Appendix
The pressure drop across the test section was calculated for water a t 200°F. for each flow rate by considering three effects: contraction loss a t the entrance, frictional loss, and pressure recovery a t the exit. These are expressed in the equation
uz = velocity of mixture after small amount of flash,
ft./sec.
v = specific volume of mixture, cu. ft./lb., v# = specific volume of saturated vapor, cu. ft./lb.m L‘I = specific volume of saturated liquid, cu. ft.ilb.,,, u~ = work done by flowing stream, ft.-lb.,/lb., y = fraction of stream vaporized after small amount of flash z = elevation above datum state, ft. literature Cited
The Fanning friction factor. f, and the contraction coefficient, K , were taken from standard correlations for flow in smooth tubes (Perry. 1963). T h e ratio L D was 228. At 200°F. the flow could be seen to expand smoothly a t the outlet of the test section. This observation, coupled with the fit of the calculated and experimental pressure drops (see Figure 4 ) , justifies the assumption that K , = 1. At the higher temperatures the water flashes a t the test section exit,. The value of K , used in calculating the theoretical curves in Figures 5 and 6 was taken as zero, since it seems unlikely that there is any pressure recovery when flashing occurs. The pressures a t the throat outlet listed in Table I were estimated by subtracting the value of APj calculated by Equation A-1 in each case from the average of the experimentally observed pressures a t P-1 a t a flow velocity of 15 feet per second. Nomenclature
C, = heat capacity of liquid, B.t.u. ’lb.m-oF . F = frictional dissipation, ft.-lb., lb.”]
f =
Fanning friction factor
g, = gravitational constant, 32.2 ft.-lb., /sec.’-lb.,
H = specific enthalpy of mixture, B.t.u./lb., Hi = AHt = K, = P =
specific enthalpy of saturated liquid, B.t.u./lb., specific enthalpy of vaporization, B.t.u./lb., contraction coefficient pressure, lb.,,’sq. it. 6 2 = heat flux to flowing stream, B.t.u./lb.m T = temperature, O F. u = velocity in direction of flow, ft./sec. u1 = velocity of liquid ahead of flash zone, ft./sec.
Allen, W. F., Trans. A S M E 73, 257 (1951). Bailey, .J. F., Trans. A S M E 73, 1109 (1951). Benjamin, M.W., Miller, J. G., Trans. A S M E 64, 657 (1942). Bottomley, W. T., Trans. Northeast Coast Inst. Engrs. Shipbuilders 53, 65 (1936). Brown, R., York, J. L., A.I.Ch.E. J . 8, 149 (1962). Cruver, ,I. E., Moulton, R. W., A.I.Ch.E. J . 13, 52 (1967). Faletti, D. W., Moulton, R . W., A.1.Ch.E. J . 9, 247 (1963). Fauske, H. K., ”Critical Two-Phase Steam-Water Flows,” Heat Transfer and Fluid Mechanics Institute, Stanford University Press, Stanford, Calif., 1961. Isbin, H . S., Moy, J. E., DaCruz, A. J. R., A.Z.Ch.E. J . 3, 361 (1957). Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, Xew York, 1967. Levy, S., J . Heat Transfer, Trans. A S M E , Ser. C 87, 53 (1965). Linning, D. L., Proc. Inst. Mech. Engrs., Ser. B 18, 64 (1952). Moody, F . J., J . Heat Transfer, Trans. A S M E , Ser. C 87, 134 (1965). Perry, J. H., ”Chemical Engineers’ Handbook,” 4th ed., Sec. 5, McGraw-Hill, New York, 1963. Starkman, E. S., Schrock, V. E., Neusen, K. F., Maneely, D. J., J . Basic Eng., Trans. A S M E , Ser. D 86, 247 (1964). Zivi, S. M., J . Heat Transfer, Trans. A S M E , Ser. C. 86, 247 (1964).
RECEIVED for review August 12, 1968 ACCEPTED June 19, 1969
DYNAMIC SIMULATION OF A N LPG VAPORIZER R I C H A R D
A .
E C K H A R T
Simulation Sciences, Znc., Fullerton, Calif. 92632
IN a mathematical model for describing the dynamic characteristics of a liquefied petroleum gas (LPG) vaporizer, the vaporizer pressure is controlled by adjusting the rate of condensate removal from the vaporizer heating coil. A more common arrangement is to have the pressure controller adjust the rate of steam entering the vaporizer heating coil. This, however, requires a larger control valve on the steam end of the coil than for condensate adjustment on the condensate end. Thus, the use of condensate rate adjustment requires a smaller and less expensive con-
trol valve. The ability of such a control arrangement to maintain the vaporizer pressure a t a level adequate to meet process requirements is of major importance. The model presented here was used to simulate a vaporizer with condensate adjustment on an analog computer, in order to determine the transient response characteristics of the system. Specifically, it was desired to determine the ability of the control system to maintain an adequate supply line pressure when subjecting the vaporizer to increases of 10 and 2 0 5 in the demand for vaporized LPG. VOL. 8 NO. 4 OCTOBER 1969
491
A mathematical model describes the dynamic characteristics of a n LPG vaporizer in which the vaporizer pressure is controlled by adjustment of the rate of condensate removal from the vaporizer heating coil. The model was used to simulate a vaporizer with condensate adjustment on a n analog computer to determine the ability of the control system to maintain an adequate supply line pressure when subjecting the vaporizer to increases of 10 and 20% in the demand rate for vaporized LPG. The simulation indicated rapid enough system response to maintain the supply line pressure essentially constant for demand rates up to at least 20% above the design rate. The simulated vaporizer is now in service and performing satisfactorily.
and
Vaporizer Model
A schematic diagram of the vaporizer is shown in Figure 1. The unit consists of the vaporizer drum, a steam heating coil, and the controls and associated lines indicated. Liquid feed enters the vaporizer drum under liquid level control, which is assumed to function ideally-Le., the liquid level in the drum is assumed constant. Steam enters the heating coil a t a constant pressure, while the pressure in the vaporizer drum is controlled by a proportional-plusintegral controller which adjusts the rate a t which condensate is removed from the heating coil. This varies the heat transfer area of the heating coil by exposing more or less coil area to condensing steam. I n the model presented here, it is assumed that condensate remaining in the coil gives up a negligible amount of sensible heat to the vaporizing liquid, in comparison with the heat released by the condensing steam. Thus the effect of condensate in the coil is to remove effective heat transfer area from the coil. The vaporized LPG passes overhead through a supply line which is under pressure control. The supply line is treated as if all resistance to flow is concentrated in the control valve while the capacitance of the line is lumped into a single capacative (volume) element. The equations describing the system may be derived as follows. In this development, the LPG is assumed to have the same physical and thermal properties as n-butane. Since ideal, or perfect, liquid level control in the vaporizer drum is assumed,
-dML =dto
(1)
w,= w,
(2)
Also, by material balance on the vapor in the drum,
dM, = w,- wo dt
(3)
Applying the ideal gas law to the vapor in the drum,
PV, =
10.72 ~
58 M , T
(4)
For the LPG which is considered a pure componentnamely, n-C4-the equilibrium temperature-pressure relationship for the vaporizing liquid may be represented by a straight line (Maxwell, 1950). I t is assumed here that the liquid LPG in the vaporizer is a t uniform temperature T-Le., the liquid is well mixed so that there are no temperature gradients within the liquid phase-and is always in equilibrium with the vapor in the drum a t pressure P, such that
P = 1.4T - 597.3
(5)
Furthermore, enthalpies of the liquid and vapor LPG, respectively, may be represented by
h = 0.633T - 196
(6)
H = 0.633T - 61
(7)
and assuming that the heat of vaporization and heat capacity of the LPG are independent of temperature. An energy balance about the steam coil metal is given by
while for the liquid LPG in the vaporizer an energy balance yields
MLCP
LPG(1 Wi
-bPD CONDENSATE
WC
Figure 1. Schematic diagram of LPG vaporizer 492
18EC PROCESS DESIGN A N D DEVELOPMENT
dT
dt = U.vA0(T.v- TI + WLhj- WLH
(9)
where heat capacities CU.vand C p Lare assumed independent of temperature. For the heat released by the condensing steam, it is assumed that the steam condenses a t the saturation temperature, T,, corresponding to the steam supply pressure,
P,, and that all heat is released at that temperature; sensible heat effects of any superheated steam are negligible. Since sensible heat release from the condensate in the coil is also neglected, the effective length of the steam coil (and thus the effective heat transfer area) can be represented by the equation
The effective mass of the heating coil metal and the effective inside and outside heat transfer areas are then given by
where D,4Lwas taken as the arithmetic average of D , and Do. T o complete the analysis, equations are needed to describe the operation of the two pressure controllers, the flow of condensate from the heating coil, the flow of vapor from the vaporizer drum, and the vapor supply line pressure. A simple mass balance yields the line pressure equation as
P
58Vi d
10.72dt (M = W" - w1 Flow of condensate from the heating coil is given by
W' =
c, (PF-
PD)l
Because of vapor blanketing considerations, the vaporizer was designed for the maximum heat flux of 15,000 B.t.u.1 hr.-sq. ft. The simulation was carried out in a fashion to limit the heat flux a t any time to no more than 15,000 B.t.u. / hr.-sq. ft. The design values of valve coefficients C, and C, , as well as the design heat transfer area, were taken as 80'4 of the maximum values available. The two pressure controller settings were obtained by the open loop technique of Ziegler-Xichols (Eckman, 1958) on the analog computer. For the overhead supply line pressure controller a proportional gain, K,, of 21.0 and a reset time, T ~ of , 0.21 second were used, while for the vaporizer drum pressure controller the proportional gain was 25.2 and the reset time was 10 minutes. These controller settings correspond to values of
Kl Kl K? KJ
= 21.0 = 100 = 25.2 = 0.042
in Equations 17 and 18. Figures 2 through 11 show the response characteristics of the system for step increases in the vapor demand rate, W1, of two different magnitudes. Figures 2 through 6 show the responses for a 105 increase in W1 from its steady-state design value, while Figures 7 through 11 show the responses for a 20% increase in W , .
1
J4671
1
1
1
1
1
50
60
(15)
while the flow of vapor from the vaporizer drum is given by
w, e,
= P (16) where there is critical flow of the vapor-Le., where P I P I < 1.89. Finally, the two pressure controllers can be represented as proportional plus integral controllers by the equations
C,
=
Klel + K I J e l d t
(17)
and 0
where
e! = Pp - P I
10
20 X , 40 TIME- MINUTES
Figure 2. System response to 10% increase in vaporizer demand rate
and
e2 = Pa - P Application to Real System
The mathematical model presented above was used to simulate an LPG vaporizer which had the following design characteristics and operating conditions.
CpM = 0.113 B.t.u./lb.-" F. D, = 0.0444 ft. T = D o = 0.0614 ft. T, = L = 794 ft. UM = ML = 25,550 lb. Us = P = 144.7 p.s.i.a. Vi = PI = 64.7 p.s.i.a. V, = PD = 50 p.s.i.a. W, = P, = 150 p.s.i.a. PM =
530"R. 520"R. 125 B.t.u./hr.-sq. ft.-OF. 500 B.t.u.Jhr.-sq. ft.-"F. 175 CU. ft. 469 cu. ft. 17,000 lb./hr. 491 lb./cu. ft.
l
0
/
10
1
1
20 33
'
l
40 50
1
60
TIME- MINUTES
Figure 3. System response to 10% increase in vaporizer demand rate VOL. 8 NO. 4 OCTOBER 1 9 6 9
493
I-
7 1200
I
I
l 0
I
I
I
I
I
3 141.7
l l l l l l 10 20 30 40 50 60
I I
TIME- MINUTES
0l
IO 1
Figure 4. System response to 10% increase in vaporizer demand rate
20l
30 l 40I
50 ! 60l
TIME- MINUTES
Figure 7. System response to 20% increase in vaporizer demand rate
!=if&m
2 a
2637
I
I
TIME- SECONDS
Figure 5. System response to 10% increase in vaporizer demand rate
I
l
1
1
1
1
1
0 1 0 2 0 3 0 4 0 5 0 6 0 TIME- MINUTES
Figure 8. System response to 20% increase in vaporizer demand rate
I
0
IO
20
30
40
TIME- SECONDS Figure 6. System response to 10% increase in vaporizer demand rate 494
I&EC PROCESS DESIGN AND DEVELOPMENT
5 65.7 m
a
1
I
I
I
0
IO 20 TIME- SECONDS
I
I
30
40
Figure 10. System response to 20% increase in vaporizer demand rate
of the physical components, and the vaporizer drum pressure transients had disappeared within about 35 minutes. When the demand rate was increased by 20‘; above the design rate, the drum pressure control valve and heat transfer area reached their saturation values for a period of time before coming to their new steady-state values. As a result of the temporary component saturation, a longer period (approximately 50 minutes) was required for the drum pressure transients to disappear. I n any case, the ratio of the vaporizer drum pressure to the supply line pressure, P PI,was always greater than 1.89Le., up to demand rates of a t least 20% above the design rate, critical flow into the supply line was maintained. The system responded rapidly enough to changes in the demand rate to maintain the supply line pressure essentially constant for demand rates up to at least 2 0 5 above the design rate. Although no actual operating data are presented here, the vaporizer simulated is in operation and giving satisfactory performance. Nomenclature
A = area, cu. ft.
c, c,
= heat capacity, B.t.u./lb.-’ F. = valve coefficient
D = diameter, ft.
L
0
10
20
30
40
TIME -SECONDS
Figure 1 1 . System response to 20% increase in vaporizer demand rate
e = error h = liquid enthalpy, B.t.u./lb. H = vapor enthalpy, B.t.u./lb. K = constant L = length, ft. M = mass, lb. P = pressure, p.s.i.a. t = time, hr. T = temperature, R. heat transfer coefficient, B.t.u./hr.-”F.-sq. ft. volume, cu. ft. w = flow rate, lb./hr. AX = wall thickness, f t . density, cu. ft. P = A = heat of vaporization, B.t.u./lb.
u= v =
SUBSCRIPTS I n both cases, the dynamics of the supply line pressure are instantaneous in comparison with the dynamics of the vaporizer drum pressure. The supply line pressure transients die out within about 10 seconds following the step increase of the demand rate, while the decay of the vaporizer drum pressure requires more than 30 minutes. I n the case of the 20% increase in W1 above the design rate, the system saturates dynamically, as shown in Figures 8, 9, and 11, though the full capacity of the system when it comes to its new steady state has not yet been reached. Actually, the system reached dynamic saturation for a step increase of 11.8% in W1. Thus, an increase of 11.8% above the design value for W , is the maximum change that the system can tolerate without causing system components to reach their physical limits. The figures also illustrate the effect of saturation of the system components. I n the first instance, where the demand rate was increased by lo%, there was no saturation
i D o L
= = = =
u = s = c = 1 = M =
inside, in discharge out, outside liquid vapor steam, steam-to-metal condensate supply line metal, metal-to-liquid
SUPERSCRIPT = set point Literature Cited
Eckman, D. P., “Automatic Process Control,” pp. 11421, Wiley, New York, 1958. Maxwell, J. B., “Data Book on Hydrocarbons,” Van Nostrand, Princeton, N. J., 1950. RECEIVED for review September 11, 1968 ACCEPTED May 24, 1969
VOL. 8 N O . 4 OCTOBER 1 9 6 9
495
