Radiative-convective equilibrium

FIGURE 5 Pure-radiative (dashed lines) and radiative-convective equilibrium (solid lines) temperature profiles for four different optical depths, and for f =0.30. Open circles represent the 1976 standard atmosphere. The radiative-convective adjustment procedure is discussed in the text. 

D. Mihalas, Stellar Atmospheres, W. H. Freeman Co., San Francisco, 1970, 1978. The first edition of this classic text deals with radiative and convective equilibrium and line formation in normal stellar atmospheres, while the second treats non-LTE effects in more detail. 

Table 21.1. Bottom half of Table O.l s 3rd catalyst bed heatup path-equilibrium curve intercept worksheet. Input and output gas enthalpies are shown in rows 43 and 44. Note that they are the same. This is because our heatup path calculations assume no convective, conductive or radiative heat loss during catalytic SO2+V2O2 —> SO3 oxidation, Section 11.9. 1st and 2nd catalyst bed enthalpies are calculated similarly - using Tables J.2 and M.2. 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

In the centre of main-sequence stars, the radiation flux l/4nr2 can become very large, whilst up remains small. Thus the temperature gradient dlnT/dlnP required for radiative equilibrium (Eq. (30)) becomes large, and the material becomes convectively unstable. This gives rise to nuclear-driven convective cores in massive stars, and also convective zones in helium-burning stars. 

In ionisation zones, the adiabatic exponent 7 becomes smaller approaching unity, and k becomes large, so that radiative equilibrium may be violated for small values of the temperature gradient. This gives rise to opacity-driven convection in the envelopes of cool stars. 

Equations (30) and (32) give the temperature gradient V in radiative and convective equilibrium respectively. Where there are situations when it is clear which equation to use, the situation frequently arises when material may be naturally convective, but the convective efficiency is sufficiently low that radiation carries a substantial fraction of the flux. In these cases, must be derived from a suitable theory of convection. 

If C2 is the thermal capacity and p2 the mass of layer 2 per area unit, neglecting convection one can derive for the temperature 2 the equation with the starting temperature 2o in the radiative equilibrium ... 

The final assumption, (6), is that of radiative equilibrium. Here, we assume that the volume absorption rate of IR radiation ial) is equal to the volume rate of emission iactB In). This was discussed in Section IV in terms of the net heating rate H, which is zero in radiative equilibrium (RE). This assumption requires that radiation alone heats or cools the atmosphere. It ignores the important process of convection, which we will include later. 

For the transient region the reader is referred to any heat transfer text for Heisler charts. Alternately, the transient region can be solved numerically by first guessing the location of the interface and iterating the procedure until the heat loss between the cyclic equilibrium region equals that of the steady state region. The heat transfer coefficients include that of convection and radiation, which we will evaluate after treating radiative heat transfer. 

The first boundary condition, Equation 8.9a, implies that we are assuming the wall and bed temperatures are equal at the initial point of contact. Also, the mean free path for the gas at the contact wall is sufficiently large that convective heat exchange by the gas is not in local equilibrium with the conduction through the bed. However, radiative heat transfer can play a vifal role within the penetration layer (Perron and Singh, 1991). As we did for the freeboard, the most practical approach is not only to solve the differential equation but to establish a heat transfer coefficient that can be used for practical calculations. The heat transfer coefficient per unit contact area may be written in terms of the overall heat balance using Newton s law of cooling. 

Steady-state heat transfer Unsteady-state heat transfer Convective heat transfer (heat transfer coefficient) Convective heat transfer (heat transfer coefficient) Radiative heat transfer (not analogous with other transfer processes) Steady-state molecular diffusion Unsteady-state molecular diffusion Convective mass transfer (mass transfer coefficients) Equilibrium staged operations (convective mass transfer using departure from equilibrium as a driving force) Mechanical separations (not analogous with other transfer processes) ... 

Figure 18.10 Effect of conductive, convective plus radiative heat loss and nonattainment of equilibrium on a first catalyst bed s final % SO2 oxidized. The two effects are seen to offset each other. (The heat loss heatup path is steeper because less heat is available to warm the gas.)...

At fuel manifold inlets, gaseous species concentrations are specified as equilibrium compositions of the town gas reformate at 650°C. Steam-to-carbon ratio is kept as 3.06 for this particular steady-state analysis. Both fuel and air gas manifold inlet conditions are summarized in Table 9.5. Mixed convective and radiative heat transfer boundary conditions are applied to the side surfaces of the stack to accurately model the heat exchange with the balance of plant components. Top and bottom surfaces, on the other hand, are assigned with... 

X) the thermal discontinuity goes to zero. On the other hand, if the internal heat source dominates and the atmosphere is very deep, turbulent convection is likely throughout most of the troposphere. This depends critically upon the infrared opacity, and whether the magnitude of the thermal gradient under conditions of radiative equilibrium exceeds that for convective stabihty. 

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