Conduction, heat transfer sphere

Contact Drying. Contact drying occurs when wet material contacts a warm surface in an indirect-heat dryer (15—18). A sphere resting on a flat heated surface is a simple model. The heat-transfer mechanisms across the gap between the surface and the sphere are conduction and radiation. Conduction heat transfer is calculated, approximately, by recognizing that the effective conductivity of a gas approaches 0, as the gap width approaches 0. The gas is no longer a continuum and the rarified gas effect is accounted for in a formula that also defines the conduction heat-transfer coefficient ... 

Figure 5 shows conduction heat transfer as a function of the projected radius of a 6-mm diameter sphere. Assuming an accommodation coefficient of 0.8, h 0) = 3370 W/(m -K) the average coefficient for the entire sphere is 72 W/(m -K). This variation in heat transfer over the spherical surface causes extreme non-uniformities in local vaporization rates and if contact time is too long, wet spherical surface near the contact point dries. The temperature profile penetrates the sphere and it becomes a continuum to which Fourier s law of nonsteady-state conduction appfies. 

Fig. 5. Profile of conduction heat transfer across the gap between a sphere and a flat plate vs projected radius, R = 3 mm, of the sphere at 40°C and 2.1...

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by... 

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

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

Figure 9-2. Contours of constant temperature for pure conduction heat transfer from a heated (or cooled) sphere. For constant surface temperature, 9 = 1, the isotherms are spherically symmetric, as indicated by Eq. (9-19).

It is seen that the Nusselt numbers for BFBs fall below those for convection from a single sphere, for Reynolds numbers less than 20. In fact, the magnitude of Nup for fluidized beds drops below the value of 2.0, which represents the lower limit of conduction heat transfer. The cause of this is the bubbling phenomenon. Low Reynolds numbers correspond to beds of fine particles (small flip and C/g), wherein bubbles tend to be clouded with entrained particles. This diminishes the efficiency of particle-gas contact below that represented by idealized plug flow, resulting in reduced values of Nup. As particle diameter increases (coarse particle beds), bubbles are relatively cloudless and gas particle contact improves. This is shown in Fig. 2 where the Nusselt numbers of fluidized beds are seen to increase with... 

An important application of heat transfer to a sphere is that of conduction through a stationary fluid surrounding a spherical particle or droplet of radius r as encountered for example in fluidised beds, rotary kilns, spray dryers and plasma devices. If the temperature difference T[ T2 is spread over a very large distance so that r2 = oo and 7 t is the temperature of the surface of the drop, then ... 

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 ... 

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) ... 

Experimental data on heat transfer from spheres to an air stream are shown in Fig. 5.20. Despite the large number of studies over the years, the amount of reliable data is limited. The data plotted correspond to a turbulence intensity less than 3%, negligible effect of natural convection (i.e., Gr/Re <0.1 see Chapter 10), rear support or freefloating, wind tunnel area blockage less than 10%, and either a guard heater on the support or a correction for conduction down the support. Only recently has the effect of support position and guard heating been appreciated a side support causes about a 10% increase in Nu... 

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 ... 

Second, the cold-plate underside consists of highly porous media composed of silver-bonded copper spheres that transfer thermal energy through conduction and aid convective heat transfer to the coolant. Heat-transfer capability is greatly... 

Figure 1736. Effective thermal conductivity and wall heat transfer coefficient of packed beds. Re = dpG/fi, dp = 6Vp/Ap, s -porosity, (a) Effective thermal conductivity in terms of particle Reynolds number. Most of the investigations were with air of approx. kf = 0.026, so that in general k elk f = 38.5k [Froment, Adv. Chem. Ser. 109, (1970)]. (b) Heat transfer coefficient at the wall. Recommendations for L/dp above 50 by Doraiswamy and Sharma are line H for cylinders, line J for spheres, (c) Correlation of Gnielinski (cited by Schlilnder, 1978) of coefficient of heat transfer between particle and fluid. The wall coefficient may be taken as hw = 0.8hp.

We assume that the adsorbent mass used in the kinetic test consists of a sphere of radius R. It may be composed of several microsize particles (such as zeolite crystals) bonded together as in a commercial zeolite bead or simply an assemblage of the microparticles. It may also be composed of a noncrystalline material such as gels or aluminas or activated carbons. The resistance to mass transfer may occur at the surface of the sphere or at the surface of each microparticle. The heat transfer inside the adsorbent mass is controlled by its effective thermal conductivity. Each microparticle is at a uniform temperature dependent on time and its position in the sphere. 

The work of Furnas above described and those of others were reexamined by Lovell and Karnofsky (1943). Their mathematical studies were confined to packings composed of spheres of uniform diameter and made allowance for resistance to heat transfer by conduction within the solids. The equation obtained by Lovell and Karnofsky, without correcting for resistance, is of the form encountered in the design of continuous heat exchangers,... 

In these parameters s designates some characteristic dimensions of the body for the plate it is the half-thickness, whereas for the cylinders and sphere it is the radius. The Biot number compares the relative magnitudes of surface-convection and internal-conduction resistances to heat transfer. The Fourier modulus compares a characteristic body dimension with an approximate temperature-wave penetration depth for a given time r. 

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. ... 

The inclusive quantity h is at least as large as the convective heat-transfer coefficient, so that a lower bound for the conduction term is given by calculating it using h instead of h. A good illustration is afforded by the fact that the term calculated in this way for a bed of steel spheres with air flowing through it at a Reynolds number of 500... 

---