than the increase in kinetic energy force. This leads to higher drop volumes with the increase in total flow rate. It is evident that in spite of the approximate nature of the drag terms used, the agreement between the experimental and calculated drop volumes has turned out to be good. In conclusion, it may be said that the existing concepts on drop formation from single nozzles can be extended to drop formation from multiple orifices, and sizes of drops formed from a distributor of sieve plate type can be predicted in the absence of coalescence.
Nomenclature A, B, C, D = substitutions given in the text CD = drag coefficient for single sphere moving with same velocity as done by plane assembly, dimensionless CD = drag coefficient per sphere in an assembly, dimensionless D = diameter of the orifice, cm D I = inter-orifice distance measured center to center, cm g = acceleration due to gravity, cm/sec2 it = power law parameter, g/cm sec2-n K1, K2 = substitutions given in the text n = power law parameter, dimensionless Q = flow rate through each orifice, cm3/sec rE = radius of the force balance drop, cm Rc = radius of the expanding drop, cm t = time,sec t c = time of detachment, sec V = volume of the drop, cm3 Vg = volume of the drop that is completing formation on multiple orifice plate, cm3
V c = volume of the growing drop on multiple orifice plate, cm3
VE = volume of the drop at the end of first stage, cm3 V F = final volume of the drop, cm3
X,
= function of the parameter n , dimensionless Greek Letters a = correction factor, dimensionless y = interfacial tension, dyn/cm 1 = viscosity of continuous phase, g/cm sec pc = density of continuous phase, g/cm3 Pd = density of dispersed phase, g/cm3 pS = Pd + ( l % d ~ g/cm3 c,
Literature Cited Hayworth, C. B.. Treybal, R. E., Ind. Eng. Chem., 42, 1174 (1950). Heertjes. P. M., de Nie, L. H., de Vries, H. J., Chem. Eng. Sci., 26, 451 (1971). Johnson, H. F., Bliss, H.. Trans. Am. Inst. Chem. Eng., 42, 331 (1946). Kumar, R., Chem. Eng. Sci., 26, 177 (1971). Kumar, R., Saradhy, Y. P., hd. Eng. Chem., Fundam., 11, 307 (1972). Kumar, R., Kuloor, N. R., Adv. Chem. Eng., 8, 255 (1970). Markowitz, A., Bergles, A. E., Chem. Eng. Progr. Symp. Ser., 66, 63 (1970). Null, H. R., Johnson, H. F., AIChEJ., 4, 275 (1958). Ramakrlshnan, S., Kumar, R., Kuloor, N. R.. Chem. Eng. Sci., 24, 731 (1969). Rao, E. V. L. N , Kumar, R., Kuioor, N. R., Chem. Eng. Sci., 21, 867 (1966). Rowe, P. N., Henwood, G. A,, Trans. Inst. Chem. Eng. (London), 39, 43 (196 1). Sankara Sreenivas, N., Kumar, R., Kuioor, N. R., Hydrocarbon Process., 48, 216 (1969). Scheele. U. F.. Meister, E. J., AIChEJ., 14, 9 (1966). Wassermann, M. L., Slattery, J. C., AlChE J., I O , 383 (1964).
Received for review January 24, 1975 Accepted July 28,1975
Eddy Thermal Diffusivities in Liquid-Fluidized Beds K. Ragunathan and K. Subba Raju’ Department of Chemical Engineering, Indian institute of Technology, Madras-600036, India
Time-averaged temperature profiles are measured downstream in liquid-fluidized beds by injecting hot water at the axis. Eddy thermal diffusivities are computed using Taylor’s theory of eddy diffusion. Correlations are proposed to predict eddy diffusivities in liquid-fluidized beds based on the present and literature data.
Introduction Recently attention has been devoted to understanding the phenomena of mixing of fluids in pipes and packed and fluidized beds. Particulate fluidization represents among various modes of fluidization an ideal behavior. A study of diffusion in this type of systems provides a basis for understanding more complex systems. Previous investigations on mixing in empty tubes and fluidized beds have been reviewed in the literature (Groenhof, 1970; Davidson and Harrison, 1971). When a tracer with physical properties similar to the fluid is injected continuously on the axis, the spread of matter can be described by Taylor’s theory of turbulent diffusion. Eddy diffusivities can be linked with theory. Most investigators (Hanratty et al., 1956; Cairns and Prausnitz, 1960; Wicke and Trawinski, 1953) used the above technique to determine radial eddy mass diffusivities 82
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
in liquid-fluidized beds and no systematic data are available on their thermal equivalents. Hence the purposes of the present work are (1) to determine radial eddy thermal diffusivities in liquid-fluidized beds by continuously injecting hot fluid a t the axis and (2) to present a generalized correlation for predicting eddy diffusivities in liquid-fluidized beds.
Theory Taylor’s theory of turbulent diffusion has been extensively treated in the literature (Batchelor, 1960; Hinze, 1959). In the case of empty tubes and liquid-fluidized beds, the velocity profile is nearly flat in the core and hence the field is assumed to be homogeneous and isotropic. Consider a fixed source located at 3: = 0. The displacement of a fluid element after time t is given by
X=LtlJdt Due to the random motion of the fluid elements, a plus or minus value of X is possible and hence for large number of is zero, but x'l yields a particles the average value (i.e. finite value and is a measure of the spread for a given diffusion time. Consider an instantaneous source or sink located a t the center of a pipe. A source which emits heat could be described by the following differential equation
x)
where 1d F @ ( t )= -2 dt
The eddy diffusivity is related to equation
(3)
3 by
the following
Unlike molecular diffusion from a fixed point source, E H will vary with time, but for larger times of diffusion, E H levels o g t o a constant value. Hence AX2/Az = slope of the plot of X2 vs. z a t larger values of z . For the case in which the contribution of the axial term is negligible, eq 2 becomes
aT
a2T 1 aT
[-+-- r ar ]
@(t)
-= at
N
eXP
[-
2~ J'@(t) d t
r2 2 L t 2 @ ( t )d t
When the diffusion time is long enough so that the material reaches the container walls, the boundary conditions become, for t > 0, T = T,, at z = + a , T = T F a t z = -a, and aTlar = 0 at r = a. The solution of eq 7 with the above boundary conditions is
where a&, are the eigenvalues of Jl(aPn) =
0
20, valves; 4, 5, centrifugal pumps; 8, 9, 10, control valves; 11, 12, 13, 26, rotameters; 15, 16, calming sections; 17, supporting screen; 18, test section; 19, expansion section; 21, sQhd feed; 22, discharge line; 23, tracer tube; 24, tracer tube assembly; 25, tracer side thermocouple; 27, needle valve; 28, 35, bypass lines; 29, thermostat; 30, column side thermocouple; 31, traverse mechanism; 32, selector switch; 33, reference junction; 34, microvoltmeter.
(7)
ar2
Let N be the amount of heat input per unit length for the two-dimensional case emitted into the field at zero time from r = 0 and z = 0. If the wall reflections are neglected the boundary conditions become, for t > 0, T = T Fat r = + a and T = T F a t z = -a. The solution of eq 7 with the above boundary conditions is
T-Tp=
Figure 1. Experimental setup: 1, feed tank; 2, 6, 14, drains; 3, 7,
(10)
Equations 8 and 9 may be used to represent a continuous source by putting t = z/U,. Substituting for N (Appendix A) and using eq 4, eq 8 and 9 reduce to
Apparatus and Experimental Procedure The general layout and the essential features of the apparatus used in the present investigations are illustrated schematically in Figure 1. The fluidization column consisted of an equalizing section, a test section, and an expansion section. The test section (50 mm i.d.) was made up of Perspex for visual observation. The calming section was packed with 12.5-mm glass spheres. This ensured uniform distribution of flow and helped in breaking large-scale eddies that might have been formed. The bed was supported on a 0.3-mm stainless steel screen reinforced on the underside by a sieve plate held in position between two flanges. Solids were introduced into the column through a 25.4-mm connection at the top of the column. The tracer tube was 3.4 mm i.d. and positioned a t the center of the pipe. The tip of the tracer tube was slightly tapered to minimize the entrance effects at the injection point. A thermostat provided hot water and was regulated through a rotameter using a needle valve and bypass arrangement. The tracer temperature was measured using a thermocouple fixed to the tube. Radial temperature distributions were measured using a traversing mechanism. All the temperature measurements were made using a 1-mm nickel-nickel chrome thermocouple. Preliminary experiments were conducted in an empty tube and the temperature profiles were recorded at z = 1to 5 cm in steps of 1 cm. Since the fluid was under turbulent condition, tracer temperature had a great effect on measurement. When the thermocouple tip was very close to the source, higher heat input caused rapid fluctuations in measurement. Based on trial runs. the thermostat was set at Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
83
Table I Diameter of the particle, m m 4.94 2.43 1.255 0.78
;.1\_:f_; 20 r
Galileo no. Porosity
2, cm
x 10-4
0.950.45 0.92-0.45 0.9 2-0.52 0.92-0.59
1.5-3.0 1.5-3.0 1.5-2.5 1.5-2.5
335 35 4.7 1.36
X)r
Re: 35000
-z :Goussmn *o
!I+
z: 2.0 o Experimental Goussmn
I-
-
0 0
60°C when temperature profiles were measured a t z = 1 to 3 cm and at 80°C when z = 4 and 5 cm.
Water was circulated until the inlet water temperature attained a steady value. The flow rate of the column fluid was then adjusted to give the desired Reynolds number. The tracer was then allowed to flow and its flow rate was adjusted to the value calculated from the central line velocity which in turn was calcualted using Nikuradse’s logarithmic velocity distribution for a smooth pipe. As soon as the tracer temperature attained a steady value, temperature measurements were made. In the case of liquid-fluidized beds, temperature measurements were not made a t z = 1.0 cm because of rapid fluctuations. At z = 1.5 cm fluctuations were observed when the thermocouple tip was far away from the axis of the tube and were damped out as soon as the tip approached the center. The thermostat was set a t 60°C when the temperature measurements were made a t z = 1.5 cm and at 80°C when z = 2.0, 2.5, and 3.0 cm. Temperature measurements were not made a t z = 3.0 cm for the particles 1.255 and 0.78 mm because of fluctuations. The tracer was allowed to flow and the velocity was adjusted slightly above the settling velocity of the particle. This prevented the particles from entering the tracer tube when the column was packed with the particles. A known amount of particles, predetermined from the height of the bed to be maintained (60 cm) and the desired porosity, were put in the column and the bed was expanded by circulating water. As soon as the inlet water temperature attained a steady value, the tracer flow rate was adjusted . equivalent to the average velocity of U s / € Measurements were made after the tracer temperature attained a steady value. The range of variables covered is shown in Table I. Reduction of D a t a Empty Tube. Temperature profiles extended only up to 1.0 cm from the flow axis, indicating that the presence of solid boundaries haveKo effect on diffusion. Thus eq 11 was used to compute X 2 from the measured temperature profiles using Newton-Raphson root-seeking technique (Appendix B). Values of were plotted against z and from the constant slope, E H was calculated using eq 6. Fluidized Beds. The value of (U,,/U,) is nearly equal to 1 in the case of liquid-fluidized bed in the central region. Thus a t r = 0, eq 11 becomes
04
r
1o
08
0
cm
04 r
, cm
0-8
r R e i loo00
aV.
F L
,
I-
,
0
Experimental
- Gaussian
5
Re =15ooo z =40 o Experimental
Y
0
0
04
r, cm
r,cm
Figure 2. Typical plots of temperature distribution in empty tube.
Z ,cm
x2
To - - T F--b2
(13)
T T - T F 2x2 Equation 13 was used to compute 9 from the measured central line temperature a t z = 2 and 2.5 cm and eq 4 was used to compute E H . Values of E H computed from the measured central line temperatures at t = 2.0 and 2.5 cm were averaged to compute Peclet number.
Results a n d Discussion Empty Tube. Radial temperature profiles were obtained downstream from a hot water source located at the axis of a tube through which water was flowing in the Reynolds 84
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
Figure 3. Typical plots of x’l vs. z for empty tube.
number range 5000 to 45000, which is in the range for flow of water through the beds of fluidized particles investigated in the present studies. Typical plots of radial temperature profiles are shown in Figure 2. The temperature profiles follow a Gaussian distribution confirming the validity of the assumed model. Computed values of are plotted against z and such typical plots areshown in F i g u r e L T h e source is finite a t z = 0 and hence X 2 # 0 at z = 0. Xo2 is initial displacement at source which can be derived (Hanratty et al., 1956) as
r2
z:I 50
a Present work
0
P
Experimentol
d p = 0.243 cm
3.0
AM E, u-
.u,
t
I-
m m
01
Z =2.0~177 6 = 0.7053
LL
(I
I
1
,
I
1
1
0 1
1
1
0
1
06
0.3
0
0.9
06
0.3
r , cm
C4
r,cm
- Goussion I
0
c
0
0
d
a
0
0.3
a
e
e . a e
06 r, c r n
0 9 0
e Present work o G r o e n h d (1970)
_.
I
- -.-
I K,
20
40 ~e
0.9
Figure 6. Typical plots of temperature distribution in fluidized bed.
a
0
06 r, crn
0.3
60
1 1
80
h
x ~5~
Figure 5. Plot of Peclet number vs. Reynolds number for empty tube.
15
-
* 2.43mm
Q
-a
s
1.255 mm
______a:~78mm
\
c 10 -Hence -
b2
Xo2 = - = 0.0072 4
5-
(16)
I t can be seen from Figure 3 that E c a t t a i n s constant value (i.e., linear relationship between X 2 and z ) , after a certain distance, justifying the assumption of constant eddy diffusion coefficient. However, a t higher Reynolds and number (Re > 20,000), linear relationship between z is observed only between z = 2 and z = 4 cm and the value at z = 5 cm is much lower than that predicted from the constant slope of the line. This may be due to the fact that the temperature measurements are susceptible to larger errors a t z = 5 cm where the temperature differences are in the order of 2°C. Eddy diffusivities and Peclet numbers computed in the present investigation and those reported by Groenhof (1970) from mass diffusion studies are shown in Figures 4 and 5 , respectively, and it can be seen that the orders of magnitudes are the same. The values of reduced eddy diffusivities, U H are nearly constant (Re > 10,000) and could This avbe represented by an average value of 6.8 X erage value of U H is higher than those reported in literature (Groenhof, 1970). This discrepancy may be due to the following reasons. In the case of mass diffusion studies, eddy diffusivities are computed from the measured concentration profiles far away from the source ( z = 40 cm) and hence the disturbances caused by the injector damps out as the diffusion time is very large. But in the present investigations temper-
4
0
S
OB
I
04
'
I
05
1
I
06
*
I
0.7
n
1
08
8
'
09
"
'
?Q
Figure
ature profiles are detected only up to z = 5 cm. Further, in the case of mass diffusion, the turbulent field is not subjected to temperature effects, whereas such effects cannot be eliminated in the present investigation, as the tracer temperature is quite above the temperature of the column fluid. Fluidized Beds. Typical plots of radial temperature profiles measured in liquid-fluidized beds are shown in Figure 6 and it can be seen that they follow Gaussian distribution. The Peclet numbers are plotted as a function of t as shown in Figure 7. The Peclet number decreases with increasing porosity, attains a minimum a t t = 0.75 and again increases with porosity. The existence of,minimum Peclet number is in agreement with the value ( t = 0.7) reported by Hanratty et al. (1956) and Cairns and Prausnitz (1960) from mass diffusion studies. The variation of Peclet number with particle Reynolds number is shown in Figure 8. The minimum Peclet number Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
85
;i
L
-
M-
Literature d a b Hanrotty et a1 (1956: e 0 9 3 mm glass v 0 4 7 m m glass
r c 02-
8
k
o 4.94mm 0 2.43 mrn
B
I
1.255mm
I
I
I
1
1
1
1
1
I
1
d
lo2
04:
005-
e 070 m m 1 1 1 0’
Present doto
o 1.255mm gbss P 0.78mm glass
L
Rep
OM-
Figure 8. Variation of Peclet number with particle Reynolds number.
Figure 10. Correlation of data on radial eddy diffusivities, Ga 105.
---L
P e = 0.0226(Rep/Rep,)-1~266Ga0.4825 (Rep/Rep, S 1) (19) Equations 18 and 19 could correlate the present and litera-’ ture data (Hanratty et al., 1956) with a standard deviation of 56.0% and 15.0%, respectively. (b) For systems whose Galileo number is greater than lo5
(17)
Pe = 4.675(Re~/Rep,)O.~~~(Repmep, 3 1) (20)
Equation 17 represents the present data and those reported by Hanratty et al. (1956) and Cairns and Prausnitz (1960) with a standard deviation of 30.0%. Galileo number and Rep, are system properties. Since the value of Rep/Rep, is equal to unity at the critical particle Reynolds number, the Peclet number is assumed to be a function of Ga and Rep/Rep,. The eddy thermal diffusivities for the systems investigated and eddy mass diffusivities reported in the literature (Hanratty et al., 1954; Cairns and Prausnitz 1960; Wicke and Trawinski, 1953) could be represented by the following equations as shown in Figures 10 and 11. (a) For systems whose Galileo number is less than lo5
Pe = 4 . 6 7 5 ( R e ~ / R e p , ) - l . ~ ~ ~(Rep/Rep, 6 1) (21)
Rep, = 0.12Ga0.617
Pe = 0.0226(Rep/Rep,)2~0Gao~4~z5(Rep/Rep, >, 1) (18) 86
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
Equations 20 and 21 correlate the present and literature data (Hanratty et al., 1956; Cairns and Prausnitz, 1960; Wicke and Trawinski, 1954) with a standard deviation of 16.0% and 13.8%, respectively.
Acknowledgment The first author is grateful to the Council of Scientific and Industrial Resrch, India, for the financial support rendered. Appendix A Equation for Source Strength. Taking energy balance for the tracer and the main fluid
aa2PFCpUav[Tav- T F ]= rb2Uc[TT - TavlP~Cp (AI) In the case of liquid fluidized bed, Ua, = Uc. Hence the above equation becomes
a2[Tav- T F ]= b2[TT- Tav]
(A21
Solving for Ta,
(b2/a2is very small compared to 1).Hence TT - Ta, = T T -
(5)
T T - T F=
T T [ I - (b2/a2)]- T F= TT - T F (A41 Substituting eq A4 in A2, eq A2 becomes
a2[Tav- T F ]= b2[TT- T F ]
(A51
Consider a differential area 2 r r d r Flux of diffusing material a t any point = UCPFC,(T- T F ) 046) Flux Q = HU2UavpFCp(Tav- T F )
(A7)
Then the probability of finding a particle of the diffusing material is equal to
Hence
N = TU'
(2)
- TF)
XZ
Substituting eq A5 in A10, eq A10 becomes
N = xb2 ( U ~ ) ~ F C , (T TF )
(All)
Since p~ and C, are unity for water, the above equation becomes
N = ab2
(%) ( T T - T F )
Appendix B Newton-Raphson Method. Equation 11 can be written as r=a T r=O
u,2
9
r=O
Nomenclature a = internal radius of the test section, cm b = internal radius of the tracer tube, cm U H = reduced eddy diffusivity of heat, EH/u*D, dimensionless C, = specific heat, cal/g°C D = internal diameter of the test section, cm d p = diameter of the particle, cm E = eddy diffusivity ED,E H or E M ,cm2/sec E D = eddy diffusivity of mass, cm2/sec E H = eddy thermal diffusivity, cm2/sec E M = eddy diffusivity of momentum, cm2/sec f = fanning friction factor, dimensionless Ga = Galileo number, [dp3(ps - p F ) p F g ] / p F 2 , dimensionless g = gravitational acceleration, cm/sec2 JO = Bessel function of first kind, zero order J I = Bessel function of first kind, first order N = source strength, cal/cm P e = Peclet number, DUJE or d,U,/E, dimensionless Q = source strength, cal/cm2 sec r = radial position, cm Ro2 = as defined by eq 14, cm2 p~, Re = tube Reynolds number, D U a V p ~ /dimensionless Rep = particle Reynolds number, dpUspF/pF,dimensionless Rep, = critical particle Reynolds number, dimehonless T = temperature of the diffusing material, "C Ta, = mixed average temperature, O C T F = temperature of the inlet water, "C To = temperature of the fluid a t r = 0, O C T T = tracer temperature, OC t = time,sec U = time averaged local velocity, cm/sec Uav = average velocity, cm/sec U , = velocity associated with the tracer material in the core, cm/sec U s = superficial velocity, cm/sec u* = friction velocity, Ua,+,cm/sec X = displacement, cm = mean squared traverse displacement of a number of fluid particles caused by eddy diffusion, cm2 = value o f F at z = 0 , cm2 YoL = initial displacement in y direction, cm2 z = axial distance, cm
TT - T F
0
Value must be such &at F ( F ) = 0. Let X n 2 is the value of X 2 for the nth iteration, then the value of for ( n 1)th iteration is given by
r2 +
where F'($) is thederivativeof F ( % ) and the subscript n indicates that F ( X 2 )and F ' ( X 2 )are calculated using X T .
Greek Symbols upn = positive roots of Bessel function of first kind, first order t = porosity p~ = viscosity of the fluid, g/cm sec p s = density of the particles, g/cm3 p~ = density of the fluid, g/cm3
Literature Cited Batcheior, G. K., "The Scientific Papers of Sir Geoffrey ingram Taylor", Voi. II, p 172, Cambridge University Press, 1960. Cairns, E. J., Prausnitz, J. M., AlChEJ., 6, 554 (1960). Davidson, J. F., Harrison, D., "Fluidization", p 291, Academic Press, London 1971. Groenhof. H. C., Chern. Eng. Sci., 25, 1005 (1970). Hanratty, T. J., Lattnen, G., Wilheim, R . H., AICMJ., 2, 372 (1956). Hinze, J. O,,"Turbulence", McGraw-Hill. New York, N. Y., 1959. Ramamurthy, K., Subba Raju, K., "Heat Transfer in Annular Liquid-Fluidized Beds", paper presented at First National Heat and Mass Transfer Conference, I.I.T., Madras, India, 1971, Paper No. HMT-55-71. Wicke, E., Trawinski, H., Chern. hgr. Tech., 25, 114 (1953).
Receiued f o r reuzew January 30,1975 Accepted July 3, 1975
Ind. Eng. Chern., Process Des. Dev., Vol. 15, No. 1, 1976
87
