Analogy Diffusion with heat conduction

In conjunction with heat conduction we will also investigate mass diffusion. As a result of the analogy between these two molecular transport processes many results from heat conduction can be applied to mass diffusion. In particular the mathematical methods for the evaluation of concentration fields agree to a large extent with the solution methods for heat conduction problems. 

A host of solutions of this problem are known from the analogous case of heat conduction and are easily adapted by substituting concentration and diffusivity for temperature and thermal conductivity. Many of these solutions, particularly those dealing with finite geometries, are forbidding in form and difficult to apply in practice. The intent here is to draw the reader s attention to some simple solutions, which are particularly useful in an environmental and biological context. 

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

The equation for the conservation of energy is similar to that for mass conservation. The equation is obtained following similar steps as the diffusion equation starting from the equation for the conservation of energy, combining it with the constitutive heat conduction law (Fourier s law), which is similar to Pick s law (in fact. Pick s law was proposed by analogy to Fourier s law), the following heat conduction equation (Equation 3-1 lb) is derived ... 

Pick s Law. Pick s law is a physically meaningful mathematical description of diffusion that is based on the analogy to heat conduction (Pick, 1855). Let us consider one side of our control volume, normal to the x-axis, with an area 4r, shown in Figure 2.3. Pick s law describes the diffusive flux rate as... 

Fick s first law of diffusion (analogous with the equation of heat conduction) states that the mass of substance dm diffusing in the x direction in a time df across an area A is proportional to the concentration gradient dc/dx at the plane in question ... 

Assuming some simplifications, analytical solutions for the transport equation may be inferred from arguments by analogy with the basic equations of heat conduction and diffusion (e.g. Lau et al. (1959), Sauty (1980), Kinzelbach (1983), and Kinzelbach (1987)). 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

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). 

Transient mass difliision in a stationary medium is analogous to transient heat transfer provided that the solution is dilute and thus the density of the medium p is constant. In Chapter 4 we presented analytical and graphical solutions for one-dimensional transient heat conduction problems in solids with constant properties, no heat generation, and uniform initial temperature. The analogous one-dimensional transient mass diffusion problems satisfy these requirements ... 

As a result of this many solutions to the heat conduction equation can be transferred to the analogous mass diffusion problems, provided that not only the differential equations but also the initial and boundary conditions agree. Numerous solutions of the differential equation (2.342) can be found in Crank s book [2.78]. Analogous to heat conduction, the initial condition prescribes a concentration at every position in the body at a certain time. Timekeeping begins with this time, such that... 

The DE (3-95) is identical in form to the familiar heat equation for radial conduction of heat in a circular cylindrical geometry. Thus we see that the evolution in time of the steady Poiseuille velocity profile is completely analogous to the conduction of heat starting with an initial parabolic temperature profile -(1 - r2)/4. In our problem, the final steady velocity profile is established by diffusion of momentum from the wall of the tube so that the initial profile for w eventually evolves to the asymptotic value uT - 0 as 1 oo. The characteristic time scale for any diffusion process (whether it is molecular diffusion, heat conduction, or the present process) is (f y cli fl iisivity ), where tc is the characteristic distance over which diffusion occurs. In the present process, tc = R and the kinematic viscosity v plays the role of the diffusivity so that... 

Diffusion of gases, vapors, and liquids in solids, however, is a more complex process than the diffusion in fluids because of the heterogeneous structure of the solid and its interactions with the diffusing components. As a result, it has not yet been possible to develop an effective theory for the diffusion in solids. Usually, diffusion in solids is handled by the researchers in a manner analogous to heat conduction. In the following paragraphs typical methods are described for the development of semiempirical correlations for diffusivity. 

Models Based on a Desorption-Dissolution-Diffusion Mechanism in a Porous Sphere. The precursor of these models was the application by Bartle et. al [20] of the Pick s law of diflusion (or the heat conduction equation, i.e. the Fourier equation) to SFE of spherical particles. In doing so they had to assume an initial uniform distribution of the material extracted (in this specific case 1-8 cineole) from rosemary particles. Since Pick s law of difiusion from a sphere is analogous to a cooling hot ball (Crank [21] vs Carslaw and Jaeger [22]), this type of models have been considered to be analogous to heat transfer. This model was also used by Reverchon and his co-workers [23] and [24] to SFE of basil, rosemary and marjoram with some degree of success. 

In his first diffusion paper [3], Pick interpreted the experiments from Graham with interesting theories, analogies, and quantitative experiments. He showed that diffusion could be described on the same mathematical basis as Pourier s law of heat conduction [5] and Ohm s law of electricity. Pick s first law of diffusion can be written as... 

In a static system where there is only molecular diffusion and chemical reaction, we have an analog to steady-state heat conduction with heat generation [see equations (5-17) through (5-20)]. Hence the applicable form of equation (10-14) is... 

Thus far, the diffusion process has been entirely analogous to that of heat conduction in solids, with the exception that for both polymers and polymeric composites D is of order 10 mm /s which is about 6 orders of magnitude smaller than the thermal diffusivity k. 

Transport of heat by conduction, and of matter by diffusion, follows analogous mathematical principles, namely Fourier s law of heat conduction and Fick s first law of diffusion. Both are differential equations. If we simplify the problem and start with the one-dimensional form, we consider two ordinary differential equations. 

In order to understand and model the effect of absorbed moisture on adhesives we need to investigate moisture transport mechanisms and how they can be represented mathematically. Mass (or molecular) diffusion describes the transport of molecules from a region of higher concentration to one of lower concentration. This is a stochastic process, driven by the random motion of the molecules. This results in a time-dependent mixing of material, with eventual complete mixing as equilibrium is reached. Mass diffusion is analogous to other types of diffusion, such as heat diffusion, which describes conduction of heat in a solid material, and hence similar mathematical representations can be made. 

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