9.3 Analogy Between Heat and Mass Transfer
4 min read•august 13, 2024
Heat and mass transfer share striking similarities in their underlying principles and equations. This connection allows us to apply heat transfer knowledge to mass transfer problems, simplifying complex calculations and analysis.
The analogy between heat and mass transfer extends to key concepts like laws, convective coefficients, and dimensionless numbers. By swapping variables, we can adapt heat transfer correlations for mass transfer applications across various scenarios.
Heat vs Mass Transfer Analogies
Similarities in Transport Phenomena
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- Heat transfer and mass transfer are analogous transport phenomena
- Heat transfer involves the transport of energy
- Mass transfer involves the transport of mass or species
- The driving force for heat transfer is the temperature gradient
- The driving force for mass transfer is the concentration gradient
Analogous Laws and Coefficients
- of diffusion for mass transfer is analogous to of heat
- Mass flux is proportional to the concentration gradient
- Heat flux is proportional to the temperature gradient
- The convective is analogous to the convective
- Relates the mass flux to the concentration difference between the surface and the bulk fluid
- Relates the heat flux to the temperature difference between the surface and the bulk fluid
Dimensionless Numbers
- The (Sh) in mass transfer is analogous to the (Nu) in heat transfer
- Represents the ratio of convective mass transfer to diffusive mass transfer
- Represents the ratio of convective heat transfer to conductive heat transfer
- The Schmidt number (Sc) in mass transfer is analogous to the (Pr) in heat transfer
- Represents the ratio of momentum diffusivity to mass diffusivity
- Represents the ratio of momentum diffusivity to thermal diffusivity
Applying Heat Transfer Correlations for Mass Transfer
Variable Substitution
- The heat-mass transfer analogy allows the use of well-established heat transfer correlations to solve mass transfer problems by replacing the relevant variables and dimensionless numbers
- Replace the Nusselt number (Nu) with the Sherwood number (Sh)
- Replace the Prandtl number (Pr) with the Schmidt number (Sc)
- Replace the thermal diffusivity (α) with the mass diffusivity (D)
Adapting Correlations
- The Dittus-Boelter correlation for turbulent flow in tubes can be adapted for mass transfer
- Original correlation:
- Adapted correlation:
- The Chilton-Colburn analogy relates the heat transfer Stanton number (St) to the mass transfer Stanton number (St_m)
- Allows the use of heat transfer correlations for mass transfer problems
Applicability to Various Geometries and Flow Conditions
- The heat-mass transfer analogy can be applied to various geometries and flow conditions
- Flow over flat plates (boundary layer flow)
- Flow in packed beds (porous media)
- Natural (buoyancy-driven flow)
- Use the appropriate heat transfer correlations and replace the relevant variables and dimensionless numbers
Limitations of the Heat-Mass Transfer Analogy
Assumptions on Dimensionless Numbers
- The heat-mass transfer analogy assumes that the Prandtl number (Pr) and Schmidt number (Sc) are of similar magnitude
- Typically within the range of 0.6 to 60
- The analogy may not be accurate for fluids with very high or low Pr or Sc numbers (e.g., liquid metals, highly viscous fluids)
Boundary Layer Assumptions
- The analogy assumes that the velocity and temperature/concentration boundary layers are similar and fully developed
- This assumption may not hold for complex geometries or flow conditions with significant boundary layer interactions (e.g., flow separation, recirculation zones)
- The analogy may be less accurate for developing boundary layers or entrance regions
Constant Fluid Properties
- The analogy assumes that the fluid properties are constant and not affected by the heat or mass transfer process
- This assumption may not be valid for systems with large temperature or concentration gradients
- Fluid properties can vary significantly with temperature or concentration (e.g., viscosity, density, diffusivity)
Neglecting Chemical Reactions and Phase Changes
- The analogy does not account for the effects of chemical reactions or phase changes
- Chemical reactions can significantly influence the mass transfer process (e.g., reaction kinetics, catalysis)
- Phase changes (e.g., , ) can alter the mass transfer rates and mechanisms
Diffusion-Convection Dominance
- The analogy assumes that the mass transfer process is governed by diffusion and convection
- Neglects the effects of other transport mechanisms (e.g., migration in electrolyte solutions, surface reactions)
- The analogy may be less accurate when these additional transport mechanisms are significant
Coupling Effects
- The accuracy of the heat-mass transfer analogy may be limited in cases where the Soret effect (thermal diffusion) or Dufour effect (diffusion thermo) are significant
- The Soret effect is the mass transfer caused by a temperature gradient
- The Dufour effect is the heat transfer caused by a concentration gradient
- These phenomena couple heat and mass transfer processes and are not accounted for in the standard heat-mass transfer analogy
Key Terms to Review (19)
Analytical solutions: Analytical solutions refer to exact mathematical expressions that solve a given problem in heat and mass transfer, providing precise values for temperature or concentration distributions under specific conditions. These solutions are typically derived using mathematical techniques and assumptions, allowing for a clear understanding of the physical phenomena at play. They are crucial for validating numerical methods and for gaining insights into the behavior of systems during transient diffusion and heat transfer processes.
Condensation: Condensation is the process by which vapor transforms into liquid, usually occurring when the vapor cools and loses energy. This process is critical in heat and mass transfer, as it involves the release of latent heat and can significantly influence system performance, especially in scenarios where heat and mass transfer occur simultaneously, the analogy between heat and mass transfer is established, and phase changes are involved.
Conduction: Conduction is the process of heat transfer through a material without any movement of the material itself, occurring when two bodies at different temperatures are in direct contact. It is essential in understanding how thermal energy moves through solids, affecting various engineering and natural processes, such as thermal resistance and the efficiency of heat exchangers.
Convection: Convection is the process of heat transfer through the movement of fluids (liquids or gases) caused by temperature differences within the fluid. This movement helps distribute thermal energy, making convection essential in various applications such as heat exchangers and natural heating systems.
Diffusion: Diffusion is the process by which molecules move from an area of higher concentration to an area of lower concentration, driven by the concentration gradient. This natural tendency of substances to spread out plays a vital role in various processes, including the transport of mass in fluids, the formation of boundary layers, and even in biological systems where nutrient and gas exchange occurs.
Evaporation: Evaporation is the process by which liquid molecules gain enough energy to transition into the gas phase. This phenomenon is crucial in many natural and engineered processes, as it involves heat transfer from a substance to its surroundings, often influencing temperature and humidity levels in various environments.
Fick's Law: Fick's Law describes the diffusion process of mass transfer, stating that the flux of a species is proportional to the concentration gradient. This principle helps explain how substances move from areas of higher concentration to lower concentration, which is crucial in various phenomena including heat and mass transfer interactions.
Fourier's Law: Fourier's Law states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the area through which heat is flowing. This principle is fundamental in understanding conduction as it quantitatively describes how heat moves through different materials and forms the basis for thermal analysis in various engineering applications.
Heat and mass balance equations: Heat and mass balance equations are mathematical expressions used to quantify the transfer of heat and mass in a system, ensuring that the energy and mass entering a system equals the energy and mass leaving it. These equations are fundamental in understanding how thermal and concentration changes occur within various processes, linking the concepts of heat transfer and mass transfer through analogous principles.
Heat Transfer Coefficient: The heat transfer coefficient is a measure of the heat transfer rate per unit area per unit temperature difference between a solid surface and a fluid in contact with it. It plays a crucial role in determining how effectively heat is transferred during conduction, convection, and radiation processes. Understanding this coefficient is vital for designing systems that involve heat exchangers and for applying methods that estimate temperature differences across boundaries.
Mass boundary layer: The mass boundary layer is a region adjacent to a surface where the concentration of a mass species, such as a gas or vapor, changes due to diffusion and convection processes. In this layer, the effects of mass transfer are significant and can greatly influence the overall transport phenomena, especially in scenarios involving simultaneous heat and mass transfer, as well as in the analogy between heat and mass transfer.
Mass transfer coefficient: The mass transfer coefficient is a measure of the mass transfer rate per unit area per unit concentration difference, which quantifies how effectively mass is transferred between phases or within a single phase. It plays a vital role in understanding processes such as convection, diffusion, and phase change, providing insights into how efficiently substances move from one location to another under different conditions.
Numerical methods: Numerical methods are techniques used to obtain approximate solutions to mathematical problems that may be difficult or impossible to solve analytically. They involve algorithms and computational approaches, making them essential for practical applications in various fields, including fluid dynamics, heat transfer, and mass transfer. These methods provide a way to analyze complex systems and simulate real-world scenarios where analytical solutions fall short.
Nusselt Number: The Nusselt number is a dimensionless quantity used in heat transfer that represents the ratio of convective to conductive heat transfer across a boundary. It helps to characterize the efficiency of convective heat transfer in fluid flows, making it essential for understanding processes involving both heat and mass transfer.
Prandtl Number: The Prandtl number is a dimensionless number that measures the relative thickness of the momentum boundary layer to the thermal boundary layer in a fluid. It helps characterize the heat transfer and fluid flow properties in convection processes, highlighting the relationship between momentum diffusivity (viscosity) and thermal diffusivity (heat conduction). Understanding the Prandtl number is crucial for analyzing various heat transfer scenarios, especially in both forced and natural convection.
Reynolds Number: Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations by comparing inertial forces to viscous forces. It is a critical factor in determining whether the flow is laminar or turbulent, influencing heat and mass transfer rates in various contexts.
Sherwood Number: The Sherwood number is a dimensionless quantity used in mass transfer operations that characterizes the relative importance of convective mass transfer to diffusive mass transfer. It is essential in understanding how effectively mass is transferred between phases, particularly in processes involving fluids, and it relates to mass transfer coefficients, boundary layer behavior, and the comparison of mass and heat transfer phenomena.
Similarity Solutions: Similarity solutions are a mathematical technique used to reduce partial differential equations into ordinary differential equations by exploiting the inherent scaling properties of the equations. This approach is especially useful in heat and mass transfer problems, where it helps simplify complex physical situations into more manageable forms, often revealing self-similar behavior in the solutions.
Thermal boundary layer: The thermal boundary layer is the region in a fluid where temperature changes from the value of the fluid away from a surface to the temperature of that surface. This layer is crucial in understanding heat transfer, as it influences convection and the effectiveness of heat exchange between a solid and a fluid. The characteristics of this layer can significantly affect the heat transfer coefficients and the overall thermal performance of systems.