Heat conduction equation spheres

The equations for the diffusion profile can be obtained from the heat-conduction equations of Carslaw and Jaeger [Ref. 3, Eq. 9.4 (10)] by using the substitutions we have indicated. The subject is discussed from a different approach by Adams, Quan, and Balkwell (I). The profiles can then be integrated over the volume of the sphere to obtain the uptake as a function of time. 

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... 

Now consider a sphere, with density p, specific heat c, and outer radius R. The area of the sphere normal to the direclion of heat transfer at any location is A — 4vrr where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case also, and thus it varies with location. By considering a thin spherical shell element of thickness Ar and repeating tile approach described above for the cylinder by using A = 4 rrr instead of A = InrrL, the one-dimensional transient heat conduction equation for a sphere is determined to be (Fig. 2-17)... 

An examination of the one dimensional tiansient heat conduction equations for the plane wall, cylinder, and sphere reveals that all tlu-ee equations can be expressed in a compact form as... 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

If A and eg change with temperature, cf. section 2.1.4, a closed solution to the heat conduction equation cannot generally be found, which only leaves the possibility of using a numerical solution method. We will show how temperature dependent properties are accounted for by using the example of the plate, m = 0 in (2.274). The transfer of the solution to a cylinder or sphere (m = 1 or 2 respectively) is... 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

Models Based on a Desorption-Dissolution-Diffusion Mechanism in a Porous Sphere. The precursor of these models was the application by Bartle et. al [20] of the Pick s law of diflusion (or the heat conduction equation, i.e. the Fourier equation) to SFE of spherical particles. In doing so they had to assume an initial uniform distribution of the material extracted (in this specific case 1-8 cineole) from rosemary particles. Since Pick s law of difiusion from a sphere is analogous to a cooling hot ball (Crank [21] vs Carslaw and Jaeger [22]), this type of models have been considered to be analogous to heat transfer. This model was also used by Reverchon and his co-workers [23] and [24] to SFE of basil, rosemary and marjoram with some degree of success. 

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) ... 

In the case K > fi, the usual diffusion determines the kinetics for any gel shapes. Here the deviation of the stress tensor is nearly equal to — K(V u)8ij since the shear stress is small, so that V u should be held at a constant at the boundary from the zero osmotic pressure condition. Because -u obeys the diffusion equation (4.18), the problem is trivially reduced to that of heat conduction under a constant boundary temperature. The slowest relaxation rate fi0 is hence n2D/R2 for spheres with radius R, 6D/R2 for cylinders with radius R (see the sentences below Eq. (6.49)), and n2D/L2 for disks with thickness L. However, in the case K < [i, the process is more intriguing, where the macroscopic critical mode slows down as exp(- Q0t) with Q0 oc K. 

Consider the hot spot (at time zero) to be a tiny sphere of material at a uniform temp greater than that of the surroundings. If the material were inert and produced no heat by reaction, the cooling of the sphere in subsequent periods of time would be represented by the temp distribution curves in Fig 7 (Ref 6), These are derived from the classical treatment of the differential equation for heat conduction and may be found in Carslaw and Jaeger (Ref 5). Time is depicted in terms of a dimensionless parameter crt/a2, where a is the radius of the sphere and a. is its thermal diffusivity [a = X/(pc) j... 

Solve the one-dimensional, unsteady conductive heat transfer equation for a homogeneous solid sphere. (Hint (1) Let = R T so that... 

The amount of heat conducted across the boundary layer of a sphere can be calculated using Equation 7.16 ... 

Aside from this, the data on burning velocities seem to be in almost quantitative accord with the conduction equation (XIV. 10.23) when adapted to flames in finite systems such as cylinders and spheres. The velocity of flame propagation in tubes is complicated by the viscous drag exerted by the walls on the flowing gas, together with the heat losses at the walls. The resulting Poiseuille type of flow tends to make the flame fronts parabolic in these systems. 

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. 

Under conditions of laminar flow, the usual natural convection equations can be used. Reference (K2) gives a table of heat transfer equations for spheres and cylinders recommended for use when molecular conduction is a factor, and a second table applicable to natural convection under laminar flow conditions. 

General solutions of unsteady-state conduction equations are available for certain simple shapes such as the infinite slab, the infinitely long cylinder, and the sphere. For example, the integration of Eq. (10.16) for the heating or cooling of an infinite slab of known thickness from both sides by a medium at constant... 

Investigator Type of correlation Phases involved Model associated Model equation Kunii and Smith [29] Effective thermal conductivity of packed bed Fluid-solid One-dimensional heat transfer model Spheres in cubic array = <°- W>(K K) (in - ) .21 ... 

A real calorimeter is composed of many parts made from materials with different heat conductivities. Between these parts one can expect the existence of heat resistances and heat bridges. To describe the heat transfer in such a system, a new mathematical model of the calorimeter was elaborated [8, 19] based on the assumption that constant temperatures are ascribed to particular parts of the calorimeter. In the system discussed, temperature gradients can occur a priori. Before defining the general heat balance equation, let us consider the particular solutions of the equation of conduction of heat for a rod and sphere. For a body treated as a rod in which the process takes place under isobaric conditions, without mass exchange, the Fourier-Kirchhoff equation may be written as... 

Figure 10, Heat conductivity A,o of packed beds of equaUsized ceramic spheres at various pressures and room temperature according to Ref, 60 and Equation 23, Bios — 0, = 0, 39, pK 3,5 10 (a) Nitrogen y = 0J9 (b) R 12 y — 1,0 (c) nitrogen y 0.9 (d)...

Analytical solutions of equation 9.44 in the form of infinite series are available for some simple regular shapes of particles, such as rectangular slabs, long cylinders and spheres, for conditions where there is heat transfer by conduction or convection to or from the surrounding fluid. These solutions tend to be quite complex, even for simple shapes. The heat transfer process may be characterised by the value of the Biot number Bi where ... 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

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