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,
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 = mbt* -Cu*/bZ = -t*/b = area of collection plates b = lno C = aU2b (4, = particle diameter d, = mass median diameter
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
base of natural logarithms mb; charging electrical field intensity = collection electrical field intensity = infinite integral in Equation 5 J = integral I with variable upper limit k = collection parameter
To whom correspondence should be addressed. Present address, University of California, Santa Barbara, Calif.
e
E E, E, I
=
= =
Esso Research and Engineering Co. Florham Park, N . J . 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
LOO
-
I
-
I
I
w
100
.
85
v
P
80
e
0
.6
24
0, kw
8
b
12
20
16
Figure 2. Assumed profile of air temperature during the day
I), h w
Figure 1. Assumed intensity of direct radiation, clear sky with total insolation = 2675 Btu/ft2
the ground surface. T1 may be looked upon as a depthaveraged temperature of the brine, which may be used in place of the actual local temperature if the brine is fairly well mixed. The terms in the right hand expression of Equation 1 represent, in order, direct and indirect radiations absorbed by the brine, re-radiation from the brine, convective heat exchange with the air, heat used for evaporation, and heat exchange with the ground. The boundary conditions are,
(3)
Equation 2 can now be solved to satisfy condition 3 and expansion 8. Extending the result given by Carslaw and Jaeger (1949),
+Cn-
OD
Tz(8,x) = T , Fourior Analysis
The governing Equations 1 and 2 can be solved analytically by the Fourier transform method provided that each term occurring in the equations is periodic with respect to one of the independent variables, in this case e. For this purpose, we assume that the meteorological conditions do not change from day to day and we are only concerned with the longtime solution (initial conditions need not be specified). When all the terms in Equations 1 and 2 are periodic in e (with a period of 24 hr), they may be replaced by suitable Fourier expansions. Since Equation 1 is in general nonlinear, we proceed by linearizing the required terms: &I
= er(T1
+ 460)'
p*(Ti)
'V
&Ti
R
+ ST1
- Bo
(5)
[ ( a , cos vnx
A
=
(8f hG
+ hk~Bi)/D~iCpi
and
F(e) =
[a&
+ a'&'
-R
+ hGTa + X ~ G ( B O+ pa)I/D~iCp1
+
cos (d)
+ b, cos v,x) sin (d)] (12)
with v , = (m/2p2)1/2. The heat flux a t the ground surface is then obtained as
-kzrs)
z4
[vn(Un
= K n isl X~
+ bn)
COS
+
(d) vn(bn
- an) sin (me)]
(13)
When cya = 0, Le., no radiation reaches the pond bottom (the bottom may also be completely reflecting to the shortwave radiation penetrating the brine), condition 4 along with Equations 8 and 13 gives
Ti
=
T,
e, = [ l f, = [ l
If we assume further that the transport coefficients are
where
- b, sin v,x)
(a, sin v,x
(6)
independent of time (time average wind speed being used), Equation 1 reduces to
x
1
+ r(n)l"+in + r(n)1/2bn + r(n)"2]bn-
r(7t)'~2an
where
The case when cy, # 0 can be handled by simply expanding cy,& in a Fourier series before applying condition 4. Substituting the expansions 8-10 into Equation 7, using Equation 14 to replace e, and fn, and comparing the coefficients of like terms on both sides, we finally get
Ti
=
B/A
Then introduce the Fourier expansions, with w = 2 r/24 hr-1
TdB) E Tz(6,O) =
T, + m
n-1
[a,
+ b, sin (me)]
cos (nu@)
(8) lnd. Eng. Chem. Procerr Der. Develop., Vol. 1 1, No. 4, 1972
627
85 F T :
75
I
I
0
0
I
0, hr
1
I
I
Figure 4. Vapor pressure of water; dashed line is p* =
rl - 54.0
24
16
95
Figure 3. F(8) and Fourier expansion with five harmonics (dashed curve)
where
The complete solution for Tl(8) and T&z) is thus given by the expansions 10 and 12 where the Fourier coefficients are related to those of F(8) determined from Equation 11.
I
Rat. of Evaporation
0
The average rate of evaporation is given by
w
-
- P,I
(16)
The sine and cosine terms drop out on integration over a 24-hr period and the average evaporation rate is only related to the average temperature TI. Since P , ( = B / A ) does not depend on the brine depth, the daily evaporation would also be independent of brine depth. The expression for can be obtained by substituting for p * ( T 1 ) from Equations 6 and 15 in the above relation:
w
p = a3 + (Y” - Q,* 4- (Pu* - ?%)ho/@1
+ m(1 + fi/hQ)/Bll
(1,)
+ +
where, m = ho/hkc, pu* = p*(Ta), and Qr* = R STu*, Tu* being the average dew point of the air [ = (BO p u ) / B l 1 . A similar but less general relationship was derived by Penman (1948) from an overall energy balance for 24 hr. When ( Y ~# 0 we have
=
Ti
+ GG/hL
and R is given by Equation 17 after replacing absorption (a a,).
+
(19) (Y
by the total
Results and Discussion
Numerical results have been obtained to check the validity of the linearized analysis. The meteorological conditions and other data have been given below and in Figures 1-4. Figure 3 shows the closeness between F(8) and its Fourier expansion containing five harmonics; the maximum deviation was 628 Ind.
I
0
1
I
16
I
I
24
0, hr
= & L 2 4 ~ o ( p * pu)de = k o [ p * ( ~ 1 )
T,
I
Eng. Cham. Process Der. Develop., Vol. 1 1 , No. 4, 1972
Figure 5. Comparison of water temperatures obtained by linear Fourier analysis (dashed) and numerical solution (solid). The two results are almost identical for pond depth of 2 and 3 ft
about 0.5%. In Figure 5 the variation of water temperature during the day has been shown for pond depths of 1,2, and 3 ft. For pond depths of 2 and 3 ft, the water temperature obtained by linear analysis was within 0,5% of that obtained by numerical integration of nonlinear equations (described in the previous paper). Ground temperatures obtained by the two methods differ by even smaller extent because of the damping of higher harmonics (Carslaw and Jaeger, 1949). However, for pond depths less than 2 ft, the water temperature during 24 hr fluctuates over a wide range which cannot be adequately handled by linear analysis, and the deviation from numerical solution becomes apparent. The effect of brine depth on gross daily evaporation is interesting to study. Just after the sunrise, the brine is relatively cold and evaporation is slow, and most of the net heat gain is consumed in raising the brine temperature. Because of its lower heat capacity a shallow pond heats up faster and attains higher maximum temperature than a deep pond. However, in the late afternoon the available energy decreases rapidly, and the shallow pond cools down faster to reach lower minimum temperature in the early morning. If the heat losses and vapor pressure were exactly linear functions of brine temperature, the gain in evaporation a t high temperatures would be offset by a similar loss when the temperature is lower, I n reality, the vapor pressuretemperature relationship is consideraly more nonlinear than
0 . 300
l
I
a
L
3
1
D,f C
Figure 6. Effect of pond depth on total daily evaporation. Solid curve, numerical solution; dashed line, linear analysis
the dependence of heat losses on temperature. As a result the overall effect is an increase in evaporation as pond depth is decreased. Figure 6 shows the inadequacy of linear analysis to account for this effect. The error in daily evaporation obtained from linear analysis decreases as the brine depth is increased, the error being less than 0.3% for brine depth greater than 2 ft. Of course, many other factors have not, been considered here, such as variation of wind speed during the day, decrease in absorption of radiation as pond depth is decreased, and the possible dependence of transport coefficients on pond depth. Validity of Assumptions
The assumptions regarding the mixing of brine and the effect of evaporation on the ambient conditions form the basis of this, as well as most of the previous, work concerning solar evaporation. These assumptions have been discussed by the author (1972), and they were considered to be acceptable when the wind speed exceeds 5 miles/hr (measured 4 ft aboveground) and when the pond size is greater than about 100 X 100 ft. Many other assumptions made in the present analysis seem to restrict the range of useful applications. These are as follows: Linearization of re-radiation from the brine (&,)-this involves little error since over the range of TI ordinarily encountered (70-10O0F), the nonlinear terms deleted in Equation 5 contribute less than 0.5%. Linearization of vapor pressure, p*( TI)-this can lead to significant errors since the actual relationship is exponential. However, if the pond depth is greater than about 1 ft, the range of brine temperatures to be considered under a given set of meteorological conditions is not too great and a linear approximation centered around the average temperature (TI) is quite satisfactory. I n reality, the process of finding 2’1 and adjusting the p*(T) relationship must be repeated a few times. As seen in Figure 4, a spread of 10-15’F can be handled adequately by a linear approximation. The assumption of time independent transport coefficients is not essential for this treatment since the equations remain linear even otherwise. The analysis of time-dependent wind speed is, however, quite tedious and the results do not differ significantly except in the case of very concentrated brines having vapor pressures much lower than that of water, and when the pond depth is less than about 1 ft. The fundamental requirement, of Fourier analysis is that the meteorological conditions be periodic in O-i.e., the conditions should not change from day to day, conditions rarely found in practice owing to the seasonal cycle of an amplitude dependent on the geographical location, and the day to day random fluctuations of relatively small amplitude. Hence
the assumption is not to be taken literally; the only purpose of the assumption is to overcome the lack of knowledge of the initial conditions (temperatures of the brine and the ground at a specific time). When the initial conditions are known, as during field experiments, and when they do not correspond to the predetermined periodic solution, the problem is best handled by numerical integration. However, the “response time” of the pond (= 1/A, hr) is only 4-6 hr depending on the brine depth, and except for the temperature deep in the ground, the pond has very little “memory” of previous weather conditions. Hence the long-time solution applied to successive days (taken individually with the actual weather conditions) produces acceptable results when the initial conditions are unknown. Although it would seem desirable to record the instantaneous meteorological conditions for every day of the year before using any model, the amount of information to be processed would be quite enormous, and averaging the conditions over short periods such as 1-2 weeks would introduce little error. Conclusion
The nonlinear equations describing the transient behavior of a solar pond can be solved in a linearized form using Fourier analysis. The solution leads to a well-known semiempirical relationship for total daily evaporation which lacks the dependence on the brine depth. The use of this relationship and other relationships derived here is limited to pond depths exceeding about 2 ft when the vapor pressure and the heat loss due to radiation can be well approximated by linear functions of brine temperature. The transport coefficients are also required to be independent of time. Acknowledgment
The author is indebted to George M. Homsy for suggesting this simplified approach to the problem. Numerical Data
Meteorological: Tu, Figure 1; &, Figure 2; p a = 20.0 mm Hg (constant); V = 10.0 miles/hr (4 ft, aboveground) Properties of brine (pure water): P I = 62.4 lb/fta; C,I = 1.0 Btu/lb, O F p* (Figure 4), linearized: p*(Tl) = TI - 54.0, mm Hg involves less than 0.5% error in the range 80-90°F CY = 0.85 for sun’s altitude = 90’. Time dependence of CY as given by author (1972), CY’ = 0.95 e = 0.95; linearized &, = 51.5 1.1 TI involves less than 0.5% error in the range 80-90°F Properties of ground: K 2 = 1.0 Btu/hr ft, OF; p 2 = 0.02 ftz/hr a* = 0.0 Transport coefficients: hL = 30 Btu/hr f t k , O F h~ = 1.0 0.3 V = 4.0 Btu/hr ft2,O F = lOWa(l.9 0.476 V ) = 6.66 X lo-* lb HzO/hr ft* mm Hg
+
+
+
Nomenclature
A a b B c
C, d
D e
f
= =
= = =
= =
= = =
coefficient of TI, defined for Equation 7, hr-l Fourier coefficient of T,,Equation 8, O F Fourier coefficient of T,, Equation 8, O F mean value of F(O), Equation 11, “F/hr Fourier coefficient of F(O), Equation 11, OF/hr specific heat capacity, Btu/lb, O F Fourier coefficient of F(O), Equation 11, OF/hr brine depth, ft Fourier coefficient of TI, Equation 10, OF Fourier coefficient of TI,Equation 10, O F
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
629
= heat transfer coefficient, Btu/hr ft2, O F K = thermal conductivity, Btu/hr ft, O F k = mass transfer coefficient, lb H2O/hr ft2mm Hg
h
= Stefan-Boltamann constant, 1.73 X l o W 9Btu/hr ft2(O R) 8 = time, hr w = 2 ~ / 2 4hr-* u
m = ho/hKo, mm Hg/OF p* = saturation vapor pressure of brine, mm Hg p , = partial pressure of water in air, mm Hg q = abbreviation defined for Equation 15 Q = intensity of direct radiation on horizontal surface, Btu/hr ft2 Q’ = intensity of indirect (“scattered”) radiation, Btu/hr ft2 Qr = re-radiation from the pond, Btu/hr ft2 r = ( k 2 / h ~d)u / 2 p ~ dimensionless , R = linearization constant for Qr, Equation 5, Btu/hr ft2 s = abbreviation defined for Equation 15 S = linearization constant for Qr, Equation 5, Btu/hr ft2,O F T = temperature, O F V = wind speed, miles/hr W = rate of evaporation, lb HnO/hr ft2 z = vertical coordinate measured below soil surface, ft
SUBSCRIPTS a = air G = gas film (brine-air) L = liquid film (brine-soil) n = summing index s = soil surface 1 = brine 2 = ground ’ = indirect radiation over bar = integrated average for 24 hr literature Cited
Cawlaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed., pp 62-68, Oxford Univ. Press, 1949. Ferguson, J., Auslr. J. 9ci. Res., 5, 315-30 (1952). Pancharatnam, S., Ind. Eng. Chem. Process Des. Develop., 11, 287 (1972).
Penman, H . L., Proc. Roy. BOC.(London), A193, 120-45 (1948).
GREEKLETTERS LY = fractional absorption of radiation bo, p1 = linearization constants for p*, Equation 6 e = spectral average emissivity of brine = heat of vaporization of water from brine, Btu/lb p = thermal diffusivity, (KIpC,), ft2/hr Y = a wave number defined for Equation 12, ft-’ p = density, lb/ft*
SUBRAMANIAN PANCHARATNAM Department of Chemical Engineering Stanford University Stanford, Calif. 94906 RECEIVED for review February 7, 1972 ACCEPTEDMay 19, 1972
Dynamic Models for pH and Other Fast Equilibrium Systems A general principle is presented to be used to formulate dynamic models for systems in which fast equilibrium reactions occur.
I n a recent article (McAvoy et al., 1972), the derivation of dynamic equations for pH-controlled stirred tank reactors (CSTR’s) was examined. The use of electroneutrality and water equilibrium in deriving the equations was discussed in detail. However, one additional principle was used but not specifically pointed out. This principle concerns what to do about making dynamic mass balances on species involved in fast equilibrium reactions. To illustrate the principle and the problem which generates it, it is convenient first to consider a trivial case. Consider a stirred vessel in which the simple equilibrium reactions A B occur. The forward and reverse reactions are assumed to be instantaneous. The feed to the tank is lOOyo A which changes to lOOyo B a t t = 0. The overall mass in the system remains constant with time. Two formulations of the governing equations can be given. Correct Formulation.
Incorrect Formulation : Balance on A
It
V -dCA at
+ Cg = constant
Equilibrium
CB 630
K ~ ~ C A
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
(1)
=
- FCA
- RI + R,
or
Balance on B 0
/I
dCB V - at
Overall balance
CA
0
or
=
FCej
- FCB + R , - R,
(4)
