11.2 Thermal and concentration boundary layers

Thermal and concentration boundary layers form when fluid flows over surfaces with different temperatures or concentrations. These layers affect heat and mass transfer rates, influencing engineering processes and designs.

Understanding boundary layer development helps predict heat and mass transfer in various applications. Factors like fluid properties, flow conditions, and surface characteristics impact boundary layer growth and transfer rates.

Boundary Layer Formation and Development

Thermal and Concentration Boundary Layer Formation

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• Thermal and concentration boundary layers form when a fluid flows over a surface with a different temperature or concentration than the bulk fluid
• The velocity boundary layer is a region near the surface where the fluid velocity changes from zero at the surface to the free-stream velocity
• The thermal boundary layer develops when there is a temperature difference between the surface and the bulk fluid, causing heat transfer and a temperature gradient within the boundary layer
• The concentration boundary layer forms when there is a difference in the concentration of a species between the surface and the bulk fluid, leading to mass transfer and a concentration gradient within the boundary layer

Boundary Layer Growth and Influencing Factors

• The thickness of the thermal and concentration boundary layers increases in the direction of fluid flow due to the diffusion of heat and mass
• The Prandtl number (Pr) is a dimensionless parameter that relates the thickness of the thermal boundary layer to the velocity boundary layer
  • Pr is defined as the ratio of momentum diffusivity to thermal diffusivity, $Pr = \frac{\nu}{\alpha}$, where $\nu$ is the kinematic viscosity and $\alpha$ is the thermal diffusivity
  • For Pr > 1 (e.g., oils), the thermal boundary layer is thinner than the velocity boundary layer, while for Pr < 1 (e.g., liquid metals), the thermal boundary layer is thicker than the velocity boundary layer
• The Schmidt number (Sc) is a dimensionless parameter that relates the thickness of the concentration boundary layer to the velocity boundary layer
  • Sc is defined as the ratio of momentum diffusivity to mass diffusivity, $Sc = \frac{\nu}{D}$, where $D$ is the mass diffusivity
  • For Sc > 1 (e.g., glycerin), the concentration boundary layer is thinner than the velocity boundary layer, while for Sc < 1 (e.g., hydrogen gas), the concentration boundary layer is thicker than the velocity boundary layer

Temperature and Concentration Profiles

Laminar Flow Profiles

• The temperature profile within the thermal boundary layer is determined by solving the energy equation, which accounts for convection and conduction heat transfer
• The concentration profile within the concentration boundary layer is determined by solving the species conservation equation, which accounts for convection and diffusion mass transfer
• For laminar flow, the temperature and concentration profiles are typically modeled using polynomial functions or similarity solutions
  • Example: The temperature profile in a laminar boundary layer over a flat plate can be approximated by a third-order polynomial, $\frac{T - T_\infty}{T_s - T_\infty} = a_0 + a_1 \left(\frac{y}{\delta_t}\right) + a_2 \left(\frac{y}{\delta_t}\right)^2 + a_3 \left(\frac{y}{\delta_t}\right)^3$, where $T$ is the temperature, $T_s$ is the surface temperature, $T_\infty$ is the free-stream temperature, $y$ is the distance from the surface, $\delta_t$ is the thermal boundary layer thickness, and $a_0$, $a_1$, $a_2$, and $a_3$ are coefficients determined by boundary conditions
• The boundary conditions at the surface and the edge of the boundary layer are used to solve for the temperature and concentration profiles

Turbulent Flow Profiles and Boundary Layer Thickness

• In turbulent flow, the temperature and concentration profiles are more complex due to the presence of eddies and fluctuations, requiring the use of turbulence models or empirical correlations
  • Example: The temperature profile in a turbulent boundary layer can be described by the logarithmic law, $\frac{T - T_s}{T_\tau} = \frac{1}{\kappa} \ln \left(\frac{y u_\tau}{\nu}\right) + B$, where $T_\tau$ is the friction temperature, $u_\tau$ is the friction velocity, $\kappa$ is the von Kármán constant, and $B$ is a constant dependent on the Prandtl number
• The thermal and concentration boundary layer thicknesses can be defined as the distance from the surface where the temperature or concentration reaches a specified fraction (e.g., 99%) of the free-stream value
  • Example: The thermal boundary layer thickness $\delta_t$ can be defined as the distance from the surface where $\frac{T - T_s}{T_\infty - T_s} = 0.99$, and similarly, the concentration boundary layer thickness $\delta_c$ can be defined as the distance where $\frac{C - C_s}{C_\infty - C_s} = 0.99$, with $C$ being the concentration, $C_s$ the surface concentration, and $C_\infty$ the free-stream concentration

Heat and Mass Transfer Rates

Heat and Mass Transfer Coefficients

• The heat transfer rate can be determined by applying Fourier's law at the surface, using the temperature gradient obtained from the thermal boundary layer analysis
  • The heat transfer rate per unit area (heat flux) is given by $q'' = -k \left.\frac{\partial T}{\partial y}\right|_{y=0}$, where $k$ is the thermal conductivity and $\left.\frac{\partial T}{\partial y}\right|_{y=0}$ is the temperature gradient at the surface
• The mass transfer rate can be calculated using Fick's law at the surface, utilizing the concentration gradient obtained from the concentration boundary layer analysis
  • The mass transfer rate per unit area (mass flux) is given by $j'' = -D \left.\frac{\partial C}{\partial y}\right|_{y=0}$, where $D$ is the mass diffusivity and $\left.\frac{\partial C}{\partial y}\right|_{y=0}$ is the concentration gradient at the surface
• The convective heat transfer coefficient (h) relates the heat transfer rate to the temperature difference between the surface and the bulk fluid, and it depends on the thermal boundary layer characteristics
  • The convective heat transfer coefficient is defined as $h = \frac{q''}{T_s - T_\infty}$, where $q''$ is the heat flux, $T_s$ is the surface temperature, and $T_\infty$ is the free-stream temperature
• The mass transfer coefficient (hm) relates the mass transfer rate to the concentration difference between the surface and the bulk fluid, and it depends on the concentration boundary layer characteristics
  • The mass transfer coefficient is defined as $h_m = \frac{j''}{C_s - C_\infty}$, where $j''$ is the mass flux, $C_s$ is the surface concentration, and $C_\infty$ is the free-stream concentration

Dimensionless Parameters and Empirical Correlations

• Dimensionless parameters such as the Nusselt number (Nu) and the Sherwood number (Sh) are used to characterize the heat and mass transfer rates, respectively, in relation to the boundary layer thicknesses and fluid properties
  • The Nusselt number is defined as $Nu = \frac{hL}{k}$, where $h$ is the convective heat transfer coefficient, $L$ is a characteristic length, and $k$ is the thermal conductivity
  • The Sherwood number is defined as $Sh = \frac{h_mL}{D}$, where $h_m$ is the mass transfer coefficient, $L$ is a characteristic length, and $D$ is the mass diffusivity
• Empirical correlations based on experimental data or numerical simulations can be used to estimate the heat and mass transfer coefficients for various flow configurations and boundary conditions
  • Example: For laminar flow over a flat plate, the average Nusselt number can be estimated using the correlation $\overline{Nu}_L = 0.664 Re_L^{1/2} Pr^{1/3}$, where $Re_L$ is the Reynolds number based on the plate length $L$, and $Pr$ is the Prandtl number
  • Similarly, for laminar flow over a flat plate, the average Sherwood number can be estimated using the correlation $\overline{Sh}_L = 0.664 Re_L^{1/2} Sc^{1/3}$, where $Sc$ is the Schmidt number

Momentum, Thermal, and Concentration Boundary Layers

Similarities and Differences in Boundary Layer Development

• Momentum, thermal, and concentration boundary layers all develop when a fluid flows over a surface, and they are characterized by gradients in velocity, temperature, and concentration, respectively
• The thicknesses of the momentum, thermal, and concentration boundary layers are related to the dimensionless parameters Reynolds number (Re), Prandtl number (Pr), and Schmidt number (Sc), respectively
  • The Reynolds number is defined as $Re = \frac{UL}{\nu}$, where $U$ is the characteristic velocity, $L$ is the characteristic length, and $\nu$ is the kinematic viscosity
  • The Prandtl number relates the thickness of the thermal boundary layer to the velocity boundary layer, while the Schmidt number relates the thickness of the concentration boundary layer to the velocity boundary layer
• In laminar flow, the velocity, temperature, and concentration profiles within their respective boundary layers can be described by similar mathematical equations and boundary conditions

Turbulent Flow and Boundary Layer Interactions

• In turbulent flow, the momentum boundary layer is characterized by the presence of eddies and fluctuations, which enhance the mixing and transport of heat and mass, leading to thinner thermal and concentration boundary layers compared to the velocity boundary layer
• The analogy between heat and mass transfer allows for the use of similar mathematical treatments and empirical correlations for both thermal and concentration boundary layers, with appropriate modifications based on the Prandtl and Schmidt numbers
  • Example: The Reynolds analogy relates the heat transfer coefficient to the friction coefficient (skin friction) in turbulent flow, $\frac{h}{\rho c_p U_\infty} = \frac{C_f}{2}$, where $\rho$ is the fluid density, $c_p$ is the specific heat capacity, $U_\infty$ is the free-stream velocity, and $C_f$ is the friction coefficient
  • Similarly, the Chilton-Colburn analogy relates the mass transfer coefficient to the friction coefficient in turbulent flow, $\frac{h_m}{U_\infty} = \frac{C_f}{2} Sc^{-2/3}$
• While the momentum boundary layer is influenced by pressure gradients and surface roughness, the thermal and concentration boundary layers are primarily affected by the surface temperature and concentration, respectively, as well as the fluid properties