Coupled Heat and Mass Transfer Processes

Introduction and Context

Many industrial operations involve heat and mass transfer simultaneously. Evaporation removes solvent as vapor; humidification adds vapor to a gas; dehumidification removes vapor by cooling or absorption; drying removes moisture from solids; cooling towers transfer heat from warm water to atmospheric air; and membranes selectively transport species while often carrying sensible or latent heat. These processes cannot be predicted reliably by treating heat and mass transfer as entirely separate phenomena, because temperature affects equilibrium vapor pressure and composition affects energy transport.

A central principle is interfacial equilibrium. At an interface, a volatile component tends to move from the phase in which its chemical potential is higher toward the phase in which it is lower. The transfer rate depends on the difference between the bulk state and the equilibrium state at the interface. Heat transfer is coupled to this mass flux through latent heat and through the temperature dependence of equilibrium.

This chapter develops practical models based on psychrometrics, interfacial balances, differential balances, and overall transfer coefficients. The objective is not merely to calculate transfer rates, but to select appropriate assumptions, identify controlling resistances, and judge whether a design is physically and operationally credible.

Learning Objectives

After studying this chapter, a student should be able to:

1. Explain how interfacial equilibrium couples heat and mass transfer.
2. Use humidity ratio, relative humidity, dew point, and enthalpy in gas–vapor systems.
3. Formulate differential heat and mass balances for evaporators, humidifiers, dryers, and cooling towers.
4. Distinguish individual and overall heat- and mass-transfer coefficients.
5. Apply the logarithmic mean driving-force method and effectiveness–NTU concepts where appropriate.
6. Analyze membrane transport using solution–diffusion and convective descriptions.
7. Identify assumptions, limitations, safety issues, and design trade-offs in coupled processes.

1. Thermodynamic and Transport Foundations

1.1 Interfacial equilibrium and driving force

Consider a volatile species A transferring between a liquid and a gas. If the interface is at equilibrium, the gas-side partial pressure of A is related to the liquid composition and temperature. For a dilute ideal solution, Raoult’s law gives

where pA,i is the interfacial partial pressure, xA,i is the liquid mole fraction, pA^sat is the saturation pressure of pure A at interfacial temperature Ti, and the subscript i denotes the interface. For a nonideal solution, activity replaces mole fraction:

Here γA is the activity coefficient. These relations apply when the interface is locally at thermodynamic equilibrium and the gas behaves approximately ideally. They are not valid for strongly nonequilibrium interfaces or when chemical reaction changes the interfacial composition.

A gas-side mass-transfer expression is commonly written as

where NA is molar flux of A, ky is a gas-phase mass-transfer coefficient based on mole fraction, yA,i is the interfacial mole fraction, and yA,b is the bulk-gas mole fraction. The sign convention is positive from the interface to the bulk gas. An equivalent partial-pressure form is N_A = k_p (p_A,i - p_A,b), with kp having compatible units.

The liquid-side expression is

where kx is a liquid-side coefficient. The interfacial composition is usually unknown, so the two films must be combined through an overall coefficient.

1.2 Overall mass-transfer coefficients

For a dilute system with a linear equilibrium relation y_A^* = m x_A, resistances in series give

and

where Kx is based on liquid mole-fraction driving force, m is the slope of the equilibrium line, and xA* is the liquid composition in equilibrium with the bulk gas. The analogous gas-based form is

with N_A = K_y (y_A^* - y_A,b). These formulas assume steady one-dimensional transfer, negligible interfacial accumulation, a locally linear equilibrium relation, and compatible definitions of coefficients and area.

For equipment of area A, the molar rate is often written dot n_A = K_x a V Δx_lm, where a is interfacial area per equipment volume, V is packed volume, and Δxlm is a logarithmic mean composition difference. For two terminal driving forces Δx1 and Δx2,

provided both driving forces have the same sign and are nonzero. If they are nearly equal, Δxlm is approximately their arithmetic mean.

1.3 Coupling through energy balance

When A evaporates, the departing vapor carries latent energy. For a control volume at steady state,

where Q̇ is heat supplied to the control volume, Ẇ is shaft work, ṁ is mass flow rate, and h is specific enthalpy. In many evaporators, shaft work and kinetic-energy changes are negligible, so heat supplied is approximately the sensible-energy change plus the latent energy of vaporization:

The first term represents vaporization of mass ṁv and the second represents sensible heating of a stream with mass flow ṁs and specific heat cp. The exact balance must include all inlet and outlet streams and any heat loss.

2. Psychrometrics of Moist-Air Systems

Psychrometrics describes mixtures of dry air and water vapor. It is essential for humidifiers, cooling towers, air conditioners, and dryers.

Let p be total pressure, pw be water-vapor partial pressure, and psat(T) be saturation pressure at temperature T. The humidity ratio is

where ω is kilograms of water vapor per kilogram of dry air. The factor 0.622 is the ratio of the molecular mass of water to that of dry air. The formula assumes ideal-gas behavior.

Relative humidity is

where φ ranges from 0 to 1, or 0–100 percent. Combining the relations gives

Moist-air enthalpy per kilogram of dry air is commonly approximated by

where h is in kJ/kg dry air and dry-bulb temperature Tdb is in °C. The constants are approximate over ordinary HVAC conditions: 1.005 kJ/(kg·K) is the specific heat of dry air, 1.88 kJ/(kg·K) is the vapor specific heat, and 2500 kJ/kg approximates the latent enthalpy at 0 °C. For high temperatures, high pressures, or high accuracy, property tables or a validated equation of state should be used.

The dew-point temperature is the temperature at which the current vapor partial pressure equals saturation pressure. If air is cooled below its dew point, condensation begins. This explains why dehumidification by cooling is inherently a simultaneous sensible-cooling and latent-removal process.

3. Evaporation and Humidification

3.1 Evaporation from a liquid surface

For a liquid evaporating into a gas, the evaporation rate may be estimated from

where km is a concentration-based mass-transfer coefficient, A is area, CA is molar concentration, and MA is molecular mass. The interface concentration is determined from equilibrium at the interface temperature. The interface temperature itself is found from an energy balance, not assumed automatically equal to the bulk-liquid temperature.

If external heat transfer supplies the latent heat and sensible heating is negligible,

where λA is latent heat of vaporization. If evaporation is supplied only by cooling of the liquid, the liquid temperature falls until the sensible-energy loss balances latent demand. This is the basis of evaporative cooling.

3.2 Humidification

In direct-contact humidification, water contacts unsaturated air. Water evaporates, increasing ω, while the air temperature may decrease because evaporation consumes latent heat. For an adiabatic humidifier with negligible heat loss and negligible liquid sensible change, the air process is approximately constant enthalpy:

This is an approximation, not a universal law. It is most reasonable when the liquid-water temperature approaches the adiabatic-saturation temperature and external heat transfer is small.

A differential balance on dry air gives

where ṁda is dry-air flow, a is interfacial area per volume, and dV is an incremental packed volume. The associated energy balance is

where q̇′ is heat transferred to the differential volume and hl is liquid-water enthalpy. The form must be adapted to the chosen enthalpy reference and sign convention.

4. Dehumidification and Condensation

When moist air contacts a surface colder than its dew point, water condenses. The total cooling rate is

and the condensate rate is

These equations are based on a steady control volume with no leakage and consistent dry-air flow. The sensible cooling load can be separated approximately from the latent load:

The remaining enthalpy change is associated mainly with sensible cooling, although exact partitioning depends on property references and condensate temperature.

A coil surface may not be at one uniform temperature. The apparatus is often modeled by a contact factor or bypass factor. If the apparatus dew point is TADP and the entering and leaving dry-bulb temperatures are Tin and Tout,

This relation is an empirical coil model and should not be used outside the operating range for which the apparatus dew point and surface assumptions are meaningful.

5. Drying of Wet Solids

Drying combines external gas-side transfer with internal moisture movement through a porous solid. The drying-rate curve commonly contains a warm-up period, a constant-rate period, and a falling-rate period.

During the constant-rate period, the surface remains approximately wet and the rate is controlled mainly by external transfer. A simplified expression is

where Y is a humidity ratio or another consistent gas composition coordinate. The surface condition is near saturation at the surface temperature.

During the falling-rate period, the wet surface recedes or internal diffusion becomes controlling. A simple lumped model is

where X is moisture content on a chosen dry-solid basis, Xe is equilibrium moisture content, and kd is an effective drying constant. Integration gives

This model assumes kd and Xe are constant over the interval. Real materials may require a moisture-dependent diffusivity, shrinkage model, or coupled heat-and-moisture partial differential equations.

Drying design must protect product quality. Excessive temperature may cause case hardening, oxidation, cracking, nutrient loss, or undesirable color changes. The optimum gas temperature is therefore a constrained economic and quality decision rather than simply the maximum temperature achievable.

6. Cooling Towers

A cooling tower rejects heat from warm circulating water to air through partial evaporation. The transferred heat includes sensible heating of air and latent energy of evaporated water.

For a counterflow tower, the Merkel method combines heat and mass transfer by assuming the gas-side resistance dominates and the interface is saturated at the local water temperature. A commonly used design relation is

where K a V/L is the volumetric mass-transfer characteristic divided by water mass flux, cp,w is water specific heat, h* is saturated-air enthalpy at local water temperature, and ha is bulk-air enthalpy. The integral is evaluated along the tower operating line.

The water range is

and the approach is

where Twb,in is entering-air wet-bulb temperature. A smaller approach usually requires greater tower volume, airflow, or power.

Water consumption includes evaporation, drift, and blowdown. Drift is entrained liquid carried out with exhaust air; blowdown is intentional water discharge used to control dissolved-solids concentration. Water-treatment programs must control scaling, corrosion, and biological growth, including Legionella risk in poorly maintained systems.

7. Membrane Heat and Mass Transport

Membranes provide selective barriers between two phases. In the solution–diffusion model, a species dissolves into the membrane, diffuses through it, and desorbs on the permeate side. For a dense membrane,

where JA is molar flux, PA is permeability, l is membrane thickness, and pA,feed and pA,perm are partial pressures. Permeability incorporates solubility and diffusivity. This expression assumes steady state, a homogeneous membrane, negligible defects, and a driving force represented adequately by partial pressure.

For porous membranes, transport may be Knudsen, molecular, viscous, or transitional. The controlling mechanism depends on pore size, mean free path, pressure, and membrane wetting. In membrane distillation, a temperature difference creates a vapor-pressure difference across a hydrophobic membrane. Heat transfer and mass transfer are then strongly coupled: conductive heat leakage reduces the useful vapor-pressure driving force, while evaporation and condensation consume latent heat.

Membrane systems require attention to concentration polarization, fouling, plasticization, pinholes, and module pressure drop. A high permeability is not sufficient if selectivity, mechanical strength, lifetime, and cleanability are inadequate.

8. Simultaneous Heat–Mass Exchange: General Design Framework

A robust design procedure is:

1. Define the system boundary and identify all phases and streams.
2. Select independent composition variables, such as humidity ratio, mole fraction, or mass fraction.
3. Establish equilibrium relations at the interface.
4. Estimate individual heat- and mass-transfer coefficients using correlations appropriate to geometry and flow regime.
5. Combine resistances into overall coefficients only when the equilibrium relation and coefficient basis are consistent.
6. Write total and differential material and energy balances.
7. Integrate along the equipment or use a stagewise model.
8. Check pressure drop, flooding, entrainment, fouling, product constraints, and safety.
9. Perform sensitivity analysis on ambient conditions, flow rates, transfer coefficients, and equilibrium data.

Dimensionless groups help select correlations. The Reynolds number compares inertia with viscosity, the Schmidt number compares momentum and mass diffusivity, and the Sherwood number characterizes convective mass transfer:

Here ρ is density, u velocity, L characteristic length, μ dynamic viscosity, DAB binary diffusivity, and km a mass-transfer coefficient. Correlations such as Sh = C Re^m Scⁿ are empirical and must be used only within their stated geometry and parameter ranges.

Heat and mass transfer coefficients are also related by analogies, but analogy-based estimates can fail when natural convection, phase change, radiation, nonconstant properties, or strong blowing effects are present. Computational fluid dynamics may resolve local fields, but its predictions remain dependent on turbulence, multiphase, and wall-condition models.

Worked Example: Humidification of Air

Air enters an adiabatic humidifier at 30 °C dry-bulb temperature, 20 °C wet-bulb temperature, and standard pressure of 101.3 kPa. It leaves approximately saturated. Estimate the outlet temperature and humidity ratio using the constant-enthalpy approximation.

At 30 °C, saturation pressure is approximately 4.24 kPa. At 20 °C, it is approximately 2.34 kPa. The entering humidity ratio is estimated from the wet-bulb condition. A psychrometric approximation gives an entering humidity ratio of about 0.011 kg water/kg dry air and enthalpy

At saturation, the outlet satisfies

Trial calculation gives T_out ≈ 23.5 °C. The saturation pressure near 23.5 °C is approximately 2.90 kPa, so

Thus the illustrative humidifier increases humidity ratio from approximately 0.011 to 0.018 kg/kg dry air while reducing dry-bulb temperature from 30 to about 23.5 °C. The result is approximate because wet-bulb data, saturation-pressure values, and the constant-enthalpy assumption are approximate. A real design must include water temperature, packing performance, pressure drop, carryover, and sanitation.

Applied Case: Industrial Cooling Tower Upgrade

Background

An illustrative food-processing plant recirculates water from a condenser at 38 °C. The existing counterflow tower returns water at 29 °C when entering air has a 24 °C wet-bulb temperature. Summer production increases the heat load by 20 percent, but the available footprint cannot be expanded.

Problem

The plant must maintain the condenser inlet temperature without excessive fan power or water consumption. Increasing water flow alone would reduce the water range but might exceed pump capacity and would not necessarily improve the approach.

Analysis and solution

Engineers first verify the water and air enthalpy balances, then inspect fill condition, nozzle distribution, fan performance, and drift eliminators. The tower characteristic is recalculated from measured inlet and outlet temperatures and airflow. The analysis shows that fouled fill has reduced effective interfacial area. Cleaning and replacing damaged fill increase KaV without increasing footprint. Variable-speed fan control is added so airflow rises only during high wet-bulb conditions. A conductivity controller optimizes blowdown, while online sensors monitor basin temperature, conductivity, vibration, and disinfectant concentration.

Outcome

The upgraded tower achieves the required heat rejection at design summer conditions with lower water loss than a strategy based solely on increased blowdown and continuous maximum fan speed. The principal gain comes from restoring transfer area and controlling operation according to ambient wet-bulb temperature.

Learning points

The case demonstrates that coupled-process design is not just a heat-load calculation. Equilibrium conditions, transfer area, airflow, water chemistry, control strategy, maintenance, and public-health requirements jointly determine performance.

Limitations, Safety, and Professional Practice

Equilibrium data are often the largest source of uncertainty. Activity coefficients, saturation pressures, sorption isotherms, and membrane permeabilities may vary substantially with temperature and composition. Coefficients measured in laboratory equipment may not scale directly because industrial devices have maldistribution, channeling, surface renewal, and fouling.

Designers should maintain consistent bases: per kilogram dry air versus per kilogram moist air, molar versus mass fractions, and local versus overall area. Energy balances must use one enthalpy reference throughout. Negative or impossible humidity ratios, outlet temperatures below physical limits, and unreasonably high transfer coefficients are warning signs of inconsistent calculations.

Safety concerns include flammable vapors, toxic solvents, pressurized steam, hot surfaces, biological contamination in cooling systems, membrane failure, and chemical exposure during cleaning. Hazard analysis, relief protection, ventilation, emissions control, lockout procedures, and applicable environmental and occupational regulations are essential. Water scarcity and greenhouse-gas emissions also influence technology selection. Heat recovery, closed-loop operation, efficient fans and pumps, low-drift towers, and solvent recovery can improve sustainability.

Modern systems increasingly use distributed sensors, digital twins, model-predictive control, and machine-learning fault detection. Artificial intelligence is useful for detecting fouling, predicting approach temperature, and optimizing fan speed, but it does not replace thermodynamic constraints, calibrated sensors, or engineering review. Models should be validated against independent operating data and monitored for drift.

Chapter Summary

Coupled heat and mass transfer occurs whenever phase change, species diffusion, or selective transport alters both composition and energy. Interfacial equilibrium determines the local composition or vapor pressure, while film resistances determine how rapidly the system approaches equilibrium. Overall coefficients combine resistances only when their bases and equilibrium relations are consistent.

Psychrometric variables—humidity ratio, relative humidity, dew point, and moist-air enthalpy—are central to humidification, dehumidification, drying, and cooling towers. Evaporation consumes latent heat; condensation releases it. Drying changes from external-transfer control to internal-moisture control as the surface dries. Cooling-tower performance depends strongly on wet-bulb temperature, approach, transfer area, airflow, water chemistry, and maintenance. Membrane transport depends on permeability, selectivity, thickness, concentration polarization, fouling, and heat leakage.

Reliable design requires material and energy balances, validated property data, appropriate correlations, sensitivity checks, and attention to safety, quality, environmental impact, and control. The most effective engineering solution balances thermodynamic performance with equipment cost, reliability, maintainability, and resource use.

Key Terms

Interfacial equilibrium: Condition in which the chemical potential of a species is balanced across an interface.

Humidity ratio: Mass of water vapor per unit mass of dry air.

Relative humidity: Ratio of actual vapor partial pressure to saturation vapor pressure.

Dew point: Temperature at which condensation begins during cooling at constant vapor content.

Latent heat: Energy associated with a phase change at approximately constant temperature.

Film resistance: Resistance to transfer within a fluid boundary layer adjacent to an interface.

Overall transfer coefficient: Coefficient combining individual transfer resistances.

Logarithmic mean driving force: Integrated average driving force for a varying-gradient exchanger or contactor.

Humidification: Addition of vapor to a gas stream.

Dehumidification: Removal of vapor from a gas stream.

Drying: Removal of moisture from a solid or liquid product.

Constant-rate period: Drying period in which a wet surface permits an approximately steady rate.

Falling-rate period: Drying period in which internal resistance causes the rate to decline.

Cooling tower approach: Difference between cooled-water temperature and entering-air wet-bulb temperature.

Merkel method: Cooling-tower method based on enthalpy driving force and saturated interfacial air.

Permeability: Membrane transport property combining solubility and diffusivity.

Concentration polarization: Boundary-layer accumulation or depletion that reduces the effective membrane driving force.

Sensible heat: Energy associated primarily with temperature change.

Blowdown: Intentional discharge used to control dissolved solids in recirculating water.

Psychrometrics: Study of thermodynamic properties of moist-air mixtures.

Review Questions

Short Answer Questions

1. Why does evaporation couple mass transfer to heat transfer?
2. Define humidity ratio and relative humidity.
3. What physical meaning is assigned to an overall mass-transfer coefficient?
4. Distinguish the constant-rate and falling-rate periods of drying.
5. Why is cooling-tower approach an important performance measure?

Long Answer Questions

1. Derive the resistance-in-series expression for an overall mass-transfer coefficient for a linear equilibrium relation.
2. Explain the psychrometric changes that occur when moist air is cooled below its dew point.
3. Describe the Merkel method for counterflow cooling towers and state its principal assumptions.
4. Compare solution–diffusion transport in dense membranes with transport in porous membranes.
5. Explain how material balances, energy balances, equilibrium data, and transfer coefficients are integrated in the design of a humidifier.

Analytical / Application Questions

1. Moist air enters a dehumidifying coil at a known humidity ratio and leaves at a lower humidity ratio. Formulate the condensate-rate and total-cooling-load equations on a dry-air basis, identifying all required measurements.
2. A dryer shows a rapid decline in drying rate after the surface becomes visibly dry. Decide whether external or internal resistance is likely controlling, and propose two measurements that could distinguish the mechanisms.
3. A cooling tower fails to achieve its specified approach after several months of operation. Develop a diagnostic sequence involving water distribution, airflow, fill condition, wet-bulb temperature, water chemistry, and instrumentation.

Multiple Choice Questions with Answers

Question 1

Which quantity is defined as the mass of water vapor per unit mass of dry air?

A. Relative humidity

B. Humidity ratio

C. Dew-point temperature

D. Wet-bulb depression

Correct Answer: B. Humidity ratio

Question 2

For a linear equilibrium relation y_A^* = m x_A, which expression represents the liquid-basis overall mass-transfer resistance?

A. 1/K_x = 1/k_x + m/k_y

B. 1/K_x = 1/k_x + k_y/m

C. 1/K_x = 1/k_y + m/k_x

D. K_x = k_x + m k_y

Correct Answer: A. 1/K_x = 1/k_x + m/k_y

Question 3

When moist air is cooled below its dew point, what occurs?

A. The humidity ratio increases without heat transfer

B. Water vapor begins to condense

C. Relative humidity becomes zero

D. The dry-air mass flow disappears

Correct Answer: B. Water vapor begins to condense

Question 4

In an approximately adiabatic humidifier, the moist-air process is often modeled as constant:

A. Pressure drop

B. Humidity ratio

C. Enthalpy

D. Relative humidity

Correct Answer: C. Enthalpy

Question 5

The cooling-tower approach is defined as:

A. Entering-water temperature minus leaving-water temperature

B. Leaving-water temperature minus entering-air wet-bulb temperature

C. Entering-air dry-bulb temperature minus wet-bulb temperature

D. Leaving-air temperature minus entering-water temperature

Correct Answer: B. Leaving-water temperature minus entering-air wet-bulb temperature

Question 6

A membrane has permeability PA, thickness l, and a partial-pressure difference across it. Under the solution–diffusion model, flux is proportional to:

A. P_A l (p_A,feed-p_A,perm)

B. l/P_A (p_A,feed-p_A,perm)

C. P_A/l (p_A,feed-p_A,perm)

D. P_A/l (p_A,feed+p_A,perm)

Correct Answer: C. P_A/l (p_A,feed-p_A,perm)

Question 7

Which drying period is most associated with increasing internal resistance as moisture moves through a partially dry solid?

A. Warm-up period

B. Constant-rate period

C. Falling-rate period

D. Condensation period

Correct Answer: C. Falling-rate period

Question 8

If 10 kg/s of dry air changes humidity ratio from 0.010 to 0.016 kg/kg dry air, the water-vapor transfer rate is:

A. 0.006 kg/s

B. 0.060 kg/s

C. 0.160 kg/s

D. 1.67 kg/s

Correct Answer: B. 0.060 kg/s

Question 9

Which term represents intentional discharge from a recirculating cooling-water system to control dissolved solids?

A. Drift

B. Blowdown

C. Carryover vapor

D. Condensate return

Correct Answer: B. Blowdown

Question 10

In the approximate moist-air enthalpy relation, the enthalpy is normally expressed per unit mass of:

A. Moist air

B. Water vapor only

C. Dry air

D. Liquid water only

Correct Answer: C. Dry air