Adiabatic boundary condition

Equation (5.19) can also be obtained from the unsteady counterpart of Equation (5.5). If internal conduction is taken as zero in Equation (5.5), this adiabatic equation arises. Equation (5.19) also becomes the basis of an alternative method for determining the bulk kinetic properties. The method is called the adiabatic furnace used by Gross and Robertson [5]. The furnace is controlled to make its temperature equal to the surface temperature of the material, thus producing an adiabatic boundary condition. This method relies on the measurement of the time for ignition by varying either ra or Too, the initial temperature. 

As mentioned above, in this simulation, one repeating unit located in the middle of the center stack is selected and modeled. In the middle of the center stack, the boundaries between the single unit and adjacent units are regarded to be thermally adiabatic. Hence, for the thermal boundary conditions at the surfaces of the solid part connecting the next units, adiabatic boundary conditions are employed. In a practical cell stack, the fuel and air are introduced into the channel through man-... 

Since we dropped the last term in the equation, we are satisfying the adiabatic boundary condition (Neumann), q(L) = 0. On the other hand, we still must consider the Dirichlet boundary condition, T(0) = T0. Since the Neumann boundary conditions is automatically satisfied, while the Dirichlet must be enforced, in the finite element language they are usually referred to as natural and essential boundary conditions. 

The effect of viscous dissipation is illustrated here for the orifice problem solved in Chapter 10. There are two options for the temperature boundary conditions. In the first option, the system can be adiabatic, where all walls use the boundary condition of no flux. In the second option, the walls can be made from metal, with plenty of coolant outside, and then the boundary condition would use a constant temperature. Here, adiabatic boundary conditions are used. Thus, the temperature out of the device (minus the inlet temperature) gives the temperature rise due to viscous dissipation. 

The isothermal or adiabatic boundary conditions for the heat transfer can be specified with the wall temperature T or the normal heat flux q , = — Tgrad T n. Isothermal boundary conditiOTis in ccmjunctimi with the law-of-the-wall are modeled as... 

Adiabatic boundary conditions are applied at all other sides of the walls ... 

The simplicity of the method described above is often offset by the difficulty in satisfying the required adiabatic boundary conditions. In order for this solution to be valid, ffie radiant energy incident on the front surface is required to be uniform, and the duration of the flash must be sufficiently short compared with the thermal characteristic time of the sample. In addition, it assumes that the sample is homogeneous, isotropic, and opaque, and that the thermal properties of the sample do not vary considerably with temperature. 

Because the die wall material usually has a thermal conductivity much higher than polymer melts, adiabatic conditions are not likely to be achieved. On the other hand, it is also not likely that the wall temperature will remain constant. In this case, the heat flux through the wall would be such as to maintain a perfectly constant temperature along the wall. This is referred to as an isothermal wall boundary condition. Because of the high thermal conductivity of the wall, the isothermal boundary condition is more likely to occur than the adiabatic boundary condition. Adiabatic conditions can be approached if the die is very well insulated. In most actual cases, the true thermal boundary condition will be somewhere between isothermal and adiabatic, depending on the design of the die and external conditions around the die. A typical temperature profile resulting from the velocity profiles shown in Fig. 7.106 is shown in Fig. 7.108. 

The predicted pressure profile is obviously a direct result of the assumptions made in the calculations. Winter assumed isothermal conditions at the barrel wall and adiabatic conditions at the flight tip. With stock temperature increases in the order of 100°C and more, it is unlikely that the isothermal boundary condition is valid for the barrel. For the same reason, it is unlikely that the adiabatic boundary condition is valid for the flight tip, particularly since the rest of the screw will be at much lower temperature. Unfortunately, it is difficult to measure actual temperature and pressure profiles. Thus, the predicted temperature and pressure profiles have not been compared to experimental results. 

Another option is to model a single cell in a stack taking into account the stack environment by using periodic or adiabatic boundary conditions... 

The first condition means that heat flux in the membrane is zero. The second condition fixes the temperature at the CCL/GDL interface. It should be emphasized that although Equations 4.285 determine the temperature shape in the CL, they do not affect the equation for the total heat flux qtot, leaving the CL. Physically, qtot is determined only by the rate of heat energy production inside the CCL, and it does not depend on boundary conditions. The adiabatic boundary condition at x = 0 simply means that q leaves the CCL at the CCL/GDL interface. 

Shear stress versus parent shear rate for two capillaries of different diameter but the same L/R = 60. ABS polymer melt squares, R = 0.26 mm, 230°C barrel circles, R = 1.6 mm, 232°C barrel. The solid line represents isothermal flow estimated at high Y from the viscosity master curve (Hg-ure 2.6.1) the dashed line, adiabatic boundary condition, viscous dissipation with no heat transfer to the wall, for / = 1.6 mm. Replotted from Cox and Macosko (1974a) and Cox (1973). 

The pressure vessel outer wall temperature is strongly coupled to the bulk temperature and heat transfer coefficient of the convective flow in the downcomer, even for cases of adiabatic boundary conditions. 

---