One-Dimensional, Unsteady Conduction Calculation

Example 3 One-Dimensional, Unsteady Conduction Calculation As an example of the use of Eq. (5-21), Taole 5-1, and Table 5-2, consider the cooking time required to raise the center of a spherical, 8-cm-diameter dumpling from 20 to 80°C. The initial temperature is uniform. The dumpling is heated with saturated steam at 95°C. The heat capacity, density, and thermal conductivity are estimated to be c = 3500 J/(kg K), p = 1000 kg/m3, and k = 0.5 W/(m K), respectively. 

Because the heat-transfer coefficient for condensing steam is of order 104, the Bi — limit in Table 5-2 is a good choice and dj = k. Because we know the desired temperature at the center, we can calculate 0/0 and then solve (5-21) for the time. 

Example 4 Rule of Thumb for Time Required to Diffuse a Distance R A general rule of thumb for estimating the time required to diffuse a distance R is obtained from the one-term approximations. Consider the equation for the temperature of a flat plate of thickness 2R in the limit as Bi — From Table 5-2, the first eigenvalue is 8 = k/2, and from Table 5-1, 

When t = R2/a, the temperature ratio at the center of the plate (f = 0) has decayed to exp( —irV4), or 8 percent of its initial value. We conclude that diffusion through a distance R takes roughly R2/a units of time, or alternatively, the distance diffused in time t is about (at).  

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Two approaches can be used for calculating interparticle and particle surface collision heat transfer (Amritkar et al., 2014). The first approach is based on the quasi-steady state solution of the coUisional heat transfer between two spheres (Vargas and McCarthy, 2002). The other approach is based on the analytical solution of the one-dimensional unsteady heat conduction between two semi-infinite objects. This approach was proposed by Sun and Chen (1988) based on the analysis of the elastic deformation of the spheres in contact. 

So far, we have considered the explicit scheme of finite-difference formulations and its stability criterion for an illustrative example. The use of the explicit scheme becomes somewhat cumbersome when a rather small Ax is selected to eliminate the truncation error for accuracy. The Ai allowed then by the stability criterion may be so small that an enormous amount of calculations may be required. We now intend to eliminate this difficulty by giving different forms to the equations resulting from the finite-difference formulation. Let us take the case of one-dimensional conduction in unsteady problems, for which we obtained the difference equation given by Eq. (4.50). Consider a formulation of the problem in terms of backward rather than forward differences in time. That is, decrease the time from f +i = (n + l)Ai to tn = nAt. Thus we obtain... 

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