Acknowledgment
The authors are grateful to the Computer Center, University of California, Berkeley, for the use of its facilities. literature Cited
Boone, W. J., DeVaney, W. E., Stroud, L., U. S. Bur. Mines, Rept. Invest. 6178 (1963). Buzyna, G., Macriss, R . A., Ellington, R. T., Chem. Eng. Progr. Symp. Ser. 59, No. 44, 101 (1963). Chueh, P. L., Prausnitz, J. M., A.1.Ch.E. J . 13, 1099 (1967a). Chueh, P. L., Prausnitz, J. M., Ind. Eng. Chem. Fundamentals 6, 492 (196713). Gunn, R. D., Chueh, P. L., Prausnitz, J. M., A.1.Ch.E. J . 12, 937 (1966). Heck, C. K., dissertation, University of Colorado, Boulder, 1968. Heck, C. K., Barrick, P. L., Aduan. Cryog. Eng. 12, 714 (1967). Heck, C. K., Hiza, M. J., A.Z.Ch.E. J . 13, 593 (1967). Herring, R . N., Barrick, P. L., “International Advances in Cryogenic Engineering,” K . D. Timerhaus, ed., p. 151, Plenum Press, New York, 1965. Knorn, M., Cryogenics 7, 177 (1967).
MacKendrick, R. F., Heck, C. K., Barrick, P. L., J . Chem. Erg. Data 13, 352 (1968). Mullins, J. C., Ziegler, W. T., “International Advances in Cryogenic Engineering,” K . D . Timmerhaus, ed., p. 171, Plenum Press, New York, 1965. Orentlicher, M., Prausnitz, J. M., Chem. Eng. Sci. 19, 775 (1964). Prausnitz, J. M., “Molecular Thermodynamics of FluidPhase Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1969. Prausnitz, J. M., Chueh, P. L., “Computer Calculations for High-pressure Vapor-Liquid Equilibria,” PrenticeHall, Englewood Cliffs, N. J., 1968. Rodewald, N. C., Davis, J. A., Kurata, F., A.1.Ch.E. J . 10, 937 (1964). Schindler, D. L., Swift, G. W., Kurata, F., Hydrocarbon Process Petrol. Refining 45, No. 11, 206 (1966). Streett, W. B., J . Chem. Erg. Data 13, 210 (1968). Streett, W. B., Sonntag, R. E., Van Wylen, G. J., J . Chem. Phys. 40, 1390 (1964). RECEIVED for review July 10, 1969 ACCEPTED December 3, 1969 Supported in part by the National Science Foundation and the donors of the Petroleum Research Fund.
THIN-FILM EVAPORATION ENHANCEMENT BY FINNED SURFACES DAVID
G .
THOMAS
AND
GALE
Y O U N G
Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830
Evaporative heat-transfer coefficients of thin films of water flowing down the inside of vertical tubes were markedly increased by clamping longitudinal rectangular fins to the tube surface. At a heat flux of lo4 Btu/hr.ft*, ten 0.013inch-widefins on the inside of a tube 4 2 l ) ~inches long and 0.44-inch i.d. gave a n evaporative coefficient of 8000 Btu/hr.ft’.’ F compared with 600 for the same tube without fins, a h / h , ratio of 13.3. The enhancement of the film evaporation coefficient decreases as the feed rate i s increased or decreased from the optimum, as the heat flux increases and as the number of fins decrease.
HEATTRANSFER to thin liquid films flowing on vertical surfaces is commonly encountered in shell and tube heat exchanges in which no appreciable vaporization takes place (McAdams et al., 1940; Sack, 1967), in long tube vertical evaporators (Sinek and Young, 1962), and in the annular flow region when flow boiling occurs in a tube (Tong, 1965a,b). Under these circumstances film heattransfer coefficients for the evaporating side are often in the range (Sinek and Young, 1962) 100 to 2000 Btu/ hr-ft2e0F,although values as large as 9000 B t u / h r . f t * . ” F have been reported (Norman and McIntyre, 1960) for short heated sections (3% inches) a t very low flow rates (r = 30 lb/hr.ft). The object of the present study was to determine whether high evaporative heat-transfer coefficients from thin
liquid films could be achieved using vertical longitudinally finned tubes in a manner analogous to the production of high film condensation coefficients by finned tubes observed in previous studies (Thomas, 1967, 1968). Surface Evaporation
Smooth Tubes. Vapor production from a thin film of liquid on a vertical heated surface may occur as bubble formation a t the liquid-solid interface or directly by evaporation from the liquid-vapor interface. Available observations (Collier and Hewitt, 1961; Isbin et al., 1961; Jakob, 1963) support the latter mechanism for modest temperature difference between the solid and vapor (AT < 16“F). If bubble formation makes negligible contribution to vapor evolution, the principal resistance to heat transfer Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970 317
will occur within the liquid film; several empirical and theoretical relations have been presented for heat transfer by conduction through this film of fluid. McAdams et al. (1940) showed that heat transfer to subcooled falling water films was a function of the water flow rate substantially independent of tube length, tube diameter, temperature difference, and inlet fluid temperature. The recommended correlation was
h = 120
r1
(1)
for values of film Reynolds number, from 400 to 5000 and maximum water temperatures of 200" F. Dukler (1960) numerically integrated the equations for the transfer of heat and momentum across the falling film. Results for film thickness and local and average heat-transfer coefficients were presented graphically. Elliot and Dukler (1965) recommended that the same results be used for both film condensation and surface evaporation. Chen (1963) correlated available data on flow boiling in tubes; for high flow rates he postulated that nucleate boiling was suppressed and vapor production occurred a t the vapor-liquid interface. For this condition Chen's correlation reduced to
hi
h = F (0.023) (N,,)P.' (Npr)0'4 -
De
where the Reynolds number was evaluated assuming the liquid filled the cross section of the tube. Values of the empirical correction factor, F , were presented graphically (Tong, 1965b). Wilke (1962) conducted an extensive study of heat transfer to falling liquid films and recommended correlations of the form N,, = A ( N R , ) a(Np,)0'344 (3) Values of the coefficient, A , and exponent, a, depended on the film Reynolds number-i.e., for (N,,)u < 400, A = 0.0614 and a = 8/15; for 400 < NRe < 800, A = 0.00112 and a = 6/5; and for N,, < 800, A = 0.0066 and a = 14/ 15. Wallgren (1967) integrated an empirical equation for the local heat-transfer coefficient in liquid film flow presented by Penman and Tait (1965) and obtained the following expression for the average heat-transfer coefficient:
Enhanced Evaporation from Special Surfaces. Among surface treatments which markedly enhance the evaporative film coefficient are the use of sharp parallel scratches on the surface (Bonilla et al., 1963), Teflon spots on the surface (Young and Hummel, 1964), and a tube with vertical grooves in the surface (fluted tube) (Carnavos, 1965). The mechanism responsible for the enhancement of the heat-transfer coefficient in the first two cases is believed to be an increase in the number of nucleate boiling sites. For fluted tubes, Carnavos (1965) postulated that, even though the feed on the evaporative side tends to be drained into the grooves by surface tension forces, boiling takes place both on the crest and in the grooves and that the crests are kept wet by the splash from the boiling action in the grooves. As yet, there is no theoretical model which predicts the amount of enhance31 8
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
ment of the rate of evaporation which may be expected from these treatments, and recourse must be made to experimental studies for all conditions of interest. Equipment and Procedure
Equipment used in previous studies (Thomas, 1967, 1968) of effect of finned tubes on film condensation coefficient of steam was modified to permit evaluation of the effect of fins on the evaporation heat-transfer coefficient of thin water films falling down the inside of vertical tubes. The principal modifications were the addition of a feed preheater to permit heating the feed to saturation temperature and the addition of a condenser to the discharge side of the tube to condense the evaporated water and permit measurement of the feed flow rate. The test section consisted of two vertical concentric tubes, a 1-inch-i.d. by 43-inch-long glass pipe with removable insulation to permit visual inspection of the mode of condensation on the outside of the tube, and a % inch-0.d. by 4-foot-long aluminum condenser-evaporator tube with a wall thickness of 0.028 inch. The over-all cooled length of the condenser-evaporator tube was 42% inches. The upper end of the condenser side of the tube was connected to a steam chest maintained at a pressure between 1 and 20 psig. Liquid flow rates were measured by collecting condensate from both evaporating and condensing sides. Because of the presence of fins on the thin liquid film evaporating side (and the attendant rivulet of water flowing down along the fins), the local heat-transfer coefficient would show wide variation both circumferentially and longitudinally. Thus accurate measurement of the evaporative-film coefficient would require a large number of surface temperature measurements. Since the object of this study was the evaluation of optimum feed flow rate and fin spacing, we chose to measure over-all heattransfer coefficients and evaluate film coefficients assuming additivity of resistances. The steam pressure was used to determine the steam temperature. Feed temperature was measured with an iron-constantan thermocouple inserted in a baffled mixing chamber a t the entrance of the test section. The feed stream was heated to within 3" C of saturation temperature before introduction to the inside of the evaporator tube. The over-all heat-transfer coefficient was measured as a function of feed flow rate for six different ranges of mean temperature difference, A T , between the condensing steam and evaporating feed. Values of A T varied from 1.7" to 21.7"F. I n all cases the evaporative film heattransfer coefficient was based on the area of the tube alone-Le., the area of the fin was not included in the tube area when calculating the heat flux. I n view of the poor thermal contact between the fins and the tube, this is believed to be a reasonable procedure. Finned Tubes. As in the previous studies (Thomas, 1967, 1968) of the effect of fins on the condensing side, conduction through the fins was minimized by manufacturing the stainless steel fins as a unit which could be inserted into the aluminum condenser-evaporator. Although the outside diameter of the fin assemblies was within 0.001 inch of the inside diameter of the tube, a water film of this thickness minimized heat conduction from the tube to the fin; this and the difference in conductivity between stainless steel and aluminum substantially eliminated "fin
effects” due to conduction and permitted evaluation of the effect of the fin on the liquid distribution and resultant thin-film evaporation coefficient. Finnei3 inserts were fabricated in two ways: from strips or srainiess steel 0.013 by )/s by 42 inches long which were welded to internal rings to maintain the strips perpendicular to the tube wall, and web fins in which the 0.013-inch-thick metal fins intersected a t the center of the t uihe. A photograph of typical fin assemblies of the two d esigns is shown in Figure 1. Once uniform water distribution t o each side of the fin was achieved, little difference could he detected between the performance of the tlvo fin-insert designs. Fabrication of the fins as remov able inserts was dictated by the nature of the optimi zation studies. The preferred method of manufacturi? would result in fins as an integral part of the surfaci3. Feec1 Distributors. Considerable care had to be exercised 2--~ LO allsure uniform feed distribution over the inside of the tube between the fins. Two distributors which gave adequate control were a hollow cone spray nozzle for the %-inch-high strip fins, and a plastic plug with a $48 inch-diameter hole centered over the intersection of the web fins. Although these devices apparently gave satisfactory feed distribution over a rather wide range of feed flow rates, thev are not necessarilv. oDtimum. and addi. tional studies are required before general design criteria can be specified. I)
.
.
15
I
r z
w
u $
“
5
a
w
z
I
I
02
01
!
(
!
( ( I (
J
05
2
Figure 2. Effect of feed rate on evaporative heat-transfer coefficient of web-fin tube Tube dimensions = 0.44-inch-i.d. 48 inches long
30
Y r
a
Experimental Results
3
9
The effect of feed rate on the evaporative side film heat-transfer coefficient of a tube with 10 web fins is illustrated in Figure 2. The film heat-transfer coefficient is given as a function of feed rate (in gallons per minnte per foot of tube perimeterj with the over-all temperature difference as a parameter. At the smallest over-all temperature difference the evaporative heat-transfer coefficient was rather insensitive to feed rate, although a broad maximum apparently occurred. As the temperature difference was increased, the evaporative heat-transfer coefficient showed more dependence on feed rate and the optimum shifted toward higher values of the feed fiow rate.
Figure I . Typical fin designs used for enhancement of falling film evaporative heat-transfer Coefficients A. B.
Web fins Normal fins
10
IL
w
-
z 0
05 w I 3
zca 0
02
3
xio3
5
loa
?
5
io5
HE4T FLUX IBtuJhr ft21
Figure 3. Optimum evaporative side flow rate as a function of heat flux for 0.44-inch-i.d. tubes containing six and ten fins Figure 3 shows the optimum value of the evaporative side flow rate a t the top of the tube as a function of heat flux for tubes containing 6 and 10 fins. It also shows the calculated value of the flow rate a t the bottom of the tube after accounting for the amount of feed evaporated as i t flowed down the tube. The optimum value of the fiow rate a t the bottom of the tube which contained 10 fins was substantially constant for the conditions covered in this study. This constant value is strikingly similar to the observation of Norman and McIntyre (1960). They reported that maximum evaporative film heat-transfer coefficients with short smooth tubes occurred a t flow rates just greater than those for dry spot formation and that this flow rate was the same a t the bottom of the tube for two different tube lengths-i.e., after correction for the amount of feed vaporized. The effect of heat flux on thin film evaporation in the presence and absence of fins a t the optimum feed flow rate is shown in Figure 4. Differently shaped points refer to check runs made on different days with different Ind. Eng. Chem. Process Der. Develop.,Vol.9, No. 2, 1970 319
finned tdbes. The evaporative film coefficient in the presence of fins is inversely proportional to the heat flux; the maximum value observed was 8000 Btu/ hr ft2- o F + 25% at a heat flux of l o 4 Btuihr-ft'. (These conditiont of low heat flux-low over-all AT resulted in maximum uncertainty in the measured values; all film coefficient measurements a t greater heat flux were substantially more precise. Estimated uncertainties in film coefficient are shown on Figure 4 as bars.) I n contrast with the finned surface results, the evaporative film coefficient on smooth tubes a t the optimum feed rate was directly proportional to the % power of the heat flux. The smooth tube data were in good agreement with the predictions of McAdams et al. (1940), Wilke (19621, Chen (1963), and Wallgren (1967), but were about a factor of 2 lower than predicted from Dukler's (1960) correlation. The effect of number of fins on the enhancement of the evaporative film heat-transfer coefficient is shown in Figure 5 . (Values of N w / * D of 0.028, 0.056, and 0.093 correspond to 3, 6, and 10 fins with a width of 0.013 inch on the inside of a 0.444-inch-i.d. tube.) Maximum enhancement was observed a t lowest heat flux with the largest number of fins, a result in qualitative agreement with our effect of fins on the film condensation coefficient (Thomas, 1968). The maximum value of the evaporative film heat-transfer coefficient with fins, 13 times the coefficient without fins, was observed a t a heat flux of lo4 Btu/ hr ft'. In the condensation studies the maximum film coefficient with fins was 9.2 times the coefficient without fins. Although the behavior of finned surfaces is qualitatively the same for evaporation and condensation, there are quane
134
I
I
U
W
8
8
0
002 FRACTION
004
006
008
010
012
OF SURFACE COVERED BY FINS (Nw/rrD)
Figure 5. Effect of rectangular fins on film evaporation coefficient Maximum values observed at optimum feed rate
titative differences in the effect of number of fins and heat flux. This may be illustrated by comparing an empirical expression for the effect of fins on the evaporative film heat-transfer coefficient:
h / h , = 1 + 1.25 x 10' ( N w / ~ D( Q ) ~/ A ) - '
(5)
with a similar expression for the condensing heat-transfer coefficient with fins (Thomas, 1968):
+
h,'h, = 1 1.40 x lo4 ( N w / r D ) ( Q / A ) - '* (6) Both Equations 5 and 6 apply for 0.02 < ( N w / x D ) < 0.1 and lo4 < Q / A < lo5 Btuihr-ft'. Comparison of Equations 5 and 6 shows that the evaporative film coefficients are more strongly dependent on both number of fins (or fin spacing) and heat flux than were the condensing coefficients of the previous study (Thomas, 1968).
5
c
2
LI -
Design Considerations
5
103
L
z 07
c e
PLAIN TUBE F NS O Y CONDEYSING S 3 E McADAMS, DREW AiiiD BAYS (19401 WlLKE (19621
w > -x-x
CHEN (19631 NALLGREN (19671
2
5
x
133
233
2
5
~ 0 4
HEAT F-UX ( B t u / h r 11'1
Figure 4. Effect of heat flux on evaporative heat-transfer coefficient of smooth a n d web-finned surfaces
320 Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
The principal factors affecting the magnitude of the increase in evaporative film heat-transfer coefficients are the fin spacing, heat flux, feed flow rate, and feed physical properties. Since feed physical properties were not varied over a wide range, the present results are directly applicable only to water or dilute aqueous solutions a t their atmospheric pressure boiling point. Based on present results, the spacing between fins should be '/s inch, independent of tube diameter, tube length, and heat flux. From Figure 3, the optimum flow rate of the bottom of the tube should be 0.5 gallon per minute per foot of tube perimeter independent of heat flux. The effect of heat flux on the amount of enhancement due to fins may be calculated from Equation 5; for fin spacing of Ye inch the value of N w l r D may be taken as 0.093. The principal uncertainty in applying the present results to feeds other than water arises from uncertainty on the role played by surface tension. Additional studies are required to resolve this question.
Discussion of Results
Fins loosely attached to the inside of a smooth vertical tube increased the evaporative side heat-transfer coefficient of an 0.44-inch-i.d. tube from 600 to 8000 Btu/ hr.ft2 at a heat flux of l o 4 Btu/hr.ft*. When the heat flux was increased to 5 x 10” Btu/hr.ft2, the fins increased the evaporative side coefficient from 900 to 2500 Btu/ hr.ft2.”F. For constant over-all temperature difference between the condensing and evaporating side there was an optimum feed rate to the evaporative side that gave maximum evaporative heat-transfer coefficient. This contrasts with the effect of fins on the condensing coefficient, where it was observed (Thomas, 1967, 1968) that decrease in flow rate of condensate always resulted in higher condensing film coefficient. The evaporative side film heat-transfer coefficient increased with decreased spacing between fins (for spacings from 0.46 to 0.12 inch) and decreased heat flux. Although qualitatively similar, the dependence on both spacing and heat flux was much more pronounced for evaporation than for condensation. Maintenance of a continuous liquid film is a necessary prerequisite for the production of large heat-transfer coefficients in thin film evaporation. The results of condensation studies (Thomas, 1967, 1968) with finned tubes indicate that surface tension forces tend to draw the liquid in thin films toward the junction of the fin and the surface, producing a rivulet which rapidly drains from the surface. I n thin film evaporation, as in condensation, the bulk of the fluid flows as a rivulet along the fins. A major difference is that for evaporation, liquid must be supplied from the rivulet to the thin film to balance vaporization, whereas in condensation the condensate is drawn from the thin film to the rivulet. We postulate the following mechanism for the effect of fins on the evaporative side thin film heat-transfer coefficient. The fluid must wet the metal tube surface. The bulk of the feed stream is channeled along the fins as a rivulet. The optimum fin spacing occurs when fins are sufficiently far apart to permit some amplification of random disturbances in the fluid film but not so far apart as to permit dry spot formation. The small amplification waves (“ropes” of fluid) sweep across the surface of the tube, acting as “windshield wipers” to provide a very thin film of fluid over the tube surface. “Roping” and other flow instabilities at very low flow rates have been observed by others in studies of thin film evaporation from vertical surfaces (Xorman and Binns, 1960; Norman and McIntyre, 1960; Unterberg and Edwards, 1965). Unterberg and Edwards (1965) studied thin film evaporation of 0 to 14% NaCl solution on the outer surface of a 4-inch-diameter by 16-inch-long tube. Continuity of the falling liquid film was observed to be dependent on the presence or absence of evaporation as well as of salt in the feed. When no heat was being transferred, introduction of cold water feed resulted in a falling film with a smooth, shiny surface. When heat was applied to the tube under these conditions, ripples were observed with the amplitude of the ripples proportional to the temperature difference. At the highest temperature difference studied, the falling film broke into irregular channels. [Apparently this behavior is similar to the “roping” described by Norman and McIntyre (1960).] When saline solution was substituted for pure water feed, the amplitude
of the ripples was markedly decreased and roping was not observed (Unterberg and Edwards, 1965). These results were attributed to the effect of local temperature and salinity gradients on surface tension gradients which either stabilized or destabilized the thin liquid film. Norman and Binns (1960) offered similar arguments to explain “dry patch” formation during thin film evaporation. Norman and McIntyre (1960) observed that evaporation coefficients of the order of 10,000 Btu/hr.ft’ could be obtained at feed rates of 30 to 40 lb/hr.ft on extremely short tubes (3Ya inches). These high heat-transfer coefficients were observed a t feed rates just greater than the minimum wetting rate which preceded dry spot formation. The most notable event which preceded dry spot formation was the occurrence of roping, in which the film became thicker in several vertical bands spaced around the periphery of the heated section. Reduction of the flow rate below the flow rate for which roping occurred resulted in dry patch formation between the bands of fluid, but the dry patches were quickly covered with thin film again as the ropes swept across them (Norman and McIntyre, 1960). The low flow rates a t which roping occurred ( r i p < 50) are well below the experimentally observed value for transition from laminar to turbulent flow (Fulford, 1964). However, theoretical studies by Benjamin (1957) predict that flow in isothermal thin films is always unstable but that rates of amplification of unstable waves become very small and this wavelength becomes very large when the Reynolds number is small. I n addition, surface tensiondriven instabilities are known to occur in thin films of fluid when there are concentration or temperature gradients in the direction of flow (Ludviksson and Lightfoot, 1968) or when there is a local temperature gradient due to a point source of heat (Mitchel and Quinn, 1968). The postulated mechanism for the effect of fins on enhancement of evaporative heat transfer requires that maximum evaporation occurs with thin films of marginal stability. Therefore, it would be of interest to calculate the distance over which amplification of waves is still small from Benjamin’s (1957) theory or the wavelength for neutral stability from Ludviksson and Lightfoot’s (1968) theory for surface tension driven flows. Benjamin’s (1957) analysis showed that the amplification factor is given approximately by c7/=
exp [ l a m (cl),/2a]
(7)
where CY’,
= 46 u: pI5gcls
(8)
(9) c L = CY NRe/ 10 Because of surface tension, flow in thin films on vertically finned surfaces is predominantly normal to the fins. I n the vicinity of the fins, a rivulet forms which flows rapidly along the fin. If the flow of the evaporating fluid normal to the fins is assumed to be influenced by the same factors governing the flow of condensate normal to the fins, then relations for film thickness, h, and mean velocity, u,, determined in previous studies of film condensation (Thomas, 1967, 1968) may be substituted into Equations 7 through 9 and the Reynolds number for large amplification may be estimated. In the model developed to describe the effects of fins on film condensation (Thomas, 1968), the film thickness was given by Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970 321
and the surface velocity normal to the fin by
with a1= empirical constant = 4 x lo-*. Combining Equations 7 through 11 gives the amplification factor for flow normal to a fin:
-
N
Acknowledgment
Stabilization of the flow will occur when the distance for wave amplification, 1, or the film Reynolds number is small. Fin spacing sets an upper limit on I ; consequently, the term l / ( r D / N ) will be assumed to be unity. The very high heat-transfer coefficients imply a thin water film over most of the heat-transfer surface; a coefficient of 5000 Btu/hr.ft2 requires a film 0.00066 inch thick. Assuming a value of the radius of curvature, R , of a “rope” on such a film of 0.002 inch and effective tube length between fins for the amplification of a random disturbance of L = 101, Equation 1 2 becomes
d = exp [13 Ni,]
titative differences, with the evaporative coefficient more strongly dependent on these two parameters than the condensing coefficient. It is postulated that, as with condensing studies using finned surfaces, the fins channeled the bulk of the liquid down the tube as a rivulet flowing along the junction of the fin and the tube; further that the optimum fin spacing occurs when fins are sufficiently far apart to permit some amplification of random disturbances but not so far apart as to permit dry spot formation. Calculations of the amplification factor for surface waves from Benjamin’s theory and of the wavelength and film thickness for surface tension-driven flows were in qualitative agreement with the postulated mechanism.
(13)
The amplification factor becomes very large for N,, > 0.6 but has relatively small values for NRe < 0.4. A Reynolds number of 0.5 corresponds to a laminar film with thickness giving a heat-transfer coefficient of 7000. When this coefficient is corrected for the portion of surface covered by fins and rivulet, assuming negligible heat transfer in these regions, the result is an average film coefficient of 5500, a value not inconsistent with the assumed value of 5000. Ludviksson and Lightfoot’s (1968) results of the stability analysis for surface tension-driven flows are presented graphically as wave number for neutral stability us. surface tension gradient for water films of various thicknesses. As an example of the expected wavelength of neutral stability for surface tension-driven flows, a film of water of thickness 2.4 x cm with a surface tension gradient 10 dynes per cm2 has a wavelength of 0.06 inch, the same order as the fin spacing used in this study. A film of the above thickness has an equivalent conductance of 5000 Btu/ hr ft2 F. Thicker films result in larger wavelength; this is the trend shown in Figure 2, where larger optimum flow rates for maximum heat-transfer coefficients were required for the smaller number of fins. Conclusions
Thin longitudinal fins spaced % inch apart on the inside of a vertical tube markedly enhanced the evaporative heat-transfer coefficient of thin water films. For a given heat flux, there was an optimum feed rate for maximum heat-transfer coefficient; it increased with heat flux and with a decrease in the number of fins. At the optimum feed rate, the relative increase in heat-transfer coefficient was directly proportional to the cube of a number of fins and inversely proportional to the heat flux. Although this behavior is qualitatively similar to the effect of fins on the condensing heat-transfer coefficient, there are quan322 Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
The authors thank W. A. Wilburn for his ingenuity and skill in fabricating the finned surface, P. H. Hayes for assistance in conducting the experiments, and K. A. Kraus for his support, suggestions, and criticism. Nomenclature
a = exponent, Equation 3 A = coefficient, Equation 3 d = amplification, dimensionless c, = specific heat, B t u / h r m eF 0 D = diameter, f t F = correction factor, Equation 2 gc = conversion factor, (lb,/lb,)ft/ hr2 h = film heat-transfer coefficient, Btu/hr.ft2.”F hf, = latent heat of vaporization, Btujlb k = thermal conductivity, Btu/hr.ft2 (” F / f t ) I = length for wave amplification, ft L = tube length, ft N = number of fins, dimensionless N P r = Prandtl number, c,p/ k , dimensionless NR, = Reynolds number, rip, dimensionless Q / A = heat flux, Btu/ hr ft2 R = radius of curvature of rivulet or “rope,” ft AT = temperature difference, F u = velocity, ft/hr width of fins, ft & I =
GREEKLETTERS ffCI ff1
= amplification factor, dimensionless = coefficient, Equation 10, 4 x l o - *
r = 6 = !J=
v = P = u =
flow rate, lb, / hr ft film thickness, ft viscosity, lb, / hr .ft kinematic viscosity, ft2/hr density, lb,/ft3 surface tension, lb,/ft
SUBSCRIPTS
e 1 m o
= equivalent = liquid = maximum
= no fins
u = vapor Literature Cited
Benjamin, T. B., J . Fluid Mech. 2, 554 (1957). Bonilla, C. F., Grady, J. S., Avery, G. W., “Pool Boiling Heat Transfer from Scored Surfaces,” 6th National Heat Transfer Conference AIChE-ASME, August 1963. Carnavos, T. C., “Thin Film Distillation,” First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965.
Chen, J. C., ASME Paper 63-HT-34, 1963. Collier, J. G., Hewitt, G. F., Trans. Inst. Chem. Eng. 39, 127 (1961). Dukler, A. E., C h m . Eng. Progr. Symp. Ser. 56 (30), 1-10 (1960). Elliot, L. C., Dukler, A. E., paper SWDj21, First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965. Fulford, G. D., Aduan. Chem. Eng. 5 , 151-236 (1964). Isbin, H. S., et al., Trans. A S M E J . Heat Transfer 83, 149 (1961). Jakob, Max, “Heat Transfer,” Vol. 1, pp. 618-20, Wiley, New York, 1953. Ludviksson, V., Lightfoot, E . N., A.I.Ch.E. J . 14, 620 (1968). McAdams, W. H., Drew, T. B., Bays, G. S., Jr., Trans. A S M E 62, 627 (1940). Mitchel, W. T., Quinn, J. A., Chem. Eng. Sci. 23, 503 (1968). Norman, W. S., Binns, D. T., Trans. Inst. Chem. Engrs. 38, 294 (1960). Norman, W. S., McIntyre, V., Trans. Inst. Chem. Engrs. 38, 301 (1960). Penman, T. O., Tait, R. W. F., Ind. Eng. Chem. Fundamentals 4, 407 (1965).
Sack, Melvin, Chem. Eng. Progr. 63 (7), 55 (1967). Sinek, J. R., Young, E . H., Chem. Eng. Progr. 58 (In), 74 (1962). Thomas, D. G., A.Z.Ch.E. J . 14, 644 (1968). Thomas, D. G., Ind. Eng. Chem. Fundumentals 6, 97 (1967). Tong, L. S.,“Boiling Heat Transfer and Two-Phase Flow,” p. 48, Wiley, Kew York, 1965a. Tong, L. W., “Boiling Heat Transfer and Two-Phase Flow,” p. 123, Wiley, New York, 1965b. Unterberg, Walter, Edwards, D. K., A I.Ch.E. J . 11, 1073 (1965). Wilke, W., V D I Forsch. No. 490 (1962). Wallgren. B. O., Ind. Eng. Chem. Fundamentals 6, 479 (1967). Young, R. K., Hummel, R . L., “Higher Coefficients for Heat Transfer with Nucleate Boiling,” 7th National Heat Transfer Conference AIChE-ASME, August 1964. RECEIVED for review July 23, 1969 ACCEPTED January 19, 1970 Research jointly sponsored by the Office of Saline Water, U. S. Department of the Interior, and U. S. Atomic Energy Commission under contract with the Union Carbide Corp.
DYNAMIC RESPONSE TO CONCENTRATION CHANGES OF A PARTLY FILLED HORIZONTAL TANK R O B E R T
M . H U B B A R D ,
G E R A R D
C .
L A H N ’ ,
J O S E P H
A N D
1.
W I L L I A M
DALFERES W .
I l l ’ ,
Z A L E W S K 1 3
Department o f Chemical Engineering, University of Virginia, Charlottesuille, Vu. 22901
Natural mixing in a partly filled horizontal tank due to flow rate and relative position of inlet and outlet w a s studied on three tanks from 5.7 to 36 inches in diameter. Length-diameter ratio w a s 3 to 1. Three liquid levels were maintained and flow rate was varied. Concentration transfer functions usually included a small dead time or delay before any change in output occurred. Dead time increased as tank diameter increased. Bypass or short-circuiting from inlet to outlet without mixing occurred when outlet w a s directly under inlet. Time constants are related to mean residence time. Relations are recommended for concentration transfer functions for many arrangements of inlet and outlet.
THEchemical plant often includes a surge tank between major process units. Such tanks are useful in smoothing out concentration and level changes, preventing excessive deviations in feed to subsequent process equipment. One object of this study was to determine concentration dynamics of partly filled horizontal tanks in which mixing occurred only as the result of inflow rate and the relative Present address, Union Carbide Chemicals Co., Taft, La. Present address, Esse Research and Engineering co., Flor$am Park, N. J. Present address, General Electric Co., Schenectady, N. Y. 1
2
positions of inlet and outlet nozzles. The principal object was to do this on several tanks of progressively larger diameters to permit extrapolation of results to surge tanks of commercial size. The principles of dynamic response and pioneering work in this field have been described in the important monograph by Hougen (1964). Considerable work has been done, particularly by Danckwerts (1953, 1954), on flow systems in which mixing was described by the distribution of residence times. Levenspiel (1962) and Levenspiel and Bischoff (1963) proposed several models to represent flow Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
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