Gray diffuse surface

In addition to the assumptions stated above, we shall also assume that the radiosity and irradiation are uniform over each surface. This assumption is not strictly correct, even for ideal gray diffuse surfaces, but the problems become exceedingly complex when this analytical restriction is not imposed. Sparrow and Cess [10] give a discussion of such problems. The radiosity is the sum of... 

The network method which we have used to analyze radiation problems is an effective artifice for visualizing radiant exchange between surfaces. For simple problems which do not involve too many surfaces the network method affords a solution that can be obtained quite easily. When many heat-transfer surfaces are involved, it is to our advantage to formalize the procedure for writing the nodal equations. For this procedure we consider only opaque, gray, diffuse surfaces. The reader should consult Ref. 10 for information on transmitting and specular surfaces. The radiant-energy balance on a particular opaque surface can be written... 

Gray diffuse surface emissivity Gas emissivity with path length r Monochromatic, unidirectional, surface emissivity Wave number in vacuum, cm" ... 

Property Approximations. Because of the lack of accurate radiative properties in many situations, it is common practice to invoke certain approximations for the property behavior. The most common assumptions are that the surface properties are independent of wavelength (a gray surface), independent of direction (a diffuse surface), that the surface behaves as an ideal mirror (a specular surface), or that the surface is black. The assumption of a gray-diffuse surface is the most commonly invoked. For a surface that is truly both gray and diffuse, Kirchhoff s law applies for all of the property sets that is, a = e, and the computation of radia-... 

When an enclosure can be assumed to be made up of a set of black or gray diffuse surfaces, well-developed techniques are available for determining the exchange among the surfaces. 

In general, one boundary condition (either surface temperature or heat flux) must be specified for every surface of the enclosure. It is possible by more advanced techniques to obtain solutions for the case when some surfaces have both conditions prescribed and others have neither condition specified. This situation causes the solution methods described here to fail, as the equation set is then ill-conditioned. So-called inverse solution methods must be invoked. Methods of handling such a problem are given for black-surfaced enclosures in Ref. 25 and for enclosures with black or gray diffuse surfaces in Refs. 26, 27, and 28. 

Uniform Surface Radiosity. Here, first, we limit consideration to gray diffuse surfaces with uniform radiosity. In that case, Eqs. 7.71 and 7.72 apply. Because of the assumption of gray diffuse surfaces, Eq. 7.71 can be rewritten as... 

Surfaces with Uniform Radiosity. For the case of gray diffuse surfaces with uniform radiosities, the solution of either Eq. 7.82 or 7.83 for q0,k or Eq. 7.86 for the unknown temperatures and heat fluxes requires the solution of a set of simultaneous equations. These are of the form... 

Radiative heat exchange between opaque solid surfaces through a nonparticipating fluid can be accounted for under the assumption of gray-diffusive surface radiation. Computation of the configuration factors (view factors) with account for the shadowing effect is described in detail in [41]. The total radiative flux incoming to the elementary surface element i(i = 1, N, where Ne is the total number of elementary surfaces on the boundary) is... 

For radiating surfaces, ANSYS assumes gray diffuse surfaces (independent of wavelength and direction). 

Gebhart B. Surface temperature calculations in radiant surroundings of arbitrary complexity—for gray, Diffuse Radiation. Int.. Heat Mitss Transfer, vol. 3, no. 4, 19iil. 

The surfaces are opaque, gray, diffuse emitters and have uniform emissivity, e. The thermal conductivity k is uniform through the plates. 

Subject to the restrictions of no scatter and diffuse surface emission and reflection, the above equations are the most general matrix statement possible for the zone method. When P = 1, the directed exchange areas all reduce to the total exchange areas for a single gray gas. If, in addition, K = 0, the much simpler case of radiative transfer in a transparent medium results. If, in addition, all surface zones are black, the direct, total, and directed exchange areas are all identical. 

This important result is applicable to any two gray, diffuse, and opaque surfaces that-fqrm an enclosure. The view factor f 2 depends on the geometry and must hj "determined first. Simplified forms ofRq. 13-36 for some familiar arrangements that fonn a two-surface enclosure are given in Table 13-3. Note that f, 2 = I for of these special cases. 

The net rate of radiation transfer between any two gray, diffuse, opaque surfaces that form an enclosure is given by... 

Consider an enclosure of N diffuse-gray, opaque surfaces, each surface being isothermal and having a uniform radiosity and irradiation. The radiative energy balance for the ith surface gives = A J - G,), where J = + p G = + (1- j)Gj. Combining these two equations to... 

First, take the case of determining the radiative transfer between two infinite parallel gray diffuse directly opposed plates separated by a distance D and of width W, with known temperatures T, and T2 and equal emissivities e (Fig. 7.18). No i attenuating medium is present between the surfaces. 

For non-black surfaces, the net interchange must account for the reflected radiant energies from other surfaces. For refractory material surfaces encountered in rotary kilns, perhaps one can assume that all surfaces are gray, diffuse, and opaque, that is,... 

Hence, knowledge of the emissivity completely characterizes the surface. Of course, non-gray and non-diffuse surfaces bring several degrees of complication into the calculations. By employing the concept of radiosity and irradiation illustrated in Figure 7.10, the net thermal radiation heat exchange between surfaces / can be computed by... 

Evaluation of the AS" s that charac terize an enclosure involves solution of a system of radiation balances on the surfaces. If the assumption is made that all the zones of the enclosure a re gray and emit and reflec t diffusely, then the direct-exchange area ij, as evaluated for the black-siirface pair A and Aj, applies to emission and reflections between them. If at a surface the total leaving-flnx density, emitted phis reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form ... 

Fig. 3. Schematic diagram of the spot photobleaching method of FRAP. (A) Darkened circles represent fluorescently labeled molecules evenly distributed over a two-dimensional surface (assumed to be an infinite plane). (B) White and light gray circles represent the initial postbleach distribution of photobleached molecules within a 1-pm diameter spot. (C) Redistribution of photobleached and unbleached molecules as a consequence of random diffusion over time. (D) Curve representing the fluorescence intensity within the l-pm diameter spot monitored over time arrows a, b, and c indicate the time-points that correspond to their respective panels. The rate of recovery from point b to point c is used to determine the diffusion constant. The magnitude of the recovery is determined by comparing the fluorescence intensity at point c with the initial intensity at point a, and is used to determine the mobile fraction.

Atherton et al. (153) have extended the calculations to include diffuse-gray radiation between components of the enclosure and reached essentially the same conclusions regarding the stability of the process. However, they discovered a new mechanism for the damped oscillation of the crystal radius caused by the radiative interaction between the crystal surface just above the melt level and the hot crucible wall. These oscillations are especially apparent when the vertical temperature gradient in the crystal is low, so that radiative heat transport has a dominating influence. 

Stafford, U. Gray, K. A. Kamat, P. V. Varma, A. An in situ diffuse reflectance FTIR investigation of photocatalytic degradation of 4-chlorophenol on a Ti02 powder surface, Chem. Phys. Lett. 1993, 205, 55. 

Very long, thin fins of thickness b, width W are attached to a black base that is maintained at a constant temperature Tb, as shown in the figure. There is a larger number of fins. The fin surface is diffuse-gray, and they are in a vacuum at temperature, Te = 0 K. Write the equation that describes the local fin temperature. 

In summary, we outline the computational procedure to be followed for numerical solution of radiation heat transfer between diffuse, gray surfaces. This basic procedure is the same for a hand computation, calculation with a minicomputer, or a large computer. 

Related Calculations. If the six surfaces are not black but gray (in the radiation sense), it is nominally necessary to set up and solve six simultaneous equations in six unknowns. In practice, however, the network can be simplified by combining two or more surfaces (the two smaller end walls, for instance) into one node. Once this is done and the configuration factors are calculated, the next step is to construct a radiosity network (since each surface is assumed diffuse, all energy leaving it is equally distributed directionally and can therefore be taken as the radiosity of the surface rather than its emissive power). Then, using standard mathematical network-solution techniques, create and solve an equivalent network with direct connections between nodes representing the surfaces. For details, see Oppenheim [8],... 

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