Boundary conditions constant surface temperature

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

The most important boundary condition in heat transfer problems encountered in polymer processing is the constant surface temperature. This can be generalized to a prescribed surface temperature condition, that is, the surface temperature may be an arbitrary function of time T 0, t). Such a boundary condition can be obtained by direct contact with an external temperature-controlled surface, or with a fluid having a large heat transfer coefficient. The former occurs frequently in the heating or melting step in most... 

Whereas the circumferential variations of the local wall shear stress (i.e., the momentum flux) in itself are not of interest in the study of the BSR, the analogous variations in mass flux or surface concentration are indeed. In Ref. 15 a graph is presented of the local heat flux relative to the circumferential average, for the constant-temperature boundary condition, as a function of a and s/dp. These data are based on a semianalytical solution of the governing PDE, following the procedure described by Ref. 8 (see Section II.B.2). At a relative pitch of 1.2 the local flux at a = 0 is ca. 64% lower than the circumferential average at a relative pitch of 1.5 the flux at a = 0 is still ca. 20% lower than the circumferential average. In the case of a constant surface temperature, the local heat fluxes are directly proportional to the local Nusselt (or Sherwood) numbers. 

Of course, if our objective were to calculate the complete temperature distribution in the fluid, we would never pose the problem in this form because 9S (x) is unknown and thus not satisfactory as a boundary condition. However, our goal here is to determine 9S (x) for cases in which dO/dY y 0 or [(90/97) + (kij k( l)0]Y=0 is specified, and the objective in writing the problem in the form (11 99) (11 100) is to show that we can evaluate 0S (x) directly from the asymptotic solutions obtained earlier for a constant surface temperature, 0S = 1. The key to transforming from our original solutions to a formula for 0S (x) satisfying (11—99)— (11-100) is that the governing equations and boundary conditions for 0 are all linear. 

Uniform inlet velocity or constant pressure is often used at the inlet along with no-slip velocity at all walls. At the outlet, mass flow rate is specified. The mass flow outlet adjusts the exit pressure such that a target mass flow rate (i.e., mass flow at the inlet) is obtained at convergence. The top wall of the chaimel is maintained at a constant surface temperature or surface gas concentration and the rest of the walls are subjected to adiabatic condition. Such a boundary condition is used just to analyze the gas flow channel in... 

Figure 11.10. Body with boundary condition of first kind (constant surface temperature). The problem is to find temperature at (T), a distance (y) below the surface, time (/) after the surface temperature equals (7 ).

Here, the concentration at x = 0 always remains constant contrary to the previous example, where a fixed concentration is introduced once. This problem is analogous to transient conduction in semi-infinite solid with constant surface temperature boundary condition. The detailed solution procedure can be found in regular heat transfer book. The solution for the above problem can be obtained by using f The governing equation in partial differential form... 

The solutions of Eqs. 5.76 and 5.77 are subject to various boundary conditions. For a slab of finite thickness (thickness 2b, with the axis at the center of the slab) the boundary conditions are usually given as constant surface temperatures or a step change in the surface temperature due to convection at the free surfaces. Mathematically for the first case the boundary and initial conditions are given as... 

Ta here is the temperature of the cooling fluid. In the case of the cylindrical geometry the corresponding boundary and initial conditions for the constant surface temperature (step change in temperature) or the step change in surface temperature due to convection are written, respectively, as... 

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

At the solid walls, the boundary conditions state that the velocity is zero (i.e. no slip). Also at the walls, the temperature is either fixed or a zero-gradient condition is applied. At the surface of the spinning disk the gas moves with the disk velocity and it has the disk temperature, which is constant. The inlet fiow is considered a plug fiow of fixed temperature, and the outlet is modeled by a zero gradient condition on all dependent variables, except pressure, which is determined from the solution. 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

Simple component exchange between solid phases is accomplished by diffusion. If only two components (such as Fe " and Mg) are exchanging, the diffusion is binary. The boundary condition is often such that the exchange coefficient between the surfaces of two phases is constant at constant temperature and pressure. The concentrations of the components on the adjacent surfaces may be constant assuming interface equilibrium. The solution to the diffusion equation... 

This leads to a powerful method when the densities, specific heat, and thermal conductivity can be assumed to have constant average values throughout the entire solid-liquid region. In this case, the moving boundary can be considered to be a source surface. As an example, consider the freezing of a semi-infinite region occupied by liquid with arbitrary initial and surface temperature and conditions. In this case Eq. (159) becomes... 

Here, the position x is the location of the streamline located near the barrel, and xc the position of the same streamline located near the root of the screw. If the flow is fast enough, we can assume that the temperature along the streamline is constant as it travels near the barrel and returns to near the root of the screw. Hence, for the barrel surface we must take the boundary condition... 

Time-dependent Temperature Boundary Conditions (a) Consider the heat-transfer problem involved inside a semi-infinite solid of constant properties with a varying surface temperature ... 

The process of mathematical fitting is error-prone, and especially two different issues have to be considered, the first one dealing with the boundary conditions of the fitting procedure itself A pure diffusion process is considered here as the only transport mechanism for fluorine in the sample. A constant value for the diffusion constant D, invariant soil temperatures and a constant supply of fluorine (e.g. a constant soil humidity) are assumed, the latter effect theoretically resulting in a constant surface fluorine concentration for samples collected at the same burial site. In mathematical terms, Dt is influenced by the spatial resolution of the scanning beam, the definition of the exact position of the bone surface, which usually coincides with the maximum fluorine concentration, and by the original fluorine concentration in the bulk of the object, which in most cases is still detectable. A detailed description on... 

We have thus far obtained the distribution functions for the incoming and reflected particles of reduced mass in different regions in terms of the unknown constants A and B and the unknown dimensionless surface temperature 8. The surface temperature was then related to the internal energy and the number density of the fictitious particle at the reflecting surface. Now, in order to determine the constants A and B, wc must specify the boundary conditions for the mass and the energy flux at the sphere of influence. 

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