They determine the dynamical response of the process to both manipulation and disturbance. Their relative size also determines the precision with which the liquid temperature T can be regulated and the speed with which the controlled process can recover from an upset. Figure 2B shows that the combined chemical reaction and thermal process can be manipulated or disturbed by several variables, As before, the heat inflow Qiand the outside temperature T o are process manipulators. They can be used to alter the temperature T which in turn alters the rate of chemical reaction. Another manipulation variable is the catalyst inflow rate Qc. Changes in Qc can produce changes in the catalyst concentration C which in turn alters the reaction rate R. The heat generation rate Q,. may be regarded as a uniformly distributed heat source in the vessel. The heat flow Qp must therefore be added to the heat flow summation point so that Z& which raises the liquid temperature T takes into account heat released due to the reaction. An important consideration in process design and in process control occurs in connection with the signal flow path ZQ T CT R &, - ZQ. This constitutes a positive feedback or regenerative signal loop. It gives rise to the possibility of sustained oscillation or even a runaway condition of the liquid temperature T. A careful study of the stability of the process is needed to determine how to counteract by means of control or by means of process redesign an inherent runaway tendency. When considering the process design and the effectiveness of control over the chemical-thermal process, the mixing lag associated with operator A , the time constant C/G, and the time con-
- -
- -
stant Ct/Gt, figure prominently in all decisions. A large mixing lag tends to interfere with process control by means of manipulating the catalyst inflow rate Qc. A large thermal lag in the vessel tends to render jacket-type heat exchange to control liquid temperature ineffective. A large mixing and thermal lag in the liquid makes temperature control difficult when heat inflow rate becomes the manipulator.
Conclusion This paper has presented in brief the concept of the signal flow diagram. The examples of the thermal and thermal-chemical processes were chosen particularly to demonstrate: Dynamical behavior of a process, as it is pertinent to regulation and control, begins with the process design. The signal flow diagram is not an analog; rather it is as exact a mathematical model as one can set down for a particular problem. The resulting primitive diagram, however, can be converted to any convenient analog form. The signal flow diagram technique is a powerful analytical tool for studying almost any kind of physical process. I t s greatest value seems to be the striking focus it puts on lack of information or hazy concepts about process dynamics. References ( I ) Brown, G . S., and Campbell D. P., “Principles of Sorvomecha-
nisms,” Wiley, New York, 1948. (2) Mason, S. J., Proc. I.R.E., No. R363.23 (September 1952). (3) Stout, T. M., Am. Inst. Elec. Ewrs., Paper No. 52-254 (No-
vember 1952).
RECEIVED for review September 28, 1964.
ACCEPTEDJanuary 31, 1955.
Analysis of Unsteady Fluid Flow Using Direct Electrical Analogs S. E. ISAKOFF,
Engineering Reseorch Department,
E. 1. du font de Nemourr & Co., Inc., Wilmington, Del.
A
direct analogy between pressure and flow fluctuations in fluid systems and voltage and current oscillations in electrical networks is discussed. The validity o f this analogy i s illustrated b y comparing the results o f experimental studies of pulsation dampers located at the discharge of a reciprocating compressor with tests of equivalent electrical filter networks, Advantages, limitations, and practical considerations involved in using electrical networks for solving analogous unsteady fluid flow problems are given. Examples are cited of use of electrical analogs in the analysis of factors affecting the tendency of a system to oscillate, in development of frequency response data to aid in instrumentation and control, in study of pressure transmission in process and measurement systems, and as an aid in determining fluid meter requirements.
T
HE visualiaation, formulation, and solution of technical problems through the use of familiar analogous systems is a technique long known in the physical sciences. Because the mathematical equations which define many different physical systems in pature are similar, it is possible to simulate the behavior of a system in a field in which computations are tedious, models expensive, and experimentation difficult, by transforming to an analogous system which can be readily assembled and tested. Results obtained with the analog can then be translated into the terms of the original physical system.
March 1955
Although many different physical Bystems have been used as analogs, electrical circuits have proved the most generally applicable. Widespread knowledge of electrical circuit theory and measurement techniques, coupled with the availability of high quality, easily assembled, moderate-cost components, makes the electrical analog computer a convenient working tool. Chemical engineers, as a group, have made little use of electrical analogs, however, and articles in the technical journals describing their application t o chemical engineering problems are few. This is due, in part, to a lack of familiarity with analog techniques and
INDUSTRIAL AND ENGINEERING CHEMISTRY
413
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT FLUID FLOW P
8
0
0
ELECTRICAL
pressure volumetric flow
GAS
v
I =current
LIQUID
ELECTRICAL
Surge Chamber
Tank
adequate. The time behavior of the fluid under external variations must be understood. I n automatir control, and even manual control, the transmission nf flow and pressure transients must be analyzed. Obviously, in rase3 where flow and pressure fluctuations lead to excessive pressure stresses, to water-hammer, to mechanical vibration, or to material fatigue failure, the time-dependent relations in fluid flow must be considered.
* voltage
Capacitance
0
1
- c
I 1
Ground
1 _I
Short Pipe
r , L
&
L
b
Analogy
o
Resistance and Inductance
A fluid system has the elements common to all mechanical systems.
TR L- - j y - - Q C O
Figure 1.
Fluid flow-electrical
T iT
Trowmission Line
analogy
their potentialities. If increased use is to be made of analogs in chemical process studies, and their pot,ential benefits obtained, more information is clearly needed to indicate the types of problems that can be handled by an analog approach and to explain in some detail the methods employed. I n this paper, the direct analogy between unsteady fluid florv and electrical circuits is discussed. The term "direct" implies a one-to-one correspondence between elements and variables in the electrical and physical systems and is used to distinguish this analog from operational analog computers (12). The latter, which are also called electronic differential analyzers, use feedback amplifiers and successive integration of voltage to form circuits which represent specific mathematical operations. The analogy itself is not new, since it has been described in detail b y Olson (19) and is in common use by acoustical engineers. Murphy ( 1 7 ) has employed t,he analogy to determine natural frequencies of engine and compressor manifold systems arid h a s described the network calculntor at, the Illinois Institute 3f Technology. Paynter (21) describes the relations betiwcxn hydraulic and electrical systems in connection with water-hammer problems. Chilton and Handley ( 5 ) discussed the analogy in cnnnection with compressor systems but did not use it, to s o l w thr relations formulated. Hughes and coworkers ( I O ) , Baird and Rechtold ( I ) , Hirschorn ( 9 ) , Mc3Iaster and coworkers (16j,and others have discussed it qualitatively. The construction and use of simple direct analog computers for determining pressure and flow amplitude and phase relations in pulsating flow systems, however, appear t o be reasonably new. It is the int,ent of this paper to show how the analogy can be employed, by chemical engineers, to obtain both qualitative and quantitative answers t,o a number of familiar design and process control problems.
Unsteady Fluid Flow
3
In principle, the relations governing the timedependent flows and pressures a t all points in a. fluid system can be obtained from force and material balancrs on sections of the flowing fluid. The solution of the resulting simultaneous differential equations is quite another matter. Manual methods for extracting answers from the equat,ions that define even the simpler systems encountered in process applications can be quite t,edious. Electrical analogs prove va,luable in solving t,hese simultaneous differential equations automatically and instantaneously. While a number of analogies bet'ween fluid flow and clectrical systems are possible and have been used (6, I S ) , the analogy discussed in this paper is the usual one employed by acoustical engineers (19 j. The corresponding terms and equivalent elrments in the analogy are given in Figure 1. From this chart, one could set, down a schematic electrical circuit corrrsponding to a given physical flow system without recourse to thr defining mathrmatical equations. Actual electrical values can be assigned t,) all the componrnt,s appearing in the schematic clectrical circuit in a purely systematic way by using the scaling procedure, d ~ scrihpd later, and the following definitions: Fluid Flow
9,'
Q
CF
414
Electrical
VoliiinPtric Flow = pressure .- o, 1'd.c. = voltage = voirimetric flow z t , Id.c. = oiirrent rate =
= PC
LF =
Rp =
5 AP
capacitance
= inertanoe
=
= resistance
c
= capacitance
'I
= inductanre
R
= resistance
M a s s Flow = pressure - 7 1 , 7 - d . r . = voltage = weight f l o x rate E i , Id.c. = c n r r e n t
P2 p vi, W CF =
The movement Si fluids through pipes, ducts, and process equipment is perhaps the most prevalent single operatiqn in a chemical plant. As a result of the inherent alternating or interrupting action of pumps, compressors, and bloiTers, or as a result of periodic vortex-shedding from obstacles in the fluid stream, flow and pressure pulsations are almost universally preqent These pulsations can often be tolerated, and, in many cases, traditional steady-state methods of analvsis and design (3j can be used without serious consequcnres. But if the fluid system is subjected to sudden upsets or large-amplitude periodic fluctuations, and if it is important to measure and control the pressure and flow for process purposes, steady-state analysis praves in-
1. Mass which can be displaced or accelerated and possesses kinetic energy 2. Elasticity due to fluid compressibility, and thus the capacity for storing potential energy 3. Frirtional dissipation or resistance
'4
c
= capacitance
= casacitance
LF = 1 =inertance a ,A
E L
= induetanre
RF
-
= resistnncp
=
W
= resistance
R
In order t o eee hon these equivalences come about, and to understand the assumptions implicit in Figure 1, hovever, it is instructive to work through the relations for a specific example, in terms familiar to engineers. The example chosen involves the damping or smoothing of flow and pressure pulses, nhich has considerable practical plant application intei est, and providep a
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41,No. 3
ANALOGS convenient opportunity to show the methods employed in direct analogs. Low-Pass Pulsation Damper-Electrical Filter Equations. Consider the gas flow system shown schematically in Figure 2, in which the combination of two tanks and an interconnecting pipe is connected to a source of periodic pulsating flow, such as a compressor. When the volume of each chamber, and the length and diameter of the interconnecting pipe are sized properly, the combination makes a highly effective pulsation damper. The damper, which depends on pressure-wave reflection and cancellation for its effectiveness, substantially reduces or eliminates flow and pressure pulsations a t frequencies above a design-set hwer limit. This lower limit is called the cutoff frequency, fc. Steady flow passes through with little pressure loss. I n this example, the time-average absolute pressures in each tank of equal volume, V , are PI and Pz; p l and pz represent the instantaneous pressure fluctuations, which are, of course, functions of time; p I and p2 are the corresponding average gas densities. The flow from the pulsating source entering the first chamber is given by Q1 ql, where Q1is the time-average volumetric flow rate, and pi defines the instantaneous deviation from the average flow. Assuming that the gas in the interconnecting pipe behaves as a single lump (like a solid plug), a dynamic force balance on this lump can be made. That is, setting the instantaneous forces on the fluid lump equal to its mass times its acceleration yields
The reciprocal of the constant terms outside the square brackets is the capacitance of gas contained in a tank of volume, V , and is designated by the symbol CF. Equation 6 then becomes
-
(7) Similarly, a material balance on the second tank yields
Similar to Equation 3, a force balance on the fluid contained in the pipe leading to the atmosphere yields
+
PZ - 93(RF)3
1- J
rl+PI
(9)
p2+ 4 4
q2
Pulsation Source-,
dq3
= (LF)3dt
A2
q3
I
PATM.
Wl
where A P p is the instantaneous pressure loss caused by fluid friction, p is the average gas density, Az is the pipe area, 2% is the pipe length, and u2 is the linear velocity in the pipe. It is assumed for the sake of simplicity and as a compromise between convenience and accuracy that a t a given mean flow rate, the constant resistance coefficient, RFP,that defines the relation between average pressure and flow, also defines the relation between instantaneous pressure drop and instantaneous flow. That is
LOW- PASS
The average pressures and flows drop from the equations and only the time dependent portions are retained. The terms in the brackets on the right-hand side of the equation define the inertance, or inertia properties, of the gas contained in the interconnecting pipe, which is represented by the symbol LpZ. The time-dependent pressures, p l and pa, are obtained from material balances on their respective chambers (4)
For practical ranges of pulsation frequencies and tank sizes, the compression and expansion of gas occurs isentropically. Introducing the dimensional constant, gc, rearranging, and integrating both sides of Equation 4
The terms in the square brackets define the square of the velocity of sound in the gas contained in chamber l, c12. It is assumed that c2 is constant over the range of pulsations in any one chamber or tank. For an ideal gas, c* is given by Pygc/p, where y is the specific heat ratJio. Simplifying Equation 5
March 1955
PULSATION DAMPER
unction
LOW-PASS
Substituting Equation 2 in Equation 1, replacing u2 by qz/Az, and simplifying
GAS
Figure 2.
ELECTRICAL
FILTER
Gas pulsation damper and direct electrical analog
Analogous Electrical Network. If in these equations the symbol p is electrical current flow, p voltage, L electrical inductance, R resistance, and C capacitance, the equations can be shown to define the electrical circuit in Figure 2. The circuit is thus the direct analog of the physical system and can be used to simulate the operation of the pulsation damper. Actually, the circuit represents a well-known low-pass electrical filter, which performs a function analogous to that of a pulsation damper. That is, it attenuates electrical pulses above a set frequency with low resistance loss. The conditions under which the analogy applies are 1. The instantaneous pressure loss due to friction (resulting from the flow fluctuations) is linearly related to the resistance coefficient, RP. This is strictly correct for streamline flow. Even in turbulent flow, if the amplitudes of the flow fluctuations are small compared to the mean flow, this assumption will introduce negligible error in the determination of pressure fluctuations. Where the resistance term is small compared with the other terms representing the flow system, this assumption is of little consequence. If the flow fluctuations are large compared to the mean flow, however, and the resistance term is important, a power-law relation between flow and fluid friction applies, and the representation of fluid-flow resistance by a linear electrical resistor can be considered only qualitatively correct. The error introduced by the linearization of the resistance term depends on the wave shape of the flow fluctuation as well as its amplitude. That is, the presence of steep wave shapes or rapid
INDUSTRIAL AND ENGINEERING CHEMISTRY
415
ENGINEERING, DESIGN, AND
Figure 3.
PROCESS DEVELOPMENT
Pulsation damper test apparatus
fluid accelerations causes a greater error .than when the flowversus-time relation is sinusoidal, for example. 2. The gas contained in a pipe or restriction was treated as a single, solid lump, possessing only mass or inertance, n-hile the gas in the tank was assumed to be a pure capacitance. Actually, each element of fluid possesses mass, resistance, and elasticity, and this lumping procedure is not justified in all cases. If long pipelines are involved, the pipe must be considered as a series of segments each possessing inertance, resistance, and capacitance. While information is not yet available for determining the error associated with the degree of lumping for all situations, some cases have been considered ( 1 8 ) and some rules-of-thumb may be stated. The length of any lump should not be greater than approximately the wave length of sound in the gas. 'The wave length is defined as elf, where j' is the sine-wave frequency of the oscillation under consideration. -4complex xave can be synthesized from its fundamental frequency and ita harmonics. The sharper the pulse or the steeper the wave form, the greater is the contribution of the higher harmonics, and t.hu#, for these wave shapes, the number of lumps,required to represent a given pipe length is greater than for sine waves. 3. The inertance of the fluid in the pipes was computed from the mass of the fluid in the conduit. However, the fluid mass associated with a conduit or restriction extends somewhat beyond the confines of the element. That is, the effective lengt'h of an element from an inertance viewpoint is greater than its actual length by a n amount approximately equal to 1,7D, where D is the pipe diameter ( 2 1 , $3, 24). For most practical pipe lengths, this correction will he negligible. For orifices, however, the term is quite significant. 4. Propagation of pressure waves in only one direction was considered. If the diameter of pipes or process equipment is greater t h a n approximately '/4 !vave length, the radial mode of pressure fluctuations will also exist ( 1 6 ) . 5. If t,he absolute pressure or fluid composition changes markedly in a fluid flow system, the fluid density and volumetric flow rate will also change. The analogy between electric current and volumetric flow would then prove inconvenient. The set of equivalences based on mass flow rate analogous to electric current is just as valid as those based on volumetric flow and should be used in these cases. Experimental Results
Pulsation Damper. I n order t o verify expenmentdly the utility of this direct analog approach, tests were made on the two-tank pulsation damper discussed previously and compared with the results obtained with the analogous electrical network. Figure 3 shows the experimental apparatus. A small singlecylinder air compressor, driven by an electric motor through a variable-speed hydraulic drive, served as the source of pulsations. Pulsation frequencies to 20 cycles per second were produced by regulating the compresFor speed. Pressure pulsations a t the inlet and exit of the damper were picked up by two Statham pressure transducers, and the resulting voltage signals were amplified by Brush Universal Analyzers and recorded on a Brush twochannel oscillograph. Calibration of the pressure measurement system yielded a linear relation between static pressure and dis416
placement of the oscillograph pen from its zero position. Since the response of the combination of transducer and electronics t electrical, and pneumatic controller elements to obtain the over-all control system response and optimize its design. Figure 9 illustrates a compressor speed cont'rol system for maintaining a desired gas flov- rate to a process. The flow through the orifice meter was sensed by a differential pressure element, and a pneumatic signal transmitted to the cont,roller, which in turn, regulated the position of a t,hrottling valve in the steam supply to the steam-driven compressor. Information was available for estimating the transfer function of the compressor and its drive. The transfer function of the piping, pulsation damper, and orifice meter combination was required t o complete the dynamic analysis of the process equipment. Using the volumetric flovi equivalences and the scaling procedure, a direct' analog network iT-as set up to represent the fluidflow part of the system. A sine-wave generator was connected to the circuit a t the point representing t'he compressor discharge piping. The vokages a t the input and across the analog of the orifice meter TTere recorded on a double-beam oscilloscope and photographed. The results of the analog tests-the transfer function graphs-are plotted in Figure 10. The sine-wave frequency-response tests on the electrical circuit are directly analogous to obtaining the same information (openloop response) in the plant. In the latter case, pressure pulsat,ions would have to be introduced a t the compressor discharge by a pneumatic sine-wave generator of sufficient amplitude t o permit accurate recording of the flax and pressure pulsations. These tests are more costly and difficult to perform in the plant than on the electrical model, and t,hey do require the existence of the plant. Often, it is desirable t o predict the controllability of the plant prior to construction, and this obviously precludes measurement on plant systems. Once the necessary frequency-response information has been developed, it is possible, through the use of frequency-response theory, to select suitahle instrumentation for the control loop, in which the dynamics of the instrument loop is then properly matched t o the dynamics of the part of the process under consideration. Another application of fluid-flow analogs is in predicting whether the magnitude of pulsations at a given point will cause a serious error in flonmet,ering. Almost all flowmcters. and cert,ainly the conventional head-tvpe meters, are subject t o large errors when metering high amplitude pulsating flow ( 8 . 14. 20).
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 3
ANALOGS For head meters, the error is a function of the flow system geometrg, meter instrument lines, and the fluid properties, as well as the pulsation amplitude, frequency, and wave form. Thus, i t is not possible to work out completely general corrections to the reading of a head meter. Where the amplitude of flow pulsations is small compared to the mean flow rate (less than 5%), and the pulsation wave form approximates a sinusoid, the error is usually insignificant. In systems which have pulsating sources, it is often advantageous, therefore, to determine, in advance of construction or operation, whether flowmetering difficulties will occur, and what should be done about them. This can be studied quite readily with the electrical analog. Various design changes (in the piping, equipment, or pulsation source) to reduce pulsation amplitudes can be simulated quickly with the analog. Using the techniques described, the amplitude of flow pulsation in any part of the system can be determined by measuring the electrical current flow through its electrical counterpart, or pressure pulsations map be determined by measuring voltage a t appropriate points. Obviously, it is easier to twist dials and read meters in the electrical system than to make the corresponding changes and tests in the plant after construction. A study of this type is also useful in determining variations in reactant compositions where one pulsating stream is mixed with another stream prior to reaction. Often, going through the steps of setting up the equivalent circuit and approximating the values of the circuit elements is as far as one has to go. For if one has some familiarity with electrical circuits, it mav become evident on mere inspection of the
A
4
Liquid
Surge Tank PILOT
PLANT
LAYOUT
B WlRlNQ DIAGRAM 'OF
ELECTRICAL
ANALOG
equivalent circuit which elements are important in a given situation and which may be neglected, or whether a change in one part or element of the system can significantly alter the currentvoltage relationships in the circuit. Figure 11 illustrates the conceptual or qualitative aspect of the direct analog approach applied to the interpretation of measurements of rapid pressure transients. In pilot plant studies of a certain type of chemical process, it might be necessary to determine experimentally the maximum pressures which would be reached if a sudden deflagration of the materials in an autoclave reactor occurred. Information on the effect of the size and location of rupture disks and the arrangement of vent lines on the peak pressures reached in the reactor vessel might be required for the structural design of the full scale reactor. To measure these rapid pressure transients, it would be necessary to use a pressure transducer with high frequency response, such as the type mentioned in the pulsation damper tests. The measurement of transient or pulsating pressures involves factors which are not ordinarily encountered in steady-state measurements. The tubing connecting the pressure transducer to the source of the pressure, and the size and location of other elements in the piping system and equipment may profoundly affect the pressure sensed by the transducer. For example, if the pressure pickup had been located a t position 2 near the barricade, which would be the usual location for a pressure gage during steady-state tests, the pressure recorded would have been considerably less than the peak pressure reached in the autoclave. A schematic analog circuit of the autoclave instrument tubing, gages, and vent lines would quickly show why this could happen. When the rupture disk opens, the impedance to flow through the vent line is considerably less than through the instrument line to position 2. Flow through this path could drop the pressure in the autoclave before the signal rezches the transducer. Without making any detailed computations of equivalent electrical elements, it would be apparent from the circuit, nevertheless, that the impedance of the electrical transmission line between the autoclave and position 2 would influence the magnitude and phase of the pressure pulse measured 'at position 2. Even the capacitance of the Bourdon gage located adjacent to the pickup would affect the transient pressure recorded. Actually, an analog of this sytem was set up for demonstration purposes. Imposing a steep voltage pulse a t the input of the electrical circuit to simulate the rapid pressure rise in the autoclave gave a voltage trace at the transducer which bore out these predictions. Relocation of the pickup to position 1 gave the proper result. Scaling the Direct Analog
The selection of the numerical equivalences between the electrical components and variables and their fluid flow counterparte depends principally on two factors:
P H O T O G R A P H OF ELECTRICAL A N A L O G
Figure
8.
March 1955
Analog for analyzing pilot plant process stability difficulties
1. The frequencv, current, and voltage ranges which will be convenient to apply and to measure in the analog circuit 2. The ranges of the electrical components available
INDUSTRIAL AND ENGINEERING CHEMISTRY
419
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT While there is considerable flexibilitv in the selection of scale factors, some important limitations exist. Inductors and capacitors deviate significantly from their nominal values in henries and farads a t frequencies above approximately 5000 cycles per second The distributed inductance of wire-wound resistors also becomes a factor a t these frequencies. The amount of deviation is a function of the nominal value of the component and its design. Information on these deviations is generally obtainable from the manufacturer of the componentsfor example, General Radio Co., 275 RIassachusetts Ave., Cambridge, MasK., catalog N, 1954. The upper frequency limit for operation of the analog is set then by the properties of the components. The lower limit may be governed by the rate a t which charge, or voltage, leaks off a capacitor, More likely, the availability of low frequency function generators will set the lower frequency limit.
3. Select the factor, k?, between physical inertance in the proper units and electrical inductance in henries. This completely defines the scale re1at)ion between the impedance of all physical elements and the impedance of the electrical components in ohms. The impedance of an inductor, X L , is 2trfL, of a capacitor, X,,is l/zrrfC, and of a resistor, is R. Hence, the scale factor betn.een phyPical capacitance and electrical capacitance in farads, and betneen the physical resistance and electrical resistance in ohms, are determined by the previous choices of ICIand kz. 4. The scale factors are
Electrical
Fluid Flox
XL =
x, =
-4
Steam-Driven Reciprocating Compressor
5. Select the scale relation between electrical current in amperes and fluid-flow rate, or between voltage and pressure, The dependent relationship is accordingly defined by the factor selected and by the impedance proportionality factor discussed in 4. That is, if v = ksp> the riirrent proportionahty factor is
A-
i
, Differentiol D.^-I,I.e
=
v/Z = ( k l k J / k p ) p / Z F
= (klka/kn)q
Con el usion s ---,
o Reactor Pulsation
Figure
Orifice Meter
Damper
9. Flow
control system
The scaling procedure may be considered as comprised of tivo main steps. First, the proportionality factors betv;een time or its reciprocal, frequency, inductance, capacitance, and resistance in the electrical system, and time or frequency, and the corresponding elements in the fluid f l o ~ system must be set. Of the four scale factors required in this step, two are independently chosen; the other two are dependent factors. Because of the frequency limitations, it is generally best to choose the time scale factor first. Because inductors and capacitors are generally not available in as wide ranges as resistors, it is \Tell to specify next the relationship between inertance and electrical inductance in henries, or between fluid capacitance and electrical capacitance in farads. The second step involves selection of suitable currents and voltages for the analog circuit. One scale factor may be chosen arbitrarilv between volts and pressure units, or betxeen amperes and fluid flow rates; the other scale factor is then fixed. Here, it is necessary only to keep the electrical current low enough to avoid damage to the components, and the voltages high enough to obtain the requisite measurement accuracy with the available meters. A stepwise procedure for scaling the analog may now he set forth. 1. Determine the numerical values of all fluid-flow elements appearing in the schematic flow diagram from the relations shown in Figure 1 . Although any consistent set of units may be chosen, it is convenient t o express preEsure in (lb. force) per (sq. ft.), flow rate in (cu. ft.) per see. or (lb. mass) per see., all equipment dimensions in feet, density in (lb. mass) per (cu. ft.), velocity of sound in ft. per see., and the dimensional constant, g e , in (lb. mass)(ft.) per (lb. force) (see.*). The components in the volumetric flow analog, for example, are then evpressed in the following units: resistance in (lb. force per sq. ft.) per (cu. ft. per see.). inertance in (lb. force per sq. ft.)(sec.)per (cu. ft. per sec.); capabitance in cu ft. per (lb. force per sq. ft.). 2. Select the factor, k,, between physical time in seconds and analog time in seconds. The relation between frequencies will then be I/h.
420
The existence of analogies among the laws of nature makes it possible to obtain rapid engineering solutions to complex un-
I
I
X
I
9
180 0.01
Figure 10,
0.10 1 .o FREQUENCY, CYCLES/SEC.
10
Transfer function of fluid-flow components
steady-state fluid-flow problems in terms of eaeily assembled and tested, low cost electrical models. While the foundation of this approach is in the similarity of the mathematical equations defining the two systems, one can make direct substitution of electrical variables and components for a fluid-flow system in a purel7. systematic may, and obtain solutions to problems without concern for the mathematical formulation a t all. Applications n hich were discussed include the performance and application of pulsation dampers, the analysis of factors affecting the tendency of a system to oscillate, the development of frequency response data for fluid-flou systems to aid in instrumentation and control, the investigation of pulses that could cause flowmeter errors, and
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 3
ANALOGS
R U P T U R E DISK
O B S E R V A T I O N WINDOW P R E S S U R E GAUGE
PRESSURE PICKUP P O S I T I O N No. I
L
PRESSURE PICKUP P O S I T I O N No.2
= pipe length, ft. = capacitive reactance, ohms ( X c ) p = fluid capacitive r e a c t a n c e , ( l b .
z
X,
force/sq. ft.)/(cu. ft./sec.), or (lb. force/sq. ft.)/(lb. mass/sec.) X L = inductive reactance, ohms ( X L ) F= fluid inertive reactance, (lb. force/ sq. ft.)/(cu. f t . / s e c . ) , o r ( l b . force/sq. ft.)/(lb. mass/sec.) Z = electrical impedance, ohms ZF = fluid impedance, ( l b . f o r c e / s q . ft.)/(cu. ft./sec.) or (Ih. force/sq. ft.)/(lb. mass/sec.) p = fluid density, lb. mass/cu. ft. y = specific heat ratio, dimensionless
TO RECORDER BARRICADE
OIL
\HOT
Figure 1 1.
Transient pressure measurements
the transmission of pressure waves in process and measurement systems. Nomenclature
A
= cross-sectional area, sq. ft.
C
= velocity of sound, ft./sec.
C CF
= electrical capacitance, farads
fluid capacitance, cu. ft./(lb. force/sq. ft.), or lb. mass/(lb. force/sq. ft.) D = pipe diameter, ft. f = frequency of pulsation, cycles/sec. fo = cut-off frequency, cycles/sec. g, = conversion factor, 32.17 (lb. mass/lb. force)[(ft./sec.)/ sec.] Id.c. = av. electrical current, amp. i = instantaneous deviation from average current, amp. kl, 2 , a = arbitrary analog scale factors L = electrical inductance, henries LF = fluid inertance, (lb. force/sq. ft.)(sec.)/(cu. ft./sec.), or (lb. force/sq. ft.)(sec.)/lb. mass/sec.) ‘P = av. pressure, lb. force/sq. ft. p = instantaneous deviation from average pressure, lb. force/sq. f t . Q = av. flow rate, cu. ft./sec. q = instantaneous deviation from average flow rate, cu. ft./sec. R = electrical resistance, ohms RF = fluid resistance, (lb. force/sq. ft.)/(cu. ft./sec.) or (lb. force/sq. ft.)/(lb. mass/sec.) 1 = time, sec. V = volume, cu. f t . Vd.c, = av. electrical voltage, volts = instantaneous deviation from average voltage, volts 2, = av. flow rate, lb. mass/sec. W w = instantaneous deviation from average flow rate, lb. mass/sec. =
-
Literature Cited (1) Baird, R. C., and Bechtold, I. C., Trans. Am. SOC. Mech. Engrs., 74, 1381-7 (1952). . . (2) Binder, R. C., and Hall, A. S., J. A p p l . Mechanics, 14, A183-7 (1947). (3) Chemical Engineers’ Handbook, 3rd ed., Sect. 6, McGraw-Hill Book Co., New York, 1950.
(4) Chestnut, H., and Mayer, R. W., “Servomechanisms and Regulating System Designs,” Wiley, New York, 1951. (5) Chilton, E. G., and Handley, L. R., Trans. Am. SOC.Mech. Engrs., 74, 931-43 (1952). (6) Firestone, F. A., J . Acoust. SOC.Amer., 4 , 249-67 (1933). (7) Gardner, M. F., and Barnes, J. L., “Transients in Linear Systems,” Wiley, New York, 1952. (8) Hall, N. A., Trans. Am. SOC.Mech. Engrs., 74, 925-30 (1952). (9) Hirschorn, M., Chem. Eng., 56, 94-5 (September 1949). (10) Hughes, R. R., Handlos, A. E., Wans, H. D., and Maycook,
R. L., Heat Transfer and Fluid Mechanics Institute, Paper No. 11, Stanford Univ. Press, Stanford, Calif., 1953. (11) Ingard, K. U., Ph.D. thesis in Physics, Massachusetts Institute of Technology, 1950. (12) Korn, G. A., and Korn, T. NI., “Electronic Analog Computers,” McGraw-Hill Book Co.. New York. 1952. (13) Le Corbeiller, P., and Yeung, Y. W., J . Acoust. Soc. Amer., 6 , 643-8 (1952). (14) Lindahl, E. J., Trans. Am. SOC.Mech. Engrs., 68, 8.83-94 (1946). (15) McMaster, R. C., Merrill, R. L., and List, B. H., Product Eng., 24, 184-95 (1953). (16) Morse, P. M., “Vibration and Sound,” 2nd ed., McGraw-Hill Book Co., New York, 1948. (17) Murphy, E. F., Petroleum Refiner, 28, 151-6 (1949). (18) Oldenbourg, R. C., and Sartorius, H., “Dynamics of Automatic Control,” Am. Soc. Mech. Engrs., New York, 1948. (19) Olson, H. F., “Dynamical Analogies,” Van Nostrand, New York, 1943. (20) Oppenheim, A. K., and Chilton, E. G., A n . SOC.Mech. Engrs., Paper No. 53-A-157, December 1953. (21) Paynter, H. M., Trans. Am. SOC.Civil Engrs., 118, 962-89 (1953). (22) Sunstein, D. E., Electronics, 22, 100-3 (1949). (23) Thurston, G. B., J . Acoust. Soc. Amer., 24, 653-6 (1952). (24) Thurston, G. B., and Martin, C. E.. Ibid., 25, 26-31 (1953). RECEIYED for review September 28, 1954.
ACCEPTED January 26, 1955.
END OF ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT SECTION
March 1955
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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