Ind. Eng. Chem. Process Des. Dev. 1904, 23, 472-479
472
be possible to design the i loop tightly enough to minimize the interaction, but again it would have to be done more than would be expected from the single input, single output designs alone. I t has been shown that analysis of the Bristol array is not sufficient to guarantee that control loops will be relatively noninteracting, even in the steady state. Under the condition that the relative gains gl,/gll or (gll/g,l)/(g,l/g+k) are quite small there may still be interaction. Such circumstances are expected to be unusual, especially in view of the apparent practical success of using the Bristol array. Therefore, just an additional step in the design procedure need be added. The Bristol array should be used as usual. Once variable pairings are made, the relative gains gll/gl, and (gll/gJ/ (gll/gdJk)should be checked. If they are significantly smaller than unity some otherwise unanticipated interaction may be encountered. It may be possible to eliminate the interaction by using tighter control although normally that would not be possible if the single input, single output loops were designed to as small gain and phase margins as safe. Otherwise, some form of compensation may have to be used.
rdu = gain ratio, gdu/gii s = Laplace transform parameter TI; = controller integral times t = time u = control variable vector x = state or output variable vector
Acknowledgment
Cohen, G. H.; Coon, G. A. Trans. A S M 19bS, 75, 827. Gagnepaln, J. P.;Seborg, D. E. Ind. Eng. Chem. Process Des. Dev. lg82, 27, 5 . Jafarey, A.; McAvoy, T. J. I d . Eng. Chem. Process Des. D e v . 197& 17,
The revision of this paper was aided by D. Grant Fisher, who provided prior to its publication a manuscript based on the work of Jensen et al. (1980). Nomenclature
A = matrix in eq 6 B = matrix in eq 6 D = matrix in eq 6 d = disturbance or load variable vector G = transfer function matrix g , = elements of the steady-state matrix, G(0) gca,= inverse of elements of the steady-state matrix [G-l(0)lT = elements of the steady state matrix, Gd(0) Z = identity matrix K = diagonal matrix on controller gains k, = controller gains rll = gain ratio, gll/gll
Greek Letters e = element of Bristol array defined in eq 7 Bi = offset in single input, single output loop i A = Bristol array X i j = element of Bristol array Subscripts i j , k = elements of matrices or vectors d = disturbance or load Special Symbols (-)T = transpose 6) = Laplace transformed variable (7) = steady-state value Literature Cited Bristol, E. H. IEEE Trans. Autom. Control 1966, AC-77. 133. Brlstol, E. H. Paper presented at AIChE Annual Meeting, Mlaml Beach, Nov
1978.
485. Jafarey, A.; McAvoy, T. J.; Douglas, J. Ind. Eng. Chem. Fundam. 1979, 78,
181. Jensen, N.; Fisher, D. G.; Shah, S. L. Paper presented at AIChE Annual Meeting, Chicago, Nov 1980. McAvoy, T. J. Ind. Eng. Chem. Fundem. 1979, 78, 269. McAvoy, T. J. A I C E J . 1981. 27, 613. Nlsenfeld, A. E.; Stravhskl, C. Chem. Eng. 1968, 75(20). 227. Nlsenfeld, A. E. chem. ~ n g prog. . s ~ p~.e r 197% . 69(9), 227. Ray, W. H. "Advanced Process Control"; McGraw-HiU New York. 1981. Shlnskev, F. G. "Dlstlliatlon Control"; McGraw-Hill: New York. 1977; DD ..
289-293.
Shinskey, F. G. "Process Control Systems"; McGraw-Hill: New York, 2nd ed.; 1977;pp 196-204. Tung, L. S.; Edgar, T. F. AIChE J . 1981, 27, 690. Witcher, M. F.; McAvoy, T. J. ISA Trans. 1877, 16(3),35.
Received for review August 6 , 1982 Accepted August 5, 1983
Breathing Losses from Fixed-Roof Tanks by Heat and Mass Transfer Diffusion James R. Beckman Department of Chemkal and Bio Englneerlng, College of Englneering and Applied Sciences. Arizona State Universm, Tempe, Arizona 85287
A simultaneous heat and mass transfer diffusion model was developed that predicts volatile breathing losses at terminal state for isobaric fixed-roof tanks experiencing a sinusoid temperature perturbation on the top dome. Typically thermal penetration into the stagnant gas phase appears to be Ilmbd to 2-3 m for a 2441 thermal cycle. The predicted breathing emissions compared closely with the emisskn data from the 1977 WOGA tests of industrial tanks. The diffuslon model may be extremely beneficial In the prediction of hydrocarbon losses from flxed-roof storage tanks.
Introduction
Hydrocarbon emissions from fixed volume tanks (i.e., fixed-roof storage tanks) along with accurate predictions of those emissions are some of the current problems facing the petroleum industry. Not only do hydrocarbon emis0198-4305/84/1123-0472$01.50/0
sions from tanks deplete gasoline and crude oil supplies but they can contribute to photochemical smog. In 1977 the Western Oil and Gas Association (WOGA) appointed a task force to oversee a study of hydrocarbon emissions from fixed-roof storage tanks as reported by 0 1984 American Chemical Society
Ind. Eng. Chem.
Engineering-Science, Inc. (1977). Some of the results showed that hydrocarbon vapor losses from a single storage tank amounted to 160 m3 of liquid/year (1000 bbl/year). The hydrocarbon losses occurred from standing storage and were due solely to the sun's diurnal heating and cooling of the vapor space within the storage tank. Others also have discussed standing storage hydrocarbon emissions from fmed-roof tanks (see API Bulletin 2513,1959; 2518, 1962; Harrer, 1978; Danielson, 1973). The American Petroleum Institute in 1956 established the Evaporation Loss Committee to advance the basic knowledge of hydrocarbon evaporation loss from fmed-roof tanks as reported in API Bulletin 2512 (1957). The committee concluded that tank breathing losses were more or less directly proportional to true liquid vapor pressure and the loss rate is probably less than directly proportional to vapor pressure, outage, and daily atmospheric temperature change. These conclusions were the results of Boardman (1952) and Bridgeman (1955). Their work was based on equilibrium considerations assuming complete saturation of the vapor space with hydrocarbon which led to the following theoretical equation for breathing loss
G = 0.517Vm[
(
P 1 + P2
1380 - 8M
)(
T2- " Tl 273
( pp2 2 --1P2 )
+
-
)+
("-)I
p2 - P2
(1)
where G = m3 of liquid gasoline lost in one breathing cycle; PI = absolute pressure at which tank vacuum vent opens, kPa; P2= absolute pressure at which tank pressure vent opens, kPa; p1 = vapor pressure at minimum liquid-surface temperature, kPa; p 2 = vapor pressure at maximum liquid-surface temperature, kPa; T I = minimum average vapor-space temperature, "C; T2 = maximum average vapor-space temperature, "C; Vm = minimum volume of vapor space, m3; and M = molecular weight of the hydrocarbon vapor. Testing and use of eq 1is difficult due to the problem of measuring the temperature of the vapor mixture immediately above the liquid surface as noted in API Bulletin 2513 (1959). In 1962, API Bulletin 2518 published by the API Evaporation Loss Committee presented methods for estimating breathing losses resulting from storage of either gasoline or crude oil in fixed-roof tanks. Equation 2 is an empirical equation based on data from 64 tanks containing gasoline and 15 tanks containing crude oil.
Process Des. Dev., Vol. 23, No. 3, 1984
473
improved the prediction of hydrocarbon breathing losses by posing a gas-phase mass diffusion controlled model for hydrocarbon movement inside a storage tank. Their analysis was more successful in predicting breathing losses than eq 2 for the three cases considered. If vapor emission controls are to be both effective and cost efficient, they must be based on accurate estimates of emissions as well as a knowledge of the mechanisms at work in the production of these emissions. Internal Model A complete model for predicting standing storage losses from fixed-roof storage tanks was divided into tank external and tank internal environments. The external model was used to predict tank roof temperature changes from the diurnal solar movement taking into account solar insolation, ambient air temperatures, tank surface emissivities and wind velocities. The internal diffusion model accepts sinusoidal temperature changes of the tank top dome as a thermal driving force which heats up or cools down stagnant gas within the tank. Both the thermal diffusion from the top dome into the stagnant gas along with the mass diffusion of volatile species from the liquid surface to the top vent are accounted for in the internal model. Emissions from the tank top vent are predicted from the rate of total off gas and the off gas composition of volatile species. In order to solve the coupled problem of thermal and mass diffusion, the energy (eq 3), continuity (eq 4), and mass species (eq 5) are needed and are listed below.
dpC,T
o = - - at
8277
k-+a22
o = -a P at
aci
+ -az
a2ci aciu
dpuC,T az
(3) (4)
DiA+(1 5 i -< N) (5) a22 az at Equations 3, 4, and 5 show that mass and energy transport are considered to be unidirectional in the vertical z direction (Bird et al., 1960). The mass equation (eq 5) is valid for each diffusing species. In most tank storage situations low vapor pressures are involved. For instance, crude oils may exert 10 kPa to 20 kPa vapor pressure and pure materials may be even less. Since the saturation compositions are small, then eq 5 does not have interdependency among all the species. The diffusivity, DiA, can be simply taken as the diffusivity of the pure component with respect to air. Also since the velocity is dominated by the thermal gradients and not by the small mass fluxes L = 0.16Kc( 1000 100- P" x L)1"3 0.305 x of all species, then the set of eq 3,4, and 5 need be solved for only one component (say the ith species). Total emissions can then be the s u m of>all individual components analyzed separately. where L = hydrocarbon loss, liq. m3/year; K, = 1 for The boundary conditions for eq 3 are (1)a sinusoid at gasoline and 0.58 for crude oil; P, = true vapor pressure the top dome ( z = 0) and (2) constant temperature of the a t bulk liquid temperature, kPa; D = tank diameter, m; gas deep inside the tank. H = average outage, m; T,= average daily ambient temTH - TL perature change, "C; F = paint factor; and S, = adjustTlz=O= TA + -sin ut (3a) 2 ment factor for small Jiameter tanks. Equation 2 was modified by a fador of 0.6 after the 1977 TIz-, = T A (3b) WOGA study since eq 2 overpredicted emissions measured In boundary equations (3a, 3b), TAis the average temfrom the 21 industrial storage tanks involved in the study perature of the tank top (TH + T L ) / 2where TH is the (AP-42, 1981). highest attained temperature and TLis the lowest temHowever, eq 2 is completely empirical (not even in the perature that the top dome experiences. The boundary form of dimensionless groups) and as such remains highly condition at infinite distance (eq 3b) was selected to ease suspect in its ability to predict hydrocarbon breathing losses from fixed-roof tanks. Beckman and Gilmer (1981) the perturbation solution. It will be seen that the thermal
&)( L ) 6 ( s
o=--
474
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
penetration is not infinite and is in fact somewhat shallow, which allows the form of eq 3b. The boundary condition for eq 4 (continuity) is that at an infinite distance from the top dome the velocity is zero. VIz-, = 0 (44 The boundary condition 4a is not strictly valid at the liquid surface since there must be some finite diffusive flux of species leaving the liquid surface. However, in the region of finite z the velocities are dominated by the thermal gradients, and the inclusion of the mass flux of diffusing species is negligible. Equation 4a is actually stating that since the gas temperature is unchanging as z m, the velocity which depends on the thermal gradient is zero. The infinite boundary condition for eq 5 is that as distance goes to infinity, the species composition in the gas is saturated, Cis Cil2-m = Cia (54
-
The boundary condition at the top dome ( z = 0) for the species equation comes from observing the species flux during inhale and exhale conditions. During inhale conditions when the gas velocity is positive (v+), the maw flux of a species is zero at the top dome (that is the tank does not emit vapors during inhale). This leads to
- Cilz=o 4 z = 0 - DiA
212=,;.-.+
(5b)
During exhale condition when the gas velocity is negative (v-), the flux of species i at the top dome is due solely to bulk flow not to diffusion. Since there is no driving force a t the top dome during exhale to sustain a composition gradient, the composition gradient is zero.
21z=o =0
"=d
Combining eq 9 and 10 resulted in a single nonlinear equation needing boundary conditions 3a and 6 for solution
The term a is the gas phase thermal diffisivity at temperature TA. Introducing dimensionless variables into eq 11 gave
with
ele=o = 1 + e sin r ,,,el
=1
(124 Wb)
Here the dimensionless temperature 0 is T/TA,dimensionless distance 5 is ( w / ~ ) ' / ~r zis, ut, and t is (THTL)/(2TA)which is very small (c
