HEAT TRANSFER T O SOLID-LIQUID SUSPENSIONS IN AN AGITATED VESSEL F R A N T I S E K
F R A N T I S A K ' ,
J.
W.
S M I T H ,
A N D
J l R l
DOHNALZ
Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Canada
Theoretical and experimental studies of heat transfer coefficients at the wall surface of a conical bottomed, baffled, centrally agitated, cylindrical vessel to nonsettling suspensions are discussed. Heat transfer coefficients are presented as functions of dimensionless groups for impellers of the propeller type. The results are given for Reynolds numbers between 270,000 and 2,000,000, Prandtl numbers between 1.9 and 6.2, and a ratio of impeller to vessel diameter between 2 . 3 7 and 4.74. The ratio of particle to liquid density ranged from 1.1 5 to 2.85 and of heat capacity from 0.1 92 to 0.4. The volumetric concentration of solids particles was up to 15%. A dimensionless, empirical correlation has been developed from over 300 measurements of heat transfer coefficients to Newtonian slurries which correlates the data with a probable Comparison of the correlation with correlations of heat transfer to pure liquids and to error of *6%. conveyed slurries shows reasonable agreement in the limiting cases.
agitated cylindrical vessel is one of the most frequently encountered items of process equipment. Agitation can result in great increases in the heat transfer, mass transfer, and simultaneous heat and mass transfer in the vessel. All these processes can be important in the design of equipment, but especially in the case of heterogeneous, solid-liquid stirred reactors in which the reaction is strongly exothermic or endothermic, the rate of heat transfer is most important. Crystallizers are an example of such a mixing reactor. This work, therefore, has been concentrated on heat transfer in two-phase liquid-solid systems in agitated vessels for which no quantitative data are available at present. The slurries selected were Newtonian in behavior. HE
Previous Work
Heat Transfer in Agitated Vessels. Heat transfer in agitated two-phase systems involving liquids and solids has not yet been treated satisfactorily in a theoretical and experimental study. Cummings and West (1950), Hixon (1944), and Aiba (1956) give only qualitative results from very few measurements; the presence of the particles decreases the rate of heat transfer from the wall into agitated slurries, in comparison with heat transfer into pure liquids under the same thermal, hydraulic, and geometric conditions. Many studies have been made of heat transfer in agitated vessels containing pure single-component systems. This work can be classified according to the type of impellers used and the shape of the heat transfer surfaces. Work in this field has been reviewed by Chapman and Holland (1965a, 1965b), Strek (1960), Frantisak (1966), and Uhl and Gray (1967). Most of the results of these investigators have been correlated in dimensionless equations using only fundamental dimensionless groups in the following form :
(3
NNu = const(NRe)"(Npr)b
(1)
I n some cases Equation 1 has been complicated with further ratios, such as d/D, H I D , etc. A number of studies have been made of heat transfer to
conveyed slurries, which are defined as a mechanical suspension of solid particles in a fluid. These systems are usually characterized by the concentration of the particle as well as by the diameters and properties of the particles, and the physical properties of the fluid. Those fundamental properties which can influence heat transfer to the slurry are viscosity, thermal conductivity, density, and heat capacity. These properties must be defined consistently for slurries. Viscosity depends on temperature, the volume concentration of the particles, the shape and size of the particles, and the viscosity of the fluid medium. The problem of characterizing the influence of the particles, and of the dispersing liquids, on the viscosity of the slurry has been studied by many workers, including Einstein (1906). A particularly useful study was made by Vand (1948a, 1948b, 1948c), whose work was later confirmed by Thomas (1965). The relative viscosity is a function only of volume particle concentration, 6, and is given by p7
=
!?
= (1
+ 2.5 @ + 7.54 &)
(2)
Pf
Hamilton has used electrical conductivity analogy to determine the thermal conductivity of the slurry:
(3) The validity of Equation 3 was confirmed by Orr and Dalla Valle (1954). The heat capacity and density are usually determined from the properties of the individual phases in the ratio of the concentrations of the phases. Measurements of heat transfer to nonsettling slurries flowing in pipes have been made by Bonilla et al. (1953), Orr and Dalla Valle (1954), Salamone and Newman (1955), Miller et al. (1956), and Kofanov (1962). I n general, their results have been correlated by dimensionless groups of various types, including Dittus-Boelter correlations, and by equations of the Dittus-Boelter type plus additional dimensionless groups which are intended to define the relationships among the physical properties of the particles and the liquids. Theoretical
Post-doctoral fellow on leave from Institute of Inorganic Chemistry, Usti-nad-labem, Czechoslovakia 2 Technical University of Prague, Prague, Czechoslovakia 188
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Fundamental Equations. Solid particles dispersed in a liquid will make fundamental changes in the behavior of the liquid and its properties. The two-phase system, like any
fluid system, can be defined by fundamental equations, which have been derived by Frantisak (1966) and Valchar (1962). CONTINUITY EQUATION (4)
4 pf 1
-4
(- - -
- + +
U S V l J S ) (5)
Ps g
PI
PI
ENERGY EQUATION
with the boundary condition:
diffusion coefficients, Prandtl number, and velocity profile, are unknown. More measurements are required before Equations 9 and 10 can even be solved for slurries in open pipes. The problem in mixers is more complicated than in the open pipe case, and data required for solution of the equations are not available. Many authors, such as Voncken (1965) and Nagata (1965), have tried to define fluid flow in mixing systems. Their descriptions, however, have been largely qualitative. I n these circumstances, therefore, no solutions for Equations 9 and 10 for the case of mixing in two-phase liquid solid systems are possible at present. General Expression of Dimensionless Groups. Since solution of the analytical expressions for heat transfer to mixing systems is impossible, recourse must be made to dimensional analysis in which the preceding equations may be used as a basis for an analysis of the dimensions. Equations 4, 5, 6, and 7 may be used to obtain the following general function for heat transfer in mixing systems in terms of the following dimensionless groups:
T h e assumptions which are made in the derivation of these equations are : The solid particles are dispersed uniformly and evenly in the liquid (4 = constantd@/dr = 0 ; d$/dx = 0 ) . T h e density of the particles and liquids is constant in space and in time ( p a = constant, p / = constant). Solid-solid interactions are negligible. T h e flow is steady. These equations can be reduced to boundary layer equations if it is assumed that (Frantisak, 1966) : The flow is two-dimensional. Mass forces are negligible. The flow is isotropic, as follows:
___ 9
1 - 4
”>
+ as ba, - + Ds ax B
pn; C ),
(12)
T h e energy equation can be derived in the same way if the assumption is made that t,’ = t f ’ :
(5 Q, + ”D.)
+ (1 - 9 ) C f p f (”
bY
bX
a,
+ -bt Of)
0
(8)
The left side and the first three terms on the right side of Equation 8 are identical to those in the momentum equation for boundary layer of pure liquids. The fourth term of the right-hand side is the modification required to account for the presence of the solid particles. If it is assumed that us’ = u,’ and us’ = vf’-in other words, that the slip between the solid particles and the liquid is negligible (as in the case when the density of the fluid equals the density of the solid)-Equation 8 becomes Equation 9 :
dX
h = F(d; n; r; h‘; D ; vfW; k,; ps; Cs; g; VI; k,;
For nonspherical particles, the usual “shape factor” correction may be used. The desired dimensionless groups then are:
(’::I
Ps
This analysis contains no groups describing the geometrical relationships in the system and further recourse to dimensional analysis must be made to define the system completely. The literature and the fundamental equations previously derived show that heat transfer into mixing slurries of spherical particles will be influenced by the following variables:
(10)
Y’
Unfortunately, Equations 9 and 10 have not been solved in any realistic system because many of the variables, such as
(13)
The number of linearly independent dimensionless groups which must be taken into account in determining heat transfer into mixing slurries is 14. These are:
This very large number may fortunately be reduced by the following means: u , / u / . This group, the ratio of solid to fluid velocity, is a function of py/p, and Reynolds number, as determined by Friedlander (1957). I n the present work, the ratio was always very nearly unity because of the high iVRe and relatively low solids density materials used. h’/d. Zwietering (1958) has proved that the main influence of h’/d in slurries is on homogeneity. The influence on pure liquids of this group is not important. I n all the present studies, we have worked in the region where homogeneity was predicted by Zwietering’s (1 958) results, and hence h’/d may be neglected. r/d. From Kofanov’s (1962) and Salamone and Newmans (1955) articles, it follows that the influence of r/d on heat transfer into open pipe flowing slurries is negligible. Kofanov has shown that the r / d group varies to the 0.02 power for the r/d range of 0.7 X lo-’ to 0.5 X 10-4. Salamone determined the exponent to be 0.05 for r/d in the range 0.35 X lo-’ to 10-4. Because of the slight dependence of the expression on this group it has been neglected. k,!k,. For the same reason as in the case of r/d, k t / k , was not included in Equation 13. v y / v f . This group is important especially for highly viscous fluids. Since only aqueous slurries have been used in the present study, the variation of this group has been neglected. VOL. 7
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APRIL 1968
189
NE". For the turbulent region the Euler number, NE", equals a constant and is determined by the type and size of the impeller, as shown by Rushton (1953) and Van de Vusse (1955). NFr. The Froude group, N F r , is important only in the case of vortex formation in mixers, and since the present studies were all made in forced flow in the turbulent region this group is of little or no importance. I t can now be assumed that the remaining groups may be expressed in the usual exponential form, whence
(14) Equation 14 would be expected to apply only under the conditions of the above assumptions.
Table I. Physical Properties Glass Po[yvinyl Sand Dolomite Chloride
Density, kg./cm. meter 2600 2850 Heat capacity, kcal./kg. deg. 0 1920 0.222. Thermal conductivity, kcal./m. deg. hour 0.725 1.5= Shape Sphere Irregular Size, mm. 0.05 0 29.2 0.45 54,02 0.05-0.1 0.1 -0.2 6.45 15.13 0 . 2 -0.3 37.95 1.01 0 0 . 3 -0.4 55.1 a
At 50' C.
b
At 20' C.
c
Diakon
MG
1380
1150
0.24b
0.46
0.13* Sphere
Sphere
31.4 60.0 8.6
0.7 10.0 82.0 7.3
0 0
O.lc
0
At 40' C.
Experimental
The apparatus, shown in Figure 1, consisted of a cylindrical vessel with a cone-shaped bottom. The vessel was furnished with an external heating jacket; the dimensions are shown in the figure in millimeters. The electrically driven variablespeed (60 to 1, 600 r.p.m.) agitator was mounted with a 20mm. diameter impeller shaft vertically in the vessel center. Four propellers of 125-, 160-, 200-, and 250-mm. diameter were used. The number of blades in the propeller was three, the number of 60-mm. baffles was four, the ratio of s'/D = 1, and the fluid was flowing in a downward direction at the center. The distance between the impeller and the bottom of the tank was less than 1 diameter of the impeller. The volume of the tank was 255 liters, and the heat transfer surface was 1.135 sq. meters. 'The vessel, cover, and bottom were insulated with 4-inch thick glass fiber and heat losses were shown to be negligible. Temperature Measurements. The heat flux was measured by mercury thermometers with a probable accuracy of C. Six calibrated FE-KO thermocouples were *0.05' mounted on the vessel wall. These were used to measure the temperature difference between the wall and the bulk of the fluid, and the cold junction of the thermocouples was placed in the bulk of the fluid. The temperature differences were indicated on a six-point compensating electronic recorder. The accuracy over the whole range was *0.5%, scale length being 250 mm. The average indicated temperature differences were used in the calculations.
Solid Particles. Glass, sand (silica), dolomite, poly(viny1 chloride), and Diakon MG (polystyrene) were used as solids in the two-phase systems. The physical properties are shown in Table I. The properties were either measured or obtained from the literature. Only water was used as a carrier fluid. Method of Operation. Heat transfer measurements in the present work were conducted in the unsteady state. Heat fluxes were obtained for the mixing slurries from the following equation : (15) where AT is the time of measurement, Atb is the difference of the bulk temperature at the beginning and end of the time element, A T . The heat transfer coefficient is then calculated from h = q / A T , where AT is the logarithmic mean temperature difference between AT 1 and 4 T 2, and 4 T 1 and AT 2 are the differences between the wall and the bulk temperatures a t the beginning and at the end of the time increment, 4 ~ . I n all cases the starting temperature was about 10' to 20' C. less than the mean temperature at which the heat transfer rate was to be measured. A simultaneous record of the bulk temperature as well as the temperature difference between the wall and the bulk fluid was kept. Thus, two graphs are obtained, the bulk temperature as a function of time and the temperature difference as a function of time. The incremental time interval A T , was chosen to correspond to a temperature difference of about 5 2 ' to 4' C. about the mean temperatures selected for the experiments. This time interval is small enough to make possible the linear interpolation of all physical properties which depend on temperature. Experimental Measurements. The same program of experiments required to characterize the system was conducted for each slurry independently. In addition, further measurements were made to define the effect of Reynolds number better. A total of 363 measurements were made in the following ranges: 2.7 X lo5 4 NRe 5 2 X 108; 1.15 1.9 2.37 h' -
Figure 1. 190
Schematic diagram of experimental equipment
l & E C PROCESS DESIGN AND DEVELOPMENT
5
Npr 5 6.2;
5D 5 d
I 0.9;
n-
4.74;
5 PI C
5
2.85
5
0.4
0.192
52
0.017
5 _"_ 5
d
c/
I-'
= 0.0028
0.125
The assumption of uniform suspension of the solid particles in the slurry was checked with the equation of Zwietering (1 958). Viscosity measurements showed that the slurries were Newtonian over the entire concentration range. Results
All previous investigators except Chapman (1956a) obtained the parameters in Equation 1 by the method of crossplotting grgphs. I n the present work the parameters required for Equation 14 were obtained from a computer program by linear regression analysis, as were Chapman's. T h e parameters obtained are shown in Equation 16.
B. Dimensionless groups which describe the physical properties of the solid phase in proportion to the concentration of the solids. A comparison can be made between those dimensionless groups under Part A with the results for pure liquids and those dimensionless groups under Part B with the results for heat transfer to slurries in open pipes. Before a comparison between the groups described under Part A can be made with groups which apply only to heat transfer to pure liquids in mixed systems, the following conditions must be imposed : The density ratio p s / p I = 1; the specific heat ratio C,/C1 = 1 ; and + 0. If we put these conditions into Equation 16, the equation becomes indeterminate. I t is, therefore, necessary to assume that the function NNu= f($) is continuous in the experimental range $I from 0 to 0.125, which means that d ( N N , ) / d @exists, and therefore, the limit
-
lim
T h e probable error in the measurements is 26.07%. The experimental results are also shown graphically in Figure 2, where the measurements and the computer results by Equation 16 are compared. Discussion
No comparison is possible with the results of previous workers, because these are the first obtained for heat transfer to systems consisting of two phases (solid-liquid) displaying Newtonian behavior. I t is apparent that the right side of Equation 16 consists of two parts.
*+'
Nusselt number for pure liquids = 1 Nusselt number for slurries
This is equivalent to stating that the Nusselt number for slurries is not much different from the Nusselt number for pure liquids when the concentration of solids is small enough. I t can be seen from Figure 3 that the influence of solids concentration is not important when 6 is less than 1%. If the value 6 = 0.01 is used in the dimensionless group (+/l +) *J4, Equation 16 becomes:
-
The appropriate form when the viscosity ratio is included becomes
A. Dimensionless groups which describe the whole system and are the same as those for the pure liquid: N R e ; NP,; Dld.
I o4
8
t6
z
0
i = 4
4
W a E 0 0
E c
a
2 2
io3 io3
2 ( NN"),.,,
Figure 2.
4 MEASURED
-
6
8
IO4
Comparison of measured and calculated modified Nusselt number VOL. 7
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191
4000 '
0
4
7
3500
z -
E
za
3000
0 d
9
2.96
A . BROWN ( 1 9 4 7 ) B I STREK ( 1 9 6 2 ) C , KAPUSTINll9631 D , PRESENT WORk
2500
0
+
I
0.15
0.1
0.05
I
I
lo5
-
(NRe)f
I-+
Figure 3. Effect of volume Concentration of solid particles on ( N K ~ L
For most mixers, D / d is usually equal to 3.0 and hence
Expressions which are commonly used for determining heat transfer to pure liquids in stirred tanks are given below for comparison Brown et al. (1947).
I
I
I 1 1 1
-
5
I O6
Figure 4. Comparison of previous and present work for pure liquid systems
of heat transfer to slurries conveyed in pipes, because no previous work has been published on heat transfer to solid-liquid slurries in mixers. Those correlations which appear to be most appropriate for comparison because they apply to the case of heat transfer to fine nonsettling suspensions, are those of Salamone, whose correlation is
(;)"'"" Strek (1962)
(23)
and the work of Kofanov
with s'/d = 1. Kapustin (1963)
A graphical comparison of Equations 19, 20, 21, and 22 is made in Figure 4, in which are plotted (N~~)I/(N~,),*
x
(k)
against
( N R A ~
I
for arbitrary values of Prandtl number in the range 2 to 6, Reynolds number in the range 105 to 106, and viscosity ratio p / p w . The values of D / d = 3, s'/d = 1, were chosen. This figure shows that the agreement of Equation 19 with the results of Brown and Strek is excellent. However, the results of Kapustin lie considerably below the other three equations. Unfortunately, it has not been possible to find the reason for the deviation. The comparison between those parts of the present correlation which are affected by the presence of the solids and other work must necessarily be made with results 192
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Table I1 compares the exponents on the comparable dimensionless groups for Salamone's work, Kofanov's work, and the present work. The exponents are in reasonable agreement except in the case of the exponent on the group, $/(l $). Hixson has shown that the particles must affect the coefficient on this group in such a way that it is negative. This result is confirmed in Figure 3, where the Nusselt number is plotted against the dimensionless group $/(l - +), where Reynolds number, Prandtl number, diameter ratio, density ratio, and heat capacity ratio are all constant.
-
Table II. Exponents PJPt
Salamone Kofanov Present work
... -0.12 - 0 .16
CJCf 0.35 -0.15 0.13
kdkj 0.05
...
...
4/(1
- 4)
01015 -0.04
Acknowledgment
SUBSCRIPTS
The experimental part of this program was carried out in the Institute of Inorganic Chemistry, Usti-nad-Labem, Czechoslovakia. The authors are grateful to the University of Toronto for a Ford Foundation Postdoctoral Fellowship, which was held by F. F., and to the University of Toronto for the use of IBM 7094 computer facilities.
m
Nomenclature
A C
D F G
H T
G
Q
a
b C
d e
f g
h
h’
k n
P f? T
s’
t tb
--
u, u, u, 0 , u ’ , u ‘ , t’ x , Y, P Y
7
P -$
= surface, cross section = heat capacity = diameter of vessel
= mathematical function = weight = height of liquid level = temperature = velocity vector = heat amount = constant = constant, width of impeller blade = constant = impeller diameter = constant = constant = acceleration due to gravity = heat transfer coefficient = submersion of impeller = constant, thermal conductivity coefficient = r.p.m., normal direction = pressure = heat flux = particles diameter = rise of impeller = temperature = bulk temperature = coordinate parts of velocity vector = average values of velocity parts = fluctuating parts of u, u, t = coordinate system = dynamic viscosity = kinematic viscosity = time = density = volume concentrations of solid particles
= = = = =
T
S W
dispersing phase slurries relative value solid phase wall
DIMENSIONLESS GROUPS = Reynolds number for mixing systems 1VRe NPr = Prandtl number “U = Nusselt number fiFr = Froude number 1VE“ = Euler number literature Cited
Aiba, Shurichi, Chem. Eng. (Japan) 20, 593 (1956). Bonilla, C. F., Cervi, A., Jr., Colven, T. J., Jr., Wang, S.J., Chem. Eng. Progr. Symp, Ser. 49, 127 (1953).
Brown, R. W., et al., Trans. Znst. Chem. Engrs. 25, 181 (1947). Chapman, F. S., Holland, F. A., Chem. Eng. 7 2 , 2 , 1 5 3 (1965a). Chapman, F. S., Holland, F. A., Chem. Eng. 7 2 , 4 , 1 7 5 (196513). Cummings, G. H., West, A. S., Znd. Eng. Chem. 42, 2303 (1950). Einstein, Albert, Anal. Phys. 19, 289 (1906). Frantisak, Frantisek, Ph.D. thesis, Technical University of Prague, 1966.
Friedlander, S. K., A.Z.Ch.E. J. 3, 381 (1957). Hixson, A. W., Znd. Eng. Chem. 36,488 (1944). Kapustin, A. S., Intern. Chem. Eng. 3, 514 (1963). Kofanov, V. J., Inzh. Fir. Zh. 5, 27 (1962). Miller, A. P., et al., Trend Eng. Univ. Wash. 8, 15 (1956). Nagata, Shingi, Chisa International Chemical Engineering Meeting, Vedeckotechicka Spolecnost, Prague, Czechoslovakia, D4.2, 1965.
Orr, Clyde, Dalla Valle, J. M., Chem. Eng. Progr. Symp. Ser. 50, 29 ( 1 954).
Rushton, J. H., Chem. Eng. Progr. 49,161 (1953). Salamone, J. J., Newman, Morris, Znd. Eng. Chem. 47, 283 (1955). Strek, Frederick, Chem. Stosowana 6, 3, 329 (1962). Strek, Frederick, Zeszyty Nauk. Politech. Szezecinskiej 14, 77 (1960). Thomas, D. G., J. ColloidSci. 20, 267 (1965). Uhl, V. W., Gray, J. B., “Mechanically Aided Heat Transfer. Mixing. Theory and Practice,” Vol. 1, Academic Press, New York, 1967.
Valchar, Jiri, Res. Report SVUTT, Prague, Czechoslovakia 6205015 (1962).
Van de Vusse, J. G., Chem. Eng. Sci. 4, 178-200 (1955). Vand, Vladimir, J. Phys. Colloid Chem. 52, 277 (1948a). Vand, Vladimir, J . Phys. Colloid Chem. 52, 300 (194813). Vand, Vladimir, J . Phys. Colloid Chem. 52, 314 ( 1 9 4 8 ~ ) . Voncken, R. M., Brit. Chem. Eng. 1 0 , 1 2 (1965). Zwietering, T. N., Chem. Eng. Sci. 8, 244 (1958). RECEIVED for review May 1, 1967 ACCEPTEDNovember 9, 1967
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