THERMAL CONDUCTIVITY OF BEDS
OF SPHERICAL PARTICLES SHINOBU MASAMUNE AND J.
M. S M I T H
University o j California, Davis, Calif.
Effective thermal conductivities, k,, were measured for beds of spherical glass beads and steel shot, 2 9 to 470 microns in diameter. Data were oblained a t low air pressures ( 1 O-* mm. of Hg) to evaluate heat transfer rates through the contact regions of adjacent solid particles, and at pressure from 1 OP2 to 760 mm. of Hg, to ascertain the effects of free molecule conduction. Geometrical considerations were used to develop equations for predicting the effect of pressure on effective conductivity. This approach, which involves no arbitrary constants, agreed well with available data. The solid-to-solid contact heai transfer was found to b e a function of area of contact, characteristics of the surfaces of the particles, and void fraction, E, as well as the solid conductivity, k,. Published data were used to evaluate &, a characteristic parameier of the system which i s not a function of k, or E. EAT T R A N S F E R in
heterogeneous fluid-solid systems is imporexample, catalytic reactors, heat exchangers, and thermal methods of oil recovery. This study is concerned with beds of fine spherical particles in which the void spaces contain air at pressures from to 760 mm. of Hg. The specific objective was to determine the effect of gas pressure and. hence, fluid conductivity, on the effective thermal conductivity of the bed. Previous investigations of heat transfer in beds of fine particles (7. 2, 9, 70. 73, 78) have been primarilv at atmospheric pressure or with flowing fluids. In attempting to write a mathematical description for the heat transfer rate in beds containing stagnant fluid it has been generally agreed that the transfer of energy occurs by three mechanisms:
H tant in many types of processing-for
Mechanism 1. Through the void fraction by conduction and radiation hfechanism 2. Through a series path consisting of an effective solid-path length and gas-path length Mechanism 3. Through the solid phase, the energ)- flowing from one particle to the next through the area of contact Because of the geometry problems involved, it is necessary to consider that the heat flow is in one direction. This assumption, which amounts to postulating that there is no bending of the heat flux lines, means that the three mechanisms operate in a parallel fashion and that their separate contributions may be added to obtain the total heat flow. With this simplification the problem of predicting the heat transfer rate is reduced to determining the series mechanism path lengths in the gas and solid phases, and evaluating the effective areas, perpendicular to heat flow, for each mechanism. Deissler ( Z ) , Yagi (79). Kunii ( 9 ) , and Baddour (7), in particular, have used this concept to develop equations for the effective conductivity, k,, of the bed. In terms of dimensionless parameters $, CY, and 6 an expression for k, may be written: k, = aek,
-+ (1dJ- a e )1( 1- d-J 6) + (I - a e ) G k , KO ".
(1
ks
I&EC FUNDAMENTALS
I t is convenient to define the three volume (or area) fractions in terms of the dimensionless parameters CY and 6, so that €1 = €2
= €8
I n this equation, radiation is neglected and k, is based upon a unit area of the bed. Each term on the right side represents the contribution of one of the three mechanisms. The effec136
tive area for heat flow for Mechanism 1 is CYE. for Mechanism 2 it is (1 - C Y E ) ( ~ - 6). and for Mechanism 3 it is (1 - C Y E ) ~ . These areas are depicted in Figure 1 , which represents the model used to derive Equation 1. The quantities $($ = L,/ On)and ( 1 - 0 ) are proportional to the path lengths. L , and L,. Parameters 0 and a should be expressible as functions of the void fraction. e . These functions are developed below and used in Equation 1 to obtain a relationship among k,, E, and the several thermal The latter quantity is the conductivities. k,, k,, and k,'. appropriate conductivity in the gas spaces near the contact areas of adjacent solid particles. Here the mean free path may be of the same magnitude as the size of the gas space. Hence k,* depends upon the pressure, k,, and a linear dimension of the gas space. The final result is then compared with aLailable experimental data. Derivation of Equation 1. In an assemblv of spheres l an element which is repeated in identical form in there ~ i l be making up the bed Figure 1 depicts the three-mechanism model for heat transfer in such an element The area fraction is assumed equal to the corresponding volume fraction, and these are designated as el. ez, and € 3 for the three mechanisms. The length of the bed in the direction of heat transfer is 4 L and the temperature change is Af. The total heat transfer rate may be expressed in terms of k , and also as the sum of the contributions of the three mechanisms-that is
(1
-
=
(1
(A21
ut
O(€)(l- 6 ) -
(-43)
(-44)
CYE)6
Then Equation 1 may be written At
ke A L
ueko AL "
+ (1 - o/~)(l
-
At 6)kz AL
+ (1 -
At AL
CYE)~*-
(A5)
Here kp is the conductivity of the series mechanism which represents heat flow through a gas path and solid path whose total length is AL. The gas-path length associated with a total is, 9 = L,/D, is length of D, is evaluated below-that determined. Then the corresponding solid-path length, per particle diameter, is L , = (1 - 9 ) D p . For cubical packing (of spheres (Lo L J , which is D,, is also equal to AL. However, for a random arrangement of spheres L , L , is not equal to I L , the deviation depending upon the porosity. If this relationship is P = f ( e ) ,
+
+
AL
=
6 D,
one-dimensional heat flow, as depicted in Figure 1, this volume will have a cross-sectional area equal to the projected area of the particle, d, and a thickness Lo. Hence,
L,
=
nA V , mo2
(5)
Applying Equations 2 to 4 to Equation 5 gives a n expression for L, in terms of void fraction. Then the required parameter, $, is 6.93 - 5.51
s] x
(.Ab)
Then the gas-path length for the element is P Lo. or @ D,P, as shown in Figure 1. Similarly, the solid-path length is (1 - @)DpP. Kunii and Smith (9) have studied the variations of P with types of packing and found maximum and minimum values of 1.0 and 0.895 as long as the spheres are in contact with each other. Using these path lengths the conductivity of the series mechanism is obtained by equating the energy transferred through each path. The result in terms of the gas phase conductivity, k,*, and solid phase conductivity, k,, is
{(sec e2 - 1 ) 2 [I
(5 -
-
where n and Oq can be evaluated from and 4.
E
02) tan 821
1
(6)
using Equations 3
Area Fraction for Mechanism 1
I n evaluating area fractions frequent use is made of the assumption that volume and area fractions are equal. The HEAT FLOW
Substituting this expression into Equation A5 yields Equation l . Effective Gas-Path length (L,
= $DD)
Figure 2 depicts two spherical particles in contact. The shaded area represents the gas phase as a pendular ring with radii r I and r z . It is asumed that the gas phase part of the series mechanism is determined by the volume of such pendular rings. From the geometry of the system Fischer (4)and Rose (72) have determined the volume of the ring. in terms of the particle radius. ro, to be. L V ~= 2rrro3(sec
or
-
112
[I -
(i
-
e.) tan
Mechanism I 0 2 1
(2)
Figure 1 .
Mechanism 2
M2chanism 3
Model for heat transfer mechanisms
To determine the total volume of the rings surrounding any one particle it is assumed that all the surface of the spherical particle is in contact w ~ t hgas in pendular rings. If 2n represents the average number of such rings. the fraction of the surface of the particle occupied by one ring is '/2 n. This fraction is related to angle 0 2 by the expression
1
or
Kunii and Smith (9) obtained a semitheoretical expression for n in terms of the w i d fraction by considering the possible orientations of spherical particles. The result is n
=
6.93 - 5.51
E -
0.476
0.260 0.260
-
1
(4)
uhich gives n = 1.42 for the most open packing ( E = 0.476) of spheres and n = 6.93 f i x the most dense packing (E = 0.260). The total heat flux per particle in the series mechanism will pass through the number of rings associated with a hemispherical surface of a particle, and hence, a volume n(AV,). For
I 2r0= ~p
ti = ro( I + tan r2= ro(sece2 Figure 2.
e, - s e c 8,) -11
Contact angles for fluid and solid VOL. 2
NO. 2
MAY 1963
137
fractions el and €2, like 4, are determined from geometry considerations alone. The area available for heat transfer by Mechanism 1 is €1. This is not equal to E, because part of the void space is involved in the series path. In terms of LY
parameter for each type of particle. By applying Equation 1 as the pressure approaches zero, 6 can be related, approximately, to the effective conductivity under vacuum conditions -i.e., k,". Thus 1
=
where €2 is the void fraction for Mechanism 2. From the viewpoint of the volume of the pendular rings, ( l / 2 AV,)2n gives the void volume per particle for the mechanism. Hence
Using Equations 5 and 8 a can be expressed in terms of 4 as follows: (Y
=
1
- 53
1 (+)
(9)
q5
Area Fractions for Mechanisms 2 and 3
The sum of the area fractions for Mechanisms 2 and 3 will be €2
+
It is convenient to define
€3
€3
= 1-
(10)
LYE
in terms of the parameter 6, so that
Then from Equation 10 €2
=
(1 - a e ) ( l - 6) and
€3
= (1
- as)6
Figure 3. 138
l&EC FUNDAMENTALS
This expression is an approximate one because the assumption of no bending of the flux lines from the vertical direction (Figure 1) is a weak one a t zero pressure. Also the first term of Equation 1, aek,, is not strictly zero except a t p = 0. At low pressures k, becomes a function of pressure and the distance AL,since heat transfer in the gas space is by free-molecule conduction. Actually, the term is small a t low pressures with respect to the heat conduction through the solid. For soft, rough particles the area of contact between two particles would be expected to be larger than for hard, smooth spheres. This variation presumably could be measured by the difference in angle of contact 01 shown in Figure 2. This angle should be a function of only the characteristics of the solid particles. The value of € 3 and 6 would also reflect the number of particles in contact and be a function of void fraction. This relationship among 6, 6 : and $1 can be approximated by considering the area of contact between two particles to be a circle in the plane perpendicular to the radius, ro. This area would be aro2 sin2 81 for one contact and n 7rro2 sin2 O1 on the hemispherical surface of a particle. I n Equation 11 € 3 is small with respect to € 2 . Hence, 6 =
(12)
These area fractions were used in developing Equation 1 as illustrated above. If € 3 , and hence 6, could be evaluated as a function of void fraction, as has been done for $ and a , Equation 1 would provide a general expression for k, containing no arbitrary constants. However, all the data available indicate that the solid-to-solid heat transfer is a function of the surface characteristics of the spheres and not solely a geometrical quantity. Hence 6 must be regarded as a specific
g
(ZJ
n r r ? sin2 81 = n sin2 el rro2
Equations 13 and 14 provide a basis for evaluating 81 from measurements of the vacuum conductivity, k,' ; thus,
(15) L
-I
where n is given by Equation 4. Equation 15 is used beIow to evaluate 01 for various kinds of particles.
Data for stainless steel shot
0.0I
0.05 0.1
a5 I 5 IO P , PRESSURE ( m m . H g 1
Figure 4.
Scope. Effective thermal conductivities were measured for four sizes of glass beads (29 to 470 microns in diameter) and one sample of steel shot (71 microns in diameter) in beds whose properties are summarized in Table I. T h e average temperature of the bed was 42' i 2' C. Measurements w r e made with the void space containing air a t pressures from to 760 mm. of Hg. Method. Conductivities were determined by a comparative method using the same type of apparatus as for studying silver catalyst pellets ( 7 7 ) . Lucite was used as a reference material, since its conductivity, 0.118 B.t.u./(hr. ft. O F.) (77), is within the range of those evaluated for the steel and glass beds [0.01 to 0.15 B.t.u./(hr. ft. O F.)]. The method involves the flow of energy as heat through the sample and Lucite in series, each in the form of a disk, 1.50 inches in diameter and 0.5 inch thick. The heat flow was obtained by using hot and cold water. a t constant temperature, as energy source and sink on the ends of the packed bed and the Lucite. For accuracy, it is essential to reduce radial heat loss from the cylindrical surfaces of the sample and reference substance. T o achieve this the space around the test section was evacuated to l o p 3 mm. of H g and the inner surface of the surrounding vessel plated with aluminum. Preliminary tests Ivith knoivn materials ( 7 7 ) indicated that a n accuracy of about 2% could be expected with the apparatus.
Table 1.
100
500
1000
Data for glass beads
Experimental Work
Thermal Conductimty, Solid Material k s , B.t.u.1 (Sfiheres) ( I l r . Ft. ' F.) .4~. Diam.. Glass (this work) 0.604 2.9 x 0.604 8.0 x 0.604 20.0 x 0.604 47.0 x Steel shot ithis work) 9.40 7.1 x MgO ( 2 ) 14.0 20.4 x Steel shot ( 4 ) 15.1 127 x a Obtained from sieve analysis data.
50
Results. The thermal conductivity \vas calculated from the following equation, using the measured temperatures and dimensions of the sample and Lucite:
The results are shoivn in Figure 3 for steel spheres and in Figure 4 for glass beads, plotted as k , us. pressure. Effect of Pressure
The sharp rise in k, noted in Figures 3 and 4 is due to the influence of free molecule conduction where the gas conductivity, k,*, is directly proportional to pressure. The general relationship between the normal gas conductivity and the pressure has been shown-for example, by Kennard (7)-to be
Intersection Point Pressures
Bulk Density of Bed, D,, Cm. p ~ G./Cc. , 10-3 1.51 10-3 1.49 10-3 1.49 10-3 1.50 10-3a 5.77 10-3a 10-3
Void Fraction of Bed, €
0.38 0.38 0.38 0.38 0,264 0.42 0,365
Intersect2on Poznt Pressure, PO,M m . Hg 5 3 . 5 (air) 2 1 . 5 (air) 1 0 . 2 (air) 4 . 0 (air) 4 0 . 0 (air) 330 (He) 0 . 3 7 (H?)
L,. Cm. Exptl. Theoi. (Eq. 6 ) 4 . 3 x 10-4 3 . 3 x 10-4 1 , I x 10-3 9 . 2 x 10-4 2 . 3 x 10-3 2 . 3 x 10-3 5.8 X 5 . 4 x 10-3 5 8 X 5 . 4 x 10-4 3 . 3 x 10-3 2 . 9 x 10-3 1 . 2 x 10-2 1 . 3 X lo-*
VOL. 2
NO. 2
M A Y 1963
139
-
Figure
Table II.
Symbol (Fig. 6 ) -0-
@
0
Glass bead
Quartz
Steel ball
0
Steel Steel shot
0
Q
-0-
e .a 0
Stainless steel shot Lead
Lead Glass MgO Sintered Cu-Sn alloy particles
Estimated b y Wilhelm e t al. ( 7 7 ) .
140
E
Sic
0
Q
Void Fraction,
Solid Material (Spheres)
I&EC FUNDAMENTALS
b
0.429 0.410 0.425 0.38 0.38 0.38 0.38 0.438 0.416 0.296 0,241 0.413 0.402 0.394 0.391 0.390 0,423 0.406 0.394 0.380 0.38 0.365 0.366 0.265 0,420 0.439 0.416 0.433 0.450 0.400 0.350 0.420 0.206 0.333 0,392 0.364 0,356 0.395 0.345 0.365 0,338 Data for
5.
Intersection pressure, po
Thermal Properties of Beds of Spherical Particles Vacuum Solid Conductivity. Conductiuity , Diameter, k,', B.t.u./ k,, B.t.u./ D,, Cm. (Hr. Ft. F . ) (H7. Ft. O F.) 0.0358" 10.4 0 ,0372a 10.4 0 . 0360a 10.4 2 . 9 x 10-3 0.03 0,604 8 . 0 x 10-3 0.03 0.604 20 x 10-3 0.03 0.604 47 x 10-3 0.03 0,604 0 , 0483a 6.35 0 .0507a 6.35 0.0884" 6.35 0,111= 6.35 0.113n 3 . 1 8 X lo-' 25.9 3 . 9 6 X 10-1 0.118. 25.9 4 . 7 5 x 10-1 0.123a 25.9 0.125O 5 . 5 3 x 10-1 25.9 0.125. 6.35 X lo-' 25.9 0.108a 7 . 9 1 X lo-' 25.9 0.116a 3 . 1 8 X lo-' 25.9 0.124. 4 . 7 5 x 10-1 25.9 0.131. 4 . 7 5 x 10-1 25.9 0,0975° 3 . 8 X 10-I 22.2 1.27 X 0.0169 15.1 0.0169 15.1 1.27 X 7 . 1 x 10-3 0.01 9.40 1 . 5 9 X lo-' 0.0708 19.8 19.8 2 . 3 8 X 10-1 0.0669 3 . 1 8 X 10-1 0.0720 19.8 0.0678 6 . 3 5 X 10-1 19.8 0,0645 7 . 9 4 x 10-1 19.8 0.0408 2.62 X 20.4 0.628 0.0183 2.04 X 0.009 14.0 15.1jb 120 4 . 9 3 x 10-3 8,22b 120 1.28 X 5,0@ 120 120 6 . 53b 1.28 X 1 0 F 120 2.11 x 10-2 7,025 2 . 1 1 x 10-2 120 4.84* 120 4.00 X 7 . 64b 120 4.00 X 6 , GOb 120 4.00 X 7.w 7-atm. pressure.
k, / k , 3 . 4 4 x 10-3 3 . 5 8 x 10-3 3.40 x 10-3 5 x 10-2 5 x 10-2 5 x 10-2 5 X 7 . 6 0 x 10-3 7 98 X io-3 1.39 X 1.75 X 4 . 3 6 x 10-3 4 . 5 5 x 10-3 4 . 7 6 x 10-3 4 . 8 3 x 10-3 4 . 8 3 x 10-3 4 . 1 6 x 10-3 4 . 4 7 x 10-3 4 . 7 6 x 10-3 5 . 0 5 x 10-3 4 . 4 0 x 10-3 1 . 1 1 x 10-3 1.11 x 10-3 1.0 3.58 3.38 3.63 3.43 3 26 2.00 2.92 6.43 1.26 1.85 4.23 5.44 5.85 4.03 6.36 5.50 6.65
x
10-3
x 10-3 X 10-3 x 10-3 x 10-3 x 10-3
x
10-3
X
x
10-3
x 10-1 x
x io-2 x 10-2 X 10+ X
x
X 10F X 10F
This expression was developed for heat flow in the gas space bet\veen t\vo parallel plates separated by distance do. I n beds of spherical particles, the path length for heat flow in the gas phase, by 5lechanisn2 2> varies from zero to a value somewhat less than the parlkle diameter. However, length L , \vas evaluated from Equation 6 so as to represent the plate to plate distance corresponding to the average path length betLveen particles in the bed. Hence d, is the same as L,. The experimental data can be used? along with Equation 17, to ascertain ..experimenial'! values of Lo for comparison with those predicted from Equation 6. This comparison can be made most easily by evaluating the intersection pressure, p ~ This pressure can be related to the experimental data by replotting Figures 3 and 4: using normal coordinates. Figure 5 illustrates the curves for the beds of glass beads. At low pressures, Lo*, and hence k,, should be directly proportional to the pressure, accordin!: to Equation 17. In this region heat transfer is by free mo1t:cule conduction. Straight lines are shoivn in Figure 5 for each size of glass bead. At high pressures (about 760 mm. of Hg) the k , curves in Figure 4 become flat. indicating that k," is a constant equal to k , and that heat flow is by molecular collisions. The intersection of the slanting straight line and the dotted, horizontal line (Figure 5) determines /IO. The equation of the slanting lines is Equation 1 with kQ* given by Equation 17. The expression for the horizontal dotted line is Equation 1 with k , substituted for k,*. The contribution to k , of the first mechanism, a cky, is less than 4% of the total a t the most severe conditions and hence may be neglected. Equating the two expressions for k , gives (1
- a e ) ( l~- 6)
L+L* i:,*
/Le
- (1 - o r e ) ( l - 6) -0 + -1 - 4 k,
k,
Particle-to-Particle Heat Transfer
.
The absolute value of k , depends upon Mechanism 3, and this is difficult to predict. However, the experimental data can be used to evaluate the values of 6 and 6'1 corresponding to various t)-pes of particles. Equation 15 postulates that the particle-particle contribution to heat transfer, as measured by keo/k,>, should not be a function of the particle diameter, but only of void fraction and contact angle 6'1. The data in Figure 4 indicate the same result? since k,' is essentially the same for bead diameters from 29 to 470 microns. There are available (Z?5>6, 8: 75) considerable data on vacuum conductivities covering a range of D,,void fractions, and solid material (Table 11). In Figure 6 the solid and dashed lines shoiv k,"/k, us. rhe solid fraction, for various contact angles, as calculated from Equation 15. The data in Table I1 are shown as experimental points on the figure. For each solid material the experimental points fit the equation, so that a single contact angle is determined. T h e dashed curves are the theoretical curves from Equation 15 using the particular 01 which best fits the data. The data include a range of D, values from 2.8 X to 7.9 X lo-' cm. and solid fractions from 0.55 to 0.80. Figure 6 suggests that the contact angle, 6'1, rises from about 1.0" for steels up to about IOo for copper-tin alloy spheres. The latter system (75) had been sintered to a certain extent. Hence 6'1 would be expected to be comparatively high, even though the particles might be expected t o be smooth and hard. In addition to the possibility of sintering, the character of the surface, perhaps the microscopic variations in roughness, could influence the heat transfer from particle to particle. Hence 6'' is a function of the surface characteristics as well as area of contact between particles. This means that a sound means for predicting 8, and hence 6> is a very difficult problem.
In this expression (1 -- o ) , l k , is small with respect to either @/k,* or Q ' X , . Then using Equation 17 for ko* gives the follon ing expression for the intersection pressure :
I Q5
I The accommodation coefficient, a. is a function of both the gas and the solid surface. A bide range of values has been reported-for example (3, 76). for air and polished metal. a = 0.8 to 1.0. and for helium 0.3 to 0.7. For the comparatively rough surface of the glass and steel beads studied, the value for air should be nearl) unity. Using this quantity and representing the Prandtl number by 4 7 / ( 9 7 - 5), Equation 18 reduces to
_ _ _ _Equation -
I
15
I
ai a05
0.01
aoo: (0
w
From curves similar to those in Figure 5, po was determined and then used in Equation 19 to obtain Lo. In addition to this ~ v o r kk~, 28s. presswe data are available (2, 74) from other sources. The values of Lo so obtained are given in Table I (experimental L 6 ) . The theoretical values of L, were determined from Equation 6 and are also sho\vn in Table I. Since there are no arbitrary constants in the equation, the agreement with the experimental results is reasonab1)- good. Hence, Equation 6 can be used to predict the effect of pressure on the effective conductivity for beds of spherical particles.
-6 I
2
Y
om
\
La 0.0009
o*ooo Figure
5
1 1 0.6 0.7 I - € , SOLID FRACTION
0.8
6. Solid-solid conductivily from vacuum data VOL 2
NO. 2 M A Y 1 9 6 3
141
k,
Figure 7.
, (CALCULATED),
Effective thermal conductivity in transition region
Effective Conductivity at Finite Pressures
The 01 values shown in Figure 6 can be used to evaluate ke0/k, and then 6 from Equation 13. With the expressions for a and 4, Equation 1 can be used to predict the effective thermal conductivity in beds of spherical particles, for which 81 is known. Thus the conductivity can be predicted from: (1) k , and k,, (2) void fraction and particle diameter, and (3) the conductivity a t zero pressure (to determine el). Figure 7 compares predicted and experimental data for pressure ranges where the heat transfer in the gas phase is in the transition range between free-molecule and molecular conduction. The experimental data where the k , us. pressure curves are changing shape-e.g., Figures 3 and 4-are limited but include several gases and solids as shown in the legend on Figure 7 . The values of k,, D,,and temperature (mean value for bed) were taken from the original papers. The accommodation coefficient is assumed to be unity. Nomenclature a
= thermal accommodation coefficient
D,
= diameter of spherical particles, feet
dB
= = = = =
do dL
kB k,
ke0 k, k,*
= = =
kL
=
142
diameter of packed bed, feet distance between parallel plates, Equation 17, feet diameter of Lucite reference cylinder, feet Boltzman constant, ft. lb./O R. effective thermal conductivity of bed, B.t.u./(hr. ft. O F.) k, value a t zero pressure, B.t.u./(hr. ft. O F.) normal thermal conductivity of gas, B.t.u./(hr. ft. O F.) thermal conductivity of gas between parallel plates separated by distance do, B.t.u./(hr. ft. O F.2 thermal conductivity of Lucite, B.t.u./(hr. ft. F.)
l&EC FUNDAMENTALS
B.tw/hr.ft.aF:
k,
= thermal conductivity of solid particle, B.t.u./(hr. ft.
O F.) AL = length of element of packed bed, feet Lo = effective path length (Figure 1) between adjacent solid
particles, feet ALB = height of packed bed, feet L u L = height of Lucite cylinder, feet L, = effective path length (Figure 1) for solid particles, feet n = average number of contact points on hemispherical surface of one solid particle Y = pressure, Ib./sq. ft. Po = intersection pressure, given by Equation 18, lb./sq. ft. P r = Prandtl number q = heat transfer rate, B.t.u./hr. ro = radius of spherical particle, feet T = absolute temperature, O R. AVO = volume of one pendular ring of gas around point of contact of tlvo spherical particles (Figure 2): cu. feet At = temperature change
GREEK a
=
P
=
6 Y
= =
e2
= contact angle associated with pendular ring of gas
el e u
= contact angle between solid particles (Figure = void fraction = molecular collision diameter, feet
9
= L,/D,
(Figure 2)
SUBSCRIPTS 1, 2, 3 = Mechanisms 1, 2, and 3 for heat transfer B = bed of spherical particles L = Lucite
2)
literature Cited
(1) Baddour, R. F., Yoon, C. Y., Chem. Eng. Progr. Symp. Ser., No. 32, 57, 35 (1962). (2) Deissler, R. G., Eian, C.S., Natl. Aeron. Space Admin. NACA RM E52C05 (1952). (3) Eckert, E. R. G., Dra.ke, R. M., Jr., “Heat and Mass Transfer,” McGraw-Hill, New York, 1959. (4) Fischer, R. A,: J . A~gr.Sci. 16, 492 (1926). (5) Grootenhuis, P.: Mackworth, R. C. A., Saunders, 0. A,, “Proceedings of Genera.1 Discussion on Heat Transfer,” p. 363, Inst. Mech. Engrs., London, 1951. (6) Kannuluick, LV. G.: Martin, L. H., Proc. Roy. SOL. (London) A141. 144 (1933). (7) Kennard. E. 8.. “Kinetic Theory of Gases,” McGraw-Hill, New York, 1938. (8) Kling, G.. Forsch. Gebiete Zngenieurw. 9, 28 (1938). (9) Kunii, Daizo, Smith, J. M., A.Z.Ch.E. J . 6, 71 (1960). (10) zbzd., 7 , 29 (1961).
(11) Masamune, Shinobu, Smith, J. M., J . Chem. Eng. Datu 8, 54 (1963). (12) Rose, Walter, J . Appl. Phys. 29, 687 (1958). (13) Schotte, W., A.Z.Ch.E. J . 6, 63 (1960). (14) Schumann, T. E., Voss, V., Fuel 13, 249 (1934). (15) W,addems, A. L., J . Soc. Chem. Znd. 63,337 (1944). (16) heidmann, M. L., Trumpler, P. R., Trans A S M E 68, 57 (1946). (17) Wilhelm, R. H., Wynkoop, W. C., Collier, D. W., Chem. Eng. Progr. 44, 105 (1948). (18) Willhite, G. P., Kunii, Daizo, Smith, J. M., A.Z.Ch.E.J. 8, 340 (1962). (19) Yagi, Sakae, Kunii, Daizo, Zbid., 3, 373 (1957). RECEIVED for review August 13, 1962 ACCEPTEDDecember 26, 1962 Financial assistance provided by National Science Foundation Grant G-17765.
VELOCITY PROFILES IN TURBULENT PIPE FLOW Newtonian and Non-NewtonianFluids D . C. BOGUEI AND A. B. M E T Z N E R
Universit.v o j Delaware, Newark, Del. Velocity profiles were measured in the turbulent core region of viscous Newtonian and non-Newtonian fluids flowing through smooth round tubes. Prior art data for Newtonian velocity profiles were comprehensively reviewed, and an empirical correction to the traditional correlation was devised. The nonNewtonian data were obtained using fluids free of significant viscoelastic effects. Reynolds numbers ranged from transitional values to beyond 100,000, and flow behavior indices were varied between 0.45 and 0.90. Turbulent core profiles for the non-Newtonian fluids were essentially the same as those for Newtonian flulids when normalized with respect to the mean velocity or, equivalently, when compared on the basis of the velocity defect parameter. Recasting the correlation in terms of generalized u + - y + parameters leads to a difference between the Newtonian and nowNewtonian cases because of differences between frictiion factors.
the classical work of Nikuradse (20). velocity profiles for the turbulent flow of fluids through smooth round tubes have been correlated by the equation (26) : ATING mob1
21+
5.75 logy+
=
+ 5.5
(1)
T h e form of this correlation is wrong in one respect: it fails to predict a zero velocity gradient a t the centerline. I n addition, there are small, but consistent, deviations of the data from the correlation; in the case of Sikuradse’s data it is found that in each run the points are higher than the curve near the centerline. These deviations are accounted for in treatments by Millikan (79): Reichardt (24),and Hinze ( 9 ) . I n connection with the present work, .a comprehensive review of the data was undertaken, and a n empirical correction function, similar to that used by these investigators but including a small effect of Reynolds number (or friction factor), was introduced ( 7 ) . A summary of the various correlations follows : Millikan: Reichardt:
u+ U+
.-
C~W(.$) =
-
C R ( ~ )=
5.75 l o g y + 5.75 logy+
+ 5.0 + 5.5
(2; (3)
Present address, Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxville, Tenn.
+
Hinze: u + - GH(E) = 5.61 logy” constant 5.57 Present work: u + - c ( E , f ) = 5.57 logy+
+
(4) (5)
The empirical correction functions are shown for the four cases in Figure 1. T h e curves for G(, f) can be represented by the equation:
&f)
=
0.05 4 - e ~ ~
- ( E - 0.8)’ o.15
(6)
The constants in Equation 5 were so selected that this equation would, upon integration, yield the usual friction factor correlation for smooth pipes (12, 20) :
dT
= 4.0 log [NR$’~]
- 0.4
(7)
The prior a r t data of Nikuradse (20) and Laufer (74) are displayed in Figure 2 in support of Equation 5. Nikuradse’s data a t low values of y+, which are in dispute (78), are not shown. Reference to any of the several common texts which display the original data of Laufer and Nikuradse reveals the significant improvement in the correlation obtained by introduction of the correction function. Subsequent discussion makes use of Equation 5 as the best available Newtonian VOL. 2
NO. 2 M A Y 1 9 6 3
143
