Transport Phenomena: Heat and Mass Transfer

Transport phenomena describes how heat and chemical species move through and between materials. In engineering practice, heat transfer and mass transfer are often treated together because they share the same underlying ideas: a driving force (a temperature or concentration difference), a resistance (within a fluid, solid, or boundary layer), and a resulting flux. Understanding conduction, convection, and diffusion is essential for designing heat exchangers, reactors, distillation columns, dryers, membranes, and many other unit operations.

This article lays out the core mechanisms, the most-used engineering correlations, and the practical analogies that connect heat and mass transfer.

The common framework: flux, driving force, resistance

Many transport problems can be expressed in the same pattern:

• Flux: how much heat or mass crosses a unit area per unit time.
• Driving force: a gradient in temperature or concentration, or a difference between bulk and surface values.
• Transport coefficient: a material property or an effective coefficient that bundles complex physics into a usable form.

For heat transfer by convection, the engineering form is Newton’s law of cooling:

$$q^{\prime \prime }=h(T_s-T_{\infty })$$

For mass transfer by convection (species transfer), an analogous form is used:

$$N_A^{\prime \prime }=k_c(C_{A,s}-C_{A,\infty })$$

Here, $h$ is the heat transfer coefficient and $k_c$ is the mass transfer coefficient (concentration-based). Similar forms exist using partial pressure or mole fraction as the driving force.

Conduction: heat transfer through solids and stagnant fluids

Conduction is transport driven by a temperature gradient within a material. Fourier’s law for one-dimensional conduction is:

$$q^{\prime \prime }=-k\frac{dT}{dx}$$

• $k$ is thermal conductivity, a property of the material.
• The negative sign reflects flow from high to low temperature.

In practice, conduction problems are often turned into thermal resistance networks. For a plane wall of thickness $L$:

$$q^{\prime \prime }=\frac{k(T_1-T_2)}{L}\Rightarrow R_{cond}=\frac{L}{kA}$$

This resistance viewpoint becomes valuable when conduction is combined with convection on one or both sides, such as insulation on a hot pipe or a wall separating two fluids.

What conduction looks like in real equipment

• In a heat exchanger tube, conduction through the tube wall can be minor compared with convection, but fouling layers can create a large additional resistance.
• In refrigeration or furnace linings, conduction through insulation dominates the heat loss design.

Convection: transport between a surface and a moving fluid

Convection is the combined effect of conduction within the fluid boundary layer and bulk fluid motion. Engineers separate convection into:

• Forced convection: flow driven by a fan, pump, or external pressure gradient.
• Natural (free) convection: flow driven by buoyancy due to temperature or concentration differences.

The key challenge is that $h$ (and similarly $k_c$ for mass) is not a pure material property. It depends on flow regime, geometry, and fluid properties. That is why engineering design relies heavily on dimensionless correlations.

Dimensionless groups used in correlations

For heat transfer:

• Reynolds number: $Re=\frac{\rho uL}{\mu }$ (flow regime and inertia vs viscosity)
• Prandtl number: $Pr=\frac{\mu c_p}{k}$ (momentum vs thermal diffusivity)
• Nusselt number: $Nu=\frac{hL}{k}$ (dimensionless heat transfer coefficient)

Typical forced convection correlations take the form:

$$Nu=CR{e}^{m}P{r}^n$$

The constants and exponents depend on geometry (flat plate, pipe, cylinder) and whether the boundary layer is laminar or turbulent.

Convection in design terms

• Increasing velocity usually increases $h$ (higher turbulence, thinner boundary layers), but at the cost of higher pressure drop and pumping power.
• Fluids with higher thermal conductivity tend to have higher $h$ for the same flow conditions, which is why gases often have lower convective performance than liquids.

Diffusion: mass transfer driven by concentration gradients

Diffusion is the molecular movement of species due to concentration gradients. Fick’s first law for one-dimensional diffusion is:

$$J_A=-D_{AB}\frac{d{C}_A}{dx}$$

• $D_{AB}$ is the binary diffusivity of species $A$ in $B$.
• $J_A$ is the diffusive flux relative to the molar-average motion.

As with conduction, diffusion can be treated with a resistance view for simple cases. For a stagnant film of thickness $\delta$ with a linear concentration profile:

$$N_A^{\prime \prime }\approx \frac{D_{AB}}{\delta }(C_{A,s}-C_{A,\infty })$$

In real flows, however, convection and diffusion occur simultaneously, and the effective mass transfer rate depends strongly on hydrodynamics and turbulence.

Mass transfer coefficients and the idea of a boundary layer

In many operations, the limiting resistance is not in the bulk phases but in thin boundary layers near an interface. This is captured by the film model, where the mass transfer coefficient $k$ represents transport through an imagined film adjacent to the surface.

Common coefficient forms include:

• Concentration-based: $N_A^{\prime \prime }=k_c(C_{A,s}-C_{A,\infty })$
• Partial-pressure-based (gas phase): $N_A^{\prime \prime }=k_p(p_{A,s}-p_{A,\infty })$

Because interfacial concentrations are often unknown, designers frequently use equilibrium relationships at the interface (Henry’s law, Raoult’s law, or partition coefficients) to relate gas-side and liquid-side driving forces.

Dimensionless groups for mass transfer

• Schmidt number: $Sc=\frac{\mu }{\rho D_{AB}}$ (momentum vs mass diffusivity)
• Sherwood number: $Sh=\frac{k_{c}L}{D_{AB}}$ (dimensionless mass transfer coefficient)

Mass transfer correlations often mirror heat transfer correlations:

$$Sh=CR{e}^{m}S{c}^n$$

Analogies between heat and mass transfer

Because heat and mass transfer share similar boundary-layer physics, engineers use analogies to estimate one coefficient from the other when direct data is limited.

Reynolds analogy (conceptual link)

In turbulent flow, enhanced mixing increases transport of momentum, heat, and species. This motivates relationships between friction factors and transfer coefficients, though simple forms work best under specific conditions (often turbulent flow with similar diffusivities).

Chilton–Colburn analogy (widely used)

A practical engineering form uses the __MATH_INLINE_30__-factors:

• Heat transfer: $j_H=\frac{StP{r}^{2/3}}{}$
• Mass transfer: $j_D=\frac{S{t}_{m}S{c}^{2/3}}{}$

Where:

• $St=\frac{h}{\rho u{c}_p}$ is the Stanton number for heat
• $S{t}_m=\frac{k_c}{u}$ (with consistent definitions) is a Stanton-type number for mass

The key result in many turbulent flows is:

$$j_H\approx j_D\approx \frac{f}{2}$$

Here $f$ is a friction factor tied to momentum transfer. The strength of the Chilton–Colburn analogy is that it links three measurements or models: pressure drop (momentum), heat transfer, and mass transfer. In design work, this can reduce experimental burden and provide a consistent set of coefficients.

Practical examples that show the connections

Evaporation and drying

Drying a wet surface involves both heat transfer (to supply latent heat) and mass transfer (to remove vapor). If air velocity increases, both $h$ and $k_c$ typically rise, accelerating drying. The controlling step may shift depending on humidity, temperature, and airflow.

Gas absorption

In a packed column removing a contaminant gas into a solvent, the overall rate depends on gas-side and liquid-side mass transfer coefficients and the equilibrium relationship. Turbulence, wetting, and diffusivity all matter, which is why $Sh$ correlations and $j$-factor approaches are common.

Heat exchanger fouling and scaling

A layer of scale adds conduction resistance and can also alter the near-wall flow, affecting convection. In systems where dissolved species precipitate, mass transfer and reaction at the wall combine to set deposition rates. The transport framework helps separate whether the limitation is diffusion to the wall or surface kinetics.

What engineers look for when applying correlations

1. Flow regime and geometry match: A pipe correlation is not automatically valid for a flat plate or a packed bed.
2. Property evaluation: Viscosity, thermal conductivity, and diffusivity should be evaluated at appropriate film or bulk conditions.
3. Dominant resistance identification: Often the smallest coefficient dominates performance, so improving the wrong side yields little benefit.
4. Consistency across heat and mass transfer: Using analogies can help, but only when assumptions (turbulence, similar boundary conditions, negligible property variation) are reasonable.

Closing perspective

Heat and mass transfer are not separate silos. Conduction mirrors diffusion, convection coefficients behave similarly for heat and species, and the same boundary-layer ideas underpin both. The payoff of learning transport phenomena is