Heat transfer penetration theory

Weekman and Myers (W3) measured wall-to-bed heat-transfer coefficients for downward cocurrent flow of air and water in the column used in the experiments referred to in Section V,A,4. The transition from homogeneous to pulsing flow corresponds to an increase of several hundred percent of the radial heat-transfer rate. The heat-transfer coefficients are much higher than those observed for single-phase liquid flow. Correlations were developed on the basis of a radial-transport model, and the penetration theory could be applied for the pulsing-flow pattern. 

An expression for the net rate of phase change can also be derived by assuming that the phase-change process is controlled solely by the rate at which heat can be transferred between the bulk liquid and the liquid surface. In this penetration theory approach, the liquid surface temperature is assumed to equal the gas-phase temperature. The heat transfer within a liquid element is assumed to occur by pure conduction, and therefore... 

The validity of the penetration theory points out that heat transfer in an agitated viscous thin film (even in the case of evaporation) and the mass transfer are mainly effected by forced convection and continuous surface renewal. 

The particle convective heat transfer component is usually treated on the basis of the penetration or packet theory originally proposed by Mickley and Fairbanks (1955) assuming that the clusters are formed next to the immersed surface (e.g., Subbarao and Basu, 1986 Basu and Nag, 1987 Zhang et ai, 1987 Liu et ai, 1990). In that case, the clusters of solids and voids or dispersed phase are assumed to come into contact with the heat transfer surface alternatively, and the heat transfer coefficient can be given as follows ... 

Equations (I3-II5) to (13-117) contain terms, e, for rates of heat transTer from the vapor phase to the liquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theory for the liquid phase. 

An analogue heat transfer coefficient relation has been derived by the penetration concept. The heat transfer theory is reviewed by Thomson [153]. 

The various forms of the penetration theory can be classified as surface-renewal models, implying either formation of new surfaee at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time or its reciprocal, the average rate of renewal, are functions of the fluid velocity, the fluid properties, the the geometry of the system and can be accurately predicted in only a few special cases. However, even if tj must be determined empirically, the surface-renewal models give a sound basis for correlation of mass-transfer data in many situations, particularly for transfer to drops and bubbles. The similarity between Eqs. (21.44) and (15,20) is an example of the close analogy between heat and mass transfer. It is often reasonable to assume that tj-is the same for both processes and thus to estimate rates of heat transfer from measured mass-transfer rates or vice versa. 

If potential flow and constant surface temperature are assumed, an equation analogous to Eq. (18) is obtained for the internal Nusselt number. Note, however, that the reference velocity in the internal Peclet number is the drop velocity. Similar results will be obtained from the penetration theory, according to which the film is assumed infinite with respect to the depth of heat penetration during the short contact time of a fluid element sliding over the interface. Licht and Pansing (L13) report West s equation, based on the transient film concept, for the case of mass transfer through the combined film resistance. In terms of the overall heat-transfer resistance, l/U (= l//jj + I/he) and if the contact time is that required for the drop to traverse a distance equal to its diameter. West s equations yield... 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

With an exothermic reaction in the bed, clusters or packets of hot solid come in contact with the cooler surface of the wall or the tubes, and they give up some of their heat in the short time before they are swept away. A model based on the penetration theory of unsteady-state heat transfer and some supporting data were presented by Mickley and Fairbanks [26]. The average coefficient is predicted to vary with the square root of the thermal conductivity, density, and heat capacity of the clusters and inversely with the square root of the average contact time. The fraction of the surface in contact with clusters is taken to be (1 — a), where a is the volume fraction bubbles in the bed. Heat transfer to the bare surface... 

The absorption of a gas during condensation of water vapor on a cold water droplet Is a complex process characterized by unsteady state mass and heat transfer [6]. In the classical development of absorption of a gas In a liquids three theoretical models have ensued The film theory, the penetration theory, and the boundary layer theory. Each model Invokes different assumptions which result in different conclusions. 

However, as with the penetration theory analysis, the difference in magnitude of the mass and thermal diffusivities with cx 100 D, means that the heat transfer film is an order of magnitude thicker than the mass transfer film. This is depicted schematically in Fig. 8, The fall in temperature from T over the distance x is (if a = 100 D) about 0% of the overall interface excess temperature above the datum temperature T.. Furthermore, in considering the location of heat release oue to reaction in the mass transfer film, this is bound to be greatest closest to the interface, and this is especially the case when the reaction becomes fast. Therefore, two simplifications can be introduced as a result of this (i) the release of heat of reaction can be treated as am interfacial heat flux and (ii) the reaction can be assumed to take place at the interfacial temperature T. The differential equation for diffusion and reaction can therefore be written... 

Combining Highbie s Penetration theory and Kolmogoroff s theory on isotropic turbulence, it has been shown recently [34l that experimental data on aerated liquids in stirred tank reactors can be correlated for coil as well as wall heat transfer by the equation... 

Other relationships based on conventional penetration theory for packed bed have deduced that Nu = 2 i -JPe and for low Peclet numbers to the order of 10, Tscheng and Watkinson (1979) deduced an empirical correlation, where Nu = 11.6 x7e°. Any of these would suffice in estimating the wall-to-bed heat transfer coefficient as functions of the kiln s rotational speed, w, and the dynamic angle of repose, The calculated values of Nusselt numbers using Perron and Singh s... 

High mass transfer rates will influence not only the mass transfer coefficient but also the heat transfer coefficients and friction factor. Analysis of film theory penetration theory and boundary layer theory (21) show that the relation of the various coefficients at high (k ) and low mass transfer (kj ) can be given by 0 s ... 

The basics of charge transfer may also be presented in the form of two analogies. One involves using equations that describe the collision mechanics between particles and the wall, as presented by Timoshenko (1951) and developed by Soo (1967). This is quite similar to the basic heat-transfer analysis. The second approach is to use the penetration theory as given by Higbie (1935) and Danckwerts (1951) for heat, mass, and momentum transfer for the analysis of charge transfer. 

Contact time in surface renewal and penetration theories (s) overall heat transfer coeflftcient J/m sK ... 

Numerous models have been proposed to explain the high heat transfer rates attainable in fluidized bed operation. Mickley et al. [56] and coworkers proposed that heat transfer between the bed and a surface takes place in unsteady pulses during a brief residence time of gas-solid pockets in the vicinity of the wall. In many ways this model is analogous to the penetration theory for mass transfer. 

The classic Thiele-Damkohler theory accounts for these effects, but is restricted to isothermal behavior and intraparticle mass transfer only by diffusion. If the reaction is highly exothermic and the particle is a poor heat conductor, the temperature in the particle center may rise above that in the contacting fluid and cause the overall rate to be higher than in the absence of heat- and mass-transfer limitations. Moreover, gas-phase reactions with change in mole number cause forced inward or outward convection that assists or counteracts reactant penetration into the particle and so enhances or depresses the rate. 

To this end, there are many theories which attempt to interpret or explain the behavior of mass-transfer coefficients, such as the film, penetration, surface-renewal, and other theories. They are all speculations, and are continually being revised. It is helpful to keep in mind that transfer coefficients, for both heat and mass transfer, are expedients used to deal with situations which are not fully understood. They include in one quantity effects which are the result of both molecular and turbulent diffusion. The relative contribution of these effects, and indeed the detailed character of the turbulent diffusion itself, differs from one situation to another. The ultimate interpretation or explanation of the transfer coefficients will come only when the problems of the fluid mechanics are solved, at which time it will be possible to abandon the concept of the transfer coefficient. 

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