Rarefaction Shock Waves: a contribution to experiments and simulations

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in collaboration with

Delft University of Technology

Process & Energy Department

Master of Science in

Mechanical Engineering

Rarefaction Shock Waves:

a contribution to experiments

and simulations

Thesis

Sergio Segreto

October 8, 2008 Supervisors:

Prof. P. Colonna

Prof. P. Di Marco

N. R. Nannan, MSc

Academic Year 2007 - 2008

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all the people I met

We have the responsibility, as engineers,

to do the best we can with what we have

So different, but all with the same stupid problems..

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Abstract Sergio Segreto

Rarefaction Shock Waves:

a contribution to experiments and simulations

Abstract

Complex thermodynamic models show that compressible flows of fluids having a sufficiently large heat capacity may exhibit rarefaction shock waves, also referred to as nonclassical phenomena. Apart from the numerous theoretical studies available in the scientific literature, no experimental evidence of RSWs in the dense vapor thermodynamic region is, however, currently available.

In order to prove their existence, a new facility has been built at the Process&Energy Department of the Delft University of Technology. The present work mainly aims at contributing to the commissioning of the shock tube. Computational fluid dynamics simulations, employing a real gas equation of state, are performed to understand the complex unsteady phenomena occuring inside the shock tube during experimental runs. The numerical solution found shows significant departure from the theoreti-cally expected solution. Though based on various assumptions, the achieved results provide insight into the real flow field that the experimenter may generate in shock tube facility.

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Abstract i

Contents ii

1 Introduction 1

2 Theoretical Aspects of BZT Effects in Dense Gases 4

2.1 Overview . . . 4

2.2 Siloxanes: working fluids suitable for experimenting BZT effects . . . . 8

2.3 Admissibility conditions for shock waves . . . 11

2.3.1 The conservation equations . . . 11

2.3.2 The entropy condition . . . 15

2.3.3 The mechanical stability condition . . . 16

2.4 Expansion waves through the BZT region . . . 19

2.4.1 Shock waves . . . 19

2.4.2 Mixed waves . . . 22

2.4.3 RSW with maximum supersonic Mach number: the Maximum Mach Locus (MML) . . . 25

2.5 Shock formation in a general fluid . . . 28

3 Commissioning of the Flexible Asymmetric Shock Tube (FAST) 32 3.1 Overview . . . 32

3.2 Mechanical components . . . 35

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CONTENTS

3.3.1 Pressure instruments . . . 42

3.3.2 Temperature instruments . . . 43

3.3.3 Other devices . . . 43

3.4 Data Acquisition and Control . . . 44

3.4.1 DAQ system . . . 44

3.4.2 Control system . . . 45

3.5 LabView . . . 46

3.5.1 The control loop VIs . . . 47

3.5.2 The operator procedure VIs . . . 50

3.5.3 The monitoring VIs . . . 55

4 Simulations 57 4.1 Overview . . . 57

4.2 CSW in a simple shock tube: the Sod problem . . . 58

4.2.1 Theoretical description of the problem . . . 58

4.2.2 Modeling of the problem . . . 60

4.2.3 Numerical results . . . 62

4.3 RSW in a simple shock tube: the triple discontinuity problem . . . 67

4.4 RSW in a simplified geometry of the FAST: an inviscid solution . . . . 76

4.5 Unsteady simulation of the RSW formation during the opening of the valve: an Eulerian approach with a sliding mesh . . . 90

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A Mathematical proofs 108

A.1 The chain rule . . . 108 A.2 The triple product rule . . . 109 A.3 Entropy change across a weak shock wave . . . 109

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Chapter 1 Introduction

A shock wave is a thin region across which there is flow of matter such that in the direction of the flow, the substance experiences an abrupt change of state. As an exact result of the Euler equations, all fluid properties - pressure, velocity, density, etc. - are discontinuous across the shock, meaning that the state variation occurs in an infinitesimally thin region. The pressure jump is the thermodynamic property that characterizes a shock. If the pre-shock pressure is less than the post-shock pressure, the wave is named compression shock wave (CSW); conversely, if the pre-shock pressure is greater than the post-shock pressure, the wave is called a rarefaction shock wave (RSW).

The theory on gasdynamics is mostly based on the ideal gas law. It is well known, however, that gaseous substances can behave differently from their ideal gas counterpart, even at moderate pressures, under both a qualitative and quantitative point of view. Therefore, in a relatively large thermodynamic region, the application of more complex equations of state (EoS) is necessary. The utilization of more complex thermodynamic models shows, for example, that compressible flows of fluids having a sufficiently large heat capacity may exhibit various interesting phenomena - referred to as nonclassical phenomena - including rarefaction shock waves, that are contrary to the predictions of the classical theory of gasdynamics.

Apart from the numerous theoretical studies available in the scientific literature, no experimental evidence of nonclassical gasdynamics phenomena in the dense

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va-por region is, however, currently available. In 1983, Borisov and co-workers claimed to have observed a negative shock wave in chlorotrifluoromethane in close proximity to the critical point, where the initial state of the fluid and the state of the substance in the wave itself were always in the single-phase region. The obtained results were confuted by Fergason. Another experiment was conducted by Thompson to generate rarefaction shocks with phase transition. The last attempt known was an experiment using a shock tube [8]; the experiment aimed at generating, in fluid PP10, a triple discontinuity flow field with a nonclassical rarefaction shock . Many problems were encountered however, due to incomplete rupture of the diaphragm, and thermal de-composition of the working fluid with the formation of highly corrosive hydrofluoric acid.

Many technical problems, such as the identification of fluids supposed to show nonclassical phenomena among molecularly complex substances, and the required high temperatures for experimental runs, that could eventually lead to thermal de-composition and formation of toxic and flammable gases, are the main causes for the lack of experimental proofs of nonclassical phenomena in the vapor phase.

Due to their nontoxicity, excellent thermal and chemical stability, and limited flammability, siloxanes are possibly the most suitable fluids from an engineering point of view, if compared to other classes of fluids. An added benefit of siloxanes is that these substances are already employed as working media in oganic Rankine cycle engines (ORCEs) and are proposed for promising ORC applications.

As treated in [7] and [3], ORCEs emloying heavy working fluids are an appeal-ing option for low-power applications, since they have superior efficiency compared to steam Rankine cycle engines for heat source temperatures below around 900 K. Moreover they typically require only a single-stage expander. This reflects in a much simpler structure than multistage steam systems. For the past several decades, thou-sands of ORCEs have been built and used for terrestrial applications with power outputs ranging from 1 to 1000 kW. A few examples of applications that have used efficient and reliable ORCE power sources include communication stations, satellite communication power supplies, irrigation pumps, air conditioners and turbogenera-tors.

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The ORC engine is one of the applications that can take advantage from non-classical phenomena. The isentropic efficiency of these turbines can in principle be increased if they are operated with fluids that exhibit nonclassical behavior, and if the expansion process is carried out, even partially, in the nonclassical gasdynamic region. One of the reasons that lead to loss of efficiency in thermodynamic cycles is the generation of steady compression shock waves at the outlet of stator turbine blades. Due to the strong adverse pressure gradient across the compression shock, boundary layer detachment occurs. As a consequence, significant loss of lift and potential to extract work from the working fluid is encountered. A shock-free flow in the stator could be in principle obtained, if the thermodynamic state at the stator inlet is appropriately chosen such that the expansion process is carried out, even partially, inside the nonclassical region of the working fluid.

A new facility has been built at the Process&Energy Department of the Delft Uni-versity of Technology to give experimental evidence of the existence of expansion shock waves. The main aim of the present work is to contribute to the commissioning of the shock tube. An attempt is made to provide insight into the flow field that the experimenter may generate in shock tube facility. Computational fluid dynamics (CFD) simulations employing real gas EoS are performed to understand the com-plex unsteady phenomena occuring inside the shock tube during experimental runs, focusing on how the finite opening time of the valve and the presence of a convergent-divergent nozzle influence the strength and the formation of the nonclassical shock wave. In detail, chapter 2 deals with theoretical aspects of nonclassical phenomena, chapter 3 describes the set up of the experimental facility and the control and monitor-ing logic of all the components, and chapter 4 presents the numerical results obtained from the simulations. Finally chapter 5 outlines the concluding remarks.

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Theoretical Aspects of BZT E

ffects in

Dense Gases

2.1 Overview

Much of the theory concerning gasdynamic flows is based on the perfect gas assumption, i.e., gases are often supposed to correspond to the ideal gas EoS and moreover to have temperature-independent isochoric specific heat Cv. The perfect

gas model however has a limited range of applicability and cannot take into account dense gas effects and phase changes. To account for these effects, more complex thermodynamic models are necessary. The simplest nonideal-gas state equation that can be used to model nonclassical gasdynamic effects is the van der Waals equation of state (vdW EoS) p+ an 2 V2 ! (V − nb)= nRT, (2.1)

where p is the pressure of the gas, n is the number of moles, V the volume of the gas, R the universal gas constant, T the absolute temperature, a and b the van der Waals constants which depend on the material. Differently from the ideal gas EoS, equation 2.1 considers the finite volume of the particles and the electromagnetic

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Overview interactions among them, to a certain extent.

Numerous theoretical and numerical studies on nonclassical gasdynamics are based on the simple van der Waals EoS. However, it is well known that thermo-dynamic data obtained from the vdW EoS are highly inaccurate, even more if their evaluation requires the computation of first and second order derivatives of the equa-tion of state. Recent studies on nonclassical fluids therefore rely on more complex EoS, such as the Martin-Hou EoS, or the Span-Wagner EoS.

As shown in [21], in dense gases with large isochoric specific heat capacity, the isentropes tend to superimpose upon the isotherms in a (v, p) thermodynamic plane because, if Cv → ∞, ∂p ∂v _s= ∂p ∂v _T − T Cv ∂p ∂T 2 v → ∂p ∂v _T, (2.2) where v is the specific volume, and s the entropy. Equation 2.2 shows that in a limited region close to the critical point, where the isotherms of all fluids must have a downward curvature, the isentropes of fluids of sufficient molecular complexity display two inflection points which bound a region of negative concavity.

By employing the vdW EoS, Bethe and Zel’dovich showed independently that substances whose ratio between the universal gas constant and the isochoric specific heat, i.e.,δ = R

Cv is smaller than approximately 0.060, have a finite region where the

concavity of the isentropes in the (v, p) thermodynamic plane is negative (downward). The result obtained by Bethe and Zel’dovich can be expressed using the fundamental derivative of gasdynamic, as introduced by Hayes,

Γ ≡ −v 2 ∂2_p ∂v2 _s ∂p ∂v _s (2.3)

whereΓ is the fundamental derivative. Γ is an important thermodynamic property, since it represents a nondimensional measure of the curvature of the isentropes and it therefore determines the qualitative behavior of compressible flows, as extensively explained in the following. Since the denominator

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∂p ∂v _T ≤ 0

everywhere from the requirement of thermodynamic stability, then ∂2_p ∂v2 _s> 0 → Γ > 0,

which is valid, for example, for dilute gases far from the critical point, while ∂2_p ∂v2 _s< 0 → Γ < 0,

which is instead valid for dense gases near the critical point. Here, the sign of∂

2_p ∂v2 _s represents the curvature of an isentrope in a (v, p) thermodynamic plane. Therefore, in the finite region of negative curvature of the isentropes, the value ofΓ is negative. A fluid that exhibits aΓ < 0 region in the vapor phase is defined as a BZT fluid, in honor of Bethe, Zel’dovich and Thompson for their contribution in the field of nonclassical gasdynamics.

Limiting this study to the treatment of single-phase flows, the BZT region is delim-ited by the dew line and by theΓ = 0 locus, which represents the locus of the points where the isentropes change their curvature, as shown in figure 2.1.

By using the more sophisticated Martin-Hou EoS, there have been identified sev-eral fluids that have an embedded region of negative nonlinearity in the dense gas phase, as in [3].

The fundamental derivative might be rewritten as Γ ≡ 1 + ρ c ∂c ∂ρ _s, as reported in [19], with ρ = 1

v the density and c the adiabatic speed of sound, defined as c ≡ s −_v2∂p ∂v _s. (2.4)

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Overview T =_T c Satu rati_on cur_ve Γ = 0 Γ> 0 s₌ cons t. s = con_st. s = con st. Γ< 0 v / v_c P / Pc 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2 Liquid-vapor critical point Liquid phase Gas phase Two-phase region

Figure 2.1: Liquid-vapor saturation curve andΓ < 0 region (shaded region) for a BZT fluid. Thermodynamic properties computed from the van der Waals model. Picture taken from [22].

Therefore, in the region whereΓ < 1, that embeds the region of negative nonlin-earity, the speed of sound increases over isentropic expansions and decreases over isentropic compressions. This implies that the variation of the sound speed with density is opposite with respect to the classical behavior of compressible gas flows. The consequence of this behavior is that inside the BZT region a dense gas can admit rarefaction shock waves (RSW) and compression fans (CF).

Moreover, other unusual (nonclassical) mixed flow fields are possible due to the finite size of the BZT region. For instance, expansion waves that traverse the Γ < 0 region can display wave splitting, and a fan-shock-fan flow field can generate. Conversely, during compression a shock-fan-shock mixed wave field is also possible.

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2.2 Siloxanes: working fluids suitable for

experiment-ing BZT e

ffects

Nonclassical gasdynamic effects can occur in a flow field if the ratio of the universal gas constant and the isochoric specific heat δ is small enough, depending on the equation of state used. Nevertheless, if nonclassical gasdynamics is to be proven, other important considerations which need to be taken into account are the availability of thermophysical property data, fluid characteristics, e.g., thermal decomposition temperature, fluid availability and price, and safety concerns such as toxicity and flammability.

According to [5], currently three classes of fluids are believed to exhibit a Γ < 0 region in the vapor phase, namely, hydrocarbons, perfluorocarbons, and siloxanes. Nevertheless, any application exploiting nonclassical gasdynamics effects would in-volve temperatures of the order of the critical temperature Tc of the explored fluid.

Therefore, complex hydrocarbons would pose difficult technical problems, due to their flammability. Perfluorocarbons, known to be extremely thermally stable among organic fluids, are characterized by a very complex molecular structure, which entails high values ofδ and low values of Γ. Moreover, their critical temperature, lower with respect to siloxanes, implies that the BZT region is located at lower temperatures. For these reasons, complex perfluorocarbons are potentially very attractive. On the other hand, decomposition products of perfluorocarbons, such as hydrofluoric acid (HF), are highly aggressive and toxic compounds, whereas thermal decomposition of siloxanes leads to polymerization and the polymer product is not toxic. Taking also into account the nowadays knowledge of the thermophysical properties, perfluoro-carbons are at a far lower stage if compared to siloxanes. This reflects in difficulties when designing or setting the operating conditions of any application. Furthermore, flammability of siloxanes is much lower than that of hydrocarbons. Remarkably enough, siloxanes are currently utilized in organic Rankine cycle turbines, which is the first application proposed for the exploitation of BZT effects. Thus, this family of fluids is the most suitable from an engineering point of view.

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Siloxanes: working fluids suitable for experimenting BZT effects possibly the best class of fluids for experiments and applications involving nonclassi-cal gasdynamic phenomena. Siloxanes are a family of fluids composed of molecules containing alternate silicon and oxygen atoms in either a linear or cyclic arrangement usually with two or three methyl groups CH3attached to each silicon atom, as shown

in figure 2.2.

Figure 2.2: Molecular structure of two cyclic (D4on the left side; D5in the middle) and one linear (MD5M) siloxane. Picture taken from [5].

At ambient conditions, siloxanes are liquid, odorless, transparent and appear slightly viscous. In table 2.1, relevant thermophysical properties of the cyclic molecules D4 (octamethylcyclotetrasiloxane, C8H24O4Si4), D5 (decamethylcyclopentasiloxane,

C10H30O5Si5), and D6 (dodecamethylcycloexasiloxane, C12H36O6Si6) together with

the linear molecules MDM (octamethyltrisiloxane, C8H24O2Si3), MD2M

(decamethyl-tetrasiloxane, C10H30O3Si4), MD3M (dodecamethylpentasiloxane, C12H36O4Si5), and

MD4M (tetradecamethylexasiloxane, C14H42O5Si6) are reported.

To comply with the demanding constraints on the accuracy of the thermodynamic model, equations of state in the Span-Wagner functional form, as reported in [5], are adopted. The correlation developed by Span and Wagner is a multiparameter function where the reduced Helmholtz energy, i.e.,ψ ≡ Ψ

RT, is expressed as a function of the reduced density,γ ≡ ρ

ρc

, and the inverse of the reduced temperatureτ ≡ Tc T. The functional form depends on 12 substance-specific parameters n1, . . . , n12and reads

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MW Tc pc vc TTP

Fluid name [kg/kmol] [K] [kPa] [m3_/kmol] _[K]

MDM 236.53 564.1 1415.2 0.921 187.15 MD2M 310.69 599.4 1179.4 1.196 205.15 MD3M 384.84 628.4 945.0 1.458 192.00 MD4M 458.99 653.2 877.5 1.607 214.15 D4 296.62 586.5 1332.0 0.958 290.25 D5 370.77 619.2 1160.0 1.267 235.15 D6 444.92 645.8 961.0 1.594 270.15

Table 2.1: Relevant thermophysical property data for the linear and cyclic siloxanes. Data taken from [5]. ψ(τ, γ) = ψ0_{(τ, γ) + ψ}r_{(τ, γ) =} = ψ0_{(τ, γ) + n} 1γτ0.250+ n2γτ1.125+ n3γτ1.500+ n4γ2τ1.375+ n5γ3τ0.250 +n6γ7τ0.875+ n7γ2τ0.625e −γ_{+ n} 8γ5τ1.750e −γ_{+ n} 9γτ3.625e −γ2 +n10γ4τ3.625e −_γ2 + n11γ3τ14.5e −_γ3 + n12γ4τ12.0e −_γ3 . (2.5)

Here, ψ0 _{represents the ideal gas contribution to the Helmholtz free energy and}

it is obtained from an ideal gas heat capacity correlation, and ψr _{accounts for the}

difference between real gas and ideal gas behavior. Equation 2.5 is valid for many classes of nonpolar and weakly polar fluids and it was developed by means of an optimization algorithm that considered data sets for different fluids and fluid families simultaneously. According to the adopted thermodynamic model, several fluids of the siloxanes class, namely, D6, MD3M and MD4M, exhibit a thermodynamic region

where the fundamental derivative of gasdynamics is negative. However uncertainties in the estimation ofΓ remain large and difficult to evaluate, because experimental data for these complex molecules are still scarce, especially in the region of interest.

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Admissibility conditions for shock waves

2.3 Admissibility conditions for shock waves

2.3.1 The conservation equations

The treatment of shock waves as discontinuities, or surfaces of zero thickness, follows as an exact solution to the Euler equations, and therefore it is an idealization of inviscid gasdynamics. By using two different reference frames, that is, an inertial reference frame and a shock related reference frame, a compression shock wave (CSW) and a rarefaction shock wave (RSW) can be represented as in figure 2.3, where U is the velocity of a fluid particle with respect to the inertial reference frame (absolute velocity), W is the velocity of the shock wave with respect to the inertial reference frame and u is the velocity of a fluid particle with respect to the shock related reference frame (relative velocity).

To determine the behaviour of the fluid, that is, the thermodynamic states and the velocity of each particle of the fluid upstream and downstream the shock, the mass, momentum and energy balance equations must be applied. These well known governing equations for an unsteady, three-dimensional, compressible, viscous flow are, in their conservation forms,

continuity equation

∂ρ

∂t + ∇ · (ρ Vabs)= 0 (2.6)

momentum equation, x component ∂ ρ U ∂t + ∂ ∂x·ρ U Vabs = − ∂p ∂x+ ∂τxx ∂x + ∂τyx ∂y + ∂τzx ∂z + ρ fx (2.7)

momentum equation, y component ∂ ρ V ∂t + ∂ ∂y ·ρ V Vabs = − ∂p ∂y + ∂τxy ∂x + ∂τyy ∂y + ∂τzy ∂z + ρ fy (2.8)

momentum equation, z component ∂ ρ W ∂t + ∂ ∂z·ρ W Vabs = − ∂p ∂z+ ∂τxz ∂x + ∂τyz ∂y + ∂τzz ∂z + ρ fz (2.9)

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(a)CSW in inertial reference frame (b)RSW in inertial reference frame

(c)CSW in shock related reference frame (d)RSW in shock related reference frame

Figure 2.3: A pressure (density) discontinuity, represented in an inertial reference frame (a, b) and in a shock related reference frame (c, d). The terms subsonic or supersonic are referred to the reference frame in use. It is assumed that the flow is perpendicular to the shock front, i.e., the shock is a normal shock. Picture taken from [21].

energy equation ∂ ∂t      ρ      e+ |_Vabs|2 2            + ∇ ·      ρ      e+ |_Vabs|2 2      V abs      = = ρ ˙q + _∂x∂ k∂T ∂x ! + _∂y∂ k∂T ∂y ! + _∂z∂ k∂T ∂z ! −∂ Up ∂x − ∂ Vp ∂y + −∂ Wp ∂z + ∂ (Uτxx) ∂x + ∂ Uτyx ∂y + ∂ (Uτzx) ∂z + ∂ Vτxy ∂x + ∂ Vτyy ∂y + +∂ Vτzy ∂z + ∂ (Wτxz) ∂x + ∂ Wτyz ∂y + ∂ (Wτzz) ∂z + ρ f · Vabs. (2.10)

Assuming that the fluid is a newtonian fluid and then neglecting the dissipative, transport phenomena of viscosity (τi j = 0), mass diffusion and thermal conductivity

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Admissibility conditions for shock waves (k = 0), as well as body forces ( f = 0) and heat generation ( ˙q = 0) (due to absorption or emission of radiation), the resulting balance equations for an unsteady, three-dimensional, compressible, inviscid flow, also known as the Euler equations, are

continuity equation

∂ρ

∂t + ∇ · (ρ Vabs)= 0 momentum equation, x component

∂ ρ U ∂t + ∂ ∂x ·ρ U Vabs = − ∂p ∂x momentum equation, y component

∂ ρ V ∂t + ∂ ∂y ·ρ V Vabs = − ∂p ∂y momentum equation, z component

∂ ρ W ∂t + ∂ ∂z·ρ W Vabs = − ∂p ∂z energy equation ∂ ∂t      ρ      e+ |_Vabs|2 2            + ∇ ·      ρ      e+ |_Vabs|2 2      V abs      = − ∂ Up ∂x − ∂ Vp ∂y − ∂ Wp ∂z . When using a moving reference frame, the absolute velocity of a fluid particle can be evaluated through the relation

Vabs = Vrel+ Vf r. (2.11)

Applying these equations to a control volume that surrounds the shock wave and that moves at a constant speed with it, and considering a normal shock so that the velocity variation only occurs along the direction of propagation of the shock, for example the x direction, and moreover introducing a notation for the states separated

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by the shock, namely 1 to indicate the pre-shock state and 2 for the post-shock state, the conservation equations for a steady, one-dimensional, compressible, inviscid flow, become ρ1u1 = ρ2u2 (2.12) p1+ ρ1u21 = p2+ ρ2u22 (2.13) e1+ p1 ρ1 + u2 1 2 = e2+ p2 ρ2 + u2 2 2, (2.14)

where e is the internal energy of the fluid. Equation 2.14 can also be expressed, through some algebraic manipulation, as

h2(p2, v2) − h1(p1, v1)=

1

2(p2−p1)(v1+ v2), (2.15) where the enthalpy h is computed from the pressure and the specific volume via the thermodynamic EoS as h= h(p, v). The jump relations 2.12, 2.13 and 2.15 link the two states across the shock and are also known as the Hugoniot-Rankine relations. Equation 2.15, often called the Hugoniot-Rankine shock adiabat (HR), is a purely thermodynamic relation, since only thermodynamic properties appear in it, which states that the total enthalpy is conserved. Once the pre-shock state has been fixed, equation 2.15 allows one to easily draw the Hugoniot-Rankine curve in a (v, p) plane, that represents the process line linking all the possible post-shock thermodynamic states to the set pre-shock thermodynamic state 1 (HR1).

The Hugoniot-Rankine relations 2.12, 2.13, 2.14 form a nonlinear system of equa-tions. Three admissibility conditions pose restrictions on its solution. One or more real solutions might exist and valid solutions are only the ones that satisfy the admis-sibility conditions. Since no assumption has been made about the sign of the pressure jump across the shock wave, the system of equations is valid for a compression shock wave as well as for a rarefaction shock wave. Thus, the type of shock depends on the equation of state chosen to model the fluid in use.

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Admissibility conditions for shock waves

2.3.2 The entropy condition

BZT phenomena can be explained through the second law of thermodynamics in conjunction with the shock theory. The second thermodynamic law inequality,

s2−s1≥ 0, (2.16)

determines if the shock is possible along the HR shock adiabat relating states 1 (pre-shock state) and 2 (post-shock state). Equation 2.16 states that a shock is possible only if across it the entropy increases. The equality sign in equation 2.16 is applicable for ”infinitely” weak shocks (these are also referred to as acoustic discontinuities).

The relationship between the entropy change and the pressure jump across a weak shock is, as shown in appendix A,

∆s = 1 6Γ1 v3 1 T1c4₁ p2−p13+ O p2−p14. (2.17)

Equation 2.17 has been derived by Taylor series expansion centred on state 1 and it is only applicable to weak shock waves. The weak shock theory therefore identifies compressive (∆p > 0) shocks as physically admissible shocks (∆s > 0) if Γ1 > 0,

whereas rarefaction shock waves (∆p < 0) are possible only if Γ1 < 0. This implies

that expansion waves will spread into fans and compression waves will steepen into shocks if the fundamental derivative Γ1 is positive, and that expansion waves will

steepen into shocks and compression waves will spread into fans, that is, the reverse behavior of classical flows, when the fundamental derivative Γ1 is negative. The

case when disturbances steepen backward to form expansion shocks (also known as rarefaction shock waves) is called a nonclassical gasdynamics effect.

Notice that the Hugoniot-Rankine relations only link the thermodynamic states upstream and downstream of the shock without focusing on what happens in be-tween, and they assume that those states have steady properties. This hypothesis is only valid if the pre- and post-shock states are chosen ”far enough” from the shock front.

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2.3.3 The mechanical stability condition

Admissible shocks must satisfy a second condition, referred to as the mechanical stability condition, that comes out, from purely mathematical considerations on the Hugoniot-Rankine relations, as

Mr₁≥ 1 ≥ Mr₂, (2.18)

where Mr_{represents the Mach number relative to a shock related reference frame,}

defined as

Mr≡ |u|

c . (2.19)

Equation 2.18 states that the upstream flow relative to the shock is (super)sonic, while the downstream flow relative to the shock is (sub)sonic. Thus the shock allows the (super)sonic-(sub)sonic transition. If thermodynamic states 1 and 2 are located on a HR shock adiabat in such a way that the mechanical stability condition cannot be satisfied, then the formation of a shock wave is impossible. In this case, a fan wave field smoothly connects the pre-shock and the post-shock state.

The stability condition 2.18 can be expressed in terms of thermodynamic variables v and p upstream and downstream of the shock wave. In the infinitesimal vicinity of upstream state 1, the HR shock adiabat and the isentrope have the same slope, since

∂s ∂p

_HR1,1 = 0 (see equation A.5). In this case, dp dρ _HR1,1 = ∂p ∂ρ _s,1, and consequently dp dv _HR1,1 = −ρ 2 1 ∂p ∂ρ _s,1 = −ρ 2 1c 2 1,

where c1 is the thermodynamic speed of sound at state 1. Furthermore, from the

continuity and momentum equations, it can be written that p2−p1 v2−v1 = −ρ2 1u 2 1.

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Admissibility conditions for shock waves By using then the left-hand side of the mechanical stability condition 2.18 it is found that p2−p1 v2−v1 dp dv _HR1,1 =u1 c1 2 = Mr 1 2 _{≥ 1.} (2.20)

It can be shown that an inequality similar to 2.20 also applies for the downstream state, i.e., state 2. Finally, an equivalent form of the mechanical stability condition can be written as dp dv _HR1,1 ≤ p1−p2 v1−v2 (2.21) dp dv _HR1,2 ≥ p1 −_p₂ v1−v2 (2.22) The term on the right side of the inequalities is the slope of a straight line which connects the pre-shock thermodynamic state with the post-shock thermodynamic state. This line is also known as the Rayleigh line.

Equations 2.21 and 2.22 allow a geometrical interpretation, as explained in [22]. With reference to figure 2.4, the classical gasdynamics case is analysed first. Consider point AC in the (v, p) plane. The isentrope through AC is concave up, since Γ > 0

everywhere. Correspondingly, the Hugoniot-Rankine curve from point ACis concave

up, since ∂2_s ∂p2 _HRA_,A = 0

which means that the Hugoniot-Rankine curve and the isentrope through A are osculatory at A (see equation A.6).

Two possible post-shock states BC1and BC2 are now considered. Point BC1, that is

located at higher pressure and would therefore correspond to a compression shock, is admissible since it satisfies the admissibility criterions. The slope of the Hugoniot-Rankine curve at point ACis larger than the slope of the Rayleigh line AC-BC1, which in

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Figure 2.4:Geometrical interpretation of the admissibility conditions. Picture taken from [22].

turn is greater than the slope of the Hugoniot-Rankine line at point BC1, as prescribed

by the mechanical stability condition. Conversely, at point BC2, which is located at

lower pressure and would lead to a rarefaction shock, the slope of the Rayleigh line AC-BC2 is greater than the slope of the HRAC. Thus, the rarefaction shock AC-BC2

is not admissible. The nonclassical case, which occurs in the region where both the isentrope and the Hugoniot-Rankine curve have negative curvature, shows a reversed situation, and therefore admissible shocks are solely of the rarefaction type.

The entropy condition as well as the shock stability condition, together with the Hugoniot-Rankine relations, can be considered as the admissibility conditions for the occurring of a shock wave. Inequalities 2.16 and 2.18 are based on the assumption that the curvature of the process line HR does not change across the shock wave, i.e., that the shock adiabat is either curved up as in the classical case or curved down -as in the noncl-assical c-ase - between states 1 and 2. However, if across the shock wave the shock adiabat changes its curvature, the two inequalities are no longer equivalent.

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Expansion waves through the BZT region

2.4 Expansion waves through the BZT region

2.4.1 Shock waves

If the flow evolves from high to low pressure and crosses the BZT thermodynamic region, RSW’s are admissible. Figure 2.5 shows four different thermodynamic states, namely points A1, A2, A3 and A4, in order to identify the different scenarios for the

occurrence of an RSW. A₄ Satu ratio n cu rve Γ =₀ Γ< 0 τ A₁ A₂A3 s =_s A s = s τ v / v_c P / Pc 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2

Figure 2.5: Selected pre-shock state points along a generic isentrope s= sA< s_τ. Picture taken from [22].

All these points are located along the isentrope s= sA< sτ, where sτis the entropy

value for which the isentrope is tangent to the Γ = 0 line. Point A4 is characterized

by Γ < 0, whereas Γ is positive for all remaining points. Each state point is taken as the pre-shock state to construct the corresponding Hugoniot-Rankine curves in figures 2.6, 2.7, 2.8 and 2.9, in which candidate post-shock points B are also indicated. Despite the qualitative trend of the curves, they can be used to investigate all the possible conditions for the formation of an RSW.

State A1in figure 2.6 is considered first. SinceΓ > 0 at point A1, both the isentrope

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Figure 2.6: Hugoniot-Rankine curve through point A1 and candidate post-shock states B11, B12, B13, B14and B15. Picture taken from [22].

Post-shock state B11, as well as all states along the Hugoniot-Rankine line through A1

and characterized by vB < vA1 can be connected to A1by a compression shock wave,

since they fulfil the admissibility conditions. Conversely, post-shock states B12, B13,

B14and B15, which would lead to an RSW, do not satisfy the admissibility conditions.

For these reasons, any shock originating from state A1can be of the compressive type

only.

Point A2 in figure 2.7 is now considered. As in the previous case, post-shock

state B21 is associated with an admissible compression shock wave. All the states

characterized by vB > vA2 fulfil the entropy inequality; nevertheless, the mechanical

stability condition is only satisfied by point B24. Therefore, the only admissible RSW

is A2−B24, which is a special one: at both A2 and B24, the Rayleigh line is tangent to

the Hugoniot-Rankine curve, namely dp dv _HRA2,A2 = pA2 −pB24 vA2 −vB24 dp dv _HRA_2,B24 = pA2 −pB24 vA2 −vB24 ,

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Expansion waves through the BZT region B24 P v A B B B 2 22 23 21

Tangent to the Rankine−Hugoniot curve Rankine−Hugoniot curve from A2

Rayleigh line from A2

Isentrope through A2

Figure 2.7: Hugoniot-Rankine curve through point A2 and candidate post-shock states B21, B22, B23and B24. Picture taken from [22].

or MA2 = 1 = MB24, that is, both the pre- and post-shock states are sonic. This

special shock wave is named the double sonic shock. All post-shock states to the right of point B24are not admissible.

With reference to figure 2.8, admissible RSW originating from point A3are bounded

by the two special shocks A3−B32and A3−B34. The former is characterized by a sonic

pre-shock state, since the Rayleigh line connecting points A3and B32is tangent to the

Hugoniot-Rankine curve in A3 and hence MA3 = 1. The latter exhibits a post-shock

sonic point, with the Rayleigh line being tangent to the Hugoniot-Rankine curve in B34, where MB34 = 1. Rarefaction shock waves connecting point A3 with a post-shock

state located between points A3 and B32, as well as those having states with v > vB34

are not admissible. The only admissible post-shock states allowing RSW are therefore located in between points B32and B34along the Hugoniot-Rankine curve, such as e.g.

point B33.

Point A4 is finally considered. Since Γ < 0 in A4, the curvature of the

Hugoniot-Rankine curve is negative in A4, as in figure 2.9. All post-shock states located between

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P

v

A B

B

Tangent to the Rankine−Hugoniot curve

B 3 32 33 34 31 B

Rankine−Hugoniot curve from A3

Rayleigh line from A3

Isentrope from A3

Summarizing, from point A1 no rarefaction shock is admissible, only a double

sonic RSW can originate from A2 and a pre- and post-shock sonic RSW as well as

non-sonic RSW can originate from point A3. Finally, from point A4, only post-shock

sonic and non-sonic RSW can arise.

Defining the strength of a generic rarefaction shock as ΠRSW ≡

pupstream−pdownstream

pupstream

, (2.23)

it can be concluded that, once the pre-shock state is chosen, i.e., once the Hugoniot-Rankine curve is established, the double sonic shock, if admissible, is the RSW with maximal strength, since the pressure jump across it is maximum.

2.4.2 Mixed waves

If the flow evolves from high to low pressure in the superheated region, and the pre- and post-shock states are far enough from theΓ < 0 region, the expansion process is made up of mixed waves. With reference to figure 2.10, an isentropic expansion

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Expansion waves through the BZT region P v A B B B 4

Rankine−Hugoniot curve from A Rayleigh line from A

Isentrope from A

Tangent to the Rankine−Hugoniot curve

41 42 43 B 44 4 4 4

process occurs from state 1 to state AD, according to the classical gasdynamics theory.

Because of the downward curvature of the isentrope through AD, a double sonic shock

from ADto BDforms, according to the nonclassical behavior. If the post-shock state is

such that p2 < pBD, a tailing isentropic rarefaction wave takes place, connecting state

BD to state 2. During the expansion process, s1 = sAD and s2= sBD, but s1, s2because

of the entropy jump across the double sonic shock.

Keeping states AD, BD and 2 fixed, even if the initial state 1 is further or closer to

the state AD, the structure of the mixed wave does not change, that is, the expansion

process occurs through a fan-shock-fan combination, where the shock part of the process is localized between states AD and BD. As soon as state 1 approaches state

AD, the upstream fan will become weaker and weaker, till it disappears when point

1 overlaps point AD. If now point 1 ≡ AD moves downward approaching point BD,

the pressure jump across the shock becomes smaller, that is, the strength of the shock is reduced. In this case the process does not start with a fan, but it occurs through a shock-fan combination. Similarly, keeping states 1, AD and BD fixed, if state 2

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AD AD BD BD P v v₂ 1 t 2 x 1

Double Sonic Shock

2

Figure 2.10: Rarefaction fan-shock-fan combination including a double sonic rarefaction shock. Picture taken from [22].

when state 2 overlaps state BD. In the latter case, and even when state BD ≡ 2 moves

upward reducing the strength of the shock wave, the process is made up of a fan-shock combination.

Summarizing, mixed waves, such as for example a mixed rarefaction shock-fan combination, can occur if theΓ = 0 boundary is crossed during flow evolution, a likely situation due to the limited extent of the Γ < 0 region. Nevertheless, an admissible RSW may also occur between two states characterized by Γ > 0, provided that the corresponding Hugoniot-Rankine curve crosses theΓ = 0 boundary.

Briefly, an unsteady expansion fan is a body of waves that move at different speeds without causing an increase in entropy. In detail, the head of the fan moves at the upstream sound speed, while the following waves move at lower speeds. As the closer to the front runner wave of the fan is the investigated wave, the higher is its speed of propagation, the waves spread and the expansion process evolves in time including bigger portions of fluid and causing a more and more gradual change in thermodynamic properties between the upstream and the downstream

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Expansion waves through the BZT region states. Therefore, a fan can be represented in the (x, t) plane as a certain number of straight lines - assuming that each wave travels at a constant speed - which tend to diverge.

An unsteady rarefaction shock is, instead, a clustered body of waves - ideally spread over an infinitesimally thin region - which causes an abrupt change of state and that moves at the same speed. Therefore, in the (x, t) plane, a shock can be represented by one straight line, assuming that it propagates at a constant speed.

The wave structure for a process that involves a rarefaction fan-shock-fan combi-nation is displayed in the (x, t) diagram in figure 2.10.

Notice that in presence of mixed waves, the Hugoniot-Rankine relations (2.12), (2.13) and (2.14) and the admissibility inequalities 2.16 and 2.18 are only valid to model the shock part of the process.

2.4.3 RSW with maximum supersonic Mach number: the Maximum

Mach Locus (MML)

Due to viscous effects, the shock thickness might increase. Therefore, the pressure drop across a moving RSW would be recorded as a ramp pressure signal rather than a step signal. Under these conditions, it is difficult to discriminate whether the wave represents a rarefaction shock or an expansion fan. Therefore, in an experiment that intends to generate and detect the presence of a moving RSW, the detection has to occur by measuring the speed of the shock. In this case, it is more important to maximize the upstream Mach number than the pressure drop across the shock.

An experiment where the pressure drop across the shock is maximized is named a ”double sonic experiment” (DSE). An experiment in which the relative Mach number upstream of the travelling RSW is maximized is instead named a ”maximum Mach experiment” (MME). Unlike a DSE, in an MME the upstream Mach number is super-sonic while the downstream Mach number is super-sonic, with respect to a shock related reference frame; moreover, the RSW is recognized by measuring the upstream Mach number. Even though the strength of the shock is reduced, since the pressure drop across it is smaller than that of an RSW in a DSE, the detection of the wave speed can

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occur by means of pressure measurement, as extensively treated in [21].

Figure 2.11: Isentropes s1. . . s5 in the (vr, pr) plane for a vdW fluid having δ = 0.02. Picture taken from [21].

Figure 2.11 shows the BZT region, the vapor-liquid equilibrium line and the double sonic locus (DSL), defined as the locus, in the (v, p) plane, of the upstream thermo-dynamic states from which a double sonic shock can originate, together with the corresponding downstream sonic states. Five isentropes denoted with s1. . . s5 are

also shown. They intersect the DSL and theΓ = 0 locus in points D1. . . D5, and points

G1. . . G5, respectively. On isentrope s3, the upstream state for producing RSWs must

be located between points D3and G3. If the upstream state is D3, then a double sonic

shock is admissible, i.e., Mr

upstream = Mrdownstream = 1. If the upstream state is instead

G3, an expansion process evolves as an isentropic fan. For any thermodynamic state

located in between points D3and G3, the upstream Mach number relative to the shock

is supersonic; this is illustrated by line D3−M3−G3in figure 2.12 for all the isentropes

s1. . . s5 in the (vr, M) plane, where M3 identifies an upstream thermodynamic state

which maximizes Mr

upstream along isentrope s3. The procedure used to obtain these

plots is detailed in [21].

Starting from the same upstream state (vupstream, pupstream), an infinite number of

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Expansion waves through the BZT region

Figure 2.12: Isentropes of figure 2.11 in the (vr, M) plane for a van der Waals fluid having

δ = 0.02; the locus of the maximum upstream Mach number states is highlighted. Picture taken from [21].

sound cupstream does not vary. Therefore, the maximum value of Mr_upstream =

uupstream

cupstream

is obtained when its numerator is maximized. Since the numerator represents the slope of the Rayleigh line, as shown in equation 2.20, it is maximized when the downstream state corresponds to a sonic shock, i.e., when the Rayleigh line is tangent to the HR curve at the downstream state. In conclusion, as far as the upstream state is fixed, the maximum supersonic Mach number of a RSW corresponds to a downstream sonic shock.

The optimization procedure, explained in detail for isentrope s3, can be repeated

for all isentropes crossing the BZT region. It is found that the maximum Mach number increases as one moves far from the tangency point T; in the tangency point, the Mach number is one, and corresponds to an isetropic (sonic) expansion fan.

The upstream thermodynamic states for an RSW characterized by a maximum Mach number on an isentrope in the superheated region are located on a geometrical locus named the Maximum Mach Locus (MML). A ”maximum-maximorum” of the Mach number in dense gases, for a given BZT fluid in its superheated region,

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corre-Figure 2.13: The Maximum Mach Locus for a vdW fluid havingδ = 0.02 in the (vr, pr) plane. Picture taken from [21].

sponds to the upstream state further from the tangency point T, i.e., at the intersection of the MML with the VLE line. This point is indicated as 1M in figure 2.13.

2.5 Shock formation in a general fluid

It has been assumed that the shock is instantaneously fully formed. Nevertheless, a shock - of the rarefaction type as well as of the compression type - could only form when a body of disturbance waves propagating into the fluid shows a peculiar behavior. Disturbance waves arise for example when a uniform flow is subject to perturbations, as depicted in figure 2.14.

By applying the balance equations to disturbance waves and by solving them using the method of characteristics, it is possible to understand the complex phenomena that lead to the formation of a shock. Neglecting the body forces acting on the fluid, and assuming that the flow is confined to a one-dimensional duct of constant cross-sectional area, so that all variables of interest depend only on x and t, and moreover

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Shock formation in a general fluid

Figure 2.14: Physical illustration of disturbance waves.

that the flow is homentropic, the continuity and the momentum equations can be combined to yield " ∂ ∂t+ (U ± c) ∂ ∂x # (U ± F)= 0, (2.24)

as extensively explained in [18]. Here, U is the absolute velocity of a fluid particle, and F is a thermodynamic function, defined as

F ≡ Z p

p0

dp ρc,

introduced to simplify the solution of the conservation equations. The terms U+ c and U − c are the velocities of waves travelling in the positive and negative direction, respectively. The paths tracked by the waves appear as lines on the (x, t) diagram. Such lines are called characteristics. The quantities U+ F and U − F, named Riemann invariants, do not vary along their respective characteristic. For an observer moving, for example, with one positive wave, the quantity U + F does not change in time, though it changes from one characteristic to another. If the disturbance waves travel toward a region of uniform flow, the characteristics are straight lines. In general, the characteristics will not be parallel but will either converge or diverge in the moving perturbed region; where they first intersect, a shock forms, as shown in figure 2.15.

By introducing the fundamental derivative as in [18], the variation of the speed of positive waves along one negative characteristic line is

d(U+ c) = ΓdF = Γ

ρcdp. (2.25)

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Figure 2.15:Wave diagram and insertion of a shock discontinuity from the point s, where the characteristics first intersect.

Γ > 0 → dp > 0, i.e., compressive disturbance waves will steepen to form a shock (as in the case of the perfect gas), and ifΓ < 0 → dp < 0, e.g., expansive disturbances will steepen to form a shock. Finally, if the hypothetical case ofΓ = 0 is considered, then the characteristics will be parallel and the waves will proceed undistorted. The three possible cases are shown in figure 2.16.

Figure 2.16: Behavior of the waves for various values of the fundamental derivative. Picture taken from [18].

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con-Shock formation in a general fluid dition 2.17, found for weak shocks. Thus, either compression shocks or rarefaction shocks are allowable, depending on whetherΓ is positive or negative. The case Γ = 0 leads instead to undistorted waves, not because of the constancy of c but because of the tendency of increases in U to just balance decreases in c so that U+ c remains constant. Γ > 1 ∂c ∂p _s> 0 Compression shocks Γ = 1 ∂c ∂p _s= 0 0< Γ < 1 ∂c ∂p _s< 0 No shocks Γ = 0 ∂c ∂p _s< 0 Rarefaction shocks Γ < 0 ∂c ∂p _s< 0

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Commissioning of the Flexible

Asymmetric Shock Tube (FAST)

3.1 Overview

A 9 m long duct of constant cross-sectional area, named charge tube (CT) and composed of tube segments, is connected, via a fast-opening valve (FOV), to a low-pressure plenum (LPP). A heated tank provides the vapor of the working fluid for the experiment and is connected to the pipe via a so-called reference-tube, a short tube segment that is used to provide the fluid reference temperature for the experiment. A condenser connected to the low-pressure plenum on one side and to the heated tank on the other side allows for the recovery of the fluid. All the components are in stainless steel 316L and are heated by special heating jackets. An isometric view and a real view of the FAST are given in figure 3.1.

The FAST is similar to a Ludwieg tube and operates in a similar way, though the experimental goal is different. In 1955, Ludwieg developed a shock tube of a peculiar kind, able to generate short duration steady supersonic flows. A Ludwieg tube, shown in figure 3.2, consists of a long cylindrical charge tube, a convergent-divergent nozzle, either a diaphragm or a fast-opening valve, and a low-pressure plenum, also known as damp tank. Initially, a pressure difference is created across the

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diaphragm/fast-Overview

Figure 3.1:Isometric view and real view of the Flexible Asymmetric Shock.

opening valve. The pressure in the charge tube is greater than the pressure in the dump tank. If the high-pressure and the low-pressure initial thermodynamic states are properly chosen, the fast opening (4 ms) of the valve initiates a flow consisting of compression waves moving into the LPP and of expansion waves that will in principle coalesce in the CT to form an RSW. Due to the peculiar shape of the FOV, a convergent-divergent nozzle is present between the tube and the reservoir. Since the RSW induces a steady subsonic flow at the inlet of the nozzle, the flow is accelerated to supersonic velocity. Therefore, choked sonic conditions are set at the nozzle throat, this way avoiding that disturbances arising in the LPP propagate into the CT disturbing the shock formation. w Reservoir Nozzle Charge tube state 2 state 1 RW

Figure 3.2: Schematic representation of a Ludwieg tube, showing the main components and the propagation of a rarefaction wave into the charge tube.

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The FAST aims at generating and detecting a rarefaction wave which propagates in the form of a nonclassical shock front into the charge tube. To achieve this type of flow field, the test fluid has to possess special thermophysical characteristics. Require-ments related to safety and costs are also to be taken into account. Among various fluids claimed to exhibit nonclassical behavior, a member of the siloxanes family, e.g., dodecamethylcyclohexasiloxane D6, is chosen as test fluid for experimental runs.

For the experiments which are to be conducted, the FAST is operated as follows, as described in [13]. The charge tube is filled with the dense gas D6, which is superheated

via heating jackets by about 0.5 ◦

C with respect to the saturation condition of the heated tank. The pressure inside the pipe is the same as the saturation pressure in the heated tank in equilibrium. The pressure in the low-pressure plenum is kept at a prescribed lower value. Also the low-pressure plenum is kept at the same high temperature by heating jackets, to avoid heat transfer from the tube. The temperature non uniformity along the tube must be very limited, i.e., within the accuracy of the reference temperature sensor (0.01% full scale, therefore approximately 0.04 K), used to measure the temperature of the fluid. Vapor condensation must be avoided. Once the desired stable initial pressure and temperature are obtained in the charge tube and in the low-pressure plenum, the fast-opening valve can be opened and the experiment can start. A supersonic flow from the charge tube to the low-pressure plenum is generated. The nozzle between the FOV and the low-pressure plenum is chocked, thus avoiding the propagation of disturbances in the charge tube. According to the nonclassical gasdynamics theory, a rarefaction shock wave should be fully formed at a distance of approximately half of the tube length. Four dynamic pressure instruments, which can detect the passing wave, are placed along the tube, two of them at half of the pipe length, and other two toward the end of the charge tube. The estimate of the speed of the wave is then made possible by correlating their signals, if their relative distance is exactly known. If the speed of the wave is supersonic, it is proven that a rarefaction shock wave has been detected.

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Mechanical components

3.2 Mechanical components

As it can be seen from the Process and Instrumentation Diagram (P&ID) (figure A.1) and from the general assembly drawing (figure A.2), the FAST is made up of the following main components, as taken from [13]:

• the Low-Pressure Plenum (LPP)

It is a 100-liter vessel of stainless steel AISI 316L. The plenum is designed to operate at pressures of up to 20 bar and temperatures of up to 400◦C. Part of the internal volume is occupied by the fast opening valve (FOV). The vessel is split into two parts, a body and a cap; the cap allows for mounting or dismounting the FOV and for cleaning the vessel. Eight flanged connections are welded to LPP: two are for the actuation of the FOV, one for the vacuum pump and static pressure instrument PIT6, one for the condenser, one for drain, one for temper-ature instrument TE4, one for the pressure safety valve PSV1. The connection between the LPP and the first segment of the charge tube occurs by means of a specially designed adapter flange.

Figure 3.3:View of the LPP.

The function of the LPP is to create and maintain the required back-pressure during the short duration experiment. Prior to each test run, the pressure in LPP is set to the desired value by discharging some vapor into the condenser VCT

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through the pneumatic valve PV1. For safety reasons a pressure valve (PSV1) is mounted on the plenum. Moreover, the pressure in LPP can be checked by looking at a static pressure transducer (PIT6), and recorded by the data acquisition and control system (DAQ&C). The pressure sensor PIT6 is protected against high pressure through manual valve MV5; protection of the sensor is needed to avoid de-calibration due to overpressure. At the bottom of LPP, manual valve MV7 is located. This is used for draining both the charge tube and the LPP. Sealing of the LPP is achieved by using a gasket compressed between the two flat surfaces of the flange.

• the Fast Opening Valve (FOV) and its mechanical actuation system

It connects the charge tube and the LPP. The FOV is custom-designed to meet the following requirements, namely a guaranteed opening time of at most 4 ms and a seal between the high pressure vapor in the charge tube and the low pressure vapor in the LPP. The justification for using a FOV instead of a diaphragm (which is commonly used in Ludwieg tubes) is related to the fact that, due to the small pressure difference between the charge tube and the reservoir, it is expected that a conventional diaphragm may rupture only partially and disturb the flow field which is of interest. Furthermore, after every experimental run, the entire facility would need to be cooled down and emptied to replace the diaphragm. A diaphragmless setup greatly increases the availability of the setup and allows for repeated and repeatable experiments. The conceptual design of the FOV contains a nozzle. Moreover, the novelty of the design is that the throat area of the nozzle can be varied by means of a gear that can be turned from outside the LPP. The ratio of the throat area of the nozzle with respect to the area of the pipe is a crucial parameter for the flow field under study. The mechanical drawing of the FOV is represented in figure 3.4.

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Mechanical components

Figure 3.4: Conceptual design of the FOV with adjustable area of the nozzle throat. Note that the figure shows both the open and closed positions of the valve.

Item number Description Item number Description

1 adapter flange 8 lever

2 nozzle section 9 closure

3 outer-valve part 10 1st spring

4 adjustable nozzle plug 11 2nd spring

5 locking mechanism 12 2nd actuator

6 sliding sleeve 13 bellow

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• the Charge Tube (CT)

It is assemble out of six pipe segments of exactly 1520 mm in length. Each seg-ment has an outside diameter of 70 mm and an inside diameter of 40 mm. To get a straight pipe segment, each part is machined from a solid block. A hole with the specified diameter is drilled into the material and afterwards, the outside is machined to the required specifications. The inner surface of the pipe segments is electrolytically polished to nearly a mirror finish (0.05 µm). Due to the large heat capacity of each segment, as a consequence of the large thickness of the pipe (15 mm) and of the low thermal conductivity of the material

≈ 20 W

m K

, temperature fluctuations on the outside are attenuated on the inside. A small slit is machined on the outer surface of the pipe to accomodate a k-type thermo-couple used by the temperature control system. Each pipe segment has a male to female connection with the adjacent segment. The male side contains a groove in which an O-ring of red copper is placed. Red copper seals are compatible with siloxanes and offer tight sealing at high temperatures both under pressure and vacuum conditions. The pipe segments are connected to each other by means of custom-made clamped couplings. These have been designed aiming at obtaining tightness for both overpressure and vacuum operating conditions. Flanges were ruled out given the requirements on straightness and smoothness of the total inner surface.

Figure 3.5: View of the pipe segment of the charge tube which is connected to the reference tube. Notice the two dynamic pressure transducers.

The use of couplings however, complicates the requirement of tight sealing. At the temperatures at which the experiments in the FAST are conducted, i.e.,

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Mechanical components ≈ 360 ◦

C, thermochemical decomposition of the test fluid is a point of con-cern. Although the process temperature is lower than the temperature where decomposition of the fluid may occur, i.e., ≈ 400 ◦

C, with a safety margin, the probability that the test fluid decomposes is greater if acidic or alkaline impuri-ties and air and water are present. These implies that, apart from degassing the test sample thoroughly, precautions must be taken in order to minimize leakage of air into the FAST. Leakage of air into the shock tube during the experimental session is not a point of concern because the process pressure is greater than the atmospheric pressure. During the charging phase however, the entire facility is under vacuum before the siloxane is charged into the FAST.

Two of the six tube segments are equipped with fast-responding dynamic pres-sure instruments which are used for wave-speed meapres-surement. Additionally, one of the segments has a side connection at the leftmost extremity which is used for filling the shock tube, as shown in figure 3.5.

The charge tube is supported by props mounted on sliding bearings that allow the charge tube, which elongates due to thermal expansion, to slide in axial direction.

• the Reference Tube (RT)

It is made of the same material as the charge tube segments and has the same inner diameter and wall thickness. The purpose of RT is setting and controlling the temperature of the charge tube. The vapor in the reference tube is maintained at a constant/set value with the heating jackets. The control is based on the temperature in the HFT, i.e., the temperature of RT has to be a few tenths of a degree greater than that of the fluid in the vapor generator so that the vapor is slightly superheated. The control of temperature along the charge tube is based on differential measurements: the signal from thermocouples placed on the exterior surface of the reference tube is compared to that of the thermocouples placed on the charge tube wall and the temperature difference is minimized. Therefore, RT contains a small, narrow slit on the outside in which all the thermocouples are placed.

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Figure 3.6: View of the reference tube and the heating jacket.

• the Heated Fluid Tank (HFT)

It has the purpose of producing the vapor to fill the CT and LPP at the set pressure. Vapor generation is obtained via appropriate electrical heaters, shaped to fit the complex geometry of this component. The vapor generator also acts as a fluid collector and is connected for drainage to a manual valve MV3, located at the lowermost part of the facility. A pressure safety valve, PSV3, is connected to HFT. Additionally, the pressure is continuously measured and monitored by PIT5, which is a high temperature pressure transducer.

Figure 3.7:View of the vapor generator.

The HFT is connected to the condenser via a slightly inclined pipe containing a pneumatic valve (PV2) and it is connected to RT via the manual valve MV4. The liquid level and the temperature of the liquid in the vapor generator are measured respectively by LIT1 and TE1. To account for the thermal expansion of the charge tube assembly, the HFT is supported by sliding bearings.

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Mechanical components • the Vapor Condenser Tank (VCT)

It is a fan-cooled cylindrical reservoir placed beneath the LPP. To enhance the heat transfer surface, cooling fins are placed on the exterior of the cylinder and ambient air is blown over the shell using a fan, M1. The condenser is equipped with a static pressure transducer PIT7. Because the cooling time is not critical, it was decided not to use a water-cooled condenser. As soon as the fluid condenses, it flows via the inclined pipe into the HFT.

Figure 3.8:View of the vapor condenser.

• the heating jackets system

Heating of the complete system is achieved using jackets with electrical wires embedded in a silk glass thermal insulation layer. The heating wires are spooled at the inner part of the silk-glass blanket, so that they are close to the surface to be heated. The heaters are divided in segments and controlled separately via thyristor units. The temperature on the outside surface of the heating jackets must be maintained at an acceptable value (below 60 ◦C) to avoid excessive heat loss and for safety reasons.

• the special bottle and valves to charge the FAST

To charge the FAST with the siloxane, the fluid needs to be prepared, i.e., volatilities need to be removed. To do this, the siloxane is first filled into

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a special bottle. The sample is then frozen by immersing the bottle in liquid nitrogen. This is followed by pumping on the bottle to deep vacuum conditions. This procedure is repeated 3-4 times or at least as many times as necessary to reduce the pressure of volatilities to an acceptably low value. Next, the bottle is connected to the vapor generator and MV1 and MV2 are opened, with the valve of the bottle still closed. The FAST must then be pumped to deep vacuum conditions using the vacuum pump before the test fluid is gravitationally filled into the HFT.

• the vacuum Pump (P)

It is connected to LPP via MV5. • the main support structure

It is made up of concrete blocks. To dampen mechanical vibrations from other equipment in the main P&E laboratory, the concrete blocks are placed on rubber foam.

3.3 Instrumentation

3.3.1 Pressure instruments

Two types of pressure measurements are conducted, namely static and dynamic pressure measurements.

The instruments for accurate static pressure measurements, together with data from the temperature sensors, allow for the knowledge of the thermodynamic state of the test fluid in the FAST. This is necessary for setting the right conditions in the FAST for the generation of nonclassical gasdynamic effects, i.e., a rarefaction shock wave.

The dynamic pressure instruments are used for accurate wave speed measure-ments. The main requirement that the dynamic pressure instruments must meet is

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Instrumentation the fast response time to a dynamic perturbation. Although the accuracy of the pres-sure meapres-surement should be as high as possible, this is not a stringent requirement. That is, since the goal of the FAST is to generate a nonclassical rarefaction shock wave, the fact that a pressure drop occurs as the wave passes a dynamic pressure sensor is employed only as a qualitative check. The quantitative validation - that is, that the wave is indeed a shock wave which propagates supersonically and not a fan which propagates at a sonic speed - is provided by the wave speed measurement, the ac-curacy of which is dependent upon the response time of the instruments to pressure changes, which must be the same.

Additional requirements, valid for both the static and the dynamic pressure in-struments, include resistance to high temperature (400 ◦

C) and high repeatibility. Moreover, the material of the sensor must not cause or enhance thermochemical decomposition of the test fluid.

The position of the pressure sensors is illustrated in the process and instrumenta-tion diagram A.1.

3.3.2 Temperature instruments

For accurate measurements of the temperature of the test fluid, four resistance temperature detectors (RTD’s) are used (PT100). The position is indicated in the general assembly drawing A.2 and the process and instrumentation diagram A.1.

3.3.3 Other devices

The level of liquid in the vapor generator is continuously monitored using a liquid level indicator. Since the instrument uses radar, the measurement is not influenced by fluid properties such as thermal conductivity, density and dielectric constant.

Thyristors are solid-state semiconductor devices with four layers of alternating N-(abundance of negative charge carriers) and P- N-(abundance of positive charge carriers) material. They can rectify alternating currents into large direct currents and can be automatically triggered off for a required time. For controlling the power supply to all the heaters, thyristors manufactured by RKC Instrument were purchased.

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A detailed treatment on the selection of the purchased instruments as well as on their specifications is reported in [13].

3.4 Data Acquisition and Control

The Data Acquisition and Control (DAQ&C) system is composed of National In-struments hardware, arranged in a common cabinet, as shown by figure 3.9. National Instruments software (LabView) is used for monitoring and control programming.

Figure 3.9: Data Acquisition and Control cabinet.

3.4.1 DAQ system

The DAQ system is dedicated to the monitoring of the data relevant for the ex-periment, that is, the initial thermodynamic states (p and T) in the charge tube and the low-pressure plenum, and the dynamic pressure signals to be correlated to obtain the speed of the wave. The DAQ is realized with 2 boards embedded in a dedicated chassis.