Specified Heat Flux Boundary Condition

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... 

Prognostic models reproduce the process of evolution of the initial condition of the current, temperature, and salinity (density) fields under the action of the boundary conditions (momentum, heat, moisture, and mass fluxes) without any correction for the observational data. Usually, the climatic annual cycle of the variabilities in the circulation and thermohaline water structure is modeled. The calculations are performed until the parameters of this cycle stabilize, i.e. the differences between two successive become lower than a certain specified value. Then, the results obtained (model current, temperature, and salinity fields energy, dynamic, and thermodynamic budgets, etc.) un-... 

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

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. 

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. 

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... 

For the statement of boundary conditions we begin with regular boundaries and consider the finite-difference systems shown in Fig. 4.10(a) for a corner node and a side node. We consider three types of boundary conditions specified temperature, specified heat flux, and heat transfer to ambient. 

For the temperature component of the modeling framework, the initial temperature is set to be the ambient temperature (300 K). Thermal boundary conditions on the tooling surfaces are considered in the forms of a specified boundary temperature or heat flux. The temperatures of the two extreme upper and lower surfaces of the external spacers are assumed to be constant of 300 K. Heat loss due to radiation is considered at the lateral surfaces to be / = — Wq), where/is the heat flux... 

Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 6.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector, and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature, and concentration values, which represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet, or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for 2D system or on a plane for 3D system. In general, there are several types of boundary conditions where the Dirichlet and Neumann boundary conditions are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature, or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the 2D or 3D model, the set of the conservation equations is closed and the computational model can be executed. 

Recall the mathematical classification of boundary conditions summarized in Table 3.5. For example, in energy transport, the first type corresponds to the specified temperature at the boundary the second type corresponds to the specified heat flux at the boundary and the third type corresponds to the interfacial heat transport governed by a heat transfer coefficient. 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

In order to complete the problem, the initial and boundary conditions must be given. The temperature and degree of cine or crystallinity must initially (at time zero) be specified at every point inside the composite and the mandrel. For the latter only the temperature is required. As boundary conditions, the temperatures or heat fluxes at the composite outside diameter and mandrel inner diameter must be specified. 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

Finally, boundary conditions for the inlet and outlet thermal flux has to be specified, as well as the heat flux continuity at the channel/electrode and chan-nel/interconnect interfaces. 

In the above discussion, it was assumed that the surface temperature variation of the plate was specified. The procedure is easily extended to deal with other thermal boundary conditions at the surface. For example, if the heat flux distribution at the surface is specified, it is convenient to define the following dimensionless temperature ... 

To proceed further with the solution, the wall boundary conditions on temperature must be specified. Consideration will first be given to the case where the heat flux at the wall, qw, is uniform and specified. Some discussion of the solution for the case where the wall temperature is kept uniform will be given later. 

This equation allows the 0,. values to be found. The boundary conditions give 0,1 = 0, 2 and either 0,jy = 1 or0,jv = 0, -1 + AY depending on whether the wall temperature or the wall heat flux is specified. Once these values are determined, the heat transfer rate can be found. The local heat transfer rate at the wall at any value of Z is given by using Fourier s law as ... 

The heat flux at the upper and lower surfaces specifies the temperature gradient at the wall, and the necessary boundary conditions... 

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