Heat Transfer in Bubble-Agitated Systems. A General Correlation Wallace F. Hart John Zink Company, Tulsa. Oklahoma 74 105
Experimental data and a general correlation are presented for heat transfer between a vessel wall and a liquid agitated by rising air bubbles. Water and ethylene glycol were tested in a 3.9 in. i.d. column using superficial air velocities from 0.001 to 0.067 Wsec. The general correlation is tested against these data and those of other authors. Some bubble holdup data are also reported.
Introduction Gas bubbles rising- through a liauid are caDable of Droducing considerable turbulence and promoting high rates of heat and mass transfer. This technique of “bubble agitation” can offer considerable advantages over mechanical stirring-especially a t high pressures where shaft-sealing is a problem, and in vessels of large length/diameter ratios where mechanical stirring is awkward to arrange. Many reactions of industrial importance are naturally suited to the use of bubble agitation, e.g., slurry-phase hydrogenations, air oxidation of organic liquids, the Oxo reaction, polymerization of low molecular weight olefins, etc. In all of these cases it is convenient to provide agitation by “sparging” an excess of the reactant gas up through the liquid phase. There are a number of aspects of bubble agitation which one might study. On a distillation tray bubble agitation is used to promote heat and mass transfer between the vapor and liquid phases. This problem has been studied rather extensively. In other cases, however, one is primarily interested in promoting heat transfer between the liquid phase and a solid surface-where the solid surface may be the vessel wall or a tube bundle or coil submerged in the liquid. This problem has not been studied very extensively, and no reliable correlation for predicting the heat transfer coefficients has been available. This is the subject of this present paper. The approach used in this work has been one of dimensional analysis of experimental data, resulting in the development of a general correlation for bubble-agitated heat transfer. The heat transfer work led to a limited study of gas holdup which is also important in the design of bubbleagitated reactors. I
Other Investigators This present work was done in 1965, and the findings were presented a t a National AIChE meeting in Houston in 1967, but it has never been published (Hart, 1965). Both before and since that time, several papers have been published on the subject. A thorough review of these other works is a fit subject for another paper. Rather than clutter this paper with such detail, I would prefer to summarize the situation, list the references, and allow the diligent reader to verify it for himself if he is so inclined. Attempts a t correlation of the heat transfer in gas-agitated systems can be classified as: (a) those which are dimensionless (hence potentially general), but which involve only the Nusselt, Reynolds, and Prandtl numbers. Study reveals they did not and could not fit the facts (Novosad, 1954; Kolbel et al., 1958a,b, 1960); (b) those which are obviously dimensional (hence not potentially general), but which may be useful for very limited design purposes (Fair, 1962, 1967); (c) those which appear to be dimensionless (hence
potentially general), but which, in fact, are not, (Yoshitome, 1965; Konsetov, 1966) but which do incorporate the gravitational constant, and this a t least represents a step in the right direction, but again, the facts ar,e not fitted; and (d) that one which is dimensionless and incorporates the gravitational constant, but is applied in such a way that it does not fit the facts accurately (Kast, 1962). The sum of it all is that there is no reliable general correlation for predicting the heat transfer between a bubbleagitated liquid and a solid surface (coil or jacket). There are, however, valuable data contained in these other papers. Though some of the experimental procedures are of questionable accuracy and most of the data are for water, several facts seem irrefutably established; i.e., the heat transfer coefficient between the liquid and a solid surface: (a) is not a function of column diameter, liquid height, or apparently of any other characteristic dimension of the system (data obtained in 2 in. diameter (Novosad, 1954) through 48 in. diameter (Fair, 1962) vessels support this); (b) is not a significant function of the location and geometry of the solid surface, i.e., the vessel wall or tube bundle are equally effective as heat transfer surfaces (Fair, 1962); (c) is not significantly affected by the type of gas distributor used (Fair, 1962; Yoshitome, 1965); and (d) is proportional to the superficial gas velocity to about the 0.25 power (Novosad, 1954; Kolbel et al., 1958a,b, 1960; Fair, 1962). Given these facts and my own experimental results, I offer a generalized correlation which I feel is reliable and extrapolatable and fundamentally correct.
Theoretical Discussion Hydrodynamics. Visual observation of a bubble-agitated liquid reveals that a t the lower gas rates, the bubbles remain fairly distinct from one another, are of a fairly uniform size, and rise in nearly single file up the column. Furthermore, a t very low gas rates-say 0.001 fthec-the bubbles are almost spherical in shape. At higher rates, however, considerable distortion is evident and visual observation reveals that the bubbles oscillate violently as they rise. At even higher gas rates, bubbles exist throughout the cross section of the column, considerable breakup and coalescence occurs, and it becomes impossible to identify any representative bubble size or shape. It is evident that considerable turbulence can be produced by bubble agitation. As a bubble rises, it exerts some drag on the liquid adjacent to it. Because of this drag, a series of bubbles rising successively will effect an upward flow of liquid in the vicinity of the bubble stream. Continuity then requires that an equivalent amount of liquid flow downward in regions outside the bubble stream “boundary layer”. This gives rise to a continuous axial circulation of liquid within the vessel. Obviously, a t high gas rates, the liquid flow pattern becomes tremendously complex. Ind. Eng. Chem.. Process Des. Dev., Vol. 15, No. 1 , 1976
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There must also exist a pulsating radial flow of liquid because a bubble stream is not a continuous thing. That is, liquid must move aside to make room for an approaching bubble and return to fill the bubble space when the bubble moves on. This radial flow is undoubtedly very important in promoting heat transfer at the wall. Heat Transfer. Owing to the complexity of bubble agitated systems, a completely analytical treatment of the heat. transfer problem is out of the question. Therefore, I have resorted to dimensional analysis of experimental data while using some semitheoretical tactics. The crux of the problem, of course, is to determine which variables are important to the hydrodynamics of the system. Toward this end it is always useful in dealing with forced-convection systems to consider the power dissipation per unit volume, because this quantity is a measure of the turbulence of the system. In a bubble-agitated system the power input is equal to the product of the gas flow rate, V , and the pressure drop across the column of liquid. The pressure drop in turn is equal to pHLg/gc, where H L is the static liquid height. If A is the cross sectional area of the column, then H L A is the volume of the system, and the power dissipation per unit volume is
This result is particularly enlightening. It exposes three important variables and offers some insight into why they are important. For one thing, it explains why the superficial gas velocity is the characteristic velocity of bubble-agitated systems. It also says that gravity influences the turbulence of the system. One might have expected gravity to influence the hydrodynamics only insofar as it influences bubble velocity. However, the appearance of g in eq 1 has nothing whatever to do with bubble velocity, but is the result of another fundamental role played by gravity. This is probably why correlations involving only the Nusselt, Reynolds, and Prandtl numbers cannot describe bubble-agitated heat transfer. Equation 1 offers some additional negative information: the overall turbulence is not a function of any dimension of the system. This is consistent with the experimental evidence that bubble-agitated systems have no characteristic dimension for heat transfer. A fourth variable which obviously affects the hydrodynamics is liquid viscosity, and this can be included in our dimensional analysis without requiring any discussion. Now, there are several other variables which one might suspect of being important; Le., bubble diameter ( D B ) ,surface tension, density difference between the gas and liquid phases ( A p ) , distributor design, and bubble velocity ( U B ) . Of these, however, only D B and U B can affect the hydrodynamics directly; the other variables affect the system only insofar as they affect D e or U B themselves. (In fact, the main effect of bubble diameter is the influence it in turn has on bubble velocity.) Thus, we can ignore surface tension, density difference, and distributor design and concentrate on D B and UB. It turns out that we can go even further and ignore both D B and U B in our dimensional analysis without incurring any serious limitations in our heat transfer correlation. A partial explanation for this is that both are rather restricted in how much they can be varied. This is especially true of UB. I t is not much affected by anything except a t very low bubble Reynolds numbers where Stoke’s law applies ( N R 1.0). ~ In this region D B , Ap, and p all have strong effects on U B , but at moderate to high bubble Reynolds numbers-which would apply in most practical situationsU B is nearly independent of all these variables. It is also 110
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pretty well independent of superficial gas velocity as will be shown later. (It should be pointed out here that the bubble velocity referred to throughout this paper is measured relative to the vessel. It is not the “slip” velocity relative to a so-called stagnant fluid which one often reads about.) The average bubble diameter does not vary a great deal either. This is especially true at moderate to high gas rates (say 0.05 ft/sec or greater) where the effect of distributor design diminishes considerably, and breakup and coalescence has an equalizing effect on bubble size. Actually, it would be useless to include D B in the dimensional analysis anyhow, because one could never satisfactorily determine it. Perhaps the best justification for leaving it out is that the correlation is successful without it. Based on preceding discussion it would seem that the basic limitation of our correlation will be that it may not be valid at very low bubble Reynolds numbers such as in fermentation processes where D B is very small, or at pressures approaching the critical where Ap approaches zero, or in liquids of extremely high viscosity. (Note, however, that has been included in other grounds and this would partially account for its effect on UB.) One may argue with good reason that the invariance of a quantity (such as D B or U B )is not justification for ignoring it in dimensional analysis. Indeed, the present author would himself insist that the gravitational constant, for instance, cannot be ignored. The answer to this paradox is that some parameters change the fundamental structure of a dimensionless correlation where others do not. The presence of g influences the basic structure of the correlating dimensionless groups. On the other hand, a variable such as Ap leads only to the group Ap/p, and U B leads only to the group UB/US.Groups such as these do not change the form of the correlation, but rather are correction factors (such as the Sieder-Tate factor ( p / p w ) ) . I t is offered, then, that the hydrodynamics of bubble-agitated systems can be sufficiently characterized by U S ,g , p , and p. The heat transfer can be related to the hydrodynamics by means of C, and k . With this choice of variables we have for predicting the film coefficient, h
Assuming the usual power function form and proceeding by usual methods gives
(3) where CY, 0,and y are constants to be evaluated from the data. Defining a j H factor in conventional fashion JH
h = CPUSP
(-)
(Y)O
(4)
Equation 3 can be written
(5) The method of determining the constants and exponents will be shown later when eq 5 is applied to the experimental results. Gas Holdup. A knowledge of the gas holdup in a bubbleagitated system is useful from a practical standpoint in that it allows one to calculate the average gas residence time. It also allows determination of the average bubble velocity (again, relative to the vessel and not the liquid). The fractional gas holdup, @, is defined
For a column of constant cross-sectional area, the volumes can be replaced by the corresponding heights above the gas inlet nozzle. The gas holdup relates the bubble velocity, UB, to the superficial gas velocity, Us. Consider a single bubble rising a t an average velocity of U B through a.liquid of height HT. Its holdup time is
where A is the column cross-sectional area. The holdup time can also be expressed as a function of the volumetric gas flow rate, V, and the volume occupied by the bubbles, VB
Eliminating t from eq 7 and 8 results in (9) or
The practical value of eq 10 is that it allows one to determine the average velocity of a complex array of bubbles, since a plot of 4 vs. U s yields a line of slope 1IUB. This, in turn, allows one to characterize the effect of other variables on U B .
Apparatus A flow diagram of the apparatus used in this work is presented in Figure 1. The study was conducted under steadystate conditions: liquid was fed continuously to a vessel supplied electrically with a uniform heat flux. The bubble agitation was provided by introducing air to the bottom of the column through a single vertical nozzle. A constant liquid level was maintained in the vessel by means of a sealleg overflow line equipped with a vent to prevent siphoning. The heated liquid overflowed into a surge tank, from which it was pumped through a jacketed cooler into a constant head tank. The cooled liquid was then recycled through a rotameter to the heated vessel. Temperature and flow rate measurements determined the heat flux, and the AT between the liquid and the vessel wall. Thus, the heat transfer coefficient could be determined. A drawing of the heated vessel is shown in Figure 2. It consists essentially of a 2-ft section of 3.9-in. i.d. copper pipe with butt-joined Fiberglas ends. The Fiberglas ends were installed so as to reduce the axial heat leak from the copper section, thus allowing a more accurate determination of the heated area involved. The copper section was wrapped with glass tape to electrically insulate it from the heating element. The heating element was 0.5 in. x 0.002 in. chromel-A tape which was wrapped around the glasscovered copper section with a spacing of about 0.25 in. between wraps. With a 10-A Powerstat, this heating element could deliver a heat flux up to about 3000 B t u h r ft2 through the inside area of the copper pipe. Nine thermocouples were installed a t equal intervals along the pipe wall. They were inserted into small holes bored to within about 1/64 in. of the inside pipe wall. The holes were then filled with lead shot which was tapped gently with a punch, causing the lead to “flow” into every cavity. Soldering could not be used because of the Fiberglas ends. For measuring the liquid temperature, five thermocouples were mounted on a lh-in. copper probe. They were
Figure 1. Apparatus for bubble-agitated heat transfer studies: 1, liquid feed rotameter; 2, air feed rotameter, same as 1;3, air supply pressure regulator; 4, mercury manometer; 5, glass vessel for photographic and visual study, 4 in. i.d. X 42 in. total length; 6, heated vessel, 3.9 in. i.d. X 42 in. total length; 24 in. copper section with butt-joined Fiberglas ends; 7, recycle tank, 2 1.; 8, recycle pump, Eastern centrifugal, Model D-10; 9, double-pipe exchanger for cooling recycle stream with tap water; 10, constant-head tank for liquid feed system; 11, probe for mounting liquid-side thermocouples, %-in. copper rod; 12, thermocouple selector switch to cold junction and potentiometer; 13, potentiometer, Leeds and Northrup Cat. No. 8687; 14, Powerstat for regulating heat flux, 220 V, 10 A; 15 volt-ammeter; 16, ice cold-junction.
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