CLOSE-CLEARANCE AGITATORS
of evaporation. I t appears that the dry wall condition can be circumvented by proper agitator design and proper liquid flow rate. An important factor for both thin-film and liquid-full operation, which is seldom mentioned in the literature, is the different effects of the various variables in different flow regimes. One must be constantly aware that there is a laminar regime and a turbulent regime with some rather ill-defined transition regime separating them. The most important consequence of different flow regimes here is that in laminar flow the heat transfer is independent of fluid viscosity, which is not true for turbulent flow. This is a very useful generalization for heat or mass transfer in any flow system.
pAy9 h
W. Roy Penney
. Bell
ur overall intent here is the same as for Part I on power requirements: namely, we aim to summarize and critically evaluate heat transfer data and correlations in close-clearance equipment. We shall scrutinize every general correlation to determine if it is consistent with the phenomenological bases of fluid dynamics and energy transport; this the design engineer needs because generally he has neither time nor desire to check every correlation he uses. We shall also attempt to find, from the many experimental and analytical investigations which this review embraces, the common thread which allows generalization. This article will be restricted primarily to the two variables which one must predict to size a heat exchanger; namely, the heat transfer coefficient and the mean temperature difference (MTD). In general, these two variables are insufficient for complete design of equipment used in distillation, crystallization, and chemical reaction where mass transfer and chemical kinetics are important. We shall not be concerned with m a s transfer and kinetics here; we caution the reader to be careful to consider all factors which affect equipment operation. Two modes of operation, which are expected to be very important from a predictive standpoint, are used in close-clearance equipment: (1) liquid-full and (2) thin-film. Liquid-full needs no explanation. Thinfilm operation refers to the condition where the liquid flowsby gravity or by agitator action (in tapered units) in a thin film down the vessel wall. Not only is the operation in the two modes different from a physical standpoint but also the applications are different. Thin-film operation is almost always used in evaporation service, whereas liquid-full operation is almost always used in sensible heating or cooling service. In thintilm operation the heat transfer surface can become dry with a resulting decrease in heat transfer coefficient. This occurs particularly in evaporators because the liquid film can evaporate to dryness: however, photographic evidence shows it may also occur in the absence
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W.Rcy Penney is N D E A FeUow, and K. J. BeII is Assmiate Professm in th School of Chmical Engincming ot Oklahoma State Univmsiy. AUTHORS
THE EFFECT OF AXIAL DISPERSION HEAT ON THE MTD
OF
In evaluating the mean temperature difference (MTD) for a heat exchanger with parallel streams, it is common practice to consider only the two extremes of (1) perfect mixiug of one or both streams which results in constant temperature throughout each mixed stream, or (2) no axial dispersion of heat either by backmixing or molecular conduction, commonly called plug flow. For perfect mixing, the temperature in the stream is equal to the stream outlet temperature. For no backmixing, constant heat transfer coefficient,and negligible conduction within and along the channel walls, the temperature distribution of a stream is logarithmic and the M T D is equal to the logarithmic mean temperature difference (LMTD). Complete mixing gives the absolute minimum M T D ; no axial dispersion gives the absolute maximum MTD. These two extremes cannot be achieved-only approached; every heat exchanger operates between these two extremes to a degree governed by axial dispersion of heat in the exchanger. Previous investigators who have attempted to develop correlating methods have neglected the effect of axial dispersion of heat on the temperature distribution in agitated exchangers, except for pot-type equipment where complete backmixing is generally assumed. All correlations have been based on the logarithmic mean temperature difference (LMTD), which assumes (among other things) no axial dispersion. At present the quantitative effect of axial dispersion on the M T D cannot be predicted. In general, as axial flow rate decreases and rotation speed increases, relative axial dispersion increases and the true M T D decreases. We shall cover the work which pertains to this very important aspect of heat exchanger performance. No work has been directed toward developing quantitative methods for predicting the influence of backmixing on the M T D in agitated exchangers. However, considerable work has been done to characterize axial mixing as a function of the system parameters. Most of this work has related to predicting the effect of axial mixing on diffusion and chemical reaction. It is beyond the scope of this article to go into detail here; we refer the reader to chemical kinetics texts such as that of Levenspiel (28) and to an article by Smoot and Bahb VOL 5 9
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(36) which investigates and predicts the effect of backmixing on mass transfer in a pulsed column. The great majority of work on mixing has been directed toward using the dispersion model, which is analogous to Fick’s law of diffusion
for characterizing both molecular and turbulent dispersion. This model has been used successfully, particularly for flow in pipes, for which the Peclet number has been shown experimentally to be a function of Reynolds number only in completely turbulent flow and a function of Reynolds number and Schmidt number in laminar flow. O n the basis of the analogy between the transfer of heat and mass, one would expect that the dispersion model would also suffice for heat transfer. White and Churchill (39) have given the differential energy equation for a stream with heat transfer by axial dispersion per the dispersion model :
Smoot and Babb (36) have solved the mass transfer analog of this equation for diffusion between parallel flowing streams. We are currently engaged in obtaining practical solutions which are applicable to predicting the M T D from this model and in checking this model with experimental observation. Blaisdell and Zahradnik (3) have conducted the only experimental investigation aimed at determining the axial temperature distribution in an agitated exchanger. They measured the temperature of water being heated in a Votator by steam condensing in helical passages in the jacket. Axial flow rate, steam temperature, and rotation speed were varied. The logarithm of the reduced temperature in the Votator was plotted us. distance along the exchanger. The resulting curves were not straight lines at low flow rates, indicating that some assumption in the L M T D derivation was violated. They concluded : “Further work is required in order to determine the importance of the change in slope of the semi-logarithmic heating curves observed a t low flows and moderate to high temperature rises. If the change in slope is caused by heat being conducted along the heat transfer surface and the mutator, or due to condensate buildup in the jacket, the effect will not be important in a well-drained commercial unit if the canonical temperature is large. . . Supplemental test data together with the literature suggest that a change in the mixing effectiveness was a probable cause” (of the nonlinearity of the semilogarithmic heating curves). They have noted that axial variations in jacket heat transfer coefficient and axial conduction in the exchanger walls can affect the temperature distribution in the exchanger. The data are not sufficiently accurate, however, to draw any quantitative conclusions concern48
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
ing the effect of axial dispersion on the temperature distribution in the Votator. Data must be taken under much more carefully controlled conditions and with much greater accuracy to develop quantitative methods for predicting the effect of axial dispersion on the MTD. At present we know that axial dispersion resulting from backmixing will certainly affect the M T D and that axial dispersion resulting from conduction in the liquid and in the agitator and vessel walls may be important; however, there are no predictive methods and there are probably not sufficient data to develop predictive methods. We do know certain qualitative facts which should prove helpful to designers and experimenters even if these facts only serve to make them aware of axial dispersion : (1) For constant heat transfer coefficients, the M T D is always intermediate between that for complete mixing and for no axial dispersion. (2) Axial dispersion in a stream will have no effect on the M T D if the stream is isothermal in the exchanger. Axial dispersion has the greatest effect on the M T D as the ratio of inlet AT to outlet A T departs increasingly from unity. (3) Axial dispersion as a result of backmixing has an increasing effect on M T D as ( a ) axial flow rate decreases, (6) rotation speed increases, and (c) equivalent diameter of the flow passage increases. (4) As a decreased axial flow rate will increase the effect of backmixing, thereby lowering the MTD, data reduced, using LMTD, often show a false dependence of the heat transfer coefficient on the axial flow rate. This point will be developed further in the review of experimental work on heat transfer.
HEATTRANSFER IN LIQUID-FULLSYSTEMS Early work by Huggins (79) demonstrated the effectiveness of scrapers for improving heat transfer. He compared the performance of a multiple paddle agitator with that of scrapers in batch cooling tests. For water the difference was negligible. However, for a number of considerably more viscous materials, the cooling time with scrapers was considerably less than for the paddle agitator (for a mixture of aluminum oleate and oil which was viscous and non-Newtonian, the transfer rate with scrapers was about five times that with paddles). The scrapers gave the greatest improvement for thick pasty materials. Laughlin (26) presented plant operating data for the use of scrapers in a batch operation where the processed solution changed from a thin liquid to a thick pasty liquid, and finally to a powder. Much research has been directed toward developing design methods for the Votator. Houlton (78) in 1940 was the first to conduct tests with a Votator. Hot water was cooled from about 175’ F. to about 100’ F. with 65’ F. cooling water. The water flow rates were about twice the maximum rates used by Blaisdell and Zahradnik (3); therefore, backmixing probably was negligible. Bolanowski and Lineberry ( 4 ) in 1952 conducted tests with the Votator on a number of food products. They
give a good discussion of the operation and uses of the Votator and give a n overall heat transfer coefficient for each food tested. From 1958 to 1962 Skelland and coworkers (33-35) conducted a lengthy investigation of heat transfer in the Votator with the aim of developing a general design correlation. The work culminated with the following correlation which includes the effects of rotation, axial flow, mutator diameter, number of blades, and fluid physical properties :
(3) For cooling viscous liquids, a = 0.014 and /3 = 0.96; for cooling thin mobile liquids, a = 0.039 and /3 = 0.70. No criterion is given for “viscous.” The functional dependence of the heat transfer coefficient on the parameters of the correlation is as follows:
I t is, immediately apparent that the correlation can a t most be applicable for the range of the experimental data upon which it is based because h should not approach zero as D,,(D - D,),and v approach zero. A more basic objection is that for viscous fluids the correlation predicts h 0: k0s04,which is most unlikely. The axial flow velocities were in the range of 0.25 to 1.5 ft./min. (1- to 4-min. residence time), whereas the rotation speed varied from 100 to 750 rev./min., giving a rotational R e about two orders of magnitude higher than Re’. It is likely that any potential effect of the axial flow on the heat transfer coefficient would be completely overshadowed by the effect of the rotation. We have already noted that axial dispersion (whether by molecular conduction or convection) can show u p as an effect of flow rate if the data are reduced using the LMTD. The axial velocities were about 25YG of those used by Blaisdell ( 3 ) , and the annulus width was 609;’, greater, but Re was much less (80 to 200 compared with 80,000 to 200,000). An R e of 80 to 200 is in the lower transition regime where turbulence would not be expected; however, this does not rule out the possibility of axial convective transport because Taylor vortices ( 7 , 13, 77) can occur before the flow becomes turbulent. The onset of Taylor vortices can be predicted for concentric cylinders; however, the blades on the inner cylinder here would be expected to alter the critical speed. Harriott (75) found for flow rates 9 to 23 times the minimum rates of Skelland and 2 to 5 times Skelland’s maximum flow rates, there was no dependence of the heat transfer coefficient on axial flow rate; he had to go about 15 times the maximum rates of Skelland (33, 35) before he found a n axial flow rate dependence. This indicates that the flow rate dependence reported by Skelland was not due to axial flow but to axial dispersion of heat.
The dependence of h on ko.O4 arises from the independence of h and p in the laminar regime. The exponent on R e was determined first and it was 1.O; therecore, for h to be independent of p, the exponent on Pr must be 1.O. Cross-plotting the data gave 1.04 as the exponent on Pr with the resulting incorrect dependence of h on k. Kool ( 2 4 , Harriott (75), and Latinen (25) have suggested that the penetration theory should be applicable to the prediction of heat transfer coefficients in the Votator. The penetration theory is based on the model of transient conduction into a semi-infinite solid; convection in the liquid is neglected except that the liquid is assumed to mix perfectly immediately after a scraper passes. I t is applied to scraped-surface heat transfer by assuming that, as a scraper passes, the surface is scraped clean and fluid comes back to the surface a t its bulk temperature. Heat transfer occurs only by conduction until the scraper passes again, repeating the process. The solution of Fourier’s equation averaged over the contact time, e, gives for the time average coefficient
Latinen (25) notes that for a two-bladed agitator, Equation 5 can be put in dimensionless form: Nu = 1.6 4 R e Pr Latinen (25) checked the penetration theory with the data of Houlton (78) and Skelland (33). The data of Houlton checked well, but those of Skelland did not. The data of Houlton were in the turbulent regime (Re > 10,000) and those of Skelland in the transition regime (10 < R e < 10,000). Latinen concluded that the heat transfer mechanism in the transition regime must be different from that implied by assumptions of the penetration theory. The effects of axial dispersion undoubtedly confounded the check of Skelland’s data. Harriott (15) took data on carrot puree and oil in a Votator and checked these data, along with Houlton’s (18) data, with the penetration theory. The penetration theory predicted coefficients up to 50YG too high for the puree and oil. Harriott (76) suggested that the penetration theory may have predicted high coefficients because the viscous fluids did not return to the heat transfer surface at their bulk temperature. The Votator blades do not wipe the wall clean of fluid. The hydrodynamic lift force keeps the blade off the wall resulting in a residual liquid film after the wiper passes. This residual film would be expected to affect heat transfer. Kool (24) and Jepson (20) have suggested that this film might be considered for predictive purposes as a solid layer on the exchanger wall. They both show graphically how a stagnant film of liquid would affect the heat transfer coefficient predicted by the penetration theory. Bell (2) noted that the stagnant film-penetration theory model could be approximately expressed in closed form as VOL. 5 9
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scraping. Then they have the following to say concerning the effect of clearance on the heat transfer rate: Apparently, at small clearances the liquid layer between the blade and wall is in strearnlinc flow and the blade serves to scrape the material up to the outer edge of the blade and mix it with the bulk of the fluid. Therefore, as the clearance increases the rate of heat transfcrs (
