The temperature distribution

A simple algorithm can be developed (see App. E) to target the minimum total number of shells (as a real, i.e., noninteger number) for a stream set based on the temperature distribution of the composite curves. The algorithm starts by dividing the composite... 

With this technology even boreholes, up to 2mm underneath the surface, can be identified, A remarkable borehole is represented in illustration 10, For the elucidation of the temperature contrast, a three-dimensional temperature distribution of the entire blade is shown beside the infrared picture (the similarity of the temperature distribution with the actual blade airfoil is purely coincidental). 

I0-38Z ) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38Z ) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. ln.st. Chem. Eng. J., 10, 269(1964)]. 

The temperature distributions for film and convection cooling design are shown in Fig. 29-32. From the coohng distribution diagram, the hottest section can be seen to be the trailing edge. The web, which is the most highly stressed blade pai t, is also the coolest part of the blade. 

To find the temperature distribution, the first procedure requires a change of variables to the variable, z, defined by the equation... 

In a recuperative heat exchanger, each element of heat-transferring surface has a constant temperature and, by arranging the gas paths in contra-flow, the temperature distribution in the matrix in the direction of flow is that giving optimum performance for the given heat-transfer conditions. This optimum temperature distribution can be achieved ideally in a con-tra-flow regenerator and approached very closely in a cross-flow regenerator. 

This design has a strut-supported porous shell (Figure 9-19). The shell attached to the strut is of wire from porous material. Cooling air flows up the central plenum of the strut, which is hollow with various-size metered holes on the strut surface. The metered air then passes through the porous shell. The shell material is cooled by a combination of convection and film cooling. This process is effective due to the infinite number of pores on the blade surface. The temperature distribution is shown in Figure 9-20. 

With this particular design, primary cooling is achieved by film cooling with cold air injected through small holes over the airfoil surface (Figure 9-21). The temperature distribution is shown in Figure 9-22. 

According to Abramovich,Regenscheit, " and Shepelev, the relation between the velocity distribution and the temperature distribution in the cross-section of nonisothermal compact, linear, or radial jets within Zone 3 can be expressed as... 

The temperature distribution within the annular fin is given by the differential equation... 

I FIGURE 11.27 Heat conduction through an external wall. The temperature distribution over die wall thickness is linear only under steady-state conditions. 

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). 

The response factors are characteristic for the layer buildup of the selected wall and are calculated before (by a preprocessor program) or at the beginning ol the simulation. Numerical reasons limit the time step to approximately 10 to 60 min, depending on the thickness and material properties of the wall layers. The method allows the calculation of surface temperatures and heat fluxes bur not the determination of the temperature distribution within the wall. Due to the precalculation of these response factors, the computer time for the simulation might be significantly reduced. 

Equations (12.40) to (12.45) describe the velocities u, v, w, the temperature distribution T, the concentration distribution c (mass of gas per unit ma.ss of mixture, particles per volume, droplet number density, etc.) and pressure distribution p. These variables can also be used for the calculation of air volume flow, convective air movement, and contaminant transport. 

S. Boschert, P. Dold, K. W. Benz. Modelling of the temperature distribution in a three-zone resistance furnace influence of furnace configuration and ampoule position. J Cryst Growth 7S7 140, 1998. 

Degras (65a) studied the temperature distribution of the surface of a molybdenum disk 0.2 mm thick and 20 mm in diameter, heated by electron bombardment. Two thin thermocouples spot-welded on the back of the sample 2 and 8 mm from its center, respectively, showed a temperature difference of less than 2% at 500°K, and less than 3% at 1200°K. 

In the problems which have been considered so far, it has been assumed that the conditions at any point in the system remain constant with respect to time. The case of heat transfer by conduction in a medium in which the temperature is changing with time is now considered. This problem is of importance in the calculation of the temperature distribution in a body which is being heated or cooled. If, in an element of dimensions dr by dy by dr (Figure 9.9), the temperature at the point (x, y, z) is 9 and at the point (x + dx, y + dy, r. + dr) is (9 4- d6>), then assuming that the thermal conductivity k is constant and that no heat is generated in the medium, the rate of conduction of heat through the element is ... 

A general method of estimating the temperature distribution in a body of any shape consists of replacing the heat flow problem by the analogous electrical situation and measuring the electrical potentials at various points. The heat capacity per unit volume C.,p is represented by an electrical capacitance, and the thermal conductivity k by an... 

Thus, if the temperature distribution at time t, is known, the corresponding distribution at time t + At can be calculated by the application of equation 9.41 over the whole extent of the body in question. The intervals Ax and At are so chosen that the required degree of accuracy is obtained. 

A graphical method of procedure has been proposed by SCHMlDT(7h If the temperature distribution at time t is represented by the curve shown in Figure 9.11 and the points representing the temperatures at x — Ax and x + Ax are joined by a straight line, then... 

If this simple construction is carried out over the whole of the body, the temperature distribution after time At is obtained. The temperature distribution after an interval 2At is then obtained by repeating this procedure. 

The most general method of tackling the problem is the use of the finite-element technique 8 to determine the temperature distribution at any time by using the finite difference equation in the form of equation 9.40. 

When a body of characteristic linear dimension L, initially at a uniform temperature 6(). is exposed suddenly to surroundings at a temperature O, the temperature distribution at any time t is found from dimensional analysis to be ... 

If an electric current flows through a wire, ihe heat generated internally will result in a temperature distribution between the central axis and the surface of the wire. This type of problem will also arise in chemical or nuclear reactors where heat is generated internally. It is necessary to determine the temperature distribution in such a system and the maximum temperature which will occur. 

These expressions are valid provided that the cross-section for heat flow remains constant. When it is not constant, as with a radial or tapered fin, for example, the temperature distribution is in the form of a Bessel function i26). 

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 /... 

From the start, we knew we needed large anodes to meet the challenge of inexpensive fluorine these calculations clearly show the need for a better design for large anodes. The obvious solution is to put a metal conductor down the middle of the anode. Figure 20 shows the results from a finite-element model of the temperature distribution in such an improved large anode with a central metal conductor. 

---