Cold boundary problem

The problem identified here is the one in which the cold-boundary difficulty, introduced in Section 2.2.2, first arose. It is clear that the trouble lies not in the mathematics but instead in the mathematical model of the physical situation the mathematical problem defined by equations (42) and (43) with the stated boundary conditions is ill-posed. A further indication of this fact may be obtained by differentiating equation (19) with respect to and using the result dejd = (de/dx)(dxldO = Am, derived from equations (42) and (19). We thereby see that... 

The problem that has been defined here possesses the cold-boundary difficulty discussed in Section 5.3.2 and can be approached by the same variety of methods presented in Section 5.3. The asymptotic approach of Section 5.3.6 has been seen to be most attractive and will be adopted here. Thus we shall treat the Zel dovich number... 

The boundary conditions for equation (35) at x = -f (X) for the burning solid are the same as those for the gaseous flame namely, = 1 and T = 1 [see equations (31) and (33)]. However, as we have already stated, the cold-boundary conditions differ in the two problems. For the burning solid, the definition of A 2 implies that... 

A number of approaches have been suggested to cope with this (somewhat artificial) problem created by the cold boundary . Typical of these is the adaptation of the Arrhenius temperature dependence. It is generally the case that the reaction rate is effectively zero for temperatures up to some ignition temperature Ti > Tq. We can choose to work with a modified temperature dependence f 0) such that f 9) = 0 forO < 9 < 6i and f 0) = f 9) for 9i < 9 <. This introduces a discontinuity and a somewhat arbitrary new... 

The temperature boundary condition is in many problems of buoyant flow, e.g., heat sources (machines) or heat sinks (cold glazings), is of great importance. 

Solution The first problem is that a different value of A Tmi is required for different matches. The problem table algorithm is easily adapted to accommodate this. This is achieved by assigning A Tmin contributions to streams. If the process streams are assigned a contribution of 5°C and flue gas a contribution of 25°C, then a process/process match has a ATmin of (5 + 5) = 10°C and a process/flue gas match has a ATmin of (5 + 25) = 30°C. When setting up the interval temperatures in the problem table algorithm, the interval boundaries are now set at hot stream temperatures minus their Arm contribution, rather than half the global ATmin. Similarly, boundaries are now set on the basis of cold stream temperatures plus their A Tmin contribution. 

We cannot make direct use of Fig. 1 for our problem of ignition of a cold gas by one heated surface, since we restricted ourselves in this drawing to functions which have a maximum at 0 = 0, i.e., those which axe symmetrical with respect to the ordinate axis, which is possible only for symmetric boundary conditions. 

The results of the numerical solution of the conjugation boundary-value problem for the mass exchange (3.106), (3.107) are shown in Fig. 3.20. The initially homogeneously cold air gets the latent heat from the droplet layer and becomes warmer and warmer from cross section 1 to 5, etc. (family (I) of curves in Fig. 3.20,A). In turn, the droplets become cooler their temperature is reduced especially intensively at the EPR entrance (the dashed curve 0) but less and less intensively in subsequent cross sections (the family of curves (II)). The curves TE(z) and t(z) attract each other and meet at the theoretical infinity z —> -oo. The complex relation between the two profiles is testified by the local maxima on TE(z) at early cross sections (curves 1 and 2) and by negative values of the mass flow (3.97) presented in Fig. 3.20,B. A dilative thermal boundary layer grows over the droplet EPR. Its width dE(x) grows theoretically to infinity over the EPR, but the internal portions of all the variables tend to a certain final position within the EPR, 0 < z < 1. 

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