1654
Ind. Eng. Chem. Res. 1987,26, 1654-1658
Heat Transfer to Concentrated Suspensions in Agitated Systems Dominic M. Kungt and Peter Harriott* School of Chemical Engineering, Cornell University, Ithaca, New York 14853
Heat-transfer coefficients were measured for suspensions of fine quartz sand or spherical polymer particles in a jacketed, baffled kettle equipped with a cooling coil and a standard turbine, propeller, or pitched-turbine agitator. For the polymer suspensions, the wall-film coefficients decreased gradually with increased solids concentration to about 0.4 times the original value a t 40% solids. T h e fractional decrease was independent of agitator type and stirrer speed and agreed within 10% with that predicted from the physical properties of the suspensions. The outside film coefficients for the cooling coil also decreased with increasing concentration of polymer, but the decrease was not as much as for the wall film. The jacket and coil coefficients for sand suspensions were greater than for polymer suspensions but less than values predicted from the properties of the sand suspensions. In many processes such as suspension polymerization, crystallization, coal liquefaction, and catalytic hydrogenation, heat is transferred to or from a suspension of particles in an agitated vessel. The suspensions may have from 1% to as much as 50 vol % particles and be Newtonian or non-Newtonian in character. There have been several studies of heat transfer to suspensions in stirred tanks, and various empirical correlations have been proposed. However, most of the experiments have been carried out with dilute suspensions (less than 15% solids), where the solids have only a small effect on the heattransfer coefficient. A few of these studies are reviewed to illustrate the lack of agreement on the proper form for the heat-transfer correlation. Heat-transfer coefficients for water slurries of sand or polymer particles in a jacketed tank with a propeller stirrer were measured by Frantisak et al. (1968). They used predicted values of the thermal conductivity, viscosity, density, and heat capacity of the slurries in evaluating their data, but their final dimensionless correlation still had three more terms than the equation for simple fluids. In addition to its effect on the physical properties of the suspension, the volume fraction c $was ~ included as a term (r&/l- 4,)-0.04. Over the range studied, 4, = 0.017-0.11, this term contributes up to a 15% decrease in Nusselt number. In a similar study with CaC03-H20 slurries and a turbine or anchor agitator, Martone and Sandall (1971) fitted their data to the equation for simple fluids with only one extra form, ( & / l -4v)0.065. The maximum value of 4" was 0.11 with the turbine agitator, so the difference between their results and those of Frantisak is not large, in spite of the different signs for the exponent. Martone and Sandall determined the effective viscosity from viscometer data and power measurements for the impeller in order to allow for the shear-thinning behavior of the slurry. However, the average shear rate determined from power measurements may not be the correct value to use for heat transfer to the wall, and incorrect viscosity values could lead to a different exponent for some other term. Moderately concentrated suspensions of CaCO, in ) investigated by ethylene glycol (25 and 44 ~ 0 1 % were Bourne et al. (1981). With an anchor agitator, the data for process-side heat-transfer coefficients agreed well with the previously determined correlation when the physical properties of the suspension were used. No extra term for 'Presently at Union Carbide Corporation, Somerset,NJ 08873.
the volume fraction solids were included. However, with a Pfaudler impeller, the measured coefficients were lower than predicted at low Reynolds numbers. The authors pointed out that the values of shear rate and effective viscosity are different near the wall than near a central impeller, and they suggested that the apparent shear rate would be a function of vessel size as well as stirrer speed. Most workers have used Maxwell's (1881) equation to predict the thermal conductivity of the suspensions
k , - kp/kf + 2 - 24J1 - k P / M _ & kp/kf + 2 + - hP/M
(1)
However, this equation was derived for stationary particles, and particles rotating in a shear field will cause microconvection currents that enhance the rate of heat transfer. The enhancement depends on the volume fraction particles and a Peclet number based on the particle size and rate of shear
k,
-=
1 + b4,Pe"
k8,O
where Pe = ya2//cu. The exponent for the Peclet number was predicted by Leal (1973) t o be 3 / 2 for dilute suspensions and Pe > 1. Using a different approach for concentrated suspensions, Sohn and Chen (1981) derived an exponent of for Pe. Their measurements in a Couette flow apparatus give an exponent of about 1/2, with values of the enhancement factor, k,/k,,o, as high as 5 for 4v = 0.30. Although the enhancement factor increased with @,, as predicted, the data for a 15% suspension of polystyrene particles did not agree with those for a 15% suspension of polyethylene particles. The pronounced increase in apparent thermal conductivity reported by Sohn and Chen is not realized for heat transfer to suspensions in stirred tanks, even though the Peclet numbers based on the average shear rate are probably comparable. For example, a 5-fold increase in thermal conductivity at 4, = 0.30 would increase the heat-transfer coefficient by a factor of 52/3or 2.9, more than enough to offset the effects of an increase in viscosity and a decrease in heat capacity of the suspension. The coefficients predicted by using suspension properties in a standard correlation plus Sohn and Chen's enhancement factor for k , would increase with particle concentration at first, go through a maximum, and then decline as the effect of high suspension viscosity became dominant. In the
0888-5885/87/2626-1654$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1655 Table I. Properties of Materials Used
polystyrene beads RH 1911 ROMAXL-195 quartz sand Agsco No10
."
1
d,, wn
k,,
C,
P,
J/(m s K)
J/(kg K)
kg/m3
190 680
0.084 0.084
1340 1340
1050 1050
45
1.73
710
2650
-o-
Polymer Surpcnsions
-a-
Sand
_...V i s c o s i t y
I
,
, , I
Range, Thomas I
,,
7 t
I
Figure 1. Jacketed tank with cooling coil.
references cited here, the raw data are not presented to permit a direct check of this prediction, but in general the coefficients appear to decrease as I#Jv is increased or else be nearly independent of &. Furthermore, correlations based on the stationary properties of the suspension work fairly well, indicating any additional effect of I#Jv is not large. However, there should be some effect of particle motion or rotation in the thermal boundary layer, and experiments with different particle sizes, concentrations, and agitation conditions might clarify the situation. In this study heat-transfer Coefficients are reported for two sizes of polystyrene beads suspended in water in a stirred tank. Polymer beads were chosen to minimize the difference between particle and fluid densities and permit nearly uniform suspensions over a wide range of agitation conditions. The lower thermal conductivity and heat capacity and higher viscosity of the suspension all act in the same direction to decrease the heat-transfer coefficient. From a practical standpoint, the concentrated suspensions are similar to those encountered in suspension polymerization. Some tests were made with sand in water, a system where the higher thermal conductivity of the suspension tends to offset the lower heat capacity and higher viscosity. For both systems, film coefficients were obtained for the vessel wall, which was heated by steam, and for a cooling coil surrounding the impeller.
Apparatus and Procedure The heat-transfer tests were carried out in an open stainless steel tank 15 in. (0.38 m) in diameter and 18 in. tall, which is shown in Figure 1. The vessel bottom was flat and slightly sloped for drainage. The steam jacket was made of five half-tubes that nearly encircled the tank and were welded to the outer wall. The tubes were connected to vertical inlet and exit headers, and the jacket heattransfer area was 2.19 ft2 (0.0024 m2). The vessel also had a helical cooling coil 7.5 in. in diameter with eight turns of 'l2-in. copper tubing to give an area of 2.16 ft2 (0.0023 m2). The coolant was a blend of hot and cold water, and the flow was measured with a rotameter. The tank had a centrally mounted agitator driven by a 1/3-hpac motor with a speed control. The impellers used included a 4-in.-diameter, six-blade flat-blade turbine, a 4-in. 45O pitched blade turbine, and a 3.8-in. marine-type propeller. There were four radial baffles 2 in. wide with 'Is-in. clearance at the wall. Thermocouples were used to measure the temperature of the inlet and exit cooling
0
" 0
"
"
IO
"
"
'
20
"
'
30
40
50
60
70
0.. Figure 2. Relative viscosity of suspensions.
water, the steam and condensate, and the water in the tank. There were no measurable temperature gradients in the tank when the stirrer was operating. Runs were made at steady state, with the heat supplied from steam in the jacket balanced by the heat removed by water in the coil. The tank temperature could be changed by adjusting the steam pressure or by changing the the temperature or flow rate of the cooling water. Most runs were made at about 60 "C.For each agitator preliminary tests were made with just water at different stirrer speeds and different coolant flows. Then a small amount of solids was added, and the runs were repeated. As more solid was added, some water was removed to keep the liquid level at 15 in. Runs with no solids were repeated at intervals to check for fouling of the heat-transfer surfaces. The solids used were low density polystyrene beads supplied by Rohm and Haas and quartz sand from Agsco. The average sizes and physical properties are given in Table I. The suspension viscosities were measured using a Brookfield Synchrolectric viscometer with an ultralow adapter. The relative viscosities were independent of shear rate for volumetric concentrations up to about 20% and decreased with shear for higher concentrations. The viscosities measured at the highest shear rate of 73 s-l (60 rpm) are given in Figure 2. The relative viscosities for suspensions of sand fall within the range reported by Thomas (1966), and the highest value is 5.5 at I#Jv = 0.40. The viscosities of polymer suspensions are about 20% lower, probably because of the difference in particle shape.
Results for Cooling Coil The overall heat-transfer coefficients for the coil, U,, were calculated from the heat gained by the water in the coil, which was generally within 5% of the heat given up by the condensing steam. The individual coefficients for the outside film, h,, were determined from Wilson plots, linear graphs of 1/U, vs. l/uo.8,where u is the cooling water velocity. A typical graph is shown in Figure 3, which has results for the turbine impeller and several concentrations of polymer beads at 540 rpm. The intercept for each case
1656 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 0.7
U
0.1
1
predicted'
5401650
5401190 4w 190
r
"
I,
0 0
0 2
04
0
0 6
R
1108
Figure 3. Wilson plots for cooling coil. Turbine a t 540 rpm. I
IO' I ,,-'
Y
N
E
,. .
s
predicted
..
2
11 100
4
10 2
5
6
7
n
vlO1
N,KPM Figure 4. Outside coil coefficients for turbine, propeller, and pitched turbine for water and a 40% suspension of polymer beads.
is the external film resistance plus the resistance of the metal wall or l/hc x/k,. The lines were drawn parallel, since the slope should depend only on the properties of the cooling water in the coil. The slopes for individual runs were calculated by least squares, and an average slope was used for the plots. For the runs used to determine h,, the average tank temperature was about 60 "C, and the average cooling water temperature was about 30 "C. However, as the coolant flow was increased, the exit water temperature decreased, and the tank temperature also decreased. Although these temperature changes were usually less than 5 "C, they were linked to changes in flow rate, and direct use of these data would have led to errors in the slopes and significant changes in the intercepts of the Wilson plots. Therefore, all the data were corrected to a tank temperature of 60 "C and a coolant temperature of 30 "C by using published correlations and the estimated fraction of the resistance in each film. As a check on the Wilson plots, the coefficients with just water in the tank were compared with predicted values. As shown in Figure 4, the outside coil coefficients with the turbine were about 10% less than those calculated from the modified correlation of Oldshue and Gretton (1954).
+
The exponent of 0.24 on the viscosity ratio was chosen to be consistent with other correlations and replaces the variable exponent proposed by Oldshue and Gretton. The slightly lower coefficients found in this work could be due to uncertainty in the viscosity exponent in eq 3 or to the
1'1
?n
Jn
50
50
9,
Figure 6. Relative coil coefficients for propeller.
wider spacing between the coils. The inside coil coefficient was obtained from the slope of the Wilson plots. For a velocity of 2 m/s at 30 "C, hi was 9.1 kW/(m2 K), which is 1.15 times the value predicted for straight pipe by using the Colburn equation (1933). This is in reasonable agreement with the factors of 1.18-1.24 predicted by using the correlation of Mori and Nakayama (1967). The coil coefficients for the three impellers are compared in Figure 4 for just water in the tank and for a 40% suspension of 680-km polymer beads. The coefficients increase with about 0.6 power of the stirrer speed, but the data do not cover a wide enough range for accurate determination of this exponent. For water, the coefficients for the pitched turbine a t the same speed are about 0.85 times and those for the propeller 0.70 times the values for the turbine. At 40% solids, the corresponding ratios are 0.93 and 0.77. Correcting to the same power input using power numbers of Rushton et al. (1950),the three stirrers give about the same coefficients for water, but for concentrated suspensions, the propeller and pitched turbine have slightly higher coefficients than the standard turbine. The effect of solids concentration on the film coefficient for different stirrers is shown in Figures 5-7. The values of h,: the coefficient for water, were taken from the straight lines in Figure 4. The predicted curves are based on the physical properties of the suspensions and published correlations for heat transfer to the jacket, which generally have the form
5)
0.14
Nu = 0Re0.67Pr0.33(
(4)
The difference between and (eq 3) is not significant, and it was convenient to use the same form of the
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1657 1.4
_i___l
1.2
.
f
1.0
5
0.8
z
0.6
E
e
.c
2
0
o,*/: m
";"
.*
4001680
,'
predicted, polymer
,
,
,
4001190 sand
,
,
,
?+
,
9, 0
0.4
4%
$,
0.0 0
10
20
30
40
50
=
volume 9% solids
0 0
0.2
equation for both coil and jacket calculations. If the change in the term (p/pW)O.l4is neglected, the ratio of coefficients depends on four physical property ratios.
8% 16%
25% 0
0" Figure 7. Relative coil coefficients for pitched turbine.
0%
40%
I
0.0
3
1
0
~ G G / N O ' 67
Figure 8. Wilson plot for jacket, Propeller agitator and polymer suspensions. Table 11. Jacket Coefficients for Water at 60 "C ~
The suspension conductivities were calculated from eq 1, and the experimental viscosities were used (Figure 2). The heat capacities and densities of the suspensions were calculated directly. For the polymer suspensions, the thermal conductivity, heat capacity, and viscosity terms all make h, decrease as & increases, and the density term is too close to 1.0 to have a significant effect. The data show a pronounced decrease in h,Jh: with but not as much as predicted. For the standard turbine, the average h,/h: is 0.50 at = 0.40 compared to a predicted value of 0.40. For the pitched blade turbine, the corresponding value of h,/h,O is 0.54, and for the propeller it is 0.59. There is a tendency to lower values of h,/h,O at the lowest stirrer speed, but the difference may not be significant. Because of particle rotation, the coefficients should increase with particle size for the same volumetric concentration. However, there was hardly any difference in h, for suspensions of 680-pm beads compared to 190-pm beads. For the most concentrated suspension, = 0.40, the coefficients for the larger beads were, on the average, only 1.02 times those for the small beads. The suspensions of quartz sand were expected to show only a slight decrease in h, with increasing concentration, since the thermal conductivity and density terms tend to offset the effects of lower heat capacity and higher viscosity. As Figure 7 shows, the values of h, were 5 1 0 % less than predicted in contrast to the higher than predicted coefficients for the polymer beads. Thus, no simple modification of eq 5 would fit both sets of results. The difference may be a shape factor, since the sand was angular and the polymer beads spherical. In a concentrated suspension, the spheres could rotate more easily to enhance heat transfer in the boundary layer.
Results for Jacket The overall coefficients for the jacket, Uj, were calculated from the steam condensate rate and the temperature difference between the inlet steam and the fluid in the tank. The film coefficients on the tank side, h., were obtained by a variation of the Wilson plot, graphs of 1/Uj vs. l/fl.67, which were extrapolated to infinite stirrer speed. Since the intercept is the resistance of the wall and the steam film, it should be about the same for all runs. The intercept was determined by least squares for each
stirrer turbine turbine propeller propeller pitched blade turbine
rpm
540 1000
540 1000
540 1000
" Brooks and Su, 1959.
measd 4.09 6.36 2.56 3.75 3.61 5.65
pred 6.32" 9.53 4.10b 6.18
measd/pred 0.65 0.67 0.62 0.61
Strek et al., 1965.
run, and the intercepts were averaged to give a value of 0.27 (m2K)/kW. This value was then subtracted from the 1J Uj value for each run to get l / h j . This procedure may have a small inherent error, since the steam film coefficient probably varies-with the heat flux. However, there are no data for the condensing coefficients in the half-tube jacket, and the lines did give about the same intercept, as shown by the typical plots in Figure 8. The tests to determine hj covered a range of stirrer speeds from 300 to 1200 rpm. The tank temperature was generally kept at 60 " C by adjusting the steam pressure or cooling water temperature. This was easy to do since increasing stirrer speed increased both U, and Uj. A few runs at 55 "C were corrected to 60 "C. With only water in the tank, the jacket coefficient increased with 0.6-0.7 power of the stirrer speed. Typical values taken from smoothed plots of the data are given in Table XI. The coefficient for the pitched blade turbine is 0.89 times that for the standard turbine and that for the propeller is 0.6 times that for the turbine. The values of hj0 for the turbine and propeller are 30-40% below those predicted from the work of Brooks and Su (1959),whereas the values of h,O are in reasonable agreement with expected values. The low coefficients for the jacket may be due to the shape of the jacket sections, since condensate collecting at the bottom of each half-pipe could significantly reduce the effective area for heat transfer. The effect of solids concentration on the jacket coefficient is shown in Figures 9 and 10. The hjJhyratios were averaged for several stirrer speeds, since there was no apparent effect of N on this ratio. The results for polymer beads fall quite close to the predicted curve for all three stirrers. The jacket coefficients for the 680-pm beads are generally a few percent higher than for 190-pm beads, as was the case for the coil coefficients. The coefficients for sand suspensions are 10-2070 lower than predicted, as shown in Figure 10, and the ratio hjlhj"
1658 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 I .o \
predicted from the physical property ratios. \9
A I
.
\
1 r-
..-
-
0.4
redicted
A 0
-.
fi8Qualpplun
turbine propeller
m
0 A
&
I so
I
0.0
0
IO
20
30
40
9,
Figure 9. Relative jacket coefficients for turbine and propeller. 1.0
0.8
o
\
\
.
m
\
\
\ \
o\
e
e
0.6
.
s-
k.
s-
0.4
. f
/
o
m
0.2
=
\.< o
-.__
Greek Symbols a = thermal diffusivity = constant in eq 4 = shear rate p = fluid density p = viscosity
i predicted for polymer
dp
680, polymer 190,polymer
+
45,~ad
0.0 0
IO
30
20
Nomenclature a = characteristic length b = constant in eq 2 c = heat capacity D, = agitator diameter D, = coil diameter h, = film coefficient outside of coil h,O = film coefficient with no solids hi = inside coefficient for coil hj, h: = jacket film coefficient k , kf = thermal conductivity of fluid k , = thermal conductivity of metal k , = thermal conductivity of particle k , = thermal conductivity of suspension ks0 = thermal conductivity of suspension at zero shear n = exponent in eq 2 N = agitator speed Pe = Peclet number, + a 2 / a Pr = Prandtl number, c p l k Re = Reynolds number, pND:/p T = tank diameter U , = overall coefficient for coil Uj = overall coefficient for jacket u = water velocity in coil x = wall thickness
40
50
pLw=
viscosity at the wall
= volume fraction dispersed phase
Q" Figure 10. Relative jacket coefficient for pitched turbine.
Literature Cited
is less than the corresponding ratio for the cooling coil.
Bourne, J. R.; Buerli, M.; Regenass, W. Chem. Eng. Sci. 1981, 36, 782. Brooks, G.; Su, G. J. Chem. Eng. Prog. 1959, 55 (lo), 54. Colburn, A. P. Trans. AIChE 1933,29,174. Frantisak, F.;Smith, J. W.; Dohnal, J. Znd. Eng. Chem. Process Des. Deu. 1968, 7, 188. Leal, L. G. Chem. Eng. Commun. 1973,1, 21. Martone, J. A.; Sandall, 0. C. Ind. Eng. Chem. Process Des. Dev.
Summary From a practical standpoint, the film coefficients for a cooling coil in a solid suspension can be predicted with reasonable accuracy by using the data for liquids and the physical property ratios for the suspension as given in eq 5. The coefficients for polystyrene suspensions (and similar polymers) could also be predicted by using the straight lines in Figures 5-7. turbine
h,/h: = 1 - 1.254,
(6)
propeller
h,/h; = 1 - 1.034,
(7)
pitched turbine
h,/h: = 1 - 1.15&
(8)
There is only a small if any effect of particle size in the range 200-700 pm. The film coefficients at the tank wall decrease somewhat more rapidly with solids concentrations than the coil coefficients but are still reasonably close to the values
1971, 10, 86.
Maxwell, J. C. A Treatise on Electricity and Magnetism, 3rd ed.; Clarendon: London, 1881; Vol. 1. Mori, Y.; Nakayama, W. Int. J . Heat Mass Transfer 1967, 10, 37. Nir, N.; Acrivos, A. J. Fluid Mech. 1976, 78, 33. Oldshue, J. Y.; Gretton, A. T. Chem. Eng. Prog. 1954, 50, 615. Rushton, J. H.; Costich, E. W.; Everett, H. J. Chem. Eng. Prog. 1950, 46, 395, 467. Sohn, C. W.; Chen, M. M. J. Heat Transfer 1981, 103, 47. Strek, F.; Masiuk, S.; Gawor, G.; Jagiello, R. Znt. Chem. Eng. 1965, 5 , 695. Thomas, D. G. J. Colloid Sci. 1966,20, 267.
Received for review December 22, 1986 Revised manuscript received April 20, 1987 Accepted April 27, 1987
