Convective Mass Transfer and Transport Analogies

Introduction

Convective mass transfer is the transport of a chemical species between a surface and a moving fluid, or within a flowing fluid. It occurs in drying, evaporation, absorption, distillation, membrane separation, combustion, humidification, pollution control, cooling-tower operation, and biological systems. The analysis resembles convective heat transfer, but temperature differences are replaced by concentration or mass-fraction differences.

The central engineering task is to determine the rate at which a species crosses a boundary layer. This requires understanding concentration gradients, diffusion, convection, mass-transfer coefficients, and the dimensionless groups that govern similarity between systems.

Learning Objectives

After studying this chapter, the reader should be able to:

• explain concentration boundary layers and convective mass transfer;
• distinguish molecular diffusion from bulk convection;
• use mass-transfer coefficients and appropriate driving forces;
• calculate Reynolds, Schmidt, and Sherwood numbers;
• select correlations for external and internal flows;
• apply Reynolds, Chilton–Colburn, and heat–mass transfer analogies;
• recognize assumptions, limitations, and experimental uncertainties in correlations;
• analyze a practical mass-transfer design problem.

1. Fundamentals of Convective Mass Transfer

Consider species A at a surface exposed to a fluid containing A. If the surface concentration differs from the concentration in the bulk fluid, molecular diffusion carries A through the fluid. Fluid motion simultaneously transports A by convection. Near the surface, the fluid velocity falls to zero because of the no-slip condition, and a concentration boundary layer develops.

The local molar flux of A normal to a stationary surface is commonly represented by

where N_A is the molar flux of A, mol·m⁻²·s⁻¹; k_c is a mass-transfer coefficient, m·s⁻¹; C_{A,s} is the concentration of A at the surface, mol·m⁻³; and C_{A,∞} is the bulk-fluid concentration, mol·m⁻³. The sign depends on the selected positive direction. In design calculations, the magnitude of the flux is often written as k_c|C_{A,s} − C_{A,∞}|.

This equation is a macroscopic rate expression, not a fundamental law that makes the coefficient universal. The value of k_c incorporates velocity, geometry, fluid properties, diffusion, and boundary conditions.

For a finite surface area A_s, the total molar transfer rate is

For mass concentration, the analogous expression is

where k_m has units of m·s⁻¹ and ρ_A is mass concentration in kg·m⁻³. The concentration basis must be used consistently. Coefficients based on molar concentration, mass concentration, mole fraction, and mass fraction are not numerically interchangeable without conversion.

1.1 Diffusion and convection

Molecular diffusion results from random molecular motion and concentration gradients. For dilute species A diffusing through a stationary or slowly moving fluid, Fick’s law is

where J_A is the diffusive molar flux relative to the molar-average velocity, D_AB is the binary diffusion coefficient, m²·s⁻¹, and ∇C_A is the concentration gradient. The negative sign indicates transport toward decreasing concentration.

The total flux relative to a fixed coordinate system may include bulk motion:

where v is the molar-average velocity. In convective mass-transfer correlations, the effects of diffusion and bulk fluid motion are generally combined into k_c.

1.2 Concentration boundary layer

At a solid surface, the fluid velocity is zero at the wall and increases toward the free stream. If the wall concentration differs from the free-stream concentration, the species concentration changes rapidly near the wall and gradually approaches the bulk value farther away. The concentration boundary-layer thickness is conventionally the distance at which the concentration has reached approximately 99% of its bulk-flow change.

Figure: Development of velocity and concentration profiles above a surface.

The concentration boundary layer may be thinner or thicker than the velocity boundary layer. Their relative thickness is controlled primarily by the Schmidt number.

2. Dimensionless Groups

2.1 Reynolds number

The Reynolds number compares inertial and viscous effects:

Here ρ is density, kg·m⁻³; U is characteristic velocity, m·s⁻¹; L is characteristic length, m; μ is dynamic viscosity, Pa·s; and ν = μ/ρ is kinematic viscosity, m²·s⁻¹. Low Reynolds number indicates viscous-dominated flow; high Reynolds number indicates stronger inertial effects and possible transition or turbulence.

2.2 Schmidt number

The Schmidt number compares momentum diffusivity with mass diffusivity:

For gases, Sc is often near unity. For many liquids, Sc may range from several hundred to several thousand because mass diffusivity is much smaller than momentum diffusivity.

If Sc > 1, momentum diffuses more rapidly than species, and the concentration boundary layer is generally thinner than the velocity boundary layer. If Sc < 1, species diffusion is relatively rapid.

2.3 Sherwood number

The Sherwood number is the dimensionless mass-transfer coefficient:

For internal flow, the characteristic length is commonly the hydraulic diameter D_h. Sherwood number correlations are the mass-transfer counterparts of Nusselt number correlations in heat transfer.

2.4 Schmidt–Sherwood interpretation

The relationship among k_c, Sh, D_AB, and L is

Thus a larger Sherwood number indicates more effective convective transport relative to molecular diffusion. Correlations usually express Sh as a function of Re, Sc, geometry, and sometimes axial position or surface condition.

3. External Convective Mass Transfer

External flow occurs when a fluid moves over a plate, cylinder, sphere, or other body. The boundary layer begins at the leading edge and grows downstream. Local mass transfer is usually highest near the leading edge because the concentration boundary layer is thin there. The average coefficient over a length is lower than the leading-edge value.

3.1 Laminar flow over a flat plate

For a smooth flat plate with uniform surface concentration, constant properties, negligible buoyancy, and negligible mass transfer through the external edge, a commonly used local correlation is

where Re_x = Ux/ν and x is distance from the leading edge. This relation is appropriate for a laminar boundary layer and is most reliable when concentration differences are not so large that properties change substantially.

The average value from the leading edge to length L is

The coefficient follows from k̄_c = Sh_L D_AB/L. The factor of two between local and average expressions results from integrating the local coefficient along the plate.

For a turbulent boundary layer, one frequently used engineering approximation is

for a plate whose boundary layer is turbulent over most of its length. The subtraction term accounts approximately for the laminar leading-edge region. Correlations differ in their transition assumptions; therefore the source and validity range must always be checked.

3.2 Flow over cylinders and spheres

For cylinders in crossflow, the characteristic length is the outside diameter. Correlations generally depend on Reynolds number based on diameter, Schmidt number, and sometimes the angular position around the cylinder. Separation, wake formation, and unsteady vortices can strongly affect transfer.

For spheres, a widely used form is the Frössling correlation:

The constant 2 represents the limiting contribution associated with diffusion around a stationary sphere, while the second term represents forced convection. This correlation is an approximation for an isolated sphere in a surrounding fluid and should not be applied blindly to packed beds or concentrated suspensions.

4. Internal Convective Mass Transfer

Internal flow occurs in pipes, ducts, channels, and packed conduits. The hydraulic diameter is

where A_c is cross-sectional flow area and P_w is wetted perimeter. For a circular tube, D_h equals the tube diameter.

The mean velocity is U_m = volumetric flow rate divided by cross-sectional area. Reynolds number is usually based on D_h:

4.1 Fully developed laminar flow

In fully developed laminar flow through a circular tube with constant wall concentration, negligible axial diffusion, constant properties, and no chemical reaction, the fully developed Sherwood number is

For constant wall flux, the corresponding value is

These values should not be confused: the boundary condition determines the concentration profile and therefore the coefficient.

In the developing region, a useful engineering form is

for a sufficiently long entrance region under specified assumptions. More complete correlations may include both entrance and fully developed effects.

4.2 Turbulent flow in tubes

A common turbulent correlation is the Chilton–Colburn form

This is often used for smooth tubes over an appropriate turbulent range and moderate property variation. A more general analogy-based expression is

where f is the Fanning friction factor. If a Darcy friction factor is used instead, the coefficient must be adjusted because f_D = 4f. Mixing Fanning and Darcy conventions is a common source of factor-of-four errors.

5. Transfer Coefficients and Driving Forces

A mass-transfer coefficient can be based on concentration, mole fraction, partial pressure, or another thermodynamic variable. For dilute gases, concentration and partial pressure are related by the ideal-gas equation:

where p_A is partial pressure, R is the universal gas constant, and T is absolute temperature. If temperature is constant, a pressure-based coefficient can be defined consistently.

For a two-film gas–liquid system, the overall resistance is the sum of gas-film and liquid-film resistances. Using mole fractions and a linear equilibrium relation y_A^* = m x_A, a typical overall resistance expression is

where k_y and k_x are individual gas- and liquid-side coefficients on compatible bases, K_y is the overall gas-side coefficient, and m is the slope of the equilibrium relation. This equation assumes that interfacial equilibrium is established and that resistances act in series.

The overall flux may then be written as

where y_{A,b} is the bulk-gas mole fraction and y_A^* is the gas-side equilibrium mole fraction corresponding to the bulk-liquid composition. The exact form depends on the selected reference phase and driving-force basis.

6. Transport Analogies

Transport analogies are useful because momentum, heat, and mass transfer share boundary-layer structures. They are not identities; they depend on similar boundary conditions, fluid properties, and mechanisms of transport.

6.1 Reynolds analogy

For turbulent flow with negligible pressure gradient, comparable diffusivities, and similar wall conditions, the Reynolds analogy gives approximately

where St_m is the mass-transfer Stanton number and C_f is the local skin-friction coefficient. The mass Stanton number is

when the coefficient and concentration basis are consistently defined. The local skin-friction coefficient is C_f = τ_w/(½ρU²), where τ_w is wall shear stress.

The analogy is most credible when Sc is near unity. It becomes less accurate when heat or mass diffusivity differs substantially from momentum diffusivity.

6.2 Chilton–Colburn analogy

The Chilton–Colburn analogy introduces j-factors:

For turbulent flow, it is commonly approximated by

where f is the Fanning friction factor. The corresponding heat-transfer form is j_H = St_h Pr^{2/3}. The analogy is valuable because friction data are often more readily available than mass-transfer data.

The resulting coefficient estimate is

This expression requires consistent definitions of velocity, friction factor, and mass-transfer coefficient. It is an engineering estimate, not a substitute for a correlation when reliable geometry-specific data exist.

6.3 Heat–mass transfer analogy

The mathematical similarity between energy and species equations permits replacement of thermal variables by concentration variables:

Thus a heat-transfer correlation such as Nu = f(Re, Pr) may, under comparable assumptions, suggest Sh = f(Re, Sc). The analogy fails or requires correction when radiation, strong property variation, chemical reaction, buoyancy, multicomponent diffusion, or unequal boundary conditions is important.

7. Worked Example: Evaporation from a Wetted Plate

Air flows at U = 5 m·s⁻¹ over a wetted flat plate of length L = 0.50 m. At the mean film temperature, take ν = 1.6 × 10⁻⁵ m²·s⁻¹ and D_AB = 2.5 × 10⁻⁵ m²·s⁻¹. The surface concentration of vapor is C_{A,s} = 0.80 mol·m⁻³ and the free-stream concentration is C_{A,∞} = 0.20 mol·m⁻³. Estimate the average mass-transfer coefficient and molar flux using a laminar flat-plate correlation. Assume constant properties and a laminar boundary layer.

First,

This Reynolds number is below a commonly used transition estimate of about 5 × 10⁵, so the laminar correlation is appropriate as an idealization.

The Schmidt number is

The average Sherwood number is

Using Re_L^{1/2} ≈ 395 and Sc^{1/3} ≈ 0.862,

Therefore,

The molar flux is

The result is an average value over the plate. The local flux is larger near the leading edge and smaller near the trailing edge.

8. Applied Case: Design of an Air-Drying Section

Background

An illustrative food-processing plant dries a thin wet coating on a moving web by passing warm air over its surface. Product quality depends on removing water rapidly without producing surface hardening or excessive temperature rise.

Problem

The design team initially estimates drying from air velocity alone. However, measured drying rates vary substantially with humidity, web length, and air-side turbulence. A defensible model must account for concentration driving force and boundary-layer development.

Analysis

Water vapor transfer from the wet surface is estimated from

The air-side coefficient is obtained from an external-flow Sherwood correlation based on web length. Relative humidity determines C_{A,∞}, while surface temperature determines the saturation value C_{A,s}. Since drying cools the surface, temperature and mass transfer are coupled; an isothermal calculation is therefore a first design estimate rather than a complete prediction.

Increasing air velocity raises Re and therefore Sh, reducing the concentration boundary-layer resistance. Lowering inlet humidity increases the driving force. Excessive velocity, however, may increase energy consumption, entrainment, noise, and mechanical disturbance of the coating.

Approach and solution

The design procedure is:

1. determine air properties at the expected film temperature;
2. calculate Re and Sc using the web length as characteristic length;
3. select a laminar, transitional, or turbulent correlation;
4. calculate k_c from ShD_AB/L;
5. determine vapor concentrations from temperature and humidity data;
6. calculate local or average evaporation rate;
7. validate the model with pilot measurements;
8. revise the design if surface cooling or changing moisture content is significant.

Outcome

The revised design uses moderate air velocity and controlled inlet humidity rather than maximizing velocity. Measurements are used to calibrate the effective coefficient, while energy use and product quality are included in the final decision.

Learning points

• A high velocity does not guarantee a high drying rate if the concentration driving force is small.
• Average correlations cannot describe local defects or leading-edge behavior.
• Surface temperature and humidity must be measured or modeled consistently.
• Correlations should be validated under the actual geometry and operating range.

9. Limitations, Safety, and Professional Practice

Mass-transfer correlations are empirical or semi-empirical. Their uncertainty may arise from geometry, surface roughness, property variation, flow transition, concentration polarization, and measurement error. Engineers should report the correlation, parameter range, characteristic length, friction-factor convention, and concentration basis.

For large concentration differences, the simple dilute-species equation may be inadequate. Stefan flow, in which evaporation creates a bulk motion normal to the surface, can modify the flux. Multicomponent diffusion may require Maxwell–Stefan theory rather than a single binary diffusivity. Chemical reaction can either enhance or limit transfer, and buoyancy can produce mixed convection.

Safety considerations include flammable vapors, toxic absorbates, pressure excursions, solvent exposure, and emissions. Equipment must comply with applicable process-safety, environmental, and occupational regulations. Computational models and AI-based surrogate models can assist correlation selection and parameter estimation, but they should not replace dimensional checks, experimental validation, or professional engineering judgment.

Best-Practice Design Procedure

1. Define the transported species and choose a consistent concentration basis.
2. Identify whether the flow is external or internal and whether it is laminar, transitional, or turbulent.
3. Evaluate properties at representative bulk or film conditions.
4. Calculate Re, Sc, and the relevant geometric parameters.
5. Select a correlation whose assumptions match the system.
6. Convert Sh into k_c only after confirming units and characteristic length.
7. Use the correct surface or bulk driving force, including equilibrium when phases interact.
8. Check whether buoyancy, reaction, Stefan flow, or property variation is important.
9. Compare predictions with measurements or validated design data.
10. Perform sensitivity analysis on velocity, diffusivity, humidity, temperature, and surface area.

Chapter Summary

Convective mass transfer combines molecular diffusion with fluid motion and is commonly described through a mass-transfer coefficient and concentration driving force. The concentration boundary layer controls the resistance between a surface and a moving fluid. Reynolds number characterizes flow regime, Schmidt number compares momentum and mass diffusivities, and Sherwood number expresses the dimensionless mass-transfer coefficient.

External-flow correlations describe transfer over plates, cylinders, and spheres, while internal-flow correlations apply to tubes and ducts using hydraulic diameter where necessary. Boundary conditions such as constant wall concentration and constant wall flux can produce different results. Reynolds and Chilton–Colburn analogies connect mass transfer with momentum and heat transfer, but their validity depends on comparable transport mechanisms and property ratios.

Reliable design requires consistent units, appropriate characteristic lengths, correct friction-factor definitions, awareness of correlation limits, and validation when conditions depart from ideal assumptions.

Key Terms

Concentration boundary layer: Region near a surface where species concentration changes rapidly.

Convective mass transfer: Species transport caused by combined fluid motion and molecular diffusion.

Diffusion coefficient: Measure of molecular mass diffusivity, usually denoted D_AB.

Fick’s law: Relation between diffusive flux and concentration gradient.

Mass-transfer coefficient: Proportionality factor relating flux to a concentration driving force.

Molar flux: Amount of species transferred per unit area and time.

Reynolds number: Ratio of inertial to viscous effects.

Schmidt number: Ratio of momentum diffusivity to mass diffusivity.

Sherwood number: Dimensionless mass-transfer coefficient.

External flow: Flow over an object or surface.

Internal flow: Flow confined within a duct, tube, or channel.

Hydraulic diameter: Equivalent diameter based on flow area and wetted perimeter.

Local coefficient: Transfer coefficient at a particular position.

Average coefficient: Area- or length-averaged transfer coefficient.

Driving force: Difference between surface and bulk or equilibrium concentration.

Fanning friction factor: Wall-friction parameter used in transport analogies.

Reynolds analogy: Approximate relation between momentum and mass transfer.

Chilton–Colburn analogy: j-factor relation connecting heat, mass, and momentum transfer.

Stefan flow: Bulk motion caused by evaporation or condensation through an interface.

Overall coefficient: Coefficient incorporating resistances from multiple phases or films.

Review Questions

Short Answer Questions

1. What physical processes are represented by convective mass transfer?
2. Define the Schmidt number and explain its significance.
3. Why is the local mass-transfer coefficient generally highest near a flat-plate leading edge?
4. Distinguish between a concentration-based coefficient and an overall mass-transfer coefficient.
5. State two limitations of the Chilton–Colburn analogy.

Long Answer Questions

1. Explain the formation and development of velocity and concentration boundary layers over a flat plate.
2. Derive the relationship between Sherwood number and mass-transfer coefficient, including units.
3. Compare laminar external flow, turbulent external flow, and fully developed internal flow correlations.
4. Explain how the Reynolds analogy and Chilton–Colburn analogy relate mass transfer to fluid friction.
5. Discuss the assumptions underlying the two-film model for gas–liquid mass transfer.

Analytical / Application Questions

1. Air flows over a 0.80 m plate at 4 m·s⁻¹. Given ν = 1.5 × 10⁻⁵ m²·s⁻¹ and D_AB = 2.2 × 10⁻⁵ m²·s⁻¹, determine Re_L and Sc, select an appropriate idealized flat-plate regime, and estimate the average Sherwood number.
2. A tubular absorber operates at a Reynolds number of 20,000 and Schmidt number of 600. Explain why a heat-transfer correlation based only on Prandtl number cannot be transferred directly without replacing the diffusivity ratio and checking the correlation range.
3. A drying process shows little improvement when air velocity is doubled. Use the mass-transfer-rate equation to identify at least three possible physical explanations and propose measurements to distinguish them.

Multiple Choice Questions with Answers

Question 1

Which dimensionless number compares momentum diffusivity with mass diffusivity?

A. Reynolds number

B. Schmidt number

C. Sherwood number

D. Nusselt number

Correct Answer: B. Schmidt number

Question 2

For a flat plate in laminar flow with uniform surface concentration, the average correlation is which of the following?

A. Sh_L = 0.332 Re_L^{1/2} Sc^{1/3}

B. Sh_L = 0.664 Re_L^{1/2} Sc^{1/3}

C. Sh_L = 0.023 Re_L^{0.8} Sc^{1/3}

D. Sh_L = 3.66

Correct Answer: B. Sh_L = 0.664 Re_L^{1/2} Sc^{1/3}

Question 3

If Sh_L, D_AB, and L are known, the mass-transfer coefficient is calculated from:

A. k_c = Sh_L L/D_AB

B. k_c = D_AB/(Sh_L L)

C. k_c = Sh_L D_AB/L

D. k_c = L/(Sh_L D_AB)

Correct Answer: C. k_c = Sh_L D_AB/L

Question 4

For fully developed laminar flow in a circular tube with constant wall concentration, the commonly used Sherwood number is:

A. 1.00

B. 2.00

C. 3.66

D. 23.0

Correct Answer: C. 3.66

Question 5

The Schmidt number is defined as:

A. ν/D_AB

B. D_AB/ν

C. UL/ν

D. k_cL/D_AB

Correct Answer: A. ν/D_AB

Question 6

In the flux equation N_A = k_c(C_A,s-C_A,∞), k_c has SI units of:

A. m·s⁻¹

B. m²·s⁻¹

C. mol·m⁻²·s⁻¹

D. Pa·s

Correct Answer: A. m·s⁻¹

Question 7

The hydraulic diameter of a noncircular duct is:

A. A_c/P_w

B. P_w/A_c

C. 4A_c/P_w

D. 4P_w/A_c

Correct Answer: C. 4A_c/P_w

Question 8

The Chilton–Colburn mass-transfer j-factor is commonly defined as:

A. j_D = St_m Sc^{2/3}

B. j_D = St_m/Sc^{2/3}

C. j_D = Sh/Re

D. j_D = Re/Sh

Correct Answer: A. j_D = St_m Sc^{2/3}

Question 9

For a sphere, the term 2 in the correlation Sh = 2 + 0.6Re^1/2Sc^1/3 represents approximately:

A. The turbulent contribution only

B. A diffusion-limited contribution for a stationary sphere

C. The Schmidt number

D. The pressure-drop contribution

Correct Answer: B. A diffusion-limited contribution for a stationary sphere

Question 10

If Sc is greater than one, which statement is generally correct?

A. Mass diffusivity is greater than momentum diffusivity

B. Momentum diffusivity is greater than mass diffusivity

C. The flow must be turbulent

D. The Sherwood number must equal one

Correct Answer: B. Momentum diffusivity is greater than mass diffusivity