Heat transfer similarity with diffusion problems

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

By storing the energy for evaporation both in the liquid film and in the disk material, the diffusion still operates in a manner similar to a flash evaporator. It does not rely entirely on the transfer of energy through a heat transfer surface. As a result, the scaling problems encountered are similar to those with flash equipment. Scale formation on the disk surface has no measurable effect on performance. Formation of scale within the condenser results in a slight decrease in performance. However, this reduction is small, since the over-all condenser performance is initially low in the diffusion still. 

The difflisional mass transport is driven by a concentration gradient, as described by Pick s law. This is very familiar with Pourier s law, which relates heat transport to a temperature gradient. It is also very similar to Newton s law that relates momentum transport to a velocity gradient. Because of the similarities of these three laws, many problems in diffusion are described with similar equations. Also several of the dimensionless numbers used in heat transfer problems are also used in diffusion mass transfer problems. 

At first sight mathematically these problems are not completely new similar ones have been met in different fields and one can count one s weapons before attacking the formulation and add the necessary complements examples of diffusion of heat or matter (diffusion-absorptions) (diffusion-reactions) [51], heat transfer with reactions c r propagation of vivid heat ( chaleur vive ), diffusion of heat through media including... 

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 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. 

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). 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

The stability problem for an exothermic reaction in a catalyst particle is similar to that for a reaction in a CSTR, in that multiple solutions of the heat and mass balance equations are possible. A typical plot of heat generation and removal rates is shown in Figure 5.11. The values of Qg and Qr are in cal/sec, g, and a is the external area in cm /g. The plot differs from the one for a CSTR (Fig. 5.2) in that the highest possible value for Qg is a mass transfer limit corresponding to Cj = 0 and not to complete conversion. The mass transfer limit increases with temperature because of the increase in diffusivity, and the limit also increases with gas velocity. The heat removal... 

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