COMMUNlCATlONS
Analytical Relationships for Influence of Particle Size on Precipitator Design This paper presents simple analytical expressions for precipitator efficiency arising from the integration of the Deutsch equation with a log-normal particle size distribution. The results have wide applicability to many other types of particulate collection devices gs well.
T h e realistic design of many types of particulate collection devices involves consideration of the influence of particle size distribution. For an electrostatic precipitator, this influence is appropriately expressed by the integration of the Deutsch equation for a log-normal particle size distribution. However, the integral which must be evaluated depends on several parameters and cannot be evaluated exactly in terms of elementary functions. To express this integral simply and accurately, asymptotic methods are employed in this paper. These methods produce analytical approximations which are expressed in terms of well-known functions and the solution of a certain nonlinear algebraic equation. Comparison with numerical calculations shows that the results of the asymptotic analysis are applicable for virtually all cases of practical interest. Electrostatic Precipitator Theory
I n an electrostatic precipitator, suspended dust particles in a gas are electrically charged and migrate to collecting surfaces where they are captured. A simple expression for the migration velocity is :
by a precipitation rate constant based on experience (Nichols and Oglesby, 1970). By lumping all nonsize related quantities into one parameter
(3) Equation 2 can be written tl = 1
- e-kd~
Equation 4 is exponential in character although k , like w , is empirical. A plot of this function provides an example of the grade-efficiency curve discussed by Stairmand (1956, 1965, 1970). The collection efficiency for each particle size in a distribution is defined by the grade-efficiency function. Collection Efficiency log-Normal Distribution
When an exponential grade-efficiency function is applied to a log-normal particle size distribution] the overall efficiency is given as follows 1 tl=l--
+m
4LS-m
and the collection efficiency of a monodisperse particulate by a precipitator in turbulent flow can be calculated using the Deutsch equation
The simple migration velocity based on Stokes law and nondielectric particles can be refined as desired; however, the value of w from Equation 1 may still be several times too high because of effects unaccounted for in its derivation. These include multiple particles] uneven gas distribution, particle reentrainment, and high dust resistivity (White, 1955, 1963). Therefore] to predict the operation of an actual precipitator, the theoretical migration velocity is replaced
(4)
-CTa&t
e
dt
(5)
where a kd, and b In CT. Although attention has been focused on electrostatic precipitators, the technique to be outlined below applies as well to other types of particulate collection devices whenever their grade-efficiency curves can be approximated by Equation 4. Typical grade-efficiencies replotted on semilogarithmic coordinates from Stairmand’s tabulations and curves are shown in Figure 1 for several such devices. Increasingly efficient collection is reflected by a larger value of k . This plot is meant only to indicate an order of magnitude of k for different types of equipment. These k values are strongly influenced by operating conditions and may vary Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No.
4, 1972 623
c
1C 21
31
3
4c
I
I
1
2
I
I
I
4
5
51
6C V W K
8
z
7c
2
w u k
2 8C
W
2
% a
0
+
V
W J -1 0 V
90 91 92
1
93 94 95 PARTICLE DIAMETER (MICRONS)
0
Figure 1. Typical grade-efficiencies for several collecting devices. Data from Stairrnand (1956, 1965, 1971). For illustration only, not for design
widely for each device. Moreover, the grade-efficiency curve itself may deviate from an exponential function. The decision regarding whether certain grade-efficiency data are adequately approximated by an exponential is best made on an individual basis. Approximate Representation of Integral
The infinite integral in Equation 5 is seen to converge for all values of a and b. The integrand -0 for both t -., - a and t and thus, since the integrand is always positive, it must have a maximum value a t some point t*. At the maximum point t*, the usual condition for an interior maximum requires that
-
t*
+ debt* = o
(6)
It is easily verified that there can be but one maximum point t* for any given set of values of a and b. Thus t* depends on both a and 6. However, by changing the variables to t* E a1%* and C 3 a1i2b,Equation 6 is reduced to a form involving the single parameter C as follows u*
+ CeCu* = o
(7)
Since the integrand involves exponentials, it is natural to hope that by investigating the maximum of the integral and suitably changing variables, the original integral might be transformed to one of the Laplace type, which could then be evaluated approximately for certain limiting values of a or b (Erdeleyi, 1956). However, in this problem a transformation of this type does not yield the desired Laplace integral, but the transformation is nevertheless extremely useful since it does lead ultimately to a resolution of the original problem. Some details of the analysis involved in approximating the integral are given elsewhere (Kunz and Hanna, 1971). Here we will present the essential results of the analysis. 614 Ind.
Eng. Chem. Process Der. Develop., Vol. 1 1 , No. 4,
1972
3 C
E
b
Figure 2. Solution of nonlinear Equation 7
The most useful approximation of the original integral 1 is as follows
where the quantity A is related to the solution of nonlinear Equation 7. The solution of this equation is indicated graphically in Figure 2. I n addition, approximate analytical solutions of this equation accurate to better than 5% are developed in the cited reference. These approximations may be used directly or as first guesses for iteration schemes for use on the digital computer. Suitable procedures include the Newton method or the ordinary iteration method accelerated by the 6* process (Henrici, 1964). Equation 8 is a simple expression which approximates the original integral very well, as shown in Table I. It is easily proved that this equation is valid in the limit of either a -P C O , a -P 0, or b 0. The fact that it applies to so many limiting cases gives some reason for its excellent agreement with numerical results over such a wide range of a and b values, except perhaps in the impractical case of large b (log-normal probability particle size distribution approaching a vertical line).
-
Approximate Representation of Incomplete Integral
Frequently an estimate of the particle size distribution of the uncollected material is required in addition to an overall efficiency prediction. This would be important where several collecting devices are to be installed in series. For example, it is not uncommon for cyclones to be installed upstream of a precipitator. I n this case, the quantity and distribution of the particulates escaping from the cyclones would be used as the feed to the precipitator, and the calculation would be repeated. To calculate the particle size distribution, it is desirable to approximate the above integral where, instead of a , the
nlf
A 0 o o o o o oo o o o o o o
Rl
z , 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Table II. Asymptotic Size Distribution of Escaping Material Checked by Two Other Methods Particle size, fi
Cum. wt. Numerical
Asymptotic"
1 2 3 4 5 6 7 8 9 10 15
8.5 32.0 53.1 72.0 82.1 88.5 92.8 95.4 97.1 98.2
...
...
m
100.0
100.0
% under
8.6 33.5 56.9 73.6 84.1 90.5 94.5 96.6 98.0 98.8
Stairmand
9.7 34.9 57.7 74.1 84.9 91 .o 94.9 97.1 98.5 99.3 100.0
a In this example, d, S 2.96 corresponds to z = t * and is therefore where we switch from Equation 9 t o Equation 10.
upper limit z = In (d,/d,)/b to a and b. The result is -
7 2% - E
is now a parameter in addition
+ + Eb)%/2(1+ b2E) (E
r= where E a-e+bz. Equation 9 holds only for e --E2- E
0 2
e-"ds
(9)
> t*, we have instead + + bE)*/2(1+ bZE)
5
t*. For z (E
sa '
e-*' d s
(10)
r f b E = 2112(1 + b2E)LIZ
The approximations given for J in Equations 9 and 10 involve the well-known error function which is widely tabulated and for which there are simple computer approximations (hbramowitz and Stegun, 1964). These equations are shown in the example following to approximate well to the integral J. Typical Application of Results
+oooooooooooooo
It is desired to compute the overall efficiency of a precipitator with an exponential grade-efficiency characteristic operating on a typical log-normal inlet dust distribution and to evaluate the particle size distribution of the uncollected material. By plotting the tabular distribution given by Finney (1968), one obtains 2, = 12 p and u = 2.8. From Figure 1, k = 0.46. Then a = kd, = 5.52, b = In u = 1.03, and C = alizb = 2.42. From Figure 2, Ab2 = 1.42 giving A = 1.34. From Equation 8, 7 = 93.4%. h refinement of these calculations, including a computer solution of Equation 7 , gives an efficiency of 1-0.065436 compared to the above figure. An efficiency of 1-0.065370 is obtained by direct numerical integration of Equation 5. The particle size distribution in terms of the fraction lost below a given size can be found by considering the integral J approximated by Equations 9 or 10, depending on the value of z. To obtain the cumulative particle size distribution, J must be divided by I, the infinite integral. A comparison between these asymptotic approximations for the uncollected Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
625
particle size distribution and numerical integration is presented in columns 2 and 3 of Table 11. The preceding calculations can also be compared with a more commonly employed method in which the inlet dust distribution is broken up into a finite number of narrow size ranges, the amount collected for each range is determined, and these values are summed to find overall collection efficiency. An example of this procedure is given by Stairmand (1956, 1970). The overall efficiency is again 93.4%, and the distribution of escaping material is given in column 4 of Table 11. The particle size distributions from all three methods are similar, with differences of the order of a few percent. The “exact” numerical integration distribution, with d, of 2.7 p and u of 1.9 lies between this approximate distribution and the one obtained using the asymptotic expressians. As expected, the distribution of escaping material is much finer than the inlet distribution because of preferential collection of the larger particles. The asymptotic functions are especially attractive for hand calculation of overall efficiency. For a complete size distribution, Equations 9 and 10 are more tedious to use by hand than the Stairmand method. However, with machine computation, both methods are competitive and represent a savings in computer time over numerical integration. Nomenclature
a A A,
= = = b = C = (4, = d, =
e
E E, E, I
J k
=
= = = =
= =
Q
= gas volumetric flow s = variable of integration t = variable of integration t* = location of maximum point
u*
=
t*/a1/2
w = effective migration velocity z = variable upper limit of integral J = In (d,/d,)/b
GREEKLETTERS = collection efficiency = gasviscosity T = numerical constant u = geometric standard deviation
q p
literature Cited
Abramowitz, M., Stegun, I. A., “Handbook of Mathematical Functions,” p 299, NBS AMS 55, U.S. Government Printing Office. Washington. D.C. (1964). Erdeleyi, A., “Lymptotic ‘Expansions,” p 36, Dover, New York, N.Y., 1956. Finney, J. A., Jr., Power Eng., 72 (12), 26 (1968). Henrici, P., “Elements of Numerical Analysis,” p 72, Wiley, New York. N.Y.. 1964. Kunz. R. G.. Hanna. 0. T.. PaDer Dresented at the 162nd National geeting, ACS, “Sympbsiui on Combustion,” Preprint 23 Div. Fuel Chem., 15 (2), 121 (1971). Nkhols, G. B., Oglesby, S., Jr., Paper presented at the 63rd Annual Meeting AIChE, “Symposium on Particulate Control Device Measurement,” Paper 4a, 1970. Stairmand, C. J., J.Inst. Fuel (London), 29, 58 (1956). Stairmand, C. J., The Chemical Engineer (London), 194, CE310 (19fiiil
kdp
-Cu*/bZ = -t*/b area of collection plates lno aU2b particle diameter mass median diameter base of natural logarithms mbt*
mb; charging electrical field intensity collection electrical field intensity infinite integral in Equation 5 integral I with variable upper limit collection parameter
Stair-mand, C. J., Filtr. Separ., 7 (l), 53 (1970). White, H. J., Ind. Eng. Chem., 47, 932 (1955). White, H. J., “Industrial Electrostatic PreciDitation,” , vv -. 155-95, Addison-Wesley, Reading, Mass., 1963. ROBERT G. KUNZ’ OWEN T. HANNA2 Esso Research and Engineering Co. Florham Park, N . J . To whom correspondence should be addressed. Present address, University of California, Santa Barbara, Calif. 1
2
RECEIVED for review November 4, 1971 ACCEPTEDJuly 10, 1972
Transient Behavior of Solar Pond Through Fourier Analysis The nonlinear equations describing the transient behavior of a solar pond have been solved in linearized form using Fourier analysis. The solution forms the basis of well-known semiempirical relationships used for predicting the rate of solar evaporation from large shallow ponds. The limitations of linear analysis as applied to solar ponds have been discussed.
THE
transient behavior of a solar pond was analyzed by Ferguson (1952) using a single energy balance for the brine, and extended by the author (1972) toinclude conduction of heat in the ground. This article presents an exact solution to the linearized form of the model described by the author. A clear basis and restrictions have been established for using the semiempirical relationship developed by Penman (1948) for predicting the rates of solar evaporation.
resistance, and neglect the changes in the air temperature and humidity caused by the pond, the following equations can be derived from transient energy balance.
D Pl CPI
dT
-’a!e
= a&
+ a’&’ - Q, - h ~ ( T 1- T,) - pa) - ~ L ( T-I T A
(1)
Basic Equations
If we assume that the heat and mass transfer between the brine and surrounding air are controlled by the gas-film 626 Ind. Eng. Chem. Process Des.
Develop., Vol. 1 1 , No. 4, 1972
where, T I @ )and T2(8,x)are brine and ground temperatures, 8 is the time, and x is the vertical coordinate measured below
