c
TU Delft, The Netherlands, 2006
NUMERICAL SIMULATION OF THE FAST
DENSE GAS EXPERIMENT
C. Zamfirescu∗, A. Guardone† and P. Colonna∗
∗Energy Technology Section, Process and Energy Department, Delft University of Technology
Leeghwaterstraat 44, 2628 CA Delft, The Netherlands, e-mail: P.Colonna@3mE.TUDelft.nl
†Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano
Via La Masa, 34, 20159 Milano, Italy
Key words: Rarefaction shock waves, dense gas flows, Ludwieg tube, siloxanes
Abstract. The preliminary design of a Ludwieg tube experiment for the verification of the existence of nonclassical rarefaction shock waves in dense vapors is here critically analyzed by means of real gas numerical simulations of the experimental setup. The Flexible Asym-metric Shock Tube (FAST) setup is a dense gas Ludwieg-type tube in which a rarefaction shock is produced after the opening of the valve separating the reservoir from the charge tube. Quasi-onedimensional simulations of the FAST experiment are presented to support the current design of the experiment. Simulations are performed using the CFD software zFlow linked to the appropriate thermodynamic libraries that implement newly developed thermodynamic models. Off-design situations are also investigated numerically and in-clude uncertainties related to thermodynamic modeling, the possibility of using different fluids and the sensitivity to a small change in the nozzle throat section. The preliminary design is confirmed to be feasible and constructions requirements are found to be well within technological limits.
1 INTRODUCTION
vr Pr 1 1.5 2 0.8 0.9 1 S atu ra tion cu rv e Γ =0 BZT region Liquid phase Vapour phase Two-phase region Critical point
Figure 1: Saturation and Γ = 0 curves for siloxane D6 in the vr-Pr plane of reduced specific volume
and pressure, as computed by the PRSV thermodynamic model. Reduced thermodynamic variables are made dimensionless by their critical point values. The nonclassical region (BZT region) is bounded by the saturation curve and the Γ = 0 curve.
i.e., the Flexible Asymmetric Shock Tube (FAST), aimed at demonstrating the existence of nonclassical gasdynamic phenomena in the dense gas flow of complex molecules [33].
Nonclassical flowfields [2, 34, 27, 21, 19] have been already observed experimentally for fluid states encompassing the liquid-vapor [30, 29, 17, 28, 24] and solid-solid [18] transition curve. These exotic gasdynamic phenomena can possibly occur provided that the fundamental derivative of gasdynamics[27], namely
Γ(s, v) = −1 2 ∂2P ∂v2 s . ∂P ∂v s , (1)
modified by Stryjeck and Vera [26] (PRSV) to achieve high accuracy near the liquid-vapor saturation curve [10].
From basic gasdynamic theory [27], an expansion wave entirely embedded in the Γ < 0 region necessarily evolves as a discontinuity, namely as a nonclassical rarefaction shock, whereas a compression wave disintegrates into an isentropic nonclassical compression wave. The first attempt to provide experimental evidence of the existence of nonclassical rararefaction shock waves in the vapor phase was made in 1987 by Borisov et al. [3, 20] in the USSR. A steep rarefaction wave in Freon-13 was observed to propagate with no dis-tortion and it was therefore claimed to be a nonclassical rarefaction shock. However, their results has been challenged by many authors [12, 20, 28, 14, 13] and their measurements are now believed to have been influenced by critical point and two-phase phenomena. In 2003, a nonclassical shock-tube facility [13] has been constructed and tested at the Uni-versity of Colorado at Boulder, but many technological problems, including the imperfect burst of the shock-tube diaphragm and the thermal decomposition of the working fluid, prevented the observation of a rarefaction shock wave in fluid PP10 (Pf - perhydrofluorene, C13F22).
The current research effort is movitated by the lack of experimental evidence of the existence of nonclassical rarefaction shock wave. In [33], the preliminary design of the FAST facility has been presented. The FAST is a dense gas Ludwieg tube in which a rar-efaction shock wave in siloxane D6 is produced and, similarly to the Boulder experiment, its nonclassical character is to be proven by observing that its propagation speed is higher than the local unperturbed speed of sound. In other words this would mean that the rar-efaction wave is indeed a nonclassical shock wave because it moves at supersonic speed. Differently from the Boulder setup, a diaphragmless configuration has been preferred, to increase the repetabeality and reduce the elapsed time between experimental trials. More importantly, the selected working fluid is D6, a fluid belonging to the siloxane fluid series, for which, differently from fluid PP10 used in the Boulder experiment, thermal stability studies and more accurate thermodynamic data are available.
2 PRELIMINARY DESIGN
In the present section, the preliminary design of the FAST setup is briefly described. For a detailed description of the experimental setup and of the preliminary design method the reader is referred to [33]. The FAST facility is in fact a Ludwieg tube made of an high-pressure charge tube connected to a low pressure reservoir, see figure 2. The charge tube and the reservoir are separated by a Fast Opening Valve (FOV). The fluid is initially at rest and the temperature is kept uniform by a suitable thermal control system.
The experiment starts when the FOV is opened, thus connecting the charge tube to the reservoir. For suitable initial states A (charge tube) and R (reservoir), a Rarefaction Shock Wave (RSW) is expected to form and to propagate into the charge tube. Past the RSW, the fluid is accelerated from rest conditions A to post-shock conditions B and it flows into the reservoir through a nozzle. The nozzle is designed to work in chocked conditions, namely, at sonic conditions S at the nozzle throat, to prevent disturbances from propagating from the reservoir into the charge tube, see figure 3.
Close to the end-wall of the charge tube, two static pressure transducers are positioned on the tube surface to measure the incident RSW. The pressure transducers are separated by an axial distance ∆x = x2 − x1, with x1 and x2 axial position of sensor 1 and 2, respectively, so that two different pressure profiles of the RSW are measured at time t1, at which the RSW is at location x1, and at time t2 = t1+ ∆t, namely, when the RSW reaches the second transducer. The wave velocity is computed as uW = ∆x/∆t and compared to the speed of sound cA in the unperturbed state A. If uW > cA, then the wave moves at supersonic speed with respect to unperturbed conditions and it is indeed a nonclassical RSW.
If on one hand the gasdynamics phenomena occurring in the FAST apparatus are indeed easy to understand and quite similar to standard Ludwieg tube applications, on the other hand very complex and accurate thermodynamic models are to be used in the computations, due to the high non-ideality of the operating conditions, see figure 1, and to the fact that the prediction of nonclassical gasdynamics phenomena is very sensitive to the accuracy of the underlying thermodynamic model [16]. Due to the strict constraints on the actual pressure and temperature intervals in which nonclassical gasdynamic phenomena can possibly occur, with maximum pressure and temperature differences across a RSW in the order of 1 bar and of five to ten degree Kelvin, respectively, the intial conditions for the experiments are to be carefully selected—and controlled during the experiment—to maximize the strength, and hence the detectability, of the phenomena to be observed.
Charge tube Nozzle Reservoir W State B State A RSW P x x RSW x i t e Pressure probes 2 1 A P PB PS xW x
Figure 2: The FAST setup: concept and expected pressure profile after the opening of the FOV separating the charge tube and the reservoir (qualitative). A RSW propagates into the charge tube at supersonic speed. Past the RSW, the fluid is accelerated from rest conditions A to post-shock conditions B and flows into the resevoir through the nozzle. At the nozzle throat, sonic conditions S are attained.
vr Pr 1.5 2 2.5 3 3.5 4 0.6 0.7 0.8 0.9 1 Ise ntrop ic ex pa nsio n (n oz zle) Satu ra tion curv e A S Γ =0 Two-phase region RSW B x / L M ac h (l abor at or y re fe re nc e) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Solution at time t = t I Rarefaction shock wave Nozzle throat (sonic) Isentropic expansion through the nozzle State A State B
Figure 3: Left: Expansion in the charge tube up to the nozzle throat in the vr-Pr plane. Right: Mach
Quantity Value Units Description
RSW uW 35.03 m/s Wave speed
MW 1.023 - Wave Mach number
State A PA 9.122 bar Pressure
ρA 188.19 kg/m3 Density TA 368.96 ◦ C Temperature ΓA -0.1233 - Fundamental derivative uA 0.0 m/s Velocity cA 34.23 m/s Speed of sound MA 0 - Mach number
State B PB 8.017 bar Pressure
ρB 127.27 kg/m3 Density TB 363.70 ◦ C Temperature ΓB 0.1388 - Fundamental derivative uB 16.78 m/s Velocity cB 51.79 m/s Speed of sound MB 0.324 - Mach number
State S PS 5.572 bar Pressure
ρS 66.85 kg/m3 Density TS 356.69 ◦ C Temperature ΓS 0.6177 - Fundamental derivative uS 75.50 m/s Velocity cS 75.50 m/s Speed of sound MS 1 - Mach number ˙
mS 2.681 kg/s Mass flow rate
State R PR 1.00 bar Pressure
Initial conditions ρR 8.89 kg/m3 Density
TR 368.96 ◦ C Temperature
State R PR 1.796 bar Pressure
Final conditions ρR 16.4 kg/m3 Density
TR 361.81 ◦ C Temperature
Table 1: Initial, post-shock and nozzle throat conditions for the FAST experiment for fluid D6 under the
Quantity Value Units Description
Characteristic dimensions Dct 40 mm Charge tube internal diameter
Act 1256 mm2 Charge tube cross-sectional area
Lct 10 m Charge tube length
Vct 0.01256 m3 Charge tube internal volume
Dnt 26.01 mm Nozzle throat internal diameter
Ant 531.2 mm2 Nozzle throat cross-sectional area
Ln 150 mm Nozzle length
Lre 0.1 m3 Reservoir volume
Characteristic times tfov 3.3091 ms Minimum valve opening time
tI 146.0 ms Shock formation time at xI= 5 m
tend 288.8 ms Overall experimental time
Table 2: Characteristic dimensions and times for the FAST experiment according to the PRSV model of D6. The overall experimental time tendis the elapsed time between the FOV opening and the instant in
which the RSW reaches the charge tube end-wall.
that ΓB > 0, namely, past the RSW classical gasdynamic phenomena are expected. To avoid the presence of temperature gradients along the charge tube and hence to simplify the temperature control system, the initial reservoir (state R) temperature is kept equal to the charge tube temperature, whereas the pressure is chosen in such a way that it is always lower than that at the sonic throat of the nozzle, which is therefore assumed to work in choked conditions during the experiment—an hyphothesis whose correctness is verified in the next section. Relevant setup dimensions and experimental characteristic times are listed in table 2. The minimum opening time tIfor the FOV has been computed as follows. The shock wave is assumed to be completely formed at the intersection point xI at which the first first C− characteristic line (moving at speed −cA) intersect with the last C− characteristic line (moving at speed u
B − cB) of an ideally smooth rarefaction profile generated at the FOV location starting from the initial time t0 = 0 at which the FOV starts opening till time tI, namely, when the FOV is completely opened. The opening time is computed here by imposing xI= 5 m, see [33].
3 NUMERICAL SIMULATIONS OF THE FAST EXPERIMENT
t [ms] P [ba r] 240 245 250 255 260 265 270 8 8.2 8.4 8.6 8.8 9 9.2 Pressure sensor 1 PA PB t [ms] P [ba r] 0 2 4 6 8 5 6 7 8 9 Simulation (835 nodes) Simulation (1669 nodes) Simulation (3335 nodes) Exact solution Nozzle throat PS
Figure 4: Pressure signals at the nozzle throat (left) and at measurement station 1 (right) as computed using three different grid resolutions. Numerical results are compared to the “exact” solution, in which the flow inside the nozzle is assumed to be steady.
the velocity lying in a plane normal to the tube axis are assumed to be zero and the Euler equations describing compressible inviscid flows with zero thermal conductivity reduces to a system of three equations in the three unknowns density ρ, axial momentum ρu, with u axial velocity, and total energy per unit volume Et as follows, see e.g. [1]
∂ρ ∂t + ∂(ρu) ∂x = − ρu A dA dx, ∂(ρu) ∂t + ∂ ∂x ρu 2+ P = −ρu2+ P A dA dx, ∂Et ∂t + ∂ ∂x E t+ P = −uEt+ P A dA dx, (2)
the reservoir: the cross-sectional area is assumed to vary smoothly from the nozzle throat value of 531.2 mm2 to a value of 1 m3 over a distance of 0.5 m.
The numerical results are presented in figure 4, where the pressure signal at the throat location and at measurement station 1 are shown as a function of the elapsed time for the three considered grids. Numerical results are found to be almost independent from the grid resolution and to converge to the “exact” solution. The “exact” solution is in fact computed by assuming that the flow in the nozzle connecting the charge tube to the reservoir is steady and chocked for the entire duration of the experiment. Moreover, the RSW is assumed to be completely formed at the exit of the nozzle, with no perturbances resulting from the interaction with the nozzle itself. The above assumptions are relaxed in the simulation to investigate their correcteness. In fact, the pressure at the nozzle throat reaches its steady-state chocked value almost immediately, namely, about 4 ms after the FOV opening—which is assumed here to open instantaneously—and the influence of the interaction with the nozzle on the RSW propagation time is found to be negligible, cf. figure 4 (right).
The preliminary design is therefore confirmed by the present numerical simulations, in terms of both initial conditions leading to the occurrence of a RSW propagating in the charge tube and to the characteristic dimensions and times of the experiment, see table 1 and 2. The intermediate grid made of 1669 nodes is therefore used in the following computations to investigate off-design operating conditions.
4 OFF-DESIGN CONDITIONS
x [m] P [ba r] 4.5 5 5.5 6 6.5 7 7.5 0.6 0.7 0.8 0.9
75% Design throat area 100% Design throat area 125% Design throat area
x [m] M 10 10.1 10.2 0 0.5 1 1.5 2 2.5 N oz zl e thr oa t M = 1 N oz zl e out le t N oz zl e inl et
Figure 5: Pressure (left) and Mach number (right) distribution for the nominal nozzle throat cross-sectional area and for smaller and larger areas (nominal ± 25 %).
4.1 Sensitivity to the nozzle throat cross-sectional area
In figure 5, the computed pressure signal at measurement station 1 is plotted as a function of time for two different nozzle throat cross-sectional areas, to investigate the sensitivity of the measurements to this design parameter. The pressure signal for the nominal cross-sectional area of 531.2 mm2, see table 2, is also shown for comparison.
The first simulation refers to a reduced nozzle cross-sectional area equal to 75 % of its nominal value. This situation can possibly occur due to the presence of a viscous boundary layer which reduces the effective inviscid cross-sectional area. Note that the boundary layer forms at the RSW location, where the fluid is set into motion towards the reservoir, so, differently from standard gasdynamic nozzles discharging from a reservoir, its thickness is not negligible at the nozzle inlet. For the reduced value of the throat area, the pressure jump across the RSW is seen to be reduced by approximately one third. The reduction of intensity of the pressure jump is due to the fact that the inlet to throat area ratio of the nozzle is augmented and therefore a stronger expansion takes place in the nozzle itself, thus reducing the expansion ratio of the RSW. However, the qualitative picture is left unchanged, with a (weaker) RSW moving towards the end-wall of the charge tube.
mixed wave made of a leading RSW plus a tailing isentropic rarefaction wave taking place in the Γ > 0 region.
4.2 Use of different working fluids
The working fluid D6 has been chosen for the present experiment as a compromise between thermal stability and the maximum strength of the envisaged RSW. The latter has been obtained via state-of-the-art thermodynamic models, whose accuracy however still remains to be verified. In particular, it is remarkable that the very existence of a nonclassical region, namely, the fact that a given fluid is indeed a BZT fluid, strongly depends on the thermodynamic model used in the computations, see [15]. It is therefore advisable to investigate the possibility of using different fluids in the Flexible Asimmetric Shock Tube facility, which should indeed be “Flexible” enough to accomodate almost all the heavier siloxanes fluids and possibly other candidate BZT fluids.
To this purpose, in table 3 and 4, significant gas states and characteristic dimensions and times of the experiments are computed for different fluids belonging to the siloxane series. In particular, the lighter D4 (octacamethylcyclotetrasiloxane, C8H24O4Si4) and D5 (decamethylcyclopentasiloxane, C10H30O5Si5) and the heavier MD4M (tetradecamethyl-hexasiloxane, C14H42O5Si6) siloxane fluids are considered in addition to fluid D6.
The initial pressure in the charge tube (state A) is higher for less complex fluids, with a maximum value of 11.998 bar for D4, to be compared with the design value of 9.122 bar and the minimum value of 8.385 for the more complex MD4M. Converserly, the temperature in state A is lower for lighter fluids, with the lowest value of 305.79◦ C for D
4 (368.96◦ C for D
6 and 376.83◦ C for MD4M). The above also holds for the reservoir (state R), which is kept at a pressure of 1 bar for all fluids and at the same temperature with respect to the charge tube. Note also that the mass flow across the nozzle is maximum for the heaviest MD4M fluid. These differences are well within the design requirements for the FAST experiment.
As far a the strength and speed of propagation of the RSW, both the pressure difference across the wave and the wave Mach number diminish with molecular complexity to a minimum value of 0.238 bar and 1.0003, respectively, for fluid D4. Due to the very small size (in terms of pressure and temperature ranges) of the BZT region for less complex fluid, the RSW tends to become an acoustic wave moving at sonic speed. More importantly, see table 4, due to the proximity of the wave states to the Γ = 0 line, where the acoustic wave speed is stationary [5], the valve opening time becomes very small for the lighter molecules, down to a value of 0.0314 ms for fluid D4, which is unrealistic for a FOV and a diaphragm as well.
Quantity D4 D5 D6 MD4M Units RSW uW 48.14 42.32 35.04 33.75 m/s MW 1.0003 1.003 1.023 1.033 -State A PA 11.998 10.613 9 .122 8.385 bar ρA 165.86 177.36 188.19 179.22 kg/m3 TA 305.79 339.84 368.96 376.83 ◦ C ΓA -0.01833 -0.0535 -0.1233 -0.1485 -cA 48.13 42.20 34.23 32.68 m/s State B PB 11.76 10.056 8.017 7.248 bar ρB 156.32 150.92 127.27 115.12 kg/m3 TB 304.75 337.32 363.70 371.29 ◦ C ΓB 0.01026 0.0430 0.1388 0.1627 -uB 2.94 7.41 16.78 18.79 m/s cB 51.08 49.73 51.79 52.54 m/s MB 0.057 0.149 0.324 0.358 -State S PS 8.072 6.903 5.572 5.062 bar ρS 74.86 74.14 66.85 61.50 kg/m3 TS 293.51 327.98 356.69 365.0 ◦ C ΓS 0.5775 0.5920 0.6177 0.6224 -cS 84.20 78.80 75.50 75.21 m/s ˙ mS 0.577 1.406 2.681 2.718 kg/s State R PR 1.000 1.000 1.000 1.000 bar Initial conditions ρR 6.35 7.53 8.69 8.89 kg/m3 TR 305.79 339.84 368.96 376.83 ◦ C State R PR 1.262 1.406 1.796 1.816 bar Final conditions ρR 8.15 10.86 16.43 17.08 kg/m3 TR 301.68 334.71 361.81 370.03 ◦ C
Table 3: Fluid states under the PRSV thermodynamic model for siloxane fluids D4, D5, D6 and MD4M.
Quantity D4 D5 D6 MD4M Units
Characteristic dimensions Dnt 10.80 17.51 26.01 27.36 mm
Ant 91.6 240.7 531.2 587.76 mm2
Characteristic times tfov 0.0314 0.3329 3.3091 4.8255 ms
tI 103.9 118.5 146.0 152.9 ms
tend 207.8 236.6 288.8 301.1 ms
Table 4: Characteristic dimensions and times for the FAST experiment according to the PRSV models of D4, D5, D6 and MD4M. Only fluid-dependent dimensions/times are reported. See table 2 for the
nomenclature.
or by mixture of linear siloxanes, which are characterised by a high thermal stability limit. A state-of-the-art thermodynamic model for mixtures of siloxanes is currently under development.
4.3 Uncertainties due to the thermodynamic model
To conclude the present analysis on off-design conditions and on the sensitivity of the preliminary setup on the uncertainties of the considered models, the PRSV model is validated against the more complex Span-Wagner (SW) model of D6 [25, 7]. The latter is still in a preliminary development phase and therefore the present results should be viewed as preliminary as well.
For the sake of the present discussion, the SW model is assumed to provide the exact thermodynamic properties of D6, of which the PRSV provides an approximation. It should be recalled that both models are indeed unvalidated models as far as the determination of the value of the fundamental derivative of gasdynamics Γ is concerned and therefore the present analysis is to be intended only as a rough evaluation of the criticalities due to the thermodynamic models, similarly to the study presented in [16].
t [ms] P [ba r] 0 50 100 150 200 250 300 5.5 6 6.5 7 7.5 8 8.5 9 9.5 RSW (PRVS) Mixed RW (SW) APRSV, SW BSW BPRSV v [m3/ kg] P [ba r] 0.01 0.02 4 5 6 7 8 9 10 PRSV EOS SW EOS S atu ra tion cu rve PR S V Γ = 0 SSW Two-phase region BPRSV BSW ASW SPRSV APRSV
Figure 6: FAST experiment according to the PRSV and SW models. The saturation curve and the Γ = 0 line computed by the SW model are not shown.
As anticipated, these are only preliminary results based on a draft version of the SW model for D6 and are reported here only to put into evidence the strong dependence of the presented results on the chosen thermodynamic model.
5 CONCLUSIONS
The preliminary design of the Flexible Asymmetric Shock Tube (FAST) setup has been validated by means of numerical simulations of the experiment. The numerical results confirm the choice of the initial conditions for the experiment; in particular, the assumption of considering a steady flow inside the nozzle connecting the chage tube to the reservoir has been found to be realistic.
for example MD4M, is instead possible since operating pressures and temperatures remain close to design values. The question on whether or not the working fluid will be thermally stable in such demanding operating conditions remains open.
Finally, a rough estimate of the sensitivity of the present design with respect to the accuracy of the thermodynamic model is provided by comparing the results obtained by two different thermodynamic models of D6. Large differences in the (computed) experi-mental output have been found in this case and they point to the accurate determination of the thermodynamic properties of the working fluids as probably the most critical issue of the FAST experiment. Further work is currently under way to improve the SW model and to validate it against experimental results. In particular, a measurement campaign aimed at filling the gaps in the knowledge of the thermodynamic properties of siloxanes has been recently carried out and includes measurements of speed of sound as well as ab initio computations of the specific heat in the dilute gas state [22].
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