Boundary layers thermal diffusion coefficient

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and dif-fusivity of the medium for both air and water. In air, these numbers are all of the same order of magnirnde, meaning that air provides a similar resistance to the transport of momenmm, heat, and mass. In water, there is one order of magnitude or more difference between kinematic viscosity, thermal diffusion coefficient, and mass diffusion coefficient. Also provided in Table 9.1 are the Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat and mass transport are more precise for air than for water, and the techniques applied to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are on the liquid side of the interface. 

Figure 17.2 shows SiFLj and SiH2 species profiles for three different surface temperatures. In all cases there is a boundary layer near the surface, which is about 0.75 cm thick. The boundary becomes a bit thicker at the higher temperatures, owing to the temperature-dependent increases in viscosity, thermal conductivity, and diffusion coefficients. The temperature and velocity boundary layers (not illustrated) are approximately the same thickness as the species boundary layers. 

In the case of combustion of a condensed substance, conservation of enthalpy and similarity occur only in the gas phase and only in part of the space. In the c-phase the diffusion coefficient is much smaller than the thermal diffusivity, and we have heating of the c-phase by heat conduction without dilution by diffusion. The enthalpy of the c-phase at the boundary, for x — 0 (from the side x < 0), is larger than the enthalpy of the c-phase far from the reaction zone and larger than the enthalpy of the combustion products. The advantage of the derivation given here is that the constancy of the enthalpy in the gas phase and its equality to H0 (H0 is the enthalpy of the c-phase far from the combustion zone, at x — —oo) are obtained without regard to the state of the intermediate layers of the c-phase. We should particularly emphasize that the constancy of the enthalpy in the combustion zone occurs only for a steady process. The presence of layers of the c-phase with increased enthalpy opens the possibility in a non-steady process of a temporary change in the enthalpy of the gas and the combustion temperature (on this see 5). 

In this situation, a film is grown on the hot surface (Tw), and its thickness will increase without limit as long as fresh reactants are provided and products can be removed. The gas state will be in quasiequilibrium far from the hot surface and in a strongly nonequilibrium condition close to it. The change from one to the other will occur across a boundary layer where temperature, velocity, and species concentration vary rapidly. The behavior of this boundary layer will be determined by gas transport properties such as viscosity, thermal conductivity, as well as gas-phase kinetics and diffusion coefficients. So, even if the kinetics at the surface are very fast, we must deal with quasiequilibrium phenomena where gas conditions vary rapidly over short distances. 

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

The concentration boundary layer and the mass transfer coefficient can immediately be found from the equations given previously, as the integral condition (3.167) for mass transfer corresponds to that for heat transfer. The temperature is replaced by the mass fraction A, the thermal diffusivity a by the diffusion coefficient D, and instead of the thermal boundary layer ST the concentration boundary layer Sc is used. This then gives us the concentration profile corresponding to (3.172) ... 

The semianalytical method developed earlier can be used to solve partial differential equations in composite domains also. Mass or heat transfer in composite domains involves two different diffusion coefficients or thermal conductivities in the two layers of the composite material.[6] In addition, even in case of solids with a single domain and constant physical properties, the reaction may take place mainly near the surface. This leads to the formation of boundary layer near one of the boundaries. In this section, the semianalytical method developed earlier is extended to composite domains. 

There is an extensive literalure on solutions to (3.1) for various geometries and flow regimes. Many results are given by Levich (1962). Results for heat transfer, such as those discussed by Schlichting (1979) for boundary layer flows, are applicable to mass transfer or diffusion if the diffusion coefficient, D, is substituted for the coeflidenl of thermal diffusivity, K/pCp, where k is the thermal conductivity, p is the gas density, and Cp is the heat capacity of the gas. The results are directly applicable to aerosols for point panicles, that is, iip = 0. 

It is necessary to replace A, mix inequation (11-111) for c.iocai by the thermal conductivity, which corresponds to the molecular transport property for heat transfer, to calculate the local heat transfer coefficient, by analogy. However, as mentioned above, it is necessary to replace j0A,mix in the expression for Sc by the thermal diffusivity to calculate the analogous thermal boundary layer... 

Summarizing, we should note that the methods presented in the present section can be applied without any modifications to heat exchange problems, because temperature distribution is described by an equation similar to the diffusion equation. The boundary conditions are also formulated in a similar way. One only has to replace D by the coefficient of thermal diffusivity, and the number Peo - by Pej-. The corresponding boundary layer is known as the thermal layer. Detailed solutions of heat conductivity problems can be found in [6]. 

Since the thermal difflisivity of solution, a, is less affected by variations of temperature and eoneentration over the ranges Ts(ka) < Tf < Tg, and kg < k < kg, respectively, than the binary diffusion coefficient, D, it is assumed henceforward that aj = const. Furthermore, sinee the thermal boundary layer is mueh thicker than the diffusion layer, it is appropriate to assume that within the latter D = D(k,Ta). 

To correlate these data for heat-transfer coefficients, dimensionless numbers such as the Reynolds and Prandtl numbers are used. The Prandtl number is the ratio of the shear component of diffusivity for momentum p/p to the diffusivity for heat k/pc and physically relates the relative thickness of the hydrodynamic layer and thermal boundary layer. 

Convection mass transfer coefficients are often used as convective boundary conditions for gas diffusion in a stationary media. However, while applying mass transfer correlations to describe mass species transport from the electrode-gas diffusion layer to gas flow stream in the channel, it is assumed that species mass transport rate at the wall is small and does not alter the hydrod5mamic, thermal, and concentration boundary layers like in boundary layers with wall suction or blowing. 

J = thermal expansion coefficient for gases (3 = 1/T L = characteristic length or length of travel of the fluid in the boundary layer, that is, the height of the stack, m. v = kinematic viscosity, m s" a = thermal diffusivity, m s ... 

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