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Heat and Mass Transfer
Table of Contents
Unit 6 Overview: Mass Transfer Fundamentals
6.1 Fick's Laws of Diffusion
6.2 Mass Transfer Coefficients
6.3 Convective Mass Transfer
6.4 Simultaneous Heat and Mass Transfer

Mass transfer coefficients are key to understanding how substances move between phases. They quantify the rate of mass transfer and help us design better systems for processes like absorption and extraction.

These coefficients depend on fluid properties, flow conditions, and system geometry. By using correlations and understanding their limitations, we can calculate mass transfer rates and optimize processes for various applications.

Mass transfer coefficients

Definition and interpretation

  • Mass transfer coefficients quantify the rate of mass transfer between a fluid and a surface or between two fluids in convective mass transfer processes
  • Defined as the ratio of the mass flux to the driving force (concentration difference) for mass transfer
  • The overall mass transfer coefficient (K) combines individual mass transfer coefficients on either side of the interface, analogous to the overall heat transfer coefficient in heat transfer
  • Higher values indicate faster mass transfer rates, while lower values suggest slower mass transfer rates ($h_{m,high} > h_{m,low}$)
  • Typically expressed in units of length per time (m/s) or molar flux per unit concentration difference (mol/(m^2·s·mol/m^3))

Significance and applications

  • Mass transfer coefficients are essential for designing and analyzing processes involving mass transfer, such as absorption, desorption, extraction, and membrane separation
  • Used to calculate mass transfer rates and determine the effectiveness of mass transfer equipment (packed towers, falling film absorbers)
  • Help optimize process conditions and select appropriate materials and geometries to achieve desired mass transfer performance
  • Enable the comparison of different mass transfer systems and the evaluation of mass transfer enhancements (surface modifications, flow disruptions)

Coefficients vs. system properties

Dependence on physical properties

  • Mass transfer coefficients depend on the physical properties of the fluids involved, such as density, viscosity, diffusivity, and Schmidt number (Sc)
  • The Schmidt number (Sc) relates the momentum diffusivity (kinematic viscosity) to the mass diffusivity, representing the relative importance of these two transport phenomena ($Sc = \frac{\nu}{D}$)
  • Higher Schmidt numbers indicate that momentum diffusion dominates over mass diffusion, leading to thinner concentration boundary layers and higher mass transfer coefficients
  • Lower Schmidt numbers suggest that mass diffusion is more significant, resulting in thicker concentration boundary layers and lower mass transfer coefficients

Influence of flow conditions

  • The flow regime (laminar, transition, or turbulent) significantly influences mass transfer coefficients
  • Turbulent flow generally results in higher mass transfer rates due to enhanced mixing and thinner concentration boundary layers (smaller $\delta_c$)
  • Increasing the fluid velocity or reducing the characteristic length scale of the system (smaller pipe diameter or shorter plate length) typically enhances mass transfer coefficients
  • Higher Reynolds numbers (Re) indicate more turbulent flow and generally lead to higher mass transfer coefficients ($h_m \propto Re^n$, where $n > 0$)

Calculating mass transfer rates

Correlations for mass transfer coefficients

  • Mass transfer rates can be calculated using correlations that relate mass transfer coefficients to system parameters such as Reynolds number (Re), Schmidt number (Sc), and geometry-specific dimensionless numbers
  • For external flow over flat plates, the Chilton-Colburn analogy relates the mass transfer coefficient to the friction factor and Schmidt number, allowing the use of heat transfer correlations for mass transfer calculations ($j_m = St Sc^{2/3}$)
  • For flow in pipes and ducts, correlations such as the Dittus-Boelter equation (turbulent flow) and the Sieder-Tate equation (laminar flow) can be used to estimate mass transfer coefficients based on Re, Sc, and geometry ($Sh = a Re^b Sc^c$)
  • For packed beds and other complex geometries, empirical correlations specific to the system configuration and packing characteristics are often employed (Onda correlations, Shulman correlations)

Assumptions and limitations

  • When using correlations, it is essential to ensure that the assumptions, limitations, and applicable ranges of the correlations are met for accurate mass transfer rate calculations
  • Correlations are often developed for specific geometries, flow regimes, and fluid properties, and their applicability to other systems should be carefully evaluated
  • The presence of surface reactions, non-Newtonian fluids, or other complex phenomena may require specialized correlations or numerical simulations to accurately predict mass transfer rates

Influence on mass transfer coefficients

Geometry effects

  • The geometry of the system, such as the shape and size of the interface between the fluids or the surface area available for mass transfer, significantly affects mass transfer coefficients
  • Larger surface areas and smaller characteristic lengths (smaller pipe diameters or shorter distances between plates) lead to higher mass transfer coefficients due to increased contact between the fluids and reduced resistance to mass transfer
  • Complex geometries (packed beds, structured packings) can enhance mass transfer by increasing the interfacial area and promoting turbulence and mixing

Flow regime and surface interactions

  • The flow regime (laminar, transition, or turbulent) greatly influences mass transfer coefficients
  • Turbulent flow enhances mass transfer by promoting mixing, reducing concentration boundary layer thickness, and increasing the effective surface area for mass transfer
  • In laminar flow, mass transfer coefficients are typically lower due to the presence of thicker concentration boundary layers and reduced mixing compared to turbulent flow
  • The presence of surface roughness, turbulence promoters, or other flow disruptions can enhance mass transfer coefficients by increasing turbulence and promoting mixing near the interface
  • Surface wettability and interfacial phenomena (surface tension, capillary effects) can also influence mass transfer coefficients, particularly in gas-liquid and liquid-liquid systems

Key Terms to Review (25)

Turbulent flow: Turbulent flow is a type of fluid motion characterized by chaotic changes in pressure and flow velocity. In this state, the fluid exhibits irregular fluctuations and eddies, making it quite different from laminar flow where the fluid moves in smooth layers. Turbulent flow plays a crucial role in heat transfer, mixing, and mass transfer processes, affecting the performance and efficiency of many engineering systems.
Laminar flow: Laminar flow is a type of fluid motion characterized by smooth, parallel layers of fluid that move in an orderly fashion without disruption between them. This type of flow occurs at low velocities and is typically seen in situations where fluid viscosity is high or flow area is small. The behavior of laminar flow is significant for understanding how fluids interact with surfaces, heat transfer, and mass transfer processes.
Onda correlations: Onda correlations are empirical relationships used to estimate mass transfer coefficients in gas-liquid systems. These correlations help quantify the transfer of mass between phases, which is essential for understanding how substances dissolve or react in processes like absorption and distillation.
Chilton-Colburn Analogy: The Chilton-Colburn analogy is a theoretical framework used to relate heat transfer and mass transfer phenomena by equating the dimensionless numbers that characterize each process. This analogy allows engineers to estimate mass transfer coefficients from known heat transfer coefficients, making it a useful tool in analyzing systems involving simultaneous heat and mass transfer.
Dittus-Boelter Equation: The Dittus-Boelter equation is an empirical relationship used to determine the heat transfer coefficient for turbulent flow in a circular pipe. This equation is particularly important in understanding convective heat transfer, where it provides a way to estimate how efficiently heat is transferred between a fluid and a surface. It is derived from experimental data and is applicable under specific flow conditions, making it a crucial tool for engineers dealing with thermal systems.
Sieder-Tate Equation: The Sieder-Tate equation is an empirical correlation used to estimate mass transfer coefficients in turbulent flow conditions, particularly for gas-liquid systems. It relates the mass transfer coefficient to the fluid properties and flow conditions, playing a vital role in predicting how substances transfer between phases in various engineering applications.
Shulman Correlations: Shulman correlations are empirical relationships used to estimate mass transfer coefficients in various fluid systems, particularly in cases involving turbulent flow and heat transfer. These correlations provide a way to predict how effectively a substance moves from one phase to another, which is crucial for understanding processes like evaporation, condensation, and absorption. They rely on dimensionless numbers such as the Reynolds number and Schmidt number to correlate flow characteristics with mass transfer rates.
Wilke: Wilke refers to the Wilke equation, which is used to estimate mass transfer coefficients in gas-liquid systems. This equation helps in understanding how easily a substance transfers from one phase to another, which is crucial for various processes such as absorption and distillation. The Wilke approach takes into account the physical properties of the substances involved, providing a practical method for engineers to predict and optimize mass transfer operations.
Ranz and Marshall: Ranz and Marshall refers to a correlation used to predict mass transfer coefficients for gas-liquid systems, especially in the context of mass transfer during evaporation or condensation. This correlation takes into account various factors such as the Reynolds number and Schmidt number to provide a means to estimate the effectiveness of mass transfer processes in different systems, thereby enhancing the understanding of how substances move between phases.
Experimental measurement techniques: Experimental measurement techniques refer to the various methods and tools used to obtain quantitative and qualitative data in scientific experiments. These techniques are crucial for understanding the behavior of physical systems, especially in the context of heat and mass transfer where accurate measurements are essential for determining properties like mass transfer coefficients.
Interphase Mass Transfer: Interphase mass transfer refers to the process by which mass moves between two distinct phases, such as liquid and gas, during a physical or chemical process. This concept is crucial for understanding how materials exchange in various applications like distillation, absorption, and evaporation, impacting the efficiency and effectiveness of these processes.
Lévêque approximation: The lévêque approximation is a simplified approach used in mass transfer calculations, particularly for estimating the mass transfer coefficient in systems where concentration gradients are present. This approximation assumes that the concentration profile remains linear within a boundary layer, which allows for easier mathematical modeling and quicker solutions when analyzing diffusion processes in various applications.
Absorption processes: Absorption processes involve the transfer of a substance from one phase to another, typically from a gas or liquid into a solid or liquid. This process is crucial in various industrial applications, such as gas purification, chemical reactions, and environmental control, where specific components are selectively absorbed to achieve desired outcomes.
Distillation: Distillation is a separation process that relies on the differences in boiling points of substances to separate components from a liquid mixture. This technique is commonly used in various industries to purify liquids, extract valuable components, and concentrate solutions. The process involves heating the liquid to create vapor, then cooling the vapor to collect it back into a liquid state, effectively separating the more volatile components from less volatile ones.
Diffusion Equation: The diffusion equation is a partial differential equation that describes how the concentration of a substance changes over time due to diffusion. It plays a critical role in understanding mass transfer phenomena, linking the rate of concentration change to the spatial distribution and movement of particles. This equation is essential in quantifying processes across various scenarios, such as steady-state conditions, multidimensional systems, and situations where chemical reactions may occur alongside diffusion.
Mass transfer equation: The mass transfer equation describes the movement of mass from one location to another, typically through diffusion, convection, or a combination of both. It is fundamental in understanding how substances move through different mediums, such as gases, liquids, or solids, and plays a critical role in processes like separation, absorption, and chemical reactions.
Mass transfer resistance: Mass transfer resistance refers to the hindrance or opposition encountered by a substance as it moves from one phase to another or within a single phase. This resistance can affect the overall rate of mass transfer processes, significantly influencing performance in applications such as separation processes, chemical reactions, and heat exchangers. Understanding and minimizing this resistance is crucial for optimizing systems where mass transfer plays a key role.
Fick's Laws of Diffusion: Fick's Laws of Diffusion describe how particles spread from areas of higher concentration to lower concentration over time. The first law states that the flux of a species is proportional to the concentration gradient, while the second law relates to the change in concentration over time and is derived from the first law. These principles are fundamental to understanding mass transfer phenomena in various processes, including those involving mass transfer coefficients, steady-state diffusion in multiple dimensions, and the behavior of concentration boundary layers.
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.
Coupled heat and mass transfer: Coupled heat and mass transfer refers to the interdependent processes of thermal energy and mass movement occurring simultaneously in a system, where the heat transfer influences the mass transfer and vice versa. This concept is crucial in various applications such as drying, evaporation, and chemical reactions, as changes in temperature can alter the concentration gradients driving diffusion while changes in mass can affect thermal properties. Understanding this relationship helps predict system behavior more accurately under different operating conditions.
Mass transfer coefficients: Mass transfer coefficients are crucial parameters that describe the rate at which mass moves from one phase to another, typically in a system involving diffusion or convection. They are used to quantify the efficiency of mass transfer processes and are influenced by factors such as fluid velocity, temperature, and concentration gradients. Understanding mass transfer coefficients is essential for designing and optimizing processes like distillation, absorption, and drying.
Concentration gradient: A concentration gradient refers to the gradual change in the concentration of a substance in a solution as a function of distance. It plays a crucial role in processes like diffusion, where substances move from an area of higher concentration to an area of lower concentration, ultimately driving mass transfer across various mediums.
Temperature Gradient: The temperature gradient refers to the rate at which temperature changes in a specific direction, often represented as a difference in temperature per unit distance. This concept is crucial in understanding heat transfer processes, as it drives the movement of heat from regions of higher temperature to areas of lower temperature, influencing various physical phenomena and engineering applications.
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.
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.