Ind. Eng. Chem. Res. 1992,31, 1384-1391
1384
Chroamtogr. Commun. 1985,8,684. Pedersen, K.; Thomassen, P.; Fredenslund, Aa. Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 1. Phase Envelope Calculations by Use of the Soave-Redlich-Kwong Equation of State Ind. Eng. Chem. Process Des. Dev. 1984,23, 163. Pedersen, K.S.;Thomassen, P.; Fredenslund, Aa. Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 3. Efficient Flash Calculation Procedures Using the SRK Equation of State. Ind. Eng. Chem. Process Des. Dev. 1985,24, 948. Pedersen, K. S.; Fredenslund, Aa.; Thomassen, P. Properties of Oils
and Natural Gases; Gulf: 1989a. Pedersen, K. S.; Thomaasen, P.; Fredenslund, Aa. Characterization of Gas Condensate Mixtures. Adv. Thermodyn. 1989b,1 , 137. Peneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982,8,7. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27,1197. Received for review July 17,1991 Revised manuscript received November 11, 1991 Accepted February 19,1992
Local Heat Transfer in a Mixing Vessel Using Heat Flux Sensors Seungjoo Haam and Robert S . Brodkey* Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210-1180
J u l i a n B. Fasanot Chemineer Inc., P.O. Box 1123, Dayton, Ohio 45401
A rapid, computer based, local heat-transfer sensor system has been developed. The concept was to provide a quick, accurate, and efficient technique for the determination of process side heat-transfer coefficient models through the use of heat flux sensors and PC data acquisition. The system has been thoroughly tested. System noise and typical sensor variation with time have been studied; however, only part of the noise is real: the remainder is associated and correlatable with specific physical phenomena (vortical motions) that occur in the mixing vessel. A detailed study of the variation of the local coefficients as a function of angular position relative to the baffles is presented. The influence due to the presence of baffles can be seen and explained. Some typical results are given to illustrate the effect of the primary variable, Reynolds number. The relative level of effects from other design variables are presented. Introduction Heat transfer in agitated vessels is a common operation in chemical processing. The rate of heat transfer during mixing of liquids depends on the type of the agitator, the design of the vessel, and processing conditions. In this research, the heat transfer at the wall of an agitated vessel with a dished bottom and four baffles was investigated. Three well-known impeller types were studied. The contents (water) of the vessel were slowly cooled by natural convection from the outer surface and small local cooling exchangers located directly under the sensors. The heat-transfer measurements were made by Micro-Foil heat flux sensors that are placed on the inside surface of the vessel. These sensors exhibit a potential directly proportional to the heat flux and are calibrated by the manufacturer. Thus, process side heat-transfer coefficients were determined. Determining heat-transfer Coefficients by conventional thermocouple and heat balance technique is arduous and time consuming. The use of the thermophile style heat flux sensor to measure the heat flux and temperature of the wall would be a useful and a viable means of making point to point local measurements. For data acquisition, MetraByte's DAS-8 data acquisition system was used. Two BASIC programs were developed for data acquisition and determination of the heat-trader coefficients and other important values. The goal of this research was t o create a quick, accurate, and efficient system for the determination of process side local heattransfer coefficients through the use of heat flux sensors and a PC based data acquisition system.
Literature Review Heat transfer in an agitated vessel is defined as q = UiAi(To- Ti)
where Vi = the overall heat-transfer coefficient, Ai = the heat-transfer area, To= temperature of the heat-transfer fluid, and Ti = temperature of the process fluid. If fouling factors are negligible, the overall heat-transfer coefficient can be described in terms of three separated heat-transfer resistances: Ui = 1/[(1/hJ
+ (b/kw)(Ai/Aw) + (l/hJ(Ai/AJl (2)
See, for example, Chemineer Co. (1985),Chapman and Holland (1965a,b), and Poggeemann et al. (1980). In eq 2, the first term is the inside film coefficient, hi, that is on the process liquid side of the surface. The second term is the conduction through the wall, and the third term is the outside film coefficient. Dimensional Analysis for Heat-Transfer Coefficient. The inside or process side heat-transfer coefficient can be defined from the heat flow, Q, heat-transfer area, Ai, and the temperature difference between the inside wall and the bulk of the fluid, ATi: hi
Q /Ai(
ATi)
(3)
The heat flow, q, can be determined from the temperature gradient at the inside wall of the vessel:
'Technical Director. O88S-5885/92/263~-1384$03.O0JO
(1)
0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1385
II
I
AmDlifier
Low pass filter
A/D converter
Personal computer
I/O
A Path for the temperature of the rnlxed liquid E Path f o r the heat flux c Path f o r the w a l l temperature
Figure 1. Experimental system with emphasis on data acquisition.
where S = surface area and k is the fluid thermal conductivity. Dimensional analysis leads to an equation of the form u"
= f(NFbflPr)
II
n
765crn
(5)
Correlations for Heat Transfer. A simple functional relationship to describe eq 5 is
= 4$JB(NpJY (6) The ratio of the bulk viscosity at the wall to the fluid is often included in the correlation to give u"
= h D / k = kl(Nb)"(Npr)bOL/pJC (7) Equation 7 has been found to be adequate for engineering purposes for the correlation of heat transfer in many systems, such as flow in and over pipes, etc. The scatter of data is often of the order of 20-30%, but this is considered satisfactory. Of course, the constant kl is usually different for each case; however, a, b, and c do not vary much. The constant a ranges from 2/3 to 3/4, b is usually 1/3, and c is 0.14. Another important empirical equation for i d values of the Nusselt number has been tested with data obtained using a Rushton impeller (Foit et al., 1979). The equation is based on the Chilton-Colburn analogy and is u"
NNu 0.0565Nh3/4Npr1/3 (8) where NFbis defined as u,-r-/v with u,,- being the maximum value of u, (axial component) on the radial profile in the wall region and r,, being the radical coordinate at uE,-. Equation 8 is somewhat unique but was found to work for these author's data. The Reynolds number is quite different from any used by others; thus, the 3/4 power is not surprising.
Experimental Section Experimental Equipment. The equipment for agitation involves the vessel to hold the fluid, impellers to provide the agitation, a motor/gear box/controller to drive the impeller, a tachometer to monitor the impeller speed, and standard longitudinal wall baffles to aid in the mixing. Heat flux sensors and a thermocouple are needed to measure the temperature difference between the fluid and the wall and the heat flux into the wall. An A/D converter system is used to obtain the data. Figure 1 shows the experimental apparatus. MetraByte DAS-8 boards were used. The programs were written in BASIC and used MetraByte's standard data acquisition calls. Means to average data to remove noise and to simply accumulate the raw data are available. A total of 3000 data points can be taken for the averaging mode, and up to 16 (0-15 differential inputs) channels can be used. A data conversion program was written to convert the raw data to appropriate variables and nondimensional numbers. Both engineering and SI units are used because engineering units are still
Figure Z. vessel details and dimensions.
common in the U.S. mixing industry. The cylindrical stainless steel container has an ASME 2:l elliptical dished bottom. The vessel has heat flux sensors,mounted on the inside wall exactly over small local cooling regions designed to increase the heat flux. In this way, the bulk of the fluid is predominately cooled slowly by natural convection. Local cooling in the region of the heat flux sensor is much greater to allow accurate heattransfer coefficients to be measured. The sensors are mounted a t different heights. Four baffles (at 90° each) are used. These could be moved and set in any angular position. Water is used for the fluid being mixed in the vessel as well as for cooling. The variables used to define the vessel are B = width of each baffle, C = distance from the tank bottom to the impeller center line, D = impeller diameter, T = inside diameter of tank, W = width of the impeller, and 2 = depth of fluid in the tank. The actual dimensions are shown in Figure 2. Baffles are not shown for clarity. As A-C/V*S drive by Reliance Electric is used. The drive consists of an adjustable frequency drive controller and a 2-hp induction motor. The frequency range is 3-60 Hz. A digital tachometer (TACHTROL) is used to measure the revolutions per minute (rpm) of the agitator turbine. The tachometer has a 2-30-kHz frequency range and a quoted accuracy of about f0.03%. It is of the magnetic sensor type that counts the teeth on a gear. The agitation characteristic and energy consumption strongly depend on the impeller design. Three different types of impeller were used. The six-blade Rushton impeller (disked turbine shape, D-6) produces a radial flow pattern (Ackley, 1960; Brodkey and Hershey, 1988; Brooks and Su, 1959; Chemineer Co., 1985; Kupcik, 1975, 1976; Strek and Masiuk, 1967; Uhl, 1952), as shown in Figure 3. The turbine impeller with 45' pitched blades (P-4) (Uhl,1966) and the open-style hydrodynamic fluid foil axial impeller (HE-3) give axial flow patterns (Chemineer Co., 1985; Oldshue, 1989) in baffled tanks (Figure 4). Rdf Micro-Foil heat flux sensors (bottom and sensor 1, 20457-2 type; sensor 2, 20457-1 type) are used. These sensors consist of a heat flux sensor part to measure q / A and a thermocouple part to measure T,. Typical specifications of the heat flux sensor and the thermocouple as given by the manufacture are as follows: (a) heat flux up to 30000 Btu ftP h-l, (b) typical sensitivity from 0.07 to
1386 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 Sensors are always a t 6.00 o ' c l o c k p o s i t i o n
0 DEG
15 DEG
30 DEG
60 DEG
75 GEG
87 I 5 GEG
PROBE B
PROBE B D/T
D / T = 1/3, C/T = 1/3
2 8 5 GEG
=
113, C / T = 1/4
0-6 I m p e l l e r
D-6 Impeller
Figure 3. Rushton (D-6) radial flow pattern. 45 DEG
Incremental Baffle Rotation Settings
Figure 5. Baffle position.
PROBE 2
PROBE l
FROBE B D/T
=
1 /3 CI'
=
113
D/- = 113 i/T = 1/4
Figure 4. Axial impeller (P-4at 4 5 O , HE-3)flow pattern.
40 mV/(Btu ft-2 h-l), (c) operating temperature range of -300 to +500 OF, (d) typical thermal impedance of 0.003-0.015 'F/(Btu fV2h-l); and (e) typical thermal capacitance of 0.01-0.05 Btu f f 2 OF-'. The corresponding SI values for the above area are as follows: (a) heat flux up to 95 000 W m-2, (b) typical sensitivity from 0.02 to 13 pV/(J m-2 s-l), (c) operating temperature range of 90-530 K, (d) typical thermal impedance of 0.0005-0.0025 K/(W m-2),and (e) typical thermal capacitance of 0.2-1 kJ m-2 K-l. The thermocouples were of copper-constantin construction. Additional copper-constantin (Cu/Co) thermocouples were used in the vessel to measure the fluid temperature. Experimental Procedure. In all runs the Z / T ratio was maintained at a little over 1. This value was selected because sensor 2 was just at the ratio of 1. The time associated data were taken under the following conditions: (a) DIT = 1/3, (b) C/T = 1/3, (c) Rushton impeller, (d) NRe= 50000, (e) 45' fmed baffle position, and (f) 600 data points at each channel (total = 600 X 8 = 4800 data points), Runs for the fullfield heat-transfer measurements were made at seven angular positions between the baffles for each mixing configuration and at two Reynolds numbers. These runs were made to establish a base line of experimental data to evaluate the parameters of importance, such as impeller type, C / T , DIT, and Reynolds number, and to provide initial comparisons to literature data. The runs made and the conditions were as follows: (a) three vertical sensors which have different locations, bottom (B), side (11, and near top of the surface (2); (b) DIT = 1 / 2 and 1/3; (c) C/T = 1/3 and 1/4; (d) three impellers (D-6, P-4, HE-3); (e) NRe= 50000 and 160000; and (f) seven different baffle positions. These were 2.85' (baffle just in front of sensor),each 15' between and 87.15' (baffle positions just behind sensor). These baffle positions are shown in Figure 5. The experimental procedure used was as follows: (a) The tank water was filtered and heated to 60 'C or higher. (b) The cooling water passed through
an ice bath before it was distributed to each sensor. (c) The raw data was taken at the conditions cited by using the averaging mode (one average of 3000 data points for each of eight channels was used). (d) The raw data was converted to processing variables and nondimensional numbers. The conditions for detailed angular position measurements are as follows: (a) DIT = 1/3, (b) C/T = 1/3 and 1/4, (c) Rushton (D-6) impeller, (d) NRs = 50000 and 160000, (e) a total of 29 baffle positions (3' increments from 2.85O to 87.15'), and (f) average of 2000 data points at each channel (total = 2000 X 8 = 16000 data points).
Results The results of the noise test are shown in Figure 6 for T,,q / A , and h ( = q / A A T ) at the side sensor position 1. The variability of the output, without averaging, is apparent. Averages can, of course, still be obtained. Indeed, the results, to be presented later, are based on averages of 3000 data points for each sensor, not the limited 600 used here. The difference for one time interval is about 0.05 s. The h i d temperature, T,, was measured with a thermophile made up of six thermocouples. The noise for this measurement was very small. Large scale variations would not be expected at the level of mixing used, as was observed. Close inspection of the figures shows that a periodic variation exists. We hypothesized that this low frequency period is associated with the precession of an axial vortex structure that has been previously observed between the surface and the impeller. Such a vortex has been previously described for the open impeller. It no doubt can exist for the disk impeller as well, but has not been as discussed. Using talcum powder sprinkled on the surface, we were able to observe the vortex on the surface of the water in the mixing tank. The vortex rotated around the shaft and then disappeared. It rarely made a full rotation of the tank; however, the formation, rotation around the shaft, and disappearance were repeated time and time again. The vortex did not approach the wall. Figure 7 provides a sketch of the vortex. To analyze the data shown in Figure 6, we tried a fast Fourier transform procedure; however, since the vortex motion was at very low frequency, the analysis was difficult and not successful. As an alternate, the signal was filtered by a Gaussian filter procedure to remove the high frequency noise and any very low frequency drift. The frequencies measured at each sensor are shown in Table I. The filtered data from the Gaussian method showed very
Ind. Eng. Chem. Res., Vol. 31,No. 5, 1992 1387
moo
Vortex motion i n the agitated vessel 42 1
842
TIME
1263
1883
2104
(SEC)
Figure 7. Sketch of observed vortex motion in the agitated vessel. Table I. Frequencies Measured unit sensor wall temperature B 1 2 heat flux B 1 2 heat-transfer coefficient B
0
1
2 Table 11. Effect of Vortex Passage sensor variable
B
qlA
T, h
1
2 zoo
42 1
842
TIME
1263
1883
2104
(SEC)
0
d
00
42 1
842
TIME
1283
1683
2104
(SEC)
Figure 6. Noise test for T , q / A , and h at NRs = 50000 and sensor 1.
qlA TW
h
frequency/Hz 0.119 0.128 0.128 0.119 0.13 0.13 0.119 0.133 0.138
max*% 7 3 45 9 3 54 8 8 68
clearly the periodic nature in the output. The unfiltered and filtered signals are shown in Figure 8. The heattransfer coefficient is, of course, computed from both the wall temperature and the heat flux. From the data of Figure 6 , q / A and T, are essentially in phase. The differences in measuring time of 0.24.25 s is negligible on the time scale of these plots. As q / A increases, T, increases or AT (=Tf- T,) decreases, since Tf is essentially constant. Each set of figures show this tendency. The observed variation is not really noise but is probably associated with the vortex passage. Some maximum estimates of the local vortex passage effect are shown in Table 11. Although q / A and T, differences are relatively small, and with the fact that Tf - T, is small, the difference in T, reflects a large difference in h. It must be emphasized that these are not errors, but real differences in h experienced by a sensor as a function of time. With the normal averaging we use, a satisfactory average can be obtained. The low frequencies associated with the vortex passage frequency are small when compared to the much higher frequencies associated with the impeller revolutions per second (rps) or blade passages. The results of full field heat-transfer measurements are discussed for the D-6 impeller only, and the flow patterns for the D-6 impeller are shown in Figure 3.
1388 Ind. Eng. Chem. Res., Vol. 31, No. 5,1992 DATA
RAW
0 0
h, Btu/hr-ft-2-deg F 3000
,
"
'
I
"
--
'
0
Re
I
2400
-~ * ,
Re
h, W/m"P-deg C (~1000) I
_____ Low, C I T High, C / T
_
--
*
_
1800 k--
0 0
- Re - Low. I
113 113
I
Re
-
C/T
High. C/T
--
1
1/4 114
'
I
-
15
1
- 12
I
,
t
I
.19
60
75
90
I
0
I
10 h d l
E I
E O
tig
fflo
2"
15
0
v
30
45
Angular Position, Degrees Figure 10. Reynolds number and location effects: D-6; D I T = 1/3; sensor 1.
G O 0
2
--
h, Btu/hr-ft^P-deg F
4000
3200
0 0
1263
84 2
42 1
00
TIME
FILTERED
0
0
2104
1883
2400
(SEC) DATA
1600
-1
&
Re
*
Re
-
L
-
113
High, C/T * 113
- -
,
-
Re
Low, C/T
Re
High, C I T
-
114 1/4
7 -
-~~
_.
-
~
Low, C/T
h, W/m^?-deg C (~1000)
-
-
1 1225 I
I
-~
18
135
9
~
.
0
10
G* I
E o
0.8 fflo \ "
3
v
-2 I
200 0 0
42 1
0.0
128.3
84.2
TIME
2104
1683
(SEC)
Figure 8. Unfiltered and filtered signals for h at NRe= 50 OOO and sensor 1. h, Btu/hr-ft^P-dea F 1
'
4
lBo0
1200
I
,
.
Re Re
1
--
LOW, C/T
h. W/m^P-dea C (~1000) 113
High, C I T * 113
- -*
1
Re
LOW, C I T
Re
High, C / T
t
T
, I
-
114
m
114
I
1 10
7'
400
01
0
I
15
30
I
!
I
45
60
75
0 90
Angular Position, Degrees Figure 9. Reynolds number and location effects: D-6; D I T = 1/3; bottom sensor.
t ~
4
I
1
I
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1389 h, W/m^P-deg.C. (~1000)
h, Btu/hr-ft"2-dea.F.
1 A
800
-
'
DIT
-
I 113, C I T
-
I 113
- -
I
- - - DIT
112, C I T
DIT
D I T . 112, C I T * 113
I
I
.
113, C I T
. .
1 - 4
__
-
I -
-
_-,
400
600-
-3
_--
....................
,I
800
I
.
600
h. Btu/hr-ft*P-deg F 1000 1
-6
114
114
I
I
I
__-
5
I
I
S ' .
-
. . . . . . . . .- I . ' . ' r 3
I
....
I
I
I
.. - .... . ..
400
h, W/mm2-deg C ( ~ 1 0 0 0 )
I
,
~
..... . . ,
I
i
-2
A
............
..
I
200
.
I
- 2
-
I
0
0
I ,
800
1 ,
I
, '5
I
600
J
0.8
,
3
0.6
j
2
0.4
I
I
I
*
DIT
-
112. C I T
I
-
113
*
DIT
-
112. C I T
.
1
/ /
114
1
Probe 1
-5
Probe 2
1
i4
I I
1 . .
I
-
B
-Probe
1
.
.....
.
*
- . . . .. . . . . . . .
T r n
0.2
.
.
.
a
s
.
I
I
i'
I 0~ 0
~
1
1
I
15
30
45
3
..
0 75
60
90
Angular Position, Degrees transfer coefficient to sensors 1 and 2 than 1/4 because of the relative location of the impeller to the sensor (Figures 10 and 11). This increase is about 50% for sensor 1. The effect of impeller diameter (DIT ratio) coupled with impeller position (C/T ratio) is better shown in an anternate series of plots (Figures 12-14). These figures are for NReof 50000; similar results were obtained for the higher Reynolds number. Some of the lowest heat-transfer coefficients observed were at the bottom for the larger impeller (DIT = 1/2,7.5 in.) at the lower Reynolds number (Figure 12). This was true for both vessel positions. This is no doubt a result of the bottom sensor being in a relatively stagnant area, as suggested in Figure 3. For heat transfer to the bottom, the smaller impeller is better, especially when it is positioned near the sensor (a factor of 2). When positioned higher, it is still better by about 40%. For transfer to sensor 1 (Figure 13), which is representative of most of the vessel heat transfer, the smaller impeller (higher velocity) located near the sensor (C/T = 1/3) is best. The larger (and slower) impeller (DIT = 1/2) and/or location of the impeller near the bottom, all give about the same results that are about 30% less. Finally, near the surface (sensor 2, Figure 14), using the smaller impeller at the higher position in the vessel or using the larger impeller gave the best heat transfer. The small impeller near the bottom gave nearly 40% less heat transfer. The reason for the small difference at sensor 2 is probably caused by increased turbulence caused by the large impeller at its higher location. The mixing is as good as possible, and the coefficients are near a limiting level. The detailed runs to investigate the regions close to the baffles are shown in Figures 15-18, for two Reynolds numbers for D/T = 1/3. First, the Reynolds number effects are similar to those mentioned earlier. In this facility, the baffles do not extend down into the dished
Figure 16. Baffle region results: D-6; D I T = 113; C / T = 114; low NRe. Btu/hr-ft^B-deg F 3000 2400
I
1
.
~
1
W/mA2-deg C ( ~ 1 0 0 0 ) I E
-Probe;
' P[be
1
I Probe 2
1800
I
10.5 14
. .
. . 600
3.5
,I
01
1
0
15
il o
I
I
I
45
30
75
60
90
Angular Position, Degrees Figure 17. Baffle region results: D-6; D I T = 113; C / T = 113; high NRe. Wlm-P-deg C (~1000)
Btu/hr-ft-P-deg F
3000
-
I Probe B
2400
-
I
1
Probe 1
I Probe 2
t 14
I
I
I
r 105
is00
-
1200
..
600
0 0
15
30
45
60
75
90
Angular Position, Degrees Figure 18. Baffle region results: D-6; Df T = 113; C / T = 114; high NRe.
1390 Ind. Eng. Chem. Res., Vol. 31,No. 5, 1992
*
c
h
87 15 DEG
2 8 5 DEG
Top V i e w
Table IV. Averaged - Data for the Detailed Angular Positions h, W m-2K-' (Btu h-l f f 2 OF') Nb 50000 N h = 160000 Sensor CfT = 113 CfT = 114 CfT = 113 CfT = 114 B 1925 (339) 2050 (361) 4287 (755) 5531 (974) 1 3129 (551) 2453 (432) 8194 (1443) 7155 (1260) 2 2425 (427) 2033 (358) 5207 (917) 4991 (879) 2663 (469) 6104 (1075) calculated
not be compared to the calculated values. Likewise, the bottom sensor is not in a baffled region of the flow and would not be expected to compare well with calculated values. The standard deviation (u) is about 15% at the lower Reynolds number. If one considers only the C / T = 1/3 value, the difference is 16% from the calculated value. The u value rises to 24% at the higher Reynolds number, but this is outside of the range of the empirical correlation used. The difference between the C / T values is 28%. Note that variation with position represents anywhere from 10 to 40% of that reported above. The more detailed measurements, obtained to study the effect of the baffles, were averaged. In the average, positions near the baffles were included. Table IV summarizea these results. Again, only sensor 1should be compared to the calculated value. The standard deviation ( u ) is nearly 12% at the lower Reynolds number. The C / T = 1/3 value is 17%. The value rises to 27% at the higher Reynolds number, but again this is outside of the range of the empirical correlation used. Once again, the difference between the C / T values is 24%.
'L 8 7 15
;EL.
DEG
Side View
Figure 19. Flow in the vicinity of the baffles. Table 111. Data at 45' Position for the Rushton Type Impeller h, W m-2 K-l (Btu h-' ft-2 OF') NR= 50000 N h 16OOOO -._ Sensor C I T = 113 CIT = 114 CJT = 113 CJT = 114 - ..-_ 1948 (343) 1982 (349) 4174 (735) 4849 (854) B 3094 (545) 2328 (410) 7978 (1405) 6553 (1154) 1 2868 (505) 2044 (360) 4951 (872) 4509 (794) 2 2669 (470) 6019 (1060) calculated
bottom; thus, as expected, the heat-transfer coefficients are not influenced by their position. Near the baffles, the flow can be complex (see Figure 19). One can visualize from the top view that the coefficients might be higher for 87.15' than 2.85', since the velocity over the sensor could be higher. However, if the Reynolds number is high enough, one would expect a large wake effect at 2.85'. At the lower Reynolds number, when the impeller was positioned close to sensor 1, the heat-transfer coefficient was greater near 87.15'. When the impeller was near the bottom, the coefficients behave similarly. Maybe the flow is more like the side view in Figure 19 for these conditions, Le., more flow over the sensor along the baffle rather than over the baffle. At the higher Reynolds number, the coefficients are larger for the 2.85' position. Here, one might suggest that the wake is important. Because the flow must change in the vicinity of the surface (sensor 21, the flow there muat be over the baffle at that position. In any event, the effects are restricted to the first and last 15' or of the full angular range. For the Rushton impeller, the experimental heattransfer coefficient data obtained in this study are consistent with each other and quite reasonable when compared with the calculated local heat-transfer coefficients based on the results of Fo% et al. (1979),the boundary layer equation (Brodkey and Hershey, 1988),or the overall heat-transfer coefficient for the entire vessel (Uhl,1966). Table I11 summarizes some of the results made at the single position of 45'. Only sensor 1 should be compared to the calculated value. Sensor 2 is near the surface of the fluid and could be out of the fluid at times, thus it should
Conclusions First, and foremost, the system to rapidly measure heat-transfer Coefficients on the process side during mixing has proven workable. The longer term, quasi-periodic contribution to noise can be associated with an axial vortex structure that slowly rotates a t a speed that is a small fraction of that associated with the impeller rotation. The major influence on heat transfer is the Reynolds number. Vessel geometry, within the bounds of good design, is of secondary importance, but still important. At the two Reynolds numbers used, the increase in heat transfer from the lowest to the highest was a factor of 2-3. Variations in geometry could cause up to a 50% change in heat transfer. The heat-transfer coefficient varied little with angular position with respect to the baffles. The flow near the baffles must be complex. Still, the effects are relatively small. When the Reynolds number was increased, the measurement at each of the angular positions increased. Finally, this research effort was directed to proving a technique for rapid heat transfer measurements. Future work, now underway, is directed toward obtaining detailed comprehensive data (i.e., at many values of NEb,not just two) to enaure correct correlations for scale-up purposes. Nomenclature A = area B = width of baffle C = distance from the tank bottom to the center line of the
impeller
D = impeller diameter h = heat-transfer coefficient k -- thermal conductivity NNu= Nusselt number NRe= Reynolds number NR = Prandtl number q = heat flow
Ind. Eng. Chem. Res. 1992, 31, 1391-1397
r = radius S = area T = temperature, inside tank diameter U = overall heat-transport coefficient u = velocity v = PI? W = width of impeller Z = depth of fluid in tank Greek Letters = density = viscosity
p p
Subscripts f = fluid i = inside o = outside w = wall z = axial direction
1391
Chapman, F. S.; Holland, F. A. Heat Transfer Correlations for Agitated Liquids in Process Vessels. Chem. Eng. 1965a, 72 (2), 155-162. Chapman, F. S.; Holland, F. A. Heat-Transfer Correlations in Jacketed Vessels. Chem. Eng. 1965b,72 (4), 175-184. Chemineer Co. Liquid Agitation; McGraw-Hill: New York, 1985. FOB,I.; Placek, J.; Strek, F.; Jaworski, Z.; Karcz, J. Heat and Momentum Transfer in the Wall Region of a Cylindrical Vessel Mixed by a Turbine Impeller. Collect. Czech Chem. Commun. 1979,44,684-698. Kupcik, F. Heat transfer at the bottom and at the walls of agitated vessels, Parts I & 11. Znt. Chem. Eng. 1975,15 (4), 658-664; 1976, I6 (l),91-96. Oldshue, J. Y. Fluid Mixing in 1989. Chem. Eng. B o g . 1989,s(5), 33-42.
Poggeemann, R.;Steiff, A.; Weinspach, P.-M. Heat transfer in agitated vessels with single-phase liquids. Ger. Chem. Eng. 1980,3, 162-1 - - - - 74. . -.
Literature Cited Ackley, R. J. Film Coefficients of Heat Transfer for Agitated Process Vessels. Chem. Eng. 1960,69 (17), 133-140. Brodkey, R. S.; Hershey, H. C. Transport Phenomena, A Unified Approach; McGraw-Hill: New York, 1988. Brooks, G.; Su, G.-J. Heat Transfer in Agitated Kettles. Chern.Eng. R o g . 1959,55 (lo),54-57.
Strek, F.; Masiuk, S. Heat transfer in liquid mixers. Znt. Chem. Eng. 1967,7 (4). 693-702. Uhl,V. W. Heat Transfer to Viscous Materials in Jacketed Agitated Kettles. Chem. Eng. Prog. Symp. Ser. 1952, 51 (17), 93-107. Uhl, V. W. Mechanically Aided Heat Transfer. In Mixing; Theory and Practice; Uhl, V. W., Grey, J. B.,Eds.; Academic Press: New York, 1966; Vol. 1, Chapter 5.
Received for review September 17, 1991 Revised manuscript received January 17, 1992 Accepted January 30, 1992
Molecular Dynamics Study of Solute-Solute Microstructure in Attractive and Repulsive Supercritical Mixtures Ariel A. Chialvot and Pablo G. Debenedetti* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263
Molecular dynamics simulations show enhancements in the density of solute molecules around each other in model attractive and repulsive dilute supercritical Lennard-Jones mixtures. The effect is pronounced in the former case and moderate in the latter. For the attractive mixture, the quantification of the density enhancement in terms of excess number of solute molecules suggests an increased collision frequency rather than the formation of a compact solute-solute aggregate. The simulations confiim the resulta of recent integral equation calculations for the same model mixtures, and suggest that solute-solute interactions in dilute supercritical mixtures could play an important role in situations involving a reactive solute. Introduction Fluid behavior near critical points is governed by longranged correlations in density fluctuations (pure substances) or in concentration fluctuations (mixtures). These order parameter fluctuations are correlated over a characteristic distance, the correlation length, that diverges at the critical point. The practical applications of supercritical mixtures, on the other hand, occur over the approximate range 1IT,I1.1; 1 IP,I2 (where subscript r denotes the value of a property divided by its value at the solvent’s critical point). In this region, the short-ranged structure, that is, deviations from a random distribution of molecules occurring over distances of the order of 2-3 molecular diameters around a given species, appears also *To whom correspondence should be addressed. Current address: Department of Chemical ~ ~ ~University i of~ California, Berkeley, CA 94720-9989. t Current address: Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442.
to be verv imwrtant. Long-ranged fluctuations cause the solute’s mecdanical partiaimolir properties in a mixture to increase in magnitude, but ita chemical potential is determined solely by ita local solvation environment; hence the importance of short-ranged structure, even in the presence of long-ranged fluctuations. We shall henceforth refer to such short-ranged features as microstructure, Much of what makes supercritical solutions of scientific interest originates from the coexistence of long-ranged correlations (whose length scale, though finite, is still large compared to molecular dimensions) and a microstructure that generally involves large deviationswith respect to bulk conditions. For example, fluorescence spectroscopy (Brennecke and Eckert, 1988, 1989a; Brennecke et al., 1990a,b), solvatochromic probe experiments (Kim and Johnston, 1987a,b), and computer simulations (Petsche and Debenedetti, 1989; Knutson et al., 1992) indicate a density around ~substantial ~ ~increase i in ~the local ~ solvent , -~ Fursolute molecules with respect to bulk conditions. thermore, solvatochromic probe experiments (Kim and Johnston, 1987a,b) suggest that the remarkable enhance-
0888-5885/92/2631-1391$03.00/00 1992 American Chemical Society
