Diffusion and heat conduction

This section deals with problems involving diffusion and heat conduction. Both diffusion and heat conduction are described by similar forms of equation. Pick s law for diffusion has already been met in Sec. 1.2.2 and the similarity of this to Pourier s law for heat conduction is apparent. 

In diffusional mass transfer, the transfer is always in the direction of decreasing concentration and is proportional to the magnitude of the concentration gradient the constant of proportionality being the diffusion coefficient for the system. 

The analogy also extends to Newton s equation for momentum transport, where 

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Other cases, neglecting heat effects would cause serious errors. In such cases the mathematical treatment requires the simultaneous solution of the diffusion and heat conductivity equations for the catalyst pores. 

This chapter demonstrated the computational simplification that is possible in systems consisting of a one-dimensional chain of coupled reservoirs, which arise in diffusion and heat conduction problems. In such systems each equation is coupled just to its immediate neighbors, so that much of the work involved in Gaussian elimination and back substitution can be avoided. I presented here two subroutines, GAUSSD and SLOPERD, that deal efficiently with this kind of system. 

In Chapter 7 I showed how much computational effort could be avoided in a system consisting of a chain of identical equations each coupled just to its neighboring equations. Such systems arise in linear diffusion and heat conduction problems. It is possible to save computational effort because the sleq array that describes the system of simultaneous linear algebraic equations that must be solved has elements different from zero on and immediately adjacent to the diagonal only. 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

Calcs rate of flame propagation for a specific relation between diffusion and heat conductance) 2) Ya.B. Zel dovich, Natl-AdvisoryComm Aeronaut, Tech Mem No 1282,... 

CA 42, 5229(1948)(Theory of propagation of flame. States conditions in an expl chem reaction necesssry for propagation of the flame at a const rate. Calcs this rate for a definite relationship between diffusion and heat conductance. Evaluates the effect of chain reactions on the propagation of the flame) 4) B. Karlovitz, JChemPhys 19, 541-46(1951) Sc CA 45, 9341 (1951)(Theory of turbulent flames) 5) G. Klein, Phil-TransRoySocLondon 249, 389—415 (1957)... 

Dissolution measurements under the microscope, therefore, are a powerful tool for selecting solvents or plasticizers. Measurements of elastic range extension give insight into structural changes caused by plasticizers. Thermal diffusivity and heat conductivity measurements are recommended to detect side group and other transitions in polymers and other substances. 

An increased temperature is achieved in diffusive combustion due to the fact that only part of the fuel is burned, but the heat propagates in an even smaller part of the mixture. According to the laws of diffusion and heat conduction, complete extraction of the fuel by diffusion would lead to distribution of the heat throughout the entire mixture, and the diffusive mechanism would not yield a temperature increase compared to normal propagation. 

Frank-Kamenetskii D. A. Diffuziia i teploperedacha v khimicheskoi kinetike [Diffusion and Heat Conduction in Chemical Kinetics]. Moscow Izd-vo AN SSSR, 266 p. (1947). 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The usefulness of the flow terms as common characteristics for transport processes allows them to illustrate such seemingly diverse processes as convection, momentum transport (viscosity), diffusion and heat conductance. To simplify the written expression, the flux components of the four processes are expressed in Eq. (7-3) in the direction of one axis of the coordinate system whereby, instead of the partial derivative for the function, a variable and useful form of the derivative expression is used ... 

Because of the special form of the sums which appear in the calculation of the pressure, the KrSnig model leads in this problem to the same results as the later, more-refined scheme of ClaUBius. In other problems, however, such as diffusion and heat conduction, this is no longer the case. 

Pore diffusion and heat conduction (combined effect) 5 < 65... 

So, the reaction rate is uniquely defined by the corresponding affinity, since J >eq becomes constant due to uniform concentration when a system is in the vicinity of global equilibrium with fast diffusion and heat conduction processes. Comparing Eq. (9.120) with Eq. (9.124), the coefficient L is defined by... 

J3 Diffusion and Heat Conduction in the Porous Network, Giving the Drying Rate for the Falling Rate Period... 

Let us consider a steady flow for which a Cartesian coordinate system (x, y, z) can be established such that -h x is the principal flow direction and all flow properties are independent of z. In this two-dimensional (x, y) flow, it will further be assumed that except in a layer extending parallel to the principal flow direction, all flow properties vary so slowly that transport effects are negligibly small. For convenience, the viscous, diffusive, and heat-conducting layer will be placed in the vicinity of the plane y = 0 (which, for example, may represent a stationary flat plate, or may divide two parallel... 

All axial diffusion and heat conduction terms can be ignored when compared to their radial counterparts. 

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