Ind. Eng. Chem. Res. 1987,26, 1698-1704
1698
4 = fugacity coefficient in vapor 0 = objective function
tended and was left unattended in our work. Acknowledgment
Super- and Subscripts 1, 2 = component c = calculated e = experimental s = saturated state Registry No. Benzene, 71-43-2; cyclohexane, 110-82-7.
Financial support for this work was provided by the National Science Foundation through Grant CBT-8516449. Lana Ferrick assisted in the apparatus construction. Nomenclature A , B , C = constants in Redlich-Kister equation for activity coefficient B = second virial coefficient e = exp[udp - P’))/RTI p = pressure R = universal gas constant T = absolute temperature u = molal volume x = mole fraction in liquid y = mole fraction in vapor z = compressibility factor
Literature Cited Barker, J. A. Aut. J . Chem. 1953, 6, 207. Gibbs, R. E.; Van Ness, H. C. Znd. Eng. Chem. Fundam. 1972, 11, 410.
Hayden, J. G.; O’Connell, J. P. Znd. Eng. Chem. Process Des. Dev. 1975,32, 1252. Jin, Z. L.; Lin, H. M.; Greenkorn, R. A.; Chao, K. C. AZChE Symp. Ser. 1985, 81(244), 155. Mentzer, R. A.; Greenkorn, R. A.; Chao, K. C. J. Chem. Thermodyn. 1982, 14, 817.
Greek Symbols y = activity coefficient in liquid
Received for review September 3, 1986 Accepted June 1, 1987
Performance Characteristics of a Continuous-Flow Fluidic Pump Sharon M. Robinson* Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Robert M. Councet and Gary V. Smith* Chemical Engineering Department and Mechanical and Aerospace Engineering Department, The University of Tennessee, Knoxville, Tennessee 37996
The fluidic pump is a type of positive-displacement pump in which basic fluid mechanics phenomena are utilized to eliminate valves and other moving parts that are exposed to the fluid being transferred. T h e version described here is powered by gas pressure serving as gas pistons and is virtually maintenance-free. It utilizes two displacement vessels and is designed to produce a steady and continuous liquid flow. This type of pump may be very useful for the transfer of radioactive or hazardous liquids where mechanical maintenance may be difficult or exposure of personnel to the fluid is undesirable. This paper presents experimental and model-predicted characteristics of such systems. The effects of several geometric parameters and operating conditions on the performance of the pump are briefly discussed. Introduction Many industrial situations require pumps that are highly reliable and have low tolerances for leakage. These requirements imply that packing glands or mechanical seals must be eliminated and that moving parts should be minimized. Several types of systems, including steam jets, air lifts, vacuum transfer, and gas-powered displacement pumps, meet these requirements. All of these systems are unlikely to suffer from wear-induced failure, but their other characteristics vary so much that the selection of one over the other depends on the particular requirements for the proposed application. The fluidic pump is a type of positive-displacement pump in which the conventional check valves have been replaced with less efficient “no-moving parts” fluidic Chemical Engineering Department.
* Mechanical and Aerospace Engineering Department, 0888-5885/87/2626-1698$01.50/0
components. It has the advantage of not diluting or heating the pumped liquid, and solutions containing up to 50 vol % solids can be handled without degrading the particles during transfer (Priestman and Tippetts, 1984, 1986). Such pumps are being developed for long-term maintenance-free applications for the transfer of toxic, radioactive, or highly corrosive liquids and slurries. Fluidic pumps can be grouped into the following two major categories: (1)single acting and (2) double acting (Priestman and Tippetts, 1984). Single-acting systems use a single pumping chamber similar to acid-egg pumps to give intermittent liquid output, while double-acting systems utilize two displacement vessels connected by fluidic devices to give a continuous output. Systems of both types have been used extensively in the United Kingdom and are being developed in the United States (Smith and Counce, 1986). 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1699
I
DELIVERY PIPE
FLOW MODE
FLOW MODE
Figure 3. Schematic of a venturi-like flow junction (FJ).
OUTPUT
REVERSE F
L
~
PUMP SYSTEM BOUNDARY-
r------
DIVERTER
“%:I
Figure 1. Schematic of double-acting fluidic pump.
DISPLACEMEI., DISPLACEMENT
’”‘
I
1
PLENUM
@
3
I
1
DISPLACEMENT pv2 1NO.F VESSEL
NOZZLE EXIT
(I> NOZZLE INLET
DIFFUSER
A
; OUTLET
I
‘DIFFUSER
INLET/ RECEIVER
Figure 2. Schematic of venturi-like reverse-flow diverter (RFD).
Several designs of pumps employing different fluidic devices exist (Tippetts et al., 1981); however, only the simplest components were selected for use in this study. A schematic of the simplest double-acting fluidic pump is shown in Figure 1. It consists of two gas-activated pumping chambers which are connected to feed and delivery tanks by two fluidic components, a reverse-flow diverter (RFD) and a flow junction (FJ). A venturi-like RFD (Smith and Counce, 1986),shown in Figure 2, consists of converging and divergingsections with a separation gap between them open to the plenum region which is connected to the liquid being transferred from the feed tank. The converging and diverging sections of the RFD must be symmetrical for application in double-acting fluidic pumps. Their designation as a nozzle and a diffuser, respectively (Figure 2), is determined by the direction of flow through the RFD. The nozzle inlet would be attached to the pressurized pumping chamber and the diffuser outlet to the vented chamber so that fluid flows from left to right. The FJ is a symmetrical Y junction designed for high efficiency by decreasing the diameter of each input leg at the junction point and adding a diffuser to the output as shown in Figures 3 and 4. Its purpose is to convert flow from either input leg into unidirectional flow in the output leg. These simple fluidic components have the disadvantage of not eliminating backflow which causes diversion of flow from the primary flow direction; however, this leakage can be minimized by proper design of the fluidic devices and sizing of the FJ relative to the RFD. The system (see Figure 4) is operated by alternately pressurizing and venting the pumping chambers. As a pumping chamber is pressurized, liquid is forced from the chamber, through the RFD and the FJ, to the output line (defined as output flow in this paper) and to the vented pumping chamber. A portion of the flow entering the FJ is delivered to the pump delivery line, while the remainder is trans-
FEED
Figure 4. Schematic flow sheet for double-acting fluidic pump model.
ferred via the alternate inlet leg to the vented pumping chamber. The narrowing of the flow passage in the inlet leg of the FJ (Figure 3) creates a semiconfined jet which is aimed at the output line. The increased momentum flux of the fluid, and hence the reduced pressure, increases the flow through the outlet and reduces the volume of flow to the alternate inactive inlet. Similarly, the nozzle of the RFD (Figure 2) creates a submerged jet at the nozzle exit which directs the flow of liquid across the plenum region to the diffuser. Smith and Counce (1986) developed the inviscid jet theory of the RFD which states that the cross-sectional area of the jet in the plenum region is equal to the nozzle exit area, while the cross-sectional area of the jet in the immediate vicinity of the receiver portion of the diffuser inlet is determined by the static pressure in that region. Under normal operating parameters, the static pressure in the receiver inlet (determined by the output pressure of the RFD which is related to the pressure in the vented pumping chamber) is less than that of the ambient (plenum) pressure (set by the hydrostatic head in the feed tank). The fluid is accelerated as it enters the receiver, resulting in (1) a reduction of the jet cross-sectional area and (2) induction of ambient fluid into the receiver. Fluid from the feed tank flows into the pumping system (defined as input flow in this paper) by this mechanism. Liquid flows from the system to the feed tank via the plenum region of the RFD if the pressure in the receiver inlet is greater than the plenum pressure. Under those conditions, the jet is decelerated, the area of the jet increases, and only a portion of the jet is captured by the receiver. The diffusers on the outlets of the RFD and FJ recover a portion of their respective original static pressures.
1700 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 IULET ;ilOM 38
l
PB*
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AB
/I//
I,
-,, ,-ALTERNATE LEG, pOL, A, /-ALTERNATE CONNECTION, A, - _ _
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Figure 5. Schematic of a flat-walled flow junction (FJ).
When the pressurized pumping chamber is nearly empty, the controller vents it and pressurizes the charged one. Alternately pressurizing and venting the pumping chambers theoretically produces rectangular-type flow waves which overlap to produce a continuous output flow. Unless otherwise stated, output flow refers to the system output flow throughout this paper. To achieve steady-state operation, the system must be operated under conditions such that (1)the total output flow for one pressurization cycle equals (2) the net input flow (input flow from the feed tank minus any leakage from the system to the feed tank that might occur through the RFD). Under these conditions, the total amount of liquid entering the vented pumping chamber during a cycle will equal that exiting from the pressurized chamber. The liquid in each pumping chamber will normally oscillate, with equal displacement around the center of the chamber. The purpose of this paper is to present model-predicted and experimental pressure-flow characteristics for a double-acting fluidic pump. The effects of geometric parameters and operating conditions on the performance of the pump are also briefly discussed. Mathematical Model
A steady-state mathematical model of a continuous-flow fluidic pump was developed assuming steady incompressible flow (i.e., transient effects will be neglected) through the axisymmetric system shown in Figure 4 (Robinson, 1985). The model is based on material and energy balances, using Smith and Counce's (1986) inviscid jet model for the RFD and a tee junction model to describe the FJ. The nomenclature for the model is given in Figures 4 and 5 where the primary subscripts designate pipe lines and directions of flows, while the secondary subscripts indicate flow from high-pressure points to low-pressure points. Material balances around the FJ, the RFD, the tees connecting the fluidic devices and pumping chambers, and the overall system yield five independent equations: Qo = QF QI
QB
- QD QA
(2)
+ QB
(3)
QF
(4)
= Qz
(5)
= Qc =
QA
Qo = Qi
(1)
where 2 is the average height of the liquid in each pumping chamber relative to the RFD-FJ plane. The single subscripted loss coefficients denote the total loss due to friction, valve, and fitting losses and losses due to changes in velocity. The double subscripted loss coefficients represents friction losses in the branch of the tees which connect the pumping chamber and the fluidic components. Smith and Counce's (1986) inviscid jet model was used to develop .equations which describe the flow through the nozzle of the RFD,
and flow through the diffuser,
where Cw is the discharge coefficient for the nozzle in the RFD and AN and PNare the area and pressure at the diffuser inlet, respectively. Two energy equations can be written to describe the flow through each of the legs of the flow junction by modeling the FJ as a tee junction (Miller, 1978) connected to a nozzle in the entrance leg, a sudden expansion in the alternate inlet leg, and a diffuser in the exit leg as shown in Figure 5:
(13)
L A B(2 1 - 2 ) - $ ]
and
( :;)'
P*,-PD1 = PQD~[1 + -11 - 2
AD2
Aj2
(
+ -1 1AL2
The flow between the pumping chambers and fluidic devices can be easily modeled by writing energy equations for each stream tube where KnFis the nozzle loss coefficient, CpFis the diffuser pressure recovery coefficient, and K32 and K31 are the friction loss coefficients for the outlet and alternate inlet branches of the tee. Equations 6-14 can be combined to obtain five de-
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1701 pendent equations which describe the flows from the feed tank and pumping chamber to the vented chamber and pump outlet (Robinson, 1985). Assuming the loss coefficients and the piping sizes are fixed, the unknown quantities in the resulting 10 equations are Po, PF,Pv,,Pv,, PN, Qo, QF, QA, QB, Qc, QD, 81, and Q P A unique solution may be found by specifying three parameters and solving the above equations simultaneously with the constraint that negative values for flow rates cannot occur. Model-predictedresults were obtained using friction loss coefficients for standard valves and fittings reported by the Crane Company (1976) and White (1979). Previous data for RFDs (Smith and Counce, 1986) indicate that flow through the nozzles used in the experiment is approximately ideal. The discharge coefficients and pressure-recovery coefficients for the RFD and FJ were assumed to be equal to the design values (White, 1979). The loss coefficients for the FJ tee were determined experimentally in a separate system designed for this purpose. Three experimental parameters, (1)the air supply pressure to the pressurized pumping chamber, (2) the pressure in the vented pumping chamber, and (3) the hydrostatic head in the feed tank, were inputs for the model.
Experimental System The experimental double-acting fluidic pump consisted of two gas-activated 0.16-m3pumping chambers which are connected to a 0.19-m3feed and delivery tank by a RFD and FJ as shown in Figure 1. The sizes of the components used in this experiment were based on previous experience with individual units (Smith and Counce, 1986; Tippetts et al., 1974). Since the effect of coupling fluidic units had not previously been addressed, three sizes of flat-walled FJs were tested. The venturi-like RFD consisted of two identical straight-walled diffusers with inlet diameters of 0.94 cm, exit diameters of 1.49 cm, and lengths of 14.2 cm. The gap length between the diffusers was equal to the diffuser inlet diameter, 0.94 cm. The FJs were symmetrical flat-walled Y junctions similar to the one shown in Figure 5. The angle between the inlet legs was 50°, and the diffuser had a double divergence angle of -3O. The FJ/RFD area ratios, based on minimum areas in each component, for the three systems are 2, 1, and 0.3. Solenoid valves activated by timers were used to control the air flow to the pumping chambers. Pressures within the system were monitored by pressure gauges and transducers, and input and output flows to the system were monitored by orifice meters attached to strip-chart recorders. Flow rates from the pumping chambers were computed from level changes in the vessels. Data shown in this report were obtained under the following operating conditions unless otherwise noted: (1) pumping cycle, 2 min; (2) average hydrostatic head in feed tank, 1.2 m of H,O; (3) average hydrostatic head in pumping chambers, 0.4 m of H,O; and (4) FJ/RFD ratio, 0.3. A pumping cycle is made up of two pressurization sequences during which each pumping chamber is pressurized and vented once. All data are for water pumped at room temperature. It should be noted that each fluidic pump must be designed for a specific application in order to obtain maximum efficiency. This pumping system was designed for flexibility of operation and was operated under a range of conditions. Therefore, the data presented in this paper represent the general characteristics of a continuous-flow fluidic pump but do not necessarily reflect the maximum capabilities of a well-designed unit. Discussion of Results Experimental data on the operating characteristics of
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120
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Air S u p p l y Pressure (kPa)
Figure 6. Comparison of experimental and model-predicted system output pressures: (a, top) 0.23-cmZflowjunction; (b, bottom) 0.76cm2 flow junction. -
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40
80
120
160
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Air S u p p l y Pressure (kPa)
Figure 7. Comparison of experimental and model-predicted RFD output pressures: (a, top) 0.23-cm2flow junction system; (b, bottom) 0.76-cm2 flow junction system.
the continuous-flow fluidic pump are presented and, where applicable, compared to the mathematical model previously developed. The effects of various geometric parameters and operating conditions are also discussed. The results of the mathematical model predict ideal rectangular-wavepressure cycles which overlap to produce steady, equal input and output flows. This experimental system was designed to operate under near ideal conditions at low output heads. In order to compare the model with
1702 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 -
-.
0
40
80
120
160
200
0
40
Air S u p p l y Pressure (kPa)
80 120 160 A i r S u p p l y Pressure (kPa)
200
c
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20
. 1
0
,
br
40
80
120
160
---
m d e l "bites
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-
0
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Air Supplq Pressure (kPa)
Figure 8. Comparison of experimental and model-predicted pumping chamber flow rates: (a, top) 0.23-cm2 flow junction system; (b, bottom) 0.76-cm2 flow junction system.
experimental results over a wider range of operating conditions, experimental data were taken to simulate ideal conditions for the two smaller FJs. The model predicted most flows and pressures within the system with
