Conclusions
T h e results of over-all segmental void fraction determination show that the over-all segment void fraction is 0.377 =k 0.033. T h e local void fraction determination by the present technique shows that the data lie within a band of 5l/2 standard deviations, and the over-all mean is close to the median. Radial variation of the peripheral average void fraction shows a definite cycling tendency which is damped toward the center of the model. Radial void fraction variation compares well with the results of other workers. Future work in this area may be carried out by the use of penetrating radiation coupled
with a direct radiation level counter. This would eliminate some of the inaccuracies contributed by radiophotographic processes. literature Cited (1) Benenati, R. F., Brosilow, C. B., A . I. Ch. E. J. 8, 359-61
(1962). (2) Roblee, L. H. S.,Tierney, J. W., Zbid., 4, 460-4 (1958). RECEIVED for review September 20, 1965 ACCEPTED January 31, 1966 Based on a thesis submitted to the University of Tennessee, Knoxville, Tenn., in partial fulfillment of the requirements for the M.S. degree, 1964.
VELOCITY PROFILES AND TRANSFER OPERATIONS A T T H E WALL OF A N AGITATED VESSEL WARREN S. ASKEW AND ROBERT B. BECKMANN
Department of Chemical Engineering, University of Maryland, College Park, Md. Velocity profiles have been measured in a cylindrical, flat-bottomed, baffled, agitated vessel in the flow regime near the wall. The velocity profiles at points near the wall surface were evaluated as functions of impeller type, impeller size, and impeller speed for varying vertical wall locations. The experimental velocity data are used with the results of a prior heat and mass transfer study to illustrate that the transfer of heat or mass from the wall of an agitated vessel closely resembles, in form and magnitude, the transfer in flat plate turbulent boundary layer flow.
agitated vessel has for decades been incorporated into the design of many industrial and experimental processes to mix or blend two or more materials, promote a chemical reaction in a multicomponent system, or transfer heat to or from a moving fluid. The most commonly used agitated system employs a rotating impeller positioned in a round, dished, or flat-bottomed cylindrical vessel. T h e transfer operations of heat and mass have been experimentally investigated for many years in many different variations of mixing systems, and the existing literature abounds in the results of such work. Experimental studies on heat transfer from the wall of a n agitated vessel have resulted in several average transfer coefficient correlations (4-6, 73); recent work has shown the dependence of local heat transfer coefficients on the vessel wall on the system geometric parameters (2, 3 ) . T h e interrelationship between the transfer of heat and mass in a n agitated vessel has also been investigated, and results show the analogous nature of the two operations. T h e fluid circulation patterns in a baffled agitated vessel are primarily dependent upon the type of impeller used in the system. For a radial flow impeller, in which the flow is outward toward the vessel wall in the plane of the impeller, two primary circulation loops are formed. At the wall of the vessel part of the fluid flows downward and part flows upward. For a n axial flow impeller, in which the flow is directly downward from the impeller, only one circulation loop is formed. T h e fluid flows downward from the impeller to the vessel bottom, then upward along the vessel wall, and finally downward back to the imHE
268
I & E C PROCESS D E S I G N A N D DEVELOPMENT
peller. These fluid circulation patterns and the development of impeller pumping capacities and mixing times have been studied by many investigators using several different techniques : photographic methods using tracer droplets or particles (9, 72), counting rates of a radioactive isotope (7), and use of Pitot and impact tubes (70, 7 7). All of these experimental studies have dealt with the bulk fluid in a n agitated vessel, and no determinations were made near the vessel wall. As a result, no connection between the transfer operations and fluid dynamics a t the vessel wall has been made. This research is thus directed toward the determination of the velocity distributions a t various points near the vessel wall in terms of the system physical and geometric parameters. T h e relationship between the heat and mass transfer and fluid dynamics at the vessel wall will also be determined. Operation and Experimental Equipment
The 10-inch i.d. baffled vessel and mixing system used in this study have been described in detail (2, 3 ) . Velocity measurements were made near the vessel wall using a n impact tube which extended through the wall. The impact tube was made from stainless steel hypodermic needle tubing, 0.050-inch 0.d. and 0.033-inch i d . , with a 0.020-inch diameter hole drilled through the tube wall to measure the dynamic pressure. A slant-type, differential, two-fluid manometer was used to measure the difference between the dynamic and static pressure at a given point in the vessel. T h e fluids used in the manometer were distilled water and Merriam red oil (specific gravity 0.827). T h e dynamic pressure probe was attached to a traversing mechanism (0.050-inch travel per revolution),
so that the impact opening could be positioned as a function of the distance from the vessel wall with respect to the vertical direction. T h e parameters varied in this study were: impeller type; impeller size; impeller speed; probe location ratio (distance of the probe from the vessel bottom divided by the impeller diameter) ; impact opening (distance of the dynamic pressure hole from the vessel wall); angular position of the impact opening; and the baffle position. T h e ranges of these parameters were:
Impeller type. 90' vertical blade turbine 45" turbine, and marine propeller Impeller size. 3 and 4 inches Impeller speed. 0 LO 18.25 revolutions per second Probe location ratio. 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00 Impact opening. 0.010 to 0.310 inch Angular position. 0' to 360" Baffle position. 15", 45', and 75' T h e ( H / T ) ratio was 1.00 and distilled water was the agitated fluid in all tests. T h e velocity a t a given point was calculated from the difference in dynamic and static pressures, which, after the correction factor determined by Goldman and Marchello is applied (7), is given by Equation 1.
flow in both cases is upward, or tangent to the vessel wall, with the maximum velocity occurring about 0.100 inch inward from the vessel wall. Velocity measurements were not made further than 0.500 inch from the wall, into the fluid regime, so no flow reversal toward the impeller was detected. T h e maximum vertical velocity increases with increasing impeller speed for both impellers. Figure 2 shows the relationship between the maximum velocity and the impeller speed for the propeller and the 45' turbine. T h e relationships are obviously linear and are given by : Propeller.
V,,,
= 0.44 N
(2)
45' turbine.
Vmax = 0.48 N
(3)
where velocities are expressed in feet per second and the impeller speed in revolutions per second. Effect of Impeller Size. Figure 3 shows the velocity profile data for the 3-inch diameter propeller under the same geometric conditions as shown for the 4-inch propeller in Figure 1.
V = 1.092 [ 2g,AP/p]'/2
(1) T h e velocity distributions near the vessel wall are presented as velocity profile graphs for the 90' and 45' turbine and marine propeller impellers. Experimental Results
Effect of Impeller Speed. Figure 1 shows the velocity profiles for the 4-inch diameter propeller and the 45' turbine for varying impeller speeds, and for impeller clearance and probe location ratios of 1.00 and a baffle position of 45 '. The
Figure 2. Effect of impeller speed on maximum velocity for marine propeller and 45" turbine 6 P = 45' P l = IC = 1.00 180' 0 Propeller X Turbine
VELOCITY, FT/SEC.
a=
I
I
I
0.3
0
0
I
2
3
4
5
6
t; 0.1 i! s
VELOCITY, FWSEC. Figure
'I.
Upper.
VE LOC1TY, FT 6 ECa
Velocity profiles
4-inch propeller lower 4-inch 45' turbine 6P = 45' P l = IC = 1.00 R = 180'
Figure 3.
Velocity profile for 3-inch propeller BP = 45O PL = IC = 1.33 R = 180'
VOL. 5
NO. 3
JULY 1 9 6 6
269
0.3
0 0
I
2
3
4
5
6
VELOCITY, FTISEC, Figure 6. Vertical (probe location) and radial (impact opening) variation of wall velocity for 4-inch 4 5 " turbine N = 9.0 B P = 45O IC = 1.00 c l = 180°
3 z
X
XI
I
X
I
m
0
2
11.75
N=9.0
2 0.2 w a 0.1
0
5 -
A
>h
XL AX
X 4
2
XA
0
0 0
4
3
0
>u 0
A
I
0
) I .
.A .#
I
I
5
6
VELOCITY, FTISEC 0.3 1
I
a-
1
I
I
10.7
N=6.1
z
A
A m
go.*
7.8
0
A
2 0.1 a
X
A
r
oA
4
xi
x
A*
0
*0
x.
4 Ao
r
Discussion of Results
0
x*
X
A
k-
I
I
*
0
XI
&
a
VELOCITY. FTISEC, Figure
4.
T h e results confirm the previous photographic studies, in that axial flow impellers produce only one primary flow loop while radial flow impellers produce two. I t is also noted from Figures 5 and 6 that the point of maximum velocity moves outward from the wall as the probe is moved vertically up the vessel wall, increasing the probe location ratio. T h e relationship between the point of maximum velocity and the vertical position of measurement (Figures 5, 6: and 7) is shown in Figure 8$and the results are summarized in Table I. T h e maximum velocity at the wall for the axial flow impellers, in this system, is located about 1 1 / 2 inches below the midpoint plane of the impeller. The flow in the plane of the 90' turbine is in the radial direction, with some spreading of the flow area near the wall. Above and below this plane, hoicever, the flow is tangent to the vessel wall in the vertical direction.
Effect of baffle position on velocity profile Upper. 4-inch propeller lower. 45" turbine BP. X 15," 0 45 A 75'
T h e velocity near the wall is linearly dependent on the rotational speed of the impeller. T h e surface heat and mass transfer coefficients, \vhich are functions of the impeller speed (in the impeller Reynolds number), therefore have the same dependence on the fluid velocity past the transfer surface. As the flow measured is tangential or parallel to the wall, one geometric system which can be used for comparison is that of flow over a flat plate. T h e relationship for heat transfer in turbulent boundary layer flow over a flat plate was developed by Colburn and is given by Equation 4 for point values of h and by Equation 5 for average values over the entire heated plate.
0.3
0
VELOCITY, FT/SEC. Figure 5. Vertical (probe location) and radial (impact opening) variation of wall velocity for 4-inch propeller N = 11.75 BP = 45O
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l & E C PROCESS D E S I G N AND DEVELOPMENT
IC
= 1.00
n=
180°
-
Q3 #= a25 QE
d I0,2
1.50
I0 ?
2.00 1.75
I2 0.1 I L
0
a=00
a=1800
BELOW IMPELLER
ABOVE IMPELLER
Figure 7. Vertical (probe location) and radial (impact opening) variation of wall velocity for 4 -inch 90" turbine N = 9.0 BP = 4 5 O IC = 1.00
NU, = hX - 0.0292 [XV p / p ] 4 / 5 [Pr]0.333 k
(4)
NuL = hL = 0.0366 [ L V p / p ] 4 / 5 [Pr]0.333
(5)
k
T h e agitated system used in this work more closely resembles flow over a flat plate where the heated section is preceded by a n unheated section. I n this case the hydrodynamic boundary layer starts a t the leading edge of the flat plat and the thermal boundary layer starts a t the beginning of the heated section. Experimental studies with this configuration have yielded a modification of Equation 4 for point values of the heat transfer coefficient in the heated section
hX
- = 0.0292[XVp/~]4/5[Pr]0.s33 [l
k
-
($) ] 0.975
-3.18
(6)
where X, is the length of the unheated section and X is the distance from the leading edge of the flat plate. For comparison purposes of flow past the wall in a n agitated vessel with flow over a flat plate, the hydrodynamic boundary layer can be considered to begin near the bottom of the vessel
(for upward flow and with the propeller and 45" turbine only) and the flow over the test section is turbulent. Turbulent boundary layer flow over a flat plate starts a t a flow Reynolds number ( X V p / p ) of about 105; a t the wall of the vessel the flow Reynolds number varies from about 50,000 to 150,000. T h e concentration and thermal boundary layers start at the beginning of the test section. For the correction term in Equation 6, for the system used in this work, X, = X - 0.0313 (feet). With the probe location ratio and the transfer surface location ratio having values of 1.00 (in the plane of the impeller), Equation 6 may be modified for various values of X to the following :
where
X , Foot 0.1458 0.3125 0.4792
x
0.0202 0.0201 0.0198
Thus, it is apparent that the effect of the correction factor for the unheated length is essentially constant for the range of transfer lengths considered in this work. For comparison purposes, therefore, the distance from the leading edge of the flat plate (the point at which the hydrodynamic boundary layer would start) to the point of measurement is 0.3125 foot, with the resulting equation for flow past the test sample on the wall of the vessel given by:
h = 0.0201 [V p/,.~]4/5[Pr]O.333
(8)
or, in another form:
hX - -- Nu, = 0.0447 [ X V p / p ] 4 / 5 [ P r ] 0 . 3 3 3 k
(9)
_____
DISTANCE FROhi MSSEL BOTTOM " Figure 8. Vertical location of maximum velocity for 4-inch propeller, 45" turbine, and 90" turbine A X 0
Turbine. N = 9.80. $7 = 180' Propeller. N = 11.75. $7 = 180' Paddle. N = 9.0. $7 = ' 0 for P l
> 1.00
< 1.00,
180' for PL
Table I. locations of Maximum Tangential Velocities Distance f~om Bottom, V,,,, Inches Impeller N Q,Oa Ft./Sec. Propeller 11.75 180 6.65 2.30 45' turbine 9.00 180 5.10 2.65 9.00 90' turbine Ob 3.70 2.25 7.10 180 2.85 a $2. Direction o f j o w . b 0 '. Impact opening upward.
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N O . 3 J U L Y 1 9 6 6 271
Nomenclature
BP
IC
= = = = =
IO
=
k L N
=
D gc
h
/
Nu,
/-
105
Re = X V p / p Figure 9. Comparison of experimental results with flat plate analogy 0
X
A
4-inch propeller 4-inch turbine 3-inch propeller
At a constant impeller speed the surface heat transfer coefficient and the maximum velocity past the transfer surface were experimentally determined [the heat transfer work is reported by Askew ( 2 , 3 ) ] . T h e experimental Nusselt number, h X / k , a t a given flow Reynolds number, X V p / p , is shown in Figure 9. The dashed line is Colburn's heat transfer correlation (Equation 4), assuming the momentum and thermal boundary layers start at the same point. T h e flat-plate correlation of Equation 9 and the experimental data agree closely, showing that the heat transfer from the small section of the wall of a n agitated vessel does approximate that from a flat plate for the same flow conditions. There exists, however, a n effect due to the unheated starting length, as the data and Equation 9 are about 5070 higher than predicted by Equation 4. Thus, by knowing the average velocity of the fluid past a point on the wall of a n agitated vessel, the local transfer coefficient at that point can be determined using a flat-plate analogy correlation. Acknowledgment
T h e authors thank Texaco Inc. for its financial assistance which enabled this research to be carried out.
272
I & E C PROCESS D E S I G N A N D DEVELOPMENT
=
= =
PL
=
Pr Re,
=
=
SL
=
V
= =
X
baffle position, degree impeller diameter, inch 32.2 lb. m-ft./lb.f-sec.2 heat transfer coefficient, B.t.u./hr.-sq. ft.-O F. impeller clearance ratio, distance of impeller from vessel bottom/impeller diameter impact opening, distance of impact tube opening from vessel wall, inch thermal conductivity, B.t.u./hr.-ft.-' F. plate length, feet impeller rotational speed, r.p.s. flow Nusselt number, ( h X / k ) probe location ratio, distance of probe from vessel bottom,/impeller diameter Prandtl number, C,plk flow Reynolds number, X V p / p sample location ratio, distance of transfer surface from vessel bottom/impeller diameter velocity, ft./sec. distance along flat plate, feet unheated distance, feet
Xo = GREEKLETTERS 0
=
K
= constant
p p
= = =
D
baffle position, degrees fluid viscosity, lb./ft.-sec, fluid density, lb./cu. ft. direction of flow, degrees
literature Cited
(1) .4iba, S., A.Z.Ch.E. J . 4, 485 (1958). (2) Askew, \V. S..Ph.D. thesis, University of Marvland, , . College ' Park, Md., June 1965. (3) Askew, \V. S., Beckmann, R. B., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 4, 311 (1965). (4) Brooks, G., Su, G., Third National Heat Transfer Conference, A.S.M.E.-A.I.Ch.E., 1959. (5) Chilton, T. H., Drew, T. B., Jebens, R. H., Znd. En?. - Chem. 36, 510 (1944). (6) Cummings, G. H., \Vest, A. S., Zbid., 42, 2303 (1950). (7) Goldman, I. B., Marchello, J. M., A.Z.Ch.E. J . 10, 775 (1964). (8) Kim, W. J., Manning, F. S., Zbzd., 16, 747 (1964). (9) Metzner, A. B., Taylor, J. S.,Zbid., 6, 109 (1960). (10) Nagata, S., Metn. Fac. Eng. Kyoto Univ. 20, 336 (1958). (11) Zbid., 21, 260 (1959). (12) Sachs, J. P., Rushton, J. H., Chen. Eng. Progr. 50, 597 (1954). (13) Uhl, V. W., Heat Transfer Symposium, Annual Meeting, A.I.Ch.E., 1953. I
RECEIVED for review August 25, 1965 ACCEPTED March 15, 1966
