Heat transfer to single sphere

SK Friedlander. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J 3 43-48, 1957. 

Interest extends from transfer to single particles to systems in which the particles are in the form of fixed or fluidised beds. The only case for which there is a rigorous analytical solution is that for heat by conduction and mass transfer by diffusion to a sphere. 

For heat transfer to a single sphere in turbulent flow, McAdams (1954) suggested... 

Because of the analogy between mass transfer by diffusion and heat transfer by conduction in a boundary layer, correlations for mass transfer and heat transfer to particles are similar. For mass transfer to a single isolated sphere,... 

In upflow operation the liquid to particle Sherwood number is higher than in downflow operation and increases remarkably with gas flow, indicating the large contribution of the liquid turbulence caused by bubble motion. Recently attempts were made to analyse the situation also theoretically and a unified correlation has been developed for heat and mass transfer from single spheres, packed beds as well as tube wall, taking into account the diffusion of solute into a liquid film,oscillating with response to the bub-... 

Scott SA, Davidson JF, Dennis JS, Hayhurst AN Heat transfer to a single sphere immersed in beds of particles supplied by gas at rates above and below minimum fluidization, Ind Eng Chem Res 43 5632-5644, 2004. 

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) ... 

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,1584 and Ranz and Marshall,15051 among others. The correlation recommended by Whitaker is accurate to within 30% for the range of parameter values listed. All properties except jus should be evaluated at Tm. For freely falling liquid droplets, the Ranz-Marshall correlation 505 is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account... 

In flow situations, empirical analogies between mass and heat transfer are usually employed. For single-particle mass transfer, the boundary layer analysis for mass transfer is similar to that for heat transfer and thus is used for typical applications such as sublimation of a solid (e.g., naphthalene ball) or evaporation of a liquid drop falling in air. For a single sphere of diameter dp moving in a fluid, in terms of a boundary layer analysis analogous... 

Rowe, P. N. and Claxton, K. T. (1965). Heat and Mass Transfer from a Single Sphere to a Fluid Flowing Through an Array. Trans. Instn. Chem. Engrs., 43, T321. 

The only reported-study on heat transfer is by Weekman and Myers.105 They measured wall-to-bed heat-transfer coefficients in a cocurrent air-water downward flow through a packed column. Three types of packings. 0.65-cm alumina spheres, 0.475-cm glass spheres and 0.38-cm TCC beads were examined. The heat-transfer coefficients were much higher than those observed for single-phase liquid flow. The transition from homogeneous to pulsed flow corresponded to an increase of... 

Once the upward flowing fluid has reached the minimum fluidisation velocity wml and with that the Reynolds number the value Remi = w dp/i/, point a in Fig. 3.40, a fluidised bed is formed. The heat and mass transfer coefficients hardly change with increasing fluid velocity The Nusselt and Sherwood numbers are only weakly dependent on the Reynolds number, corresponding to the slightly upwardly arched line a b in Fig. 3.40. After a certain fluid velocity has been reached, indicated here by point b in Fig. 3.40, the particles are carried upwards. At point b the heat and mass transfer coefficients are about the same as those for flow around a single sphere of diameter dP. 

MASS TRANSFER IN PACKED BEDS. There have been a great many studies of mass transfer and heat transfer from gases or liquids to particles in packed beds. The coefficients increase with about the square root of the mass velocity and the two-thirds power of the diffusivity, but the correlations presented by different workers differ appreciably, in contrast to the the close agreement found in studies of single spheres. An equation that fairly well represents most of the data is ... 

In order to use equation (8-15) for determination of T it is necessary to know the value of hGtkr known as the psychrometric ratio. Values of hG and ky can be estimated independently for the particular shape of the wetted surface by correlations like those presented in Chapter 2, using the heat- and mass-transfer analogy if necessary. Alternatively, experimental values of the psychrometric ratio can be used. Henry and Epstein (1970) have examined the data and methods of measurement and have produced some measurements of their own. For turbulent flow of gases past cylinders, such as wet-bulb thermometers, and past single spheres, the results of 18 gas-vapor systems are well correlated by... 

Rowe, P.N. and Claxton, K.T. (1965) Heat and mass transfer from a single sphere to fluid flowing through an array. Transactions of the Institution of Chemical Engineers, 43, 321-331. 

When a single sphere is being heated or cooled by a fluid flowing past it, the following equation can be used to predict the average heat-transfer coefficient for a-.VR, = Dvpjp of 1 to 70 000 and a Nf, of 0.6 to 400. 

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... 

It should be noted that the Prandtl number term in these two equations is somewhat speculative, since the experimental data did not cover a sulficient range for good correlation. Equations (12) and (13) include Pr to the 0.33 power, based on the expectation that dependence is similar to that for single spheres, as shown in Eq. (11). In applying heat transfer coefficients calculated by Eqs. (12) and (13), it is important to use a model that considers the particles to be well mixed and the gas to be in plug flow, in order to be consistent with the definition of as discussed above. 

Mass transfer between gas and particles affects gas-solids contact efficiencies in CFB risers. The mass transfer from a single particle to the suspension in CFB risers has been studied based on the sublimation of naphthalene spheres (Haider and Basu, 1988 Li et al., 1998), dehydration of 2-propanol (Masai et al., 1985), adsorption of CCI4, naphthalene, H2S, and NO (Kwauk et al., 1986 van der Ham et al., 1991, 1993 Vollert and Werther, 1994), and heat transfer between a heat pulse and suspension (Dry et al., 1987). For one-dimensional steady-state plug flow of the gas, a mass balance of the adsorbed species in a differential volume element of the reactor (Kwauk et al., 1986 Vollert and Werther, 1994) yields... 

Many industrially important chemical reactions occur in liquid-liquid systems since heat and mass transfer can be very efficient in agitated heterogeneous stirred reactors. The reaction usually takes place in the dispersed phase. Transport rates depend on the slip velocity between the phases as shown in eqs. (12-63) and (12-64). They are applicable only to single drops that are larger than the turbulent macroscale and are presented for illustrative purposes only. A tank-specific correlation is given later. The heat transfer coefficient, hx, for a single sphere is given by... 

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