Specified Heat Flux

One feature of the preceding analysis that may seem unduly restrictive is that we assume the actual temperature is known both in the far field ( the ambient value ) and at the surface of the body. Often, in reality, it is not the temperature that is specified by the physics of the problem, but rather the heat flux, i.e., for this problem 

In this case, the overall heat flux is known (specified), and there is no point in calculating a Nusselt number. Instead, it is the temperature at the surface of the body that is unknown. In addition, the boundary condition (9-2a) is now replaced with (9-65), and we may ask how the analysis of the preceding sections is changed. The job of answering this question is partly left to the problems section at the end of this chapter. However, it is worthwhile to consider some of the issues here. 

if we go back to Section A, we see that some aspects of the nondimensionalization will need to be changed. The definition of a dimensionless temperature according to (9-3) no longer makes sense, because the temperature at the surface of the body is no longer known. Instead, we can see from the form of (9-65) that an appropriate way to define a dimensionless temperature is 

Alternatively, we can think of an appropriate generalized form of the dimensionless temperature as 

when the Brinkman number is small, the governing equation is again precisely (9-7), but now with boundary conditions 

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Saturated nucleate flow boiling of ordinary liquids. To maintain nucleate boiling on the surface, it is necessary that the wall temperature exceed a critical value for a specified heat flux. The stability of nucleate boiling in the presence of a temperature gradient, as discussed in Section 4.2.1.1, is also valid for the suppres-... 

Other passive network solutions are given by Hlinka and Paschkis (H8). Otis (01) employs the Landau transformation for the problem of an ablating slab with uniform initial temperature and specified heat fluxes at the front and back faces,... 

Dewey et al. (D3) present a numerical scheme for the ablation of an annulus with specified heat fluxes at the outer (ablating) surface and at the inner surface. An implicit finite difference technique is used which permits arbitrary variation of the surface conditions with time, and which allows iterative matching of either heat flux or temperature with external chemical kinetics. The initial temperature may also be an arbitrary function of radial distance. The moving boundary is eliminated by a transformation similar to Eq. (80). In addition a new dependent variable is introduced to... 

Some materials exhibit nearly steady mass loss rates when exposed to a fixed radiant heat flux. The surface temperature for these materials reaches a steady value after a short initial transient period, and all terms in Equation 14.7 are approximately constant at a specified heat flux level. L can then be obtained by measuring steady mass loss rates at different radiant heat flux levels, and... 

Using Fourier s law, thq boundary condition on temperature at the wall in the specified heat flux case is ... 

If the problem formulation is to include a specified heat flux qJAj at one of the /th surfaces, we can solve for Eh, from Eq. (8-110) to give... 

Radiation-balance equation for surface with specified heat flux m - F ) - 2 FuJi = J (8-115a)... 

The heat conduction equations above were developed using an energy balance on a differential element inside the medium, and they remain the same regardless of the thermal conditions on tlie surfaces of the medium. That is, the differential equations do not incorporate any information related to the conditions on the surfaces such as the surface temperature or a specified heat flux. Yet we know that the heat flux and the temperature distribution in a medium depend on the conditions at the surfaces, and (he description of a heat transfer problem in a medium is not complete without a full description of the thermal conditions at the bounding surfaces of the medium. The mathematical expressions of the thermal conditions at the boundaries are called the boundat7 conditions. 

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. 

Then the boundary condition at a boundary is obtained by setting the specified heat flux equal to -k(3T/dx) at that boundary. The sign of the specified heat flux is determined by inspection positive if the heat flux is in the positive direction of the coordinate axis, and negative if it is in the opposite direction. Note that it is extremely important to have Ihe correct sign for the specified heat flux since the wrong sign will invert Ihe direction of heat transfer and cause the heat gain to be interpreted as heat loss (fig. 2-29),... 

For a plate of thickness L subjected to heat flux of 50 W/m into the medium from both sides, for example, the specified heat flux boundary conditions can be expressed as... 

FIGURE 2-29 Specified heat flux boundary conditions on both surfaces of a plane wall. 

Some surfaces are commonly insulated in practice in order to minimize heat loss (or heal gain) through them. Insulation reduces heat transfer but does not loialf eliipinate it unless its thickness is infinity. However, heat transfer through a properly insulated surface can be taken to be zero since adequate insulation reduces heat transfer through a surface to negligible levels. Therefore, a well-insulated surface can be modeled as a surface with a specified heat flux of zero, llien the boundary condition on a perfectly insulated surface (at X - 0, for example) can be expressed as (Fig. 2-30)... 

The boundary condition on the outer surface of the bottom of the pan at X = 0 can be approximated as being specified heat flux since it is staled that 90 percent of the 800 W (i.e., 720 W) is transferred to the pan at that surface. Therefore,... 

So far we have considered surfaces subjected to single inode heat transfer, such as the specified heat flux, convection, or radiation for simplicity. In general, however, a surface may involve convection, radiation, and specified heat flux simultaneously. The boundary condition in such cases is again obtained from a surface energy balance, expressed as... 

The boujidary condition on the inner surface of the wall at x = O.is a typical. convectiorj condition since it does pot involve any radiation or specified heat flux. Takihg the direction of heat transfer to be the positive x-direction, the boundary condition on the inner surface can be expressed as... 

Consider a spherical container of inner radius r, outer radius rj, and thermal conductivity k. Express the boundary condition on the inner surface of the container for steady one-dimensional conduction for the following cases (a) specified temperature of 50°C, (b) specified heat flux of 30 W/m toward the center, (c) convection to a medium at 7. with a heal transfer coefficient of/i. 

In these relations, qo i specified heat flux in W/m h is Ihe convection cocfficieiii, /i(ombin d Combined convection and radiation coefficient, T is the temperature of the surrounding medium, T un [Pg.315]

C Consider transient onc-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall ate subjected to specified heat flux, express the stability criterion for this problem in its simplest form. 

We mentioned earlier that the mass diffusion equation is analogous to the heat diffusion (conduction) equation, and thus wc need comparable boundary conditions to determine the species concentration distribution in a medium. Two common types of boundary conditions are the (1) specified species concentration, which corresponds to specified temperature, and (2) specified species flux, which corresponds to specified heat flux. 

C Write three boundary conditions for mass transfer (on a mass basis) for species A at. t — 0 that correspond to specified temperature, specified heat flux, and conveciion boundary conditions in heat transfer. 

Problem 9-7. Heat Transfer from a Sphere at Pe -C 1, with Specified Heat Flux. Consider heat transfer from a solid sphere of radius a that is immersed in an incompressible Newtonian fluid. Far from the sphere this fluid has an ambient temperature Too and is undergoing a uniform flow with a velocity U-x. At the surface of the sphere the heat flux is independent of position with a specified value q. Determine the temperature distribution on the surface of the sphere, assuming that the Peclet number is small. 

F. APPROXIMATE RESULTS FOR SURFACE TEMPERATURE WITH SPECIFIED HEAT FLUX OR MIXED BOUNDARY CONDITIONS... 

There are at least two approaches that we can take to solve problems in which either the heat flux or the mixed-type condition is specified as a boundary condition. If it is desired to determine the temperature distribution throughout the fluid, then we must return to the governing thermal boundary-layer equation (11-6)- assuming that Re, Pe / - and develop new asymptotic solutions for large and small Pr, with either dT/dr] y 0 or a condition of the mixed type specified at the body surface. The problem for a constant, specified heat flux is relatively straightforward, and such a case is posed as one of the exercises at the end of this chapter. On the other hand, in many circumstances, we might be concerned with determining only the temperature distribution on the body surface [and thus dT /dr] [v from (11 98) for the mixed-type problem], and for this there is an even simpler approach that... 

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