Transport Phenomena and Rate Processes

Conduction through composite walls

For steady one-dimensional conduction through layers placed in series, the heat flow rate is the same through every layer, so each layer acts as a thermal resistance and the resistances add. A plane layer of thickness L, conductivity k and area A has resistance R = L/(kA) in K/W. The heat flux is q = \Delta T_{total}/\sum R, and the temperature drop across any layer is \Delta T_i = qR_i, so the layer with the larger resistance takes the larger share of the total drop. An imperfect contact between two solids adds a contact (interface) resistance R_c = \Delta T_c/q, which is treated as one more element in the series chain. The temperature at a junction follows from the requirement that the flux on both sides is equal, and the interface lies closer to the temperature of the side with the higher conductivity.

When two rods or layers are joined perfectly, the temperature is continuous at the junction while its slope changes. Because the gradient in each layer is q/k, the lower-conductivity material shows the steeper profile.

For a single plane wall, q = k\,\Delta T/L. For example, a 0.2 m slab with k = 4 W m^{-1}K^{-1} and a 100 K difference carries q = 4 \times 100/0.2 = 2000 W m^{-2}.

• Series resistance — resistances in the same heat path add, like electrical resistors. Example: R_1 = 0.1/1 = 0.1 K/W and R_2 = 0.2/4 = 0.05 K/W give R_{tot} = 0.15 K/W for a 200 K drop, so q \approx 1333 W m^{-2}.

Poiseuille pipe flow

Fully developed laminar flow of a Newtonian fluid in a circular tube, driven by a constant pressure gradient, has a parabolic velocity profile with no-slip at the wall: v_z = \dfrac{\Delta P}{4\mu L}(R^2 - r^2). The radial velocity is zero and v_z depends only on r, so there is no variation along the axis. The axial momentum balance \frac{1}{r}\frac{d}{dr}\left(r\frac{dv_z}{dr}\right) = \frac{1}{\mu}\frac{dP}{dz} is the governing equation.

The shear stress is \tau_{rz} = \mu\,\dfrac{dv_z}{dr} = \dfrac{r}{2}\dfrac{\Delta P}{L}. It is zero on the centreline and grows linearly with r, so it is maximum at the wall, where the wall shear stress is \tau_w = \dfrac{\Delta P R}{2L}.

The centreline velocity v_{max} = \dfrac{\Delta P R^2}{4\mu L} is proportional to \Delta P and R^2, and inversely proportional to \mu and L. It is not proportional to 1/R^2.

The laminar friction factor for a smooth circular tube is f = 16/Re with the Fanning convention, or 64/Re with the Darcy convention, since the Darcy value is four times the Fanning value. The laminar friction factor depends only on Reynolds number and varies inversely with it.

• Fully developed — the velocity profile no longer changes with axial position, so \partial v_z/\partial z = 0.

Newtonian viscous flow

A Newtonian fluid has a shear stress that is linearly proportional to the velocity gradient, \tau = \mu\,\dfrac{du}{dy}, with a constant dynamic viscosity \mu that does not depend on the shear rate. In plane Couette flow between parallel plates with a linear velocity profile, the shear rate is U/h and the stress is \tau = \mu U/h, so \mu = \tau h/U.

For example, if a 1 mm gap is sheared at 0.5 m s^{-1} and the required stress is 2 Pa, then \mu = 2 \times 10^{-3}/0.5 = 4\times10^{-3} Pa·s. The SI unit is Pa·s, and 1 poise equals 0.1 Pa·s.

The viscosity of a liquid falls as temperature rises because thermal agitation weakens intermolecular cohesion. Andrade's empirical form \mu = A\exp(B/T), with A and B positive, describes this decrease. A linear or exponentially rising form in T does not represent liquid viscosity.

• Newtonian — stress is directly proportional to rate of strain, with no yield stress and no shear-thinning or shear-thickening behaviour.

Boundary layer thickness

The boundary layer is the thin region near a wall in which viscous effects are significant. Its edge is conventionally where the velocity reaches the free-stream value. For a polynomial velocity profile, the edge is located where the velocity reaches the free stream smoothly, so the gradient vanishes there, \partial u/\partial y = 0. Solving this condition for the given profile yields the thickness.

For laminar flow over a flat plate, the Blasius solution gives \delta = 5x/\sqrt{Re_x} with Re_x = \rho U x/\mu. Hence \delta \propto \sqrt{x}, and \delta^2 = 25\mu x/(\rho U). This relation can be rearranged to find the viscosity from a measured thickness: \mu = \rho U \delta^2/(25x).

For example, for water with \nu = 10^{-6} m^2s^{-1}, U = 0.2 m s^{-1} and x = 0.5 m, Re_x = 10^5 and \delta \approx 5(0.5)/316 = 7.9 mm.

• Free-stream velocity — the velocity outside the boundary layer, U or V_\infty.

Reynolds number and flow regimes

The Reynolds number Re = \rho u L/\mu = uL/\nu is the ratio of inertial forces to viscous forces. It governs the transition from laminar to turbulent flow: in a pipe, flow is laminar at low Re and becomes turbulent beyond a critical value of roughly 2000 to 4000. The Schmidt, Biot and Prandtl numbers are not used for this purpose.

Dynamic similarity requires equal Reynolds numbers in model and prototype. When the characteristic length is the same, \rho_1 v_1/\mu_1 = \rho_2 v_2/\mu_2, so v_2 = v_1(\rho_1/\rho_2)(\mu_2/\mu_1). A denser or more viscous fluid must therefore move at a different speed to keep the same flow regime.

• Kinematic viscosity — \nu = \mu/\rho, the momentum diffusivity.

Arrhenius activation energy

The Arrhenius equation k = A\exp(-E_a/RT) gives the temperature dependence of a rate constant, where E_a is the activation energy in J mol^{-1} and R = 8.314 J mol^{-1}K^{-1}. Plotting \ln k against 1/T gives a straight line of slope -E_a/R.

The two-temperature form is \ln\dfrac{k_2}{k_1} = \dfrac{E_a}{R}\left(\dfrac{1}{T_1} - \dfrac{1}{T_2}\right). Taking T_1 = 300 K and T_2 = 400 K, a threefold increase in k implies E_a = R\ln 3/(1/300 - 1/400) \approx 11 kJ mol^{-1}. The result must be expressed in kJ mol^{-1} when the question asks for that unit.

The same exponential form appears for the conductivity of an intrinsic semiconductor, \sigma = \sigma_0\exp(-E_g/2k_BT), so the band gap is obtained from the ratio of conductivities at two temperatures by the same method, with E_g/2k_B in place of E_a/R.

• Pre-exponential factor — the constant A in the Arrhenius equation, related to collision frequency and orientation.

Transport law constitutive equations

Each transport process is described by a linear constitutive law that links a flux to a driving gradient through a material property. Newton's law of viscosity, \tau = -\mu\,dv/dy, relates molecular momentum flux to velocity gradient. Fourier's law, q = -k\,dT/dx, gives heat conduction, and Fick's law, J = -D\,dC/dx, gives molecular mass diffusion. Darcy's law relates the superficial velocity through a porous medium to the pressure gradient through the permeability and viscosity.

The corresponding material properties are viscosity for momentum, thermal conductivity for heat, diffusivity for mass, and permeability for flow in porous media. Hooke's law, \sigma = E\varepsilon, is a mechanical law and relates stress to strain through Young's modulus.

Radiative energy transport is not a gradient law. The Stefan–Boltzmann law E = \varepsilon\sigma T^4 gives the emissive power, with \sigma = 5.67\times10^{-8} W m^{-2}K^{-4}.

• Permeability — the property in Darcy's law, with units of m^2, describing how easily a fluid passes through a porous bed.

Radiative heat exchange with view factors

The view factor F_{ij} is the fraction of radiation leaving surface i that reaches surface j. Two rules govern it: the summation rule \sum_j F_{ij} = 1 for an enclosure, and reciprocity A_iF_{ij} = A_jF_{ji}. For black surfaces, the net exchange is Q_{12} = A_1F_{12}\sigma(T_1^4 - T_2^4).

For gray surfaces, the radiation network has a surface resistance (1-\varepsilon)/(\varepsilon A) for each surface and a space resistance 1/(A_1F_{12}) between them. The net heat flow is Q = \sigma(T_1^4 - T_2^4)/R_{tot}, where R_{tot} is the sum of these resistances in series. For two parallel disks of equal area, a surface emissivity below one increases the total resistance and reduces the exchange.

A small opening in an isothermal black cavity emits as a black body at the cavity temperature. The heat loss through the opening is therefore A_{open}\sigma(T^4 - T_{surr}^4), and the power required to hold the cavity at temperature T equals this loss.

• Summation rule — \sum_j F_{ij} = 1 for each surface in a closed enclosure.

Dimensionless numbers in transport

Each dimensionless number compares two transport properties or two forces. The Reynolds number Re = \rho uL/\mu compares inertia with viscosity and marks the laminar-to-turbulent transition. The Schmidt number Sc = \nu/D compares momentum and mass diffusivity, and the Prandtl number Pr = \nu/\alpha compares momentum and thermal diffusivity.

The Biot number Bi = hL/k_s uses the solid conductivity and compares convective surface resistance with internal conduction resistance. It is a solid-side parameter, unlike the Nusselt number Nu = hL/k_f, which uses the fluid conductivity.

By heat–mass analogy, the Sherwood number Sh = k_mL/D plays the role of Nu, and Sc replaces Pr. For laminar forced convection over a flat plate, the Nusselt number depends on Re and Pr, and the Sherwood number depends on Re and Sc. The Weber number, which compares inertia with surface tension, does not enter single-phase boundary-layer mass transfer correlations.

• Sherwood number — the mass-transfer analogue of the Nusselt number, Sh = k_mL/D_{AB}.

Ergun equation for packed beds

The Ergun equation gives the pressure drop of a fluid flowing through a packed bed: \dfrac{\Delta P}{L} = 150\dfrac{\mu(1-\varepsilon)^2V}{\varepsilon^3 d_p^2} + 1.75\dfrac{\rho(1-\varepsilon)V^2}{\varepsilon^3 d_p}, where \varepsilon is the bed porosity, d_p the particle diameter and V the superficial velocity. The first term dominates at low velocity and the second at high velocity.

The equation applies to gas flow through packed or fixed solid beds, such as the burden of a blast furnace, the sinter bed and solid beds in roasting. In the LD (basic oxygen) converter, oxygen is blown onto a liquid metal bath with no packed bed, so the Ergun equation is not applied there.

The permeability of a granular bed follows the Kozeny–Carman relation k = \varepsilon^3 d^2/[180(1-\varepsilon)^2], so at fixed porosity the permeability increases with particle size. A wider size distribution lets fine particles fill the voids, which lowers the porosity and reduces the permeability.

• Superficial velocity — the flow rate per unit cross-sectional area of an empty bed.

Zeroth-order reaction kinetics

In a zeroth-order reaction, the rate is independent of the reactant concentration: -d[A]/dt = k. The concentration therefore falls linearly with time, [A] = [A]_0 - kt, and the plot of concentration against time is a straight line.

The half-life is t_{1/2} = [A]_0/(2k), which depends on the initial concentration and on k. For example, with [A]_0 = 2 M and k = 0.1 M s^{-1}, t_{1/2} = 10 s. The rate constant still depends on temperature through the Arrhenius relation.

An increase in the reactant concentration does not increase the rate of a zeroth-order reaction, because the rate is set by the surface or catalyst condition rather than by the concentration.

• Zeroth order — a reaction whose rate does not vary with reactant concentration.

Rate process time estimation

The time needed for a rate process is found by integrating the rate law between the initial and final states. For a zeroth-order process, t = ([A]_0 - [A])/k. For a first-order process, t = \ln([A]_0/[A])/k. The integrated form therefore converts a rate constant into a time for a given conversion.