Heat conduction mass transfer problem

Consider the heat conduction/mass transfer problem in a cylinder.[6] [9] [10] The governing equation in dimensionless form is... 

Procedures - Stzady State. Expz/umeMtA. To approximate a differential reactor and to avoid heat and mass transfer problems, conversions were kept below 5.0% and partial pressures of O2 and CO below 6.3 kPa were used. The total pressure was kept constant at 108.3 kPa for all the runs. For any particular set of Pqo and Pq2i runs at different values of W/F q were conducted by varying the flowrates of O2, CO and He in such a manner that the P02 and Pco fed stayed constant. The procedure was repeated for various combinations of Pq, and Pco To ascertain the effect of temperature, one particular set of Pqq and P02 was repeated at different temperatures. 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

In the preceding sections of this chapter, we considered weak convection effects in the transfer of heat from a hot or cold body. In the remainder of this chapter, we continue our study of convection effects in heat (or mass) transfer problems, but now we focus on the limit Pe I, where the convection of heat appears dominant in the governing equation (9-7) relative to conduction. We actually consider the limit Pe 1 in two steps. In this chapter, we consider problems in which Re . 1 Then, following Chap. 10, in which we consider flow problems in the limit Re 1, we return to consider the high-die and high-Pe regimes in Chap. 11. 

The analysis of heat transfer is parallel to that for diffusion because heat transfer and diffusion are described with the same mathematical equations. Indeed, many experts argue that because of this mathematical identity, the two are identical. I do not agree with this view because the two processes are so different physically. For example, heat conduction is faster in liquids than in gases, but diffusion is faster in gases than in liquids. This relation between the mathematical similarity and the physical difference affects the problems involving both heat and mass transfer, problems central to the next chapter. 

In the previous chapters, the stresses arising from relative motion within a fluid, the transfer of heat by conduction and convection, and the mechanism of mass transfer are all discussed. These three major processes of momentum, heat, and mass transfer have, however, been regarded as independent problems. 

In Volume 1, the behaviour of fluids, both liquids and gases is considered, with particular reference to their flow properties and their heat and mass transfer characteristics. Once the composition, temperature and pressure of a fluid have been specified, then its relevant physical properties, such as density, viscosity, thermal conductivity and molecular diffu-sivity, are defined. In the early chapters of this volume consideration is given to the properties and behaviour of systems containing solid particles. Such systems are generally more complicated, not only because of the complex geometrical arrangements which are possible, but also because of the basic problem of defining completely the physical state of the material. 

In addition to the similarity between the heat conduction equation and the diffusion equation, erosion is often described by an equation similar to the diffusion equation (Culling, 1960 Roering et al., 1999 Zhang, 2005a). Flow in a porous medium (Darcy s law) often leads to an equation (Turcotte and Schubert, 1982) similar to the diffusion equation with a concentration-dependent diffu-sivity. Hence, these problems can be treated similarly as mass transfer problems. 

The second part of the chapter is devoted to the effect of pressure on heat and mass transfer. After a brief survey on fundamentals the estimation of viscosity, diffusivity in dense gases, thermal conductivity and surface tension is explained. The application of these data to calculate heat transfer in different arrangements and external as well as internal mass transfer coefficients is shown. Problems at the end of the two main parts of this chapter illustrate the numerical application of the formulas and the diagrams. 

After writing mass balances, energy balances, and equilibrium relations, we need system property data to complete the formulation of the problem. Here, we divide the system property data into thermodynamic, transport, transfer, reaction properties, and economic data. Examples of thermodynamic properties are heat capacity, vapor pressure, and latent heat of vaporization. Transport properties include viscosity, thermal conductivity, and difiusivity. Corresponding to transport properties are the transfer coefficients, which are friction factor and heat and mass transfer coefficients. Chemical reaction properties are the reaction rate constant and activation energy. Finally, economic data are equipment costs, utility costs, inflation index, and other data, which were discussed in Chapter 2. 

Many practical mass transfer problems involve the diffusion of a species through a plane-parallel medium that does not involve any homogeneous chemical reactions under one-dimensional steady conditions. Such mass transfer problems are analogous to the steady one-dimensioiial heat conduction problems in a plane wall with no heal generation and can be analyzed similarly. In fact, many of the relations developed in Chapter 3 can be used for mass transfer by replacing temperature by mass (or molar) fraction, thermal conductivity by pD g (or CD ), and heat flux by mass (or molar) flux (Table 14-8). 

Then die solution of a mass ditfusion problem can be obtained directly from the analytical or graphical solution of the corresponding heat conduction problem given in Chapter 4. The analogous quantities between heat and mass transfer are summarized in Table 14-11 for easy reference. Tor the case of a seiiii-Lnfinite medium with constant surface concentration, for example, the solution can be expressed in an analogous manner to Eq. 1-45 as... 

We have already considered steady-state one-dimensional diffusion in the introductory sections 1.4.1 and 1.4.2. Chemical reactions were excluded from these discussions. We now want to consider the effect of chemical reactions, firstly the reactions that occur in a catalytic reactor. These are heterogeneous reactions, which we understand to be reactions at the contact area between a reacting medium and the catalyst. It takes place at the surface, and can therefore be formulated as a boundary condition for a mass transfer problem. In contrast homogeneous reactions take place inside the medium. Inside each volume element, depending on the temperature, composition and pressure, new chemical compounds are generated from those already present. Each volume element can therefore be seen to be a source for the production of material, corresponding to a heat source in heat conduction processes. 

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 two-dimensional method has resulted in better agreement than the simplified approach, but the computed conversions are still less than the experimental results. In view of the problems in estimating the radial heat- and mass-transfer rates, and possible uncertainties in kinetic rate data, the comparison is reasonably good. The net effect of allowing for radial heat and mass transfer is to increase the conversion. The computed results are sensitive to rather small variations in the effective thermal conductivities and diffusivities, which emphasizes the need for the best possible information concerning these quantities. 

H.-E. Jeong and J.-T. Jeong, Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel, International Journal of Heat and Mass Transfer 49, 2151-2157 (2006). 

The Lewis number appears in double diffusive problems of combined heat and mass transfer. From Table 3.5.2 it can be seen that in gases all the transport effects are of the same order, but in liquids conduction heat transfer is the controlling mechanism on a large scale. 

If the heat and mass transfer Peclet numbers are large, then it is reasonable to neglect molecular transport relative to convective transport in the primary flow direction. However, one should not invoke the same type of argument to discard molecular transport normal to the interface. Hence, diffusion and conduction are not considered in the X direction. Based on the problem description, the fluid velocity component parallel to the interface is linearized within a thin heat or mass transfer boundary layer adjacent to the high-shear interface, such that... 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

The differential form of the eneigy balance for a multicomponent mixture can be written in a variety of forms. It would contain terms representing heat conduction and radiation, body forces, viscous dissipation, reversible work, kinetic energy, and the substantial derivative of the enthalpy of the mixture. Its formulation is beyond the scope of this chapter. Certain simplified forms will be used in later chapters in problems such as simultaneous heat and mass transfer in air-water operations or thermal effects in gas absoiben. 

ABSTRACT. Most important elementary processes governing the behaviour of the converter (chemical reactions, heat and mass transfer processes to the catalyst surface, hydrodynamic, heat conduction and heat loss) are first described. Next, published models are reviewed and main modeling problems are considered. Then, the structure of a fairly general model is proposed. The behaviour of the converter at steady state and in the transient state is reported and the sensitivity of the model to various parameters is discussed. It is concluded that the lack of accurate data (especially kinetic data) makes the predictions with available models only qualitative assessments. However, many design or operating improvements can be deduced from these models. 

In addition to these kinetic steps, there are also physical processes of heat and mass transfer to be considered. The external transport problem is one of heat and species exchange through the boundary layer between the surrounding bulk fluid and the catalyst surface (Figure 5). Concentration and temperature gradients are necessarily present in this case and would have to be accounted for in the modeling equations. Also, there is often an internal transport problem of heat conduction through the catalytic material -- and in the case of porous catalyst particles, an internal diffusion problem as well. Internal transport problems are beyond the scope of this paper. It must be noted, however, that any model intended to describe real-life systems will have to account for these effects. 

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