thousand gal of acid. Passing the raffinate through a 0.5-hr holdup tank reduced this loss to less than 0.10 gal/thousand gal of acid. Chemical Reagent Costs. Total reagent costs for the process, based on the continuous bench-scale tests, are conservatively estimated to be less than $1.00 per lb of Us08 (Table 111). Since the major cost is that of the relatively expensive solvent, the economic success of the process depends on keeping solvent losses a t a tolerable level. Although solvent loss by entrainment in the aqueous raffinate in tests in the laboratory coiitinuous unit was low (less than 0.01% of the wet-process acid volume after the deentrainment tank) , it will be necessary to demonstrate that a significant loss of solvent does not occur during cleanout of the solid material that accumulates in the settlers a t the aqueouq organic interface. These losses can be more accurately determined in a small pilot plant located a t a plant site where the process can be ruii continuously with fresh acid feed.
Blake, C. A., Jr., Homer, D. E., Schmitt, J. M., U.S. At. Energy Comm., Rept. ORNL-2259, 1959. Cronan, C. S., Chenz. Eng., 66 (91, 108-11 (1959). Ferguson, D. E., US.At. Energy Comm., Rept. ORNL-4272, 1968. Ferguson, D. E., ibid., ORNL-4422, 1969. Ferguson, D. E., ibid., ORNL-4572, 1970. Greek, B. F., Allen, 0 . W., Tynan, D. E., I n d . Eng. Chem., 49, 628-38 (1957). Hurst, F. J., Crouse, D. J., U.S.At. Energy Comm., Rept. ORNL-2952, 1960. Hurst, F. J., Crouse, D. J., Brown, K. B., ibid., ORNL-TM2.522. -1969. - Kennedy, R. H., in "Processing of Low-Grade Uranium Ores," pp 220-6, International Atomic Energy Agency, Vienna, Austria, 1967. Latimer, W. M.,"Oxidation Potentials," Prentice-Hall, New York, N.Y., p 44. Long, R. S., Ellis, D. A., Bailes, R. H., Proc. Znt. Conf. Peaceful Uses At. Energy, Geneva, Switzerland, 8 , 77-80 (1955).
literature Cited
Presented at the Division of Nuclear Chemistry and Technology, 161st Meeting, ACS, Los Angeles, Calif., April 1971. Research sponsored by the U.S.Atomic Energy Commission under contract with the Union Carbide Corp.
Baes, C. F., Jr., J . Phys. Chem., 60, 805 (1936). Blake, C. 9., Jr., Bam, C. F., Jr., Brown, K. B., I n d . Eng. Chem., 50, 1763-7 (1958).
I
RECEIVED for review March 29, 1971 ACCEPTED July 19, 1971
Activity Coefficients of Strong Electrolytes in Aqueous Solutions Herman
P. Meissnerl
and Jefferson W. Tester
Deparfment of Chemical Engineering, Siassachusetts Institute of Technology, Cambridge, Mass. 02139
The value of yi,the mean activity coefficient of an electrolyte in aqueous solution, varies with the ionic strength p. If the charges on the anions and cations present are, respectively, L- and L+, then the term l/Z,Z-
yi
is readily calculated. When this term yTtl/'+'- is plotted against p for various strong electrolytes at 25"C,a curve family forms such that, given a single experimental value of y* at a known ionic strength for an electrolyte, its value of y+ a t any other ionic strength can b e approximated. For use, when no such experimental point is available, a method applicable to a number of electrolytes is described for predicting T~ at an ionic strength of 2. By this empirical procedure, approximate values of y+ can b e predicted at any concentration for many strong electrolytes at 25'C.
M u c h of t'lie experiment'al data available for Ti, the activity coefficients of strong electrolytes a t various concentrations in aqueous solutions, is well summarized in the standard references (Harned and Owen, 1958; Robinson and Stokes, 1965). The Debye-Hilckel equat'ion applies to t,hese data below about 0.01m, while the modified equations of Scatchard and others, involving empirical constants to be determined for each electrolyte, apply a t higher concentrations. Pitzer and Brewer (1961) have presented a procedure for interpolating y L values for a number of electrolytes. These activity coefficient data in general are frequently presented as in Figure 1, showing curves of y* a t 25OC for several 1
To whom correspondence should be addressed
128 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1 , 1972
arbitrarily selected electrolytes plotted against concentrations from 0. to 2.0m. Inspect,ion indicates that the change of yh with molality differs widely from one electrolyte to the next. The object here is to present a simple graphical method for representing the act,ivity coefficients of strong electrolytes in aqueous solution, applicable from low concentrations to saturated solutions. To use this method, a value of y i for the electrolyte in question must be known a t a concentration well above those to which t,he Debye-Hiickel equation applies. When such an experimental value is not available, a procedure for predicting yi a t a known concentration is presented. This procedure appears applicable to chlorides. bromides. iodides, acetates, nitrates, chlorates, and hydroxides. In combination, these relat,ionshjps make it possible to predict yh values
4 KOH
1!5
cO
0.7' I
c
XHCl OKCI 0 LizS04 OCaC12
r
I.o
2 .o
m
Figure 1. The effect of concentration on the mean activity coefficient y i for aqueous solutions ut 25°C
Figure 3. The effect of ionic strength p on at 25°C in the low concentration region with the Debye-Huckel behavior shown
T h e data of Figure 1 are replotted on Figure 2, usiag the coordinates log r vs. p , A wider range of p values is covered in Figure 2 t,han in Figure 1, while additional data (Harned and Owen, 1958; Robinson and St'okes, 1965; Rloore, 1962; Pitzer and Brewer, 1961) at' relatively low concentrations are presented in Figure 3. I n contrast to the divergence and curve crossing of Figure 1, the curves on Figures 2 and 3 fall into a single family having tmhesecharact'eristics: All curves converge a t low ionic strengths and terminate a t a value for r of unity when p is zero. Also plotted on Figure 3 is the dotted liiie representing the Deybe-Huckel limiting law (Debye and Huckel, 1923), written in terms of r as follows: log
IO.
0
20.
)I
r
Figure 2. The effect of ionic strength p on at 25°C (line identification numbers correspond to those of Figure 1)
a t any concentration for a number of pure electrolytes in aqueous solution at 25°C. Activity Coefficients vs. Ionic Strength
I n the correlations pr'oposed, extensive use is made of the term I?, defined as follows:
Here z+ and z- are the charges on the cations and anions of the electrolyte in question. Concentrations are expressed as p , the ionic strength, defined as 0.5m ( Y + Z + ~ v-z-') where Y+ and v- are the moles of cations and anions, respectively, formed upon complete clissociation of 1 mole of electrolyte, while m is the electrolyte molality in g-mol/1OOO g HzO. Thus, in an m molal solution of AlC13, z+ is 3, z- is 1, Y+ is 1, Y- is 3, and p is 0.5m (1 >< 3* 3 X 12) or 6m.
+
+
r
=
-0.510
pl/z
(2)
As expected, all curves converge on the deb ye-Kickel line as p approaches zero. With few exceptions, the curves of Figure 2 usually do not C ~ O R S ,each remaining bet'ween its neighboring curves regardless of the electrolyte involved. Included here are the 1 : l electrolytes like NaC1, 1 :2 electrolytes like Na2S04, 2 : 1 electrolytes like CaC12, 2 :2 types like CuSO4, 3: 1 types like AICI,, and 3:2 types like Al,(SOI)3. Locating a single point on Figure 2, therefore, serves to locate the entire curve. Earlier plots using the coordinates of Figure 2, such as the one of Guggenheim (1933, are restricted to ionic strengths below unitmy,making it less evident t h a t a curve family exists. Inspection of Figure 3 shows t.hat a t any p value below about 0.2, the term F is similar enough for all electrolytes so that it cannot with corifideiice be assigned to a single curve. On the other hand, these curves have diverged sufficiently at a p value of unity or greater, so that the parent curve is easily selected. Knowing yrt at' a p value of unity or greater, therefore, makes i t possible to 1ocat)e the entire curve within the family of Figure 2 or 3. For example, assume that y i is to be predicted for RlgClz iii aqueous solution a t a p of 9, knowing that when p is 2, yh is 0.5, corresponding to a r of 0.707 by Equat'ioii 1. When we locate this known point on Figure 2 and extrapolate to a p of 9, r is 1.45, corresponding to a yrt value of 2.09 which is wit'hiii 10% of the experimental value for yrt of 2.32. I n view of the consistency of the curves, it also seems reasonable to assume t h a t Figure 2 applies to supersaturated solutions. Thus, knoring yk to be 0.269 for KKOs a t a molality of 3, then by extrapolation on Figure 2, yk is 0.126 a t a p of 8, Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
129
+O.IO
+ 2.0
+ 1.6
0 + 1.2
-0.10
+0.8
c
-,g
2 +0.4
-0.20
0
-
0
0
-0.30
- 0.4
-0.40
- 0.8 -0.50 0
-1.2
0
4.0
120
80
160
P
r
Figure 4. Generalized plot of as a function of ionic strength p for use at 25°C and over an ionic strength range of 2.0 to 20.0
a concentration well beyond the 3.7 molality of a saturated K x 0 3 solution. For simplicity, curves for only 21 electrolytes are shown in Figures 1 and 2. However, the generalizations just presented apply to most of the 120 electrolytes for which experimental data of yk vs. p are summarized in the standard references (Harned aiid Omen, 1958; Robinson and Stokes, 1965; Wu and Hamer, 1969). Not surprisingly, the notable exceptions are sulfuric acid, sodium and potassium hydroxide, thorium nitrate, and most of the cadmium and zinc salts, whose lines would intersect neighboring lines on Figure 2. I n addition, lithium chloride, magnesium iodide, uranyl fluoride, lithium nitrate, and berylliuni sulfate show some curve crossing. Working Charts of l o g
r vs. p
While y+ values can be determined by extrapolation on Figure 2, as illustrated above, it is more convenient to use Figures 4 and 5 as working plots for this purpose. The lines on these working plots were located by interpolation of the data for the over 110 selected electrolytes. Figure 4 is best utilized for ionic strengths above 2, and Figure 5 for ionic strengths ranging from 0.1 to 2. A t p values below 0.1, the curves have converged so that the experimental data on Figure 3 apply with sufficient accuracy. The curves on Figures 4 aiid 5 have been chosen at intervals which appear convenient for interpolation, and have been smoothed to conform to what appears to be the general family configuration. Such smoothing recognizes that for any two neighboring curves I and I1 drawn through experimental data on Figure 2, the term log rI/log rII is reasonably linear in p , a t least ovei shorter ranges of concentration. This behavior is consistent with Scatchard’s observations (Scatchard, 1936; Pitzer and Brewer, 1961) that the term log yr/log Y I I is linear in m for two 1:1 electrolytes, for two 2: 1 electrolytes, and so forth. Closer scrutiny of Figure 1 shows that it contains several curve families, one for the 1: 1 salts, another for the 1:2 curves, another for the 2 : 1 curves, another for the 1: 3 salts, and so forth. The curve crossover which occurs on Figure 1because of the dissimilar pattern of the various curve families is largely 130 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1 , 1972
0.50
I.oo
1.50
2.00
F of r
20.0
Figure 5. Generalized plot as a function of ionic strength p for use at 25°C and over an ionic strength range of 0.1 to 2.0
eliminated b y replotting as in Figures 2 to 5. Curve extrapolation within each family on Figure 1 could of course be carried out, and would have the advantage of yielding yh directly, rather than r, but would lose the guidance afforded on Figures 2 to 5 by the neighboring curves for the other electrolyte types. Precision
Since the curves of Figures 4 and 5 diverge as p increases, extrapolation from a higher to a lower p value is always more successful than the reverse. The success attained by extrapolation also depends on the range of p involved, and on the electrolyte selected for study. Thus, extrapolation should not be undertaken for those electrolytes showing excessive deviations from the curves of Figures 4 and 5, namely the chlorides, bromides, and iodides of zinc and cadmium as well as sulfuric acid and thorium nitrate. Excluding these materials, 59 electrolytes were found for which experimental yf values were reported (Harned and Owen, 1958; Robinson and Stokes, 1965; W u and Hamer, 1969) up to a p of at least 9. Figure 4 was used to extrapolate the experimental y+ value at a p of 2 to a p of 9. The average deviation of the predicted from the experimental y* value at a p of 9 was 18%. Several of the 59 electrolytes included in this calculation showed considerable curve crossing, with differences between the experimental and predjcJed values for yi in excess of 25%, as listed: LiN03 (74%), LiCl (3i%), CaCl? (%%I, Mg(C104)~(54%), MgIz (46%), BeS04 (41%), NaOH (42%), CuCh (30%), Mg(Ac)? (32761, Ba(Ach (39%)) and UOZ(XO& (25.5%). Extrapolation over a range of p values smaller than from 2 to 9 would, of course, have resulted in smaller errors. Further consideration of these data shows greater success in prediction with the electrolytes of lower ionic charge. Thus, for the 59 electrolytes considered in the above test, the 43 1: 1, 1: 2, and 2: 1 types showed a n average deviation of 16.5% while the 15 3: 1, 2 : 1, and 3 : 2 types showed a deviation of 22%. Deviations of the experimental r curves of electrolytes having a (z+z-) product of 3 or greater from the generalized curves limit the success attained in extrapolating yh values. These deviations are not large, as is evident from inspection of
Table 1. Valuer of I’M,+cl,- at p Equals 2 MutClu-
I
\
-*-
-
I
AgCl AlC13 BaCle BeCl, CaClz CdClz CeC13 CoClz CrC13 CsCl CuClz EuC13 FeC12 HC1 KC1 LiCl
I
I
lo
20
t Figure 6. The effect of ionic strength p on I‘ a t 25°C for 19 2 : 2, 3 : 1, and 3 : i! electrolytes. Lines from Figure 4 are replotted as dotted lines
rMyLCly-
MvtClv-
rM,+Cl,-
0.550 0.673 0.630 0.710 0.676 0.292 0.639 0.690 0.666 0.496 0.641 0.646 0.679 1.009 0.573 0.921
MgC12 MnClz NaC1 NdC13 NHiCl NiCl2 PbClz PrC13 RbCl scc13 SrClz ThC14 U02Clz Yc13 ZnCls
0.707 0.670 0.668 0.639 0.570 0.691 0.599 0.638 0.546 0.657 0.660 0.717 0.733 0.646 0.612
6. For example, starting with the experimental value of r a t a of 2 for MgS04, r can be predicted a t a H of 9, using either Figure 4 or Figure 6. The difference between this predicted r and the experimental value is 29% with Figure 4 but only 2% with Figure 6, which is approximately the accuracy of reading the chart itself. p
Estimating Y& Values
As just pointed out, Figures 4 and 5 can be used t o predics a t various ionic strengths only when a known value of T~ it
,>
0.8-
+>
X
L=
0.6
-
0.4
-
0.2
-
c
u -
OO
0.2
0.4
r
0.6
0.8
I .o
wv.cAV-
Figure 7. Cross plots of I’Mv+Xv- vs. r M u + c l u - [ X and NO3-(ll)] a t 2 5 ° C and an ionic strength of 2.0
= I-
Figure 6, where experimental values of r for 2 :2, 3 :2, and 3 : 1 electrolytes are plotted on a n expanded ordinate of r, along with dotted lines taken directly from Figure 4. Errors in predicting y& caused by these deviations are magnified b y the ( z + z - ) product in the exponent of Equation 5. Thus, when extrapolating for a 2 : 2 electrolyte, the term (z+z-) equals 4, and consequently, a n error of 1Oyo in r from Figures 4 and 5 results in a 40% error in y5. This error, which obviously beconies larger as the (z+z-) product increases, is reduced by U K of the expanded ordiiiate scale of Figure 6, and so improved success can be attained by replacing Figure 4 b y Figure
available for the electrolyte in question a t some given concentration well outside the Debye-Huckel region. When such a n experimental reference value is lacking, the term log r, and consequently yi itself, can often be predicted b y the procedure illustrated in Figure 7 . This figure applies always at 25OC at a value for of 2, with the ordinate referring to the iodide or the nitrate of a particular cation, and the abscissa to the chloride of the same cation. Each of the 13 iodide points plotted, therefore, involves a different single cation, with the lowest point representing cadmium, the next lowest point cesium, the third rubidium, and so forth. The points on Figure 7 can be identified b y their abscissa values-i.e., r for Mv+C1,- from Table I. Inspection shows that the iodide points on Figure 7 not only fall on a single line, b u t that this line is straight. Consequently, from Figure 7, when we know r for the chloride of a given cation at an ionic strength of 2, the value of r for the iodide of this cation (at a n ionic strength of 2) can presumably be obtained. Similarly, when we know r for the iodide, the r value for the chloride of the same cation can be obtained when p is 2. The same type of correlation just described, relating r values for the iodides and chlorides, exists between other anions and the chlorides. Thus when I’values for bromides are plotted against F values of the chlorides of these same elements, always a t 25OC and a n ionic strength of 2, a new straight line is obtained. Similarly, r values of the sulfates, fluorides, and nitrates (excepting nitric acid, zinc, and cadmium) are plotted against the I- values of the corresponding chlorides in Figures 7 and 8. I n each case, a straight line forms, regardless of the z+ and z- values of the electrolyte involved. The lines on Figures 7 and 8 are all of the form rMv+Xv- = A r M V + C l v -
+B
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1, 1972
(3) 131
I
I
a
1
I
I
-
finds that the contributions of “charge-charge” and “chargemolecule” effects to log yrt all vary as the product (z+z), while the “molecule-molecule” effects are relatively independent of ion charge. I n other words, for a given solvent, T~ is a function of p (z+z), and noncharge factors unique to each solvent-electrolyte system. A similar dependence between variables is involved in the equation of Davies (1938) which followed the work of Guggenheim and others (Guggenheim, 1935; Guggenheim and Turgeon, 1955; Guggenheim and Stokes, 1969; Wu and Hamer, 1969) :
I n this equation, log y* is again indicated to be a function of B, which changes from one solvent- electrolyte system to the next. There appears to be no theoretical conflict between all this earlier work and the correlations of Figures 2-5, wherein I’ (which includes the effect of ion charges as is evident from Equation 5) is shown to be a function only of p and of the electrolyte itself in aqueous solutions. The curves of Figures 2-5 fit the experimental data more successfully than the analytical equations mentioned, presumably because the latter are derived on assumptions regarding the force-distance relationships between ions, while no such assumptions need be made in the graphical approach. The relationships shown in Figures 7 and 8 and by Equation 4 are again empirical. The selection of chloride as the reference ion was arbitrary, but was made because of the relative abundance of data for chloride systems. The success encountered in predicting r values a t an ionic strength of 2 by Equation 4 has already been discussed. Presumably equal success will be encountered with other anions such as the cyanides, but this remains to be established. To use Equation 4 for electrolytes involving other untested anions, values for A and B in Equation 4 must be available, requiring experimental data on a t least two compounds. I n the cyanide system, these might be KCN and NaCN. The selection of an ionic strength of 2 as a basis for Equation 4 and Figures 7 and 8 was also arbitrary, since equivalent relations exist a t other ionic strengths. Moreover, the iodidechloride data a t ionic strengths both a t unity and 4 yield points which fall approximately onto the line of Figure 7 and so have the A and B values of Table I. Other systems, however, such as the sulfate-chloride electrolytes, form different lines from those of Figure 7 for ionic strengths of unity and 4. T h e
p ( z + z - ) , and the empirical constant,
0.4 0.4
0.6
0.0
I .o
rM,,+CIv-
Figure 8. Cross plots of r M v + X v - vs. rMv.Clv- ( X - = F- and S 0 4 2 - ) at 25’C and an ionic strength of 2.0
where r M u + C l v - refers to the chloride of a given cation M while I’Mv+Xv- represents the combination of M with the anion always a t the ionic strength of 2. Table I presents values of r M v + C l u - while Table 11 presents values of A and B for various anions to be used in Equation 3. These tabulated A and B values were obtained by least square calculations, while the standard deviations a* of the best fit are listed in Table 11. From Figures 7 and 8, or Equation 3, values of I? a t an ionic strength of 2 can be predicted for many substances. Thus knowing r for Pb(NO& to be 0.426 a t a p of 2, the value of r for PbClz a t this concentration (which would represent a highly supersaturated solution) can be predicted from Figure 7, and is equal to 0.599.
x,
Discussion
Deviations from ideal solution behavior for electrolytes, leading to T* values lower or higher than unity, are the result of interactions of ions with other ions and with the solvent. Scatchard (1959) discusses these under three headings, namely the “charge-charge” (electrostatic) interactions of the DebyeHiickel theory, ‘Lcharge-molecule’’ effects such as are encountered in salting out behavior, and the “molecule-molecule” effects involved in solvation-association reactions. I n developing his final equation 2 (Scatchard, 1959), Scatchard
Table II. Values of A and B for Equation 3; p = 2.0 Anion
FC1Br-
INO*-
504’-
Cation
Na, K, Rb, Cs’
H, Na, Li, K, Cs, Rb, Mg Ca, Sr, Ba, Co, Cd, Zn Na, K, Rb, NHI, Mg, Ca, Sr, Co, UOZ Li, ”4, Rb, Cs, Na, Al, Mg, Cu, Mn, Ni, U0zb Na, Rb, K, Cs, Mg, B a
OAc(acetate) H, Li, Na, Mg, Ba, Ca C104OHK, Na, Li ~ UO2F2 omitted. * H2SO4 omitted.
132 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1, 1972
A
B
d 0.00018
No. of points
4
-1.482 1 1.259 1.648 2.248
1.521 0 -0.121 -0.313 -0.918
0 0.00078 0.00196 0.00272
...
0.100
0.443
0.00056
11
1.830
0.01565
6
-0.085 1.437
0.01065 0.00174
6 3
-1.707 1.223 -1.043
13 13 9
choice of a n ionic strength of 2 for Equation 4 and for these cross plots is therefore a compromise, since a t smaller p values the differences between r for various electrolytes diminish, thereby increasing uncertainty in locating the parent curve, while a t larger p values fewer data on y* become available. Finally, values of yh for electrolytes can be measured very accurately by methods such as freezing point depression and half cell potentials. In consequence, experimental values of yf are often reported to three significant figures. A n equal success cannot, of course, be obtained by the empirical method described here, and so this procedure should be used only when direct measurements are lacking. Nomenclature = electrolyte activity, mo1,/1000 g solvent = ykmk = mean electrolyte activity, mo1/1000g yk = mean activity coefficient’ at 25°C = (y+)b’*+z-
a
uk
solvent
r
molality, mo1/’1000 g solvent rn(v+y+v-y-)l/v moles of completely dissociated catioiis moles of completely dissociated anions
Y
= = = = =
p
=
ionic strength =
m
mi: V+
V-
Y+
+
Y-
‘/z
x(Vini)zi2 a
zt
z-
= =
charge on the cation charge on the anion
Literature Cited
lhvies, C. W., J . Chem. Soc., 1938, p 2093. Debye, P., Huckel, E., Physik. Z., 24, 185, 305 (1923). Guggenheim, E. h.,Phil. Mag., 19, 588 (1935). Guggenheim, E. A , , Stokes, R . H., “Equilibrium Pro erties of ilqueous Solutions of Single Strong Electrolytes,” $ergamon Press, New York, K.Y., 1969. Guggenheim, E. A., Turgeon, J. C., Trans. Faraday Soc., 51, 747 (1955). Harned, H . S., Owen, R. B., “Physical Chemistry of Electrolyte Solutions.” No. 95. 3rd ed.. Reinhold. New York. N.Y.. 1958. hloore, W.’ J., “ P h & d Chemistry,” 3rd ed., Prentide-Hall, Englea-ood Cliffs, N.J., 1962. Pitzer, K. S., Brewer, L., “Thermodynamics,” McGraw-Hill, Kew York, N.Y., 1961. Robinson, R. A., Stokes, It. H., “Electrolyte Solutions,” 2nd ed. rev., Academic Press, New York, N.Y., 1965. Scatchard, G., Chem. Rev., 19, 309 (1936). Scatchard, G., in “The Structure of Electrolyte Solutions,” p 9, W.J. Ilamer, Ed., Wiley, New York, N.Y., 1959. W u , Y-C., Hamer, W. J., Electrochemical Data, Part XIII, Vatl. Bur. Std., 1Lept. 10002 (1969). RECEIVED for review March 31, 1971 ACCEPTED July 22, 1971 Presented in part a t the Meeting of the Electrochemical Society, Washington, D.C., May 11, 1971.
Heat Transfer to Air-Solids Suspensions in Turbulent Flow Shafik E. Sadek Dynatech R I D Co., 99 Erie St., Cambridge, Mass. 02189
A correlation of heat transfer coefficients to air-solids suspensions based on the principles of similarity i s presented. Published data indicate that a single parameter, particle area per unit gas volume, i s sufficient to define reasonably well the relative increase in heat transfer coefficients. The scatter in the available data prevents the weak dependency on another parameter, the pipe Reynolds number, from being accurately defined. The correlation represents data in air covering pipe Reynolds numbers from 4000-80,000, particle diameters from 20-600 1.1, pipe diameters from 17-1 02 mrn, and solids loading ratios up to 300 in vertical transport.
T h e effect of particulate matter on gas-phase heat transfer has recently received widespread attention. Particles suspended in a gas were discovered to increase heat transfer rates, often significantly above those expected from the higher heat capacities of the mixtui~esand the resulting greater driving forces (Koble e t al., 1951; Farbar and Xorley, 1957). The promise of high heat transfer rates to gas-solids suspensions during transport led to a number of studies to gain an understanding of the mechanisms involved. At first, the augmentation of heat transfer mas explained on the basis of the mixture’s behaving as a homogeneous mixture (Depew and Farbar, 1963; Tien, 1961). When experimental data indicated that this theory was inadequate, i t was concluded that the particles disturbed the velocity profile and consequently the temperature profile, thus reducing the effective thickness of
the boundary layer (Depew and Farbar, 1963; Jepson et al., 1963; Leva, 1959; Peskin and Briller, 1970). Even though this is the most widely accepted theory for explaining the augmentation effect, no satisfactory correlation was developed on this basis. Theoretical developments did not correlate the data properly olving to the complexity of the details of the augmentation mechanism (Peskin and Briller, 1970), while empirical correlations fell short from being general (Danaiger, 1963; Koble et al., 1951; hlickley and Trilling, 1949). It was pointed out that a compromise between these two approaches was the most promising approach toward generating a useful correlation (Peskin and Briller, 1970). Such a compromise is offered b y the principles of similarity: Dimensionless parameters governing the process are first defined based on a simplified understanding of the augmentation mechanisms folInd. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
133
