Boundary conditions uniform surfaces temperature

To illustrate this fact, we may consider the 2D heat transfer problem of uniform flow past a heated circular cylinder with uniform surface temperature. In this case, if we look for a solution in the asymptotic form (9-15) for low Peclet numbers, the nondimensional governing equation and boundary conditions for do are again (9-16) and (9-17), but this time are expressed in cylindrical coordinates, namely,... 

Uniform Surface Temperature, Air as Coolant. When the boundary layer and coolant gases are the same, the equations controlling boundary layer behavior are Eqs. 6.6-6.8. The mass injection at the surface simply alters the boundary conditions (Eq. 6.9) at the wall to be... 

Uniform Surface Injection. Although a mass transfer distribution yielding a uniform surface temperature is most efficient, it is much easier to construct a porous surface with a uniform mass transfer distribution. Libby and Chen [34] have considered the effects of uniform foreign gas injection on the temperature distribution of a porous flat plate. For these conditions, however, boundary layer similarity does not hold. Libby and Chen extended the work of Iglisch [35] and Lew and Fanucci [36], where direct numerical solutions of the partial differential equations were employed. An example of the nonuniform surface enthalpy and coolant concentrations resulting from these calculations is shown in Fig. 6.16. 

Parallel to the boundary conditions discussed above for the species continuity equation, we consider in this book only uniform temperature on the surface of the particle, uniform temperature in the continuous phase remote from the particle and uniform initial temperatures in each phase. Hence... 

For the same uniform initial temperature distribution, we could suddenly expose the surface to a constant surface heat flux qboundary conditions on Eq. (4-7) would then become... 

A semi-infinite aluminum cylinder 5 cm in diameter is initially at a uniform temperature of 200°C. It is suddenly subjected to a convection boundary condition at 70°C with h = 525 W/m2 °C. Calculate the temperatures at the axis and surface of the cylinder 10 cm from the end 1 min after exposure to the environment. 

Here, ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid... 

The critical Reynolds number Reor is typically taken as 5 x 105, lie, < Re, < 3 x 107, and 0.7 < Pr < 400. The fluid properties are evaluated at the film temperature (7, + I )/2 where 7, is the free-stream temperature and 7 is the surface temperature. Equation (5-60) also apphes to the uniform heat flux boundary condition provided h is based on the average temperature difference between 7 and 7, ... 

A spherical metal ball of radius r<, is heated in an oven to a temperature of SOO C throughout and is then taken out of the oven and allowed to cool in ambient air at = 2T C, as shown in Fig. 2-38. The thermal conductivity of the ball material isk = 14.4 W/m K, and the average convection heat transfer coefficient on the outer surface of the ball is evaluated to be h = 25 W/m K. The emissivity of the outer surface of the ball is e = 0.6, and the average temperature of the surrounding surfaces is 7, = 290 K. Assuming ] the ball is cooled uniformly from the entire outer surface, express the initial and boundary conditions for the cooling process of Ihe ball. 

Consider a short cyUnder of radius r<, and height H in which heat is generated at a constant rate of Heat is lost from the cylindrical surface at r = r by convection to the surrounding medium at temperature with a heat transfer coefficient of /i. The bottom surface of the cylinder at z = 0 is insulated, while the top surface at z — is subjected to uniform heat flux Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not. solve. 

Consider a semi-inlinite solid with constant thermophysical properties, no internal heat generation, uniform theimal cnnditinn.s on its exposed surface, and initially a uniform temperature of Tj throughout. Heat tfansfec in this case occurs only in the direction uormal to the surface (the x direction), and thus it is one-dimensional. Differential equations are independent of the boundary or initial conditions, and thus Eq. 4—lOa for one-dimensional transient conduction in Cartesian coordinates applies. The depth of the solid is large (x expressed mathematically as a boundary condition as T x —> , 0 = T,. 

Heat convection for gaseous flow in a circular tube in the slip flow regime with uniform temperature boundary condition was solved in [23]. The effects of the rarefaction and surface accommodation coefficients were considered. They defined a fictitious extrapolated boundary where the fluid velocity does not slip by scaling the velocity profile with a new variable, the shp radius, pj = l/(l + 4p.,Kn), where is a function of the momentum accommodation coefficient, and defined as p, =(2-F,j,)/F,j,. Therefore, the velocity profile is converted to the one used for the... 

Boundary conditions are 5T/0n = 0 (n is the normal to interface surface) on the gas-solid interface and on the symmetry lines as far as on the external wall surface (for case of channel with heat production in the wall) continuity both of heat flux and temperature on solid-liquid interface and the continuity of temperature on gas-liquid interface. Gauss-Zeidel iterative procedure has been used to solve the heat problem numerically. We used a non-uniform grid pattern near the vapor-liquid interface for higher computational accuracy. 

In order to solve mass and energy balance equations, initial and boundary conditions should be given. Initially, the biomass material is in a quiescent environment at atnbient conditions and thus is specified as uniform tenqrerature and solid compositions. For 1>0, the spatial conditions at the centerline (i=0) are specified by symmetry, the two sides (r=R) are exposed to radiation by a constant temperature of heater. At these two sides, radiant heat transfer from the surface takes places. Further conditions of constant ambient pressure and zero gradient of tar and gas mass at centerline are also used. 

Low-energy boron bombardment of silicon has been simulated at room temperature by MD. Tersoff potential T3 was used in the simulation and smoothly linked up with the universal potential. The boron-silicon interaction was simulated according to Tersoff potential for SiC but modified to account for the B-Si interaction. Silicon crystal (Si-c) in the (001) direction, with (2x1) surface reconstruction, was bombarded with boron at 200 and 500 eV. Reasonably good statistics are obtained with 1000 impact points uniformly distributed over a representative surface area. The simulation size was 16x 16x 14 unit cells. Periodic boundary conditions were applied laterally. The temperature was kept at 300 K with a thermal bath applied to the more external cells in the crystal except the top surface. In these conditions the crystal was relaxed during 19 ps. In order to avoid direct channeling, the incidence was inclined 7° out of the normal, as usual in experiments, with random azimuthal direction. 

The leading edge of the plate is located at x = 0. The surface boundary conditions at y = 0 reflect the assumed conditions of zero mass transfer, a prescribed uniform temperature including the case of zero heat flux and an implied condition of a smooth surface. 

If a constant heat flux boundary condition is required, an electrical heating element, often a thin, metallic foil, can be stretched over an insulated wall. The uniform heat flux is obtained by Joule heating. If the wall is well insulated, then, under steady-state conditions, all of the energy input to the foil goes to the fluid flowing over the wall. Thermocouples attached to the wall beneath the heater can be used to measure local surface temperature. From the energy dissipation per unit time and area, the local surface temperature, and the fluid temperature, the convective heat transfer coefficient can be determined. Corrections to the total heat flow (e.g., due to radiation heat transfer or wall conduction) may have to be made. 

So far, all the experiments falling in the first category have been performed with both solid surfaces being maintained at constant uniform temperatures, this being in accord with boundary conditions (i)a, (ii)b, and (iii)a of Table III. 

Various solutions are obtained according to the assumptions made for the initial and boundary conditions. A few of them are presented, and calculation is made for the more simple system, even if it is not realistic and should be used with great care—for example, when the temperatnre is uniform initially, and the temperatures are kept constant on the surfaces. The two other cases considered are (i) when a finite coefficient of heat transfer is at the snrfaces of the solid, and (ii) when the rubber is heated by the mold on both surfaces. 

Diffusivity measurement methods based on analytical solutions to Eq. 4 have all had the same initial condition that the whole sample is at a constant uniform temperature. But three different types of boundary conditions have been employed first, the sample surface is subjected to a step change in temperature second the surface is subjected to a linear rate of temperature rise and third, the surface is subjected to a periodic temperature fluctuation. 

A method for measuring diffusivity of solid polymers based on this type of boundary condition has been developed by Berlot [49, 50], and by Gehrig et al. [51], A disc sample of thickness 2a is held at a uniform temperature and then a sinusoidal temperature fluctuation of angular frequency ta is imposed on the outer surfaces. The amplitude ratio and phase of the temperature at the center are monitored with a thermocouple. Under these conditions the amplitude ratio A and phase 0 are given by [52]... 

Mashayek and Ashgriz [98] considered effects of the heat transfer from the liquid to the surrounding ambient, the liquid thermal conductivity, and the temperature-dependent surface tension coefficient on the jet instability and the formation of satellite drops. Two different disturbances were imposed on the jet. In the first case, the jet is exposed to a spatially periodic ambient temperature. In addition to the thermal boundary condition, an initial surface disturbance with the same wave number as the thermal disturbance is also imposed on the jet. Both in-phase and out-of-phase thermal disturbances with respect to surface disturbances are considered. For the in-phase thermal disturbances, a parameter set is obtained at which capillary and thermocapiUary effects can cancel each other and the jet attains a stable configuration. No such parameter set can be obtained when the thermocapillary flows are in the same direction as the capillary flows, as in the out-of-phase thermal disturbances. In the second case, only an initial thermal disturbance is imposed on the surface of the liquid while the ambient temperature is kept spatially and temporally uniform (Fig. 1.20). 

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