Transport phenomena include three closely related topics:

• fluid dynamics (the transport of momentum)
• heat transfer (the transport of energy)
• mass transfer (the transport of mass)

Transfer can be due to:

• convection (transport by a flowing fluid, where the flow can be due to an external force or an internal difference in temperature or density)
• conduction (transfer of heat by collision of particles in fluids)
• radiation (transfer of heat by means of wave motion in space)

Flux and flow

Flux \(q\) is the rate of transfer per unit area normal to the direction of the transfer.

Flow \(\phi = q\cdot A\) is the rate of energy or mass transferred through a given surface \(A\) .

In general:

• flux = conductivity x driving force
• flow = conductivity x driving force x cross-sectional area

In a steady state, the flux is constant.

The driving force for conductive and convective heat transfer is a temperature difference.

When starting to analyze a situation, get clear about what kind of situation this is:

• conduction vs convection
• steady vs unsteady
• short time (penetration) vs long time (permeation)

Heat conduction (Fourier’s law)

In the one-dimensional case of conduction in the \(x\) dimension, the rate of conduction is: $$\phi_q = -k A \frac{dT}{dx}$$

• \(k\) [ \(J/msK\) ] is the conductivity coefficient of the material (the substance’s ability to transfer heat by conduction)
• \(\cfrac{dT}{dx}\) is the temperature gradient

If the temperature gradient is linear, it can be simplified to: $$\phi_q = -k A \frac{\Delta T}{\Delta x}$$

The heat flux is \(\cfrac{\phi_q}{A}\) .

Mass conduction (Fick’s first law)

Concentration diffusion is mass transport across a concentration gradient. The driving force is a concentration difference. In the one-dimensional case of diffusion in the \(x\) dimension, the rate of diffusion is:

$$\phi_m = -D A \frac{dc}{dx}$$

• \(D\) [ \(m^2/s\) ] is the diffusion coefficient of the material
• \(\frac{dc}{dx}\) is the concentration gradient

If the concentration gradient is linear, we can simplify Fick’s law: $$\phi_m = -D A \frac{\Delta c}{\Delta x}$$

The mass flux is \(\frac{\phi_m}{A}\) .

In order to calculate the temperature or concentration at a specific point:

1. Substitute \(A\) by the surface area of that geometry (a function of \(x\) ).
2. Solve the differential equation by separation of variables and integrating.
3. Use the boundary condition(s) to determine the integration constant.

The concentration profile \(c(x)\) varies for different geometries.

• Flat (e.g. membrane): \(c(x)=f(x)\)
• Sphere: \(c(x)=f(\frac{1}{x})\)
• Cylinder: \(c(x)=f(\text{ln}\,x)\)

Analytical solution for heat and mass conduction

Temperature conduction: \(\cfrac{dT}{dt}=k(T_e-T)\)

Where \(T\) the temperature of some substance, \(T_e\) the external temperature, and \(k\) the conductivity of whatever separates them ( \(k=0\) would be a perfect insulator). If \(T_e<T\) , the equation means that the rate of cooling of the substance is proportional to the temperature difference.

Concentration diffusion: \(\cfrac{dC}{dt}=k(C_e-C)\)

Where \(C\) the concentration in the solution under question, and \(C_e\) the concentration in the surrounding solution.

Standard form: \(\cfrac{dT}{dt}+kT=kT_e\) and \(\cfrac{dC}{dt}+kC=kC_e\)

Or, with \(y\) for \(T\) and \(C\) , and \(x\) for \(T_e\) and \(C_e\) :

$$\cfrac{dy}{dt}+ky=kx$$

General solution: \(y=x+(x(0)-x)e^{-kt}\)

Here, \(x\) is the steady state solution that \(y\) will approach for \(t\to\infty\) , and \((x(0)-x)e^{-kt}\) is the transient solution, which will approach 0 for \(t\to\infty\) .

Heat convection (Newton’s law of cooling)

Convective heat transport: $$\phi_q = U A \Delta T$$

• \(A\) is the cross-sectional area through which heat flows.
• \(\Delta T\) is the difference in temperature between the volume and the environment.
• \(U\) is the overall heat transfer coefficient ( \(\frac{1}{U}\) can be seen as the resistance to heat transfer). For multiple layers, the heat transfer coefficient can be calculated as:

$$\frac{1}{U} = \frac{1}{h_1} + \frac{1}{h_2} + ...$$

Note that the the heat transfer coefficient \(h\) [ \(W/m^2K\) ] relates to conductivity depending on the geometry:

• flat plate: \(h=\cfrac{k}{d}\)
• sphere: \(h = \cfrac{k}{r}\)
• cylinder: \(h = \cfrac{2k}{r_2\,\text{ln}(\frac{r_2}{r_1})}\)

Also, cf. Nusselt number.

Mass convection

Convective mass transport happens, for example, if a liquid flows over a solid surface that is dissolving in the liquid. The transport of material is specified by:

$$\phi_m = k A \Delta c$$

• \(k\) [ \(m/s\) ] is the mass transfer coefficient ( \(\frac{1}{k}\) can be seen as the resistance to mass transfer)