Thermal boundary conditions

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

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. 

Thermal boundary condition at constant wall heat flux... 

Hackert, G.L., Ellzey, J.L., and Ezekoye, O.A., Effect of thermal boundary conditions on flame shape and quenching in ducts. Combustion and Flame, 112, 73-84, 1998. 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

For our simulations with heat sinks in spheres, we chose not to stack runs, in order to minimize the change in temperature across the WS, and instead carried out a study of the effects of simplified boundary conditions. For most runs, simple boundary conditions were used in which the fluid entering at the base of the segment was set uniformly to Tin, and fluxes through the solid areas on the top and bottom planes were set to zero. No thermal boundary condition was required for the flow outlet boundary. 

Accurate modeling of the temperature distribution in a PEFC requires accurate information in four areas heat source, thermal properties of various components, thermal boundary conditions, and experimental temperature-distribution data for model validation. The primary mechanism of heat removal from the catalyst layers is through lateral heat conduction along the in-plane direction to the current collecting land (like a heat sink). Heat removed by gas convection inside the gas channel accounts for less than 5% under typical PEFC operating conditions. 

Solving the energy equation provides prediction of the temperature distribution and its effect on cell performance in a PEFC. Figure 12 presents a temperature distribution in the middle of the membrane for a single-channel PEFC. The maximum temperature rise in this case is 4 °C, which will only fect cell performance slightly. However, the temperature variation depends strongly on the thermal conductivities of the GDL and flow plate as well as thermal boundary conditions. 

The time since initiation of the oscillating thermal boundary condition is large, so that initial conditions can be neglected. 

The temperature and species profiles also have entry-region behavior. The fully coupled entry-region problem is easily formulated and can be solved using the method of lines. The details of the entry-region profiles depend on species and thermal boundary conditions as well as fluid properties. The entry length and the corresponding profile development also depend on the channel geometry. 

These results verified that heat transfer in the melt was conduction dominated, except at intense convection levels because of the low-Prandtl-number characteristic of semiconductor melts. The shape of the melt-crystal interface changes with convection only at these higher convection levels. The flows are cellular, with the direction and magnitude of each cell determined by the radial temperature gradients induced by the thermal boundary conditions. In the idealized system studied, the mismatch in boundary conditions at the junction of the hot zone and the adiabatic region (Figure 16) causes the temperature to increase radially and drive a flow up along the... 

Effect of Thermal Boundary Conditions. When the side walls are cooled instead of being insulated, there is no critical Rat number, and any transverse temperature gradient will lead to a buoyancy-driven secondary flow. Compared with the previous example (Figure 8b), the rolls are reversed and now rotate outward. These examples demonstrate the strong influence of the thermal boundary conditions on CVD reactor flows. 

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-... 

The boundary conditions for the momentum balance were ux = uy = 0 on the upper and lower plates, p = 105,000 Pa at the left wall (x = 0) and p = 0 at the right wall (x = 0.015m). The thermal boundary conditions were T = 200°C on the upper and lower plates, and insulated boundary conditions (dT/dx=0) on both... 

Fig. 12.11 Uncorrected shear stress versus Newtonian wall shear rate for ABS Cycolac T measured and calculated using various thermal boundary conditions. Dq — 0.319 cm L/Dq = 30 T0 — 505 K. [Reprinted by permission from H. W. Cox and C. W. Macosko, Viscous Dissipation...

Two thermal boundary condition limits were considered for the electrolyser isothermal and adiabatic. Actual electrolyser operation will generally lie between these limits. For the isothermal cases, heat from the reactor was directly supplied to the electrolyser to maintain isothermal conditions for operation below the thermal neutral voltage. Heat rejection from the electrolyser is required to maintain isothermal operation at operating voltages above thermal neutral. For the adiabatic cases, the direct electrolyser heater was not used. 

These can be solved numerically given the usual initial and boundary conditions, including the thermal boundary condition at the reactor wall, r=l ... 

WATER HELPS DEFINE THERMAL RELATIONSHIPS AND THERMAL BOUNDARY CONDITIONS OF LIFE... 

Consider fully developed flow in a pipe. A thermal boundary condition is applied, starting at a distance x = x<,. For the four different cases listed below, sketch the temperature profiles in the pipe as the thermal boundary layer develops, and the temperature profiles after the thermal boundary layer has fully developed ... 

In the next section, incompressible flow with constant properties and no body forces is discussed. Under such conditions, the governing momentum equations are decoupled from the governing energy equation. Once the flow field is known, different temperature distributions may be computed with different types of thermal boundary conditions. 

SIMILARITY SOLUTIONS FOR FLOW OVER FLAT PLATES WITH OTHER THERMAL BOUNDARY CONDITIONS... 

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 ... 

Consideration will next be given to the solution for the temperature function, G. As with fully developed pipe and plane duct flows, the solution depends on the nature of the thermal boundary conditions at the wall. In the case of flow in a rectangular duct there are a variety of possible boundary conditions, some of these being shown in Fig. 4.11. Here, attention will be restricted to the case where the wall... 

Effect of wall thermal boundary condition oh the Nusselt number for hilly developed flow in a rectangular duct (Nusselt number based on hydraulic diameter)... 

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