Heat and Mass Transfer

❤️‍🔥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.

Key Concepts and Definitions

  • 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/(m2s)kg/(m^2 \cdot 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=DdCdxJ = -D \frac{dC}{dx}, where JJ is the diffusive flux, DD is the diffusion coefficient, and dCdx\frac{dC}{dx} 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: Ct=D2Cx2\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} for one-dimensional diffusion
    • Assumes constant diffusion coefficient and no convection
  • The diffusion coefficient DD 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 Ct=0\frac{\partial C}{\partial t} = 0, simplifying Fick's second law to d2Cdx2=0\frac{d^2 C}{dx^2} = 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 kk relates the mass flux to the concentration difference between a surface and the bulk fluid: J=k(CsC)J = k(C_s - C_\infty)
    • CsC_s is the concentration at the surface, and CC_\infty 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 KK accounts for resistance to mass transfer in both the fluid and the solid phases: 1K=1kf+1ks\frac{1}{K} = \frac{1}{k_f} + \frac{1}{k_s}
    • kfk_f is the fluid-side mass transfer coefficient, and ksk_s 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: hk=ρcp(Le)2/3\frac{h}{k} = \rho c_p (Le)^{2/3}, where hh is the heat transfer coefficient, ρ\rho is the fluid density, cpc_p is the specific heat, and LeLe 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(CsC)J = k(C_s - C_\infty), where kk 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 ShSh is a dimensionless number that relates the convective mass transfer coefficient to the diffusive mass transfer coefficient: Sh=kLDSh = \frac{kL}{D}, where LL is a characteristic length
  • The Schmidt number ScSc is a dimensionless number that relates the viscous diffusion rate to the molecular diffusion rate: Sc=μρDSc = \frac{\mu}{\rho D}, where μ\mu is the dynamic viscosity
  • Empirical correlations for the Sherwood number are often expressed in terms of the Reynolds number ReRe and the Schmidt number ScSc (Dittus-Boelter correlation for turbulent flow in pipes: Sh=0.023Re0.8Sc1/3Sh = 0.023 Re^{0.8} Sc^{1/3})

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\delta_c is defined as the distance from the surface where the concentration difference is 99% of the total concentration difference
  • The mass transfer Biot number BimBi_m is a dimensionless number that relates the external mass transfer resistance to the internal diffusion resistance: Bim=kLDBi_m = \frac{kL}{D}, where LL is a characteristic length
    • For Bim1Bi_m \gg 1, the external mass transfer resistance is negligible, and the process is controlled by internal diffusion
    • For Bim1Bi_m \ll 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=0.332Rex1/2Sc1/3Sh_x = 0.332 Re_x^{1/2} Sc^{1/3} for laminar flow and Shx=0.0292Rex4/5Sc1/3Sh_x = 0.0292 Re_x^{4/5} Sc^{1/3} for turbulent flow, where ShxSh_x and RexRe_x are based on the distance xx 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)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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