Steady heat conduction spheres

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

Sphere-Flat Test Results. Kitscha [47] performed experiments on steady heat conduction through 25.4- and 50.8-mm sphere-flat contacts in an air and argon environment at pressures between 10 5 torr and atmospheric pressure. He obtained vacuum data for the 25.4-mm-diameter smooth sphere in contact with a polished flat having a surface roughness of approximately 0.13 pm RMS. The mechanical load ranged from 16 to 46 N. The mean contact temperature ranged between 321 and 316 K. The harmonic mean thermal conductivity of the sphere-flat contact was found to be 51.5 W/mK. The emissivities of the sphere and flat were estimated to be e, = 0.2 and e2 = 0.8, respectively. 

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... 

Solution of the problem for spherical growth of a single particle is simple. Let the rate of heat loss for a sphere be given by its steady-state conduction. If the diameter of the surface is d2 and that of the seed d, and the temperatures T% and Th respectively, then... 

Consider one-dimensional steady-state heat conduction in sphere that is, there is the temperature has only r dependence. The governing energy equation for a sphere with radius b, is... 

For one-dimensional steady-state heat conduction with no heat generation, in a hollow sphere of inner and outer radii a and b respectively, if the first kind boundary conditions are T r=a=T0 and... 

Conduction with Heat Source Application of the law of conservation of energy to a one-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equations for steady-state conduction in a flat plate of thickness 2R (b = 1), a cylinder of diameter 2R (b = 2), and a sphere of diameter 2R (b = 3). The parameter b is a measure of the curvature. The thermal conductivity is constant, and there is convection at the surface, with heat-transfer coefficient h and fluid temperature I. ... 

We start this chapter with a description of steady, unsteady, and multidimensional heat conduction. Then we derive the differential equation that governs heat conduction in a large plane wall, a long cylinder, and a sphere, and generalize the results to three-dimensional cases in rectangular, cylindrical, and spherical coordinates. Following a discussion of the boundary conditions, we present tlie formulation of heat conduction problems and their solutions. Finally, we consider lieat conduction problems with variable thermal conductivity. 

As a simple, but practically important application, the conduction of heat independent of time, so called steady conduction, in a flat plate, in a hollow cylinder and in a hollow sphere will be considered in this section. The assumption is made that heat flows in only one direction, perpendicular to the plate surface, and radially in the cylinder and sphere, Fig. 1.3. The temperature field is then only dependent on one geometrical coordinate. This is known as one-dimensional heat conduction. 

The three bodies — plate, very long cylinder and sphere — shall have a constant initial temperature d0 at time t = 0. For t > 0 the surface of the body is brought into contact with a fluid whose temperature ds d0 is constant with time. Heat is then transferred between the body and the fluid. If s < i90, the body is cooled and if i9s > -i90 it is heated. This transient heat conduction process runs until the body assumes the temperature i9s of the fluid. This is the steady end-state. The heat transfer coefficient a is assumed to be equal on both sides of the plate, and for the cylinder or sphere it is constant over the whole of the surface in contact with the fluid. It is independent of time for all three cases. If only half of the plate is considered, the heat conduction problem corresponds to the unidirectional heating or cooling of a plate whose other surface is insulated (adiabatic). 

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. 

Two approaches can be used for calculating interparticle and particle surface collision heat transfer (Amritkar et al., 2014). The first approach is based on the quasi-steady state solution of the coUisional heat transfer between two spheres (Vargas and McCarthy, 2002). The other approach is based on the analytical solution of the one-dimensional unsteady heat conduction between two semi-infinite objects. This approach was proposed by Sun and Chen (1988) based on the analysis of the elastic deformation of the spheres in contact. 

Liquified gases are sometimes stored in well-insulated spherical containers that are vented to the atmosphere. Examples in the industry are the storage of liquid oxygen and liquid ammonia in spheres. If the radii of the inner and outer walls are r, and r, and the temperatures at these sections are T, and T, an expression for the steady-state heat loss from the walls of the container may be obtained. A key assumption is that the thermal conductivity of the insulation varies linearly with the temperature according to the relation ... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

Here, q is the flux of heat (W m ), X is the thermal conductivity (W K ), T is temperature (K), and r is the distance from the center of the spherical heat source. Under the steady state approximation, the heat generated in the small sphere, Qj , is equal to the heat flow, Qflow. from the surface of the small sphere to the surrounding medium, as expressed by Eq. (8.7). 

The use of a wetted spherical model affords the opportunity of studying combustion under steady-state conditions. Forced convection of the ambient gas may be employed without distortion of the object. Sufficiently large models may be employed when it is desired to probe the gas zones surrounding the burning sphere. It is apparent that the method is restricted to conditions where polymerization products and carbonaceous residues are not formed. In the application of such models, the conditions of internal circulation, radiant heat transmission, and thermal conductivity of the interior are somewhat altered from those encountered in a liquid droplet. Thus the problem of breakup of the droplet as a result of internal temperature rise cannot be investigated by this method. 

C Consider steady oac-dimeosional heal conduction in a plane wall, long cylinder, and sphere with constant thermal conductivity and no heat generation. Will the temperature in any of these mediums vary linearly Explain. 

If we study the heat transfer from the surface of a sphere with r = R, that is immersed in a stagnant fluid we have a process of steady conduction from the surface of the sphere (r = R) through the fluid, radially outward to r = oo. The radial conductive flux is ... 

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. 

Here we indicate how previous effectiveness factor analyses may be extended to situations where the pellet is not isothermal. Consider the case of a spherical pellet within which a catalytic reaction is taking place. If we examine an infinitesimally thin spherical shell with internal radius r similar to that shown in Figure 12.4 and write a steady-state energy balance over the interior core of the pellet, it is obvious that the heat flow outward by conduction across the sphere of radius r must be equal to the energy transformed by reaction within the central core. The latter quantity is just... 

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