Thermal boundary layer temperature distribution

Thus the conditions for the thermal boundary layer, with respect to temperature, are the same as those for the velocity boundary layer with respect to velocity. Then, if the thickness of the thermal boundary layer is 5 the temperature distribution is given by ... 

The integral in equation 11.55 clearly has a finite value within the thermal boundary layer, although it is zero outside it. When the expression for the temperature distribution in the boundary layer is inserted, the upper limit of integration must be altered from /... 

The impact process of a 3.8 mm water droplet under the conditions experimentally studied by Chen and Hsu (1995) is simulated and the simulation results are shown in Figs. 16 and 17. Their experiments involve water-droplet impact on a heated Inconel plate with Ni coating. The surface temperature in this simulation is set as 400 °C with the initial temperature of the droplet given as 20 °C. The impact velocity is lOOcm/s, which gives a Weber number of 54. Fig. 16 shows the calculated temperature distributions within the droplet and within the solid surface. The isotherm corresponding to 21 °C is plotted inside the droplet to represent the extent of the thermal boundary layer of the droplet that is affected by the heating of the solid surface. It can be seen that, in the droplet spreading process (0-7.0 ms), the bulk of the liquid droplet remains at its initial temperature and the thermal boundary layer is very thin. As the liquid film spreads on the solid surface, the heat-transfer rate on the liquid side of the droplet-vapor interface can be evaluated by... 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

St is, of course, the thickness of the thermal boundary layer. The first of these conditions follows from the requirement that die fluid in contact with the wall must attain the same temperature as the wall. The other two conditions follow from the requirement that the boundary layer temperature profile must blend smoothly into the freestream temperature distribution at the outer edge of the boundary layer. 

The temperature distributions in the thermal boundary layer over an EPR with t0 = -0.5 (i.e. obstructions are colder than the wall) have been shown in Fig. 3.17,B. The thermal distributions in such a flow are much complicated. However, analytical estimations are possible if the structure of the boundary layer is taken into account. 

There are at least two approaches that we can take to solve problems in which either the heat flux or the mixed-type condition is specified as a boundary condition. If it is desired to determine the temperature distribution throughout the fluid, then we must return to the governing thermal boundary-layer equation (11-6)- assuming that Re, Pe / - and develop new asymptotic solutions for large and small Pr, with either dT/dr] y 0 or a condition of the mixed type specified at the body surface. The problem for a constant, specified heat flux is relatively straightforward, and such a case is posed as one of the exercises at the end of this chapter. On the other hand, in many circumstances, we might be concerned with determining only the temperature distribution on the body surface [and thus dT /dr] [v from (11 98) for the mixed-type problem], and for this there is an even simpler approach that... 

Problem 11-2. Specified Heat Flux. We have considered the development of thermal boundary-layer theory for a 2D body with a constant surface temperature 0q. We have also discussed a method to determine the surface temperature when the heat flux is specified. In this problem, we wish to solve for the leading-order approximation to the temperature distribution in the fluid for an arbitrary 2D body when the heat flux is specified as a constant. 

Flow type Hydrodynamic boundary layer Velocity distribution in the flow direction Friction factor Thermal boundary layer Dimensionless temperature distribution in the flow Nusselt number... 

Figure 6.5 shows the velocity distributions in a boundary layer of a liquid with Pr , = 100 (e.g., sulfuric acid at room temperature). For this Prandtl number, the thermal boundary layer penetration into the liquid is much less than the flow boundary layer, and the regions where viscosity variations occur are confined close to the surface. The curve corresponding to p /pe = 1 is the Blasius solution (see Fig. 6.1). The curve labeled p ,/pe = 0.23 corresponds to a heated surface where the low viscosity near the surface requires steeper velocity gradients to maintain a continuity of shear with the outer portion of the boundary layer. The heated free-stream cases reveal the opposite effects. In general, the outer portions of the flow boundary layers act similarly to the velocity distribution of the Blasius case except for their being displaced in or out by the effects that have taken place in the thermal boundary layer.

An important conclusion to be drawn from the form of Eq. (73) is that the nature of the initial temperature distribution some distance away from the bubble wall is of little consequence, so that no restriction need be made that the initial temperature variation occur entirely within a thin thermal boundary layer. For uniform initial superheat the effect of nonzero initial radius is next examined. Taking / = i (0), a simple modification of the analysis leads to... 

In this form the parametric solutions, Eqs. (73) and (74), for the spherically symmetric problem can be applied immediately. Physically this implies that a bubble growing in an initially arbitrary temperature field grows at precisely the same rate as if the initial temperature were averaged in each thin spherical shell surrounding the bubble center. Two special cases are considered in detail (7) the linear thermal boundary layer, of thickness /, next to the heating surface, outside of which the temperature is uniform and (2) the exponential boundary layer, where the temperature is assumed to decay exponentially with distance from the wall. The latter distribution is of the form... 

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius 7-power law is used for the temperature distribution. These give results that are quite similar to the experimental equations as given in Section 4.6. 

In order to determine the heat transfer rate by convection, the temperature distribution in the thermal boundary layer needs to be known. This temperature distribution depends on the nature of the fluid motion or the velocity field, and this is determined by solving the energy equation along with the mass and momentum equations for specific flow geometry. 

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. 

We shall find the temperature distribution in the gas in those layers adjacent to x — 0 in which the chemical reaction has not yet started. The evaporation heat or the heat of endothermic reaction of gas-formation, L, is equal to the jump in the thermal energy of the original substance. Thus, at x = 0 at the phase-boundary the magnitude of the thermal flux experiences a jump. Using one prime for the c-phase and two primes for the gas, we construct the equation... 

Thermal stress calculations in the five cell stack for the temperature distribution presented above were performed by Vallum (2005) using the solid modeling software ANSYS . The stack is modeled to be consisting of five cells with one air channel and gas channel in each cell. Two dimensional stress modeling was performed at six different cross-sections of the cell. The temperature in each layer obtained from the above model of Burt et al. (2005) is used as the nodal value at a single point in the corresponding layer of the model developed in ANSYS and steady state thermal analysis is done in ANSYS to re-construct a two-dimensional temperature distribution in each of the cross-sections. The reconstructed two dimensional temperature is then used for thermal stress analysis. The boundary conditions applied for calculations presented here are the bottom of the cell is fixed in v-dircction (stack direction), the node on the bottom left is fixed in x-direction (cross flow direction) and y-direction and the top part is left free to... 

The problem for the first term in an asymptotic solution for the temperature distribution 9 in the outer part of the thermal layer is thus to solve (11 66) subject to the conditions (11 67a), (11 67c), and (11 7c). Again, we see that the geometry of the body enters implicitly through the function ue (x) only. As in the high-Pr limit, a general solution of (11 66) is possible even for an arbitrary functional form for ue (x ). Before we move forward to obtain this solution, however, a few comments are probably useful about the solution (11 69) for the innermost part of the boundary layer immediately adjacent to the body surface. 

Summarizing, we should note that the methods presented in the present section can be applied without any modifications to heat exchange problems, because temperature distribution is described by an equation similar to the diffusion equation. The boundary conditions are also formulated in a similar way. One only has to replace D by the coefficient of thermal diffusivity, and the number Peo - by Pej-. The corresponding boundary layer is known as the thermal layer. Detailed solutions of heat conductivity problems can be found in [6]. 

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