As noted by Caretto and Nobe (1966), this behavior is typical of pore diffusion. Further, radial temperature gradients that become progressively larger as the temperature is increased could also affect the shape of the Arrhenius plot. In our investigation, because of the shape and the diameter of the reactor, the highest radial gradients were not in excess of 1” to 2’C. and, as such, would have a minimal effect on the Arrhenius plot. I t seems reasonable to attribute the curvature in the Arrhenius plot to the pore diffusion only. The values of constants k , and E in the equation:
k = hoe-”
(9) were obtained using the lower temperatures only. The values were: Apparent activation energy,
E = 37.2 x IO3 cal. per g. mole
Frequency factor, h, = 1.25 x 10” cc. gas/cc. catalyst-second Conclusions
The methanation of carbon monoxide a t parts per million levels was studied over a 0.5% ruthenium catalyst in a fixed-bed reactor. The rate of reaction of carbon monoxide follows simple pseudo-first-order kinetics:
where h follows the Arrhenius temperature dependence a t low temperatures. Evidence of diffusion control of the reaction rate was found in higher regions of the temperatures investigated.
k = pseudo-reaction rate constant, cc. gas/cc. cat.sec. h, = Arrhenius frequency factor, cc. gas/cc. cat.-sec. n = reaction order rA = reaction rate of species A , (p.p.m. CO/sec.) act. gas/cc. cat. R = gas constant, cal./g. rnole-OK. T = temperature, K. v, = volumetric flow rate of feed gas, cc./sec. V = volume of catalyst, cc. XCH4= moles of CH1produced per mole of CO in feed X,, = moles of CO converted per mole of CO in feed All volumes measured a t 60’ F. and atmospheric pressure. literature Cited
Akers, W. W. White, R. R., Chem. Eng. Progr. 44, 554 (1948). Caretto, L. S., Nobe, K., IND. ENG. CHEM. PROCESS DESIGNDEVELOP. 5, 217 (1966). Cohen, A. E., Nobe, K., IND.ENG.CHEM.PROCESS DESIGN 5, 214 (1966). DEVELOP. Dirksen, H. A., Linden, H. R., “Pipeline Gas from Coal by Methanation of Synthesis Gas,” Institute of Gas Technology, Res. Bull. 31 (1963). Gilkeson, M. M., White, R. R., Sliepcevich, C. M., Znd. Eng. Chem. 45, 460 (1953). Karn, F. S., Shultz, J. F., Anderson, R. G., Znd. Eng. Chem. Prod. Res. Develop. 4, 265 (1965). Levenspiel, O., “Chemical Reaction Engineering,” Chap. 14, Wiley, New York, 1962. McKee, D. W. J., Catalysis 8, 240 (1967). Nicolai, J., d’Hont, M., Jungers, J. C., Bull. SOC.Chim. (Belges) 55, 160 (1946).
Nomenclature
B, D C,, C,, E
RECEIVED for review August 8, 1968 ACCEPTED June 2, 1969
= integration constants = concentration of CO in feed, p.p.m.
= concentration of CO in effluent, p.p.m.
Institute of Gas Technology sponsored this work through its basic research program.
= activation energy, cal./g. mole
M I N I M U M CRITICAL VELOCITY FOR ONE-PHASE FLOW OF LIQUIDS ALFONSO
GUTIERREZ’
AND
S C O T T
LYNN
Department o f Chemical Engineering, University of California, Berkeley, Calif. 94720
IT IS frequently necessary when
designing chemical processing plants t o provide for transporting a relatively hot liquid from a reactor, heat exchanger, or holding tank through a pipe to a receiver where the pressure is relatively low. If flashing occurs in the pipe or in a valve, control of the liquid flow may become difficult. If the flashing is accompanied by the precipitation of dissolved solids, the designer may be faced with a decidedly vexing problem.
’ Present address, Carrera 20, No. 54-45, Bogota D.E.2, Colombia 486
1 8 E C PROCESS D E S I G N A N D D E V E L O P M E N T
Many authors have investigated the flow of flashing liquids in pipes (Allen, 1951; Benjamin and Miller, 1942; Bottomley, 1936; Starkman et al., 1964), and virtually all of the possible two-phase flow regimes have been studied. Several have noted that critical flow is frequently obtained (Cruver and Moulton, 1967; Isbin et al., 1957; Levy, 1965; Moody, 1965; Zivi, 1964)-i.e., that the flow rate in the pipe is unaffected, within limits, by variation of the pressure in the receiver. The velocities of these critical flows are always far below sonic velocity in either phase of the flowing mixture.
When a liquid is flowing at a sufficiently high velocity in a tube, it cannot flash until the outlet is reached and the streamlines can diverge. This critical flow phenomenon i s stable at velocities exceeding a certain minimum. Consideration of the energy balance at the point of incipient flashing shows this minimum to be
Experimental data obtained with water agreed well with this the ry. The water one tube diameter ahead of the outlet was significantly supersaturated and substantially above atmospheric pressure. This phenomenon may be utilized in the design of piping systems to prevent flashing in control valves, flowmeters, and elsewhere ahead of the outlet.
Brown and York (1962) studied the sprays formed by superheated liquid jets having velocities of the order of 100 feet per second. Bailey (1951) observed critical flow in short tubes and nozzles, but explained it in a manner different from that which follows. His data can be shown to be in agreement with the treatment below. In this work it is shown that a range of flow rates exists in which the liquid flashes just a t the exit of the pipe. Under these conditions the flow is truly critical, even though the velocities are much less than sonic. The minimum velocity for this one-phase critical flow is predicted from the physical properties of the liquid.
results in an increase in the velocity, ALL, because of the increase in the specific volume of the mixture, Au.
Theory
When the liquid flashes, its temperature drops. Most of the resulting decrease in enthalpy of the liquid provides the heat of vaporization of the vapor formed, while the rest appears as the increased kinetic energy of the flowing stream. This result is expressed analytically when a small fraction of vapor is formed as above. Equation 1 becomes
Consider a relatively hot liquid in turbulent flow through a long straight pipe leading to a receiver. The pressure a t the pipe inlet is above the saturation pressure of the liquid, whereas the receiver is below it. The pressure gradient due to the friction a t the wall is nearly constant until the saturation pressure is reached and the first bubbles of vapor appear. The analysis which follows is concerned with the increment of the pipe in which the flash is initiated. The length of this segment is short enough to justify neglecting heat transfer and change of elevation. Under these conditions the flow is very close t o being one-dimensional and steady. The following forms of the total energy balance and Bernoulli’s equation apply.
dH+vdp
udu
ge
+ -+ d F = 0
(2)
gc with the understanding that because the segment is short and no work is done,
dlc = dz = dq = 0
Solving for A P and substituting from Equation 5 yields
ldP1sat.
=0
udu
Remembering that AP is small and U, >> c:,
(3)
When the flash is initiated the first bubbles formed must of necessity be traveling a t the same velocity as the liquid surrounding them-Le., there is initially no slip between the flows of liquid and gas. Thus the formation of a small fraction of vapor, y, resulting from a small drop in pressure after the liquid is saturated, A P ,
Thus the assumption of no slip enables one to apply the total energy balance to calculate the fraction of the liquid which will vaporize if the pressure is dropped slightly below saturation. However, the increase in kinetic energy due t o the flashing of a small fraction of the liquid cannot exceed the energy released by the expansion. This is expressed analytically when the appropriate terms are substituted into Equation 2:
If -uiAP exceeds U J U , g, in Equation 8, a significant extent VOL. 8 N O . 4 OCTOBER 1969
487
Tap Water
1
-2
t
+ Drain
Figure 2. Sketch of apparatus Figure 1. Minimum critical velocity for one-phase flow of water
of flashing can occur. The excess energy released by the expansion is dissipated in friction, represented by the term AF. If -uiAP is smaller than ulAu/g,, the indicated expansion cannot occur, since a negative value for AF is physically impossible. Flashing can then take place only when the exit of the pipe is reached and the flow can diverge. When the two terms above are equal, AF is zero and velocity u I is the minimum necessary to maintain onephase flow in the pipe. Solving Equation 8 for AP and comparing with Equation 7 then yields
ldP1sat. Substituting the value of Au from Equation 5 and dividing through by y allows solving Equation 9 for u l :
sat.
The minimum critical velocity for water was calculated using the data of Keenan and Keyes (1967) and is plotted as a function of temperature in Figure 1. The term (dHi/dP),,,, is the change of enthalpy of saturated liquid with pressure and is usually much larger, than ut. From the thermodynamic relationship dH = C,dT + vdP it follows that
sat.
sat.
In the case of a multicomponent solution [dPidT],,,, is the slope of the vapor pressure curve a t bubble point. 488
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Experimental Apparatus
Equation 10 predicts the minimum velocity for onephase critical flow of liquids in pipes from the physical properties of the fluid concerned. At this and higher velocities flashing should not occur until the liquid reaches the end of the pipe where the flow lines can diverge. Under these conditions the pressure just inside the end of the pipe is near saturation. The flow rate is thus independent of the pressure in the receiver as long as the pressure there is well below saturation. At lower velocities flashing will occur ahead of the pipe exit. The flow in the pipe may still be critical, but it will be two-phase. To test this prediction the equipment sketched in Figure 2 was built. Water was obtained either by partially condensing steam or from the tap. I t was then metered and heated to the desired temperature in heater H-1. Tank R-1 served to disengage any entrained gas bubbles. The temperature of the water was measured just ahead of the needle valve, V-1, which controlled the flow of the water. The pressure of the water was measured immediately after the valve a t P-1 and just ahead of the test section. The test section is shown in more detail in Figure 3. I t consisted of a “pipe” made of a heavy-walled glass capillary having an inner diameter of 0.041 inch and 9 % inches long. One-inch sections of larger bore glass tubing were blown onto either end of the test section, giving smooth transitions in the tube diameter. The receiver consisted of a piece of 2-inch glass pipe. A spray of water condensed the steam which formed. Partially condensed steam was used as the source of water in an attempt to eliminate dissolved gases. These were troublesome because of the need to operate very near the saturation pressure. Even with this precaution, enough dissolved gas was present to prevent an accurate visual determination of the minimum critical velocity.
Pressure Gauge P- I
0.041" l . D . x g - i / 4 " l o n g Capillary Tube (Test Tube)
0.100"l.D.xl" long
Needle Valve
U n n t , ~W a l l
Glass Tube
a 0 (D
2. (D < _1
P
-
N
Copper Tubing Glass Tube
Figure 3. Detailed view of test section
The pressure a t P-1 was measured with a Bourdontube gage with a precision of j ~ 0 . 5p.s.i. The temperature was measured with a thermometer to h 1 " F . The flow rate of water was measured with a rotameter with an estimated precision of * 2 % .
30 28
Results
Runs were made by setting the temperature of the water a t a constant value and measuring the pressure a t P-1 as a function of the flow rate of the water. The first run was made with water at 200°F. At this temperature the saturation pressure is below 1 atm. and no flash can occur. The data obtained, plotted in Figure 4, compare well with the solid line calculated from the standard friction factor correlation for flow in smooth tubes. The method of calculation is outlined in the Appendix .
26
-.-
Runs with water at 24OOF
Calculated for I-Phase Flow
--
Runs with water a t 220°F
24
0 ln
22 20 18
16
0
2
4
6
8
IO
12
14
16
U,(ft/sec) Figure 5. Variation of pressure for critical flow of water at 220" and 240" F.
I
@ -1
15 14 0
2
i
Runs with water A at 200°F 0
f
I
4
6
8 UI (ft/sec)
IO
(TAP WATER)
12
14
16
Figure 4. Variation of pressure with flow rate of subsaturated water
Runs were also made with water a t 220", 240", 260°, and 280°F. over the same range of flow rates (Figures 5 and 6). At the higher velocities in these runs it could be ascertained visually that no flashing occurred in the glass capillary until about one tube diameter before the exit a t the point called the "throat" of the test section. The dashed curves in Figures 5 and 6 were calculated from the fluid properties and flow rates at each temperature in a manner similar to that used for the curve in Figure 4 (see Appendix). However, a t each temperature the calculated curve was translated vertically to fit the experimental data a t 15 feet per second. I n the run made a t 200°F. the throat pressure was virtually the same as the receiver pressure and the flow was one-phase a t all velocities. The pressure a t P-1 approached the throat pressure as the velocity was reduced and agreed well with the calculated curve at all velocities. At each of the higher temperatures the pressure at P-1 followed the respective dashed curve as the flow rate VOL. 8 NO. 4 OCTOBER 1 9 6 9
489
at the predicted velocities the points have diverged only 1 or 2 p.s.i. from the curves, which is only slightly greater than the scatter of the data and is probably due to a combination of three effects. The first is the presence of a very small amount of inert gas. The second arises from the fact that Equation 10 was derived by assuming a constant velocity across the tube diameter. Actually, it applies for each streamline as long as the flow is unidirectional. Hence, the minimum velocity will be reached first near the tube wall and a small amount of flashing may occur there while the core is still moving too fast to allow it. Finally, because of the supersaturation a t the throat, the total pressure a t P-1 was frequently very close to saturation, which made reproducible measurement difficult.
h
0 .-
-u)
a I
a
Discussion
Figure 6. Variation of pressure for critical flow of water at 260" and 280" F. was reduced until the minimum velocity for one-phase critical flow was approached. As the flow rate dropped below that, the pressure a t P-1 dropped below the curve, heading for 1 atm. a t zero flow. The velocity a t which the experimental points began to diverge from the dashed curve was the most sensitive indication we could obtain of the minimum velocity for one-phase critical flow. For flow rates a t which one-phase flow was maintained throughout the test section, the pressure a t the throat was estimated by subtracting the calculated pressure drop for one-phase flow through the test section from the measured pressure a t P-1. The pressures so obtained are the same as the intersections of the dashed curves with the zero-velocity axis in Figures 5 and 6, and are shown in Table I. The throat pressures appear to be 15 to 25% below saturation in each case, showing that the water is significantly superheated a t the point where the flash is initiated. As long as the pressure in the receiver is below the throat pressure, it can have no influence on the flow rate in the test section and hence the flow will be critical. The experimental points in Figures 5 and 6 begin to diverge from the dashed curves a t velocities somewhat greater than those predicted by Equation 10. However,
Table I. Estimated Throat Pressures for One-Phase Critical Flow of Water"
Temp.. F.
P S a t ,P.S.I.A. ,
220 17.2 240 25.0 260 35.4 280 49.2 Velocity of liyuid. Zt5 feetper second. ~~~
490
Throat Pressure, P.S.I.A 15.2 20.5 29.4 40.7
~
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
The purpose of the experiments reported here was to demonstrate the existence of one-phase critical flow of liquids and to obtain a t least a semiquantitative verification of the theory predicting the minimum velocity a t which it occurs. This phenomenon should prove of some utility in the design of chemical processes The application of the theory is not restricted to flow in straight pipes. Flow in a convergent nozzle may be analyzed in a similar fashion and for a well-designed nozzle the same expression for the minimum velocity is obtained. I t is thus practical to design a piping system for a hot fluid in which the control valve, flowmeter, pump, and other components of the line handle only liquid, with flashing limited to the exit of the pipe. The use of a nozzle a t the pipe exit allows relatively low velocities in the rest of the piping and avoids excessive pressure drop in such a system There is no maximum critical velocity for liquids until the speed of sound is approached. Since sonic velocity is of the order of thousands of feet per second for most liquids, it does not represent a real limitation. I n most cases of practical importance the engineer can design for a wide range of flow rates, the lowest being set by the minimum critical velocity in his nozzle and the highest by the capacity of his pump. A number of the previously cited studies of the critical flow of two-phase mixtures follow developments similar to that presented in this paper, particularly the work of Allen (1951) and Linning (1952). The primary difference in the developments lies in the assumptions which must be made regarding the slip between the two phases. In the present work the assumption of zero slip is justified because only an infinitesimal amount of the vapor phase is present. When major amounts of both phases are present, an accurate estimation of the slip becomes much more difficult. Knowledge of the slip is essential for evaluating the velocity terms in Equations 1 and 2, but one is forced to rely on empirical correlations which depend on the type of flow regime obtaining (froth, annular, slug, etc.). These correlations are based on steady-state, uniform flow and are of uncertain reliability for nonuniform flow. Furthermore, they represent the average conditions over the cross section of the pipe, and for some types of flow regimes such averages may not be appropriate for use in Equations 1 and 2. For these reasons it is in general not possible to derive an equation valid for two-phase flow which contains terms related only to the physical properties of the fluid.
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
