External Heat Transfer Correlations

In this section the correlations used to determine the heat and mass transfer rates are presented. The convection process may be either free or forced convection. In free convection fluid motion is created by buoyancy forces within the fluid. In most industrial processes, forced convection is necessary in order to achieve the most economic heat exchange. The heat transfer correlations for forced convection in external and internal flows are given in Tables 4.8 and 4.9, respectively, for different conditions and geometries. 

Although further diseussion of heat transfer correlations is no doubt worthwhile, it will not help us to determine the mass transfer coefficieni and the mass flux from the bulk fluid to the external pellet smface. However, the preceding discussion on heat transfer was not entirely futile, because, for similar geometries, the heat and mass transfer correlafions are analogous. If a heat transfer coirrelation for the Nusselt number exists, the mass transfer coefficient can be estimated by replacing the Nusselt and Prandtl numbers in this correlation by the Sherwood and Schmidt numbers, respectively ... 

Since the outer cold fluid flows through a tube grid rather than over a single tube, we will employ tube-bank correlations for the external heat transfer coefficient. The use of these correlations requires that some quantities such as (the tube spacing), ST, and SL (see Appendix A) be known. For the tube size selected, the cross sectional area is Ar = 2.16 X 10-4 m2. From the total flow rate of the hot gas inside the tubes we get, for the number of tubes,... 

Fluid mechanics and heat transfer correlations involved with internal and external forced convection and natural convection are summarized in this appendix. These correlations are organized in the following manner... 

Equations are presented in this section for evaluating the heat transfer by natural convection from the external surfaces of bodies of various shapes. The correlation equations are of the form described in the section on the heat transfer correlation method, and the orientation of the surface is given by the surface angle defined in Fig. 4.4. Supporting experimental evidence for each such equation set is outlined after each equation tabulation. The correlations are in terms of Nu, Ra, and Pr, parameters that involve physical properties, a length scale, and a reference temperature difference. Rules for the evaluation of property values are provided in the nomenclature, and the relevant length scale and reference temperature difference are provided in a separate definition sketch for each problem. 

Now we will present separately heat transfer correlations for external and internal film-wise condensation. 

Heat Transfer Correlations for External Condensation. Although the complexity of condensation heat transfer phenomena prevents a rigorous theoretical analysis, an external condensation for some simple situations and geometric configurations has been the subject of a mathematical modeling. The famous pioneering Nusselt theory of film condensation had led to a simple correlation for the determination of a heat transfer coefficient under conditions of gravity-controlled, laminar, wave-free condensation of a pure vapor on a vertical surface (either flat or tube). Modified versions of Nusselt s theory and further empirical studies have produced a list of many correlations, some of which are compiled in Table 17.23. 

TABLE 17.23 Heat Transfer Correlations for External Condensation on Vertical Surfaces... 

Heat Transfer Correlations for Internal Condensation. Internal condensation processes are complex because a simultaneous motion of both vapor and condensate takes place (in addition to phase change phenomena) in a far more complex manner than for unconfined external condensation. The flow regime can vary substantially. Characteristics of a particular flow pattern involved are extremely important in describing particular heat transfer conditions. Correspondingly, to predict with confidence the heat transfer coefficient for internal film condensation appears to be even more difficult than for external condensation. 

The external mechanisms by which heat is transferred to the environment are well known. These include convection, conduction, radiation, and evaporation. Although physiologists have been inclined to measure their own heat and mass transfer coefficients on living men and on mannikins, there is no reason to believe that standard engineering correlations are inadequate for this purpose. We will concentrate on the way in which the human body exploits various external heat transfer mechanisms in maintaining a constant central temperature. 

Heat transfer in circulating fluidised bed boilers occurs on two types of surfaces. The bulk of heat transfer from the combustion side to the water side occurs on planar membrane wall surfaces which shape the furnace. Heat may also be transferred to horizontal tubes located in external bubbling bed heat exchangers. The latter may be treated using conventional bubbling bed heat transfer correlations. These two surface locations are shown in Figure 15. This discussion focuses entirely upon heat transfer to vertical membrane wall type surfaces. These are identical to the surfaces found in more conventional boilers such as pulverised coal, oil fired and stoker systems. 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Table 4. Correlations for Convective Heat-Transfer and Friction Coefficients for External Flow...

Fig. 35. Correlations for calculating heat-transfer coefficients for (a) turbine external jackets, internal cods, and baffle cods, and (b) for close-clearance...

Correlations of heat and mass-transfer rates are fairly well developed and can be incorporated in models of a reaction process, but the chemical rate data must be determined individually. The most useful rate data are at constant temperature, under conditions where external mass transfer resistance has been avoided, and with small particles... 

The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

Correlations for friction factors and heat transfer coefficients are rated in HEDH. Some overall coefficients based on external bare tube surfaces are in Tables 8.11 and 8.12. For single passes in cross flow, temperature correction factors are represented by Figure 8.5(c) for example charts for multipass flow on the tube side are given in HEDH and by Kays and London (1984), for example. Preliminary estimates of air cooler surface requirements ram be made with the aid of Figures 8.9 and 8.10, which are applied in Example 8.9. 

Forced convection heat transfer is probably the most common mode in the process industries. Forced flows may be internal or external. This subsection briefly introduces correlations for estimating heat-transfer coefficients for flows in tubes and ducts flows across plates, cylinders, and spheres flows through tube banks and packed beds heat transfer to nonevaporating falling films and rotating surfaces. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. 

Correlations for friction factors and heat transfer coefficients are cited in HEDH. Some overall coefficients based on external bare tube surfaces are in Tables 8.11 and 8.12. For single passes in cross flow, temperature correction factors are represented by... 

Chapter 3 dealt with the problem of the reaction kinetics for different gas-solid reactions, while chapter 5 dealt with the mass and heat transfer problems for porous as well as non-porous catalyst pellets. In chapter 5 different degrees of complexities and rigor were used. In chapter 5, the analysis started with the simplest case of non-porous catalyst pellets where the only mass and heat transfer Coefficients are those at the external surface which depend mainly on the flow conditions around the catalyst pellet and the properties of the reaction mixture. It was shown clearly that j-factor correlations are adequate for the estimation of the external mass and heat transfer coefficients (k, h) associated with these resistances. For the porous catalyst pellets different models with different degrees of rigor have been used, starting from the simplest case of Fickian diffusion with constant diffusivity, to the rigorous dusty gas model based on the Stefan-Maxwell equations for multicomp>onent diffusion. 

The catalyst pellets and the processes taking place on them represent the heart of this important and troublesome reactor. Modem catalysts for this reaction are usually non-porous with active material coating the external surface of the support pellets. Only few steady state cases are presented in order to show the complexity of the behaviour of the catalyst pellets. The behaviour is quite complex although the governing mathematical equations are simple. The only mass and heat transfer resistances in this case are the external mass and heat transfer resistances which are evaluated using J-factor correlations. The effect of the different parameters on the effectiveness factors are shown on the effectiveness factors vs. bulk temperature diagrams. 

FLOW OUTSIDE TUBES PARALLEL TO AXIS. Some membrane separators have bundles of hollow fibers in a shell-and-tube arrangement with liquid or gas flowing parallel to the tube axis on the outside of the tubes. The external flow passages are irregular in shape and not uniform, since the fibers are not held in position as are the tubes in a heat exchanger. Empirical correlations for the external mass-transfer coefficient have been proposed using an equivalent diameter to calculate the Reynolds number. For a bundle of fibers with diameter d packed in a shell with c void fraction, the equivalent diameter is... 

FLOW NORMAL TO CYLINDERS. A correlation of vs. for flow of air perpendicular to single cylinders is shown in Fig. 21.4. The dashed line shows values of jg calculated from Eq. (12.69), which was based on data for liquids. The data for heat transfer to air, taken from Fig. 12.6, fall slightly below the dashed line and very close to the mass-transfer data. The good agreement shows that the analogy between mass and heat transfer holds very well for external flows as well as for flows inside pipes. 

Vertical Surfaces. Condensation heat transfer coefficients for external condensation on vertical surfaces depend on whether the vapor is saturated or supersaturated the condensate film is laminar or turbulent and the condensate film surface is wave-free or wavy. Most correlations assume a constant condensation surface temperature, but variable surface temperature conditions are correlated as well as summarized in Table 17.23. All coefficients represent mean values (over a total surface length), that is, h = (1/L) 10 hloc dx. 

Vertical Surfaces. If the laminar flow direction is downward and gravity-controlled, heat transfer coefficient for internal condensation inside vertical tubes can be predicted using the correlations for external film condensation—see Table 17.23. The condensation conditions usually occur under annular flow conditions. Discussion of modeling of the downward internal convective condensation is provided in Ref. 76. 

In this equation, q",.dA is the net radiative heat flux at the moving material surface imposed by external sources such as radiant burners/heaters or electric resistance heaters. Both parabolic, boundary layer [80], and full, elliptic [61,81] problem solutions have been reported. Because of the nature of the problem, the heat transfer results can t be given in terms of correlations. The interested reader is referred to Refs. 62 and 79 for citation of relevant references. 

The external temperature difference can be related to the reaction rate and the heat generated in a pellet using the appropriate correlation to predict the heat transfer coefficient for the gas film. At steady state, the rate of heat generation is equal to the rate of heat removal. For a first-order reaction in a spherical pellet, the heat balance is... 

Figure 13.3-6 Wender and Cooper s correlation of fluid bed to external surface heat transfer coefficients [13], (from Zenz and Othmer [10]).

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