Boundary layer analogies

It is apparent that the source and sink terms can be different between mass, heat, and momentum transport. There is another significant difference, however, related to the magnitude of the diffusion coefficient for mass, heat, and momentum. 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

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What are the boundary conditions in the environment to which the stagnant film, penetration, surface renewal theories, and laminar boundary layer analogy should be applied Briefly explain why. 

Because of the various effects noted above, the entire subject of heat transfer to fluids without phase change is complex and in practice is treated as a series of special cases rather than as a general theory. All cases considered in this chapter do, however, have a phenomenon in common in all of them the formation of a thermal boundary layer, analogous to the hydrodynamic Prandtl boundary layer described in Chap. 3, takes place it profoundly influences the temperature field and so controls the rate of heat flow. 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

Heat and Mass Transfer Differential Equations in the Boundary Layer and the Corresponding Analogy 131... 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that thermal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

Mass and heat transfer to the walls in turbulent flows is a complex mixture of molecular transport and transport by turbulent eddies. The generally assumed analogy between mass and heat transfer by assuming Sh = Nu, is not valid for turbulent flows [26]. Simulations and measurements have shown that there is a laminar film close to the surface where most of the mass transfer resistance for high Sc liquids is located. This fUm is located below y+ = 1 and for low Sc fluids, and for heat transfer the whole boundary layer is important [27]. 

An analogy exists between mass transfer (which depends on the diffusion coefficient) and momentum transfer between the sliding hquid layers (which depends on the kinematic viscosity). Calculations show that the ratio of thicknesses of the diffnsion and boundary layer can be written as... 

The (Ta - Ts)/8 term is the temperature gradient, which correlates (dTIdy through the boundary layer thickness. The fact that 8 can be correlated with the Reynolds number and that the Colburn analogy can be applied leads to the correlation of the form... 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

Attempts to obtain theoretical solutions for deformed bubbles and drops are limited, while no numerical solutions have been reported. A simplifying assumption adopted is that the bubble or drop is perfectly spheroidal. SalTman (SI) considered flow at the front of a spheroidal bubble in spiral or zig-zag motion. Results are in fair agreement with experiment. Harper (H4) tabulated energy dissipation values for potential flow past a true spheroid. Moore (Mil) applied a boundary layer approach to a spheroidal bubble analogous to that for spherical bubbles described in Chapter 5. The interface is again assumed to be completely free of contaminants. The drag is given by... 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 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 is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

By analogy to electrical systems, the resistance r can be thought of as consisting of several components. For convenience, three such components are often defined, a surface resistance (rsurl), which depends on the affinity of the surface for the species, a boundary layer resistance (rh(ulll), which depends on the molecular diffusiv-ity of the gas in air, and a gas-phase resistance (rgas), which depends on the micrometeorology that transports the gas to the surface ... 

Fallis (FI) considered thermal transport in transitional and turbulent boundary flows and supplied a reasonable analysis of this difficult problem which is in agreement with the work of Eber (El) and the theory of Eckert and Drewitz (E2). Callaghan (Cl) contributed to the analogies between thermal and material transport in turbulent flow with particular emphasis upon the behavior near and in the boundary layer. The effect... 

As stated, this equation finds limited practical application. It requires knowledge of both velocity profiles and its solution requires vorticity boundary conditions that also depend on the velocity profiles. The principal reason to write the equation is to make the point that vorticity is transported within the boundary layer by convection and diffusion in a manner analogous to momentum transport. 

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