Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
and reduces the overall reaction rate. The results were obtained using two different models of the pore structure, diffusivity profiles and three different activity profiles under optimal and nlonoptimal conditions; therefore, they are applicable to a variety of reactions and solids. These results are relevant to some new advanced catalyst coal gasification processes under development. Literature Cited Bischoff, K., Chem. Eng. Sci., 18. 711 (1963). Calvelo, A,, Cunningham, R. E., J . Catal., 17, 1 (1970). Gardner, N., Samuels, E., Wilks, K., Am. Chem. SOC.,Div. FuelChem., Prepr., 18, 217 (1973). Guzman, G., Wolf, E. E., Chem. Eng. Sci., submitted for publication, 1978.
11
Johnson, J. L., Catal. Rev. Sci. Eng., 14 (l),131 (1976). Lee, E. S.,"Quasilinearization and Invariant Imbedding," Academic Press, New York, N.Y., 1968. Luss, D., Corbett, E. W., Chem. Eng. Sci., 29, 1473 (1974). Otto, K., Shelef, M., 6th International Congress of Catalysis, Prepr. €347, London,
1976. Petersen, E., AIChE J . , 3, 443 (1957). Shadrnan-Yazdi. F., Petersen, E. E., Chem. Eng. Sci., 27, 227 (1972). Szekely, J., Evans, J. W., Sohn, H., Gas-Solid Reactions," Academic Press, New York, N.Y., 1976. Szekeiy, J., Evans, J. W., Chem. Eng. Sci., 25, 1091-1107 (1970). Walker, P. L., Rusinko, F., Austin, L. G., Adv. Caral.. 11, 178 (1959). Wolf, E. E., J . Catal., 47, 85 (1977).
Received for review November 29, 1977 Accepted September 6, 1978
The Effect of Pulsations on Heat Transfer 0. Erdal Karamercan and John
L. Gainer*
Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 2290 1
The effect of pulsating the water stream in a double-pipe heat exchanger, with steam on the shell side, was investigated. Pulsation frequencies ranged up to 300 cycles per minute, and five different displacement amplitudes were used at each flow rate investigated. The heat transfer coefficient was found to increase with pulsations, with the highest enhancement observed in the transition flow regime.
Background The effect of pulsating one of the flows in a heat exchanger has been studied previously, with conflicting conclusions as to the effects seen. This probably indicates a lack of proper understanding of the pertinent variables involved. There is evidence in the literature (Baird et al., 1966; Darling, 1959; Havemann and Rao, 1954; Keil and Baird, 1971; Lemlich, 1961; Lemlich and Hwu, 1961; Ludlow, 1975; Martinelli et al., 1943; West and Taylor, 1952) that pulsating the flow in a heat exchanger enhances the heat transfer, although there is disagreement on this point. Furthermore, most of the previous investigators considered only a small number of operating variables in their studies and usually confined their studies to relatively narrow ranges of these variables. The aim of this study, therefore, was to investigate the effects of a combination of independent variables on the performance of pulsed-flow heat exchangers. These variables included thle frequency and the amplitude of the pulsations, the Reynolds number, the location of the pulsator with respect to the heat exchanger, and the length of the heat exchanger area. This study also investigated a broader range of these operating variables than covered in most previous works. The effect of pulsed flow on heat and mass transfer has been investigated by various researchers. In the area of heat transfer, theoretical and experimental studies of Mueller (1957) showed that, for pulsating flows which encompassed a frequency range of 2.3 to 14.9 cycles/min and a Reynolds number range of 53000 to 76000, the average Nusselt number was found to be less than the corresponding steady flow Nusselt number. McMichael and Hellums (1975) have presented a theoretical development which concludes that for laminar flow, pulsations cause a decrease in the rate of heat transfer at flow am0019-7874/79/1018-0011$01 .OO/O
plitudes which do not allow flow reversal to take place. Experimental studies under laminar flow conditions were conducted by Martinelli et al. (1943), using semisinusoidal velocity disturbances on the tube side fluid in a concentric tube heat exchanger. They found that the overall heat transfer coefficient was increased over the steady flow coefficient by 10% at the most. Lemlich and Hwu (1961) observed increases of up to 51% in the Nusselt number for air flowing at Reynolds numbers of 560 to 5900 in a horizontal, double-pipe, stream-to-air heat exchanger when acoustic vibrations were imposed on the air. Experiments performed by Lemlich (1961) and later by Lemlich and Armour (1965) on a double-pipe, steam-towater heat exchanger showed increases up to 80% in the overall heat transfer coefficient when the pulsator was installed upstream. However, when installed downstream, a decrease in the overall heat transfer coefficient was observed. The Reynolds numbers investigated were between 500 and 5000 and the pulsation frequencies ranged from 30 to 200 cycles/min. Darling (1959) reported similar observations. He obtained a 90% increase in the heat transfer coefficient, at a Reynolds number of 6000 and a pulsation rate of 160 cycles/min, when pulses were introduced upstream of the heaters. No improvement in the heat transfer coefficient was observed with the interrupter value downstream of the heater. In the turbulent flow regime, Baird et al. (1966) concluded that pulsations will improve heat transfer, particularly if the flow can be made to reverse in direction for part of the cycle. The maximum improvement they found was 225% for the water-side heat transfer coefficient in a double pipe, steam-to-water vertical heat exchanger. Keil and Baird (1971) observed as much as a 100% increase in the overall heat transfer coefficient using pulsating frequencies of 24 to 66 cycles/min in a commercial shell0 1979 American Chemical Society
12
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
uater
stet
in
1
Condenldte
OYt
Figure 1. Simplified schematic diagram of the apparatus.
and-tube heat exchanger with steam in the shell. Ludlow (1975), using a double pipe heat exchanger with hot water in the annulus, obtained a 500% increase in the tube-side heat transfer coefficient in the transition flow regime. His pulsation frequencies were between 10 and 170 cyclesfmin. Experiments conducted by West and Taylor (1952) on pulsating turbulent flow showed as much as a 70% increase in water film heat transfer coefficients inside tubes at Reynolds numbers of 30000 to 55000. The pulsation frequency was 100 cycles/ min. Havemann and Narayan Rao (1954), using pulsations at 5-33 cyclesfs, obtained an increase of 42% in the heat flux over the steady flow value under turbulent flow conditions. As is seen, there are considerable variations in the quantitative results of pulsatile flow experiments. However, this is not surprising in view of the many possible combinations of pulsation variables, pulse generation mechanisms, flow regimes, etc. As a result, some investigators report increases in heat transfer from pulsatile flow, and others report little increase, no increase, and even decreases in heat transfer.
Experimental Section Principally, the equipment consisted of the flow control valves and the rotameters, a pulsator mechanism, and a horizontal, steam-to-water heat exchanger. A simplified flow sheet for the apparatus appears as Figure 1. The water entering the system was passed through flow control valves placed upstream from the three rotameters, which measured the flow rate of the water. Besides controlling the flow, these valves also served as a major pressure drop in the system. The temperature of the entering water varied from 10 to 12 "C. The pulsator mechanism was constructed of three main parts: an electric motor, a variable speed transmission, and a reciprocating pump. The output of the 3/4 hp motor was used as the input to the variable speed transmission. The output of the transmission could be adjusted manually to any value within the range of 0-300 rpm. Pulsations in the water stream were generated by a reciprocating pump, which consisted of a brass cylinder in. in diameter and 5 in. in length containing an aluminum piston inside. In order to convert the cyclic output of the transmission into a back-and-forth movement for the piston, a coupling mechanism was devised. Its mode of operation was based upon the Scotch yoke principle, and it dispensed with the need for precision ground slides and intricate machining. Two heat exchangers, which only differed in length (6 and 3 ft, respectively), were used in the experiments. They both consisted basically of a copper tube, having a nominal inside diameter of 3 / 4 in., mounted concentrically within an outer shell of 21f4 in. i.d. stainless steel. The core
carried water heated by the steam flowing countercurrently in the annulus. Inlet and outlet temperatures of the water were measured thermometrically. Pressure and temperature measurements on the steam side were provided within the end caps of the heat exchangers which also allowed for the steam to flow into the annulus as well as for the condensate to drain out. Experimental Variables As stated previously, it was intended that this investigation cover a number of independent variables which might affect the heat transfer in pulsed flow. The ranges of the variables such as the frequency of the pulsations, the amplitude of the displacement, and the Reynolds number were also to be kept as broad as possible within the limitations of the apparatus. In view of these facts, the operating variables and their respective ranges covered in this study were as indicated below. The frequency of the pulsations could be varied up to 300 rpm. Five different displacement amplitudes were used at each flow rate investigated, and the frequency and the amplitude of the pulsations could be made large enough to cause periodic reversal of the flow. The flow rates studied ranged from 0.1 to 12 gpm. The corresponding steady flow Reynolds numbers were in the range of 1000 to 50000. After making measurements with the pulsator located upstream of the 6 f t long heat exchanger, another set of data was collected with the pulsator located downstream. The experiments were then repeated using a 3 f t long heat exchanger to determine the effect of changing the length of the heat transfer area. The enhancement in heat transfer obtained under pulsed flow conditions was calculated from a knowledge of the inlet and the outlet temperatures of the water, the temperature of the steam, and the flow rate of the condensate. The expression of the overall heat transfer coefficient can be written in the form 1 Vi = 1 Dixw Di hi D~kw Doh, where Vi is the overall heat transfer coefficient based on the inside surface area, Ai, of the tube, hi is the water-side heat transfer coefficient, kwand xware the thermal conductivity and the thickness of the tube wall, h, is the steam-side heat transfer coefficient, and Di, Do, and DL are the inside, the outside, and the logarithmic mean diameters of the tube, respectively. The overall heat transfer coefficient, Vi, was obtained from a heat balance over the heat exchanger as q = UiAiATL= micPiATi (2)
-+-+-
where q is the rate of heat transfer in Btu/h, mi and cpi are the mass flow rate and the heat capacity of the water, respectively, AT, is the logarithmic mean temperature difference over the heat exchanger, and ATi is the change in water temperature between the inlet and the outlet. The steam-side heat transfer coefficient, h,, was calculated using the Nusselt equation (Sieder and Tate, 1936) for film condensation on a horizontal tube of length L (3) in which W / L is the mass rate of condensation per unit length of tube and the subscript f refers to values calculated a t the film temperature. Substitution of values obtained from eq 2 into eq 1 yields the value of the
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
2
w,
I
13
Flow Revem11 Frequency
0
1 .25"
1.1
0
1 .OO"
1.4
0
0.75"
1.9
h
0.50'
2.8
0.25"
5.5
v I I I I I I I I 0
I I I I
100
50
f.
77 6
0 2s
~
0
~
150
I
~
I
I
(
250
200
rm
Figure 5. Frequency dependence of enhancement for Re = 11500; 3-ft exchanger. tpm
f.
Figure 2. Frequency dependence of enhancement for Re = 1500; 3-ft exchanger.
5
4
$
3
0 0.75" (L 0.50"
38.1
58 0
0.25''
I 0 (L
0 50
14 8
p
0 25
29 6
A, 0 1 S
0
50
I
I
I
100
B
I
I
0
1
150 f.
50
I
I
I
I
I
I
3
8
,
I
t
250
I
I
I
I
I
I
150 f,
200
I
100
115.0
'
I
8
#
,
I
200
cpm
Figure 6. Frequency dependence of enhancement for Re = 17 000; 3-ft exchanger.
CPrn
Figure 3. Frequency dependence of enhancement for Re = 5600; 3-ft exchanger.
'r
I
152
0
I .OC'
49 1
0
0.15'
65.7
A
0.50'
98 i
'r
ilegYLnLy
50
0
150
100
t.
v I I I I I I I I I I I
8
0
1
1
I * 150
1W f.
~~~
0 25
1
zao
a
1
1
250
Figure 7. Frequency dependence of enhancement for Re = 27 200; 3-ft exchanger.
46 6
'
200
iyn
I 250
im
Figure 4. Frequency dependence of enhancement for Re = 7800; 3-ft exchanger.
water-side heat transfer coefficient, hi. The enhancement was then calculated as the ratio of the film heat transfer coefficient obtained in pulsed flow, hip, to that obtained in steady flow, his,namely
(4) 0
Dohop where the subscripts s and p refer to values calculated from steady flow and pulsatile flow temperature readings, respectively. Results Figures 2 through i' show the calculated enhancement values, obtained with the 3 ft long heat exchanger, as functions of the pulsation amplitudes and frequencies. Six different average fla'ws were investigated with corresponding Reynolds numbers of 1500, 5600, 7800, 11500, uip
10,000
20,000
30,000
Re
Figure 8. Maximum enhancement values for 3-ft exchanger.
17 000, and 27 000. In each figure, the horizontal dotted line represents the maximum enhancement obtained a t that average flow. A plot of the maximum enhancement values obtained vs. the corresponding Reynolds numbers is given in Figure 8. Similar graphs which represent the calculated enhancements as functions of pulsation amplitudes and frequencies were developed for the pulsator located up-
I
14
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 8
6
3
2
C
10,000
ZC,O?L
40,000
30.000 le
Figure 9. Maximum enhancement values: 0 , 6 ft long heat exchanger; A, 6 ft long heat exchanger (pulsator located downstream); 0 , 3 f t long heat exchanger.
1
1
1
1
1
1
10,000
1
1
1
1
1
1
20,000
1
1
1
1
30.000
1
1
1
1
1
1
1
1
,
40,000
Re
Figure 10. Comparison of maximum enhancements in heat transfer with previous work. Shaded areas are results of current investigations with top l i e for the 6-ft exchanger and lower one for the 3-ft exchanger. Other results are from: A, Baird et al. (1966); B, Keil and Baird (1971); C, Darling (1959); D, Lemlich (1961); E, Lemlich and Armour (1965); F, Ludlow (1975); G, West and Taylor, (1952).
stream, and also downstream, from the 6 f t long heat exchanger. The seven different average flows were investigated with corresponding steady-flow Reynolds numbers of 2900, 6200,9100,10 700,24 000, 37 800, and 47400. A comparison of the maximum improvements obtained with the different heat exchanger and pulsator configurations is given in Figure 9. A comparison of the results reported by various investigators working with systems similar to the ones used in this study is presented in Figure 10. The shaded area shows the enhancement values obtained in this work, and the solid lines represent the maximum enhancements reported in the literature within the corresponding ranges of Reynolds numbers. From this figure, it can be seen that the results obtained by other researchers fall into the shaded regions that represent the findings of this investigation. It would appear that, although the previously reported data have appeared to be inconsistent, the smaller range of operating variables used by these investigators may have been responsible for the differences seen. It should also be noted that in some previous investigations in which no heat transfer enhancement was seen with pulsations, the studies were done over a range of variables in which no flow reversal occurred (McMichael and Hellums, 1975;Mueller, 1957). Their findings are consistent with our results as well as those of Ludlow (1975), who showed that no increase in heat transfer should occur before the onset of flow reversal. His data were obtained, however, by means of a reciprocating pump which caused the fluid to reverse its direction by pulling on the fluid for
a portion of the pulse cycle. Thus, cavitation as well as flow reversal probably occurred in those experiments. Although the experimental studies conducted by Lemlich (19611,Darling (19591,Baird et al. (1966),Keil and Baird (19711,and Ludlow (1975)for the most part covered different values of the variables involved, they all showed that the greatest improvements in heat transfer due to flow pulsations were obtained in the transition flow regime. This, also, is in agreement with the experimental results obtained in the present study which showed the highest enhancements in the heat transfer coefficient obtained within a Reynolds number range of 7500 to 9500. Possible Causes on Enhancement, The improvement in the heat transferred to a flowing fluid by pulsing the flow has been variously ascribed to cavitation (Darling, 1954;Lemlich, 1961;Lemlich and Armour, 1965;Ludlow, 1975),to the increased level of turbulence (Lemlich, 1961; Villarroel et al., 1971),and to the introduction of forced convection in the boundary layer (Marchant, 1943). The influence of each of these factors on the enhancement in the heat transfer coefficient will be discussed below. (i) Cavitation. Bubbles may be formed in the fluid due to cavitation. These bubbles are produced during the low-pressure portion of the pulse cycle when the flow is suddenly interrupted or forced to change direction and the pressure in the liquid drops below the local vapor pressure. Since the static pressure in the tube is the same across the whole diameter, the bubbles must originate where the vapor pressure is reached first as the pressure drops. This occurs at the hottest point, namely in the boundary layer next to the tube wall. The periodic formation and collapse of these bubbles agitates the boundary layer and thus increases the rate of heat transfer. According to Lemlich and Armour (1965),the bubbles also act as carriers of latent heat by moving toward the bulk and giving up their latent heat as sensible heat upon collapsing. They observed a periodic growth and collapse of new bubbles with pulsation, growing from “virtually nothing” to perhaps 0.2 in. in diameter before collapsing. Although they did not degas their water stream before passing it through the exchanger, they noted that the relative growth and collapse of the bubbles far exceeded the comparatively small relative expansion and contraction of stray air bubbles. They felt that this indicated that the bubbles were formed by cavitation rather than merely the expansion of a gas. Like Lemlich and Armour (1965),we did not degas our entering water, and we also observed numerous bubbles leaving through the plastic tube fitted to the down-stream end of the heat exchanger. From Figure 9,it is seen that the greatest enhancement obtained with the pulsator at the downstream location is 8070 lower than the upstream location. This same phenomenon has been observed by other investigators (Darling, 1959;Lemlich, 1961;Lemlich and Armour, 1965) as well, and the improvement in the enhancement values observed when the pulsator is located upstream has been attributed to cavitation. To determine more about the effect of cavitation, a test was conducted on the 3 ft long heat exchanger a t the flow rate where the highest enhancement (5.8-fold) had been obtained. The static pressure in the heat exchanger was increased by discharing the water 20 f t above the level of the heat exchanger. With all other conditions the same, the highest enhancement obtained was 3070 lower than when the exit was level with the exchanger. This would appear to be another proof of the obvious effect of cavitation on the enhancement of heat transfer by flow pulsations.
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
(ii) Increase in the Level of Turbulence. In convective heat transfer, (axialflow through the tube is a very important parameter. Pulsations of sufficient frequency and amplitude can improve heat transfer in such a way as to increase the longitudinal flow for part of the cycle which, in turn, decreases the film thickness in the tube. The subsequent reversed axial flow then results in radial diversion of much of the kinetic energy. Hence, pulsatile flow is associated with a periodic pressure gradient reversal which causes an increase in radial and longitudinal mixing. Furthermore, each cycle of motion can be viewed as acting as a disturbance which increases the level of turbulence in the tube. Higher amplitudes and higher frequencies mean correspondingly larger or more frequent disturbances, hence improved turbulence, and a higher coefficient for heat transfer. Subsequently, the effect of pulsations on the enhancement in the heat transfer coefficient becomes less a t higher flow rates because it must compete with a higher level of turbulence already present in the fluid. The decrease in the maximum enhancement values toward higher Reynolds numbers, as observed in Figures 8 and 9, supports this view. (iii) Introduction of Forced Convection in the Boundary Layer. Another effect which may well be related to the ones already discussed is the introduction of forced convection in the boundary layer when pulsing the flow. It is generally conceded that the heat transfer through the laminar film of the water side of the tube occurs by means of conduction. In steady flow, convection plays no part in the transfer of heat through this inner boundary layer which is effectively at rest with respect to the tube wall. However, when pulsations are introduced into the flow, the periodic pressure variations produce forced circulations in the fluid and increase the effective heat transfer by promoting the formation of eddies, thus introducing convection in the boundary layer. Among all the effects discussed above, cavitation seems to be the most dominant factor in the enhancement of heat transfer by flow pulsations, as suggested by two different tests conducted in this study. First, changing the location of the pulsator from upstream to downstream of the heat exchanger, thereby decreasing the effect of cavitation,
15
caused a decrease of about 80% in the maximum enhancement. Secondly, increasing the pressure in the tube by elevating the discharge above the heat exchanger, so as to make it difficult for the cavitation bubbles to form, dropped the maximum enhancement by 30%. Note that, due to the nature of the pulsator used in the present study, it was not possible to reach high frequencies without causing flow reversals. Since periodic reversals in the direction of flow improve cavitation and also promote a higher level of turbulence in the fluid, flow reversal seemed to be the important parameter in the enhancement of heat transfer when pulsations were introduced by means of a reciprocating pump. Current studies are underway in our laboratory aimed a t investigating the effects of pulsations on the improvement in heat transfer, without introducing reversals in the direction of the fluid flow. In this respect, pulsations in the stream are produced by means of an interrupter valve, which shuts the flow off for a portion of the pulse cycle but does not cause the flow to reverse its direction. With this approach, it is intended to upcouple the effect of flow reversal and the other factors to which the enhancement in heat transfer under pulsatile flow conditions may be attributed. Literature Cited Baird, M. H. I., Duncan, G. J., Smith, J. I., Taylor, J., Chem. Eng. Sci., 21, 197 (1966). Darling, G. B., Petroleum, 22, 1977 (1959). Havemann, H. A,, Marayan Rao, N. N., Nature (London), 174, 41 (1954). Keii, R. H., Baird, M. H. I., Ind. Eng. Chem. Process Des. Dev., 10, 473 (1971). Lemlich, R., Chem. Eng., 66, 1971 (1961). Lemlich, R., Armour, J. C., Chem. Eng. Prog. Symp. Ser., 61, 83 (1965). Lemlich, R., Hwu, C. K., AIChE J . , 7, 192 (1961). Ludlow, J. C., M.S. Thesis, University of Virginia, Charlottesville, Va., 1975. Marchant, J. H., Trans. ASME, 65, 786 (1943). Martinelli, R. C., Boelter, L. M., Weinberg, E. B., Yakal, S., Trans. ASME, 65, 789 (1943). McMichael, W. J., Hellums, J. D., AIChE J . , 21, 743 (1975). Mueller, W. K., "Proceedings Fifth Midwestern Conference on Fluid Mechanics", p 146, Ann Arbor, Mich., 1957. Sieder, E. N., Tate, G. E., Ind. Eng. Chem., 26, 1429 (1936). Viilarroei, F., Lanharn, C. E., Bishoff, K. B., Regan, T. M., Calkins, J. M., Chem. Eng. Prog. Symp. Ser.. 67, 96 (1971). West, R. B., Taylor, A. T., Chem. Eng. Prog., 48, 39 (1952).
Received for reuieu! January 23, 1978 Accepted September 6, 1978
Liquid-Level Control in Single Tanks and Cascades of Tanks with Proportional-Only and Proportional-Integral Feedback Controllers Tak-F,ai Cheung and William L. Luyben' Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania
180 15
Desigri charts are given which permit the control engineer to select reasonable controller settings for liquid-level control loops by examining quantitatively the effects of these settings on damping coefficient, peak height, and rate of change of flow. The decrease in damping coefficient for a sequence of level tanks in series is quantified in an easy to use design chart.
Introduction Control of liquid level in process vessels often involves two important and conflicting considerations. First, the changes in flow rates out of the vessel should be as smooth 0019-7874/79/1018-0015$01.00/0
as possible so as not to upset the downstream process. Second, level should not be permitted to deviate too far from the desired normal operating level. These two objectives place conflicting demands on the control system. C
1979 American Chemical Society
