Heat transfer asymptote

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

Kutateladze, S. S., and A. I. Leont ev, 1966, Some Applications of the Asymptotic Theory of the T urbu-lent Boundary Layer, Proc. 3rd Int. Heat Transfer Conf, vol. 3, pp. 1-6, AIChE, New York. (5) Kutateladze, S. S., and L. G. Malenkov, 1974, Heat Transfer at Boiling and Barbotage Similarity and Dissimilarity, Proc. 5th Int. Heat Transfer Conf., Tokyo, vol. IV, p. 1. (2)... 

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

In reality, Eqs. (13) and (14) should be solved simultaneously with Eqs. (8) and (9), but no analytical solution is available. However, we can examine the asymptotic solutions to Eqs. (13) and (14) to determine the bubble growth rate when heat transfer limits the growth, i.e., when P r) — Pq and r Tb so no inertial effects are present. For this extreme,... 

Thus the initial growth is largely controlled by inertial restraints and the radius increases linearly with time [Eq. (12)], whereas at long times heat transfer considerations become predominant and the growth is given by Eq. (16). The actual variation of r with t is less than either of these two asymptotic extremes (see Theofanous, 1969). 

Relaxing the restriction of low Reynolds number, Rimmer (1968,1969) used a matched asymptotic expansion technique to develop a solution in terms of Pe and the Schmidt number Sc (or Prandtl number Pr for heat transfer), where Sc = v/D.j and Pr = v/a in which v is the kinematic viscosity of the flowing fluid. His solution, valid for Pe < 1 and Sc = 0(1), is... 

This asymptote occurs whenever the heat transferred back to the burning surface is small compared to the heat released in the gas phase, which implies that Xg/bg is sufficiently large. Though Eq. (3.48) is not precise, it indicates the burning rate behavior without introducing mathematical complexities and reaction parameters. 

In what follows, the preceding evaluation procedure is employed in a somewhat different mode, the main objective now being to obtain expressions for the heat or mass transfer coefficient in complex situations on the basis of information available for some simpler asymptotic cases. The order-of-magnitude procedure replaces the convective diffusion equation by an algebraic equation whose coefficients are determined from exact solutions available in simpler limiting cases [13,14]. Various cases involving free convection, forced convection, mixed convection, diffusion with reaction, convective diffusion with reaction, turbulent mass transfer with chemical reaction, and unsteady heat transfer are examined to demonstrate the usefulness of this simple approach. There are, of course, cases, such as the one treated earlier, in which the constants cannot be obtained because exact solutions are not available even for simpler limiting cases. In such cases, the procedure is still useful to correlate experimental data if the constants are determined on the basis of those data. 

The ratio, L/D, of length to diameter of a packed tube or vessel has been found to affect the coefficient of heat transfer. This is a dispersion phenomenon in which the Peclet number, uL/Ddisp, is involved, where D Sp is the dispersion coefficient. Some 5000 data points were examined by Schliinder (1978) from this point of view although the effect of L/D is quite pronounced, no dear pattern was deduced. Industrial reactors have LID above 50 or so Eqs. (6) and (7) of Table 17.18 are asymptotic values of the heat transfer coefficient for such situations. They are plotted in Figure 17.36(b). 

The Asymptotic Law of Heat Transfer at Small Velocities in the Finite Domain Problem ... 

An exothermal reaction is to be performed in the semi-batch mode at 80 °C in a 16 m3 water cooled stainless steel reactor with heat transfer coefficient U = 300 Wm"2 K . The reaction is known to be a bimolecular reaction of second order and follows the scheme A + B —> P. The industrial process intends to initially charge 15 000 kg of A into the reactor, which is heated to 80 °C. Then 3000 kg of B are fed at constant rate during 2 hours. This represents a stoichiometric excess of 10%.The reaction was performed under these conditions in a reaction calorimeter. The maximum heat release rate of 30Wkg 1 was reached after 45 minutes, then the measured power depleted to reach asymptotically zero after 8 hours. The reaction is exothermal with an energy of 250 kj kg-1 of final reaction mass. The specific heat capacity is 1.7kJ kg 1 K 1. After 1.8 hours the conversion is 62% and 65% at end of the feed time. The thermal stability of the final reaction mass imposes a maximum allowed temperature of 125 °C The boiling point of the reaction mass (MTT) is 180 °C, its freezing point is 50 °C. 

Measurements of heat transfer in circulating fluidized beds require use of very small heat transfer probes, in order to reduce the interference to the flow field. The dimensions of the heat transfer surface may significantly affect the heat transfer coefficient at any radial position in the riser. All the treatment of circulating fluidized bed heat transfer described is based on a small dimension for the heat transfer surface. The heat transfer coefficient decreases asymptotically with an increase in the vertical dimension of the heat transfer surface [Bi et al., 1990]. It can be stated that the large dimensions of the heat transfer surface... 

The overall heat transfer coefficient U in Eqn. (3) is based on the measured temperature difference between the central axis of the bed and the coolant. It is derived by asymptotic matching of thermal fluxes between the one-dimensional (U) and two-dimensional (kr,eff kw,eff) continuum models of heat transfer. Existing correlations are employed to describe the underlying heat transfer processes with the exception of Eqn. (7), which is a new result for the apparent solid phase conductivity (k g), including the effect of the tube wall. Its derivation is based on an analysis of stagnant bed conductivity data (8, 9), accounting for "central-core" and wall thermal resistances. 

The next table lists the asymptotic formulas for Bessel functions for large values of x. These are useful for problems involving cylindrical geometry in heat transfer. 

D. Steiner and J. Taborek. Flow boiling heat transfer in vertical mbes correlated by an asymptotic model . Heat Transfer Engineering, 13, 2, pp. 43-69 (1992). 

To investigate the role of heat transfer in OMD and measure the asymptotic temperature difference, Celere and Gostoh [86] used the flat sheet membrane module shown in Figure 19.15. This contains a few flat sheet membranes placed 1 mm apart and supported by mesh-type spacers of 2 mm thickness, leaning against polypropylene walls. One stream (feed) flows between the membranes and other (extractant) flows cocurrently through the spacers. The mass transfer zone of each membrane is 80 mm in breadth and 200 mm in length. 

This relationship serves in many publications as the definition of a thermally fully developed flow. It is, as we have already seen, a result of the fact that the heat transfer coefficient reaches its asymptotic, constant end value downstream. 

An alternative insert for tubes is the spiral insert produced by Cal Gavin. The device consists of a matrix of wire loops on a twisted wire core. The diameter of the coils is such that it is a push fit. Laboratory studies using these wire matrix inserts with an Arabian crude oil demonstrate that the steady asymptotic value of the fouling resistance is reached in only 10 hr from start up and only of the order of 2-7% of the recommended TEMA value. The presence of the wires, particularly those in contact with the heat-transfer surface, creates turbulence that is largely responsible for the beneficial effects on the fouling. The benefits are not without the potential penalty of increased pressure drop and hence increased pumping costs. By suitable design for the same duty, this penalty can be substantially reduced. 

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