Heat and mass transfer equations

This process has been used for various situations (1—14). Eor the condensation of a single component from a binary gas mixture, the gas-stream sensible heat and mass-transfer equations for a differential condenser section take the following forms ... 

The three-dimensional, fully parabolic flow approximation for momentum and heat- and mass-transfer equations has been used to demonstrate the occurrence of these longitudinal roll cells and their effect on growth rate uniformity in Si CVD from SiH4 (87) and GaAs MOCVD from Ga(CH3)3 and AsH3 (189). However, gas expansion in the entrance zone combined with flow obstructions, such as a steeply sloped susceptor, can also produce... 

Of the many extensions of the classical quasi-steady droplet combustion models, most have dealt with the nonsteady eflFects which take place during droplet ignition and combustion (3-9). Using Greens function techniques to evaluate the nonsteady heat and mass transfer equations... 

Although under certain experimental condition, step 2 or step 3 may be ignored due to its non-conspicuous influence to the flnal pyrolysis product distribution, however, a good mathematical model for biomass pyrolysis should be versatile applicable to other pyrolysis conditions, and thus it should be involved the above-mentioned three steps of process, of course heat and mass transfer equations should be included also. This paper presents this kind of mathematical model. Although the model is constructed based on sawdust pyrolysis, it is quite straightforward to apply the same approach to other cases such as straw and municipal solid waste pyrolysis even if to biomass or coal gasification or metal ore reduction. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

However, to solve the heat and mass transfer equations an additional modeling problem has to be overcome. While there are sufficient measurements of the turbulent velocity field available to validate the different i>t modeling concepts proposed in the literature, experimental difficulties have prevented the development of any direct modeling concepts for determining the turbulent conductivity at, and the turbulent diffusivity Dt parameters. Nevertheless, alternative semi-empirical modeling approaches emerged based on the hypothesis that it might be possible to calculate the turbulent conductivity and diffusivity coefficients from the turbulent viscosity provided that sufficient parameterizations were derived for Prj and Scj. 

The model was solved using orthogonal collocation on finite elements (OCFE). Orthogonal collocation on finite elements was developed by Carey and Finlayson (26) for solution of boundary layer problems. Carey and Finlayson used OCFE to solve the simultaneous heat and mass transfer equations describing a catalyst pellet and found the new method to be more efficient than finite difference techniques. They also showed that OCFE was applicable to boundary layer problems that could not be solved by global orthogonal collocation. Jain and Schultz (27)... 

The Heat Mass Transfer Factor is that portion of the heat and mass transfer equations which relates to the cellular structure. This factor is defined by use of the equation for energy flux as follows ... 

Polyanin, A. D., Zhurov, A. I., and Vyazmin, A. V, Generalized Separation of Variables in Nonlinear Heat and Mass Transfer Equations, J. Non-Equillib. Thermodyn. 25 (2000) 252-267. 

S.l. Exact Solutions of Linear Heat and Mass Transfer Equations... 

For a problem involving fluid flow and simultaneous heat and mass transfer, equations of continuity, momentum, energy, and chemical species (Eqs. 1.41, 1.44, 1.54, and 1.63) are a formidable set of partial differential equations. There are four independent variables three space coordinates (say, x, y, z) and a time coordinate t. 

First, we will introduce theory for the drying and method of modeling of the drying proeess. A drying process of liquid coated film can be simulated by setting up heat and mass transfer equations and vapor liquid equilibrium equations with appropriate heat... 

Here Q. is the sum of all heat fluxes involving particle i and p CV. is the thermal capacity of particle i. The change in mass of the solvent is computed from the coupled heat and mass transfer equation as... 

Even if the intrusion of heat transfer or bed diffusional resistance cannot be entirely eliminated, it may still be possible to derive values for the diffusional time constant by using the appropriate solution of the diffusion equation. Some relevant solutions to the coupled heat and mass transfer equations for such systems are given in Section 6.3. 

The applicability of the component balance equations with reaction terms is limited. It requires the knowledge of the reaction kinetics and then, the equations are rather part of a more complex mathematical model involving heat and mass transfer equations and the like. In balancing proper, the integral reaction rates W (n) in (4.2.11) can be known only approximately or rather, they are unknown. We will now show how they can be eliminated from the set of balance equations. 

In the original versions of the two latter balance equations, the authors also account for the compressibUity of the Uquid, which is negligible from a physical point of view but helpful for numerical stabilization.) The energy balance contains contributions from heat conduction and Uquid and soUd convection. The usual flux boundary conditions are appUed to the heat and mass transfer equations. 

It should be noted that any analysis or evaluation of the cyclic pressure freeze drying process should involve nonsteady-state heat and mass transfer equations like those presented in Seetion 11.6.1.1 and Section 11.6.1.2. The effectiveness of cyclic pressure freeze drying and the effect of cycle period and shape on drying times have been the subject of a number of investigations [1,60,61]. Litchfield and Liapis [12]... 

The theoretical formulation of heat and mass transfer in porous media is usually obtained by a change in scale. We can pass from a microscopic view where the size of tlie representative volume is small with regards to pores, to a microscopic view where the size of the representative volume CO is large with regard to the pores. Moreover, the heat and mass transfer equations can be deduced from Whitaker s theory. The macroscopic equations can be obtained by averaging the classical fluid mechanics, dififiision and transfer equations over the averaging volume co[m ]. The average of a function/is ... 

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