Transient diffusion bodies

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

We will consider transient diffusion of a substance in a semi-infinite body B. At time t = 0, substance A is stored in the body at a concentration cAa. The desired concentration profile cA = cA(x,t) satisfies the following differential equation, under the assumption cD = const... 

Transient Interdiffusion in Two Semi-Infinite Bodies The transient diffusion problem illustrated in Figure 4.8, which involves the interdiffusion of two semi-infinite bodies in contact with one another, is closely related to the previous semi-infinite transient diffusion problem. In fact, if you consider just one-half of the problem domain (e.g., consider the evolution of the diffusion profiles for species A for X > 0), diffusion proceeds exactly like the previous semi-infinite diffusion problem. The only difference is that in this case the interfacial concentration of species A is assumed to be pinned at half of its bulk (i.e., pure material A) value. 

FIGURE 4.11 Transient diffusion of a thin layer between two semi-infinite bodies. 

Problem 4.2. Equation 4.40 in the text provides the solution for the transient diffusion of a thin layer of material between two semi-infinite bodies ... 

The shape factor given by equations 40 through 42 applies to steady-state diffusion. The results are compared to F obtained from the lattice walk simulation of transient diffusion in a cubic pore in Figure 9. An effective pore outlet angle was defined for the cubic pore 0 = tan 1 (w/W) where w is the width of the pore outlet and W is the width of the pore body (recall Figure 7). F values for the cubic pore are the reciprocal of the relative rate shown in Figure 8. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Considep two-dimensional transient heat transfer in an L-shaped solid body that is initially at a uniform temperalure of 90°C and whose cross section is given in Fig. 5-51. The thermal conductivity and diffusivity of the body are k = 15 W/m C and a - 3.2 x 10 rriVs, respectively, and heat is generated in Ihe body at a rate of e = 2 x 10 W/m. The left sutface of the body is insulated, and the bottom surface is maintained at a uniform temperalure of 90°C at all times. A1 time f = 0, the entire top surface is subjected to convection to ambient air at = 25°C with a convection coefficient of h = 80 W/m C, and the right surface is subjected to heat flux at a uniform rate of r/p -5000 W/m. The nodal network of the problem consists of 15 equally spaced nodes vrith Ax = Ay = 1.2 cm, as shown in the figure, Five of the nodes are at the bottom surface, and thus their temperatures are known. Using the explicit method, determine the temperature at the top corner (node 3) of the body after 1,3, 5, 10, and 60 min. 

Transient mass diffusion in bodies of simple geometry with one-dimensional mass flow... 

Unsteady-state or transient heat conduction commonly occurs during heating or cooling of grains. It involves the accumulation or depletion of heat within a body, which results in temperature changes in the kernel with time. The rate at which heat is diffused out of or into a kernel or layer of kernels is dependent on the thermal diffusivity coefficient, a, of the grain ... 

The thermal diffusivity can also be measured directly by employing transient heat conduction. The basic differential equation (Fourier heat conduction equation) governing heat conduction in isotropic bodies is used in this method. A rectangular copper box filled with grain is placed in an ice bath (0°C), and the temperature at its center is recorded [44]. The solution of the Fourier equation for the temperature at the center of a slab is used ... 

A simplifying assumption for diffusive heat transport processes that is often invoked for materials with high thermal conductivities. Under this assumption, the transient temperature within a body subject to heating/cooling is assumed to be spatially uniform at any instant of time. 

FIGURE 4.8 Transient interdiffusion of two semi-infinite bodies. Material A on the left (t < 0), assumed to be initially composed purely of species A, is in contact with material B on the right (x > 0), which is assumed to be initially composed purely of species B. At time f = 0, species A and B begin to interdiffuse. If the initial bulk concentrations and diffusivities of A and B are equal, their concentrations at the interface will be pinned at 1/2 c. The materials will slowly diffuse into one another, but the position of the interface between them will not move. 

Solution To answer this question, it is helpful to first establish a sketch of the problem, as shown in Figure 4.9. Considering first the diffusion of A into B from left to right, we wish to determine the right-side boundary of the interdiffusion region. As detailed in the problem statement, this corresponds to the location (x value) where the concentration of species A falls to 1/10 of its initial bulk value. As shown on the schematic illustration, we will call this location x = 8. Applying this criteria to the solution for the transient interdiffusion of two semi-infinite bodies gives... 

Kirkendal Effect When we previously discussed the transient interdiffiision of two semi-infinite bodies (material A and material B, respectively), we explicitly specified that the diffusion of A in B and that of B in A were identical and could therefore be described by a single diffusion coefficient. In many solids, however, this is not true. For example, the diffusivity of zinc in copper is much larger than the diffiisivity of copper in zinc. If a block of brass (a copper-zinc alloy) and a block of pure copper are bonded together at high temperatures, the zinc atoms will diffuse out of the brass and into the copper at a much faster rate than the copper atoms diffuse into the brass block. The net result is that the effective interface between the brass and copper blocks moves toward the brass, as illustrated schematically in Figure 4.17. This phenomenon is known as the Kirkendal effect and it occurs in many solid-state systems. 

Phillips CG, Jansons KM (1990) The short-time transient of diffusion outside a conducting body. Proc R Soc Lond A 428 431-439... 

It is not straightforward to determine the transient solution that leads to the solid-body rotation of the fluid starting from rest The solution of the Navier-Stokes equations in the form ur=Q,ug r,z,t),u =0), for which the azimuthal component ug evolves in time nnder the effect of a process of vorticity diffusion from the lateral circular wall toward the tank axis, does not portray reality. A secondaiy flow Ur 0 and 0) actually occurs, which carries vorticity from the boundary layer on the horizontal bottom wall into the water layer. Boundary layers play a key part in rotating flows they allow steady-state flows to be established more rapidly. ... 

The center of diffusion is a unique point, because when the molecular coordinate system is centered at CD, the coupling tensor is symmetric. Moreover, for molecules with elements of symmetry (e.g., spheres, cylinders, ellipsoids), Dj. CD vanishes, and it is clear that for such bodies the dipole moment referred to CD is the quantity to be compared to the experimental dipole moment. Similar situations can be expected for globular proteins because their shape is close to a sphere or an ellipsoid. Brownian dynamics simulations of electrooptical relaxation experiment indicated that even for a large molecule such as tRNA, the difference is negligible between transient dichroism and derived dipole moments with and without inclusion of the coupling tensor when the coordinate system is centered at CD. Therefore it can be expected that calculated dipole moments of a molecule relative to its center of diffusion should closely correspond to those derived from orientational behavior. Some possibility for a bias of experimental dipole moments exists, however, since none of the orientation mechanisms considered in Eqs. [96] and [97] includes a possible contribution due to the translation-rotation coupling diffusion tensor. 

The computer display then shows the steady-state values for characteristics such as the thermal conductivity k [W/(mK)], thermal resistance R [m K/W] and thickness of the sample s [mm], but also the transient (non-stationary) parameters like thermal diffusivity and so called thermal absorptivity b [Ws1/2/(jti2K)], Thus it characterizes the warm-cool feeling of textile fabrics during the first short contact of human skin with a fabric. It is defined by the equation b = (Xpc)l, however, this parameter is depicted under some simplifying conditions of the level of heat flow q [ W/m2] which passes between the human skin of infinite thermal capacity and temperature T The textile fabric contact is idealized to a semi-infinite body of the finite thermal capacity and initial temperature, T, using the equation, = b (Tj - To)/(n, ... 

In case of steady state heat conduction, the material property is the conductivity which can be calculated once the heat loss from the body is known and the boundary temperature is measured. In case of transient heat flow, the main factor is the diffusivity a which is equal to the ratio of the conductivity and the heat content of the body. Hence equation 10.8 is the basic governing equation of transient heat transfer with boundary conditions relevant for textile materials. Transient state heat conduction is related to instantaneous conduction of heat fi-om the surface of the body to the clothing. Instantaneous heat transfer can be related to the warmth or coolness to touch and the warm-cool feeling of any clothing can be quantified. 

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