A low-dimensional dynamic model of severe slugging for control design and analysis

Transcription

1 A lo-diensional dynaic odel of severe slugging for control design and analysis Espen Storkaas Sigurd Skogestad, and John-Morten Godhavn Dep. Of Cheical Engineering, Noregian University of Science and Technology, N-749 Trondhei, Noray Statoil ASA, R&D, Process control, Arkitekt Ebells vei 0, Rotvoll, N-7005 Trondhei, Noray ABSTRACT We have developed a siplified dynaic odel of ultiphase flo in the type of systes that severe slugging occurs. Inside the open loop slug regie, the odel covers both the stable liit cycle knon as slug flo, and even ore iportantly, the unstable but preferred stationary flo regie, aking it suitable for controller design. We have fitted the odel to data both fro a siulation (OLGA) test case and fro ediuscale experients. We have in both cases achieved good agreeent ith the data, and controllers designed based on our odel are able to stabilize the flo for the systes odeled. INTRODUCTION Multiphase transport of gas, oil and ater has becoe increasingly iportant for the offshore oil industry. The trend toards ore satellite ells leads to longer ultiphase transport pipelines fro the ell clusters and ellhead platfors into the production platfors. In addition to the increased length, greater depths provide additional challenges for flo assurance. The flo regie knon as slug flo is characterized by interittent axial distribution of liquid and gas. The bulk of the liquid is transported as slugs, here the pipe flo consists of pure liquid. The gas is transported as bubbles beteen the liquid slugs. Slug flo can be divided into hydrodynaic and gravity induced slugging (also called terrain induced slugging). Hydrodynaic slugging occurs in near horizontal pipelines and is caused by velocity differences beteen the phases and ill not be treated here. Gravity induced slug flo are generally induced by a lo point in the pipeline topography. A typical situation is that liquid accuulates at a lo point at the botto of the riser, blocks the flo of gas and initiates the slug cycle. The prerequisites for this to occur are relatively lo pipeline pressure and flo rates. Gravity induced slugs are often long, and thus associated ith severe pressure oscillations.

2 The flo and pressure oscillations due to gravity induced slugging have several undesirable effects on the donstrea processing facilities. They ill create severe feed disturbances for the inlet separator, causing poor separation and in soe cases overfilling. The oscillations ay also affect the copressor trains, resulting in unnecessary flaring. Other adverse consequences are ear and tear on the equipent resulting in possible unplanned process shutdons. The severe slugging proble can be solved by increasing the pressure drop over the topside (or ellhead) choke valve. Hoever, the increased pressure drop ill loer oil recovery, and is not an optial econoic solution. Other proposed solutions such as installing slug catchers are given in (). Using active control to prevent severe slugging has been proposed in () and by other researchers. Control based strategies have recently received a lot of attention and are currently vieed as a proising solution. Both ipleentations and coercial products exist, as reported in (3), (4) and (5). These control systes are designed based on siulations using rigorous ultiphase siulators, process knoledge and trial and error. Using active control to stabilize an unstable operating point has several advantages. Most iportantly, one is able to operate ith even, non-oscillatory flo at a pressure drop that ould otherise give severe slugging. This ill in turn lead to less need for topside equipent, higher production, higher oil recovery and less ear and tear on the equipent. To design efficient control systes, it is advantageous to have an accurate odel of the process. When odeling for control purposes, one should keep the control objective and its associated tiescale (bandidth) in ind. One should only include those physical phenoena that are significant at the relevant tiescales in the odel, soething that allos us to use sipler odels for control purposes than for ore detailed siulations. There are to coon ethods for odeling ultiphase flo. The first is called a to-fluid odel, here one uses partial differential equations (PDE s) for conservation of ass and oentu in each phase. The other is a drift flux odel, here one has PDE s for conservation of ass in each phase, and a cobined oentu conservation equation. More details on odeling of ultiphase flo can be found in (6) and (7). These PDE-based odeling techniques are versatile, even though epirical correlations are needed, and can be adapted to a ide range of flo regies. The draback of these odels fro a control point of vie is that they are infinitediensional, thus poorly suited for conventional control design ethods. This can partially be reedied by discretizing in space, but the resulting set of ordinary differential equations (ODE s) ill be large. We originally used a PDE based to-fluid odel in our ork (8), a to fluid odel (4 PDE s) discertized ith 5 grid points yielding an ODE odel ith 00 states. This is too any state variables for efficient analysis and controller design. The control objective is to stabilize an unstable operating point, and the relevant tiescale is the speed of hich the instability evolves. Keeping that in ind, a lot of the spatial variations and fast dynaics included in the PDE based odels are unnecessary. This leads us to the conclusion that it should be possible to develop a sipler odel that describes the doinant behavior of a syste ith severe slugging.

3 MODEL DESCRIPTION To achieve the size and coplexity reduction in the odel that ould be desirable, the basic odeling principles ust be changed. We focus on describing the observed acro-scale behavior rather than the detailed physics that governs the flo. Since severe slugging basically is a pressure-gravity driven process, it is natural to try to describe it using bulk ters rather than spatial variables. The acro scale behavior that e ant to describe includes: the stability of the solutions and the operational conditions as function of choke valve opening the nature of the transition to instability the presence of an unstable stationary solution at the sae choke valve openings as those corresponding to severe slugging the aplitude/frequency of the oscillations Note that e for control purposes are ore interested in the desired (unstable) operating behavior than the open loop (no control) severe slugging behavior. This eans that the three first points on the above list of objectives are ore iportant than the last one. G, P, ρ G, α LT. Assuptions Figure : Model characteristics ith iportant paraeters We ill base our odeling on the setup depicted in Figure. Our ain assuptions are: Constant liquid velocity in feed pipeline (neglecting liquid level dynaics). This iplies: o Constant upstrea gas volue (volue variations due to liquid level at the lo point are neglected)

4 o Constant liquid feed directly into riser Only one liquid control volue (hich includes both riser and part of the feed pipeline) To gas control volues, separated by the lo point, and connected through a pressure-flo relationship. Ideal gas behavior Stationary pressure balance beteen riser and feed section Siplified choke odel for gas and liquid leaving the riser Constant syste teperature. Model fundaentals The odel has three states; the holdup of gas in the feed section, the holdup of gas in the riser, and the holdup of liquid. The corresponding ass conservation equations are: d [] L = L,in - L,out dt d [] = G,in - dt d [3] G = - G,out dt Based on the description in Figure, the coputation of ost of the syste properties such as pressures, densities and phase fractions is straight forard. We use a stationary pressure balance over the riser, here the pressure drop equals the eight of the fluids in the riser. The use of a stationary pressure balance is justified because the pressure dynaics are significantly faster than the tie scales in the control proble. For long pipelines, it ight be necessary to add soe dynaics beteen the riser-botto pressure and the easured pressure if the pressure sensor is located far fro the riser. The boundary condition at the inlet can either be constant or pressure dependent inflo. We use a siplified valve equation (eq. 4) to describe the choke valve at the outlet. ix,out =Kz ρt ( P-P 0 ) [4] The crucial part of the odel is to describe the distribution and velocities of the to phases in the riser. This is described in detail belo. The entire odel is given in detail in Appendix A. A Matlab version of the odel is available on the eb (9). As it is given above, the odel ust be solved as a differential-algebraic (DAE) syste. Hoever, through a state transforation, it is possible to transfor the syste into a set of ordinary differential equations (ODE s), hich ay be necessary in order to perfor a range of nonlinear analyses such as Lyapunov based stability calculations. The state transforation is given in Appendix B... Relationship beteen gas flo into riser and pressure drop For the gas phase, neglecting the acceleration ters, the pressure difference P=P -P is the su of the frictional pressure drop and the static pressure drop. P= P f+ρlgh [5] For turbulent flo

5 vi P f= cρ i i [6] i Where P i is the individual frictional pressure drops, and the c i s are friction coefficients. Focusing on the lo point and assuing that the pressure drop is purely frictional, e get v [7] P= ( P-P -αlρgh L ) =cρ Clearly, the friction coefficient c depends on the relative opening (H -h )/H, and assuing an inverse quadratic dependency gives eq. 8. The corresponding ass flo is given by eq. 9 here A is the gas flo area at the lo point. Note that cobining equations 8 and 9 ith f(h )=( A (H -h )/H ) yields a valve equation (eq. 0) here the functional dependency on ((H -h )/H ) is approxiately quadratic. H-h P-P-ρ LgαLH h<h K v = H ρ 0 h H [8] =vρa [9] =Kf ( h) ρ ( P-P ) [0].. Entrainent equation The final iportant eleent to deterine is the fluid distribution in the riser. This can be done in several ays. One is using a slip relation siilar to the one used in drift flux odeling to deterine the liquid velocity and fro that calculate the distribution. Attepts to do this ere not successful. Another ay is to odel the to-phase flo as entrainent of liquid by the gas. We found that this as better (and easier), and e choose to use an entrainent equation relating the fraction of liquid in the top (α LT ) to other syste variables. The to extree cases are no entrainent (liquid behaves as if in a tank) and no slip (α LT =α L ) for high flo velocities. The transition beteen the should be sooth, but ho steep the transition should be ill probably be case dependent. We ill consider a transition as depicted in Figure, here the paraeter q depends on the gas velocity in the syste.

6 Figure : Entrainent ith different slopes The entrainent in slug flo is soehat siilar to flooding in for exaple distillation coluns. The flooding velocity in a distillation colun is equal to the terinal velocity for a falling liquid drop and is given by ρl-ρg v=k f [] ρg This expression only gives a yes/no anser to hether it is flooding or not. To get a sooth transition, e use the square of the ratio of the internal gas velocity v to the flooding velocity as a paraeter (eq. 3). The entrainent equation in eq. produces the transition depicted in figure, here the paraeter n is used to tune the lope of the transition. The paraeter K 3 in eq. 3 ill shift the transition along the horizontal axis. n q α LT =α LT + n ( αl -αlt ) [] +q v K3ρv [3] q=k = vf ρl-ρ Note that e use the velocity at the lo point (v ) to obtain the entrainent of liquid through the riser. This is because the ipulse of the gas at the lo point initiates the entrainent by affecting the velocity of the incopressible liquid. By assuing that the average gas ipulse responsible for the entrainent of liquid through the riser is linearly dependent on the ipulse of gas through the lo point, the use of v in the entrainent equation is justified.

7 3 MODEL TUNING The odel paraeters K, K, K 3 and the exponent n in the entrainent equation are free paraeters that can be used to tune the odel. In addition to these, one can also vary the copressibility in a corrected ideal gas la and the upstrea gas volue to achieve better fit to the data. The upstrea gas volue can be used because this represents the effective volue available for the gas and ill depend on the aount of liquid present in the feed pipeline. The tuning of the odel ill depend on the available data. If one has a dependable, detailed odel of the syste available, one can generate the data needed. If the syste is already operational, field data is a better alternative. In this ork e have based the tuning on data fro the point here the unstable region begins, that is, the highest valve opening ith stable operation. Earlier ork (8) has shon that the syste goes through a Hopf bifurcation at this point. A Hopf bifurcation point is characterized by having a pair of coplex iaginary eigenvalues (poles). This inforation together ith to other easureents of, for instance, pressures in the syste can be used to initiate the syste to steady state. The last of the ain tunable paraeters (K, K, K 3 and n) should be found by iterating on the value of h until a satisfactory behavior is achieved, as this has to lie in the interval 0<h <H to ensure consistency. Finally, one can tune the copressibility and the upstrea gas volue V to get an acceptable fit of pressure levels, aplitudes, and frequencies for other valve openings. 4 CASE : COMPARISON TO OLGA DATA To illustrate the use of the siplified odel, e ill use data fro the test case for severe slugging in OLGA. OLGA is a coercial ultiphase siulator idely used in the oil industry. The case geoetry is given in Figure 3. It is a constant ass flo ith constant liquid fraction into the syste. The pressure after the choke valve is constant at 50 Bar.

8 Figure 3: Case geoetry Based on OLGA siulations, a bifurcation diagra for the syste as ade. The bifurcation diagra is given in Figure 4. The upstrea pressure is given on the vertical axis and the valve opening is on the horizontal axis. The bold lines are the results fro the OLGA siulations. The bifurcation diagra shos that for valve opening up to 0.3, the syste is stable, as indicated by the single bold line. For valve openings over 0.3, e ould norally have severe slugging ith axiu and iniu pressures given by the solid bold lines. Initiating OLGA to steady state gives the unstable stationary solution indicated by the dashed bold line.

9 Figure 4: Bifurcation diagra for the case The odel as tuned as described above. It is unrealistic to achieve a good fit to both the slug regie and the stationary regie over the entire range of valve openings, so soe priorities have to be ade. First, the unstable, stationary regie is ore iportant than the stable oscillatory regie. The reason for this is that e ish to avoid the slug regie, and operate at the stationary regie, hence e focus on here e ant to be and not on here e don t ant to be. A parallel to this can be found in everyday life; if you are teaching soeone to ride a bike, you are teaching the ho the bike behaves hen they have astered the balancing act (the desired unstable operating point), not ho it behaves hen it lies on the ground (the undesired slug flo). Second, e ill focus in the lo to ediu valve openings. The reason for this is that for the controller to ork, the control actuator has to have a significant ipact on the process. For the high-range valve openings, the pressure drop over the choke valve is too sall for the effect of a sall change in valve opening to be significant. This eans that e can only operate in the lo to ediu valve opening range. This is really not a restriction, because, as can be seen fro the bifurcation chart, there is no significant incentive to operate at higher valve openings. There is an insignificant difference beteen the upstrea pressures for valve openings beteen 0.4 and. As can be seen fro Figure 4, the fit of the siplified odel (thin lines) to the reference data fro OLGA is good. The fit for the stationary data is excellent, and the fit for the oscillatory data in the lo to ediu range are also good. The deviations in aplitude at higher valve openings are acceptable in light of the priorities behind the odel fitting. The frequency of the oscillations are not included in the bifurcation diagra, but coparisons beteen the

10 results sho that the slug frequency is about 0-0% higher for the siplified odel for lo to ediu range valve openings and up to about 50% higher for high valve openings. The higher frequency coes fro neglecting the liquid dynaics in the feed section. We have in this case tuned to achieve a good fit for the aplitude, and hen the upstrea gas volue is fixed, e cannot fit both frequency and aplitude siultaneously. The odel can no be used for analysis and controller design. This is not the focus of this paper, but it can be shon that the unstable stationary steady state can be stabilized easily using feedback control hen the riser-base or a reasonably close (a fe kiloeters) upstrea pressure can be easured. It is also possible to stabilize the flo using only upstrea easureent (at or close to the choke) (0). This can be done by a cobination of a flo easureent and soe other easureent, e.g. pressure using a cascade controller (5). The flo can either be easured or estiated fro density and pressure drop easureents. Figure 5: Stabilization of severe slugging using a PI controller ith upstrea pressure as easureent (Case ) Figure 5 shos a siulation ith a PI controller designed using our siplified odel stabilizing the flo. The siulations are also done using the siplified odel. The top part shos the donstrea pressure vs. tie, the loer shos the valve position. The syste is started up in open loop (controller offline) ith a valve opening of z = As predicted fro the bifurcation diagra (Figure 4), severe slugging develops. The controller is turned on after 0.5 hrs, and the controller stabilizes the syste rapidly at the set point of 70 Bara. After.5 hrs, the set point is changed to 69 Bara. The control syste anages to stabilize both operating points. After 3 hours, the controller is put back in anual, leading to the reappearance of the severe slugging.

11 The operating point ith a set point of 69 Bara corresponds to a valve opening of 0.5, hich is ell ithin the unstable region. As can be seen fro Figure 5, the valve position oves toards this value, and sees to stay constant as this value. Hoever, if one had zooed in on the graph, or studied an actual valve used for stabilizing control, one ould see that the valve is continuously aking sall corrections to hold the syste stable. The controller designed based on the siplified odel has been tested on the OLGA odel ith siilar results. 5 CASE : FIT TO EXPERIMENTAL DATA We have fitted the odel to data fro recent experients perfored by Statoil at a ediu scale loop at the SINTEF Petroleu Research Multiphase Flo Laboratory at Tiller outside Trondhei, Noray. The loop consists of a 00 eters long slightly declining feed pipeline entering a 5 eters high vertical riser ith a control valve located at the top. The fluids used are SF 6 for the gas and Exxsol D80 for the liquid. After the riser the ixture enters a gasliquid separator ith an average pressure of Bara. The inflo into the feed pipeline is pressure dependent. More inforation on these experients can be found in (5) and (0). We ere able to obtain a very good fit of our siplified odel to the experiental results. The bifurcation chart is given in figure 6, here experiental data are given as dots. More iportantly, the controllers (designed based on the siplified odel) reproduced the stability results confired experientally. In fact, the optiized controller tunings found using the odel atched the ones found to be optial fro the experiental ork.

12 Figure 6: Bifurcation diagra for the Tiller experiental data (case ) 6 CONCLUSIONS We have developed a siplified odel of severe slugging suitable for controller design. The odel has three states and is based on phenoenological odeling, here e identify the ajor characteristics of the syste at hand and develop a odel that incorporates these characteristics. The ajor characteristics are the stability of the flo as a function of choke valve position, the nature of the transition to instability (Hopf bifurcation), the presence of an unstable steady-state solution and the aplitude of the oscillations. It should be stressed that the odel, in addition to describing the (undesired) slug behavior, ust describe the (desired) steady state flo regie. Based on this odel, e are able to investigate the linear and non-linear behavior of the syste. We can study the nature of the instability, evaluate different easureents candidates for control and test control configurations. The odel is currently used both as a tool for understanding the behavior of these systes and for designing control systes that stabilizes the flo. We have fitted the odel to data both fro an OLGA test case and fro ediu-scale experients. We have in both cases achieved good agreeent ith the data, and controllers designed based on our odel are able to stabilize the flo for the systes odeled. It is also our experience that this siplified odel is easier to fit to experiental data than the ore

13 coplicated PDE odels that are based on a ore rigorous representation of the true syste. We hope to test the odel against field data in the near future. It is our fir belief that one can design better and ore robust controllers for stabilizing ultiphase flo in offshore pipeline-riser systes ith less effort using a siplified odel. 7 ACKNOWLEDGEMENTS The authors ould like to thank the ebers of the Petronics project, as ell as its sponsors in Norsk Hydro and ABB. We ould also like to thank Statoil for supplying experiental data and valuable insight. Finally, e ould like to thank the Noregian Research Council for financial support. 8 REFERENCES () Sarica, C and Tengesdal, J. A ne technique to eliinate severe slugging in pipeline / riser systes, SPE Annual Technical Conference and Exhibition 000, Dallas, Texas. SPE6385 () Hende, P. and Linga, H. Suppression of terrain slugging ith autoatic and anual riser choking, Advances in Gas-Liquid Flos, 000, pp (3) Havre, K., Stornes, K. and Stray, H. Taing slug flo in pipelines, ABB revie 4, 000, pp (4) Henriot, V., Courbot, A., Heintze, E. and Moyeux, L. Siulation of process to control severe slugging: Application to the Dunbar pipelines, SPE Annual Conference and Exhibition in Houston, Texas, 999. SPE5646 (5) Skofteland, G. and Godhavn, J.M. Suppression of slugs in ultiphase flo lines by active use of topside choke - Field experience and experiental results, Accepted for publication at Multiphase'03, San Reo, Italy, -3 June 003. (6) Bendiksen, K., Malnes, D. and Nydal, O.J. On the odeling of slug flo, Cheical Engineering Counications, 4: 7-03, 996 (7) Taitel, Y. and Barnea, D. "To phase slug flo", Advances in Heat Transfer, Hartnett J.P. and Irvine Jr. T.F. ed., vol. 0, 83-3, Acadeic Press (990). (8) Storkaas, E., Alstad, V. and Skogestad, S. Stabilization of desired flo regies in pipelines, AIChE Annual eeting 00, Reno, Nevada. Paper 87d (9) Matlab odel available at :.chebio.ntnu.no/users/skoge/softare/index.htl (0) Fard, M. and Godhavn, J.M. Modeling and Slug Control ithin OLGA Subitted to SPE Journal.

14 Appendix A: The odel presented belo is for the case of constant inflo. For pressure dependent inflo, the odel for the inflo echanis ust also be included. A. Notation Sybol Description Unit Rearks Gi Mass of gas in volue i kg State variable L Mass of liquid kg State variable V Gi Gas volue i 3 V = const. V L Volue occupies by liquid 3 V LR Volue of liquid in riser 3 V T Total volue of riser 3 Constant P i Pressure in volue i Pa ρ Gi Gas density in volue i kg/ 3 ρ L Liquid density kg/ 3 Constant ρ Average density in riser kg/ 3 ρ T Density at valve kg/ 3 v Gas velocity at lo point /s Internal ass rate of gas kg/s Total ass rate through valve kg/s ix,out G,out Mass rate of gas through valve kg/s L,out Mass rate of liquid through valve kg/s α L Average liquid fraction in riser, volue basis - α LT Liquid fraction at valve, volue basis - α Liquid fraction at valve, ass basis - L h Liquid level upstrea lo point H Critical liquid level Constant H Height of riser Constant r Radius of pipe Constant A Cross section area in horizontal plane, upstrea lo point Constant A Cross section area in horizontal plane, riser Constant A Gas flo area at lo point Constant L 3 Length of horizontal top section Constant θ Feed pipe inclination rad Constant R Gas constant J/K/kole Const=834 g Specific gravity /s Const=9.8 T Syste teperature K Constant M G Molecular eight of gas kg/kole Constant Mass rate of gas into syste kg/s Disturbance G,in L,in Mass rate of liquid into syste kg/s Disturbance P 0 Pressure after valve Pa Disturbance z Valve position - Input K Choke valve constant - Tuning paraeter K Internal gas flo constant - Tuning paraeter

15 K 3 Friction paraeter - Tuning paraeter n Exponent in friction expression - Tuning paraeter A. Equations A.. Internal equations RT P= [4] V M G ρ = [5] V L V= L [6] V ha+v =V [7] LR L V=A T ( H+L 3) [8] V G =VT -V LR [9] G ρ G = V [0] V G LR α L = [] V T RT G P= [] V G M G ρ= +V ρ V G LR L ( 3) L T [3] ρg H +H -ρ gh =P -P [4] K3ρv q= [5] ρ -ρ L n VLR -HA q VLR -HA α LT =(V LR >HA ) + n αl-(v LR >HA ) AH 3 3 +q AH 3 3 [6] ρ T=αLTρ L+-α ( LT) ρ G [7] αltρl α L = α ρ +-α ρ [8] LT L LT G A.. Transport equations H-h P-P-ρ gα H v = h <H K [9] H ρ A..3 Geoetric equations L L =vρa [30] ix,out =Kz ρt ( P-P 0 ) [3] = -α [3] G,out L ix,out G,out =αl ix,out [33]

16 (( ) ) (( ) ) r H= cos θ A A= sin θ - φ= H -h cos θ <r π-cos - ( H-h) cos( θ) ( H-h) cos( θ) r - + H -h cos θ r cos - A=r A..4 Conservation equations ( π-φ-cos ( π-φ) sin ( π-φ) ) d L = L,in - L,out dt d = G,in - dt d G = - G,out dt r [34] [35] [36] [37] [38] Appendix B: State transforation With its current states, the odel is a differential-algebraic (DAE) syste. If the odel is to be used for linearization or nonlinear analysis, it ould be preferable to transfor the syste into an ODE syste. This can be done through the state transforation given belo. L To siplify the notation, e ill scale and renae the states L, and G to x=, V ρ G x= and V 3 ρ G G x=. The quantities arked ith an asterisk are reference values for V ρ the given variable. When ( ) is used as an arguent, it is an abbreviation to siplify the odel and should be taken to represent the appropriate arguents for the given expression. With the current states, the odel is a DAE syste. The height h ill be used as the algebraic state, denoted γ. Entries that are not states and that are not given ith an arguent are constant syste paraeters. With these changes of notation, e can siplify the odel to: A x= - Ψ z αlt () -x + γ VT ()( ) ( () ) A( γ) v ( ) x L,in VTρ L αlt V-Vx+Aγ T T +VTρ G -αlt x3 G,in Vρ V () T L [39] x= - [40] ρ A γ v x= - Ψ z [4] ( ) x -αlt ( ) x3 () () 3 VTρ G αlt ρl V-Vx+Aγ T T +VTρ G -αlt x3 ()

17 ρla RT VTρ Gx 3 ρgx+ρ 3 Lx- γ g( H +H3 )-ρlgγ= ρx- VT M G ( VT-VTx +Aγ) v ( ), Ψ( ), and α LT ( ) used above is given belo. A (γ) is given by equation 6. [4] RT VTρ Gx 3 A ρ x- -ρ L gh x - γ H-γ M G VT-VTx +Aγ VT v (x,γ)= ( γ<h ) K H ρ x ix,out Ψ( x,γ,α LT () ) = = z VTρ G( -αlt() ) x 3 VTρ GRTx 3 K αlt() ρ L+ -P0 V-Vx+Aγ T T ( V-Vx+Aγ T T ) M G ( K ρ x ( v ()) ) 3 A αlt ( x,γ,v (),α LT () ) =α LT () + x n - γ-α LT () n VT ( ρl-ρx ) + K3ρx( v() ) Vx-Aγ-A T H α LT ( x,γ ) = (( VT x-aγ ) >HA ) AH 3 3 Solving eq. 40 for x MGg( H +H3 ) ρla ρlmgg VTρ Gx3 x=φ ( x,x,γ 3 ) = ρgx+ρ 3 Lx- γ - γ+ RTρ VT RTρ ρ V-Vx+Aγ T T Differentiating Φ( x,x,γ 3 ): δφ M gρ H+H V ρ x = - δx RTρ ρ V-Vx+Aγ G L 3 T G 3 T T δφ M gρ H+H V ρ = + δx RTρ ρ V-Vx+Aγ G G 3 T G 3 T T n δφ MGgρ A L H +H3 VAρ T Gx = [46] δγ RTρ VT ρ ( V-Vx+Aγ T T ) A state transforation fro x to γ yields the ne differential equation: δφ δφ x- x- x 3 δx δx3 γ= [47] δφ δγ The full expression can be found by inserting equations 39-4 and into 47. [43] [44] [45]