Sphere temperature distribution

What is the temperature distribution in the insulating material covering the sphere. 

The second model, proposed by Frank-Kamenetskii [162], applies to cases of solids and unstirred liquids. This model is often used for liquids in storage. Here, it is assumed that heat is lost by conduction through the material to tire walls (at ambient temperature) where the heat loss is infinite compared to the rate of heat conduction through the material. The thermal conductivity of the material is an important factor for calculations using this model. Shape is also important in this model and different factors are used for slabs, spheres, and cylinders. Case B in Figure 3.20 indicates a typical temperature distribution by the Frank-Kamenetskii model, showing a temperature maximum in the center of the material. 

Transient heating effects occur for a small particle over a very short time after heating commences, and then quasi-steady state is reached. The quasisteady temperature distribution within the sphere is governed by the thermal energy equation... 

Fig. 23. Temperature distribution around a sphere with material transport.

For simplicity, it is assumed that the temperature distribution throughout the sphere is radially symmetric, no heat generation occurs inside the sphere, and no thermal radiation takes place between the sphere and the surrounding environment. Thus, the transient heat balance in the particle leads to... 

Equation (4.8) indicates that the one-dimensional transient temperature distribution inside a solid sphere without internal heat generation varies with Fo and Bi. On the basis of the equation, the temperature distribution in the solid can be considered uniform with an error of less than 5 percent when Bi < 0.1, which is the condition for most gas-solid flow systems. In transient heat transfer processes where the gas-solid contact time is very short, it also requires Fo > 0.1 [Gel Perin and Einstein, 1971] for the internal thermal resistance within the particles to be neglected. In the following, unless otherwise noted, it is assumed that the temperature inside a solid particle is uniform. 

For a quasi-steady-state heat conduction between an isothermal sphere and an infinitely large and quiescent fluid, the temperature distribution in the fluid phase is governed by... 

Derive an expression for the temperature distribution in a sphere of radius r with uniform heat generation q and constant surface temperature Tw. 

The Heisler charts discussed above may be used to obtain the temperature distribution in the infinite plate of thickness 2L, in the long cylinder, or in the sphere. When a wall whose height and depth dimensions are not large compared with the thickness or a cylinder whose length is not large compared with its diameter is encountered, additional space coordinates are necessary to specify the temperature, the above charts no longer apply, and we are forced to seek another method of solution. Fortunately, it is possible to combine the solutions... 

The maximum temperature in a solid that involves uniform heat generation occurs at a locationybr//iesr away from the outer surface when the outer surface of the solid is maintained at a constant temperature T,. For example, the maximum temperature occurs at the midplane in a plane wall, at the centerline in a long cylinder, and at the midpoint in a sphere. The temperature distribution within the solid in these cases is symnetrical about the center of symmetry. 

Figure 7. Matrix-matrix unlike distribution function for poa3 = 0.2 for the low temperature (solid line and circles) and high temperature (dotted line and squares) matrix. In the latter case like and unlike correlations are identical to the uncharged hard sphere pair distribution function.

Table 2.6 Smallest eigenvalue fj,i from (2.170), (2.177) or (2.182), associated expansion coefficient C in the series for the temperature distribution +(r +,i+) as well as coefficient D in the series for the average temperature i(i+) of a plate, a very long cylinder and a sphere, cf. also (2.185) and (2.188) in section 2.3.4.5...

The temperature distribution in the sphere is obtained in the same manner, such that tf+(r+,t+) =J2ciSm l bxp(- t+). ... 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

Problem 9-7. Heat Transfer from a Sphere at Pe -C 1, with Specified Heat Flux. Consider heat transfer from a solid sphere of radius a that is immersed in an incompressible Newtonian fluid. Far from the sphere this fluid has an ambient temperature Too and is undergoing a uniform flow with a velocity U-x. At the surface of the sphere the heat flux is independent of position with a specified value q. Determine the temperature distribution on the surface of the sphere, assuming that the Peclet number is small. 

Let the rate of energy per unit volume generated in a solid cylinder or a solid sphere be u" (r) = u , the radius and the thermal conductivity of the cylinder or the sphere be R and k(T) = kr (alternate notations ur and kr are used for convenience in the following formulation). Under steady conditions, the total energy generated in the cylinder or sphere is transferred, with a heat transfer coefficient h, to an ambient at temperature Too. This cylinder could be one of the fuel rods of a reactor core, or one of the elements of an electric heater, and the cylinder or sphere could be a bare, homogeneous reactor core. We wish to determine the radial temperature distribution. 

The temperature distribution in a flat plate, solid cylinder, and solid sphere with internal energy generation given respectively by Eqs. (2.55), (2.98), and (2.100) are sketched in Fig. 2.22 for l = R. For the same v ", the temperature levels in the cylinder and sphere respectively are 1/2 and 1/3 of those in the flat plate. Increasing the effect of curvature increases the heat loss, as expected. 

A typical plot of Dtja versus t is shown in Fig. 3. At the lower temperatures the straight-line portions do not extrapolate to the origin but intersect at a finite time which becomes progressively shorter as the temperature is raised. At temperatures near 900°C the lines go through the origin. The plots were linear until the exchange was 95-98% complete. Some tailing at the end may be due to the sphere size distribution. 

Within the bed, the radiation is treated by combining the ray tracing with the Monte Carlo method. The conduction through the spheres is allowed by solving for the temperature distribution in a representative sphere for each particle layer in the bed. 

Under steady state conditions, i.e., when the temperature at any point does not change with time, the temperature distribution in the sample is governed by Laplace s equation (Eq. 6). The sample geometries are chosen so that the temperature is a function of only one coordinate, and simple analytical solutions to Laplace s equation can be used. There are two geometries that satisfy this condition the parallel-faced slab with heat flow normal to the surfaces and the hollow cylinder with radial heat flow. In the latter case, although the heat flow is now in two dimensions the temperature is a function of only one coordinate, namely the radius, because of symmetry. The condition is also satisfied by the sphere, but that is not relevant to the present discussion. The first case is illustrated in Fig. I which shows a parallel-faced section of thickness. v and cross-scctional area A. 

In his first studies of the Fourier equation for temperature distribution in a reactive system, Frank-Kamenetskii restricted his attention to three shapes, the infinite slab, the infinite cylinder, and the sphere. For these three geometries (class A) the Laplacian operator can be expressed in terms of a single co-ordinate, and the steady-state problem is reduced to solving the ordinary differential equation... 

Temperature Distribution in a Hollow Sphere. Derive Eq. (4.2-14) for the steady-state conduction of heat in a hollow sphere. Also, derive an equation which shows that the temperature varies hyperbolically with the radius r. 

In 1939 Frank-Kamenetskii considered circumstances where Newtonian cooling was only an empirical approximation, and where the escape of heat was impeded internally by the thermal properties of the medium. (This will always be the case for a large enough system.) An internal temperature-distribution with a maximum at the middle results. For stability, this central temperature may not exceed a critical value. For a sphere with its surface at T the relationship is ... 

Hunter, S. C. (1967) The transient temperature distribution in a semi-infinite viscoelastic rod, subject to longitudinal oscillations. Int. J. Eng. Sci. 5, 119-143 Hunter, S.C. (1968) The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium . Proceedings of the Edinburgh Mathematical Society 16 (Series II), Part I, pp. 55-69 Hunter, S.C. (1983) Mechanics of Continuous Media, 2nd edition (Wiley, New Ycrk)... 

Figure 1.19 (a) Sphere diameter distribution for silica particles precipitated using R5 peptide (black) and R5-EAK (white) (b) Sphere size distribution for silica particles formed with RS-EAKi at different temperatures. Reprinted with permission from Ref [67] 2008, American Chemical Society. 

Their composition induces the formation of a morphology in which polystyrene spheres are distributed with a cubic symmetry in an elastomeric matrix of polybutadiene. At the service temperature the spherical polystyrene nodules are in the glassy state, whereas the polybutadiene chains connecting them are in the elastomeric state. Figure 5.36 shows the rigid nodules of PS in their role of physical cross-links, which are responsible for the reversibility of the deformations undergone by the sample. 

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