Radiative flux

The more general use of Eq, (5-128) is to obtain the set of total interchange areas AS"- which con stitute a complete description of the effect of shape,. size, and emi.s.sivity on radiative flux, independent of the pre.sence or absence of other tran.sfer mechanisms. It may be shown that... 

We now repeat the derivation of the steady-state heat transport limited moisture uptake model for the system described by VanCampen et al. [17], The experimental geometry is shown in Figure 9, and the coordinate system of choice is spherical. It will be assumed that only conduction and radiation contribute significantly to heat transport (convective heat transport is negligible), and since radiative flux is assumed to be independent of position, the steady-state solution for the temperature profile is derived as if it were a pure conductive heat transport problem. We have already solved this problem in Section m.B, and the derivation is summarized below. At steady state we have already shown (in spherical coordinates) that... 

One way to simplify the procedure is to specify a threshold radiative flux. It is assumed that 100% fatalities will occur to anyone exposed to anything above this value. Anyone exposed to a lesser value will be unharmed. Estimate an approximate threshold radiative flux value that will result in the same number of fatalities as the detailed probit calculation. 

A radiative flux QR is imposed on a solid fuel burning in air in a stagnation film mode. The expression for the burning rate is... 

Taylor, M.)., Spectral Acquisition and Calibration Techniques for the Measurement of Radiative Flux Incident upon Propellant, Propellants, Explosives, Pyrotechnics, Vol. 28, 2003, pp. 18. 

Similarly, Chou et al. (1998) used measurements of surface radiative fluxes and satellite radiance data in the Pacific warm pool region to conclude that the effect of clouds was similar to that expected, i.e., that the excess absorption, if it exists, is small. 

Rossow, W. B., and Y.-C. Zhang, Calculation of Surface and Top of Atmosphere Radiative Fluxes from Physical Quantities Based on ISCCP Data Sets. 2. Validation and First Results, J. Geophys. Res., 100, 1167-1197 (1995). 

Introduction Recently, Kitamura and NakamuraW have found that the anomalous gravity darkening occurs in semi-detached binary systems. The exponent of gravity darkening for the secondary components, which is defined by ac = where F is the radiative flux and g is the... 

For the simplicity of analysis, we make 1-dimensional approximation. Then, from these equations, the radiative flux F is given by... 

The principal physical error is probably geometrical. Compared to this, the above assumptions are not unduly restrictive, although extremely fast high-power excursions at low pressures are ruled out by assumptions (2) and (3). Assumption (2) is more nearly fulfilled at high pressures with low liquid heads. Assumption (3) is acceptable in the vapor-film problem even when the radiative flux from the solid surface is appreciable, provided that the liquid (and, of course, vapor) is nearly transparent. 

Dudokovic and co-workers (154, 155) extended the analysis of a QSSM to include difluse-gray radiation. They computed view factors by approximating the shapes of the crystal and melt by a few standard geometrical elements and incorporating analytical approximations to the view factors. Atherton et al. (153) developed a scheme for a self-consistent calculation of view factors and radiative fluxes within the finite-element framework and implemented this scheme in the QSSM. 

Zhang Y-C. Rossow W.B. Lacis A.A. Oinas V. and Mishchenko M.I. (2004). Calculation of radiative fluxes from the surface to top of atmosphere based on ISCCP and other global data sets Refinements of the radiative transfer model and the input data. J. Geophys. Res., 109, 1 -27. 

Since the body is in radiative equilibrium, qx(T) also expresses the spectral radiative flux emitted by the body at the wavelength X. The incident radiation q (T) comes from the black walls of the enclosure at temperature T, and the emission by the walls is not influenced by the body regardless if it is a blackbody or not. Let qxb(T) be the spectral blackbody emissive flux at temperature T. Then,... 

Number of surface and volume zones in enclosure Partial pressure of species k, atm Number of WSGG gray gas spectral windows Total radiative flux originating at surface zone t, W Net radiative flux between zone t and zonej, W Temperature, K... 

Matrix characterization of the radiative energy balance at each surface zone is facilitated via definition of three M X 1 vectors the radiative surface fluxes Q = [ i], with units of watts and the vectors H = [if,] and W = [Wi] both having units of W/m2. The arrays H and W define the incident and leaving flux densities, respectively, at each surface zone. The variable W is also referred to in the literature as the radiosity or exitance. Since W = el-E + pIH, the radiative flux at each surface zone is also defined in terms of E, II, and W by three equivalent matrix relations, namely,... 

Zone Methodology and Conventions For a transparent medium, no more than E = M(M —1)/2 of the M2 elements of the ss array are unique. Further, surface zones are characterized into two generic types. Source-sink zones are defined as those for which temperature is specified and whose radiative flux Q, is to be determined. For flux zones, conversely, these conditions are reversed. When both types of zone are present in an enclosure, Eq. (5-118) may be partitioned to produce a more efficient computational algorithm. Let A/ M t M. represent the total number of surface zones where A/ is the number of source-sink zones and Mf is the number of flux zones. The flux zones are the last to be numbered. Equation (5-118) is then partitioned as follows ... 

Example 11 Calculations of Gas Emissivity and Absorptivity Consider a slab of gas confined between two infinite parallel plates with a distance of separation of L = 1 m. The gas pressure is 101.325 kPa (1 atm), and the gas temperature is 1500 K (2240°F). The gas is an equimolar mixture of C02 and H20, each with a partial pressure of 12 kPa (pc = pw = 0.12 atm). The radiative flux to one of its bounding surfaces has been calculated by using RADCAL for two cases. For case (a) the flux to the bounding surface is 68.3 kW/m2 when the emitting gas is backed by a black surface at an ambient temperature of 300 K (80°F). This (cold) back surface contributes less than 1 percent to the flux. In case (b), the flux is calculated as 106.2 kW/m2 when the gas is backed by a black surface at a temperature of 1000 K (1340°F). In this example, gas emissivity and... 

Having formulated the directed exchange areas, the governing matrix equations for the radiative flux equations at each surface zone and the radiant source term are then given as follows ... 

The total radiative flux Qy at surface Ay and the radiative source term Qi = S are given by... 

Example 12 WSGG Clear plus Gray Gas Emissivity Calculations Methane is burned to completion with 20 percent excess air (50 percent relative humidity at 298 K or 0.0088 mol water/mol dry air) in a furnace chamber of floor dimensions 3 x 10 m and height 5 m. The entire surface area of the enclosure is a gray sink with emissivity of 0.8 at temperature 1000 K. The confined gas is well stirred at a temperature of 1500 K. Evaluate the clear plus gray WSGG constants and the mean effective gas emissivity, and calculate the average radiative flux density to the enclosure surface. 

The solar concenfrafion ratio of a solar collector is defined as the ratio of the aperture area of fhe reflector to fhe area of the receiver (Rabl, 1985). This quantity gives an approximation to the number of times the radiative flux density (W m ) is increased in the surface of the receiver, as compared to the incoming solar radiation. The concentration ratio is commonly expressed as a number of suns for insfance, a collector that increases the radiative flux densify five fimes is said to have a five suns concentration ratio. Thus, the formula for the concentration ratio is... 

Another quantity of interest in the discussions to follow is the radiative flux vector qA(r) (W m sr ), which gives the net spectral flux of radiation along the preferential propagation direction in a given point. It is, in simple terms, intensity summed as a vector over all propagation directions... 

Other authors have also used approximate methods to solve the radiation problem. Li Puma and Yue (2003) used a thin film slurry model which does not include scattering effects. More recently, Li Puma et al. (2004), Brucato et al. (2006), and Li Puma and Brucato (2007) have used six flux models for different geometries. Salaices et al. (2001, 2002) used a model which allows for an adequate evaluation of the absorbed radiation in terms of macroscopic balances, based on radiometric measurements. They measured separately total transmitted radiation and nonscattered transmitted radiation, modeling the decay of both radiative fluxes with concentration by exponential fimctions. 

To solve Equation (38) boimdary conditions which describe the reflection and transmission of radiation at the boimdaries are required. In principle, boimdary conditions can only be established in a rigorous manner for the radiative intensity, not for G, because the optical properties of the interfaces depend on the direction of incidence of radiation. Because the PI approximation solves for an integrated quantity like G instead, approximate boundary conditions must be established (Modest, 2003). One possibility is the Marshak boundary condition (Marshak, 1947), which comes from considering the continuity of the radiative flux through the interface. If this continuity is considered together with the assumption (34) of the PI approximation and Equation (37), the following equation is obtained (Spott and Svaasand, 2000)... 

To model the radiation field for a tubular reactor. Equation (38) is written in cylindrical coordinates. If the tube is slender we can neglect end effects and the radiative flux is independent of the longitudinal variable z, then... 

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