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
Figure 1 . Effect of particle diameter and particle number density on heat transfer coefficients of gases during vertical transport Data of Farbar and Depew (19631 and Wilkinron and Norman ( 1 9 6 7 ) at Reynolds numbers of 15,000-22,000
0.I IO'
I0 5
IO6
Dimensionless Particle Number Density, nD3
lowed b y an empirical correlation of these parameters. This article presents a n attempt at correlating heat transfer coefficients in air-solids suspensions using the principles of similarity. Correlation
It has been postulated that particles suspended in a gas prevent full development of the velocity and thermal boundary layers inside the pipe. Even though the thermal properties of the particles will affect the temperature profile of the gas along the pipe inasmuch as they act as numerous tiny heat sinks, it was assumed that their effect on the heat transfer coefficients was largely mechanical. According to this theory, the particles affect the heat transfer coefficients because they disturb the flow patterns in the pipe, and not because of their thermal properties, even though these properties may affect the rate of heat transfer significantly. If this mechanism truly governs the augmentation, the augmentation will be independent of the thermal properties of the solids but will be a function of the rate of penetration of particles into the boundary layer-i.e., to the rate of exchange of particles between the core and the wall regions. Another mechanism explaining the increase in heat transfer coefficients to gases in the presence of particles is the convection of sensible heat carried by the particles as they are exchanged between the wall and the core regions (Wicke and Fetting, 1954) ; this mechanism will govern the augmentation if particle exchange between these regions is rapid and if the heat transfer between the particles and the gas is rapid. If this were the mechanism governing the augmentation, the augmentation will be strongly dependent on the thermal properties of the particles as well as their rate of exchange across the pipe. A general correlation expressing the increase in heat transfer coefficient must therefore include terms reflecting the rate of exchange between particles in the bulk and near the wall, the undisturbed boundary layer thickness, the particle and pipe diameters, and the thermal properties of the solids and the gas. Under conditions of vertical transport, when gas velocities are appreciably higher than the particle settling velocity, the rate of exchange of particles across the pipe (and therefore their rate of penetration into the boundary layer) will depend on the Reynolds number and on the average number of particles in 134 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
the pipe (Soo, 1962) within some characteristic distance along the t'ube. Under conditions of transport, then, three independent dimensionless parameters-in addition to the property ratios-can be defined which relate the relative increase in heat transfer coefficient to the operating variables. These are a dimensionless number of particles within the pipe, the particleto-pipe diameter ratio, and the particle-free pipe Reynolds number. Because considerable data have been published showing the effect of particles on the heat transfer coefficients to air, and because most of the data gathered under turbulent transport mas obtained while using glass spheres, silicates, or silica, the correlation presented in this note will be limited to these systems. When we omit the dependence of the increase in heat transfer coefficient on the property terms, it is then assumed that a correlation of the form expressed in Equation 1, will fit the data: Ah/ho =
CY
(nD*) a(d/D)b(Reo)
(1)
Here, n is the average number of particles in the pipe per unit volume of gas. For convenience, the dimensionless parameter nD3 can be expressed in terms of the primary quantities:
nD3 =
p(?)(:)(y)'(?)
The symbols p , p Q J and p , represent the average density of the mixture in the tube, the gas density, and the particle density, respectively. W p and W , represent the mass flow rates of particles and gas in the tube. V o is the gas velocity inside the tube, and V , is the particle velocity. Figure 1 shows data gathered by Farbar and Depew (1963) and by Wilkinson and Norman (1967) on the heat transfer coefficients to air-solids suspensions. The solids in these tests were uniform glass spheres. Two pipe diameters were used: 0.056 ft by Farbar and 0.175 ft by Wilkinson. Pipe Reynolds numbers (based on the gas properties) were 15,300for Farbar and Depew's data, and ranged between 15,000 and 22,000 for Wilkinson and Pu'orman's. I n all these tests, gas velocities were greater than about 20 ft/sec, considerably higher than
particle terminal velocities, which were estimated to be less than 2 ft/sec; in reducing these data, it was therefore assumed that average gas and particle velocities were equal. The reduced data are plotted in Figure 1 as the relative increase in heat transfer coefficient vs. nD3 for that range of Reynolds number. I n spite of the scatter, it is readily seen that the data from these two sources fall in their proper positions relative to the ratio ( d / D ) . These data and those of nlickley and Trilling (1949) under conditions of vertical transport were used to determine the constants in Equation I.. With air as the transporting medium, conveying glass spheres, this relationship may be expressed as: Ah/ho = 0.20 [ (nD3)(d/D)z]l’*’g =
(3)
0.20 (nd2D)’J9
The single parameter (nd2D),a dimensionless particle area per unit gas volume, appeared to correlate the data well. No significant effect of Reynolds number on the relative increase in heat transfer coefficient was observed: The scatter in the data covers any such effect within the operating ranges investigated. Figure 2 shows the data referred to above, that of Jepson et al. (1963) with sand and of Farbar and Morley (1957) with a silica-alumina catalyst. Since the sizes of these catalyst particles were not uniform, an area-mean diameter was cal-
0
A
c
0
Wilkinson cind Normon Farbar and Morley Mickley and Trilling
culated from the distribution which they reported. These data as a whole cover a range of Reynolds numbers from 4000 t o 80,000, average particle diameters from 20-600 p, pipe diameters from 0.056-0.33 ft and solids loading ratios up to 300 lb solids/lb gas. Conclusion
-4single parameter (do)correlates the data for the increase in heat transfer coefficients in air-solids suspensions. The solids consisted of glass spheres, sand, and a silicaalumina catalyst. More accurate data are required to express reliably the effect of the Reynolds number on the augmentation. D a t a covering a wider range of gas and solids properties are necessary to define the mechanism governing the augmentation: Lack of dependence on the thermal properties of the solids will indicate that the augmentation is due to a mechanical disruption of the boundary layer by impacting particles, while a strong dependence on these properties will indicate that the exchange of particles contributes to the convection between the wall region and the bulk of the gas. Nomenclature
a = exponent on the parameter nD3 (Equation l ) , dimen-
sionless exponent on the diameter ratio, d / D (Equation l ) , dimensionless c = exponent on the pipe Reynolds number, R e (Equation l), dimensionless d = particle diameter, ft D = pipe diameter, ft h = heat transfer coefficient, Btu/(hr-ft2 OF) n = average number of particles per unit gas volume in pipe,/ft3 R e = pipe Reynolds number, dimensionless V = velocity, ft/hr 1.t7 = mass flow rate, lb/hr
b
=
GREEKLETTERS
c Q
a = proportionality constant (Equation 1) A = increase beyond particle-free value p =
density, lb/ft3
SUBSCRIPTS g = pertaining t o gas 0 = in particle-free flow p = pertaining t o particles literature Cited
0.01 0.I
I
Figure 2. Correlation between increase in heat transfer coefficient and particle area parameter Data of Farbar and Depew ( 1 963),Reo = Data of Wilkinson and Norman (1 967),Reo = Data of Jepson et al. ( I 963),Reo = Data of Farbar and Morley ( 1 957),Reo = Data of Mickley and Trilling (1 9491,Reo =
15,300-26,500 15,000-80,000 11,000-55,000 13,000-27,000 3800-1 5,400
Danziger, W. J., Ind. Eng. Chem. Process Des. Develop., 2, 269 (1963). Depew, C. A., Farbar, L., J. Heat Transfer, Trans. A S M E , 85, 164 (1963). Farbar, L., Depew, C. A., Ind. Eng. Chem. Fundam., 2, 130 (1963). Farbar, L., Morley, h1. W., Ind. Eng. Chem., 49, 1143 (1957). Jepson, G., Poll, A,, and Smith, W., Trans. Inst. Chem. Eng., 41, 207 (1963). Koble, R. A., Ademino, J. N., Bartkus, E. P., Corrigan, T.E., Chem. Eng., 58,174 (1951). Leva, M.,“Fluidization,” pp 218, 231, McGraw-Hill, New York, N.Y. (1959). Mickley, H. S., Trilling, C. A., Ind. Eng. Chem., 41, 113; (1949). Peskin, R. L., Briller,,,R., “Augmentation of Convective Heat and Mass Transfer, Winter Annual Meeting ASME, A. E. Bergles, R. L. Webb, Eds., New York, N.Y., December 1970. Soo, S. L., Ind. Eng. Chem. Fundam., 1,33 (1962). Tien, C. L., J. Heat Transfer, Trans. ASME, 83,183 (1961). Wicke, E., Fetting, F., Chem. Ingr. Tech., 26, 301-9 (1954) (as referred to in Leva, M., “Fluidization,” p 185). Wilkinson, G. T.,Norman, J. R., Trans. Inst. Chem. Eng., 45, T314 (1967). RECEIVED for review April 2, 1971 ACCEPTED June 21, 1971 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1, 1972
135
