Summary
The concept of invariant imbedding when applied to countercurrent equilibrium stage problems yields equations which can be solved for outlet concentrations and flow rates as functions of the number of stages and some properly chosen input quantity. The advantages of this approach are at least threefold : The solution procedure is easily programmable, obviating the need for graphical procedures which are often subject to error; one solution yields outlet values for systems with any number of stages (and any value of the chosen input quantity), thus providing the designer an opportunity to make a n optimal choice of design parameters; and the existing process problem conventionally solvable by uncertain timeconsuming trial and error methods (graphical or otherwise) now becomes 110 more difficult than the common design problem. Nomenclature
A = absorption factor ( L / K V ) b, bi = inlet concentration of V stream c
= Y2N
d
= measure of error = component index
i k K Lk
(YN+.I
or y 2 ~ + 1 )
= stage index = equilibrium constant = flow rate of L stream leaving
kth stage
No. of values of bi chosen
m
=
M N
= order of polynomial = total number of stages = polynomial of order M = polynomial of order M
Phi QM r
= outlet concentration of
R
= outlet flow rate of L stream = rN(b)/b
LN
L stream
Vk = flow rate of V sheam leaving kth stage = mass fraction of the ith component in L stream leaving kth stage = mass fraction of the i t h component in stream leaving Ytk kth stage
v
literature Cited
Badger, W. L., Banchero, J. T., “Introduction to Chemical Engineering,,’ McGraw-Hill, New York, 1955. Bellman, R., Kalaba, R., Wing, G., Proc. Natl. Acad. Sci. 46, 1646-9 (1960).
Coulson, J hi. Richardson. J. F., “Chemical Engineering,’’ Vol. 11, Pergamon Press, New York, 1956. Foust, A., et al., “Principles of Unit Operatiom,” p. 77, Wiley, New York, 1960. Koenig, D. hl., Chem. Eng. 74, 181 (September 1967). Lee. E. S.. IXD.ENQ.CHEM.FUNDAMENTALS 7. 164 (19681. Rlikkley, H. S., Sherwood, T. K., Reed, C. E., “A plied Mathematics in Chemical Engineering,” p. 324, %IcGraw-Hill, New Tork, 1957. Wing, G. XI., “Introduction to Transport Theory,” Wiley, New Tork, 1962. RECEIVED for review July 10, 1968 ACCEPTED March 21, 1969
SPATIAL DISTRIBUTION OF PARTICLES IN A SUSPENSION LYNN M. LEEK,’ PRADEEP DESHPANDE, AND JAMES R. COUPER Department of Chemical Engineering, University of Arkansas, Fayetteville, Ark. 72701 The distribution of solid particles in a two-phase system was studied, using a spectrophotometer and transparent phases of different absorptivities. The solid phase consisted of uranium-impregnated borosilicate glass cylinders. A gel composed of 85 volume % glycerol and 15 volume % benzyl alcohol was used as the liquid phase. The gelling agent was 1.5 weight % hydroxyethylcellulose. The beta distribution gave an excellent fit to the data, as evidenced by a mean P r ( X 2 2 C) by chance of 0.746. The parameters of the beta distribution are a function of the size of the solid particles.
H E X solid particles are added to a homogeneous liquid, Wthe physical properties of the liquid are changed. Some properties of the resultant suspension, such as density and heat capacity, are easily calculated as the weighted average of component properties. Other properties, such as thermal and electrical conductivity, are highly dependent upon the spatial distribution of the particles in the liquid and canriot be calculated accurately from solid and liquid conductivities and bulk particle volume fraction. Inaccuracies increase with increased difference between liquid and solid conductivities and with increased particle volume fraction. Inveqtigators (Bruggeman, 1935; Jakob, 1950; Jefferson, 1 Present address, Humble Oil and Refinery Co., Baytown, Tex. 77520
540
l&EC
FUNDAMENTALS
1955; Lees, 1898; Loeb, 1954; McAllister and Orr, 1964; Maxwell, 1904; Meredith and Tobias, 1960; Nahas and Couper, 1966; Rayleigh, 1892; Russel, 1935; Woodside, 1958) who have characterized particle distribution with well-defined arrays of cubes, spheres, or cylinders have witnessed this discrepancy in every case. Others (Baxley, 1966; Baxley and Couper, 1966; Fricke, 1924; Hamilton, 1960; Hamilton and Crosser, 1962; Reymond, 1964; Roblee, 1958; Tsao, 1961) who have more realistically characterized particle spatial distribution have found that their predictive expressions contain variables that are difficult to evaluate. However, investigators who have considered particle spatial distribution have not taken data specifically on particle spatial distribution or particle description. Instead, they have proposed various particle characterizations and have incorporated these into
a predictive equation for a heterogeneous system property such as thermal conductivity. Comparisons then were made between predicted and experimentally determined property values. As a result, little is known about the spatial distribution of particles in a homogeneous fluid. There is a need for direct observations of buch distributions, so that two-phase heterogeneous conducti7iities may be predicted more accurately. The object of this study is to obtain oiie-dimensional solid particle distribution dai a for liquid-solid systems and, using these data, to test proposed theoretical models for system characterization. To obtain thebe data, a specially designed spectrophotometer is neceqsary. The model tested is a beta distribution functioii with tu-o parameters. Theoretical Background
A review of the literature revealed t,hat the characterization of heterogeneous liquid-solid tenis has been considered by several investigat,ors. Their studies, which were entirely theoretical with regard to system characterization, were concerned with either thermal or electrical conductivity. Fricke (1924), in a.ttenipting to predict the electrical conductivity of dispersed systems, chose to represent the particles as prolate and oblate ellipsoids. Since his systems included red blood cells in blood and butter fat in cream, his approximations were not unrealistic. The particle description was formulated into a parameter X , which was a funct’ion of the ellipsoidal axis ratio and the phase conductivity ratio. Evaluation of the shape factor required determinat’ion by observation of the average axis ratio for the individual parM e a . Fricke suggested that parameter X is almost independent of particle form or shape. Hamilton and Crosser (1962) sought to describe particle shape in order to evaluate thermal conductivit,y by the following equation :
where K , V , and ( d T / d X ) refer to thermal conductivity, volume fraction, and temperature gradient, respectively. Subscripts c and d refer to the continuous and discontinuous phases, respectively. The need for particle description developed froni the necessity of evaluating the temperature gradients in Equation 1. The following expression was used to determine these temperature gradients:
For oblate ellipsoids
+x
n= 1
= 3 / p
(5)
The sphericity, then, was calculated directly from the spherical and the cylindrical particles which were used. Tsao (1961) indicated that the bulk volume fraction was not adequate for characterizing of particle distribution in the the continuous phase. He suggested that a onedimensional (1-0)and two-dimensional (2-0) porosity would provide a more rigorous description. He considered the liquid-solid system of unit dimensions shown in Figure 1. The 1-D porosity, P I , was defined as the fraction of linear space occupied by solids on a line drawn through the system parallel to the 5 axis. The 2-0 porosity, P2, was defined as the fraction of area occupied by solids of a plane drawn through the system parallel to the yz plane. In his model, Tsao defined the effective thermal conductivity, K,, as follows:
Obviously a relationship between PI and P2 is required. He accomplished this with a stochastic model by slicing the system into n layers parallel to the yz plane. He suggested that PZwill vary in the n layers and that they can be rearranged in order of decreasing Pp. The system was thus transformed as shown in Figure 2. The probability of Pln out of n segments, in the 2 direction, being occupied by solid was suggested to follow the point binomial distribution. Further, as n increased in size, the point binomial distribution could be approximated by the nornial distribution. The resulting relationship between the one- and two-dimensional porosities was:
where p and respectively.
Q
are the mean and standard deviation of PI,
Y
1
1
+
where n is a function oE particle shape and is equal to 1 X , where X is Fricke’s ]parameter (1924). An attempt was made to improve the description of n by defining it in terms of sphericity, #. Thie, is the ratio of the surface area of a sphere with volume equal to that of the particle to the surface area of the particle. Hamilton and Crosser’s experimental measurements of thermal conductivity dictated that n assume the form n = 3/# (3) These investigators calculated n in terms of theory : For prolate ellipsoids n= 1
+x
=
3/42
Figure 1.
Liquid-solid system
1
2
>
#, using Fricke’s 1 Figure 2. VOL.
Sequential layer arrangement 8
NO.
3
AUGUST
1969
541
Baxley (1966) followed Tsao’s basic model but introduced a more realistic distribution for relating PI and Pz. In Baxley’s critical review of Tsao’s work, he called attention to two ideas. First, for Pln to have a binomial distribution, P1 is necessarily an integer. This, of course, is unreasonable in the physical situation. Second, the normal distribution (Equation 7 ) which relates PI and P2 has a density function which does not integrate to 1 over its sample space. Hence, it is not a legitimate density function. To overcome these shortcomings, Baxley suggested the beta distribution with parameters T and t for relating PI and
P2. f ( I - PI) = ( I / [ B ( r ,t ) ] ) (I - Pl)rlP1t-l
(8 )
The resulting expression between PI and PS is: 1
P2 = { I / [ B ( r , t ) ] ] /
(1 - P1)r-lP1l-l d ( l - P I )
p1
= 1 - zpl (r, t )
(9 1
which is 1 minus the incomplete beta function with argument P1 and parameters r and t. The beta distribution was suggested since it is the limiting case of the point binomial distribution. As a further argument for the applicability of the beta distribution, Baxley suggested that a line be drawn through Figure 2 parallel to the 5 axis. The probability that any point on the line will be in the solid phase is Pl-i.e., “success.” The probability that the same point will be in the liquid phase is 1 - P1-i.e., “failure.” In a+ b 1 trials, the probability of a or fewer successes is:
+
which, by taking b + 1 derivatives, can be shown to be exactly equal to:
However, before the beta distribution can be used for system characterization, parameters r and t must be estimated. If the technique of maximum likelihood (Mood and Graybill, 1963) is used, the estimation of r and t results in the following expressions:
h(r, t ) = -ol(r)
+ In II - PI)= (12) + o l ( r + + In II ( P i L (13)
+
U
(1
t)
P l
0
fz@,
1) =
-ol(t)
t)
-1
where l ( ) is the digamma function and CI is the number of observations of PI. The above two expressions may be solved by iteration of both simultaneously, using Newton’s iteration method. Also required for solution are a large number of measurements of PI. Solution of Equations 12 and 13 should result in maximum likelihood estimates P and f. These values may be substituted for r and t in Equation 9 with the result: P2 = ( I / [ B ( P ,f)]}
/
1
(1 - Pi)G1(Pl)i-ld(l- P1)
JPI
(14) The density function of the beta distribution cannot be integrated in closed form, but may be expressed by the 542
I&EC
FUNDAMENTALS
hypergeometric series:
P* = 1 -
zpl (7,
T)
(1.5) The physical consistency of Equation 15 may be verified by comparing the bulk volume fraction with the following:
Pa
=
[
P2 dP1=
1
P1 dPz
(16)
where Ps is defined as the fraction of the total volume which is occupied by the solid phase. To test the work of Baxley, it was necessary to devise a system with a measurable property that was dependent upon the spatial arrangement of the solid phase. Absorptivity of the fluid medium only would be expected to be different from when particles are suspended in the fluid. Therefore, measurements of light transmission through a fluid medium, with and without the solid phase present, should give information concerning the spatial arrangement of the solid phase in the suspension. This is the justification for the method used in this investigation. Description of Equipment
The light transmission was measured by a spectrophotometer. This instrument accommodated a cubical sample container approximately 4 inches on a side. Two dimensions of the cubic container could be varied. The scanning movement permitted a measurement a t any point on the face of the cube. The light source consisted of a 6.3-volt variable intensity tungsten bulb, a focusing lens, light filter, and an aperture plate. Several plates with openings of various sizes were available. The detector was a multiplier phototube with variable sensitivity. The entire assembly was housed in a light-proof cabinet. The 4-inch cubical cell (Figure 3 ) was suspended on an aluminum plate from the top of the cabinet on threaded bolts. Vertical movement was accomplished by means of four simultaneous-drive gears, operated by a ten-turn dial secured to the top of the cabinet. The light source and detector were fastened to a brass plate which was mounted on rollers. This mounting could be moved in a horizontal direction. Again, a ten-turn dial was employed and, in this case, it was secured on the front panel. The movement in the horizontal direction was 2.7 inches and in the vertical direction was 2.4 inches. The smallest detectable movement was 0.003 inch. The cell (Figure 3) was constructed of 0.25-inch aluminum plate on four sides and 3/16inch borosilicate glass plate on the ends. The glass plates fitted into recessed grooves in the aluminum. There were three sets of these recessed grooves, so that two cell lengths could be employed. Two lengths were needed so that an alternative would be available in case absorption could not be sufficiently varied a t one length. The cell was also equipped with two flexible membranes, one on each side wall. These allowed for contraction of the suspension as it cooled. The entrance to the cell, which was on the top, was fitted with a 1.50-inch-diameter brass plug, drilled and tapped so that excess liquid could be removed by overflow as the plug was being inserted. The output from the multiplier phototube was noted in microamperes direct current. The range of this meter was 0 to 50 pa., graduated in increments of 1 pa. The chassis which supported the transformers and other parts of the circuitry was secured in the rear of the cabinet. Switches, meters, and controls were, of course, mounted on the front
Equation 18 may be solved for P I , giving: PI=
C1(C,
- In I)
(19)
where C1 = [ L ( a ,- al)]-l Cz = In l o - a&
To calculate these constants, it was first necessary to evaluate the absorption coefficients, ai and an. This was accomplished using the three small glass cells. The shortest and the longest cells were filled with liquid, and the same incident beam was passed through both. This led to the simultaneous solution of the following two equations :
66
I1 = Zo exp (Pla,L1) exp [- (1 - P~)c&]
gaskef
i1 1
ZZ= Zoexp (-Ppla,L~) exp [- (1 - Pl)a!&] where ZI= transmitted beam intensity for cell 1, pa.
All grooves
milled - 7 / 3 2 " wide x I / 0 " deep
Since P1 was zero in each case, the equations reduced to:
ZI 6-6-32x i "
-.-
(21)
L1 = length of cell 1, inches. ZZ= transmitted beam intensity for cell 2 , pa. LZ= length of cell 2 , inches
(22 1
= Z ~ e x p(-a!lL1)
12 =
'11
tk-
(20)
ZOexp
(-ah)
(23)
These equations may be solved for a1 as follows:
p l _ -
az = In
FAONT
Figure 3.
Sample container
panel. A safety switch was placed on the cabinet door to protect the detector from overexposure to light. Three small glass cells for calibration purposes were made of 1.25-inch diameter borosilicate glass tubing with 3/16-inch thick borosilicate glass plates cemented to the ends. The lengths of these cells were 0.547, 2.141, and 3.391 inches from plate to plate. Since it was not necessary to fill the cells completely, openings were made by cutting slits in the walls of the tubing. Because of the instability of the line voltage, it was necessary to place a 500-va, 115-volt a.c. transformer in series with the spectrophotometer. Since the low power requirements of the instrument did not load the transformer, two 150-watt light bulbs were placed in parallel with it.
Development of E,quations. The required measurements of PI were obtained by means of the spectrophotometer. The principle upon which spectrophotometers are based is Beer's law. It may be stated as: =
ZOexp (--ah)
(17 1
where Z = transmitted beam intensity, pa. ZO = incident beam intensity, pa. a! = abf;orption coefficient, inches-' L = length of absorbing media traversed, inches
=
The absorption coefficient for the solid, a,, was determined in a similar manner. Again it was necessary to solve two equations in two unknowns. I n this case the third cell was used, and a known length of 5-mrn-diameter solid rod was immersed in the liquid parallel to the incident beam. The beam was then passed through the liquid and, in turn, through the length of solid. By this experimental procedure, PI in Equation 18 was zero in one case and was a known quantity in the other. The cell length, L , was the same in each case; therefore : I1 = ZOexp (-a!&) (25) 12
=
IOexp (-Pla!.,L) exp [- (1 - Pl)alL]
(26)
where I1 = transmitted beam intensity through liquid, pa. ZZ = transmitted beam intensity through solid, pa.
IO exp (- P1au,L)exp (- (1 - P1)alL)
+
as= [In ( I I / I ~ ) I / P I L a1
(27 )
Because of the high absorption of the suspension compared to air, it was decided to define the intensity of the beam transmitted through the cell filled with liquid as full scale deflection on the ammeter or 50pa. The incident beam intensity, l o , was then calculated:
I = 50 = IO exp (a&) In ZO= In 50
(28 )
+ crtL
(29)
If Equation 29 is substituted into Equation 19, then
Beer's law may be modified to apply to two-phase media:
Z
(24 1
Equations 25 and 26 may be solved simultaneously:
Experimental Method
Z
(Zdzz)/(L - I d
(18)
where Z = transmitted beam intensity, pa. ZO = incident becamintensity, pa. PI = fraction of beam occupied by solid phase a8= absorption coefficient for solid phase, inches-' cy1 = absorption coefficient for liquid phase, inches-' L = length of suspension traversed, inches.
P1 = Cl(1n 50 - In I )
(30 )
Thus, a simple equality was derived to relate P1 and the transmitted beam intensity. Choice of System. Of major importance in this work was the selectioii of the two phases for the suspension. The substances to be used had to meet four requirements : To avoid reflection and refraction, the refractive indices of the two substances had to be closely matched; the absorption VOL.
a NO. 3
AUGUST
1969
543
coefficients had to be significantly different to ensure measurable variation in the transmitted beam intensities; the liquid had to be capable of forming a gel with a gelling agent; and the two phases had to be noncorrosive and compatible. The only readily available solid which was transparent, colored, and had uniform absorbance was uranium-impregnated borosilicate glass. The color was to ensure an absorption difference when compared with a water-white liquid. An Abbe-3L refractometer was used to determine the refractive index of this glass, which was found to be 1.4800. Five pounds of the glass were obtained, 3 pounds in 3-mm. diameter cane and 2 pounds in 5-mm. diameter cane. The 3-mm. cane was cut into 4-mm. lengths and the 5-mm. cane into 6-mm. lengths. The liquid solution chosen was a mixture of 85% glycerol and 15% benzyl alcohol. A 1.5 weight yo solution of the Cellosize (hydroxyethylcellulose) gelling agent in the mixture formed a suitable suspension. Preparation of Suspension. The solution was prepared by adding the powdered Cellosize to the glycerol in a 1000-ml. Erlenmeyer flask with continual stirring. The flask was then placed in a mineral oil bath and heated to approximately 230°F. After a few moments a t this temperature, the Cellosize began to go into solution, and the liquid began to turn clear. The benzyl alcohol was then added, and the temperature was increased to approximately 250" F. This temperature was maintained until the liquid became colorless, usually about 20 to 30 minutes. The flask was covered during this time to prevent excessive vaporization of the benzyl alcohol. The metal cell, which had had all joints sealed with rubberbase cement, was filled about three quarters full and then suspended in another mineral oil bath a t approximately 250" F. The liquid a t this stage was too viscous to release small entrapped air bubbles; therefore the glass cylinders had to be added individually. When the desired quantity of cylinders had been added, the cell was filled with more liquid. Any bubbles which had been entrapped by the top were removed with a wire or hypodermic syringe. The calibration cells were also filled at this time. The plug was then inserted and the cell removed from the bath. As it cooled, the cell was repeatedly inverted and the solid phase allowed to fall through the liquid. This inverting action was continued until all of the glass cylinders were suspended, When the cell had cooled to room temperature, it was placed in the spectrophotometer. The schematic diagram of the assembly apparatus is shown in Figure 4. The small cells, filled previously with pure liquid, were used to determine the absorption coefficients. The small cell of medium length, since it was the same length as the metal test cell, was used to set the transmitted beam intensity a t 50 Ha. The cell containing the suspension was then placed in the instrument, and the transmitted beam intensity was noted /-TOP
// LIQHT SOURCE MOUSINQ
Figure 4. 544
l&EC
on the microammeter. When fixed in the instrument, the cell could be moved in increments of 0.3inch. This procedure allowed nine data points to be taken in the horizontal direction and eight in the vertical, for a total of 72 data points in each experimental run. A beam aperture of 0.080 inch was used in each case. Calculation from Experimental Data. -4computer program calculated P1 from Equation 30. If these values for PI were used together with Newton's iteration method, the program simultaneously solved Equations 12 and 13 for parameters 7 and f. The method of moments was employed to determine the first estimate to be used by the Newton iteration method. The variances and covariance of the two parameters were also calculated. The most efficient test for a beta distribution fit is the chi-square test (Dunn, 1967). In this test the interval between 0 and 1 is divided into k intervals such that the probability is l / k of an observation falling in any interval, if the fitted beta distribution is used to calculate the probabilities. Constant k is chosen such that n/k 2 5, where n is the number of observations. The various interval widths are evaluated by solving the following equation for r, (see also Figure 5) :
*i, 1
ra
x+1(1-
U
x)i-1
ax = k
where a / k is the cumulative area and r, is the corresponding value of x. This evaluation is accomplished by using the tables of incomplete beta function (Pearson, 1934). If 0, is the number of observations falling in the ith interval, the test statistic, C, may be calculated :
(ei - n / k ) 2
c= i-1
n/k
The hypothesis that the beta distribution gives an adequate fit to the data is rejected if
(33) where x 2 ( h - 2 ) , is the tabulated value of the chi-square distribution with k - 2 degrees of freedom and a significance level of 1 - a. Presentation of Results
The results of this investigation are shown in Table I. All 10 trials were conducted using a suspension containing 30 volume Yo glass particles. Four runs involved the small (3-mm. diameter by 4-mm. long) cylinders. In the remaining six runs, the larger cylinders (5-mm. diameter by 6 mm. long) were used.
FACE FIXED
SAMPLE CONTAINER WITH STAND
Schematic diagram
FUNDAMENTALS
of apparatus
Figure 5.
Basis for chi-square test
Parameters, Variances, and Degree of Fit for Beta Distribution
Table 1.
Trial No.
i
i
PrIx2>Cl (by Chance)
C
Var(b)
Var(l)
10.8 9.7 17.8 5.0
40.7 32.6 47.8 24.3
0.735 0.846 0.531 0.559
3.56 2.67 5.11 4.89
--
3.1 2.5 8.6 0.7
-
46.3 29.6 63.5 16.7
--
--
10.8
36.4
0.668
4.06
3.7
39.0
11.6
11.9 22.3 17.2 2.5 2.4 2.2
14.1 24.3 23.3 10.5 8.2 5.6
0.918 0.846 0.647 0,503 0.938 0.938
2.00 2.67 4.22 5.33 1.78 1.78
3.8 13.0 8.0 0.2 0.1 0.1
5.5 16.2 14.9 3.2 1.9 0.9
4.4 14.6 10.7 0.6 0.5 0.3
9.8
14.3
0.798
2.96
Volume % Solids
Particle Size
30
Elmall
1 2 3 4
Mean
Small
-
30
Large
1 2 3 4 5 6
Mean
:Large
-
_-
__
__
_-
For the smaller cylinders, the average f was 10.8 and its average variance was 3.7. The average f was 36.4, with an average variance of 39.0. The mean covariance was 11.6, while the mean Pr{x2 2 C) by chance was 0.668. The average f was 9.8 with a variance of 4.3 for the larger cylinders. The average f was 14.3 with a variance of 7.2. The mean covariance was 5.2, while the mean P r ( x 22 C) by chance was 0.798. When the hIann-Whitney procedure (Siegel, 1956) for testing medians was used, a significant difference was found for f between the two different sized glass particles. Discussion
The purpose of this study was to resolve a statistical distribution which would adequately fit data pertaining to the spatial distribution of solid particles in a suspension. Theoretically, the beta distribution with two parameters appeared t o have the characteristics to fit such data. Experimentally, the beta distribution was found to give an adequate fit to the data obtained in this investigation. To determine the degree of fit for the beta distribution, a chi-square test was used. For this test, a high P r ( x 22 C} simply means a high probability that x2 will be as large a6 or larger than the test statistic C by chance. This indicates that factors other than chance are almost negligible. In many instances (Dunn, 1967), a P r { x 22. C) equal to 0.05 is considered sufficient to assure an adequate fit. In this study, the beta distribution was found to give an excellent fit to the data, as evidenced by an over-all mean P r ( x 22 C) of 0.746. This means that it is very improbable that any factor other than chance variation contributed to the magnitude of the test statistic, C. To check the effect of the two different particle sizes, the nonparametric hfann-Whitney test (Siegel, 1956) was employed. A nonparametric test does not assume a distribution and, thus, could be applied to these data. This test indicated no significant difference for 7, but a very significant difference for i. This variation in f was thought to have resulted from the fact that, as the particle size decreases, the transmitted beam intensities vary less and less. For infinitely small particles, a nearly uniform suspension would result, and the variation would be too small to detect. This would suggest that the parameters, particularly 1, are a function of the particle size as well as the cell length--if the small cylinders were suspended in a cell of comparabively shorter length, the resulting f would be approximately the same as for the large cylinder- in the present investigation.
__
__
__
-_
4.3
-_
cov(b,
i)
11.7 8.4 23.0 3.1
__
7.2
5.2
Several problems developed as this study progressed. Only one of these, however, could not be solved. The absorption coefficient for the liquid was not always the same in the metal test cell as in the calibration cells. The two primary causes were the differences in the heating and cooling of the cells and the varying amounts of small entrapped bubbles in each. There appeared no way to avoid this, so an attempt was made to reduce this discrepancy by using average absorption coefficients and the same incident beam for each trial. This bas thought to suffice, since the distribution, not the level of beam intensities, was of interest. This absorptivity problem, along with the fact that the entire cell could not be traversed because of the flexible membranes, made it impossible to evaluate Equation 16 for P3. This mas not thought to be a shortcoming, since the integral evaluates the area under the curve, defined as the solid phase. However, the evaluation would have been a good check on the theory as well as the procedure. The spectrophotometer worked satisfactorily. Although it was difficult to return exactly to a previous position because of a small amount of play in the gear drives, the reproducibility was good. Because of unstable line voltage during the day, it was necessary to take data a t night. Even a t night there was some minor fluctuation : approximately f 0 . 2 pa., which was considered negligible. There were several other minor problems. The rubber-base cement, used to seal the cell, softened and tended to flake as the temperature was increased. This was more noticeable inside the cell than outside. The flaking was reduced by allowing the cement 2 weeks to set. The flexible membranes, which were made of latex sheeting, also deteriorated under the repeated temperature cycles. Their lifetime was approximately 8 t o 10 trials. In applying the chi-square test to the results, it was necessary to interpolate ra from the tables of the incomplete beta function. The function is not quite linear. However, because of the small intervals and the amount of work that would have been required to apply the interpolation formula, linearity was assumed. The linear interpolation was considered sufficient and well within the data error. This study is another step toward a better equation for predicting thermal conductivity of two-phase suspensions. Much work is needed to determine the effect of concentration, particle size, particle shape, and other variables on the parameters of the beta distribution. The next step will be to design a more efficient cell in such a way that the liquid absorption coefficient may be accurately calculated. This cell should also yield data by which Pa may be evaluated using Equation 16. VOL.
8
NO.
3
AUGUST
1969
545
Conclusions
The beta distribution gives an adequate fit for the patial distriblition of cylindrical solids in a suspension, and : f ! size ~ of the ’aspended cylinders has a significant effect, par4 i~*~rlarly on parameter t. Appendix.
where r,+l and t,+l are ‘‘be -r” estimates. This iteration continued until the correctio J Prm was less than 0.001; then f and E were printed. For this set of data f = 11.866
f = 14.137
Sample Calculation
For these calculations, data set 30ll-that is, 30‘$ solid phase, large cylinders, trial 1--was used. The raw data were introduced dire,tly into the computer, which used Equation 30 to evaluate a P I for each datum point.
PI = Cl(1n 50 - In Z )
(30 1
where C1 = [ L ( a 8- al)]-l = C2.140 (0.98162 - 0.23733)]-’ = 0.629 In 50 = 3.91202
The variances of f and t and the covariance of f and also calculated :
f were
- (dG/at)/[ ( d F / a r ) (dG/dt) Variance of f ==: - ( d F / a r ) / [ ( a F / d r ) (dG/at) - (aF/at)*] Covariance of f and i. = ( a F / d t ) / [ (at‘/&) (aG/at) -
Variance of f =
(aFlat 17
For this set of data, Z ranged from 18.3 to 39.7 pa. These values for PI were used to evaluate the final t ms in Equations 12 and 13, which are the Naperian logarithm of the product of 1 - P1 and PI, respectively. The sum of PI and the sum of P t were also calculated. The values of PI were also ordered and printed, to aid in the test of fit. The first estimates of the parameters were then calculated using the method of moments:
These values were found to be 3.8, 5.5, and 4.4, respectively. In applying the chi-square test, it was necessary to evaluate the x coordinates which divided the area under the curve into k equal segments. Since n was 72, it was decided to let k equal 8, so that there were 8 equrl segments of area. The tables of the incomplete beta function (Pearson, 1934) were developed for values of f greater than f. Therefore, since f was greater than 7, the values from these tables were 1 - x. The frequency of observations is shown in Table 11.
n
-If1 = ( l / n )
c (PI), i-1
-v2= (l/n)
c (Pa1
From tabulated chi-square values :
i-1
P r { x 2>_ C) = 0.918
r, = M1 (AT2 - MI)/ (M12 - M 2 ) t, = (r,/Ml)
- ri
+
n{(r+ t )
+
n
Cln (1 - PI)]=
U-1
+ n { ( r + t ) + C Cln (PI)& n
-n{(t)
a=l
+ n{’(r 4-1 )
aF/ar = - n { ’ ( r )
aFpt aG/ar
= n{’(r =
dG/dt =
-t-
Small cylinders. 40.7 32.6 47.8 24.3 Largecylinders. 14.1 24.3 23.3 10.5 8 . 2 5 . 6 These values were numbered in order:
(12)
h(r,t ) = G =
The Mann-Whitney test was applied to all of the values of f to test for significant differences between the sample medians. This test was based on rank sums as follows:
f and
Newton’s iteration method was then applied to Equations 12 and 13:
.fl(r, t ) = F = - n t ( r )
(13)
5 . 6 8 . 2 10.5 14.1 23.3 24.3 24.3 32.6 40.7 47.8 1 2 3 4 5 6.5 6.5 8 9 10 Ties were given mean values from the ranks in the tie. The test statistic, U , was then evaluated:
U=-nlnz+nl(nl+l)/::-l&
t)
where nl
= observations in first sample = 4 n2 = observations in second sample := 6 R1 = sum of the ranks occupied by the first sample =
+t) --n{’ ( t ) + n{’ (r + t )
n{’(r
33.5
where { ( r ) = ( d / d r ) In r ( r )
[ ’ ( r ) = (G/dr2)In r ( r )
Table 11.
The digamma, { ( ), and trigamma, {’( ), functions were approximated by series.
{ ( r ) = In ( r ) - 1/2r - 1/12r2
+
Cumulative Area
l/120r4 - 1/252r8 {’ ( T ) r;+l
1/r = r,
aF/& - F
- aG/dt)/[ (aF/ar) X (dG/dt)
t,+l = t,
+ ( F . aG/ar - G
I&EC
- ( a F / d t ) (aG/dr)]
a~//ar)/[:(a~/a~) x (dG/&)
546
1/240r8
+ 1/21“ + l/6$ - 1/30r“ + 1/42r7 - 1/30rg
+ (G
(32 1
= (1/9) (18) = 2.00
n
FUNDAMENTALS
- ( d F / d t ) (dG/ar)]
0 0.125 0.250 0.375
0.500 0.625 0,750 0.875
1.ooO
1 - x (from Tables)
0 0.43119 0,47803 0.51330 0.54471 0.57610 0.61057 0.65530 1.00000
Frequency of Observations
X
1.OOOOO 0.56881 0.52197 0.48670 0.45529 0.42390 0.38943 0.34770 0
No. of 0bscrvations in Each Interval. O i
10 8 12 8 10
(Bi
- n/k)r
9 8
7
1 1 4 0 1
-
-
72
18
Therefore:
U
= 4(6)
+ 4(5)/2 - 33.5
VC
= volume fraction for continuous phase, dimen-
vd
= volume fraction for discontinuous phase, dimen-
2
X
= horizontal axis = Fricke’s parameter for particle description,
a
= summation and cumulative product count num-
a
= statistical significant level = absorption coefficient for liquid phase, inches+ = absorption coefficient for solid phase, inches-’ = gamma function = digamma function = mean = cumulative product = standard deviation = statistical test
sionless
= 0.5
From tabulated values for the Mann-fhitney test (Siegel, 1956), this value of I: was found to have a Pr( U 5 0.5) equal to 0.008. The size of this number indicates the degree of interrelation between the two samples. A probability of less than 0.05 is usually considered significant to declare a difference between two samples. Thus, there was a significant difference between these two samples. When this test was applied t o the samples for 7, a Prf li 5 10) equal to 0.381 was found. Thus, in this case, the null hypothesis of‘ identical populations could not be rejected. I n other words, there was no significant difference between these two samples. Acknowledgment
The helpful suggestions offered by Orrin K. Crosser, University of Missouri at Rolla, are gratefully appreciated. Nomenclature
a n
d b
c c1
C2 i I
Io I1 I2
k K, Kd
Kt L L1 L2 n n
0%
P1 Pz P3
r 7
ra
R
s
t
t V
of hypothesized number of trials, a+6+1 = cumulative area for chi-square test = number of successes = portion of hypothesized number of trials, a+b+1 = test statistic for chi-square test constant for Equations 19 and = [L(as30 = In 10 - alL, constant for Equation 19 = summation count number = transmitted beam intensity, pa. = incident beam intensity, pa. = transmitted beam intensity through cell 1, pa. = transmitted beam intensity through cell 2, pa. = arbitrary number of intervals for chi-square test = thermal conductivity of continuous phase, B.t.u./ hour-foot-’ F. = thermal conductivity of discontinuous phase, B .t.u./hour-foot-’ F. = effective thermal conductivity of system, B.t.u./ hour-foot-’ F. = length of test cell, inches = length of cell 1, inches = length of cell 2, inches = variable which is a function of particle shape = number of observations = observation in the ith interval = one-dimensional porosity, dimensionless = two-dimensional porosity, dimensionless = three-dimensional porosity or bulk volume fraction, dimensionless = parameter for beta distribution = maximum likelihood estimate of r = 2: coordinrtte for cumulative for chi-square test = resistance t o heat flow, 1/K = dummy variable for integration = parameter for beta distribution = maximum likelihood estimate of t = number of observations of PI
sionless dimensionless ber a1 an
r t
P
n
U
;
= sphericity, dimensionless B ( T ,t ) = r ( r ) r ( t ) / r ( r + t ) ( d T / d X ) = temperature gradient Pr = probability 1-D = one-dimensional 2-0 = two-dimensional literature Cited
= portion
Baxley, A. L., Department of Chemical Engineering, University of Arkansas, Fayetteville, Ark., private communication, 1966. Baxley, A. L., Couper, J. R . , University of Arkansas Engineering Experiment Station, Research Report Series, No. 8, November 1966. Bruggeman, D. A. G., Ann. P h y i k 24, 639-79 (1935). Dunn. J. E.. DeDartment of Mathematics. Universit,v of Arkansas, Fayettehle, Ark., private communication, 1$67. Fricke, H., Phys. Rev. 24, 575-87 (1924). Hamilton, R. L., Ph.D. dissertation, University of Oklahoma, Norman, Okla., (1960). Hamilton. R. L.. Crosser. 0. K., IND.ENG. CHEM.FUKDAMENTALS 1, 187-90 (1962);, Jakob, %I.,“Heat Transfer, Vol. I, p. 88, Wiley, Xew York, 1 grin.
JeffeFion, T. B., Ph.D. dissertation, Purdue University, Lafayette, Ind., 1955. Lees, C. H., Phil. Trans. Rou. SOC. London A191, 399-400 (1898). ’ Loeb, A. L., J . A m . Ceram. Soc. 37, 96-9 (1954). McAllister, A., Orr, C., J . Chem. Eng. Data 9 , 71-4 (1964). Maxwell, J. C., “Treatise on Electricity and Magnetism,” 3rd ed., pp. 435-49, Oxford University Press, London, 1904. Meredith, R. E., Tobias, C. W., J . - 4 p p l . Phys.31, 1270-3 (1960). Mood, A. M., Graybill, F. A., “Introduction to Theory of Statistics,” 2nd ed., pp. 152-4, RIcGraw-Hill, Yew York, 1963. Nahas, N. C., Couper, J. R., “Thermal Conductivity of TwoPhase Systems,” Part 111, “Thermal Conductivity of Emulsions,” University of Arkansas Engineering Experiment Station, Research Report Series, No. 7 , March 1966. Pearson, K., ed., “Table of the Incomplete Beta-Function,” Cambridge University Press, Cambridge, England, 1934. Rayleigh, R. J. S., Phil. Mag. 34, 481-507 (1892). Reymond, B., M.S. thesis, University of Oklahoma, Norman, Okla., 1964. Roblee, L. H. S.,Baird, R. M., Tierney, J. W., A.1.Ch.E. J . 4 460-4 (1958). Russell, H. W., J . Am. Ceram. SOC.18, 1-5 (1935). Siegel, S.,,,“Nonparametric Statistics for the Behavioral Sciences, p. 118, McGraw-Hill, Kew York, 1956. Tsao, G. T., Znd. Eng. Chem. 63, 395-7 (1961). Woodside, W., Can. J . Phys. 36, 815-23 (1958). RECEIVED for review August 14, 1967 ACCEPTEDJanuary 14, 1969 Financial assistance given by the National Science Foundation through Grant GK-1151.
VOL.
0
NO. 3 A U G U S T
1969
547
