❤️🔥Heat and Mass Transfer Unit 6 – Mass Transfer Fundamentals
Mass transfer fundamentals explore how substances move from areas of high concentration to low concentration. This unit covers key concepts like diffusion, convection, and Fick's laws, which describe how molecules spread through different mediums.
Understanding mass transfer is crucial for many engineering applications. From designing chemical reactors to developing water purification systems, these principles help engineers control and optimize the movement of substances in various processes.
Mass transfer involves the transport of a substance from a region of higher concentration to a region of lower concentration
Diffusion is the movement of molecules from an area of high concentration to an area of low concentration driven by a concentration gradient
Convection is the transport of mass due to the bulk motion of a fluid (liquid or gas)
Mass flux is the rate of mass transfer per unit area perpendicular to the direction of transfer, typically expressed in units of kg/(m2⋅s)
Concentration gradient is the change in concentration of a species per unit distance in a particular direction
Steady-state diffusion occurs when the concentration at any point does not change with time
Transient diffusion occurs when the concentration at a point changes with time
Equimolar counterdiffusion is the diffusion of two species in opposite directions with no net molar flux
Mechanisms of Mass Transfer
Molecular diffusion is the transport of mass due to the random motion of molecules, driven by a concentration gradient
Occurs in gases, liquids, and solids
Governed by Fick's laws of diffusion
Convective mass transfer is the transport of mass due to the bulk motion of a fluid
Can be natural convection driven by density differences (hot air rising) or forced convection driven by external forces (wind, pumps)
Enhances mass transfer compared to molecular diffusion alone
Turbulent diffusion is the transport of mass due to the chaotic motion of fluid particles in a turbulent flow
Increases mixing and mass transfer rates compared to laminar flow
Eddy diffusion is the transport of mass due to the motion of large-scale eddies in a turbulent flow
Surface diffusion is the transport of adsorbed molecules along a solid surface
Knudsen diffusion occurs when the mean free path of molecules is comparable to or larger than the pore size in a porous medium (low-pressure gases in small pores)
Fick's Laws of Diffusion
Fick's first law relates the diffusive flux to the concentration gradient: J=−DdxdC, where J is the diffusive flux, D is the diffusion coefficient, and dxdC is the concentration gradient
The negative sign indicates that diffusion occurs in the direction of decreasing concentration
Fick's second law describes the change in concentration with time due to diffusion: ∂t∂C=D∂x2∂2C for one-dimensional diffusion
Assumes constant diffusion coefficient and no convection
The diffusion coefficient D is a measure of the ease with which a species diffuses through a medium
Depends on temperature, pressure, and the properties of the diffusing species and the medium
Can be estimated using empirical correlations or measured experimentally
Steady-state diffusion occurs when ∂t∂C=0, simplifying Fick's second law to dx2d2C=0
Transient diffusion occurs when the concentration changes with time, requiring the use of Fick's second law and appropriate initial and boundary conditions
Mass Transfer Coefficients
The mass transfer coefficient k relates the mass flux to the concentration difference between a surface and the bulk fluid: J=k(Cs−C∞)
Cs is the concentration at the surface, and C∞ is the concentration in the bulk fluid
Mass transfer coefficients are used to quantify the rate of mass transfer in convective mass transfer problems
The overall mass transfer coefficient K accounts for resistance to mass transfer in both the fluid and the solid phases: K1=kf1+ks1
kf is the fluid-side mass transfer coefficient, and ks is the solid-side mass transfer coefficient
Mass transfer coefficients can be determined experimentally or estimated using empirical correlations based on dimensionless numbers (Sherwood, Schmidt, Reynolds)
The Chilton-Colburn analogy relates mass transfer coefficients to heat transfer coefficients: kh=ρcp(Le)2/3, where h is the heat transfer coefficient, ρ is the fluid density, cp is the specific heat, and Le is the Lewis number
Convective Mass Transfer
Convective mass transfer occurs when mass is transported by the bulk motion of a fluid
The convective mass flux is given by: J=k(Cs−C∞), where k is the mass transfer coefficient
Natural convection mass transfer is driven by density differences caused by concentration gradients (salt dissolution in water)
Forced convection mass transfer is driven by external forces such as pumps, fans, or wind (drying of clothes in a dryer)
The Sherwood number Sh is a dimensionless number that relates the convective mass transfer coefficient to the diffusive mass transfer coefficient: Sh=DkL, where L is a characteristic length
The Schmidt number Sc is a dimensionless number that relates the viscous diffusion rate to the molecular diffusion rate: Sc=ρDμ, where μ is the dynamic viscosity
Empirical correlations for the Sherwood number are often expressed in terms of the Reynolds number Re and the Schmidt number Sc (Dittus-Boelter correlation for turbulent flow in pipes: Sh=0.023Re0.8Sc1/3)
Boundary Layer Theory in Mass Transfer
The boundary layer is a thin region near a surface where the velocity and concentration gradients are significant
The velocity boundary layer is the region where the fluid velocity changes from zero at the surface to the free-stream velocity
The concentration boundary layer is the region where the concentration changes from the surface concentration to the bulk concentration
The thickness of the concentration boundary layer δc is defined as the distance from the surface where the concentration difference is 99% of the total concentration difference
The mass transfer Biot number Bim is a dimensionless number that relates the external mass transfer resistance to the internal diffusion resistance: Bim=DkL, where L is a characteristic length
For Bim≫1, the external mass transfer resistance is negligible, and the process is controlled by internal diffusion
For Bim≪1, the internal diffusion resistance is negligible, and the process is controlled by external mass transfer
The analogy between heat and mass transfer allows the use of heat transfer correlations for estimating mass transfer coefficients (Chilton-Colburn analogy)
The Sherwood number can be expressed as a function of the Reynolds number and the Schmidt number for flow over a flat plate: Shx=0.332Rex1/2Sc1/3 for laminar flow and Shx=0.0292Rex4/5Sc1/3 for turbulent flow, where Shx and Rex are based on the distance x from the leading edge
Mass Transfer Equipment and Applications
Packed bed columns are used for gas-liquid mass transfer operations such as absorption and stripping
Consist of a vertical column filled with packing material (Raschig rings, Berl saddles) to increase the interfacial area for mass transfer
The height of a transfer unit (HTU) and the number of transfer units (NTU) are used to design and analyze packed bed columns
Tray columns are used for gas-liquid mass transfer operations and distillation
Consist of a vertical column with a series of horizontal trays that promote mixing and mass transfer between the gas and liquid phases
The efficiency of a tray is characterized by the Murphree tray efficiency, which compares the actual concentration change to the ideal concentration change
Membrane separations use selective membranes to separate mixtures based on differences in permeability
Examples include reverse osmosis, ultrafiltration, and gas separations (CO2 capture)
The performance of a membrane is characterized by its selectivity and permeability
Adsorption processes use solid adsorbents to selectively remove components from a fluid phase
Examples include activated carbon for water treatment and zeolites for gas purification
The adsorption capacity and selectivity of an adsorbent depend on its surface area, pore size distribution, and surface chemistry
Drying processes involve the removal of moisture from a solid material by evaporation
Can be achieved through various methods such as hot air drying, vacuum drying, and freeze-drying
The drying rate depends on the properties of the material, the drying conditions (temperature, humidity), and the mass transfer coefficients
Problem-Solving Techniques
Identify the type of mass transfer problem: steady-state or transient, one-dimensional or multi-dimensional, diffusion or convection
Determine the appropriate governing equations: Fick's laws for diffusion, convective mass transfer equations, or a combination of both
Specify the initial and boundary conditions based on the physical situation
Examples of boundary conditions include constant concentration, constant flux, or convective mass transfer at a surface
Simplify the problem by making reasonable assumptions, such as constant properties, no chemical reactions, or negligible convection
Solve the governing equations analytically or numerically
Analytical solutions are possible for simple geometries and boundary conditions (one-dimensional, steady-state diffusion with constant concentration boundaries)
Numerical methods (finite difference, finite element) are required for more complex problems
Use dimensionless numbers (Sherwood, Schmidt, Biot) to characterize the mass transfer process and to obtain mass transfer coefficients from empirical correlations
Apply the analogy between heat and mass transfer to adapt heat transfer solutions and correlations to mass transfer problems
Verify the solution by checking the units, the limiting cases (zero flux at an impermeable boundary), and the overall mass balance
Interpret the results in terms of the physical situation and the design requirements (mass transfer rate, concentration profiles, equipment size)