Towards physics based autonomous control of the ... - AIAA ARC

Towards physics based autonomous control of the cryogenic propellant loading system Ekaterina Ponizovskaya-Devine∗ SGT, Inc., Greenbelt, MD, USA

Dmitry G Luchinsky† Mission Critical Technologies, Inc., El Segundo, CA, 94035, USA

and Michael Khasin∗ SGT, Inc., Greenbelt, MD, USA

Dogan Timucin‡ Ames Research Center, Moffett Field, CA, 94035

Jared Sass§ and Jose Perroti¶ and Barbara Brownk Kennedy Space Center, Kennedy Space Center, FL, 32899, USA We report on progress in development of model-based optimization methods for the autonomous control of the propellant loading system. We briefly discuss properties of the models and demonstrate examples of their validation using the experimental data obtained at NIST and at Kennedy Space Center. We consider application of the modelbased optimization methods to the analysis of chilldown and identification and evaluation of the faults in the cryogenic transfer line. It is shown that model-based optimization provides an efficient tool for the development of autonomous control of cryogenic loading operations.

Nomenclature d l A S V l T p e h u hlg c H

Internal pipe diameter, m Internal pipe perimeter, m Cross-section area, m2 Surface area, m2 Volume, m3 Internal pipe perimeter, m Temperature, K Pressure, Pa Specific energy, J/kg Specific enthalpy, J/kg Velocity, m/s Specific heat of evaporation, J/kg/K Specific heat, J/kg/K Heat transfer coefficient, W/m2 /K

∗ Senior

Researcher, SGT, Inc., Greenbelt, MD, USA Researcher, Mission Critical Technologies, Inc., El Segundo, CA, 94035, and AIAA Senior Member. ‡ Group lead, Ames Research Center, Moffett Field, CA, 94035, and AIAA Senior Member. § Head of the Cryogenic Laboratory, Kennedy Space Center, FL, 32899, USA ¶ Program Manager, Kennedy Space Center, FL, 32899, USA k Deputy Chief Technology, Kennedy Space Center, FL, 32899, USA † Senior

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q˙ τ ρ

Heat flux, J/m2 /s Wall shear stress, N/m2 Density, kg/m3

α

Void fraction

µ

Viscosity, Pa·s

κ

Thermal conductivity, W/m/K

Subscripts w wall g gas l liquid i interface o outer value amb ambient

I.

Introduction

Interest to autonomous control and optimization of cryogenic transfer has been renewed recently in the context of NASA plans to develop advanced cryogenic systems that will increase safety and reliability of launch operations and minimize losses of expensive commodities.1 This development is part of the long term NASA strategy to rely more and more on autonomous operations and systems2, 3 in the future deep space and extraterrestrial missions. It is also expected that reliable virtual prototyping and optimization tools will reduce cost and time required to design and test future cryogenic system operating under reduced gravity conditions. Autonomous cryogenic loading operation requires that the system can perform a number of tasks without human interaction, including in particular: (i) recognition and control the state of the flow in the transfer line; (ii) detection, isolation and mitigation of faulty operational mode; (iii) learning parameters of the system in nominal and off-nominal regimes. To perform these tasks efficiently in autonomous regime the system has to include the following basic components: (i) fast and reliable solvers for two-phase cryogenic flow; (ii) optimization tools; and (iii) machine learning methods. Development of fast and reliable solvers for two-phase flows is a challenging problem on its own.4, 5 It becomes even more complicated when analysis of cryogenic fluids is concerned,6, 7 for which knowledge of required correlations remains sparse. We have demonstrated recently that a hierarchy of two-phase models8 can provide fast and accurate predictions of cryogenic flows in strongly non-equilibrium regimes including chilldown.9–13 Here we report on the progress in development and applications of the model-based optimization tools. We consider two example problems - optimization of chilldown and fault identification and evaluation in cryogenic transfer line. It the first example it is shown chilldown optimization reveals a trade-off between the amount of time and commodity losses required to chill the transfer line and that optimal solution depends on additional constraints imposed on the system. In the second example it is shown that optimization tools allowed us to correctly identify and evaluate one fault between three closely related possibilities within one ambiguity group. In the next section we discuss an example of formulation of optimization problem for cryogenic transfer line. Next, we will briefly review properties of the two-phase flow solvers and provide results of their validation. In the Section IV we present results of chilldown optimization in the cryogenic testbed (CTB). Next, we discuss an application of the optimization tools to the fault identification and evaluation during chilldown in the CTB. In conclusion, we summarize obtained results and outline future work.

II.

Example of optimization problem

Optimization of the cryogenic flow in transfer line may have multiple objectives, including e.g. virtual prototyping at the design stage, optimization of the loading regimes, fault identification and evaluation, and

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search for the optimal recovery strategy. Here we discuss in more details formulation of the optimization problem for the chilldown operation. The main objective of such an optimization1, 14 is to minimize commodity losses and reduce time required for the loading operations under constraints that flow rates and pressure and temperature gradients (i.e. local mechanical and thermal loads) are within safety margins. The set of optimization parameters includes pressure in storage and vehicle tanks, openings of the in-line and bleed valves, and parameters of the feedback control for the openings of the bleed valves. Let us consider chilldown of a unit length of the transfer line shown in Fig. 1, which is characterized by the equation for the energy conservation in the form cw ρw Aw

∂Tw = Hwl ll (Tl − Tw ) + Hwg lg (Tg − Tw ) + Hamb lo (Tamb − Tw ) . ∂t

(1)

Figure 1. Control volume of the two-phase flow: void fractions for gas
(αg ) and liquid (αl ), phasic temperatures (Tg(l) ), phasic velocities (ug(l) ), wetted perimeter for each phase lg(l) , perimeter of the interface (li ), the crosssectional area (A), the wall temperature (Tw ). Mass flow rate through: the input valve (m ˙ in ), the output valve (m ˙ out ), and the bleed valve (m ˙ bl ).

The amount of liquid required to chill the system down is mainly determined by the heat stored in the pipe walls, the volume of the pipe filled with liquid, and by the heat fluxes to the pipe walls. To estimate the cooling time of the pipe we rewrite the equation (1) in the form (assuming lo ≈ lw ) ∂Tw 1 =− (Tw − Tef f ) . ∂t τef f

(2)

Here the effective characteristic time τef f and temperature Tef f are defined as follows τef f =

cw ρw dw , Hwl ll /lw + Hwg lg /lw + Hamb

Tef f =

Hwl ll Tl /lw + Hwg lg Tg /lw + Hamb Tamb . Hwl ll /lw + Hwg lg /lw + Hamb

(3)

The values of the coefficients Hwg and Hwl can be estimated by noticing that during the chilldown the liquid flow rates are usually small and the heat transfer correlations follow closely pool boiling curve.15 In this case the pipe can be roughly divided into three regions: (i) dry region; (ii) region with the wall temperature above Leidenfrost point (Tw ≥ Tmf b ); and (iii) region with the wall temperature below Tmf b . The lowest heat transfer coefficient Hwg (of the order of tens W/m2 /K) corresponds to the dry region and can be estimated as the maximum value16, 17 corresponding to one of the possible gas flow regimes: (F-L) forced laminar, (F-T) forced turbulent, (N-L) natural laminar, and (N-T) natural turbulent convection   4.36, F-L 18 ;    0.8 0.4 κ 0.023 · Re P r , F-T 18 ; Hwg = (4) Dh  0.1 · (Gr · P r)1/3 , N-L 19 ;    0.59 · (Gr · P r)1/4 , N-T 19 . Here P r =

µCp κ

and Gr =

ρ2 gβ(Tw −Tl(g) )d3 µ2

are Prandtl and Grashof numbers respectively. 3 of 11

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An intermediate value of the heat transfer coefficient Hwl ≈ Hf b (of the order of hundred W/m2 /K) can be found in the film boiling region with Tw > Tmf b . Where the minimum film boiling temperature Tmf b can be found using e.g. Kalinin20 correlations  0.25 ! Tmf b − Ts (ρcκ)l = C1 0.16 + 2.4 (5) Tcr − Tl (ρcκ)w and film boiling heat transfer coefficient can be found using e.g. Bromley correlations21 # " ˜ lg Cpg 0.25 gρg κ2g (ρl − ρg ) h ˜ lg = hlg + 1 cpg (Tl − Ts ), , h Hf b = C 2 · d (Tw − Ts ) P rg 2

(6)

where parameters C1(2) are of the order of 1. The largest value of the heat transfer coefficient Hwl (of the order of thousands W/m2 /K) can be found in the region corresponding to the critical heat flux. The heat transfer coefficient in this region has a sharp peak for Ts < Tw < Tmf b a . The value of the Hwl ∝ q˙chf /(Tw − Ts ) can be estimated using e.g. correlations for critical heat flux Zuber23 correlations in the form 1/4   1/2 ρl σg(ρl − ρg ) π . (7) Hlg ρg q˙chf = 24 ρ2g ρl + ρg The ambient heat transfer coefficient Hamb can be estimated using e.g. radiative and convective heat transfer from the environment to the pipe surface. Using these estimations one can show that the characteristic cooling time of the unit length of the pipe by liquid nitrogen can vary between a few hundreds of seconds in the dry region to a value around ten seconds in the nucleate and transition boiling region. It can also be elucidated from the above discussion that the chilldown time is mainly determined by the velocity of the propagation of the wetting front in the pipe. The control of the cold front propagation in the cryogenic transfer line is a non-trivial task, which can be analyzed in further details using model-based optimization tools.

III.

Physics based models of the two-phase cryogenic flow

As was mentioned in the introduction, efficient optimization tools for autonomous control of cryogenic transfer should be based on accurate two-phase flow solvers. Here we briefly review properties of such solvers developed in our work.9–13 Taking into account future on-line applications of these solvers we focused our analysis on one-dimensional models and tested several of them. It was concluded that a wide range of approximations of the two-phase cryogenic flows can be covered with required fidelity using so-called Wallis24 model and a quasi-steady extension10 of the homogeneous moving-boundary model.25 A.

Separated model

The Wallis24 or separated model (SM) consists of a set of conservation equations for the non-homogeneous (ug 6= ul ) and non-equilibrium (Tg 6= Tl ) flow that can be written in the form ∂F 1 ∂U + = C + C0 , ∂t A ∂z

(8)

where the column-vectors of the conservative variables F and fluxes U are defined by the following expressions     αg ρg ug αg ρg         αl ρl ul αl ρl       αg ρg ug   αg ρg u2g    . F = ; U = A (9)   2 αl ρl ul  αl ρl ul         αg ρg eg   αg ρg eg ug  αl ρl el αl ρl el ul a the

actual onset of the peak corresponds to the onset of nucleate boiling Tonb , which close to Ts , see e.g.22

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The right hand side is divided into two column-vectors: C that involve time or space derivatives and C0 that includes source terms defined as algebraic the following expressions     0 Γg     0     −Γg      αg ρg y,z − τwg lwg − τig li + Γg uig    0 A A    . (10) C=  ; C0 =   li wl 0 αl ρl y,z − τwl lA − τil A + Γg uil         p l li  q˙wg wg  −pαg ,t − A (Aαg ug ),z  A + q˙ig A + Γig Hil + Γwg Hgs  p li wl q˙wl lA pαg ,t − A (Aαl ug ),z + q˙il A − Γig Hil − Γwg Hgs The set of equations (8) - (10) is closed by the volume conservation condition and by equations of state for each phase
αg + αl = 1, ρ(g,l) = ρ(g,l) p, e(g,l) . and is coupled to the energy conservation equation (1) for the unit length of the wall. The solution algorithm for this model belongs to the class of all-speed Implicit Continuous-Fluid Eulerian (ICE) algorithms.26 The nearly-implicit version of this algorithm developed for the analysis of the two-phase flow9 follows closely the ideas of.27 The heat transfer and pressure losses correlations are based on the flow pattern recognition introduced by Wojtan et al.28 The single phase friction factor fg(l) is approximated using Churchill formula.29 The two-phase friction pressure drop is defined using Lockhart-Martinelli correlations.30 The flow boiling correlations currently recognize stratified and dispersed flow regimes. For horizontally stratified regime we use correlations described in Section II with Griffith corrections31 to the critical heat flux and Iloeje corrections32 to the film boiling correlations. In dispersed flow regime the analysis of correlations follows results in.16, 17 The known33, 34 instability issues were mitigated using limiters, smoothers, and time step control as described in.9 The resulting algorithm is quite reliable and provides accurate predictions of 1000 sec of cryogenic loading in around 5 sec. However, the instabilities cannot be eliminated completely and may slowdown the integration and sometimes prevent convergence. To guarantee continuity and robustness of the model prediction in a wide range of parameters the SM was embedded into a hierarchy of models.8 The second key component in this hierarchy was so-called homogeneous model (HM),10, 25 which we now describe in more details. B.

Homogeneous moving boundary model

In the HM the problem is reduced to an analysis of two conservation equations for the mass and energy written for the mixture density (ρ) and enthalpy (h) of the cryogenic fluid ρ = γρv + (1 − γ) ρl ;

ρh = γρv hv + (1 − γ) ρl hl ;

h = xhv + (1 − x) hl .

(11)

The column-vectors with the conservative variables, fluxes, and source terms are now simplified to the following form (C˜ = 0) " # " # " # ρ ρu 0 ˜ =A F˜ = ; U ; C˜0 = . (12) ρe ρhu q˙w lAw The fidelity of the equations (8), (12) is increased by allowing coexistence of different flow regimes (liquid, gas, two-phase) within one control volume using moving-boundary approximation and linear approximation for mass fraction in the two-phase region.10, 25 The fidelity of the moving-boundary approximation was further improved in10 by coupling the equations (8), (12) to a quasi-steady solution of the momentum equation Aρu2


,z

+

1 (τw lw )2φ = −p,z − ρg sin θ. A

(13)

The resulting system of differential-algebraic equations was solved using explicit Euler integration of the linearized equations (8), (12), and equation (13) with respect to pressure, enthalpy, and the mass fluxes. 5 of 11 American Institute of Aeronautics and Astronautics

Figure 2. Model predictions (solid colored lines) for the wall temperature are shown in comparison with the experimental time-traces (black dashed lines with open symbols) at 4 different locations along the transfer line: (i) 6 m from the line entrance (open circles); (ii) 24 m (open squares); (iii) 42 m (open diamonds); and (iv) 60 m (open triangles).

The frictional losses f1,3 in Eq(5) were calculated using the Swami-Jain35 approximate solution of the Colebrook equation. For the two-phase flow the frictional losses were calculated according to the MuellerSteinhagen and Heck correlation.10, 35 The heat transfer correlations were modeled using Dittus-Boelter approximation with the GungorWinterton enhancement factor.10, 35 The resulting algorithm is very fast and can accurately predict 2000 sec of real time of cryogenic loading in less one second. We now provide examples of validation of these algorithms. C.

Validation

Both algorithms described above were extensively verified11, 36 and validated using experimental data obtained during chilldown of the straight horizontal transfer line at National Bureau of Standards (NBS)37 and during cryogenic transfer at CTB.1 An example of the validation of the SM using NBS data is shown in the Fig. 2. The vacuum jacketed line in this experiment was 61 m long. The internal diameter of the copper pipe was 3/4 inches. The measurements were collected at four stations located at the distance 6, 24, 42, and 60 m from the input valve. In the experiment shown in the figure the working liquid was nitrogen and the pressure in the storage tank was 4.2 atm.

Figure 3. Model predictions (red lines) of the chilldown in cryogenic transfer line are shown in comparison with experimental data (black lines): (top) temperature and (bottom) pressure at three locations alone the line.

Several features can be noticed in this figure. For an extended period of time dry wall conditions can be observed at the locations 42 and 60 m. The characteristic cooling time of the wall at these locations exceeds hundred seconds. A region with characteristic cooling time a few tens of seconds corresponding to 6 of 11 American Institute of Aeronautics and Astronautics

the film boiling heat transfer can be observed at location 24 m. The cooling time is substantially reduced indicating presence of the wetted wall at the location 6 m. It can also be seen from the figure that the colling time is further reduced to the value around 10 sec once the wall temperature approaches Tmf b ≈ 125K at every location. We note that these features can be well reproduced by the SM and agree with estimations discussed in Section II

Figure 4. Sketch of the cryogenic transfer line build at KSC. It icludes storage tank (ST) and vehicle tank (VT); the in-line control valves: CV1, CV2, CV3, and CV4; remotely controlled bleed valves: BV1, BV2, BV3, BV4, BV5, BV6, BV73, and BV8; eight temperature sensors (TT) and 7 pressure sensors (PT).

An example of validation of the HM is shown in the Fig. 3. The experimental data were obtained during chilldown of the first half (up to valve CV3) of the CTB transfer line shown in Fig. 4. The CTB line includes storage tank (on the left) and vehicle tank (on the right) connected via cryogenic transfer line. The latter has a number of in-line (CV) and bleed (BV) valves that control the flow anda set of pressure (PT) and temperature (TT) sensors that monitor it. The chilldown begins at around 400 sec with opening the main valve CV1 located near the storage tank. At this time all other valves remain closed and after a small temperature drop the liquid is bounced back out of the transfer line. At approximately 500 sec the valves CV2 and BD 1, 2, and 3 are opened allowing for the nitrogen flow through the system up to the location of the valve CV3. The chilldown of the first part of the transfer line is accomplished, when the temperature sensor TT74 indicates the presence of the liquid nitrogen. It can be seen from the figure that the HM can reproduced accurately the experimentally observed temperature and pressure dynamics during the chilldown in the CTB. Below we provide examples of applications of the homogeneous model to solutions of two problems optimization of chilldown and identification and evaluation of fault during cryogenic transfer.

IV.

Analysis of chilldown

Fast and accurate prediction of the two-phase cryogenic flow by the physics models paves a way to a development of a number of important applications including development of the model-based optimization tools for the integrated health management and control of the flow regimes. In this section we present results of recent preliminary analysis of optimization of the chilldown in the cryogenic transfer line at KSC.

Figure 5. Temperature sensitivity to opening of the valve BV1 predicted at three locations: (a) TT74, (b) TT46, and (c) TT49. Last figure shows the total mass of nitrogen used by the system as a function of time and valve opening. Different colored lines correspond to the different openings of the valve BV1 form 0.25% for the smallest total mass to 0.75% for the largest mass.

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This analysis was performed in two steps. First, we analyzed the response of the system to the variation of control parameters to determine the system sensitivity and reveal the most important trends in the system behavior. In this test the chilldown of the whole transfer line was completed when the last temperature sensor TT91 (see Fig. 4) indicated temperature below 85 K. An example of the results of the sensitivity analysis for the chilldown is shown in the Fig. 5. This analysis reveals existence of a trade-off between the amount of time and commodity losses required to chilldown the system. In particular, opening of the bleed valve BV1 beyond the nominal value (25%) causes a decrease of the chilldown time and an increase of the total mass required to chill the line. At the second step of this test we selected a few most sensitive parameters (BV1, 2, and 3) and performed optimization of the commodity losses under condition that chilldown is completed in less than 1500 sec. The cost function was defined as the total mass of the liquid nitrogen used to complete chilldown. To simplify optimization problem the analysis was limited to the chilldown of the first part of the pipe and the external heat flux at the pipe wall was neglected. The optimization was conducted as follows. First, an Table 1. Results of minimization of commodity losses during chilldown. Columns BV1, 2, and 3 show openings of the bleed valves. Total mass of liquid nitrogen removed from the storage tank to complete the chilldown is shown in the 4-th column. The chilldown time is shown in column 5. The last column shows the difference between the total mass and the mass of liquid nitrogen accumulated in the pipes by the end of chilldown.

BV1 0.90 0.21 0.01 0.50 0.30

BV2 0.45 0.40 0.22 0.00 0.00

BV3 0.945 0.001 0.001 0.001 0.001

Total Mass (kg) 283.53 215.24 156.05 196.41 150.49

Time (s) 1116 1264 1402 1471 1511

Mass (kg) 132.62 82.56 74.66 36.82 23.60

approximate location of the global minimum was found using the direct search in the parameters space. Next, unconstrained nonlinear optimization was performed to find relative openings of the valves that minimize the commodity losses. A fragment of the solution of this optimization problem is shown in the Table 1. These results include both minimum and maximum chilldown time. To draw conclusions from these results we note that to chill the pipe walls in the absence of the heat flux we need approximately 10 kg of the liquid nitrogen. To fill the pipe volume with liquid we need approximately 251 kg of nitrogen. Usually by the end of the chilldown the pipe is only partially filled with liquid, and to estimate how much liquid was actually used to chill the pipes one has to subtract the amount of liquid remaining in the pipe after chilldown from the total amount of liquid removed from the storage tank. The amount of liquid dumped through the bleed valves to complete the chilldown is shown in the east column of the Table 1. It can be seen from the table that during optimization the amount of dumped liquid was substantially reduced from 132 kg to 23 kg. We note that the latter value is close to the minimum value of approximately 10 kg. The minimization was achieved at the expense of increasing chilldown time by nearly 400 sec. As was discussed in the Section II the long chilldown time corresponds to the regime when a large area of the pipe walls remains dry and chilldown occurs via heat transfer from the cold gas. We note that this regime corresponds to the relatively small opening of only one of the three bleed valves, while two other wares remain closed. We can see from this simple example that model-based optimization analysis of the chilldown problem can provide important information required for optimal autonomous control of the cryogenic flow. We will now illustrate how model-based optimization can enhance fault detection and evaluation in the cryogenic transfer line.

V.

Application to the fault detection and evaluation

Integrated health management (IHM) of cryogenic loading involves (but not limited by) the following basic operations fault detection, isolation, and recovery (FDI&R). Model-based approach can enhance capabilities of the IHM at every step of the FDI&R.38 In this section we consider briefly application of the model-based 8 of 11 American Institute of Aeronautics and Astronautics

to the fault detection and isolation. For example to detect faults during loading operation the cryogenic system is monitored continuously to ensure that the measured sensors data remain within margins that define nominal regime of operation. The deviation of the sensors data beyond margins is reported by the IHM system as a fault. The complexity of the chilldown dynamics illustrated in two previous sections forces many existing functional fault modeling approaches to the IHM to exclude chilldown regime from the analysis. However, it is very desirable to detect faults at the earlier stages of operation, i.e. at the chilldown stage.Such an enhancement may prevent excessive commodity losses and catastrophic failures at the later stages when the system is filled with explosive propellant. Fast and accurate predictions of the two-phase flow dynamics within model-based approach can help to create and maintain digital library of the system faults at all stages of loading including chilldown regime. An example of fault detection is depicted in the Fig. 6. In this test the fault (30% CV3 stuck open) is

Figure 6. Fault detection in the transfer line when one of the bleed valves is stack closed. Top (green) and bottom (red) lines indicate margins of the nominal regime. Middle (blue) line correspond sensors readings during loading operation. The fault (CV3 stuck open) is injected at 500 sec. Fault detection (crossing the margins) is shown by arrows.

injected around 300 sec. Increased velocity of the flow causes the fluid temperature along the line to cross bottom margin at a few locations as shown in the figure. Once this crossing is observed the system signals fault detection. Numerical analysis reveals that similar response of the system will be induced by a few other faults including stuck open bleed valve BV2 and mass leak at the location of sensor TT74. These faults form so called ambiguity group. Fault identification within such group is a challenging problem. We now provide an example showing that the model-based optimization tools can further enhance the capabilities of the IHM system helping to identify and evaluate fault within one ambiguity group. We assume that the fault was identified to belong to one of the ambiguity groups and three model parameters are suspected to be off-nominal: openings of the valves BV2 and BV3 and leaking at the location of sensor

Figure 7. The results of fault identification and evaluating within one ambiguity group. (a) Inferred openings of the valves and the value of the leak as a function of number of iterations. (b) Cost function as a function of the number of iterations.

TT74 (which us modeled as a side flow through an orifice of unknown diameter). We chose the cost function S in the form of the square sum of the deviations of the predicted values (T˜k,n ) of the temperature sensors

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from the measured data (Tk,n ) S(c) =

N X K  X

T˜k,n (c) − Tk,n

2

,

n=0 k=1

where the c is the set of parameters of suspected faults and k runs through the sensors T T 02, T T 05, T T 62, and T T 74 (see Fig. 4). The sum over n corresponds to the summation over discrete set of measurements at the time instants t0 , ..., tN . To identify fault within one ambiguity group we use direct search algorithm. The results of the fault evaluation are shown in the Fig. 7. Time required for one using the HM is less than half a second It can be seen from the figure that the algorithm converges to the correct value of the fault eliminating two other possibilities in approximately 50 iterations, i.e. in less than 25 sec on a laptop.

VI.

Conclusion

To summarize, we introduced a hierarchy of two-phase models for cryogenic flow. We validated these models and demonstrated fast and accurate predictions of the chilldown dynamics during cryogenic transfer. We discussed that next important step towards autonomous control of cryogenic systems is development of the model-based optimization tools.

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23 Zuber

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