Heat Transfer Coefficients of Pseudo-Plastic Fluids

INDUSTRIAL AND ENGINEERING CHEMISTRY

1686 Nomenclature

Y

Vol. 45, No. 8

Literature Cited

= concentration in light solvent

(1) Compere, E. L., a n d Ryland, A., IXD. ENG.CHEM.,43, 239 (1951). (2) Craig, L. C., Anal. Chem., 23, 41 (1951). (3) Craig, L. C., and Craig, D., in Weissberger, A., “Technique of

X = concentration in heavv solvent D = distribution ratio, Y / x EL = extraction factor on light solvent input (heavy product) side of feed. L,D/(H, 4-H i ) E H = extraction factor on’heavy sol&t input (light product) side of feed, H,/(L, Lj)D Lo, Ho = input volumes of pure light and heavy solvent, respectivelv Lf,Hf = volumb of feed solution in light and heavy solvent, respectively T = number of stages to feed stage from light solvent input, including feed stage S = number of stages t o feed stage from heavy solvent input, including feed stage 1 r S s = total number of stages

(7) Rometsoh, R., H e h . Chim. A d a , 33, 184 (1950). (8) Scheibel, E. C., Chem. Eng. Progr., 44, 681 (1948). , 242 (1951). (9) Scheibel, E. C., IND.ENG.C H E M 43, ( I O ) Treybal, R. E., Ibid., 43, 79 (1961). (11) TT‘alker, C. A., Ihid., 42, 1226 (1950).

Subscripts n indicates any stage on heavy solvent input side of feed m indicates an stage on light solvent input side of feed p indicates va?Le in product

RECEIVED for review October 22, 1051. ACCEPTED February 27, 1963. Taken from the dissertation of Ada L. Ryland, submitted in partial fuifillment of the requirements f o r the degree of doctor of philosophy, Louisiana State University, Baton Rouge, La., August 1961.

.+

+

Organic Chemistry,” Vol. 111, Chap. IV, New York, Intelscience Publishers, 1950. (4) Craig, L. C., and Post, O., Anal. Chem., 21, 500 (1949). ( 5 ) Golumbic, C., J . Am. Chem. Soc., 71, 2627 (1949). (6) Golumbic, C., Orchin, M., and Weller, S., Ibid., 71, 2624 (1949).

0

e 0

e

Heat Transfer Coefficients of Pseudo-Plastic Fluids JU CHIN CHU, FRANK BROWN’, AND K. G. BURRIDGE’ Polyfechnic lnrtifofe o f Brooklyn, Brooklyn 2, N. Y.

A

COSSTANTLY increasing number of industrially iinpoi taiit liquids show non-Newtonian flow behavior ( 1 , 9,8, 19-21, 24-27). The viscoaity coefficients of these liquids are dependent on shear conditions. The anonlalous behavior may take vmious forms, but the most significant kind to date from the industIial standpoint is pseudo-plastic behavior. A pseudo-plastic fluid is one whose viscosity has a finite value at zero rate of shear, falling off, as the rate of shear increases, asymptotically to a lower value. Specific illustrations in industrial processes are: emulsion polymerization of butadiene and styrene to GR-S synthetic rubber; emulsion polymerization of vinyl acetate to polyvinyl acetate; solution polymerization of styrene to polystyrene; the compounding of natural and synthetic rubber latices and high polymer latices such as polyvinyl chloride latex; preparation of starch by extraction from potatoes; film casting and dipping processes involving solutions of plastics such as vinyl chloiide-vinylidene chloride copolymers and rubber hydrochloride; the processing of cellulose acetate solutions in the rayon industry; the preparation of nitrocellulose lacquers; and the preparation of adhesives such as polymethyl methacrylate solutions. The object of the present work is to investigate the heat transfer characteristics of pseudo-plastic fluids whose viscosities fall into the medium to low viscosity range-Le., less than 15 centipoises. Pseudo-plastic fluids, such as plastics ( l 7 ) ,which fall into the high viscosity range, are not included in the present work, as they are not in general handled in the standard types of heat transfer equipment used for liquids. The viscosity range covered includes the important fields of synthetic rubber latex and natural rubber latex, most of the vinyl plastic emulsions, and many of the more dilute high polymeric solutions in industrial use. Attempts to widen the range of viscosity too much in the investigation might have led to confu1

Present address, Dunlop Rubber Go., Birmingham, England.

sion if the region where rodlike flow obtained had been infringed upon. This practice of division of results according to the viscosity ranges appears to be in accordance with the practice of many investigators of the flow behavior of Newtonian liquids. The temperature range considered in the investigations is 30 to 80 ’ C. Lower temperatures are usually avoided, if possible, in industrial processes to eliminate the need for refrigeration equipment. Similarly, higher temperatures are not too frequently encountered because the emulsions and high polymer solutions which comprise the majority of the industrially occurring pseudoplastic fluids are either chemically or physically unstable a t higher temperatures. While it is not pertinent to review here all the work on heat transfer characteristics of the Newtonian fluids (6, 7 , 11, 14-16, 62), the fundamentals of the approach to the problem of the heat transfer characteristics of R pseudo-plastic fluid are similar. Applying the concept of the film coefficient of heat transfer to a fluid flowing in a tube (2, 12, 23), the following relation holds: O

In practice it is always found that the temperature difference across the fluid film varies along the length of the tube. The log mean temperature difference can be used in Equation 1, if the value of the film coefficient of heat transfer is a constant over the range of the change in temperature of the fluid during its passage through the exchanger. Heat transfer data on fluids deviating from simple Newtonian behavior are presented by Winding, Dittman, and Kranich (26), Bonilla ( 4 ) , Hoopes et al. (IO),and hlacLaren and Stair ( I S ) . The data of Winding ( 2 6 ) are for the heating and cooling of pseudo-plastic GR-S latices of several types. They are correlated by a log-log plot of

August 1953


Figure 1.

General Arrangement of Heat Transfer Apparatus

The correlation on this plot is presented as two parallel straight lines of slope approximately equal to 0.9. The upper line was inch for pipe of 1 inch nominal diameter; the lower for pipe of nominal diameter. There is considerable scattering of the data. The dafa of Bonilla ( 4 ) are for the heating of precipitated chalk in water slurries of various concentrations. The viscosities of the slurries are related to that of water by the Hatschek (8)equation:

where fie = viscosity of slurry p,,, = viscosity of water r+ = volume fraction of solids in slurry The heat transfer data are correlated as a straight line of approximate slope 0.8 by a log-log plot of j -I- A j VS.

DG

MS

where j is the Colburn factor ( 6 ) . A j is a correction applied to the Colburn j factor. I t s form is an empirical function of the slurry concentration. The slurry viscosity as derived from the

R.

rl" Figure 2.

1687

Hatschek equation is assumed constant and independent of the shearing stress, provided this is above a certain minimum value, in accordance with standard Bingham body behavior. The data of Hoopes (IO)are for the cooling of 0 to 21% FilterCel slurries, and are found to agree within 10% with the standard Dittus-Boelter equation using the 0.4 exponent for the Prandtl number. However, for the data of MacLaren and Stair ( I S ) also on Filter-Cel slurries, it was found necessary t o alter the Reynolds number exponent in the Dittus-Boelter equation from 0.8 to 0.705, and to change the constant from 0.0225 t o 0.0385. Both Hoopes et al. and MacLaren and Stair treat the slurries as showing Bingham body behavior, though MacLaren and Stair did notice that a t low fluid flow rates there appeared to be some tendency to deviate from this behavior. The findings of Kern and Van Nostrand ( 1 1 ) treated the fatty acids as having constant viscosities a t a given temperature, but in developing a correlation they found that the earlier correlations ( 6 , 16, 16, 22) all predicted heat transfer coefficients deviating above those observed. A similar effect is observed in the results of the present work, and in this latter case it has been found possible t o relate the magnitude of this deviation to the degree of pseudo-plasticity , Apparatus

A schematic general arrangement drawing of the heat transfer apparatus is presented in Figure 1. Figure 2 presents a detailed sectional drawing of the actual heat exchanger,

Sectional View of Heat Exchanger

The pseudo-p I a s t i c liquid under investigation was contained in the t h e r m o s t a t bath, A . This was a borosilicate glass bowl 12 inches in aiameter and 12 inches deep, with walls inch thick. Heat was supplied to the bath by a 1000-watt Lolag heater and temperature control to f0.01' C. was exercised by the mercuryin-glass thermostat switch, G, which energized the relay in t h e main leads to the heater. T w o a u x i l i a r y Lolag heaters, B, of 1000- and 2000-watt capacities, respectively, could be manually switched on or off, as required. The bath was stirred by a stainless steel marinetype impeller, driven by a l/le-hp. electric motor. The temperature of the

1688

Vol. 45, No. 8


b a t h was indicated by the precision grade mercury-in-glass thermometer, F , which was graduated in 0.1" C. The pseudo-plastic liquid from bath A was circulated through the tube side of the heat exchanger, K , via the stainless steel centrifugal pump, G, having a rated maximum capacity of 8 gallons per minute delivery of water against a zero head pressure, falling to zero delivery of water against a head pressure of 17 pounds per square inch. The deliver rate from G was indicated by the flowmeter, HI and thence t i e pseudo-plastic liquid passed to the heat exchanger via the tube side inlet thermometer pocket, I . After passing through the exchanger, the pseudoplastic liquid left via the outlet thermometer pocket and was piped back to A. The flowmeter was a straightforward Venturi type constructed of glass. During the calibration of the flowmeter, a full allowance ( 5 ) was m d e for the variation in viscosity with variations in fluid composition or temperature. The exchanger inlet and outlet thermometer pockets were all designed to promote turbulent flow to ensure accurate liquid temperature measurements. Beckmann thermometers (t,, T,, to, and To)reading t o 0.01' C. were used to measure the inlet and outlet temperatures of both the tube and shell sides of the exchanger. Three exchangers of exactly similar construction and dimensions, except for tube length, were used. I n each exchanger the tube was 0.500 inch in outside diameter and 0.433 inch in inside diameter. Tube lengths are given in Table I. The calming sections had internal diameters identical to that of the tube, and the inlet calming section, J , was removable to enable a study of the effect of variation in its length. The whole of the exchanger, the calming sections, and the thermometer pockets were well insulated. The water countercurrent to the flow of pseudo-plastic fluid n a s delivered to the shell side of the exchanger via the pump, 0, and the rate of delivery was indicated by the flowmeter, P.

Before a new series of heat exchange runs was started, the tube was washed inside and outside with dilute (1%) ammonia for about 20 minutes, and this was followed by a water wash. T h e condition of the exchanger surface was then checked by carrying out a tube side coefficient of heat transfer determination with water on both sides. The coefficient thus obtained was checked with t h a t predicted from the Dittus-Boelter equation for cooling within 333%. The liquid to be circulated through the tube was placed in the thermostat, A . The Beckmann thermometers ( t i , to, T,, and T o ) were adjusted t o operate in the required range. The inlet temperature of the shell-side water was adjusted to be 15' to 20' C. below the temperature of the tube-side inlet liquid. Thus a temperature difference of about 10" C. was maintained across the liquid film of the tube side to minimize the effect of any error. The water rate on the shell side was adjusted to limit the liquid temperature drop through the tube to about 4' C. a t the most. Following adjustment of the circulation rates to the desired values, it was necessary to establish the temperature equilibrium of the system. The heat supply to the bath, A , from the auxiliary heater, B, and the cooling of bath LV by means of the cold water and the overflow siphon were adjusted. I n both cases the thermostatic control was in operation a t the respective selected temperatures. The tube side could be maintained a t a constant temperature within 1 0 . 0 3 " C., and the shell side within 3=0.05"C. About 30 minutes were then allowed for the establishment of equilibrium in the exchanger; the flow rates were checked, and finally the flowmeter readings and the inlet and outlet temperatures of the shell side and tube side liquids were recorded. Second sets of temperature readings were taken to check t h a t the temperature differences between inlet and outlet thermometers showed no change.

The measured variable in determining tube wall temperature mas the change in electrical resistance of the copper exchanger tube with temperature.

Table I. Exchanger

Referring to Figure 1, the clips were of flattened copper tubing, inch in diameter, shaped to clamp around the exchanger tube. The contact areas of the tube and clip had polished surfaces. The clips were placed flush against the rubber stoppers holding the ends of the exchanger shell, and were tightened onto the tube with a nut and bolt. Two heavy-gage terminals were attached to the projecting arm of each clip and heavy copper battery leads connected from these to the ohmmeter terminals. The total resistance of the external circuit was approximately equal t o that of the heat transfer section of the tube in exchanger 1. The ohmmeter was a Leeds & Northrup Kelvin bridge ohmmeter No. 4286. The ohmmeter had both current and potential terminals; hence two leads were required to each end of t h e resistance element-Le., the exchanger tube-being measured. To obtain a true indication of tube wall temperature it was essential to maintain the ambient temperature strictly constant to eliminate changes in the resistance of the leads. A screen between the thermostat, A , and the ohmmeter leads revented errors due to local temperature disturbance. With &e above arrangements the tube wall temperature could be read to 0.3" C. jlS

The resistance method measures the average tube temperature. The drop across the tube is, however, small. For example, in the case of run B/33, the temperature difference, 17" C., between the hot and cold fluids is high; the temperature drop through the copper is only 0.25 C. The inside wall temperature will deviate from the average tube temperature b y only approximately half this amount, 0.125' C., which introduces an error of about 1$To into the calculation of the heat transfer coefficients. Experimental Runs Give Data for Heat Transfer Calculations

Calibration of Apparatus. Experimental heat transfer runs were accomplished under identical conditions, except t h a t the length of the inlet calming section was varied. The heat transfer rate was substantially the same for each of the three lengths of calming section investigated. Thus, even though the shortest length of calming section appeared t o give adequate protection against errors arising from undue turbulence, it was chosen t o use the longest calming section throughout the experimental work as a n additional safeguard.

NO.

1 2 3

T u b e Lengths, Inches 29.5 49.0

67.0

Tube Lengths Inlet Calming Section, Inches 50.5 50.0 49.0

Outlet Calming Section, Inchrs 11.0 11.0 11.0

The viscosity of the pseudo-plastic liquid was determined by the method discussed in another paper (6) a t the commencement of each liquid series, and was point checked at intervals during the course of the heat transfer runs. No significant reduction in viscosity, due to polymer breakdown, was noted in any of the three materials investigated. As an additional check on this effect, each solution was circulated through the tube for 2 houre a t 70" C. before the commencement of a series of heat transfer runs. The viscosity was checked before and after the circulation, with deviations less than 2% in all cases. The materials selected were aqueous solutions of (1) polyvinyl alcohol, ( 2 ) sodium carboxymethylcellulose, and (3) sodium polymethacrylate. These liquids were selected to meet the following conditions:

No time-shear dependence--e.g., thixotropy Pseudo-plastic behavior in var ing degrees Not subject to significant mecKanical or thermal degradation during the course of the experimental work These materials were dissolved in water a t concentrations t o yield solutions of viscosities in the range usually encountered industrially. The deviation from simple ITewtonian behavior varied; the polyvinyl alcohol solution was only slightly pseudoplastic, while the sodium polymethacrylate solution was the most pseudo-plastic fluid in this viscosity range for which data are currently available. The sodium polymethacrylate solution was prepared from a solution of ammonium polymethacrylate Vulcastab T which was supplied initially as a 10% solution in water. This 10% solution was boiled with the stoichiometric quantity of caustic soda necessary to convert the ammonium t o the sodium salt until no trace of ammonia was detectable in the vapors. The resulting thick gel was diluted with water to the required concentration.


August 1953

Table II. Wt Ws Run No. Lb./Min. Lb./&.

A/; 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

49 50 51 52 53 54 55 56 57 58 59 60 61

B/; 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 I9 20 21 22 23 24 25 26 27 28 29 30 31 32

a

Lb./Sq.'Ft./ Sea.

ti, O C .

lo,

oc.

At,

OC.

1689

Heat Transfer Data Ti,

OC.

To,

OC.

AT

tw,

OC.'

O C .

25.8 23.25 21.4 19.3 15.95 13.15 11.5 9.0 5.3 23.25 21.4 19.3 16.0 13.15 11.5 10.0 9.6 9.0 7.4 6.7 6.0 5.3 23.25 21.4 19.3 16.0 13.15 11.5 9.6 9.0 7.4 6.7 5.3 23.25 21.58 19.3 16.0 13.15 9.6 5.53 23.25 21.4 19.3 16.0 13.15 9.6 9.0 7.4 6.7 13.38 18.51 18.67 13. OS 18.8 13.58 13.06 18.42 18.75 13.41 18.32 13.1

21.2 21.2 17.05 17.05 15.8 14.9 14.9 14.9 12.6 21.2 17.05 16.2 16.5 14.9 14.6 14.4 14.65 14.4 14.4 14.4 14.4 14.4 21.2 17.05 14.2 16.5 14.9 14.6 14.6 14.4 14.4 14.4 14.4 19.5 17.05 16.0 16.5 14.9 14.65 14.4 21.2 17.05 16.2 16.5 14.9 14.6 14.4 14.4 14.4 15.41 15.52 15.88 15.71 14.05 15.4 16.1 16.1 13.98 13.67 8.22 7.96

421.5 3 80 349 3 15 260.5 215 188 147 86.5 3 80 349 315 261 215 188 163 157 147 121 109 98 86.5 380 349 315 261 215 188 157 147 121 109 86.5 380 352 315 261 215 157 90.5 380 349 315 26 1 215 157 147 121 109 218.5 302.5 304.5 213.5 307 222 213 301 306 219 299.5 214

0.25% C A R B O X Y M E T H Y L C E LSOLUTION. L U L O ~ ESERIEBA 2.02 41.0 49.76 48.09 1.67 30.14 32.16 1.87 32.07 1.99 41.0 49.98 48.11 30.07 49.94 48.24 1.70 29.80 32.02 2.22 41.7 30.13 41.4 49.95 48.19 1.74 32.10 1.97 1.75 29.95 41.3 50.03 48.28 31.70 1.75 30.12 41.7 1.75 50.07 48.34 31.68 1.46 36.6 1.90 30.15 2.57 49.79 47.22 32.05 29.93 2.55 36.3 31.49 1.55 49.53 46.98 30.07 33.2 30.98 0.91 2.37 50.01 47.64 46.92 29,03 31.27 2.24 39.1 48.89 1.97 48.90 1.86 47.04 29.28 31.63 2.35 39.9 29.19 49.80 2.03 47.77 31.53 2.34 39.9 28.36 50.27 2.35 47.92 30.57 2.21 37.9 30.22 37.2 28.17 50.61 2.44 48.17 2.05 29.83 50.19 2.29 47.90 37.2 31.65 1.82 30.42 50.58 2.36 48.22 38.2 32.04 1.62 27.02 49.54 2.44 47.10 1.58 36.6 28.60 29.81 31.27 50.99 2.33 48.60 1.46 37.0 29.20 2.25 1.10 30.30 51.04 48.78 37.0 1.06 28.61 51.30 2.35 48.95 35.4 29.67 0.84 28.68 29.52 50.46 2.00 48.46 33.7 29.34 30.12 50.44 2.20 48.24 33.7 0.78 49.01 45.91 3.10 29.26 3.34 32.60 36.80 33.10 49.97 46.85 3.12 29.28 3.82 37.50 32.56 37.75 50.13 47.79 3.34 28.62 3.94 46.23 3.32 3.22 31.85 36.40 49.65 28.63 50.47 47.01 3.46 2.96 33.30 36.40 30.34 32.70 50.05 3.47 30.00 46.58 2.70 35.70 2.53 31.62 50.35 29.09 46.55 3.80 34.00 46.90 4.02 31.33 32.60 50.92 28.57 2.76 1.82 46.20 3.34 31.85 34.00 49.54 30.03 1.85 46.60 2.40 29.55 34.40 50.00 27.70 0.94 47.10 2.78 29.60 31.50 49.88 28.66 50.49 47.45 3.04 28.77 32.37 3.60 38.5 29.27 50.35 47.35 3.00 33.09 3.82 38.5 3.20 28.94 50.40 47.20 32.88 3.94 38.0 49.63 46.57 3.06 29.88 32.80 2.94 38.0 49.68 46.53 3.15 29.55 32.34 2.79 38.0 49.74 46.52 8.22 29.31 29.43 2.12 36.0 51.34 49 01 2.33 28.90 29.70 0.80 36.0 49.86 45.60 4.26 28.20 32.88 4.66 35.9 27.67 49.75 45.63 4.12 32.79 5.11 35.2 27.81 32.85 50.07' 46.07 4.40 5.04 35.9 27.46 49.63 45.03 4.60 31.67 4.21 34.6 32.79 28.53 50.73 46.01 4.72 4.21 34.6 30.69 27.79 2.90 33.7 50.59 45.68 4.91 32.50 30.75 2.75 49.90 45.44 4.46 32.95 30.21 28.25 49.07 45.26 3.81 32.4 1.96 30.35 50.83 47.18 3.65 31.98 34.6 1.63 44.58 43.11 1.47 28.85 30.03 1.18 33.88 44.53 43.00 1.36 29.53 1.51 34.2 31.04 48.48 1.99 29.83 50.47 2.25 38.3 32.08 50.13 1.72 48.01 2.12 29.48 37.2 31.20 54.57 33.39 36.05 52.69 1.88 2.66 43.9 54.83 33.64 35.61 52.41 2.42 1.97 41.7 60.05 40.23 57.40 2.65 2.15 42.38 47.5 59.72 57.41 2.31 39.99 2.19 42.18 48.6 46.43 48.26 64.42 62.30 2.12 2.83 55.85 2.43 64.77 62,34 45.72 48.08 2.36 54.5 49.64 53.44 3.80 70.01 1.88 68 13 61.55 2.26 49.61 53.20 69.84 67.58 3.59 60.8

22.27 20.65 19.25 16.5 14.0 11.3 8.3 6.5 22.27 20.65 17.85 16.33 14.68 12.18 10.49 7.82 23.5 22.0 20.2 17.5 14,82 12.0 8.78 6.9 22.5 20.75 19.05 16.5 14.0 11.3 8.3 6.5

21.2 17.06 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 10.2 21.2 17.05 17.05 17.05 17.05 17.05 14.4 14.4

364 337 311 269.5 228.5 184.5 135.5 106.1 364 337 291.5 267 239.5 199 187.5 127.7 384 359 330 286 242 196 143.3 112.7 368 339 311 269.5 228.5 184.5 135.5 106.1

45.80 45.58 45.50 45.29 45.49 45.00 45.59 45.61 49.63 51.06 50.15 51.40 51.70 50.12 51:15 50.25 55.02 54.95 55.14 55.20 55.05 56.61 55.50 55.29 58.40 60.21 60.41 60.37 60,46 60.91 61.95 62.41

h qr q, B.t.u.jHr./ B.t.u.jHr. B.t.u./Hr. Sq. Ft./' F. 4605 4655 3875 3575 2955 2335 3145 2500 1303 4895 4245 4185 4135 3415 2795 2495 2529 2340 1743 1645 1240 1210 7735 7165 6905 5680 4860 4255 3885 3900 2615 2403 1535 7595 6945 6615 5235 4405 3295 1270 10645 9445 9105 7895 6665 5035 4285 2995 2585 2080 2680 3960 2905 4237 3464 3656 4673 4186 3427 3597 3097

4630 4560 4090 3640 2980 2490 3066 2420 1235 4900 4300 4090 3940 3315 2870 2520 2455 2270 1710 1650 1308 1215 7650 7040 6900 5740 4730 4260 3990 4260 2830 2410 1461 7600 7050 6700 5210 4495 3355 1245 10650 9600 8830 7500 6790 4505 4270 3050 2540 1970 2540 3860 2918 4040 3280 3560 4260 4270 3490 3370 3085

25% P O L Y V I N Y ALCOHOL L SOLUTION, SERIESB 43.16 2.64 2.71 42.88 2.70 3.45 43.18 2.37 2.55 42.81 2.48 2.31 2.63 42.86 2.13 42.47 2.53 2.05 42.55 3.04 1.73 41.49 3.11 1.28 3.57 4.28 3.70 3.84 3.11 3.11 2.64 2.29 50.92 4.10 30.23 34.75 4.52 50.97 3.98 30.97 34.27 6.25 50.67 4.47 28.10 33.19 5.08 50.67 4.53 29.26 33.74 4.48 50 34 4.71 29.30 33.42 4.12 51.27 5.34 28.76 33.15 4.39 50.41 5.09 30.41 32.59 3.18 50.45 4.84 30.49 33.66 3.11 46.18 2.81 48.39 3.14 48.63 2.90 47.83 2.84 47.48 2.64 2.45 47.28 46.68 2.56 46.21 2.21

6305 5985 4835 4375 3930 3045 3680 2070 7995 7795 6795 7095 5945 4945 4405 3545 10334 9384 9694 8484 7474 6854 4754 3534 6948 6128 5898 5598 5298 4533 3858 3293

6203 5820 4700 4260 3920 3190 2690 1994 8180 7880 6810 7070 5730 4840 4110 3560 10350 9660 9350 8250 7590 6820 4950 3425 6440 5790 5340 5230 4870 4515 3980 3435

38.1 37.3 37.7 37.3 35.3 35.9 35.0 33.4 40.0 40.0 38.1 37.7 37.7 35.8 37.3 35.3 44.0 43.5 42.4 42.9 42.7 41.8 41.2 40.7 50.8 51.6 51.2 50.8 60.5 49.7 50.1 50.1

1165 1160 1045 92 1 751 622 529 406 167 1111 1051 920 772 611 471 444 43 1 367 269.5 223 223 154.5 875 789 709 590 474 406 323 287 226.5 208 108.9 870 805 749 62 1 524 325 108 785 665 645 534 424 306 253.5 181 156.8 414 555 709 49 1 858 581 651 94 1 994 759 959 777 1189 1038 880 648 532 467 355 245 1205 1005 795 730 642 478 447 333 1365 1165 1090 980 881 654 485 339 1450 1015 916 842 715 566 475 405

R8,

Seo.

-1

1498 1350 1240 1120 925 763 666 52 1 308 1350 1240 1120 929 763 666 580 556 52 1 429 388 348 308 1350 1240 1120 929 763 666 556 52 1 429 3 88 308 1350 1250 1120 929 763 556 321 1350 1250 1120 929 763 556 521 429 388 775 1072 1082 756 1090 786 756 1069 1088 777 1062 760 1292 1198 1115 956 811 655 488 377 1292 1200 1035 949 850 706 609 454 1361 1275 1171 1012 860 695 510 395 1305 1202 1105 955 811 655 ,481 377

(Continued on page 1600)

I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY

1690

Table II. Run No.

G, Wt W s , Lb./Sq. Ft./ L b . / i h n . Lb./Min. See.

B/33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

23.0 20.75 19.05 16.5 14.0 11.3 8.3 6.5 22.8 21.5 19.05 16.5 14.0 11.3 8.3 6.5

21.2 17.05 17,05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4

376 339 311 269.5 228.5 184.5 135.5 106. 1 372 351 311 269.5 228.5 184.5 135.5 106.1

C/;

26.05 22.7 19.55 16.2 13.5 10.3 8.6 6.8 26.05 22.7 19.55 16.2 13.5 10.3 8.6 6.8 26.05 22.7 19.55 16.3 13.5 10.3 8.6 6.8 26.05 22.7 19.55 16.2 13.5 10.3 8.6 6.8 25.4 22.7 39.55 16.2 13.5 10.3 8.6 6.8 26.05 22.70 19.55 16.2 13.5 10.3 8.6 6.8

21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.20 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4 21.2 17.05 17.05 17.05 17.05 14.4 14.4 14.4

426 371 319.5 264.5 220.5 168.3 140.3 111 426 371 319.5 264.5 220.5 168.3 140.3 111 426 371 319.5 264.5 220.5 168.3 140.3 111 426 371 319.5 264.5 220.5 168.3 140.3 111.0 415 371 319 5 264.5 220.5 168 3 140 3 111.0 426 371 319.5 264,s 220.5 168.3 140.3 111.0

3 4 5 6 7 8 9 10

11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

tz o c .

to,

oc.

Vol. 45, No. 8

Heat Transfer Data (Continued) At,

oc.

Ti,

oc.

To. 0

c.

tT,

c.

c.

25% POLYVISYL ALCOHOLSOLUTIOS, SERIESB 64.37 60.37 4 00 43 42 47 55 4.13 53 9 3 91 43 66 54 2 64.67 60,76 48 23 4.57 4.31 4.20 44 05 48 36 53 9 65.39 61,lQ 4 50 65.97 61.47 44 36 48 45 4.09 53 9 65 93 61.17 44 33 4 76 48 20 3 87 53 5 53 5 4 54 3.73 65 01 60.47 44 06 47 79 66.31 61.07 44 83 47 91 53 5 5 24 3,08 5 53 66. 69 2.34 45 08 61. 16 47 62 53 9 66.41 3.35 53.72 57.34 69 76 3.62 62.6 70 55 66.95 3.60 52.29 56.60 4.31 61.6 69 56 66.03 61.0 3.53 52.96 56.54 3.58 57.45 3.14 61.0 70 05 66.39 3 66 54.31 66.55 4.25 53.32 56.60 70 80 3 28 59.7 71 53 66.83 4.70 52.32 55 99 3.67 59.7 52.44 55.59 3.15 59.2 71 92 66.62 5.30 71 87 65.57 4 35 53.56 56.38 2.82 59.2 ' 0.02570 SonIuhr POLYMETHACRYLATE SOLCTIOS, SERIES C 45.18 43,45 1.73 30.25 32.29 2.04 38.1 44.18 42.64 1.54 30.22 32.30 2.08 37.3 45.98 44.60 1.89 30.04 32.25 2.21 36.3 46.50 44.52 1.98 29.50 31 44 1.94 36.0 46.38 44.92 1.46 28.23 29.30 35.3 1 07 46.58 45.22 1.36 29.67 36.3 0.96 30.63 0.90 46.84 32.05 45.21 1,63 31.15 34.9 47.12 32.83 45.14 1.88 32.06 35.3 0.77 50 70 47.86 2.84 33.03 3.86 40.6 28 27 51.86 34.16 29.21 48.96 2.90 3.95 41.3 3.10 52.52 34.19 29.78 49.22 3.41 40.2 ta1.50 33,24 29.22 48,44 3.06 3.02 39.0 53.05 32.35 3.25 2 66 38.8 28 67 48,80 52.45 28.89 49.96 31.54 2.49 L65 36.7 52.62 30.36 50.20 32.63 2.42 1.27 37.0 52.31 49,50 31.16 2.81 1.26 35.6 28.90 56.06 3.37 28.41 34.16 4.25 43.1 51.69 54.61 3.33 29.07 34.69 4.62 42.8 51,28 51.32 54.78 3.46 29.65 34.65 4 00 42.4 3.58 42.0 51.28 54.95 3.67 29.77 34.35 3 20 29.43 33.63 41.6 51.08 54.86 3.78 2.53 52.33 55.98 3.65 27.97 31.50 39.1 1.92 3.28 37.4 53.33 56 61 29.74 32.66 1.79 37.4 3.80 52.48 56.28 29.81 33 60 59.85 57.23 2.62 47.25 45 56 3.31 51.0 2.55 3.36 51.4 43.10 60 23 57.68 46,46 43,85 3.09 58.13 2.67 46 94 51.4 60.80 51.4 2 80 43.62 57.15 2.92 46.42 61.07 50.7 2.47 43.69 3.14 46.16 57.32 61.46 58.81 50.3 2.51 42. Q5 62.02 3.21 45.46 43,75 48.2 1.90 45.65 59.18 62,15 2.97 44.37 47.4 1.21 2.98 58.92 45.68 61.87 3.52 42.83 47.00 4.17 54.5 64.33 60.81 54.2 46.80 60.91 3.51 42,05 4.73 64.42 54.2 48.48 61.45 3,53 44.36 4.12 64.98 54.8 48 77 66 05 62,25 3,80 45.16 3.61 54.2 44.62 3.24 65.78 47 86 61.67 4.11 65.92 54.6 48.16 61.93 3.99 45.36 2.80 54.2 2.42 65.32 48.46 61.71 3.51 46.04 46.24 1.59 50.6 65 73 62.30 3.43 44.65 53.01 3.67 62.4 67.34 3.06 56 68 70.40 4.06 56. 68 52.62 62.6 67.76 3.17 70.93 56.40 3.58 3.32 52.82 61.1 70.62 67.30 55.42 3.16 3.43 52 26 60.6 69.90 66.47 55.99 2.56 61.4 66.33 3.39 53.43 69.72 55.72 2.59 59.7 3.62 53.13 66.64 70.26 56.02 2.21 53.81 58.4 71.51 3.96 67.55 60.2 1.52 54 80 53.28 3.65 69.65 66.30

Experimental Results. The results obtained from the heat transfer experiments are presented in Tables 11, 111, and IV. The experimental runs are classified as: SERIESA. Runs 1 to 61. Material, 0.25% aqueous solution of sodium carboxymethylcellulose. SERIESB. Runs 1 t o 48. Alaterial, 2.GY0 aqueous solution of polyvinyl alcohol. SERIES C. Runs 1 t o 48. Rlaterial, 0.025Tc aqueous solution of sodium polymethacrylate. The rate of heat transfer occurring in the exchanger was calculated from tube-side data, and also in an independent manner from shell-side data. The quantity of heat transferred, as calculated from the tube-side data, had to be corrected for heat loss from the calming sections. KO correction for heat loss was applied in the calculation of the rate of heat transfer based on the shell-side data. ilny such correction was considered unnecessary, as the shell-side data were used only as a check on the tube-side data. Furthermore, the shell-side temperature was always lower than the tube-side temperature. The h.eat loss to the atmosphere

It,. See. - 1

tw,

9851 8674 8554 7921 7104 5454 4609 3784 7907 8257 7157 6437 6247 5647 4651 4347 4815 3780 3845 3420 2085 1470 1470 1225 7943 7055 6495 5305

4695 2713 2195 2017 9414 8114 7234 6420 5454 3990

2984 2721 7288 6168 5358 5028 4499 3488 2618 2083 9554 8504 7364 6534 5894 4344 3169 2424 8497 7667 7907 6897 4837 4922 3587 2377

9453 8420 7940 7550

7130 5805 4795 3960 8282 7930 6620 5790 $055

m n

4900 4405 4670 3843 4070 3575 1970 1495 1402 1200 8800 7270 6290 5560 4900 2570 1975 1960

9730 8500

7350 6590 5890 3940 2960 2790 7580 6190 5090 5060 4550 3915 2960 2040 9550

8560 7595 6650 5960 p350 3765 2475 8400 7480 6595 5820 4715 4030 3443 2365

1380 1212 1092 953 840

699 533 442 5 1436 1368 1242 1042 820 703 $46 ,134 93.5

755 515 433 2-11 167 159 138 5 1106 937 739 637 465 226 165 160 5 1080 920 820 696 570 323 207 193 919 775 647 623 505 397 5 243 183 3 1420 1210 986 840 750 562 410 227 1580 1330 1215 1090 810 675 387 399

1335 1202 iinj

955 811

662 481 377 1323 1248 1103 0.55 811

655 48 I 377 1512 1319 1132 940 781 597 499 3P1

1512 131!) 1132 940 781 597 499 394 1512 1319 1132 940 781 597 499 394 1512 1319 1132 940 781 597 499 394 1471 1318 11x2 940 78 1 597 499 391 1512 1310 1132 940 781 597 499 391

from the shell side was small. Compaiisori b e t w e n the heat transfer rates, calculated as specified above, from the tube-+le data arid from the shell-side data, showed in general an agreement within +6%. I n the subsequent development of the heat transfer corrrlation, the heat transfer rates, as calculated n ith Equation l from the tube side data, were used exclusively. While in the present work the heat transfer coefficient shoueti somr variation with temperature, it was considered that its rate of change with temperature was sufficiently small t o preclude introdurtion of any substantial error by the use of Equation 1 The viscosity determinations ( 5 ) were made in a tube-type instrument, so that in the absence of a rigorous mathematical treatment of pseudo-plastic viscometer behavior, the apparent viscosity, as determined from the viscometer, would he appropriate for use in conjunction v+iththe heat transfer data from n tubular exchanger in the establishment of a heat transfer correlation Thus the appropriate apparent viscosity for a given heat t r m + fer run was obtained by calculating the consistency variable, R,, from

August 1953


1691 ~

Table 111. R u n No.

a, Sec. -1 SERIE0

A/1 2 3 4 5

6 7 8

9

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

6080 6060 6050 6060 6050 6040 6130 6170 6100 6220 6210 6170 6130 6010 6060 6010 61019 5966 5950 5920 6000 6010 6290 6150 6150 6220 6100 6160 6150 6080 6230 6170 6140 5940 6170 6130 6200 6200 6200 5930 6250 6260 6180 6320 6170 6190 6260 6390 6070 6900 6900 6000 6050 5500 5510 5130 5140 4950 4940 4810 4820

Rs,

Sec.

-1

so0

G./Crn.)Sec.2

Viscosity Data for Heat Transfer Calculations rqr

"4'

pm,

Poise

Poise

Poise

.OSE SOLUTION A, 0.25% CARBOXYMETHYLCELLUI

1498 1350 1240 1120 925 763 666 521 308 1350 1240 1120 929 763 666 580 556 521 429 388 348 308 1350 1240 1120 929 763 666 556 521 429 388 308 1350 1250 1120 929 763 556 321 1350 1250 1120 929 763 556 521 429 388 775 1072 1082 756 1090 786 756 1069 1088 777 1062 760

135 135 134 134 134 134 137 138 136 140 140 134 136 132 133 131 138 139 128 127 181 132 143 137 137 140 136 138 137 135 141 138 137 128 138 137 140 140 140 127 142 142 138 144 137 139 142 147 134 169 169 131 134 110 111 90 91 75 74 55 56

0.01405 0.01405 0.01405 0.01405 0 0140 0.0140 0.0141 0.0141 0 0141 0.01415 0.01415 0.01405 0.0141 0.0140 0.01405 0.0140 0.0141 0.01395 0.0139 0.0139 0.01395 0.0140 0.0142 0.0141 0.0141 0.01415 0.0140 0.0141 0.0141 0.01405 0.01415 0.0141 0.0141 0.0139 0 0141 0.0141 0.01415 0.01415 0.01415 0.0139 0.0142 0.0142 0.0141 0.01425 0.0141 0.01415 0.0142 0.0143 0.01405 0.0146 0.0146 0.0140 0.0140 0.0134 0.0134 0.0125 0.01255 0.01185 0.0118 0 0111

0.01115

0.0319 0,03225 0.03245 0.03275 0.0332 0.0337 0.0343 0.03465 0.0353 0.03265 0.0332 0.03245 0,03345 0.0335 0.03385 0.0339 0.0348 0.03545 0.0340 0.0340 0.03465 0.0349 0.0329 0.03265 0,03295 0.03375 0.03385 0.0343 0.0345 0.03445 0.03525 0.0351 0.0354 0.03145 0.03265 0.0330 0.0338 0.03425 0.03485 0.0342 0.0329 0.0331 0.03305 0.03385 0.0339 0.0348 0.0351 0.0359 0.03485 0 0366 0 0358 0 0325 0 0337 0 0301 0 0310 0 0278 0 0271 0 02425 0 02475 0 02045 0 0212

.

0.03625 0.03635 0.0364 0 03623 0.0361 0.0362 0.03645 0.03645 0.03635 0.03665 0.0367 0.03575 0.03632 0.0360 0.0360 0.0358 0.0367 0.03725 0.0354 0.03535 0.03575 0.03595 0.03695 0.0364 0.0364 0.03685 0.0363 0.03645 0.0364 0.03625 0.03675 0.03645 0.0364 0.0355 0.0365 0.0365 0.03675 0.03675 0.03675 0.0353 0.0369 0.03685 0.0364 0.0370 0.0363 0.03665 0.0369 0.0373 0.03615 0 0391 0 0391 0 03585 0 03615 0 0334 0 0335 0 03005 0 03025 0 0270 0 02675 0 0225 0 02275

3 4 5 6 7 8

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

6120 6180 6170 6220 6200 6280 6220 6320 5450 5175 5350 5160 5260 5360 5200 5400 4650 4650 4660 4650 4680 4650 4650 4660 4250 4125 4110 4125 4125 4100 4060 4020

1292 1198 1115 956 811 655 488 377 1292 1200 1035 949 850 706 609 454 1361 1275 1171 1021 860 695 510 395 1305 1202 1105 955 811 655 481 377

64.8 65.7 65.3 66.0 65.8 67.1 66.0 67.5 55.5 51.8 54.5 51.8 52.5 54.5 51.8 54.6 43.1 43.1 43.1 43.1 44.0 43.1 43.1 43.1 35.0 31.5 31.0 31.7 31.7 31.0 30.0 29.0

0.0174 0.0176 0.0176 0.0177 0.0176 0.0179 0.0177 0.0180 0.0158 0.0153 0.01565 0.0153 0.0154 0.01565 0.0153 0.0157 0.0143 0.0143 0.01435 0.0143 0.0144 0.0143 0.0143 0.01435 0.01355 0.0133 0.0133 0.0133 0.0133 0.01325 0.0132 0.01315

0 0 0 0 0 0 0

02615 0265 02655 0269 0270 0277 02755 0 0281 0.02405 0.02345 0.0242 0.02375 0.0240 0,02465 0.0242 0.0250 0.02145 0.02155 0.02175 0.0219 0.02235 0.02235 0.02265 0.0229 0.01985 0.0192 0.01925 0.01955 0.0197 0.01975 0.0198 0.01975

B/33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

C/i

3 4 5 6

7 8

9

10

11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

44

SERIESB, 2.5% POLYVINYL ALCOHOLSOLUTION

l;

Run No. Sez:-1

0 0 0 0 0 0 0 0

0280 0282 0282 0283 0282 0286 0283 02865 0.0260 0.0253 0.02585 0.02535 0.0254 0.0258 0.0253 0.0258 0.02355 0.02355 0.0236 0.02355 0.0238 0.02355 0.02355 0.0236 0.0218 0.02095 0.02085 0.0210 0.0210 0.0208 0.0206 0.02035

45 46 47 48

R.,

800,

Sec-1 G./Cm./Sec.a

e;;$

Pp.>

Poise

SERIESB,2.5% POLYVINYL ALCOHOLSOLUTION 1335 25.1 0.0129 0.0177 1202 24.5 0.0129 0.01775 23.6 1105 0.0128 0.0176 955 23.0 0 01765 0.0128 23.2 811 0.0178 0.0128 655 24.5 0.01835 0.0129 481 23.0 0.0128 0.0182 377 22.7 0.0128 0.01825 1323 15.8 0.01255 0.0158 1248 14.8 0.0125 0.0156 16.5 0.01255 0.0161 1105 15.8 0.01255 955 0.01605 14.9 0.01255 0.0160 811 14.2 0.0125 0.0159 655 0.0125 0.0160 481 14.0 0.0125 0.0163 377 14.8

M'D.

Poise

0 0194 0 01925

0.0190 0,01885

0 0189

0.01925 0.01885

0.0188

0.0170 0.0167 0.0172 0.0170 0.0168 0.01655 0.0165 0.0167

SERIESC, 0.025% SODIUM POLYMETHACRYLATE SOLUTIOX 1512 96.3 0.0100 0,03205 0.0438 1319 98.0 0.0105 0.03415 0.0452 1132 94.4 0.0099 0.0334 0.0427 94.0 0.0424 940 0.0099 0.03445 93.8 0,0099 0.03545 0.0423 782 0.0420 93.2 597 0.0099 0.03655 0.00985 0.0372 93.0 499 0 0419 394 0.00985 0.0379 92 8 0.04175 2990 1512 86 1 0.00965 0.02875 0 . 0384.5 3010 1319 0,03735 83 6 0.00955 0.02885 3020 1132 0 0095 0.02O4 0.0369 82 8 3000 940 84 9 0 0096 0.03115 0.0377 3020 82 5 0 0095 0 0312 0.0368 781 81 9 0,0095 0.0321 0.0365 3030 597 499 0.0095 0.0325 0.0363 81 3 3030 3020 0,03385 0.0370 0 00955 394 82 9 1512 77 0 0.0093 0 02605 0.0342 3080 3080 1319 76 9 0.0093 0.02675 0.0343 1132 3070 77 7 0.00935 0.02785 0.03465 3070 940 77 6 0.00935 0.0287 0.03465 78 0 3070 781 0.00935 0.0296 0.03475 0.0335 75 2 3100 597 0.00925 0.02945 0.02965 0.0327 73 3 3120 499 0.0092 0.03325 3100 394 0.00925 74 6 0.0306 1512 65 0 0,00885 0.0221 0.02885 1319 64 0 0.0088 0.0228 0.0288 1132 0.0284 63 0 0.00875 0,0230 0.0088 0.0240 0.02795 63 8 940 0.00875 0.02435 0.0281 63 3 781 0.0270 61 2 597 0.00865 0,0242 499 0.0086 0.0243 60 5 0 0268 61 2 394 0,00865 0.02505 0.02695 3450 1471 56.5 0.0084 0.0199 0.02475 0.0084 0.0202 0.02475 3450 1319 56.4 0.00835 0,02025 0.0241 3500 1132 55.2 940 53.2 0.00825 0.02005 0.0232 3560 781 54.5 0.0083 0.0209 0.0238 3520 597 54.0 0.0083 0.0213 0.0235 3550 3520 499 55.0 0.00835 0.00205 0.02395 3520 394 55.0 0.00885 0.0224 0.02395 1512 42.4 0.01535 0.01815 4080 0.00775 1319 42.0 0.0153 0.0177 4190 0.0077 1132 42.2 0.01575 0.0180 4100 0.0077 4000 0.00785 0.01655 940 43.0 0.0185 3980 0.00785 0.0169 781 43.1 0.01865 0.01705 0.01845 4020 0.0078 597 42.8 41.9 4220 0.00765 499 0.01655 0.0176 394 43.2 0.01775 3980 0.00785 0.0187

where Wt = mass rate of flow through the tube side of the exchanger (pounds per minute) and p = density of tube-side Auid (pounds per cubic foot) and using the value thus obtained in the Winding equation. The constants for the Winding equation a t the temperature in question were read from Figures 3,4,and 5. The Williamson equation constants tabulated (6) were read at the log-mean temperature of the tube-side liquid. All the solutions used in the present work were relatively dilute (2.5% concentration or less), so that it was possible to use the thermal conductivity and specific heat data given for water (18). Heat Transfer Coefficient, Fluid Properties, and Equipment Dimensions Have Been Correlated

Non-Newtonian fluids are characterized by their variable viscosity dependent upon the rate of shear, or the duration of shear,


1692

I

30

I

I

i

I

40

I

5 0

I

6 0

I

I

I

I

30

Figure 4.

Constant of f i m Winding Equation with Temperature 1. 2. 3.

40

1.

I 0000

or in some cases show both of these effects. Under this general category come Bingham bodies, and general non-Newtonian and pseudo-plastic fluids. Bingham bodies show no evidence of flow until a certain minimum value of shear stress is achieved. B t this point they conimence to flow and exhibit simple Sewtonian behavior, providing the shear stress is maintained in excess of the value a t the yield point. In practice, many materials behave as a Bingham body in so far as they sholT a yield point. Thereafter the viscosity, which is the slope of the curve plotting shear stress versus rate of shear, falls off asymptotically to a lower fixed value as the rate of shear increases to infinity. S o time dependence of the viscosity on the condition of shear is shown. These fluids are considered by Hedstroni (9) as general non-Newtonian fluids. Pseudo-plastic fluid behaves like general non-Xewtonian fluid, except that it has no yield point. At the zero value of shear stress there is no rate of shear. The degree of pseudo-plasticity may be defined in terms of its viscometric behavior in several ways. Winding ( 2 7 ) and Baker (3)proposed the expression: I

- Po - P a PO

(4)

where y' = coefficient of pseudo-plasticity defined by Winding p o = Limit Sy/R, R, o

-

pa =

Limit Sg/R,

R,+

03

For purposes of the development of the present correlation, it is proposed to define the coefficient of pseudo-plasticity as:

p a , which is defined as the apparent viscosity, or the coefficient of apparent viscosity, is the ratio of the shearing stress a t the tube wall to the rate of shear, &l This definition of pseudo-plasticity

-.

R,

I

I

1

I

was chosen because of its sensitivity t o changes in pseudo-plastic behavior as well as variations in fluid flow rates. With y defined by Equation 5 , for isothermal conditions, y is equal to zero for a simple Kewtonian fluid, and greater than zero for a pseudo-plastic fluid. Thus the larger the value of y, for a given selected value of R,, the hrgher the "degree of pseudo-plasticity." The degree of pseudo-plasticity of the fluid may be regarded as a measure of its deviation from simple Kewtonian behavior. It is a reasonable inference that the degree of pseudo-plasticity mav also be a measure of the difference between the observed coefficient of heat transfer for the pseudo-plasticity fluid, and the value of the heat transfer coefficient predicted for a simple Xewtonian fluid of similar viscosity by the already established correlations. Previously published heat transfer data on fluids deviating from simple Newtonian behavior have hardly been sufficiently extensive to attempt a correlation from this approach. Thus, there is no prior indication as to which of the several established correlations (6, 11, 14-16, 2 2 ) for simple Newtonian fluids should be

,

70

0.25 70CMC solution O.O25C/o SPM solution 2 . 5 7 ~PVA solution

I

I 30

50 60 T E M P E RATURE ("C)

Constant S,g of Winding Equation vs. Temperature 2. 3.

0.2570 CMC solution 0.025 YOSPM solution 2.5Y0 PVA solution

Y --

I

I

7 0

TEMP. ("C)

Figure 3.

Vol. 45, No. 8

I

I

1

I 50

I

40

TE

I

I

M P E R ATUR E(%)

60

I

I 70

Figure 5. Constant a of Winding Equation vs. Temperatur 1. 2. 3.

0.2570 CMC solution 0.025% SPM solution 2.5% PVA solution

preferred as basis for developing a correlation for the results of the present work. The factors t o be considered in making such a choice are as follows: The correlation should have been established primarily for the transition region range of Reynolds numbers, as it is in this range that most of the data in the present work were obtained; the correlation should preferably have been shown to hold for viscosities in the range of 1 to 15 centipoises; and the correlation should not include a term involving the viscosity ratio p ! p L wwhere , = viscosity a t average fluid temperature and ptu = viscosity a t tube wall temperature. This is to be avoided, not because in the present work the tube wall temperature is unknown, but because no satisfactory expression has yet been developed for the condition of shear existing in a pseudo-plastic fluid a t the tube wall. Hence, the magnitude of the effect of shear on the pseudo-plastic viscosity a t the tube wall may not be determined. The correlation of Morris and Whitman includes no term involving a ratio of viscosity values, and it was selected as the basis for the correlation of the present data because i t also satisfies the first t\vo factors. Replacing the true viscosity of the simple Newtonian fluid by the apparent viscosity of the pseudo-plastic fluid, the plot was obtained as indicated in Figure 6. This is a plot of

J"

21s. Re,

(y)(y)

where J " = Reynolds number defined by

Re,

-0'37($)0'63

DG

= Pa

(6)

and Re, is apparent

(7)

The function of the ratio of tube length to diameter is included in the ordinate function on Figure 6 to facilitate comparison between Figure 6 and Figure 7 . The full line on Figure 6 is a curve fitting Morris and Whitman's data, Series E (14). The Morris and Whitman data selected are for the cooling of oil, the properties of the oil being such that the Prandtl number is in the same general range as that of the liquids used in the present work. Examination of Figure 6 showrs two things: that the scatter on a percentage basis is more severe for lower values of the apparent

August 1953


Table IV. Run

k B.t.u.;/Hr./

KO, Sq. Ft./Ft./O F.

A/; 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5.8 59 60

61 B/; 3

4 5 6

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

F;aiLb Hr.

Calculations for Heat Transfer Correlation

J”

0.369 0.369 0.369 0.369 0.369 0.369 0.369 0.3685 0.369 0.3685 0.3685 0.369 0.369 0.369 0.369 0.369 0.3685 0.3695 0.3695 0.3695 0.369 0.369 0.368 0.3685 0.3685 0.3685 0.369 0.3685 0.3685 0.369 0.3685 0.3685 0.3685 0.3695 0.3685 0.369 0.3685 0.3685 0.3685 0.3695 0.3685 0.3685 0.3685 0.368 0.3685 0,3685 0,3685 0.368 0.369 0.366 0.366 0.3695 0.369 0.372 0.372 0.3745 0.3745 0.377 0.3775 0.380 0.380

7.72 7.81 7.85 7.93 8.03 8.15 8.30 8.38 8.54 7.90 8.03 7.85 8.10 8.11 8.19 8.20 8.42 8.58 8.23 8.23 8.38 8.45 7.96 7.90 7.97 8.17 8.20 8.30 8.35 8.34 8.53 8.50 8.56 7.61 7.90 7.99 8.18 8.29 8.44 8.28 7.96 8.01 8.00 8.20 8.21 8.42 8.50 8.69 8.44 8.86 8.66 7.86 8.15 7.28 7.50 6.73 6.56 5,87 5.99 4.95 5.13

0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0,9991 0.9991 0.9991 0.9991 0.9992 0.9991 0.9991 0.9991 0.9991 0 9991 0 9992 0.9991 0.9991 0.9991 0.9990 0.9991 0.9991 0.9991 0.9990 0.9991 0.9988 0.9988 0.9991 0.9991 0.9995 0.9995 1.0000 1.0000 1.0004 1 ,0004 1,0012 1.0012

0.3665 0,3665 0.3665 0.3665 0.3665 0,3665 0.3665 0.3665 0.3685 0,3690 0.3685 0.369 0.369 0.3685 0.369 0.3685 0.371 0.371 0.371 0.371 0.371 0.372 0.371 0.371 0.3735 0,3745 0.3745 0.3745 0,3745 0.3745 0.375 0.375

6.33 6.41 6.43 6.51 6.53 6.70 6.67 6.80 5.82 5.68 5.86 5.77 5,Sl 5.97 5.86 6.05 5.19 5.22 5,26 5.30 5.41 5.41 5.48 5.54 4.80 4.65 4.66 4.73 4.76 4.78 4.79 4.78

0.9988 0,9988 0,9988 0.9988 0,9988 0.9988 0.9988 0.9988 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9991 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9994 0.9994 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001

1693

SERIEB A, 0.25% CARBOXYMETHYLCELLULO~E SOLUTION 68.1 7100 37.10 0.148 705 534.0 68.1 6330 36.60 0.140 634 527.0 68.1 5780 33.00 0.120 580 474.0 68.1 5160 28.90 0.110 520 417.0 68.1 4220 23.60 0.089 438 340.0 68.1 3430 19.25 343 277.0 0.072 68.1 2940 16.40 0.062 288 236.0 68.1 2280 12.50 0.052 180.0 218 1320 68.1 0.028 5.25 75.6 109 68.1 6250 35.00 0.159 626 505.0 68.1 32.90 0.108 5650 566 474.0 68.1 5210 29.00 525 418.0 0.100 68.1 4190 24.00 422 347.0 0.085 68.1 3440 19.05 343 274.5 0.072 68.1 2980 14.60 291 0.062 210.0 68 1 2580 13.75 253 197.0 0.058 2420 13.20 0.056 68.1 236 189.5 2230 11.20 212 161.0 0.050 68.1 1910 119.5 0.040 68.1 8.30 177 1720 155 0.038 68.1 6.90 99.0 1520 112 0.030 68.1 6.85 98.0 1330 4.73 109 0.028 68.1 67.9 113 6210 27.50 0.124 620 540.0 24.85 113 0.112 487.0 5750 577 5140 22.40 113 0.101 517 439.0 4150 18.30 113 0.086 417 359.0 3400 14.40 113 0.072 340 283.0 113 2940 12.55 0.063 288 246.0 2440 113 0.055 195.0 9.95 236 2290 113 0.050 173.5 8.85 218 1840 6.94 113 0.041 169 136.0 113 0.038 124.5 1665 6.35 147 0.030 1315 3.31 113 109 65.0 113 6490 27.70 0.127 544.0 648 5780 25.30 113 581 0.118 496.0 5120 23.40 113 517 0.105 459 I5 4150 19.30 113 417 0.085 370.0 3480 16.20 113 346 0.070 317.5 2420 10.00 113 236 0.052 196.0 1420 113 3.34 120 0.031 65.7 154.4 6200 24.50 0.132 225 560.0 0.112 154.4 5660 20.80 566 495.0 154.4 5110 20.10 0.100 517 478.0 154.4 4140 16.65 443 396.0 0.090 154.4 314.0 3400 13.18 340 0.071 154.4 245 323.0 2520 0.052 9.36 154.4 2250 2 12 184.5 0.050 7.75 154.4 1810 5.50 163 131.0 0.039 154.4 1290 4.82 103 112.0 0.032 68.1 3200 12.55 0.067 316 180.0 68.1 4540 0.090 243.0 16.95 457 0.100 68.1 5030 22.30 509 320.0 68.1 0.070 340 220.0 3400 15.25 68.1 0.108 5480 27.70 550 399,O 68.1 0.082 3850 18.55 359 267.0 68.1 4110 21.90 0.079 413 315.0 68.1 31.50 599 454.0 5960 0 116 68.1 34.50 496.0 6760 0.112 672 68.1 4750 366.0 26.10 0.081 577 68.1 7860 35.10 772 506.0 0.090 68.1 5420 28.30 407.0 0.072 547 SERIES B, 2.5% POLYVINYL ALCOHOL SOLUTION 113 7470 40.90 802.0 0.072 739 113 6840 35.40 0.062 694.0 681 113 6290 30.10 0.060 589.0 630 113 5380 22.00 0.052 542 432.0 113 4550 18.05 0.051 354.0 461 113 3580 15.75 309.0 0.032 357 113 2640 11.95 0.029 234.0 259 113 2130 8.20 0.019 160.8 202 113 8130 42.80 0.079 798 840.0 35.90 113 0.079 763 703.0 7710 6460 28.00 113 0.068 646 573.0 6010 25.90 113 0.067 605 508.0 113 0.058 5360 22.80 539 447.0 4340 113 0.046 16.75 438 328.0 4160 15.80 113 0.046 447 310.0 11.60 113 0.031 227.5 2740 270 113 9610 52.10 0.098 932 1020.0 113 8940 42.70 0.092 872 840.0 113 8150 40.00 0.084 801 785,O 113 7010 35.80 0.072 701.0 697 113 5810 32.10 0.062 582 630.0 113 4710 23.95 0 055 476 469.0 113 3400 17.40 341.5 0 039 340 113 2640 12.15 0.030 238.0 259 113 9960 54.50 0.100 956 1069.0 113 9470 38.60 918 755.0 0.090 113 674.0 8670 34.40 0.081 848 31.90 113 625.0 7410 0.072 735 6250 28.00 113 627 549.0 0.065 113 5020 20.80 0.052 507 408.0 368 347.0 3680 17.72 0.040 113 113 278 2830 15.30 0.030 300.0

J 608.0 591.0 540.0 478.0 396.0 330.0 283.0 220.5 90.0 579.0 535.0 477.0 358.0 325.5 255.5 241.5 232.0 198.0 154.5 133.0 127.0 94.3 630.0 572.0 520.0 433.0 351.0 308.5 255.0 228.0 184.5 170.0 108.0 635.0 584.0 542.0 455.0 383.0 252.0 106.0 696.0 600.0 577.0 490.0 396.0 291.0 251.5 193.5 164.2 228.5 301.0 380.0 268.0 474.0 321.0 365.0 517.0 560.0 429.0 562.0 457.0

588.0 284.0 333.0 392.0 500.0 370.0 325.0 303.5 920.0 372.0 263.0 415.0 314.5 399.0 773.0 488.0 387.0 578.5 1055.0 1301.0 156.3 1413.0 166.2 243.0 227.0 193.0 281.0 213.0 302.0 416.0 381,O 234.0 162.5 262.0 213.0 110.0 82.5 67.8 364.0 2070.0 72.0 156.0 56.6 112.0 58.5 80.7 169.0 384.0 64.5 1850.0 2190.0 1320.0 1255.0 740.0 656.0 561 .O 243.0 686.0 1335.0 1189.0 656.0

872.0 755.0 652.0 508.0 411.0 351.0 271.0 189.5 910.0 775.0 615.0 573.0 506.5 380.5 362.0 268.0 1102.0 916.0 859.0 771.0 693.0 528.0 348.0 278.0 1152.0 834.0 746.0 694.0 614.0 465.0 396.0 339.5

72.5 0.4 42.3 412.0 538.0 180.3 93.0 416.0 25.0 61.7 126.4 256.5 291.0 630.0 645.0 249.0 74.2 13.5 39.9 0.3 67.9 2.2 0.0 65.7 139.1 315.5 421.1 224.0 154.7 380.0 32.5 53.6

212.5 45.9 47.5 77.3 91.9 14.4 3.0 1.3 303.6 56.3 33.5 83.6 230.0 26.0 148.8 20.6 2.9 43.5 161.1 201.3 179.5 181.9 2.6 0.8 0.3 14.7 10.5 50.7 64.7 21.0 84.0 313.0 0.8 4.0 0.3 6.2 82.8 93.1 46.1 136.2 145.0 37.4 107.0 112.0 272.0 353.0 346.0 349.0 3530.0 765.0 1160.0 645.0 448.0 191.0 112.0 132,O 188.0 279.0 655.0 739.0 390.0

314.0 117.5 12.2 39.3 117.3 2.8 18.0 38.3 196.3 2.5 23.0 28.0 36.3 172.0 174.0 0.6 332.0 25.4 52.5 112.2 363.0 119.0 5.5 53.8 418.0 83.6 44.5 31.1 4.3 68.5 50.0 488.0 (Continued on page 1604)


1694

Table IV.

Vol. 45, No. 8

Calculations for Heat Transfer Correlation (Continued)

ho k

k,

B.t.u./Hr./ F8,b. Run No. Sq. Ft./Ft./' F. Hr.

CP, B.t.u./Lb./

F.

B/33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

0.376 0,377 0.377 0.3775 0,3775 0.3775 0,3775 0,3775 0.3795 0.380 0.3795 0 3795 0.380 0.380 0.380 0,380

4.28 4.30 4.26 4.27 4.31 4.44 4.40 4.42 3.82 3.78 3.89 3.88 3.87 3.85 3.87 3.55

1.0003 1.0004 1,0004 1.0005 1,0005 1,0005 1.0005 1.0005 1.0011 1.0012 1.0011 1 0011 1.0012 1.0012 1 .0012 1.0017

C/;

0.3665 0,366 0.367 0.3675 0 3675 0 3675 0 3675 0.3676 0 369 0 3655 0.3670 0 3695 0 370 0,370 0 370 0.370 0 371 0 371 0 371 0 371 0.371 0 372 0 3725 0 3725 0,3745 0.3745 0.3745 0.3745 0.3745 0.375 0.376 0.375

7.76 8.28 8.08 8.34 8.58 8.82 9.00 9.17 6.95 6.59 7.11 7 . .51 7.55 7.77 7.86 8.19 6.30 6.47 6.74 6.55 7.16 7.18 7.13 7.41 5.35 5.52 5.56 5.81 5.89 5 85 5.88 6.06 4.82 4.89 4.90 4.85 5.06 5.16 5.34 5.42 3.72 3.70 3.81 4.01 4.09 4.13 4.01 4.24

0 9988 0 9988 0.9989 0,9990 0,5990 0.9590 0.9590 0 9990 0.9951 0,9592 0.5993 0.5992 0.9953 0.9993 0.9993 0.5593 0,9994 0.9594 0.9594 0.9954 0.9994 0.9955 0,9995 0.9994 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1 0002 1.0001 1.0004 1.0004 1,0004 1,0005 1.0005 1,0005 1.0005 1.0005 1,0012 1,0012 1.0012 1.0011 1.0011 1.0011 1.0012 1.0011

3 4 5 6 7 8 5 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

0.377 0.377 0 377 0.3775 0.3775 0.3775 0.3775 0.3775 0.380 0.380 0.380 0.3795 0.3795 0.3795 0.3805 0.3795

LG

LID

(9)"'"' w =

pa J* SERIESB, 2.5% POLYVISYL ALCOHOLSOLUTION 113 11420 52.90 0,090 1089 968 113 10250 46.40 0.084 0.080 920 113 9500 42.00 36.70 0.068 806 113 8200 687 113 6900 32.00 0.061 545 113 5400 26.50 0.048 403 113 4000 20.30 0,035 16.50 0.030 308 113 3120 113 0.075 1198 12700 57.70 ' 113 1144 65.40 12100 0.070 113 997 50.20 10330 0.068 113 9020 0.058 880 42.00 0.050 757 113 7670 33.30 0.041 6220 65 1 30.80 113 461 0.031 4550 22.00 113 21.20 113 349 0.023 3490 S E R I E SC, 0.25yo SODICMP O L Y M E T H A C R YSOLCTIOS LATE 707 113 7140 29.7 0.365 23.4 0.322 585 113 5840 113 5150 16.15 0,280 519 113 4120 0.230 416 13.35 113 3340 0,190 332 8.0 242 5.1 113 2480 0.150 190 4.8 113 2020 0.126 4.17 136 113 1570 0.100 36.3 0 338 783 113 7960 30.8 0.295 685 113 6900 0 252 24.1 587 113 5840 49.5 0 210 113 4910 20.35 381 0.180 113 3800 14.85 277 7.15 0.148 113 2820 223 5.2 0 118 113 2320 158 5.0 0.092 113 1760 0 312 865 113 8790 36 9 737 0 280 113 7450 31.0 0 241 618 113 6160 27.2 0 209 498 113 4950 22 9 403 0 172 113 4000 18.?259 10.00 0 130 113 3050 250 0 109 113 2560 6.7 182 6.2 0 088 113 1950 995 41.50 0 301 113 10350 851 0.265 35.10 113 8740 736 28.20 0.232 113 7460 593 0.164 27.40 113 5910 489 21.65 0.152 113 4870 375 0.114 14.45 113 3740 304 0.101 8.75 113 3100 113 2380 6.67 0.075 228 1065 53.10 0 240 113 11200 951 45.10 0.225 113 9860 0.188 826 36.60 113 8460 701 0.158 113 7090 31.30 568 0.140 113 5660 27.45 428 20.40 0.101 113 4240 342 0,085 14.75 113 3420 258 8.15 0.070 113 2660 1378 0.180 113 14900 64.50 1222 0.155 113 13030 54.30 1042 49.30 0.142 113 10900 833 43.20 0.118 113 8570 696 0.102 32.00 113 7010 533 0,080 26.45 113 5300 460 0.063 15.40 113 4560 340 15.32 0.052 113 3400 Fa

J"

J

1034.0 910.0 824.0 721.0 627.0 520.0 398.5 323.5 1133.0 1086.0 983.0 822.0 655.0 605.0 431.0 419.0

1160.0 984.0 894.0 788.0 689.0 574.0 442.0 363.0 1205.0 1155.0 lO52,O 886.0 704,O 652.0 472.0 449,O

25.5 59.2 108.7 110.8 76.3 21.9 1.4 25.3 29.5 25.7 2.0 43.4 181.0 50.0 40.9 402.0

5.7 0.0 8.0 5.0 0.1 28.2 93.2 318.5 0.0 1.0 30.4 0.4 56.6 0 0 5,7 889.0

581.0 459.0 316.0 262.0 156.5 100.0 94.2 82.2 711.0 605.0 473.5 308.5 201.5 140.0 102.0 56.0 723.0 608.0 533.0 449.0 365.0 206.5 131.0 121.5 820.0 687.0 552.0 537.0 424.0 283.0 171.5 131.2 1040.0 885.0 696.0 614.0 537.0 419.0 289.0 155.5 1263.0 1062.0 967.0 846.0 627.0 517.0 301.0 300.0

679.0 579.0 435.0 374.0 263.0 198.0 184.0 162.5 834.0 723.0 588.0 :10.0 395,O 236.5 189.0 176.0 843 0 723 0 647.0 559.0 472.0 299.0 216.0 197.0 898.0 804.0 666.0 644.0 524.0 370.0 252.5 201.0

317.0 462.0 1535,O 1369.0 2680.0 3430.0 2620.0 1042.0 84 2 135 5 374 0 380 0 552.0 2460 0 2950 0 1540.0 261.0 304.0 189,O 97.0 71.2 956.0 2260.0 1140.0 309.5 366 0 625.0 89.1 176.8 600.0 1859.0 1650.0 5.5 48.1 248.0 153.8 29.7 5.4 240.3 1453 . 0 69.6 171.3 51.8 0.9 98.0

15.6 1.1 262.2

1198.0 1038.0 842.0 714.0 632.0 481.0 363.0 227.0 1371.0 1168.0 1062.0 940.0 108.0 606.0 365.0 357.0

9.0

1194.0 138.5

100.2

428.0 330.0 10.0 380.0 42.6 27.5 0.0 9.2 13.5 121.5 232.5 130.0 6.5 3.6 20.0 150.0 292.0 0.0 246.0 60.0 125.0 30 4 90.2 74 0 45.5 1.8

28.8 14.8 155.0 70.5 3.7 3.4 126.5 151.5 27.7 144.0 0.3 19.4 3.5

149.0 3.0 187.5 425.0 25.0

Nomenclature for Table IV was read from graph of data of Morris and Whitman at corresponding Reynolds number.

Reynolds number than a t the higher values, and that there is a general trend for the polyvinyl alcohol data t o be higher than the carboxy~thylcellulose data, and the carboxymethylcellulose data t o be higher than the sodium polymethacrylate data. This trend is in the same order as the relative degrees of pseudoplasticities of the three materials. Because of the variation of pseudo-plastic behavior with temperature, i t cannot be anticipated that the groups of points for the different materials should fall onto separate and distinct

curves. Kevertheless, it appears that use of the apparent viscosity in the Reynolds and Prandtl groups does not completely compensate for the effect of pseudo-plasticity. It was therefore necessary to develop some exact measure or coefficient of pseudoplasticit'y which could be used to modify the functional behavior of the quantity J" in such a way that the three groups of points shown in Figure 6 were superimposed. A careful examination of several possible methods of defining a coefficient of pseudoplasticity directed the choice t o the form given by Equation 5.


August 1953

1695

LEX

DATA FOR P V A tr

PI

X

CMC

2000

THE CURVE F I T T I N G

F U N C T I O N VERSUS APPARENT REYNOLDS NUMBER FOR THE

I 0

AUTHORS'

100

APPARENT REYNOLDSNUMBER

11

I1

CMC

''

I'

SPM

DATA

i I I I 2000 4000 6000 8000 10000 12000 APPARENT REYNOLDS N U M B E R , Re,

I

14020.. 1-

1400

I200 THE FULL LINE IS THE CU RVE: F IT T INC loo0

MORRIS A N D W H I T M

800 J

600

400

200

APPARENT

APPARENT REYNOLDS NUMBER FOR THE AUTHORS' DATA 8000 loo03 12000 14000 16000 REYNOLDS NUMBER k q

1

A function of the coefficient of pseudo-plasticity, as defined by Equation 5, was added t o the J" function, and the resulting function, J , was plotted against apparent Reynolds number (Figure 7 ) . As the pseudo-plastic characteristics diminish to zero, so does the coefficient of pseudo-plasticity, as defined by Equation 5 , diminish to zero. It was essential that the correction applied t o the J" function in terms of the coefficient of pseudo-plasticity should also approach zero as the fluid tended toward simple Newtonian behavior. The coordinates of Figure 7, which satisfy this requirement, are:

'

us.

(Z)

Figure 7 shows similar scatter at higher values of the apparent Reynolds number, as does Figure 6, but the scatter a t lower values of the apparent Reynolds number is much improved, so that it is no longer possible t o separate the data points into groups according to material. On the assumption t h a t the curves for different ratios of tube length to diameter could be brought into coincidence by multiplication of the J" function by the rati6 of the tube length to the tube diameter raised to some exponent, three equations resulted from measurements of the average logarithmic differences between the ordinate values for the three curves. These equations solved t o give an average value of 0.33 for this exponent. This is in good agreement with the findings of some previous workers (11,%E), but when the data of Winding (26) were also taken into consideration i t was found that a value of 0.63 was t o be preferred for the exponent if both the data of Winding and those of the authors were to be represented by a single correlation. The additional merit of the choice is that it brought both Winding's

100 -

APPARENT REYNOLm NUMBEX

data and the present data onto substantially the same curve as that fitting Morris and Whitman's data. The root mean deviation of the data points on Figure 7 from the Morris and Whitman curve is 22.2%. This deviation is reduced in the plot shown in Figure 8 t o 12.3y0 as a result of applying the correction factor in terms of the coefficient of pseudoplasticity. The equation representing the curve in Figure 8 is found t o be: (J 100) = 0.56 Re282 (9)

+

Figure 9 shows Winding's data and also those of Morris and Whitman, Series E, plotted on the same coordinates as those of Figure 8. The straight line on Figure 9 is that defined by Equation 9, and it shows that the relation thus defined is satisfactory for the correlation of the data of Winding and those of Morris and Whitman. Thus, it is shown that the data for the cooling of pseudo-plastic fluids may be correlated for (1) the Reynolds number range 1200 t o 21,000; (2) the viscosity range 1 to 13 centipoises; and (3) the coefficient of pseudo-plasticity range 0.0 t o 1.26. Three data points are given by Winding (26) for the heat transfer coefficients for the heating of latex. These three points all tend to lie above the rest of the data points, at equivalent apparent Reynolds numbers. This is in accordance with the difference between heating and cooling data points for simple Newtonian fluids when the correction factor (11, 16,29) in terms of the ratio of the fluid viscosity a t the average fluid temperature to the fluid viscosity at the tube wall temperature is omitted. It therefore appears that a similar correction would suffice t o bring the heating and cooling data to-

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

1686

gether for data on the transfer coefficients for pseudo-plastic fluids. Unfortunately, the existing data are insufficient t o enable a quantitative estimate of such a correction. I n conclusion, i t should be possible to extend the present correlation to include data for general non-Newtonian liquids. For these, Hedstrom (9) has defined the apparent viscosity

Ti

Acknowledgment

Grateful acknowledgment is made t o C. C. Winding, Cornell University, who gave his valuable advice and heat transfer data on GR-S latex; t o the PB Exchange Program of the Marshall Plan which made this study financially possible; and to E. I. du Pont de Nemours& Co., Hercules Power Co., and Rubber Reserve Corp. for their kindness in providing samples for the experimental work. The authors are indebted to D. F. Othmer, head of the Chemical Engineering Department, Polytechnic Inst,itute of Brooklyn, for his kind interest and encouragement, and to Eli Perry, Monsanto Chemical Co., and B. 0. Harris, now Chemical Co., for their valuable comments in reviewing the manuscript. Nomenclature

Ai = inside area of tube wall per unit length of tube, square feet a = a constant in the Williamson and Winding equations, see.-‘

cp

= specifi; heat of fluid a t constant pressure, B.t.u. per pound

D

= diameter of tube, feet

G

= mass velocity, pounds per square foot per hour

J”

=

J‘

=

J

=

per

F.

acceleration of gravity, centimeters per second per second R =-- film coefficient of heat transfer based on logarithmic mean temperature difference, B.t.u./hour/sq. ft./’ F.

(Y )($)-’”’ (

for Morris and Whitman data = Colburn‘ heat transfer factor k = thermal conductivity, B.t.u./hour/sq. ft./ft./’ F. L = length of heat transfer, feet M = intercept of consistency curve of generalized h’ewtonian fluid on R, coordinate, grams/cm./sec.z N = length of tube, feet P = pressure differential along tube, em. of mercury Pr = Prandtl number = quantity of heat transferred per unit time, B.t.u. per hour q = heat gained by fluid flowing through shell, B.t.u. per hour q. pt = heat lost by fluid flowing through tube, B.t.u. per hour R = inside radius of tube, feet or centimeters as indicated in text Re = Reynolds number Rea = apparent Reynolds numbers 4v R, = consistency variable rate of shear a t tube aR3 wall, sec.-1 r = inside radius of cylindrical lamina or distance from axis of tube, feet Sg = consistency variable shearing stress a t tube wall, grams/(cm.)(sec.)* S,g = a constant in Williamson and Winding equations, grams/ (em.)( J*

j

(= -)

(g!),

’ C.

T o = outlet temperature of shell-side fluid, AT = To- Ti, C. Ah - Ah, c. Atm =

O

C.

~

Ah Ah

In inlet temperature of tube-side fluid, ’ C. outlet temperature of tube-side @id, ’ C. t, average tube-wall temperature, C. 51 ti - t o or temperature difference across log mean temperature difference, fluid film, O C. A t , At2 = terminal temperature difference, ’ F. li = volume rate of flow W , = mass rate of flow through shell side, pounds per minute W t = mass rate of flow through tube side, pounds per minute y, y‘ = coefficients of pseudo-plasticity p = density, pounds per cubic foot 4 = volume fraction of solids in slurry pa = coefficient of apparent viscosity ( -Sg/RI) poise ps = viscosity of slurry, poises PLW= viscosity of simple Xewtonian fluid a t tube-wall temperature, poises pu = viscosity of water, poise PO = limiting value of Sg/R8as R, approaches zero-i.e., apparent viscosity a t zero rate of shear l a = limiting value of SglRS as R, approaches infinity. Coefficient of viscosity a t infinite rate of shear, also a constant in the Rilliamson and Winding equations 1,

to

where M is the intercept of the consistency curve of the generalized Newtonian fluid on the R. =O axis. The studies of MacLaren and Stair (IS) show that Filter-Cel slurries tend to behave as general non-Newtonian fluids, but in this case the deviation from the ideal plastic behavior was small and the viscosity data they quote are not adequate to enable an examination of their heat transfer data in terms of the present correlation.

= inlet temperature of shell-side fluid,

Vol. 45, No. 8

= = = =

Literature Cited

Andrade, E. N. da C., ”Viscosity and Plasticity,” p. 10, New York Chemical Publishing Co., 1951. Badger, W. L., and RIcCabe, W.L., “Elements of Chemical Engineering,” New York, McGraw-Hill Book Co., 1950. Baker, C. P., Ph.D. thesis in chemical engineering, Cornel1 University, 1950. Bonilla. C. F., personal communication, 1952. Brown, F., and Burridge, K. G., chemical engineering thesis, Polytechnic Institute of Brooklyn, 1952. Colburn, A. P., Trans. Am. Inst. Chem. Engrs., 29, 174-210 (1938). Dittus, F. W., and Boelter, L. XI. K., Univ. Calif. Publs. Eng., 2 , 443 (1930). Hatschek, E., Kolloid-Z., 27, 163 (1920). Hedstrom, B. 0. A , ISD. ENG.CHEW,44, 651-6 (1952). Hoopes, J. W., et al., Ch.E. S-111 Report, Columbia University, 1945. Kern, D. Q., and Van Sostrand, W.,IKD.ENG.CHEM.,41, 2209-12 (1949). McAdams, W. H., “Heat Transmission,” p, 187, New York, McGraw-Hill Book Co., 1933. MacLaren, D. D., and Stair, R. G., b1.S. thesis in chemical engineering, 1948. Morris, F. H., and Whitman, W. G., IND.ENG. C m x r . , 20, 234-40 (1928). Norris, R. H., and Sims, S . W., Trans. Am. Inst. Chem. Engrs., 38, 469-90 (1942). Othmer, D. F., and Kern, D. Q., Ibid., 39, 517-51 (1943). Perry, E., and Reese,F.E., Chem.Eng. Progr., 47,354-62 (1951). Perry, J. H., editor, “Chemical Engineers’ Handbook,” 3rd ed., New York, McGraw-Hill Book Co., 1950. Porter, A. W., and Rao, P. A. M., Trans. Faradau SOC.,23, 311 (1927). Reiner, M., “Deformation and Flow,”.London, H. K. Lewis & Co., Ltd., 1949. Reiner, M., “Twelve Lectures on Theoretical Rheology,” New York, Interscience Publishing Co., 1949. Sieder, E. N., and Tate, G. E., IND. ENG.CHEM.,28, 1429-35 (1936). Walker, W. H., Lewis, W. K., McAdams, W. H., and Gilliland, E. R., “Principles of Chemical Engineering,” pp. 109 ff., Kew York, McGraw-Hill Book Co., 1937. Williamson, R. V., IND.ESG. CEEM.,21, 1108 (1929). Winding, C. C., Baumhnn, G. P., and Kranich, W. L., Chem. Eng. Progr., 43, 527-36 (1947). Winding, C. C., Dittman, I?. W., and Kranich, W. L., “Thermal Properties of Synthetic Rubber Latices,” Report to Rubber Reserve, 1944. Winding, C. C., Kranich, W. L., and Baumann, G. P., Chem. Eng. Progr., 43, 813-22 (1947). RECEIVED for review September 16, 1952. ACCEPTED March 24, 1953. Presented before the Division of Industrial and Engineering Chemistry rtt the 122nd Meeting of the AMERICAN CHEhiIICAL SOCIETY,Atlantic City, N. J.

Heat Transfer Coefficients of Pseudo-Plastic Fluids

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