- Email: [email protected]
Automatic control systems analysis for multi-loop fluid dynamics facility with prominent distributed parameters
Automatic Control Systems Analysis for Multi-loop Fluid Dynamics Facility with Prominent Distributed Parameters J. J. LANDY and G. J. FIEDLER Introduction Plant description- The plant is the Transonic Propulsion Wind Tunnel (PWT), a part of the Arnold Engineering Development Centre built at Tullahoma, Tennessee, for the U.S. Air Force. The PWT is designed to test full-scale engines, missiles and rockets under flight environmental conditions at speeds from Mach No. 0·8 to 1·6 and at altitudes from sea level to 120,000 ft . In operation the PWT is interconnected with the Engine Test Facility (ETF) and the Engine Test Facility Addition (ETFA) for support purposes such as the scavenging of products of combustion from the test engine and the supply of conditioned make-up air. The three plants then operate as a single functional unit and must be analysed as a single dynamic entity for purposes of control system design. The size and complexity of the integrated PWT, ETF and ET FA complex is evident from Figure 1. Performance criteria- The automatic control system must hold the two controlled variables within accuracy limits and set-point ranges for both transient and steady-state testing as follows: (I) stagnation pressure within ± 0·5 per cent from 34 to 3,950 Ib. /ft. 2 abs.; (2) Mach number within ± 0·25 per cent from 0·8 to 1·6. Systems analysis- In order to make the systems analysis of this complex plant technically feasible and economical , five major simplifications are validated : (I) linearizing techniques, (2) critical flow, (3) separation of pressure and temperature systems, (4) lumped-parameter treatment of distributedparameter effects, (5) compressor and motor drive dynamics. The first two simplifications use existing engineering knowledge and are not treated in detail owing to space limitations . It is believed that simplifications 3, 4 and 5 are not previously treated in the literature; they are, therefore, developed more in detail. The use of all valid simplifications reduces the plant to a 14th-order system having 25 feedback loops as shown on Figure 3. The transfer functions used on the simplified plant Figure 3 are given; however, due to space limitations, only the principal transfer functions are developed in the paper. Synthesis and design- The entire control loop is synthesized and connected into the uncontrolled simplified plant simulation on an analogue computer to complete the systems analysis and satisfy performance criteria.
Sil1lplijicatiolls The system-wide aspects of the complex plant processes are analysed for the purpose of effecting all possible simplifications to facilitate the analysis and design work . The principal simplifications are discussed in the following paragraphs. Linearizing techniqlles-The development of system transfer functions employs the theory of small perturbations l using Laplace transformations l effecting substantial mathematical simplifications. Critical flow- Certain points in the air-flow circuit reach critical flow or sonic velocity 2. Critical flow permits only unilateral transmission of pressure and temperature variations, thereby preventing feedback effects. This permits the isolation of parts of the aerodynamic circuit with resulting simplification of the analysis. Separation of pressure and temperature rariafiolls-Disturbances due to extreme engine throttle bursts and noise cause excursions in both temperature and pressure from nominal values. The temperature and pressure changes are ampl ified by the compression process which determines the amount and mode of transmission of these changes. The magnitude of the potential simplifications realized from the separation of these effects can be seen by study of Figure 1. All of the pressuretemperature loops can be severed and all temperature transfer functions can be deleted from the pressure system diagram. In order to effect a valid separation two conditions must be proved: (a) The wave-component of temperature variation which is inseparable from the pressure variation must be very small compared with the convective-temperature variation, (b) the wave-component of temperature variation must also be greatly out of phase with the convective-component of temperature variation. Since these factors are determined by the compression process in this facility the compressor transfer functions must be developed and evaluated. Compressor dynamics-The compressor dynamics are defined by the application of the principles of fluid mechanics and turbo-machinery.3 The energy equation for gas flow is:
Systems Analysis This paper involves several branches of engineering, each having long-established upper and lower case symbols for the variables treated. This precludes strict conformity with the conventional use of lower and upper case letters for variables before and after Laplace transformation. Owing to the large number of variables used, symbols are defined with each development, for clarity.
Substituting the enthalpy equation,
II +Pl Ul+V l 2+Z l + Q = 1
J
2gJ
J
h =
II2
+PZV2+VZ2+Z2+WS
lI+ pv = J
J
C T
2gJ
J
J
(I)
(2)
and the total temperature equation, (3)
288
1389
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
and deleting the Q term, since the processes are essentially adiabatic, and the Z term because there is small difference in input and output elevations, results in:
and,
dTlOa = [1_T,oa-T,i(K . _T,oa-To(K -K)d T'i ry _K)]dT I 11 W q I W
(4) The negative sign is used for Ws to account for the compressor process where work is done on the fluid. The adiabatic efficiency or temperature rise ratio is: Wideal E,
=
Wactual
Cp(T,o - T,i ) T ,o - T'i = C (T,Oa - T,rl = T ,Oll - To
(17)
Equations 16 and 17 are expressed in terms of total temperature increments . Since the separation of temperature and pressure variations is sought, it is also necessary to express dw and dT,oa in terms of convective or wave-transmitted components. This is done by means of equations 18 and 19, which show that the total temperature increment is the sum of the convective component and the wave-transmitted component defined by the two right-hand terms in the equations. These equations also define the magnitude of the wave-transmitted components.
(5)
The use of the isentropic relationship,
(6)
PIVI Y = poroY
and the equation of state, P~' =
(7)
RT
results in:
dT,Oa
=
and,
dT'i
The modified Euler equation for an axial-flow compressor 3 is: QV,Va
= -J g-
(
) ,tanal-tana,-
dw = - [Y-I
(9)
y
=
~ Y+ I ~]dPo-
Po
Y
Kry
[(I T'iKry + Kry )W] dTi
+ [(2+ Kry)W] dl/ + [(y-I + Kry Y)II'] dpi
Combining equation 4 and equation 9 and substitution of the isentropic factor Y from equation 8, plus the relation, C p = [y/(y - 1)]( R/J) gives:
g[y/(y-I)]RYT'i QV,,( ) V2 = V tan a l-tan a 2 = '7
(18)
y-I T 'i (19) dTi +-- -dp" Y Pi Expressed in terms of convectively transmitted components equation 16 becomes:
(8)
E
y-I T ,o
dTo+---dpo y Po
Kry
11
(20)
yYPiKry
Linearizing and applying the Laplace transformation gives:
(10)
(21)
Equations 8 and 10 together with equations 11 and 12 govern compressor behaviour.
Expressed in terms of convectively transmitted components equation 17 becomes:
I
I
Compression ratio, ,.p = PO/Pi = (Y + I)y/(y-I) Output temperature, T ,oa = T,/I
+ Y/E,)
(11) (12)
The pressure-rise and temperature-rise ratios are characteristic of particular operating conditions for specific compressor units. Curves of '7 and E, caslIs volume flow with speed as a parameter are available and two dimensionless quantities are defined which make use of the slope at the operating point. These are: . 1:-'7 q/I/ Pressure-rise factor, K'I ~ - -, - /- -
.
Temperature-rise factor, K,
cq
1/
(13)
'7
CE q/I/
(22)
(14)
A .,-/' - cq 1/ E,
Linearizing and transforming gives:
To(s) = K 1I6 P/S) + K40 Ti(s) + K44 W(s) + K4 1N(s) /1l
Applying linearizing techniques to the foregoing and substituting the mass flow relationship, W
Symbols for the above developments are :
Cp E
(15)
= pq
gives:
g
Iz J 11
+ {[(y- 1)( : + 1) + YYK lW} dpi + [(2 + K~h\'] dl/ (16) YPi YKry
K~
Pi Po
11
289 1\)
1390
Specific heat, constant pressure (B.Th.U. /lb. /deg.) Energy transfer to fluid (B .Th .U ./lb.) Gravitional constant (ft. /sec 2) Static enthalpy (B.Th .U. /lb.) Mechanical heat equivalent (ft.Lb ./ B.Th.U.) Impeller speed (rev/min) Total inlet pressure (Ib./ft. 2) Total outlet pressure (Ib ./ft. 2)
(23)
COMPRESSOR & MOTOR DRIVE SYSTEM N
Vo F
Vo COMPRESSOR is MOTOR DRIVE SYSTEM
"'17
F
Wm
MAIN COMPRESSOR
"'20
"' 18 "' 19 VALVE 30
: Pp Wb
L -__________
~ "'22 ~T~t~5i--------------
L-______________~ "'22 ~"~t-----------------------
Figure 1.
290
1391
Plant trallsfer
Volume flow, inlet (ft. 3/sec) Volume flow, outlet (ft. 3/sec) Heat transfer to fluid (B.Th .V ./lb.) Q Gas constant (ft.tR) R Time (sec) Static temperature, inlet COR) TI Static temperature, outlet (OR) T2 Input temperature, convective COR) Ti Output temperature, convective COR) To Total temperature inlet, actual (OR) Tti Total temperature outlet, ideal (OR) T lo T IOa Total temperature outlet, actual CO R) Internal energy, inlet (B .Th.V. /lb .) III Internal energy, outlet (B.Th.V. /lb.) 112 Specific volume, inlet (ft. 3 jlb .) VI
V2 VI V2 Va VI
ql q2
-~~f-
~
Pp~
w Ws y ZI Z2 a l
az y
.----fKDl-
w~ ~i-~_~lTT~ttpp~~;~ ~ Ttp ~ ~f- I Ttae; . ~ST; Tfae e-sTp i--l:i}-.4j+l--""-I K 40 ~<5/)...-+----"'~e- ae
-\-----4- ~
~
K
f-<
IWh~ I--"-~I-MI XING
PL ENUM DUCT
~V. 70
T
fa
K69
N
~H
I~~ ~ K44f~M
COMPRESSOR
DUCT
~aeK 70 Tta
K
69
diagram ; original plallt
291
1392
Specific volume, outlet (ft. 3/lb.) Inlet velocity (ft./sec) Outlet velocity (ft./sec) Axial velocity (ft./sec) Peripheral velocity (ft./sec) Weight flow (Ib./sec) Total shaft work (ft.lb./lb.) Isentropic factor (numeric) Potential energy, elevation, inlet (ft .) Potential energy, elevation, outlet (ft.) Angle, blade profile tangent and entrance velocity vector (degrees) Angle, blade profile tangent and exit velocity vector (degrees) Ratio specific heats (numeric)
J . J . LANDY AND G. J. FIEDLER
"I 'r} T
Q
Temperature rise ratio (numeric) Pressure rise ratio (numeric) Time constant (seconds) Work-done factor (numeric)
sentation is made valid by the proper adjustment of controlled system cross-over frequency . A suitable frequency criterion is developed by using the mathematical analogy between the equations of fluid flow system and the equations of the electrical transmission line 4 • This criterion is developed in reference 4 as:
Equations 18 and 19 are developed by use of acoustic theory relating flow velocity and pressure. These equations are analogous to the electric transmission line equations 4 •
cp 0'\ =
_ J-. ow = Ag ot
ew
L
ow et
(24)
ep
et
The sonic velocity, tJ-c, is developed follows:
In
tJ-c
(30)
= 9~
The pertinent facility values when substituted in this equation result in a cross-over frequency Wc ~ 1 rad./sec. The equations developed in reference 4 are used to calculate data for the curves of Figure 2, which compares the response
(25)
C-
0'\
7T
Wc
terms of Land C as
~ :1_-_"_~=----
(26)
~-20-
T- ~ •
1
-
z
Also,
j
M =
~
=
-4 0 ~ --
~(RTI) l-
Ap
tJ-c
,
(27)
yg
For large Mach numbers the relationship for the isentropic condition is derived from aerodynamic relationshi ps 2 as: i --2 d T.v - y [ 1+ (y-I ) M ]- = y-I
TI
[l+yM]--yM2 dp 2 dw (28) p
IV
o ClI
Equation 28 reduces to:
Figure 2,
dp p
for
M<%; 1
a·os
0·1
0 ·5 I 5 FREQuENCY (RADIANS/SEC.)
-10
-
- ~~~-180 so
Comparisoll o/Iumped alld distribllled trails/er /ullctiolls
(29)
For the foregoing: Cross-section area duct (ft.2) Temperature, total e R) TII' Temperature, wave-transmitted (eR) M Mach number (numeric) ,\ Duct length (ft.) p Fluid density (lb./ft. 3) tJ-c Sound velocity (ft./sec) A
Tt
Other symbols are as previously defined. Calculations made by inserting the pertinent facility values in equations 16, 17, 18 and 19 show that 95 per cent of the total temperature variation is transmitted convectively and only 5 per cent by wave transmission. This establishes the first of the two conditions previously defined, for the separation of temperature and pressure effects. Compliance with the second condition for the separation is ascertained by the calculation of propagation rates. The pressure and wave-transmitted temperature variation propagation rates are calculated by application of equation 26 as 1,140 to 1,250 ft. /sec. The velocity of the fluid which carries the convective temperature component is calculated by application of the aerodynamic continuity equation as 70 to 200 ft .jsec. Consequently, it is proved that the wave-transmitted component of temperature variation is both small and greatly out of phase with the convective component, thereby satisfying in full the criteria for the separation of the temperature and pressure systems. Validatiol/ of lumped-parameter representation of distributed elemel/ts-A major simplification is realized if lumped-parameter representation is used for this system. Such repre-
of the lumped and distributed elements over the pertinent frequency spectrum. Examination of Figure 2 shows that the distributed gain oscillates and the phase angle deteriorates rapidly at frequencies greater than W = 1·5 rad. /sec. The mcillation of the gain is caused by the mis-match of the surge impedance and actual duct impedance. The point in the freqlJency spectrum where oscillations begin is determined by the value of the propagation constant. From these curves it is evident that it is not practical to attempt to operate a control system beyond the cross-over point. Thus, this technique at once defines the possible operating region and validates lumpedparameter description of the distributed elements. COlllpressor al/d 1110 tor dril'e d)'l/umics- The torsional vibrational properties of this system, which comprises one compressor, four drive motors and two brakes, are studied to determine the frequency and amplitude of each of the six modes of oscillation. The only frequencies resonant with exciting frequenci.;:s were found when the induction drive motors were operating in an abnormal unbalanced condition in their secondary circuits. This investigation also revealed that it is possible to represent this distributed system with acceptable error as the second-order lumped systems shown on Figure 1. A typical step function response of the lumped main compressor motor drive system is given by : W
=
dB = 0·026 e- O. 261 sin 1·68t dt
(31)
This shows that the oscillation frequency W = 1·68 rad .jsec is only slightly higher than the controlled cross-over frequency of 1·0 rad ./sec. An investigation revealed that the system is not excited at w = 1·68 rad. /sec. The effects of power line voltage and frequency variations are negligibly small with re3pect to the effects of other system inputs. This permits the deletion of
292
1393
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
these transfer functions. Space limitations preclude the development of this material in this paper. Equivalent simplified systems analysis The application of all five areas of simplification discussed or developed previously for Figure I, together with the deletion of ambient and other negligible effects, reduces the system to that shown on Figure 3. In connection with this work all transfer functions shown on Figure 1 were developed. The important and more interesting transfer functions on the simplified plant Figure 3 are developed in this paper and the remainder are merely defined because of space limitations. Air-coll1pressibility ejfects- This effect is present in plant components having large volumes such as the ducting sections. This development is for the case where spatial effects can be neglected and the process may be considered adiabatic and isentropic. The thermodynamic equation of state for a perfect gas is: W
=
Pi V RTti
Differentiation of this expression and substitution of the isentropic relationship gives: (33) where: C = v/(yRTr;)
Linearizing and transforming gives: W(s) = sCPi(s)
which represents the lumped transfer function of all ducting sections. Air inertia effects-For a flow path of constant cross-sectional area A and length 10 carrying a flowing gas at a rate of w Ib./sec, and being accelerated at a rate of dw/dt Ib./sec 2, the fluid velocity is determined by a combination of the equation of state and the relationships w = qp and V = q/ A. The time derivative of the result yields the acceleration, thus:
(32)
The rate-of-change of weight in a volume is the difference between the in-flow and out-flow, or
(34)
dV dt
RTtidw APi dt
(35)
The air mass, M , is given by: (36)
Figure 3.
Plant transfer diagram; simplified plant
293
1394
J. J. LANDY AND G. J. FIEDLER
Therefore, the accelerating force equated to the pressure drop causing it gives:
Equation 43 can now be re-written as 10RTIi (P'-Po) = - - w2 I 2A2pi
(37)
Solving equation 46 for w, taking its total differential and transforming gives:
Linearizing and transforming yields:
l pes) = A~ sW(s)
(47)
(38)
where:
where:
_ (2pi-PO) K31 - 2(Pi2- PiPo)l(IOR/2A2)' (T/i) '
10 = A"Ir mertance Ag
[2A2(Pi 2- PiPO)]t 2(10R)±(TIi )f
Nozzle and cOl/trol valve flow characteristics-Research conducted at the University of Michigan and monitored by Sverdrup & Parcel indicates that the air flow through a butterfly valve can be adequately described by the nozzle equation 3 • The general expression is:
The coolers have no volumetric storage capacity on the air side. Pressure ratio alld scoop flow-These transfer functions are defined as follows:
where
K48 = l/p p
(40)
K49 = -Pn/P/ Kso = A(ygRT/i)!Ma/RTIi' where Ma = Mach number
The total differential of equation 39 is: =
OW
OW
G
0Pi
Compressor transfer /UI/CtiOIlS-K23 , K24 and K 2S are developed in this paper and given in equation 20. S),stem equations-The array of equations for the simplified plant is developed from Figure 3. The solution of the uncontrolled system for a typical principal control ratio results in an expression of the form:
OW
-;;-A dA + -;c- dpi +"----T dT1i 0
li
OW [oF}.
(41)
Applying the linearizing and transforming techniques previously described gives:
Ts6 + UsS + VS4 + Ws 3 + Xs 2 + YSI + Z\O Ps/A30(S) = A S8+ B S7 + C S6+ DsS + £54+ F53+ GS2+ HSl + isO
The synthesis of the control system using the analogue computer results in a controller transfer function of the form:
where:
G (s)
K
K
(48)
Pi K33 = - 2(Pi2- PiPo)1(10R/2A2)t(TIi)t
(39)
dw
(46)
18
c
.[1'43(0!)J.4.1_ 71]} 1.71 (0!)1' p. p.
=2'055A!F+ T·t l'\ p, tl-
19 -
K 20 =
I
p,
,p,
GsCs) = (I
p,
Cooler pressure drop-The fractional drop across the cooler depends directly upon the dynamic head' q' of air flowing over the cooler. From experimental data on this cooler it was determined that: 10
(43)
q = _pV2
(44)
where: I 2
Substitution of the mass flow equation 3 and the equation of state 3 in equation 44 gives: RT· q = __ 1_, w2 2A2pi
~~s)
(50)
This together with the second-order valve system shown on Figure 3 results in the 14th-order simplified system.
2·055Api FII 2(T/i)I.S
q
(49)
and a sensing system of the form:
2FA po
LIp
K/I +72 S)(I +74 5) s(1 +7IS)(1 +735)
I
po)I.43 (pO)I'71] 2'055A . [ 1·43 ( ~ -1,71 ~ T/it
=
(45)
Control System Synthesis The uncontrolled plant equations are simulated on the analogue computer using standard procedures. The accuracy of simulation is checked by comparison of analytically-developed uncontrolled system responses with the same responses obtained on the analogue computer. This operation also checks operation of the computer. Using established frequencyresponse and root-loci methods of servo design the characteristics of the control components are established, simulated and connected into the uncontrolled plant circuitry. From a 'map' of the large number of operating points a relatively few points are carefully selected to cover adequately the plant operating envelope. Plant controller adjustments are made on the analogue computer until the required transient and steady-state plant performances are obtained for each selected operating point. To insure compliance with the lumpedparameter frequency criteria previously developed it is necessary
294
1395
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
to monitor continuously cross-over frequency during the synthesis. This is done on the computer by determining the values of each of the pertinent control ratios such as Ps/Pse(jw) at an excitation frequency of 1·0 rad. /sec. The integration gains, network time constants and compensation inflexibilities inherent in standard pneumatic and electronic controllers preclude their use in this application. The control system dynamics required to meet performance criteria are attainable only by specially designed controllers incorporating increased integration gain and flexible, wider-range lead-Iag networks.
capabilities for : (I) Testing larger engines. (2) Expanded transient testing. These capabilities probably would have remained unknown without the systems analysis because of the cost and difficulty of using the plant as an analogue.
The authors gratefully acknowledge the counsel and assistance of R. J. Kochenburger, V. B. Haas, Jr., V. E. Scottron and W. B. Heinz on the work relating to compressor dynamics and the rates of propagation of temperature and pressure variations. References
Conclusions
1
This paper demonstrates that : (a) An integrated systems analysis based on valid simplifications makes the design of electronic control systems for the largest dynamic processes technically feasible and economical. (b) The techniques used greatly facilitate the mathematical work compared with using partial differential and transcendental equations for time and space variables. (c) The analysis discloses the existence of unknown facility
2 3 4
CHESTNUT, H. and MAYER, R. W. Servomechallisms and Regulating System Design, Vol. 2, pp. 215-218. 1957. New York; McGraw-HiIl LIEPMANN, H. W. and PUCKETT, A. E. Introduction to Aerodynamics of a Compressible Fluid. 1947. New York; Wiley SHEPHERD, D. G. Principles of Turbomachinery, pp. 68, 70, 101 102. 1956. New York; Macmillan STALZER, T. R. and FIEDLER, G. J. Criteria for validity of lumpedparameter representation of ducting flow characteristics. Trans. Amer. Soc. mech. Engrs Paper No. 56-IRD- 21 May (1957)
Summary
An integrated mathematical systems analysis is used as the basis for the design of a high-performance automatic pressure and flow control system for the world's largest supersonic wind tunnel complex. The facility processes are aerodynamic and thermodynamic and consume 470 MW of power. The processes interact to the extent that the plant can be described mathematically as a 107th-order system having about 100 feedback loops. The analysis of this
complex plant is further complicated by the fact that most of the components have non-linear characteristics and there are prominent distributed-parameter effects. Valid simplifications are developed which reduce the plant to a 14th-order system having 25 feedback loops. This makes the control system analysis, synthesis and design technically feasible and economical using an analogue computer.
Sommaire
On utilise le calcul mathematique pour definir initialement les regulations de pression et de debit it hautes performance necessaires a la souffierie supersonique la plus importante du monde. Cet ensemble releve de I'aerodynamique et de la thermodynamique et consomme 470 megawatts. Les phenomenes reagissent les uns sur les autres it tel point que l'installation peut ctre consideree comme un systeme mathematique du 107eme ordre comprenant environ 100 boucles de contre reaction. L'etude de cet ensemble complexe est, en
plus, compliquee par le fait que beaucoup des elements constituants presentent des caracteristiques non lineaires et qu'il y a interaction des differents parametres. Certaines simplifications valables sont envisagees ensuite, ElIes reduisent le degre du systeme au 14emc ordre avec 25 boucles de contre-reaction. Ceci permet de mener economiquement I'etude de la regulation et sa definition technique it [,aide d'un calculateur analogique.
Zusammenfassung Eine umfassende mathematische Systemanalyse wird als Grundlage fUr die Konstruktion eines selbsttatigen Hochleistungsregelsystems fur Druck and Stromung wm Betrieb des grof3ten UberschallWindkanals verwendet. Dabei handelt es sich um aerodynamische und thermodynamische Vorgange, fUr die eine Leistung von 470 MW gebraucht wird. Die Vorgange beeinflussen sich in einem solchen Mal.le, dal.l man die Anlage bei etwa 100 Regelkreisen mathematisch als System 107. Ordnung beschreiben mul.l . Die Analyse der
gesamten Anlage wird weiter durch die Tatsache kompliziert, daf.l die meisten GJieder nichtJineare Kennlinien aufweisen und verteilte Parameter bestimmend sind . Durch wlassige Vereinfachungen wird die Anlage auf ein System 14. Ordnung mit 25 Regelkreisen reduziert, wodurch Analyse, Synthese und Konstruktion des Regelsystems mit Hilfe eines Analogrechners technisch ausfUhrbar und wirtschaftlich vertretbar werden.
295
1396
Sil1lplijicatiolls The system-wide aspects of the complex plant processes are analysed for the purpose of effecting all possible simplifications to facilitate the analysis and design work . The principal simplifications are discussed in the following paragraphs. Linearizing techniqlles-The development of system transfer functions employs the theory of small perturbations l using Laplace transformations l effecting substantial mathematical simplifications. Critical flow- Certain points in the air-flow circuit reach critical flow or sonic velocity 2. Critical flow permits only unilateral transmission of pressure and temperature variations, thereby preventing feedback effects. This permits the isolation of parts of the aerodynamic circuit with resulting simplification of the analysis. Separation of pressure and temperature rariafiolls-Disturbances due to extreme engine throttle bursts and noise cause excursions in both temperature and pressure from nominal values. The temperature and pressure changes are ampl ified by the compression process which determines the amount and mode of transmission of these changes. The magnitude of the potential simplifications realized from the separation of these effects can be seen by study of Figure 1. All of the pressuretemperature loops can be severed and all temperature transfer functions can be deleted from the pressure system diagram. In order to effect a valid separation two conditions must be proved: (a) The wave-component of temperature variation which is inseparable from the pressure variation must be very small compared with the convective-temperature variation, (b) the wave-component of temperature variation must also be greatly out of phase with the convective-component of temperature variation. Since these factors are determined by the compression process in this facility the compressor transfer functions must be developed and evaluated. Compressor dynamics-The compressor dynamics are defined by the application of the principles of fluid mechanics and turbo-machinery.3 The energy equation for gas flow is:
Systems Analysis This paper involves several branches of engineering, each having long-established upper and lower case symbols for the variables treated. This precludes strict conformity with the conventional use of lower and upper case letters for variables before and after Laplace transformation. Owing to the large number of variables used, symbols are defined with each development, for clarity.
Substituting the enthalpy equation,
II +Pl Ul+V l 2+Z l + Q = 1
J
2gJ
J
h =
II2
+PZV2+VZ2+Z2+WS
lI+ pv = J
J
C T
2gJ
J
J
(I)
(2)
and the total temperature equation, (3)
288
1389
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
and deleting the Q term, since the processes are essentially adiabatic, and the Z term because there is small difference in input and output elevations, results in:
and,
dTlOa = [1_T,oa-T,i(K . _T,oa-To(K -K)d T'i ry _K)]dT I 11 W q I W
(4) The negative sign is used for Ws to account for the compressor process where work is done on the fluid. The adiabatic efficiency or temperature rise ratio is: Wideal E,
=
Wactual
Cp(T,o - T,i ) T ,o - T'i = C (T,Oa - T,rl = T ,Oll - To
(17)
Equations 16 and 17 are expressed in terms of total temperature increments . Since the separation of temperature and pressure variations is sought, it is also necessary to express dw and dT,oa in terms of convective or wave-transmitted components. This is done by means of equations 18 and 19, which show that the total temperature increment is the sum of the convective component and the wave-transmitted component defined by the two right-hand terms in the equations. These equations also define the magnitude of the wave-transmitted components.
(5)
The use of the isentropic relationship,
(6)
PIVI Y = poroY
and the equation of state, P~' =
(7)
RT
results in:
dT,Oa
=
and,
dT'i
The modified Euler equation for an axial-flow compressor 3 is: QV,Va
= -J g-
(
) ,tanal-tana,-
dw = - [Y-I
(9)
y
=
~ Y+ I ~]dPo-
Po
Y
Kry
[(I T'iKry + Kry )W] dTi
+ [(2+ Kry)W] dl/ + [(y-I + Kry Y)II'] dpi
Combining equation 4 and equation 9 and substitution of the isentropic factor Y from equation 8, plus the relation, C p = [y/(y - 1)]( R/J) gives:
g[y/(y-I)]RYT'i QV,,( ) V2 = V tan a l-tan a 2 = '7
(18)
y-I T 'i (19) dTi +-- -dp" Y Pi Expressed in terms of convectively transmitted components equation 16 becomes:
(8)
E
y-I T ,o
dTo+---dpo y Po
Kry
11
(20)
yYPiKry
Linearizing and applying the Laplace transformation gives:
(10)
(21)
Equations 8 and 10 together with equations 11 and 12 govern compressor behaviour.
Expressed in terms of convectively transmitted components equation 17 becomes:
I
I
Compression ratio, ,.p = PO/Pi = (Y + I)y/(y-I) Output temperature, T ,oa = T,/I
+ Y/E,)
(11) (12)
The pressure-rise and temperature-rise ratios are characteristic of particular operating conditions for specific compressor units. Curves of '7 and E, caslIs volume flow with speed as a parameter are available and two dimensionless quantities are defined which make use of the slope at the operating point. These are: . 1:-'7 q/I/ Pressure-rise factor, K'I ~ - -, - /- -
.
Temperature-rise factor, K,
cq
1/
(13)
'7
CE q/I/
(22)
(14)
A .,-/' - cq 1/ E,
Linearizing and transforming gives:
To(s) = K 1I6 P/S) + K40 Ti(s) + K44 W(s) + K4 1N(s) /1l
Applying linearizing techniques to the foregoing and substituting the mass flow relationship, W
Symbols for the above developments are :
Cp E
(15)
= pq
gives:
g
Iz J 11
+ {[(y- 1)( : + 1) + YYK lW} dpi + [(2 + K~h\'] dl/ (16) YPi YKry
K~
Pi Po
11
289 1\)
1390
Specific heat, constant pressure (B.Th.U. /lb. /deg.) Energy transfer to fluid (B .Th .U ./lb.) Gravitional constant (ft. /sec 2) Static enthalpy (B.Th .U. /lb.) Mechanical heat equivalent (ft.Lb ./ B.Th.U.) Impeller speed (rev/min) Total inlet pressure (Ib./ft. 2) Total outlet pressure (Ib ./ft. 2)
(23)
COMPRESSOR & MOTOR DRIVE SYSTEM N
Vo F
Vo COMPRESSOR is MOTOR DRIVE SYSTEM
"'17
F
Wm
MAIN COMPRESSOR
"'20
"' 18 "' 19 VALVE 30
: Pp Wb
L -__________
~ "'22 ~T~t~5i--------------
L-______________~ "'22 ~"~t-----------------------
Figure 1.
290
1391
Plant trallsfer
Volume flow, inlet (ft. 3/sec) Volume flow, outlet (ft. 3/sec) Heat transfer to fluid (B.Th .V ./lb.) Q Gas constant (ft.tR) R Time (sec) Static temperature, inlet COR) TI Static temperature, outlet (OR) T2 Input temperature, convective COR) Ti Output temperature, convective COR) To Total temperature inlet, actual (OR) Tti Total temperature outlet, ideal (OR) T lo T IOa Total temperature outlet, actual CO R) Internal energy, inlet (B .Th.V. /lb .) III Internal energy, outlet (B.Th.V. /lb.) 112 Specific volume, inlet (ft. 3 jlb .) VI
V2 VI V2 Va VI
ql q2
-~~f-
~
Pp~
w Ws y ZI Z2 a l
az y
.----fKDl-
w~ ~i-~_~lTT~ttpp~~;~ ~ Ttp ~ ~f- I Ttae; . ~ST; Tfae e-sTp i--l:i}-.4j+l--""-I K 40 ~<5/)...-+----"'~e- ae
-\-----4- ~
~
K
f-<
IWh~ I--"-~I-MI XING
PL ENUM DUCT
~V. 70
T
fa
K69
N
~H
I~~ ~ K44f~M
COMPRESSOR
DUCT
~aeK 70 Tta
K
69
diagram ; original plallt
291
1392
Specific volume, outlet (ft. 3/lb.) Inlet velocity (ft./sec) Outlet velocity (ft./sec) Axial velocity (ft./sec) Peripheral velocity (ft./sec) Weight flow (Ib./sec) Total shaft work (ft.lb./lb.) Isentropic factor (numeric) Potential energy, elevation, inlet (ft .) Potential energy, elevation, outlet (ft.) Angle, blade profile tangent and entrance velocity vector (degrees) Angle, blade profile tangent and exit velocity vector (degrees) Ratio specific heats (numeric)
J . J . LANDY AND G. J. FIEDLER
"I 'r} T
Q
Temperature rise ratio (numeric) Pressure rise ratio (numeric) Time constant (seconds) Work-done factor (numeric)
sentation is made valid by the proper adjustment of controlled system cross-over frequency . A suitable frequency criterion is developed by using the mathematical analogy between the equations of fluid flow system and the equations of the electrical transmission line 4 • This criterion is developed in reference 4 as:
Equations 18 and 19 are developed by use of acoustic theory relating flow velocity and pressure. These equations are analogous to the electric transmission line equations 4 •
cp 0'\ =
_ J-. ow = Ag ot
ew
L
ow et
(24)
ep
et
The sonic velocity, tJ-c, is developed follows:
In
tJ-c
(30)
= 9~
The pertinent facility values when substituted in this equation result in a cross-over frequency Wc ~ 1 rad./sec. The equations developed in reference 4 are used to calculate data for the curves of Figure 2, which compares the response
(25)
C-
0'\
7T
Wc
terms of Land C as
~ :1_-_"_~=----
(26)
~-20-
T- ~ •
1
-
z
Also,
j
M =
~
=
-4 0 ~ --
~(RTI) l-
Ap
tJ-c
,
(27)
yg
For large Mach numbers the relationship for the isentropic condition is derived from aerodynamic relationshi ps 2 as: i --2 d T.v - y [ 1+ (y-I ) M ]- = y-I
TI
[l+yM]--yM2 dp 2 dw (28) p
IV
o ClI
Equation 28 reduces to:
Figure 2,
dp p
for
M<%; 1
a·os
0·1
0 ·5 I 5 FREQuENCY (RADIANS/SEC.)
-10
-
- ~~~-180 so
Comparisoll o/Iumped alld distribllled trails/er /ullctiolls
(29)
For the foregoing: Cross-section area duct (ft.2) Temperature, total e R) TII' Temperature, wave-transmitted (eR) M Mach number (numeric) ,\ Duct length (ft.) p Fluid density (lb./ft. 3) tJ-c Sound velocity (ft./sec) A
Tt
Other symbols are as previously defined. Calculations made by inserting the pertinent facility values in equations 16, 17, 18 and 19 show that 95 per cent of the total temperature variation is transmitted convectively and only 5 per cent by wave transmission. This establishes the first of the two conditions previously defined, for the separation of temperature and pressure effects. Compliance with the second condition for the separation is ascertained by the calculation of propagation rates. The pressure and wave-transmitted temperature variation propagation rates are calculated by application of equation 26 as 1,140 to 1,250 ft. /sec. The velocity of the fluid which carries the convective temperature component is calculated by application of the aerodynamic continuity equation as 70 to 200 ft .jsec. Consequently, it is proved that the wave-transmitted component of temperature variation is both small and greatly out of phase with the convective component, thereby satisfying in full the criteria for the separation of the temperature and pressure systems. Validatiol/ of lumped-parameter representation of distributed elemel/ts-A major simplification is realized if lumped-parameter representation is used for this system. Such repre-
of the lumped and distributed elements over the pertinent frequency spectrum. Examination of Figure 2 shows that the distributed gain oscillates and the phase angle deteriorates rapidly at frequencies greater than W = 1·5 rad. /sec. The mcillation of the gain is caused by the mis-match of the surge impedance and actual duct impedance. The point in the freqlJency spectrum where oscillations begin is determined by the value of the propagation constant. From these curves it is evident that it is not practical to attempt to operate a control system beyond the cross-over point. Thus, this technique at once defines the possible operating region and validates lumpedparameter description of the distributed elements. COlllpressor al/d 1110 tor dril'e d)'l/umics- The torsional vibrational properties of this system, which comprises one compressor, four drive motors and two brakes, are studied to determine the frequency and amplitude of each of the six modes of oscillation. The only frequencies resonant with exciting frequenci.;:s were found when the induction drive motors were operating in an abnormal unbalanced condition in their secondary circuits. This investigation also revealed that it is possible to represent this distributed system with acceptable error as the second-order lumped systems shown on Figure 1. A typical step function response of the lumped main compressor motor drive system is given by : W
=
dB = 0·026 e- O. 261 sin 1·68t dt
(31)
This shows that the oscillation frequency W = 1·68 rad .jsec is only slightly higher than the controlled cross-over frequency of 1·0 rad ./sec. An investigation revealed that the system is not excited at w = 1·68 rad. /sec. The effects of power line voltage and frequency variations are negligibly small with re3pect to the effects of other system inputs. This permits the deletion of
292
1393
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
these transfer functions. Space limitations preclude the development of this material in this paper. Equivalent simplified systems analysis The application of all five areas of simplification discussed or developed previously for Figure I, together with the deletion of ambient and other negligible effects, reduces the system to that shown on Figure 3. In connection with this work all transfer functions shown on Figure 1 were developed. The important and more interesting transfer functions on the simplified plant Figure 3 are developed in this paper and the remainder are merely defined because of space limitations. Air-coll1pressibility ejfects- This effect is present in plant components having large volumes such as the ducting sections. This development is for the case where spatial effects can be neglected and the process may be considered adiabatic and isentropic. The thermodynamic equation of state for a perfect gas is: W
=
Pi V RTti
Differentiation of this expression and substitution of the isentropic relationship gives: (33) where: C = v/(yRTr;)
Linearizing and transforming gives: W(s) = sCPi(s)
which represents the lumped transfer function of all ducting sections. Air inertia effects-For a flow path of constant cross-sectional area A and length 10 carrying a flowing gas at a rate of w Ib./sec, and being accelerated at a rate of dw/dt Ib./sec 2, the fluid velocity is determined by a combination of the equation of state and the relationships w = qp and V = q/ A. The time derivative of the result yields the acceleration, thus:
(32)
The rate-of-change of weight in a volume is the difference between the in-flow and out-flow, or
(34)
dV dt
RTtidw APi dt
(35)
The air mass, M , is given by: (36)
Figure 3.
Plant transfer diagram; simplified plant
293
1394
J. J. LANDY AND G. J. FIEDLER
Therefore, the accelerating force equated to the pressure drop causing it gives:
Equation 43 can now be re-written as 10RTIi (P'-Po) = - - w2 I 2A2pi
(37)
Solving equation 46 for w, taking its total differential and transforming gives:
Linearizing and transforming yields:
l pes) = A~ sW(s)
(47)
(38)
where:
where:
_ (2pi-PO) K31 - 2(Pi2- PiPo)l(IOR/2A2)' (T/i) '
10 = A"Ir mertance Ag
[2A2(Pi 2- PiPO)]t 2(10R)±(TIi )f
Nozzle and cOl/trol valve flow characteristics-Research conducted at the University of Michigan and monitored by Sverdrup & Parcel indicates that the air flow through a butterfly valve can be adequately described by the nozzle equation 3 • The general expression is:
The coolers have no volumetric storage capacity on the air side. Pressure ratio alld scoop flow-These transfer functions are defined as follows:
where
K48 = l/p p
(40)
K49 = -Pn/P/ Kso = A(ygRT/i)!Ma/RTIi' where Ma = Mach number
The total differential of equation 39 is: =
OW
OW
G
0Pi
Compressor transfer /UI/CtiOIlS-K23 , K24 and K 2S are developed in this paper and given in equation 20. S),stem equations-The array of equations for the simplified plant is developed from Figure 3. The solution of the uncontrolled system for a typical principal control ratio results in an expression of the form:
OW
-;;-A dA + -;c- dpi +"----T dT1i 0
li
OW [oF}.
(41)
Applying the linearizing and transforming techniques previously described gives:
Ts6 + UsS + VS4 + Ws 3 + Xs 2 + YSI + Z\O Ps/A30(S) = A S8+ B S7 + C S6+ DsS + £54+ F53+ GS2+ HSl + isO
The synthesis of the control system using the analogue computer results in a controller transfer function of the form:
where:
G (s)
K
K
(48)
Pi K33 = - 2(Pi2- PiPo)1(10R/2A2)t(TIi)t
(39)
dw
(46)
18
c
.[1'43(0!)J.4.1_ 71]} 1.71 (0!)1' p. p.
=2'055A!F+ T·t l'\ p, tl-
19 -
K 20 =
I
p,
,p,
GsCs) = (I
p,
Cooler pressure drop-The fractional drop across the cooler depends directly upon the dynamic head' q' of air flowing over the cooler. From experimental data on this cooler it was determined that: 10
(43)
q = _pV2
(44)
where: I 2
Substitution of the mass flow equation 3 and the equation of state 3 in equation 44 gives: RT· q = __ 1_, w2 2A2pi
~~s)
(50)
This together with the second-order valve system shown on Figure 3 results in the 14th-order simplified system.
2·055Api FII 2(T/i)I.S
q
(49)
and a sensing system of the form:
2FA po
LIp
K/I +72 S)(I +74 5) s(1 +7IS)(1 +735)
I
po)I.43 (pO)I'71] 2'055A . [ 1·43 ( ~ -1,71 ~ T/it
=
(45)
Control System Synthesis The uncontrolled plant equations are simulated on the analogue computer using standard procedures. The accuracy of simulation is checked by comparison of analytically-developed uncontrolled system responses with the same responses obtained on the analogue computer. This operation also checks operation of the computer. Using established frequencyresponse and root-loci methods of servo design the characteristics of the control components are established, simulated and connected into the uncontrolled plant circuitry. From a 'map' of the large number of operating points a relatively few points are carefully selected to cover adequately the plant operating envelope. Plant controller adjustments are made on the analogue computer until the required transient and steady-state plant performances are obtained for each selected operating point. To insure compliance with the lumpedparameter frequency criteria previously developed it is necessary
294
1395
ANALYSIS FOR MULTI-LOOP FLUID DYNAMICS FACILITY WITH PROMINENT DISTRIBUTED PARAMETERS
to monitor continuously cross-over frequency during the synthesis. This is done on the computer by determining the values of each of the pertinent control ratios such as Ps/Pse(jw) at an excitation frequency of 1·0 rad. /sec. The integration gains, network time constants and compensation inflexibilities inherent in standard pneumatic and electronic controllers preclude their use in this application. The control system dynamics required to meet performance criteria are attainable only by specially designed controllers incorporating increased integration gain and flexible, wider-range lead-Iag networks.
capabilities for : (I) Testing larger engines. (2) Expanded transient testing. These capabilities probably would have remained unknown without the systems analysis because of the cost and difficulty of using the plant as an analogue.
The authors gratefully acknowledge the counsel and assistance of R. J. Kochenburger, V. B. Haas, Jr., V. E. Scottron and W. B. Heinz on the work relating to compressor dynamics and the rates of propagation of temperature and pressure variations. References
Conclusions
1
This paper demonstrates that : (a) An integrated systems analysis based on valid simplifications makes the design of electronic control systems for the largest dynamic processes technically feasible and economical. (b) The techniques used greatly facilitate the mathematical work compared with using partial differential and transcendental equations for time and space variables. (c) The analysis discloses the existence of unknown facility
2 3 4
CHESTNUT, H. and MAYER, R. W. Servomechallisms and Regulating System Design, Vol. 2, pp. 215-218. 1957. New York; McGraw-HiIl LIEPMANN, H. W. and PUCKETT, A. E. Introduction to Aerodynamics of a Compressible Fluid. 1947. New York; Wiley SHEPHERD, D. G. Principles of Turbomachinery, pp. 68, 70, 101 102. 1956. New York; Macmillan STALZER, T. R. and FIEDLER, G. J. Criteria for validity of lumpedparameter representation of ducting flow characteristics. Trans. Amer. Soc. mech. Engrs Paper No. 56-IRD- 21 May (1957)
Summary
An integrated mathematical systems analysis is used as the basis for the design of a high-performance automatic pressure and flow control system for the world's largest supersonic wind tunnel complex. The facility processes are aerodynamic and thermodynamic and consume 470 MW of power. The processes interact to the extent that the plant can be described mathematically as a 107th-order system having about 100 feedback loops. The analysis of this
complex plant is further complicated by the fact that most of the components have non-linear characteristics and there are prominent distributed-parameter effects. Valid simplifications are developed which reduce the plant to a 14th-order system having 25 feedback loops. This makes the control system analysis, synthesis and design technically feasible and economical using an analogue computer.
Sommaire
On utilise le calcul mathematique pour definir initialement les regulations de pression et de debit it hautes performance necessaires a la souffierie supersonique la plus importante du monde. Cet ensemble releve de I'aerodynamique et de la thermodynamique et consomme 470 megawatts. Les phenomenes reagissent les uns sur les autres it tel point que l'installation peut ctre consideree comme un systeme mathematique du 107eme ordre comprenant environ 100 boucles de contre reaction. L'etude de cet ensemble complexe est, en
plus, compliquee par le fait que beaucoup des elements constituants presentent des caracteristiques non lineaires et qu'il y a interaction des differents parametres. Certaines simplifications valables sont envisagees ensuite, ElIes reduisent le degre du systeme au 14emc ordre avec 25 boucles de contre-reaction. Ceci permet de mener economiquement I'etude de la regulation et sa definition technique it [,aide d'un calculateur analogique.
Zusammenfassung Eine umfassende mathematische Systemanalyse wird als Grundlage fUr die Konstruktion eines selbsttatigen Hochleistungsregelsystems fur Druck and Stromung wm Betrieb des grof3ten UberschallWindkanals verwendet. Dabei handelt es sich um aerodynamische und thermodynamische Vorgange, fUr die eine Leistung von 470 MW gebraucht wird. Die Vorgange beeinflussen sich in einem solchen Mal.le, dal.l man die Anlage bei etwa 100 Regelkreisen mathematisch als System 107. Ordnung beschreiben mul.l . Die Analyse der
gesamten Anlage wird weiter durch die Tatsache kompliziert, daf.l die meisten GJieder nichtJineare Kennlinien aufweisen und verteilte Parameter bestimmend sind . Durch wlassige Vereinfachungen wird die Anlage auf ein System 14. Ordnung mit 25 Regelkreisen reduziert, wodurch Analyse, Synthese und Konstruktion des Regelsystems mit Hilfe eines Analogrechners technisch ausfUhrbar und wirtschaftlich vertretbar werden.
295
1396