrac{\partial^2 v_z}{\partial x^2} + \dfrac{\partial^2 v_z}{\partial y^2}\right)$
No-slip condition $F(x, y)$ is equation of conduit perimeter	$v_z(x, y) = 0 for F(x, y) = 0$
Velocity profile	$v_z(x, y) = \dfrac{\Delta p}{\mu L} F(x, y)$
Volumetric flow rate	$Q = \dfrac{\Delta p}{\mu L} \displaystyle\iint F(x, y) \ dy\ dx$

Hydraulic resistance in micro-channels

Description	Equations
Flow equation	$\Delta p = \mathcal{R}_{\text{hyd}}Q$
Volumetric flow rate	$Q = \dfrac{\Delta p}{\mathcal{R}_{\text{hyd}}}$

Capillary driving force and wicking phenomena

Description	Equations
Pressure difference	$\Delta p = \sigma\kappa = \dfrac{2\sigma}{R}$
Wicking velocity	$v = \dfrac{r^2}{8\mu}\dfrac{\Delta P}{x} = \dfrac{r\sigma \cos\theta}{4\mu x}$
Washburn equation	$x = \sqrt{\dfrac{r\sigma\cos\theta}{2\mu} t} \propto \sqrt{t}$
Wicking into porous media	$h = \sqrt{\dfrac{r_e\sigma\cos\theta}{2\mu} t} \propto \sqrt{t}$