CHEM E 330 Transport Processes I
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
- Driving force
| 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 | $h = \sqrt{\dfrac{r_e\sigma\cos\theta}{2\mu} t} \propto \sqrt{t}$ |