Convection Boundary Condition

The Fesults aFe stFongly influenced by the boundary conditions. Which values of the heat transfer coefficients are appropriate depends on the movement of the air surrounding the pipe, which is a stochastic quantity. It is recommended to use more realistic boundary conditions (convective heat transfer) and to provide for a safety factor. Additionally one has to be aware of the fact that the boundary conditions and values of coefficients may be modified by random events such as... 

Outlet Boundary Conditions Convection dominates the heat transport at the outlet side and thus we assume no convective heat flux defined as... 

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

In the absence of convective effect, the profiles of > between any two adjacent bubbles exhibits an extremum value midway between the bubbles. Therefore, there exists around each bubble a surface on which d jdr = 3(C )/3r = 0, and hence the fluxes are zero. Using the cell model [Eqs. (158) or (172)] one obtains the following boundary conditions For t > 0... 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

The boundary conditions for engineering problems usually include some surfaces on which values of the problem unknowns are specified, for instance points of known temperature or initial species concentration. Some other surfaces may have constraints on the gradients of these variables, as on convective thermal boundaries where the rate of heat transport by convection away from the surface must match the rate of conductive transport to the surface from within the body. Such a temperature constraint might be written ... 

Figure 3. Finite element simulation of plane Couette flow with thermal dissipation and conductive heat transfer. (f) — fixed temperature condition (c) — convective boundary condition.

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

Together with the boundary condition (5.4.5) and relationship (5.4.6), this yields the partial differential equation (2.5.3) for linear diffusion and Eq. (2.7.16) for convective diffusion to a growing sphere, where D = D0x and = Cqx/[1 + A(D0x/T>Red)12]- As for linear diffusion, the limiting diffusion current density is given by the Cottrell equation... 

Solving Eq. (16) with the boundary conditions of saturation solubility at the solid surface, Cs(0), sink conditions in the bulk solution, and assuming no convection or reaction contributions, yields... 

In fact, since the unsteady-state transport equation for forced convection is linear, it is possible in principle to derive solutions for time-dependent boundary conditions, starting from the available step response solutions, by applying the superposition (Duhamel) theorem. If the applied current density varies with time as i(t), then the local surface concentration at any time c0(x, t) is given by... 

Farmer (6) reviewed the various diffusion models for soil and developed solutions for several of these models. An appropriate model for field studies is a nonsteady state model that assumes that material is mixed into the soil to a depth L and then allowed to diffuse both to the surface and more deeply into the soil. Material diffusing to the surface is immediately removed by diffusion and convection in the air above the soil. The effect of this assumption is to make the concentration of a diffusing compound zero at the soil surface. With these boundary conditions the solution to Equation 8 can be converted to the useful form ... 

One conclusion from these results is that the axial diffusion model begins to fail as Pe, - small, when an open boundary condition is used at the outlet. The case Pe, - small means increasing backmixing, or that the diffusive flux becomes increasingly significant compared with the convective flux. For an open boundary condition, it is also questionable whether the actual response C(e) can be identified with E(B). Furthermore, regardless of the boundary conditions chosen, it is difficult to envisage that cA... 

If we can define the radiative terms a priori, we are simply adding a constant to the RHS. An expedient transformation reduces the boundary condition to its original convective form ... 

For a fully developed fire, conduction commonly overshadows convection and radiation therefore, a limiting approximation is that h hk, which implies Tw T. This result applies to structural and boundary elements that are insulated, or even to concrete structural elements. This boundary condition is conservative in that it gives the maximum possible compartment temperature. 

PROFILE is a biogeochemical model developed specially to calculate the influence of acid depositions on soil as a part of an ecosystem. The sets of chemical and biogeochemical reactions implemented in this model are (1) soil solution equilibrium, (2) mineral weathering, (3) nitrification and (4) nutrient uptake. Other biogeochemical processes affect soil chemistry via boundary conditions. However, there are many important physical soil processes and site conditions such as convective transport of solutes through the soil profile, the almost total absence of radial water flux (down through the soil profile) in mountain soils, the absence of radial runoff from the profile in soils with permafrost, etc., which are not implemented in the model and have to be taken into account in other ways. 

Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer <5, is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is ... 

Electrophoretic separations in the purely liquid phase, as used by Tiselius, were complicated by the need to keep stable boundary conditions. Problems arose from convectional and gravitational mixing. In the 1950s various attempts were made to increase the stability and thus the reproducibility of the procedure by using filter paper, or more effec-... 

Consider a droplet of radius r. If the droplet is vaporizing, the fluid will leave the surface by convection and diffusion. Since at the liquid droplet surface only A exists, the boundary condition at the surface is... 

The uniformflux of oxygen, S, into the fluid along fhe Pt surface and zero flux along the Au surface provide boundary conditions for the convection-diffusion equation. 

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