Transport analogies heat/mass

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]. 

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

Analogous equations for the unidirectional transport of heat and mass are the Fourier law and Fick s law, that are written, respectively, as ... 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

The transfer of heat in a fluid may be brought about by conduction, convection, diffusion, and radiation. In this section we shall consider the transfer of heat in fluids by conduction alone. The transfer of heat by convection does not give rise to any new transport property. It is discussed in Section 3.2 in connection with the equations of change and, in particular, in connection with the energy transport in a system resulting from work and heat added to the fluid system. Heat transfer can also take place because of the interdiffusion of various species. As with convection this phenomenon does not introduce any new transport property. It is present only in mixtures of fluids and is therefore properly discussed in connection with mass diffusion in multicomponent mixtures. The transport of heat by radiation may be ascribed to a photon gas, and a close analogy exists between such radiative transfer processes and molecular transport of heat, particularly in optically dense media. However, our primary concern is with liquid flows, so we do not consider radiative transfer because of its limited role in such systems. 

A transport of mass or diffusion of mass will take place in a fluid mixture of two or more species whenever there is a spatial gradient in the proportions of the mixture, that is, a concentration gradient. Mass diffusion is a consequence of molecular motion and is closely analogous to the transport of heat and momentum in a fluid. 

The second milestone in chemical engineering came in 1960 with the publication of Transport Phenomena, by Bird et al. [2]. Their new approach emphasized the microscale processes and the analogy among mass, heat, and momentum transfer in different processes. 

A most telling examination of radial heat transfer can be distilled from a recent series of papers by Dixon and co-workers [13> 19, 20, 21], who systematically studied the various underlying transport processes - radial mass transfer and mass transfer at the wall (from which heat transfer parameters may be estimated by heat and mass transfer analogies), together with effective radial solid conduction and solid-to-wall heat transfer. The relevant parameter values to be inserted into Eqns. (28) and (29) are summarised in Table 1 below. All the experiments were conducted on a tube of 75. mm I.D. 

Our treatment so far has made occasional reference to heat transfer, primarily to draw the reader s attention to tiie analogies that exist between the transport of heat and mass. For example, in Chapter 1 we highlighted the similarities between the rate laws governing convective and diffusive heat and mass transfer. The analogy between tiie two phenomena when dealing witii co-current or countercurrent operations has been brought out on several occasions, notably Illustration 8.7. 

In this section the analogy between heat and mass transfer is introduced and used to solve problems. The specific estimation relationships for permeants in polymers are discussed in Section 4.2 with the emphasis placed on gas-polymer systems. This section provides the necessary formulas for a first approximation of the diffusivity, solubility, and permeability, and their dependence on temperature. Non-Fickian transport, which is frequently present in high activity permeants in glassy polymers, is introduced in Section 4.3. Convective mass transfer coefficients are discussed in Section 4.4, and the analogies between mass and heat transfer are used to solve problems involving convective mass transfer. Finally, in Section 4.5 the solution to Design Problem III is presented. 

The heat and mass transport phenomena of the char gasification is not described in the literature as much as for the char combustion [11,28,78]. There are good reasons to believe that it is quite analogous to the char combustion phenomenology [79]. However, the heterogeneous gasification reactions are overall endothermic which results in some differences with respect to the intraparticle heat transport [79]. 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

Momentum can be transported in an analogous manner to mass or heat. The diffusion of momentum is described with a kinematic viscosity, v, which has SI units of m /s ... 

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. 

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