Heat Transfer in Mechanically Agitated Gas-liquid Systems K. Balakrishna Rao and P. S. Murti* Regional Research Laboratory, Hyderabad-9, India
Steady-state heat transfer experiments were carried out in jacketed unbaffled agitated cylindrical vessels using single-phase and two-phase fluids as heat transfer media. Air-water, air-transformer oil, and airclavus oil served as the two-phase fluids. The rate of heat transfer to and from two-phase gas-liquid mixtures was determined using a six-flat blade disk turbine. Correlations have been developed based on dimensionless numbers by least-square regression analysis from over 280 measurements. The resulting equations predict the experimental data with standard deviations of 10.1 and 9.8%, respectively, for the jacket and the coil.
M e c h a n i c a l l y agitated vessels, equipped with cooling coils or heating jackets are in common use throughout the chemical process industry. Heat transfer rates were studied previously in such vessels and most of t'hese investigations have been carried out with single-phase liquids (Chilton e t al., 1944), Strek (1963), and -1skew and Beckmann (1965) and twophase liquid-liquid mixtures (Bodman arid Cortea, 1967; Lavinga and Dickson, 1971). Results of these investigators are summarized in a comprehensive review by Tlhl and Gray (1966). X a n y chemical processes such as hydrogenation and sulfonatioii by sulfur trioside require the processing of twophase gas-liquid mixtures in agitated vessels. Therefore, a knowledge of the effect of a second gas phase on the rate of heat transfer in a jacket'ed vessel is of importance. Measurements of heat t'rarisfer to gas-liquid systems without a mechanical stirrer-Le., bubbling beds, have been made by Kolbel and Siemes (1958), Kast (1962, 1963), Fair (1962), Konsetova (1966), and Lelirer (1968). But no quantitative data are available a t present on the rate of heat transfer to or from two-phase gas-liquid systems in a mechaiiically agitated vessel. The purpose of the present investigation was to obtain and correlate the data which describe the rate of heat transfer to gas-liquid mixtures in stirred vessels. Experimental
h schematic diagram of the experimental apparatus is shown in Figure 1 Steady-state heat transfer experiments were conducted in dished bottom cylindrical vessels of two diameters 0.23 and 0.70 meter made out of type 304 stainless steel. -1gitation was provided by a 6-flat blade disk turbine with the shaft along the axis of the vessel. variable speed drive assembly (Kirloskar Co., India) permitted the impeller speed to be varied from 40-500 rpm. Impeller speeds were measured to =k0.5% accuracy with a self-timing Jacquet tachometer. Table I contains the important geometric parameters of the two unhaffled heat' transfer vessels, the impellers, and spargers used. Except for the coil configurations, the smaller vessel is dimensionally similar to the larger one mid is scaled dowii a t a ratio of 3: 1. The means of measurements and control are identical for the two systems. Heating was llrovided by condensing steam I
190 Ind. Eng. Chem.
Process Des. Develop., Vol. 12, No. 2, 1973
in the vessel jacket'. Cooling of the tank fluid was achieved by circulating water in the helical coils. The vessel was insulated heavily with asbestos rope and asbestos magnesia cement. Clean air from the compressor was introduced into the vessel through a ring sparger opening immediately below the impeller. The jacket and coil wall temperatures were measured by means of four 26-gauge copper constantan thermocouples positioned 180' apart and located one half and one third of the distance from the bottom of the vessel. The temperatures of the cooling water entering and leaving the cooliiig coil, the bulk liquid, and steam and condensate were measured by means of mercury in glass thermometers with an accuracy of =0.05OC. A glass thermometer placed 15 cm below the liquid surface and midway between the vessel and impeller shaft measured the temperature of the vessel liquid. Preliminary studies of temperature profiles at various locations ill the vessel using thermometers showed no measurable temperature gradients within the agitated liquid. A blank heat loss value was obtained by measuring the heat required to keep the jacket hot with no liquid in the vessel. This heat loss was so small (20 kcal/hr) in comparison with the act#ual heat transferred that it was neglected in the calculation. The impeller was operated a t the highest speed with no steam in the jacket to ascertain the heat effects due to mechanical energy of the rotating impeller. These were negligible; in this case the temperature of the liquid did not show any appreciable rise. The flow rates of water and air were measured using calibrated rotameters. Each run was made on a given batch of liquid, keeping constant the speed of agitation, rate of air flow, and rate of water flow i n the coil. Hoivever, neither the sparger geometry nor the properties of gas could be varied in this work. Results and Discussion
Single-phase Heat Transfer with Water. T h e first p a r t of the experimental program was to standardize the experimental setup in accordance with the procedure described above. Steady-state tests were first made with water in the vessel 4multaneously heated Isy steam in the jacket and cooled by water in the coil. The Reynolds number was varied
WATER
Figure 1. Schematic flow diagram
from 1 x l o 4 to 5 x 1oj to keep the flow in fully developed turbulerit region. The film heat transfer coefficients for both the jacket and the coil were calculated using the equation, h=-
Y '4. AT
(1)
Ahthe apparatus was well lagged, heat losses were ignored and q , the total heat transferred, mas calculated from the temperature rise of the coolirig water using the equation, Y = W,Cp(t,o - t c d
(2)
The experimental heat transfer coefficients for tlie runs using water as vessel liquid were compared with the values calculated by tlie equation proposed by Brooks and Su (1959) for tlie jacket heat transfer and Cummings and West (1950) equation for coil heat transfer who reported results using a somewhat similar vessel without baffles: Jacket Coil
Siyu =
0 . 5 4 ( S ~ , ) ~ ~ ~ ' ( S p , ) ~ . ~ ~ ( S , . i , ) (3)
LySu = l . o ~ ( ~ ~ R ~ ) o ' 6 z ( ~ V I' ~ S )-''I~ 4 ~ ) 0 ' 3 3 ((4) ~ ~
Figures 2 and 3 show a comparison between the calculated and oljserved values. As can be seen the single-phase results compare well with Equation 3 for the jacket and Equation 4 for the coil, with a n average deviation of 9.6 and 8%, respectively. Two-Phase H e a t Transfer Wthout Stirring. Twophase heat transfer d a t a without stirring were taken with water in a 0.23-meter diameter vessel. T h e air rate was varied from 0.02-0.083 m/sec. It was observed by Tmyell et al. (1965) and IIeNevers (1968) that when a gas is sparged as a swarm of bubbles in a pool of liquid, the liquid will have a regular flow pattern-upward in the middle and downward near the walls. the gas rate is increased the downward liquid velocit,y eveiitually exceeds the rising velocity of the bubbles giving bubble d o \ n i f l o ~near the walls. Recently, Freetimati and Davitlsoii (1969) and Reitema and Ottengraf (1970) presented their mathematical treatment of the liquid circaulatioti in a bubble column in comparison with the ex1)erimetitnl results. This downward circulation of the liquid near the will accounts for the increased heat transfer. coefficient in a sparged vessel. The experimental values of heat transfer for the jacket side were compared with the sparged syst,emdiscussed by F a i r e t al. (1962) for which (5)
Table 1. Geometrical Characteristics of Experimental Equipment
Heat transfer vessel Vessel diameter, m (D) Shaue of base Material of construction Liquid height, m (Hi) Heat transfer area, m2(.la,) Turbine impeller Diameter, m ( L ) Number of blades (na) Blade length, m (1) 13lade nidth, m (w) Turbine position, H , from bottom of the tank
I
I1
0.70 Dished S.S. 304 0.89 2.40
0.226 Dished 9.s. 304 0.226 0.204
0.233 6 0,059 0,047 0.233
0,075 6 0,019 0.015 0 , 075
roll Material of construction Outside diameter of coil tube, n1, ( d o ) Mean diameter of coil helix, m. ( d F l Pitcii, ;1 ( p ) Kumber of turns (n) Heat transfer area, m2 (3,)
0.51 0 054 15'1'2 1 26
0.20 0 025 10 0.26
Sparger geometry (Discharge dom-iiward) Ring diameter, m Hole diameter, mm Pitch, mm Xumber of holes
0 233 3 9 60
0 075 3 9 20
Copper 0.016
S.S.
304
0,0127
the results are plotted in Figure 4 and compare well with Equation 5. Mechanically Agitated Gas-Liquid H e a t Transfer. T h e rate of heat transfer from jacket and coil to two-phase gasliquid mixtures in uiibaffled agitated vessels was studied. For the two vessels used in this investigation, a total of 280 data points were obt'aiiied. Steady-state tests were made with air-water, air-transformer oil (Shell-Diala Oil 13) and air-clavus oil (Shell clavus 33 oil). No experimental results on heat transfer to two-phase gas-liquid systems iii mechanically agitated vessels are known to have been reported in the literature and therefore the present data are the first to be reported without a comparison. Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 2, 1973
191
L
4-
-
2 -
4 IO'=
-
8 b -
8rO' Om
Experimentaldata:
Brook8 and Su q:-
I
21
2
I
I
I
IO'
6 8
4
1
I IIll
I
2
1 4
0.05
0.02
0.1
V&, m h c
I I l l 1
6 8 IO'
Figure 4. Jacket heat transfer coefficient vs. superficial gas velocity (air-water)
%a
Figure 2. Single-phase heat transfer correlation (jacket)
i
f
4
2 x IO'
0.02
0.0)
0.05
0.1
V,, m/rrc 2xld
4
2
IO'
6
8
Figure 5. Jacket heat transfer coefficient vs. superficial gas velocity (air-water)
IO'
NRe
--L
Figure 3. Single-phase heat transfer correlation (coil)
Table II. Range of Variables
40-500 0.002-0.083 273-1090 700-5000 1 X lo3-5 X lo5 2-266 0.18 -0.98 1.02 -1.50 0.008-0.53
Impeller spred, rpm Air rate, m/sec Water rate, kg/hr Nusselt number, NNTJ Reynolds number, N R ~ * Prandtl number, N p , Viscosity ratio, p w / p Viscosity ratio, p c / p Froude number, N F ,
&=
t
4 0.02 0.0s 0.1
2 x IO2 0.01
V,, m/rrc
Figure 6. Jacket heat transfer coefficient vs. superficial gas velocity (air-transformer oil)
Table I1 summarizes the range of experimental variables studied. From experimental measurements, heat transfer coefficients were calculated using the equations
i.e., Q
=
WcCp(tc,-
tci)
+ (V,a,~,)[Cp'(t,o - tail + (H, -
Jacket
(8)
and the rate of heat transfer for the coil was obtained from 4 = WCCP(t,
Coil
(7)
The rate of heat transfer from the jacket was calculated from
Q 192 Ind.
WXI
qwater
+
yair
Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
- tci)
(9)
It was assumed that under steady-state conditions the vessel liquid and outlet air will have the same temperature. The error introduced by this assumption is negligible. Equations 8 and 9 neglect the rate of heat transfer to the impeller and impeller shaft. The experimental data and results are
2 x 1 02 0.01
s
6.02
5,m/sw
0.05
1.0
2
0.02
0.05
C
V,, m/wc
Figure 7. Jacket heat transfer coefficient vs. superficial gas velocity (air-clavus 33 oil)
i"
2 x 1 02 0.01
Figure 10. Coil heat transfer coefficient vs. superficial gas velocity (air-clavus 33 oil)
p \
+ .. 2
0.01
#
I
0.02
O
8
0.W
S
I2
CO
V,, drro Figure 8. Coil heat transfer coefficient vs. superficial gas velocity (air-water)
0.01
0.05
0.02
Figure 11. Ratio of two-phase to single-phase jacket heat transfer coefficient vs. superficial gas velocity (air-water)
0.1
4, m/src Figure 9. Coil heat transfer coefficient vs. superficial gas velocity (air-transformer oil)
Figure 12. Ratio of two-phase to single-phase coil heat transfer coefficient vs. superficial gas velocity (air-water)
presented in Tables 5 7 deposited with the American Chemical Society Microfilm Depository Service. Figure 5-7 for jacket heat transfer and Figures 8-10 for coil heat transfer, respectively, for water, transformer oil, and clavus 33 oil show the dependence of film heat transfer coefficients on the gas rate. A study of these figures indicates three points: Sparging of a gas into a n agitated liquid improves the heat transfer coefficient a t any impeller speed and for any vessel liquid. This may be explained on the same reasoning as that extended to bubble columns namely, that rising bubbles
create circulatory currents in the liquid near the transfer surface in addition to those already created by the rotating impeller. The intensity of the existing turbulence is thus enhanced by the presence of rising bubbles. However, the increase in turbulence cannot be in proportion to the gas velocity b u t depends upon the value of the intensity of turbulence which existed as a result of rotating impeller. Thus, a t higher speeds of the impeller, the intensity is already high, and sparging of gas marginally enhances its value; therefore, the improvement in heat transfer coefficient or reduction in heat transfer resistance is rather small (Figures 11 and 12). Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
193
Table 111.
Constants for Heat Transfer Correlations Mean dev., %
Std dev., %
No. data
d
-0.02
-0.03
6.3
7.8
208
0.33
-0.14
-0.10
7.4
10.1
208
0.48
0.21
-0.48
-0.05
7.2
8.7
183
0.64
0.33
-0.14
-0.10
8.1
9.8
183
b
System
Method
C
Jacket heat transfer
All constants fitted Assuming b and c A11 constants fitted Assuming b and c
6.8
0.48
0.24
1.35
0.59
7.6
0.87
Coil heat transfer
a
For example, in the case of air-water system, the jacket side heat transfer coefficient a t a stirrer speed of 100 rpm and a t a n air velocity of 0.063 mjsec is 4413 kcaljhr m2 "C, indicating a n improvement of 280%, whereas a t a stirrer speed of 500 rpm, the improvement is only 740/,-true with the other viscous liquids also. The coil side heat transfer coefficients, in general, are lower than the jacket side values except in the case of air-water system. I n single-phase (Newtonian or non-Newtonian) heat transfer studies, this behavior was reported by Chilton et al. (1944) and Cummings and F e s t (1950). This result is due to the mode of heat transfer from the jacket to the coil through the vessel liquid, resulting in lower viscosities of the liquid a t the jacket. When the gas is sparged into the liquid, it is likely that the concentration of the gas bubbles across the vessel cross section is uneven (except a t very high impeller speed) with higher values of concentration away from the wall. When we assume for the moment that the concentration in the vicinity of the coil is higher than a t the jacket wall, the turbulence caused by these rising bubbles is higher near the coil, sufficient enough to counter the effect of increased viscosity because of lower coil wall temperature, thus resulting in higher heat transfer coefficients. The same phenomena can be assumed to take place with more viscous liquids, but in this case, the higher viscosity is still the controlling factor for the reduction in the heat transfer coefficient. Although i t became impossible to operate the unit in this work with much higher gas velocities because of the entrainment of liquid, it is to be expected that increased velocities teiid to increase the heat transfer rate up to a value beyond which the rate should fall because of the gas-blanketing of the heat transfer surface. Correlation of Data
Basic Considerations. Recently, heat transfer to t'wophase liquid-liquid mixtures in agitated vessels has received some attention. However, no work has been reported on heat transfer to mechanically stirred gas-liquid systems. The choice of a proper similarity criterion is difficult if a comparative solution is desired. The existence of large divergences in correlations published on heat transfer to pure liquids in agitated vessels further indicates that corresponding studies on two-phase gas-liquid systems will involve more difficulties. X o s t of the investigators using single-phase liquids and two-phase liquid mixtures agree on the general form of the correlations using dimensionless numbers, but some variations exist in the coefficients. Rushton et al. (1950) and Oldshue and Gretton (1954) used a similar approach in arriving a t a correlation for agitator power and single-phase heat transfer, and this can be extended to the present work on two-phase gas-liquid heat transfer. 194 Ind.
Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
C
points
The literature and experimental observations showed that heat transfer to mechanically stirred gas-liquid systems will be influenced by the following variables: Fluid properties of liquid, k , p , p , and C, Superficial gas velocity, V , Degree of mixing Driving force, At, and temperature distribution Vessel geometry and impeller dimensions Acceleration due to gravity, g For geometrically similar systems, the Buckingham Pi theorem gives the following general function for heat transfer in gas-liquid systems in terms of the following dimensionless groups among film heat transfer coefficient, h , and the abovementioned variables,
and a solution of this equation is l\'Xu
=
.f~ (SRe*)f2 (-k7Pr)f8(*j7v~8)f4(.L'Fr)
(12)
expressing the groups in the usual exponential form, whence
The introduction of Froude number in the analysis is supported by the vortex formation noticed during the visual Observations made in the early stages of the work. A three-stage digital computer program prepared by Rao (1972) was used to analyze the data taken in this heat transfer study. The first stage deals with the calculation of the heat transfer coefficients, the second stage u+h the calculation of the dimensionless groups used in the correlation, and the third and final stage iyith the estimation of the parameters C, a, b, c, and d i n Equation 13 using linear regression analysis. Two cases were considered : in the first all constants in Equation 13 were fitted and in the other, b and c were taken to be 0.33 and -0.14, respectively. The values chosen for b and c have been fairly well established for heat transfer to pipe flow and to some extent for heat transfer to Xewtonian fluids in jacketed agitated vessels. I n both cases, the value of the constant, e , in the modified Reynolds number is evaluated by trial and error such that the resulting equations predict with a minimum of average deviations. X value of 4 for e has thus been secured. Table I11 summarizes the results of the data fitted. When all the constants were independently fitted, the exponents on the Reynolds number and the Prandtl number are in fair agreement with the usual values found for these exponents. However, the exponent on the viscosity ratio
sx,dK*
-1
5x1 0
'
1
'
I
I0 '
k
1
I
4
I
I
6
,
I
C
8 IO4
Nureell No., calculated
Nurtelt No., calculated
Figure 13. Comparison of measured and calculated Nusselt number (jacket side)
Figure 14. Comparison of measured and calculated Nusselt number (coil side)
number is varying from the generally reported value of -0.14. There is, however, some disagreement in the literature on the exact value of this exponent. Uhl (1955) finds -0.24 for turbine agitators and -0.18 for anchor agitators. Chapman e t al. (1964) also reported -0.24 for turhiiie agitators. There values are all higher than the commonly reported value of -0.14 as initially obtained by Sieder and T a t e (1936) for heat transfer in pipe flow. On the other harid, Ckristiansen and Peterson (1966) report the value of c to be -0.10 for pseudoplastic fluids in pipes. hlalina arid Sparrow (1964) obtained a value of -0.05 for pure liquids iii turbulent pipe f l o ~ Sandal1 . and Pate1 (1970) obtained the value of c a t -0.02 for turbine agitators in pseudoplastic fluids. A proper evaluation of velocity profiles will permit a better understanding of this viscosity ratio number and of influence on heat transfer. When b aiid c are assumed to be 0.33 and -0.14, respectively, the average and standard deviations of the experimental data (Table 111) from the correlation remain approximately the same as for the case when all the constants are evaluated by the regression analysis. Since the exponents on the l'raridtl number and viscosity ratio number have been fairly ne11 established for heat transfer to pipe flow and to some extent for heat transfer to single-phase fluids in agitated vessels, i t is desirable to retain this form of the equation for mechanically stirred gas-liquid heat transfer. Thus, the recommended equations are:
The two equxtionr were used to predict the Susselt number for each of the data point4. The results were compared with the measured valueq and tile average and standard deviations were calculated for each correlatioii ab:
Jacket
h_2_4
-
li
and coil
h,D --
k
-
The experimental data for jacket and coil are plotted according to these equations in Figures 13 and 14, respectively.
Standard deviatioii
=
Equat'ion 14 for the jacket describes the data of about 208 points with a n average deviation of +7.4% and staridard deviation of 10.1%. Similarly, Equation 15 for t h e coil describes the data of about 183 points with a n average deviation of 7.2% arid st'andard deviation of 9.8'%. 130th the errors compare favorably with the expected experimental error in the measurement of physical properties, temperature, and flow. Conclusions and Recommendations
The following five conclusions were drawn from the present investigation: The heat transfer rate from coildelising steam to a gasliquid mixture iii a mechanically agitated vessel and from the mixture to the fluid in the coil increases with the gas velocity for a given impeller speed. The increase in heat transfer coeficient was mainly attributed to a n increase in turbulence near the heat'ed wall caused by gas bubbles resulting in disturbances and decrease of effective thickness of the laminar sublayer. The influence of gas velocity is more proiiounced a t lower impeller speeds than a t higher values and is apparently due to the increase in turbulence being margillally more than the irnpeller could create a t higher speeds. The coil heat transfer coefficients are slightly higher in magnitude than the jacket heat transfer coefficients in the air-water system and lower with air-oil systems. Two opposing factors, the higher viscosity because of lojyer coil m l l temperature aiid higher intensity of turbulence because of higher population of gas bubbles near the coil, could arInd. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
195
Table IV.
Equations for Physical Properties of Oils
(Temperature range 30-90°C)
Transformer oil (ShellDiala oil B) Clavus 33 oil (Shell clavus 33 oil)
Thermal conductivity, k, k g cal/hr m2 OC/m
Heat capacity, Cp, kg cal/kg 'C
Oils
+ 0.11 k = - 0 . 0 0 0 0 ~ 9 Tb + 0.11
C, = 0.001 To -I- 0.414 k C p
= 0.001
Tb
+ 0.408
=
-0.000067
count for this behavior. Coil side coefficients are always reported lower than the jacket for this mode of heat transfer. I n general, the heat transfer coefficient in two-phase gas liquid systems are higher than the single-phase liquids in agitated vessels, although part of the transferred heat is used to saturate the exit gas, often only a n undesirable side effect. The experimental data for jacket were correlated with average and standard deviations of 1-7.4 and 10.lyo,respectively. Similarly, the experimental data for coil were correlated with average and standard deviations of ~ k 7 . 2and 9.8%, respectively. The proposed equations can be used in the design for geometrically similar systems and in the range of variables studied in this work. The physical properties of liquids used are presented in Table IV in a form suitable for the computer calculations. Densities of the two oils were determined by weighing a known volume. Viscosity data were determined using a Brookfield Synchrolectric Viscometer (Model LVF). The thermal conductivity and specific heat of the oils were taken from the book by Kern (1950). The physical properties of water were obtained from Perry's Chemical Engineer's Handbook (1963). Nomenclature
a a,
A A,
= exponent of the Reynolds number group = vessel cross-sectional area, m2 = heat transfer area, m 2 = outside heat transfer area of helical coil, m2 = area of the inside wall of a mixing vessel in
contact with liquid, m 2 b = exponent of t'he Prandtl number group c = exponent of the viscosity ratio number group C = constant C, = heat capacity of water per unit mass, kcal/kg "C C,' = heat capaciby of air per unit mass, kcal/kg "C D = inside diameter of vessel, m d = exponent of the Froude number group d, = mean diameter of the helical coil, m do = coil tube outside diameter, m e = constant in modified Reynolds number g = acceleration due to gravity, m/hr2 h = individual film heat transfer coefficient, kcal/hr m2 "C h j = film coefficient of heat transfer, inside vessel wall t o liquid, kcal/hr m2 "C h, = film coefficient of heat transfer, vessel liquid to coil wall, kcal/hr m2 "C H = humidity of air, kg water/kg d r y air H ( = impeller height from vessel bottom, m HL = working height of vessel or liquid height, m H , = saturated humidity of air a t liquid temp., kg water/ kg dry air j, = j-factor, ( . ~ ~ ~ ) ( p , ~ p c ) o . l ~ j ( S p ~ ) 0 . 3 a j = number of parameters t o be fibted k = thermal conductivity of agitated liquid a t the bulk temperature, kcal m/hr m2 "C 1 = blade length, m
-4,i
Viscosity, 1, kg/hr m
196 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
Tb
- 0 . 6 1 Tb
p =
68.2
p =
136.1
- 1 19 Tb
Density, p, b/m8
p =
-0.63 To
p =
-0.72
Tb
+ 929 + 947
L n' n nb
= impeller diameter, m = number of turns in coil = number of points = number of turbine blades AT = agitator speed, rev./min L V ~ r= Froude number (dimensionless) = N 2 L / g N N " = Susselt number (dimensionless) = h,D/k or h,D/k LVpr = Prandtl number (dimensionless) = C,w/k N ..R- ~= Reynolds number (dimensionless)L*Xo/u Y R ~ *= modified Reynolds number (dimensionless) ( L p j p )
(LV + eV,)
N,,,
viscosity ratio number (dimensionless) p J p or p c / p pitch, m Q rate of heat transfer based on cooling water and air flow rates, kcal/hr p = rate of heat transfer based on cooling water flow rate, kcal/hr tat = inlet air temperature, "C t,, = outlet air temperature, "C t,, = inlet cooling water temperature, "C t,, = outlet cooling water temperature, "C A T = temperature difference, "C T b = bulk liquid temperature, "C T , = coil tube wall temperature, "C T,, = inside vessel d l temperature, "C V , = superficial gas velocity based on empty cross-sectional area, m/hr W , = mass flow rate of cooling water, kg/hr w = blade width, m p
=
= =
G R E E KLETTERS p
density of liquid a t bulk temperature, kg/m3 density of gas a t bulk liquid temperature, kg/m3 viscosity of agitated liquid a t bulk temperature, kg/m
=
po = p =
hr = viscosity of agitated liquid a t inside vessel wall temperature, kg/m hr = viscosity of agitated liquid a t coil wall temp., kg,'m hr p, X = latent heat of water in air, kcal/kg pu
References
Askew, W. S., Beckmann, R. B., Ind. Eng. Chem. Process Des. DelJPlo?I r
. 4. 8 - 7 -1- 1- (196,;)
I
\ - - - -
Bodman, S. W., Cortez, D. H., ibid., 6 , 127 (1967): Brooks, G., Su, G. J., Chem. Eng. Progr., 55 (lo), 34 (1959). ChaDman. F. S., Dallenbach, H. R., Holland, F. A., Trans. Inst. Chem. Engrs.,'42, T398 (1964). Chilton. T. H.. Drew. T. B., Jebens, R. H., Znd. Eng. Chem., 36, 510 (1944). Chriitiansen, E. B., Peterson, -4. W., A.Z.Ch.E. J . , 12, 221 (1966). Cummings, G. H., West, A. S., Ind. Eng. Chem., 42, 2303 (1930). DeXevers, N., A.I.Ch.E. J., 14, 222 (1968). Dunlop, Jr., Rushton, J. H., Chem. Eng. Progr. Symp. Ser., 49, No. 5 , 137 (1953). Fair, F. R., Lambright, A. J., Andersen, J . W., lncl. Eng. Chem. Process Des. Develop., 1 , 3 3 (1962). Freedman, W., Davidson. J. F., Trans. Inst. Chenz. Eng., 47, T25l (1969). Kast, W., Chem. Inp.-Tech., 35, 785 (1963). Kast, W., Int. J . Heat Mass Transfer, 5,329 (1962). Kern, D. Q., "Process Heat Transfer," 1st ed., pp. 803-10, McGraw-Hill, New York, ?;. Y., 1950. Chem. Zng.-Tech., 30, 400 (1958). Kolbel, H., Siemes, W.,
Konsetova, V. V., Znt. J . Heat X a s s Transfer, 9, 1103 (1966). Lavinga, P. J., Dickson, P. F., Znd. Eng. Chem. Process Des. Dmelop., 10, 206 (1971). Lehrer, L. H., ibid.,7,226 (1968). Llalina, J. A , , Sparrow, E. M., Chem. Eng. Sci., 19, 957 (1964). Oldshue, J. Y., Gret,ton, A. T . , Chem. Eng. Progr., 50, 615 (1954). Perry, J. H., Chemical Engineers' Handbook, 4th ed., RIcGrawHill, New zork, N.Y., 1963. liao, K. B;: Heat Transfer in mechanically agitated gas-liquid systems, PhD thesis submitted to Andhra University, India, 1972.
Rietema, K., Ottengraf, S. P. P., Trans. Znst. Chem. Eng., 48, T54 ( 197 0).
Ruditon, J. H., Costich, E. W., Evertte, H. J., Chem. Eng. Progr., 46, 39.5, 467 (19.70). Sandall, 0 . C., Patel, K. G., Znd. Eng. Chem. Process Des. Dwelop., 9 , 139 (1970).
Sieder, E. N., Tate, G. E., Znd. Eng. Chem., 28, 1429 (1936). Strek, F., Znt. Chem. Eng., 3,533 (1963). Towell, G. D., Strand, C. P., Ackerman, G. H., paper presented at AIChE London Meeting, England, 1965. Uhl, V. W., Chem. Eng. Progr. Symp. Ser., 51, (17) 93 (1955). Uhl, V. W., Gray, J. B., "Mixing Theory and Practice," Vol. 1, Academic Press, New York, N.Y., 1966. RECEIVED for review July 10, 1972 ACCEPTEDDecember 8, 1972 Tables 5-7 with two-phase heat transfer data will appear following these pages in the microfilm edition of this volume of the Journal. Single copies may be obtained from the Production Department, Books and Journals Division, American Chemical Society, 1155 Sixteenth St., K.W., Washington, D.C. 20036. Refer to the following code number PROC-73-190. Remit by check or money order $4.00 for photocopy or $2.00 for microfiche.
Number of Steady-State Operating Points and Local Stability of Open-Loop Fluid Catalytic Cracker Wooyoung Lee" and Alan M. Kugelmanl Mobil Research and Development Corp., Research Department, Paulsboro, LV.J.08066
The existence of multiple steady states and local stability of an open-loop fluid catalytic cracker (FCC) were studied. Based on a simplified mathematical model developed, we found ihai the FCC had no multiple steady states over a wide range of the operating space of practical interest, and that the open-loop FCC was locally stable under normal operating conditions. However, the conclusion on the uniqueness and stability was extremely sensitive to both model and parameters.
T h e fluid cat'alytic cracking (FCC) unit, owing t o the tight interaction b e b e e n t h e regenerator and the reactor, is dynamically sensitive. It is difficult t o accomplish satisfactory control of t h e F C C operation under frequent disturbances. Since t h e F C C unit is one of t'he principal refinery processes for gasoline production, a n understanding of its dynamic characteristics is extremely important. I t is, therefore, not surprising that a number of investigators have studied the dynamics and control of the F C C (Iscol, 19iO; Kane e t al., 1962; Kurihara, 1967; Pohlenz, 1963; Snodgrass et al., 1968). I n this paper we are concerned with two specific aspects of F C C dynamics. First, we examine whether multiple steady states exist over a large region of practical importance. Second, we study the local stability of the uncontrolled F C C (open loop). .-I simple mathematical model of the F C C was first developed and used in the subsequent analysis. Steady-state equations were solved tiumerically to obtain steady-state operating points. The dynamic equations were linearized about known steady states, and conventional techniques for linear stability analysis were applied. Basing our conclusions 011 the mathematical model developed and employed here, we found t'hat the F C C had a unique steady state over a large region of practical interest. I n addition, we found t h a t t'he open-loop F C C was locally stable about the steady-st'ate
points studied. However, instability was encountered for certain unrealistic situations; for example, a case in which excess oxygen concentration 111 the regenerator was extremely high.
Present address, Chemical Engineering Department, The University of Uaryland, College Park, Lid. 20742.
h simplified mathematical model of the F C C corisiits of the four ordinary differential equations (Equations 1-4) listed
Process Description
The catalytic cracking process (Figure I) has been widely employed in converting heavy hydrocarbons into more valuable lighter products such as gasoline. The catalytic section of the F C C system consists of two vessels: a reactor and a regenerator. Gas oil feed is introduced at the bottom of t h e riser, where it meets hot catalyst from the regenerator. The cracking reaction starts immediately while the mixture of catalyst and gas oil passes through the riser into the reactor. During the encounter of catalyst with gas oil, carbonaceous material deposits on the catalyst surface, which coiisequently deactivates the catalyst. The product vapor is transferred t o t h e fractionation section, and the spent catalyst flows down through the reactor standpipe into the regenerator, where the carbonaceous deposits on t h e catalyst surface are burned off. The regenerated catalyst passes through the regenerator standpipe into the riser. This completes the catalyst circulation circuit. Mathematical Model
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