Radiation boundary conditions

This power law decay is captured in MPC dynamics simulations of the reacting system. The rate coefficient kf t) can be computed from — dnA t)/dt)/nA t), which can be determined directly from the simulation. Figure 18 plots kf t) versus t and confirms the power law decay arising from diffusive dynamics [17]. Comparison with the theoretical estimate shows that the diffusion equation approach with the radiation boundary condition provides a good approximation to the simulation results. 

Another virtue of the procedure is that it can explicitly take into account a partially diffusion-controlled recombination reaction in the form of Collins-Kimball radiation boundary condition—namely, j(R, t) = -m(R, t) where j(R, t) is the current density at the reaction radius and K is the reaction velocity k— < > implies a fully diffusion-controlled reaction. Thus, the time dependence of e-ion recombination in high-mobility liquids can also be calculated by the Hong-Noolandi treatment. 

As will be shown below, writing this equation as a matrix equation is rather straightforward. Similar techniques can be used for the so-called radiation boundary conditions which involve linear combinations of concentration and concentration gradient and appear in connection with elemental fractionation between adjacent phases. 

This is called the partially reflecting or radiation boundary condition by analogy to the heat conduction equation analyses. It is also variously called a mixed, an inhomogeneous or the Robbins boundary condition because it mixes the value of the dependent variable p and the first... 

Monchick [36, 273] has used the diffusion equation and radiation boundary conditions [eqns. (122) and (127)] to discuss photodissociative recombination probabilities. His results are similar to those of Collins and Kimball [4] and Noyes [269]. However, Monchick extended the analysis to probe the effect of a time delay in the dissociation of the encounter pair. It was hoped that such an effect would mimic the caging of an encounter pair. Since the cage oscillations have periods < 1 ps, and the diffusion equation is hardly adequate over such times (see Chap. 11, Sect. 2), this is a doubtful improvement. Nor does using the telegraphers equation (Chap. 11, Sect. 3.3) help significantly as it is only valid for times longer than a few picoseconds. 

To describe quantitatively the diffusion-controlled tunnelling process, let us start from equation (4.1.23). Restricting ourselves to the tunnelling mechanism of defect recombination only (without annihilation), the boundary condition should be imposed on Y(r,t) in equation (4.1.23) at r = 0 meaning no particle flux through the coordinate origin. Another kind of boundary conditions widely used in radiation physics is the so-called radiation boundary condition (which however is not well justified theoretically) [33, 38]. The idea is to solve equation (4.1.23) in the interval r > R with the partial reflection of the particle flux from the sphere of radius R ... 

Gutowski Method. An alternative approach is given by Gutowski (G10), who considers the particular case of a radiation boundary condition for the solidification of a semi-infinite liquid mass. The ambient temperature is considered to be constant, and the liquid is initially at the fusion temperature. The heat equation in the solid phase is formulated in the integral form... 

Consider the one dimensional TISE in Eq. (24), where we allow x to vary between -L and L, i.e., in the interaction region only. In order to solve this equation, we must supplement it with some boundary conditions. Although motivated from different quantum phenomena, Siegert [30] was the first to introduce the idea of solving the TISE with outgoing BCs, also known as Siegert boundary conditions or radiation boundary conditions. In one dimension, these outgoing BCs read... 

Laminar Boundary Layer Flow Along a Flat Plate with Radiation Boundary Condition... 

Figure 14.5 Flow along a flat plate with radiation boundary condition.

From time to time we have mentioned that thermal conductivities of materials vary with temperature however, over a temperature range of 100 to 200°C the variation is not great (on the order of 5 to 10 percent) and we are justified in assuming constant values to simplify problem solutions. Convection and radiation boundary conditions are particularly notorious for their nonconstant behavior. Even worse is the fact that for many practical problems the basic uncertainty in our knowledge of convection heat-transfer coefficients may not be better than 20 percent. Uncertainties of surface-radiation properties of 10 percent are not unusual at all. For example, a highly polished aluminum plate, if allowed to oxidize heavily, will absorb as much as 300 percent more radiation than when it was polished. 

We divide the wall into five nodes as shown and must express temperatures in degrees Kelvin because of the radiation boundary condition. For node l the transient energy equation is... 

When approaching a numerical solution recognize the large uncertainties present in convection and radiation boundary conditions. Do not insist upon,... 

The heat conduction equation is first order in time, and thus the initial condition cannot involve any derivatives (it is limited to a specified temperature). However, the heal conduction equation is second order in space coordinates, and thus a boundary condition may involve first derivalives at the boundaries as well as specified values of temperature. Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions. 

In some ca.se.s, such as those encountered in space and cryogenic applications, a heat transfer surface is surrounded by an evacuated space and thus there is no convection heat transfer between a surface and the surrounding medium. In such cases, radiation becomes the only mechanism of heat transfer between the surface under consideration and llie surroundings. Using an energy balance, the radiation boundary condition on a surface can be expressed as... 

For one-dimensional heat transfer in llie jr-direction in a plate of thickness L, the radiation boundary conditions on both surfaces can be expressed as (Fig. 2-36)... 

The radiation boundary condition involves the fourth power of temperature, and thus it is a nonlinear condition. As a result, the application of this boundary condition results in powers of the unknown coefficients, which makes it difficult to determine them. Therefore, it is tempting to ignore radiation exchange at a surface during a heat transfer analysis in order to avoid the complications associated with nonlinearity. This is especially the case when heat transfer at the surface is dominated by convection, and tlie role of radiation is niinor. 

Radiation boundary conditions on both surfaces of a plane wall. 

C Why do we tiy to avoid the radiation boundary conditions in heat transfer analysis ... 

Consider a spherical shell of inner radius r outer radius Tj, thermal conductivity k, and emisslvity e. The outer surface of llie shell is subjected to radiation to surrouuding surfaces at but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell. 

Tien attempting to gel an analytical solution to a physical problem, there is always the tendency to oversimplify the problem to make the mathematical model sufficiently simple to warrant an analytical solution. Therefore, it is common practice to ignore any effects that cause mathematical complications such as nonlincarities in the differential equation or the boundary conditions. So it comes as no surprise that nonlinearities such as temperature dependence of tliernial conductivity and tlie radiation boundary conditions aie seldom considered in analytical solutions. A maihematical model intended for a numerical solution is likely to represent the actual problem belter. Therefore, the numerical solution of engineering problems has now become the norm rather than the exception even when analytical solutions are available. 

Note that thermodynamic temperatures must be used in radiation heat transfer calculations, and ail temperatures should be expressed in K or R when a boundary condition involves radiation to avoid mistakes. We usually try to avoid the radiation boundary condition even in numerical solutions since it causes the finite difference equations to be nonlinear, wlu ch are more difficult to solve. 

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