Theoretical Investigation of Gas Filling and Leaking in Inertial Conﬁnement Fusion Hohlraum

sustainability

Article Theoretical Investigation of Gas Filling and Leaking in Inertial Conﬁnement Fusion Hohlraum

Cheng Yu 1, Suchen Wu 1 and Weibo Yang 1,2,*

1 Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China; [email protected] (C.Y.); [email protected] (S.W.) 2 School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou 225127, China * Correspondence: [email protected]; Tel.: +86-25-8379-2483

Received: 28 August 2018; Accepted: 16 October 2018; Published: 18 October 2018

Abstract: The gas filling and retention of inertial confinement fusion (ICF) hohlraum is an important issue in ICF studies. In this study, a theoretical model of gas filling and leaking processes for ICF hohlraum is developed based on the unified flow theory. The effects of the fill tube size and the filling pressure on the gas filling and leaking performance are investigated. The results indicate that an increase in the variation rate of the filling/leaking pressure leads to a larger maximum pressure difference between the inside and outside of the ICF hohlraum during the filling/leaking process. The critical pressure difference of the filling process is nearly equal to that of the leaking process. Increase in fill tube diameter and decrease in its length both lead to a lower probability of the rupture of polymeric films at two ends of the hohlraum, and thus increases the security of the hohlraum. In addition, a departure in cross sectional shape of fill tube from circle to rectangle triggers an increase in pressure difference between the inside and outside of the ICF hohlraum, which raises the risk of polymeric films rupture and decreases the security of the hohlraum structure.

Keywords: inertial confinement fusion; hohlraum; filling and leaking; unified flow model

1. Introduction Inertial confinement fusion (ICF) [1–3] is considered as a promising way to achieve controlled thermonuclear fusion that can provide individuals with high-quality sustainable energy [4–6]. In inertial confinement fusion [7], there are two approaches to drive an ICF implosion, namely direct drive and indirect drive. In the direct drive approach, laser beams are directly aimed at a spherical target containing fusion fuel that is typically from the emulsion droplet prepared by the microfluidics [8–11]. In the indirect drive approach [12,13], energy from a laser is first absorbed by a typically cylindrical-shaped enclosure termed as a hohlraum to produce X-rays using the high-Z material inside the hohlraum to irradiate the target [14,15]. Given the relaxed requirements on laser-beam uniformity, the indirect drive approach is the subject of intense interest in ICF recently. To ensure the symmetry of the implosion, ICF hohlraum is initially filled with a low-density and low-Z gas as a symmetry-control technique to suppress the interaction of the plasma in the hohlraum, thereby resulting in a reduction in the energy loss of the laser [16,17]. Considering that, the gas filling and retention of ICF hohlraum has become a key issue in the ICF studies [18]. Aiming to select and optimize the process parameters of ICF hohlraum, it is of particular importance to study the gas flow behaviors during gas filling and leaking processes of the ICF hohlraum. To prevent the gas in the hohlraum from leaking, a gas barrier film (mainly a polyimide film) is placed on the diagnosis port and laser entrance port of the hohlraum. When the variation rate of the gas filling/leaking pressure is excessively high, the pressure difference across the film exceeds

Sustainability 2018, 10, 3763; doi:10.3390/su10103763 www.mdpi.com/journal/sustainability Sustainability 2018, 10, 3763 2 of 12 its pressure tolerance, thereby resulting in the rupture of the film and failure of gas filling or leaking. A few investigations focus on gas filling and leaking processes of ICF hohlraum [19]. With respect to the Nova experiments, the hydrocarbons were used as low-Z gas fill [20–22]. Unfortunately, it is not possible to provide a definitive answer with respect to the gas flow behaviors during the gas filling and leaking processes due to restrictions on measurement of the gas pressure in the hohlraum and the gas mass flow in the fill tube using existing experimental measurement devices [23]. Thus, a theoretical method is necessary to determine the gas filling and leaking processes and hence to evaluate the extreme rupture condition of the hohlraum film. During gas filling and leaking processes, the pressure in the hohlraum changes significantly in the range of 0 atm–1 atm. It should be noted that the gas flow at standard atmospheric pressure is in the continuum flow region, however the gas flow at nearly 0 atm is in the free-molecular flow region of rarefied gas [24–26]. In addition to the continuum flow region, the continuum approach of the classical Navier–Stokes equations is no longer valid in the transitional flow region and free molecular flow region, which is attributed to the gradually dominant gas rarefaction effect [27,28]. However, there is a paucity of theoretical studies on gas filling and leaking processes in the hohlraum where the multiscale gas flow involved. While several attempts have been conducted to investigate the gas filling and retention of the ICF hohlraum, it is impossible to measure the dynamic gas pressure in the hohlraum directly due to the small size of the hohlraum [29–31]. Thus, the applicability of the pressure variation rate during the gas filling and leaking processes with the required critical pressure value for safe gas filling/leaking of the hohlraum has not been well clarified, resulting in the potential risk of the hohlraum rupture. In addition, in order to optimally design the gas fill tube and the hohlraum, the gas flow behaviors as well as the dependences of the geometries of the gas fill tube and the hohlraum on the performance of the gas filling and leaking processes are still waiting to be explored. For these reasons, it is extremely important to develop a quantitative analysis and prediction method that considers the gas rarefaction effect, to provide a guideline for the parameters selection and optimization of gas filling and leaking processes in the ICF hohlraum. To provide a guideline for the parameters selection and optimization of gas filling and leaking processes in the ICF hohlraum, it is extremely important to develop a quantitative analysis and prediction method that considers the gas rarefaction effect. In this study, a theoretical model of the gas flow behaviors during gas filling and leaking processes of ICF hohlraum is developed. Furthermore, the model is numerically analyzed to investigate the effects of the gas fill tube size and the filling pressure on gas filling and leaking processes. The current study provides a deep understanding of the gas filling and leaking processes of the ICF hohlraum.

2. Mathematical Model To further insight into the gas flow behaviors in ICF hohlraum, a theoretical model of gas filling and leaking processes of ICF hohlraum are developed in this study. As shown in Figure1, the hohlraum with the shape of a hollow cylindrical cavity is placed in an infinitely large gas filling chamber and connected to the gas filling chamber through a fill tube. The initial pressure in the hohlraum is Pi, and the ambient pressure outside the hohlraum, i.e. the pressure of the gas filling chamber, is Po. The cylindrical hohlraum is of a cross-sectional diameter of dh = 2.2 mm and a length of Lh = 4.6 mm. The diameter and the length of the fill tube for the hohlraum are dx = 0.1 mm and Lx = 500 mm, respectively. The temperature of the hohlraum during the filling and leaking processes is maintained constant, i.e., Ti = To = 298.15 K. The hohlraum is filled with helium as low-Z gas, and this is considered as an ideal gas and satisfies the ideal gas state equation. The pressure difference between the inside and outside of the hohlraum is defined as ∆P = Po−Pi. It is assumed that the film of the hohlraum ruptures at a pressure difference exceeding 0.5 atm, and thus the critical pressure value for safe gas filling/ leaking of the hohlraum is set as ∆Pc = 0.5 atm [32]. Sustainability 2018, 10, x FOR PEER REVIEW 3 of 12 Sustainability 2018, 10, 3763 3 of 12

dx Gas filling chamber

To, Po Fill tube Lx

Ti, Pi Target dh hohlraum

Figure 1. Schematic diagram of the physical model of the gas ﬁllingfilling and leaking processes.

2.1. Governing Equations To evaluateevaluate the rarefaction effects during the gasgas fillingfilling and leaking processes of ICF hohlraum, the Knudsen number ((Kn)) isis introducedintroduced asas [[33,34]33,34] λ KnKn== (1) dx where ddxx denotesdenotes thethe diameterdiameter of of the the fill fill tube tube and andλ λdenotes denotes the the molecular molecular average average free free path. path. Based Based on theon the theory theory of gasof gas dynamics, dynamics, the the molecular molecular average average free free path pathλ is λ relatedis related to theto the dynamic dynamic viscosity viscosityµ, pressureμ, pressureP, andP, and temperature temperatureT as T as r µ πRT λ = (2) =P 2 (2) 2 The relationship between the Knudsen number and pressure is The relationship between the Knudsen numberr and pressure is µ πRT = Kn , (3) Pd = x 2 , (3) During the gas filling and leaking processes, the Knudsen number ranges from 0.001 to 50, During the gas filling and leaking processes, the Knudsen number ranges from 0.001 to 50, indicating that the gas flow goes through the continuum flow region, slip flow region, transition flow indicating that the gas flow goes through the continuum flow region, slip flow region, transition flow region, and free-molecular flow region [35,36]. When the Knudsen number is sufficiently large, it is region, and free-molecular flow region [35,36]. When the Knudsen number is sufficiently large, it is not possible to ignore the gas rarefaction effect, and this results in invalid Navier–Stokes equations not possible to ignore the gas rarefaction effect, and this results in invalid Navier–Stokes equations based on continuity assumption [37,38]. Therefore, Beskok, and Karniadak is proposed a unified flow based on continuity assumption [37,38]. Therefore, Beskok, and Karniadak is proposed a unified flow model [39] in the entire Knudsen regime, and this effectively predicts the velocity distribution, pressure model [39] in the entire Knudsen regime, and this effectively predicts the velocity distribution, drop, and mass flow rate in microtubules. In the model, a rarefaction coefficient is introduced to correct pressure drop, and mass flow rate in microtubules. In the model, a rarefaction coefficient is the gas velocity distribution in the transitional flow zone and free molecular flow zone. The results introduced to correct the gas velocity distribution in the transitional flow zone and free molecular are in good agreement with the analytical solutions of the linearized Boltzmann equation. Thus, flow zone. The results are in good agreement with the analytical solutions of the linearized Boltzmann the gas flow behaviors in the fill tube during the gas filling and leaking processes of ICF hohlraum are equation. Thus, the gas flow behaviors in the fill tube during the gas filling and leaking processes of described by the unified flow model in the entire Knudsen regime. This is coupled with the description ICF hohlraum are described by the unified flow model in the entire Knudsen regime. This is coupled of the gas state equation inside the hohlraum to analyze the gas flow behaviors during gas filling and with the description of the gas state equation inside the hohlraum to analyze the gas flow behaviors leaking processes of ICF hohlraum. during gas filling and leaking processes of ICF hohlraum. The differentiating gas state equation in the hohlraum PV = mRT is derived as The differentiating gas state equation in the hohlraum PV = mRT is derived as dP 1 1 dT dm = = mR ++RT (4) dt V dt dt The gas filling and leaking processes are simplified by ignoring the temperature change, so Equation (4) is modified as

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The gas filling and leaking processes are simplified by ignoring the temperature change, so Equation (4) is modified as dP RT dm RT . = = M (5) dt V dt V . . where M denotes the gas mass flow rate. When the gas flow goes into the hohlraum, M is positive and vice versa. Based on the unified flow theory model, the gas mass flow rate in the gas fill tube is calculated by

. 4 πadPavg ∆P 4Knavg M = − × 1 + αKnavg 1 + (6) 8µ0RT L 1 − bKnavg where b = −1, R denotes the gas constant of helium, T denotes the system temperature, Pavg denotes the average pressure drop, i.e., the average pressure between the inlet and outlet of the ﬁll tube in which Pavg = (Pi + Po)/2. Furthermore, µ0 denotes the dynamic viscosity that is assumed to be constant in this study, and Knavg is obtained from Pavg using Equation (3). The parameter α varies with Kn, and this is calculated by

2 β α = α tan−1 α Kn (7) 0 π 1 where α1 = 4, β = 0.4, and α0 = 1.19, and these parameters are derived from the air ﬂow experiment in the ﬁnite-length tube by Tison [40].

2.2. Boundary Conditions and Numerical Solution The pressures of the hohlraum and gas filling chamber are both 0 atm at the initial time of the gas filling process. Subsequently, the gas is filled into the gas filling chamber until the pressure of gas filling chamber increases to 15 atm. The pressure variation rate of gas filling chamber is Φ. Through the gas fill tube, the hohlraum is filled with gas gradually until the pressure inside and outside of the hohlraum are balanced. Pi|t=0 = Po|t=0 = 0 atm (8)

Pi|t→∞ = Po|t→∞ = 15 atm (9) dP o = Φ (10) dt During the gas leaking process, the pressures of the hohlraum and gas filling chamber are both initially assumed as 15 atm. Subsequently, the gas in the gas filling chamber leaks into vacuum until the pressure of gas filling chamber decreases to 0 atm. The pressure variation rate of gas filling chamber is the same as Φ. Through the gas fill tube, the gas in the hohlraum also leaks into the gas filling chamber until the pressure inside and outside of the hohlraum are balanced.

Pi|t=0 = Po|t=0 = 15 atm (11)

Pi|t→∞ = Po|t→∞ = 0 atm (12) dP o = −Φ (13) dt With above external pressure boundary conditions, Equations (2), (3), and (6) are solved using the Newton-iterative method to obtain the time evolution of pressure in the hohlraum and in the gas ﬁlling chamber. Sustainability 2018, 10, 3763 5 of 12

2.3. Case Validation

SustainabilityThe dimensionless 2018, 10, x FOR gas PEER mass REVIEW ﬂow rate in Equation (6) is 5 of 12

3π 4Kn M = − 31 + αKn 1 + 4 (14) =− 1+ 1− + (14) 64Kn64 1 b1−Kn ToTo validate validate the the model, model, a theoreticala theoretical case case of of the the linearized linearized Boltzmann Boltzmann solution solution onon thethe gasgas flowflow in in an elongatedelongated channelchannel obtainedobtained byby LoyalkaLoyalka and and Hamoodi Hamoodi [ 41[41]] is is simulated simulated and and examined examined in in the the presentpresent study. study. As As shown shown in Figurein Figure2, the 2, the present present simulation simulation results results for thefor the dimensionless dimensionless gas gas mass mass flowflow rate rate agree agree with with the the available available linearized linearized Boltzmann Boltzmann solutions solutions at differentat different scales. scales. The The agreement agreement betweenbetween the the analytic analytic solutions solutions and and simulation simulation data data indicates indicates that that the the unified unified flow flow model model predicts predicts the the gasgas flow flow characteristics characteristics during during gas gas filling filling and and leaking leaking processes processes of theof the ICF ICF hohlraum. hohlraum.

3.0 Analytical solution of Boltzmann equation[41] 2.5 Unified flow model 2.0 M 1.5

1.0

0.5 0.1 1 10 100 Kn

FigureFigure 2. Effect 2. Effect of theof the Knudsen Knudsen number number on on the the gas gas mass mass flow flow rate: rate: comparison comparison between between the the analytical analytical solutionsolution of theof the Boltzmann Boltzmann equation equation [41 ][41] and and numerical numerical simulation simulation based based on theon the unified unified flow flow model. model.

3. Results3. Results and and Discussion Discussion

3.1.3.1. Gas Gas Filling Filling and and Leaking Leaking Processes Processes ToTo more more clearly clearly understand understand the the effect effect of pressureof pressu variationre variation rate rate as wellas well as theas the extreme extreme rupture rupture conditioncondition of theof the hohlraum hohlraum film, film, this this study study investigates investigates the the gas gas flow flow behaviors behaviors of theof the gas gas filling filling and and leakingleaking processes processes in thein the hohlraum hohlraum and and the the maximum maximu ofm pressureof pressure difference difference between between the the inside inside and and outsideoutside of of the the ICF ICF hohlraum hohlraum throughthrough varying the the pressure pressure variation variation rate. rate. The The gas gasflow flow behaviors behaviors during duringthe gas the filling gas filling process process are shown areshown in Figure in 3. Figure In the3. in Initial the stage initial of stagethe gas of filling the gas process, filling the process, gas mass theflow gas rate mass quickly flow rate rises quickly to its peak rises value to its and peak rapidly value decreases and rapidly to a decreasessteady-state to value. a steady-state Furthermore, value. the Furthermore,pressure difference the pressure between difference the inside between and outside the inside of the and hohlraum outside Δ ofP theinitially hohlraum reaches∆ Pa initiallymaximum reaches a maximum value ∆P , and this is followed by a gradually decrease. This phenomenon value ΔPmax, and this is followedmax by a gradually decrease. This phenomenon indicates that the pressure indicatesin the hohlraum that the pressure and gas in filling the hohlraum chamber andtends gas to filling balanc chambere with respect tends toto balancetime. Mo withreover, respect an increase to time. in Moreover,the variation an increase rate of the in the filling/leaking variation rate pressure of the le filling/leakingads to a larger pressuremaximum leads pressure to a larger difference maximum between pressure difference between the inside and outside of the ICF hohlraum during the filling/leaking the inside and outside of the ICF hohlraum during the filling/leaking process ΔPmax. process ∆Pmax. The gas flow behaviors during the gas leaking process are shown in Figure4. The gas leaking 3 ) mass flow rate is maintained at a stable value for a long period) of time after the leaking process begins, -1 6 -1 M s s ⋅ ⋅ 2 and this is attributed to the4 extremely low pressure variation rate at this stage. At the end of the leaking ( kg (kg 10 process, the mass flow rate decreases abruptly. Furthermore, the10 pressure difference between the inside 2 1 10 M 10 × and outside of the hohlraum ∆P increases slowly during the initial× period of the leaking process and M 0 M 0 reaches a maximum ∆Pmax0 finally. Additionally,Δ ∆Pmax increases0 when the pressureΔP =0.5 atm variation rate of 10 Pc = 0.5 atm 10 0.525 atm c the gas leaking increases. 10-1 0.259 atm 10-1 ΔP* = P -P

(atm) o i (atm)

* * ΔP = P -P * P

-2 o i P -2 Δ

10 Δ 10 101 101 100 100 -1 P P 10 i 10-1 i (atm) -2 P (atm) -2 P

P o o 10 P 10 10-3 10-3 0 306090 0 5 10 15 20 t (s) t (s) (a) (b)

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3 4 =− 1+ 1 + (14) 64 1− To validate the model, a theoretical case of the linearized Boltzmann solution on the gas flow in an elongated channel obtained by Loyalka and Hamoodi [41] is simulated and examined in the present study. As shown in Figure 2, the present simulation results for the dimensionless gas mass flow rate agree with the available linearized Boltzmann solutions at different scales. The agreement between the analytic solutions and simulation data indicates that the unified flow model predicts the gas flow characteristics during gas filling and leaking processes of the ICF hohlraum.

3.0 Analytical solution of Boltzmann equation[41] 2.5 Unified flow model 2.0 M 1.5

1.0

0.5 0.1 1 10 100 Kn Figure 2. Effect of the Knudsen number on the gas mass flow rate: comparison between the analytical solution of the Boltzmann equation [41] and numerical simulation based on the unified flow model.

3. Results and Discussion

3.1. Gas Filling and Leaking Processes To more clearly understand the effect of pressure variation rate as well as the extreme rupture condition of the hohlraum film, this study investigates the gas flow behaviors of the gas filling and leaking processes in the hohlraum and the maximum of pressure difference between the inside and outside of the ICF hohlraum through varying the pressure variation rate. The gas flow behaviors during the gas filling process are shown in Figure 3. In the initial stage of the gas filling process, the gas mass flow rate quickly rises to its peak value and rapidly decreases to a steady-state value. Furthermore, the pressure difference between the inside and outside of the hohlraum ΔP initially reaches a maximum value ΔPmax, and this is followed by a gradually decrease. This phenomenon indicates that the pressure in the hohlraum and gas filling chamber tends to balance with respect to time. Moreover, an increase in the variation rate of the filling/leaking pressure leads to a larger maximum pressure difference between theSustainability inside and2018 outside, 10, 3763 of the ICF hohlraum during the filling/leaking process ΔPmax. 6 of 12

3 ) ) -1 6 -1 M s s ⋅ ⋅ 2 4 ( kg (kg 10 2 10 1 10 M 10 × × M Sustainability 2018, 10, x FOR0 PEER REVIEW M 0 6 of 12 0 Δ 0 ΔP =0.5 atm 10 Pc = 0.5 atm 10 0.525 atm c Figure 3. Gas flow10 behaviors-1 0.259 atm during the gas filling process: 10(a-1) the pressure rise ΔP* rate= P -isP 10 atm/min;

(atm) o i (atm)

* * ΔP = P -P * P

(b) the pressure rise-2 rate is 40 atm/min. o i P -2 Δ

10 Δ 10 101 101 The gas flow behaviors100 during the gas leaking process10 are0 shown in Figure 4. The gas leaking -1 P P mass flow rate is maintained10 at a stable valuei for a long 10period-1 of time after thei leaking process (atm) -2 P (atm) -2 P

P o o begins, and this is attributed10 to the extremely low pressureP 10variation rate at this stage. At the end of 10-3 10-3 the leaking process, the0 mass 306090 flow rate decreases abruptly. 0Furthermore, 5 10 15the pressure 20 difference t (s) t (s) between the inside and outside of the hohlraum ΔP increases slowly during the initial period of the (a) (b) leaking process and reaches a maximum ΔPmax finally. Additionally, ΔPmax increases when the pressureSustainabilityFigure variation 2018 3. Gas, 10, x;rate ﬂow doi: of behaviorsFOR the PEER gas duringREVIEWleaking the increases. gas ﬁlling process: (a) the pressure www.mdpi.com/journal rise rate is 10 atm/min;/sustainability (b) the pressure rise rate is 40 atm/min.

6 2.0 ) ) -1 -1 s ⋅ s 1.5 4 ⋅ kg ( 1.0 ( kg 10 2 9 10

10 0.5

× M M × M

0 M 0.0 0 ΔP = 0.5 atm 10 c 0 Δ 0.517 atm 10 Pc = 0.5 atm 10-1 0.257 atm 10-2 10-1 (atm) (atm) -3 P 10 Δ P Δ P=P -P Δ -2 Δ i o 10 P=P -P 10-4 i o 101 101 100 100 -1 -1 Pi 10 Pi 10 (atm) -2 (atm) -2 P

P o 10 Po P 10 10-3 10-3 0 20406080 0 5 10 15 20 t/s t/s (a) (b)

FigureFigure 4. 4.GasGas flow ﬂow behaviors behaviors during during the gas the leaking gas leaking process process (pressure (pressure decreases decreases from 15 from atm 15to atm0.001 to atm):0.001 (a atm):) the pressure (a) the pressure drop rate drop is 10 rate atm/min; is 10 atm/min; (b) the pressure (b) the pressure drop rate drop is 40 rate atm/min. is 40 atm/min.

ToTo avoid avoid ICF hohlraumhohlraum film film rupture, rupture, it isit necessaryis necessary to analyze to analyze the critical the critical rate of rate pressure of pressure variation variationΦc for the Φc critical for the pressure critical pressure value ∆P valuec. The Δ effectPc. The of effect the pressure of the pressure variation variation rate Φ during rate Φ gas during filling gas and fillingleaking and process leaking on process the maximum on the maximum of pressure of pres differencesure difference between between the inside the andinside outside and outside of the ICFof thehohlraum ICF hohlraum∆Pmax isΔ shownPmax is inshown Figure in5 .Figure As shown 5. As in theshown figure, in underthe figure, the same under physical the same property physical and propertyworking and condition, working increases condition, in the increases rate of pressurein the rate variation of pressureΦ lead variation to a larger Φ maximum lead to a pressure larger maximumdifference pressure∆Pmax. Additionally,difference ΔPmax the. Additionally, curves of gas the filling curves and of leaking gas filling process and nearlyleaking collapse process onnearly each collapseother, and on each the criticalother, and rates the of critical the pressure rates of variation the pressureΦc for variation the gas Φ fillingc for the and gas leaking filling processesand leaking are processesapproximately are approximately equal. equal.

1.0 filling 0.8 leaking

Φ 0.6 Δ c (atm) Pc max P

Δ 0.4

0.2

0.0 0 10203040 Φ (atm⋅min-1)

Figure 5. Effect of the critical rate of pressure variation during the gas filling and leaking processes on the maximum pressure difference between the inside and outside of the ICF hohlraum.

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Figure 3. Gas flow behaviors during the gas filling process: (a) the pressure rise rate is 10 atm/min; (b) the pressure rise rate is 40 atm/min.

The gas flow behaviors during the gas leaking process are shown in Figure 4. The gas leaking mass flow rate is maintained at a stable value for a long period of time after the leaking process begins, and this is attributed to the extremely low pressure variation rate at this stage. At the end of the leaking process, the mass flow rate decreases abruptly. Furthermore, the pressure difference between the inside and outside of the hohlraum ΔP increases slowly during the initial period of the leaking process and reaches a maximum ΔPmax finally. Additionally, ΔPmax increases when the pressure variation rate of the gas leaking increases.

6 2.0 ) ) -1 -1 s ⋅ s 1.5 4 ⋅ kg ( 1.0 ( kg 10 2 9 10

10 0.5

× M M × M

P o 10 Po P 10 10-3 10-3 0 20406080 0 5 10 15 20 t/s t/s (a) (b)

Figure 4. Gas flow behaviors during the gas leaking process (pressure decreases from 15 atm to 0.001 atm): (a) the pressure drop rate is 10 atm/min; (b) the pressure drop rate is 40 atm/min.

To avoid ICF hohlraum film rupture, it is necessary to analyze the critical rate of pressure variation Φc for the critical pressure value ΔPc. The effect of the pressure variation rate Φ during gas filling and leaking process on the maximum of pressure difference between the inside and outside of the ICF hohlraum ΔPmax is shown in Figure 5. As shown in the figure, under the same physical property and working condition, increases in the rate of pressure variation Φ lead to a larger maximum pressure difference ΔPmax. Additionally, the curves of gas filling and leaking process nearly Sustainabilitycollapse on2018 each, 10, 3763other, and the critical rates of the pressure variation Φc for the gas filling and leaking7 of 12 processes are approximately equal.

1.0 filling 0.8 leaking

Φ 0.6 Δ c (atm) Pc max P

Δ 0.4

0.2

0.0 0 10203040 Φ (atm⋅min-1)

FigureFigure 5. 5.Effect Effect of of the the critical critical rate rate of of pressure pressure variation variation during during the the gas gas filling fillingand andleaking leaking processesprocesses on theon maximum the maximum pressure pressure difference difference between between the the inside inside and and outside outside of of the the ICF ICF hohlraum. hohlraum. SustainabilitySustainability 2018 2018, 10, 10, x, xFOR FOR PEER PEER REVIEW REVIEW 7 7of of 12 12 3.2.Sustainability Influence 2018 Factor, 10, x; Analysis doi: FOR PEER REVIEW www.mdpi.com/journal/sustainability 3.2.3.2. Influence Influence Factor Factor Analysis Analysis To improve the control capability of gas filling and leaking processes, the influence factors are ToTo improve improve the the control control capability capability of of gas gas filling filling and and leaking leaking processes, processes, the the influence influence factors factors are are analyzed in this study. To elucidate the role of the structure parameters of the gas fill tube in the gas analyzedanalyzed in in this this study. study. To To elucidate elucidate the the role role of of the the structure structure parameters parameters of of the the gas gas fill fill tube tube in in the the gas gas flow behaviors during gas filling and leaking processes Figures6 and7 compare the critical rate of flowflow behaviors behaviors during during gas gas filling filling and and leaking leaking processes processes Figures Figures 6 6 and and 7 7 compare compare the the critical critical rate rate of of pressure variation Φc with different structure parameters of the gas fill tube, i.e., diameter dx and pressurepressure variation variation Φ Φc cwith with different different structure structure parameters parameters of of the the gas gas fill fill tube, tube, i.e., i.e., diameter diameter d dx xand and length Lx. With the increases in the diameter of the gas fill tube dx and decreases in the length of the lengthlength L Lx.x .With With the the increases increases in in the the diameter diameter of of the the gas gas fill fill tube tube d dx xand and decreases decreases in in the the length length of of the the gas fill tube L , the critical rate of the pressure variation Φ during the gas filling/leaking process gasgas fill fill tube tube xL Lx,x ,the the critical critical rate rate of of the the pressure pressure variation variation Φ Φc cduring during the the gas gas filling/leaking filling/leaking process process graduallygraduallygradually increases, increases, increases, thereby thereby thereby leadingleading leading toto to a a lower lower probability probability probability of of the the ICF ICF hohlraum hohlraum hohlraum film film film rupture rupture rupture and and and an an an increaseincreaseincrease in in thein the the security security security of of of the the the hohlraum. hohlraum. hohlraum.

4040 L = 500 mm Lx x = 500 mm

) 30 filling leaking ) 30 filling leaking -1 -1 min min ⋅  2020 (atm (atm c c Φ  1010

00 0.050.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0.10 0.10 dd (mm) (mm) x x FigureFigureFigure 6. 6. Effect6. Effect Effect of of of the the the diameter diameter diameter ofof of thethe the gas gas tube tube on on on the the the critical criticalcritical rate rate of of pressure pressure pressure variation variation variation during during during the the the gas gas gas ﬁllingfillingfilling and and and leaking leaking leaking processes. processes. processes.

d dx = = 0.1 0.1 mm mm 6060 x fillingfilling leakingleaking ) ) -1 -1 5050 min min ⋅  (atm (atm c 40

c 40 Φ 

3030

300300 400 400 500 500 600 600 700 700 LLx (mm) (mm) x FigureFigureFigure 7. 7. Effect7. Effect Effect of of of the the the length length length ofof of thethe the gasgas gas tube tube on on the the critical critical critical rate raterate of of pressure pressure variation variation variation during during during the the the gas gas gas ﬁllingfillingfilling and and and leaking leaking leaking processes. processes. processes.

AsAs mentioned mentioned above, above, the the cross-section cross‐section of of the the gas gas fill fill tube tube is is circular. circular. To To examine examine the the effect effect of of the the cross-sectionalcross‐sectional shape shape of of the the gas gas fill fill tube tube on on the the filling/leaking filling/leaking process, process, a a tube tube with with a a length length of of 500 500 mm mm andand a a rectangular rectangular cross cross section section is is applied applied to to the the gas gas filling filling and and leaking leaking processes. processes. Differing Differing from from the the equationequation for for the the mass mass flow flow rate rate of of the the tube tube with with circ circularular cross-section, cross‐section, the the gas gas mass mass flow flow rate rate in in tube tube withwith the the rectangular rectangular cross-section cross‐section is is calculated calculated as as follows follows [39]: [39]: ℎ𝑤ℎ𝑃 ΔΔ𝑃 66𝐾𝑛 =−C(AR) × 1 + 1+ (15) 𝑀 C(AR)12 1 𝛼𝐾𝑛11− (15) 12𝜇𝑅𝑇 𝐿 1𝑏𝐾𝑛 wherewhere w w denotes denotes the the width width of of the the rectangular rectangular cross-section, cross‐section, h h denotes denotes the the height height of of the the rectangular rectangular cross-section,cross‐section, and and b b = = − −1.1. The The parameter parameter β β = = 0.5. 0.5. C C(AR)(AR) is is a a correction correction coefficient coefficient in in which which the the value value is is relatedrelated to to the the aspect aspect ratio ratio of of a a rectangular rectangular section section as as follows: follows:

SustainabilitySustainability 2018 2018, 10, 10, x;, x; doi: doi: FOR FOR PEER PEER REVIEW REVIEW www.www.mdpi.com/journmdpi.com/journalal/sustainability/sustainability Sustainability 2018, 10, 3763 8 of 12

As mentioned above, the cross-section of the gas fill tube is circular. To examine the effect of the cross-sectional shape of the gas fill tube on the filling/leaking process, a tube with a length of 500 mm and a rectangular cross section is applied to the gas filling and leaking processes. Differing from the equation for the mass flow rate of the tube with circular cross-section, the gas mass flow rate in tube with the rectangular cross-section is calculated as follows [39]:

3 . wh Pavg ∆P 6Knavg M = −C(AR) × 1 + αKnavg 1 + (15) 12µ0RT L 1 − bKnavg where w denotes the width of the rectangular cross-section, h denotes the height of the rectangular cross-section, and b = −1. The parameter β = 0.5. C(AR) is a correction coefﬁcient in which the value is related to the aspect ratio of a rectangular section as follows:

Sustainability 2018, 10, x FOR PEER REVIEW  8 of 12 0.42173, w/h = 1 : 1  C(AR) = 0.68605, w/h = 2 : 1 (16) 0.42173, /ℎ = 1: 1  = CAR =0.84244,0.68605, w/ℎ/h = 2:4 :1 1 (16) 0.84244, /ℎ = 4: 1 As the aspect ratio of a rectangular section increases, the cross-sectional shape of the gas fill tube As the aspect ratio of a rectangular section increases, the cross-sectional shape of the gas fill tube departs from a circle to a rectangle. Therefore, the aspect ratio w/h of the gas fill tube with the same departs from a circle to a rectangle. Therefore, the aspect ratio w/h of the gas fill tube with the same rectangularrectangular cross cross section section area area is variedis varied to to understand understand the the effect effect ofof cross-sectionalcross-sectional shape shape on on the the gas gas flow behaviorsflow behaviors duringgas during filling gas and filling leaking and leaking processes. proce Figuresses. Figure8 shows 8 shows the time the revolutiontime revolution of the of gasthe mass flowrategas massΦ and flowrate the pressure Φ and the difference pressure betweendifference thebetween inside the and inside outside and outside of thehohlraum of the hohlraum∆P. Compared ΔP. withCompared the gas fill with tube the with gas fill circular tube with cross-section, circular cross-section, the pressure the differencepressure difference between between the inside the andinside outside of theand hohlraum outside of∆ theP follows hohlraum a trendΔP follows similar a trend to that similar when to that the cross-sectionalwhen the cross-sectional shape of shape the gasof the fill tube is rectangular.gas fill tube Theis rectangular. maximum The pressure maximum difference pressure also difference occurs also at the occurs initial at stagethe initial of gas stage filling of gas process or thefilling final process stage ofor gasthe leakingfinal stage process. of gas However,leaking process. increases However, in the increases aspect ratioin the gradually aspect ratio increase gradually increase the maximum pressure difference ΔPmax, and this is not conducive for the safety the maximum pressure difference ∆Pmax, and this is not conducive for the safety of the hohlraum of the hohlraum during the gas filling and leaking processes. When compared with the gas filling during the gas filling and leaking processes. When compared with the gas filling and leaking processes and leaking processes shown in Figures 3b and 4b with the same pressure rise/drop rate, the shown in Figures3b and4b with the same pressure rise/drop rate, the maximum pressure difference maximum pressure difference ΔPmax for the circular cross-section tube is lower than that for the ∆Pmaxrectangularfor the circular section cross-sectiontube. Therefore, tube a departure is lower thanin cross that sectional for the rectangularshape of fill tube section from tube. circle Therefore, to a departurerectangle in triggers crosssectional an increase shape in pressure of fill tube diff fromerence circle between to rectangle the inside triggers and outside an increase of the in ICF pressure differencehohlraum, between which raises the inside the risk and of polymeric outside of films the rupture ICF hohlraum, and decrea whichses the security raises the of the risk hohlraum of polymeric filmsstructure. rupture and decreases the security of the hohlraum structure.

) 2.0 ) -1 -1 s ⋅ s

2 ⋅ 1.5

(kg w/h = 1:1 1.0

9 w/h = 1:1 (kg 1 w/h = 2:1 9 10 w/h = 2:1

10 0.5 ×

w/h = 4:1 ×

M w/h = 4:1

M 0.0 0 ΔP = 0.5 atm -1 c ΔP = 0.5 atm 10 10-1 c w/h = 1:1 -2 w/h = 1:1 (atm) -2 10 (atm)

P w/h = 2:1 10 P w/h = 2:1 Δ w/h = 4:1 Δ -3 w/h = 4:1 10 10-3 0 5 10 15 20 05101520 t/s t (s)

(a) (b)

FigureFigure 8. Effect8. Effect of of the the rectangular rectangular tube aspect aspect ratio ratio on on (a) ( athe) the gas gas filling ﬁlling and and(b) the (b )gas the leaking gas leaking processesprocesses (both (both the the pressure pressure rise rise rate rate andand drop rate rate of of gas gas filling ﬁlling chamber chamber are are40 atm/min). 40 atm/min).

Apart from the structure parameters of the gas fill tube, the size of the hohlraum also significantly affect the gas flow behaviors during gas filling and leaking processes. Figure 9 shows the effect of the hohlraum volume V on the critical rate of the pressure variation Φc during the gas filling and leaking processes. When the hohlraum volume V increases, the critical rate of the pressure variation Φc gradually decreases, thereby resulting in high probability of hohlraum film rupture and a decrease in the security of the hohlraum. Therefore, with respect to the design of the hohlraum, the volume of the target cavity should be reduced to the maximum possible extent based on the premise that the physical requirement is satisfied.

Sustainability 2018, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/sustainability Sustainability 2018, 10, 3763 9 of 12

Apart from the structure parameters of the gas fill tube, the size of the hohlraum also significantly affect the gas flow behaviors during gas filling and leaking processes. Figure9 shows the effect of the hohlraum volume V on the critical rate of the pressure variation Φc during the gas filling and leaking processes. When the hohlraum volume V increases, the critical rate of the pressure variation Φc gradually decreases, thereby resulting in high probability of hohlraum film rupture and a decrease in the security of the hohlraum. Therefore, with respect to the design of the hohlraum, the volume of the target cavity should be reduced to the maximum possible extent based on the premise that the Sustainability 2018, 10, x FOR PEER REVIEW 9 of 12 physicalSustainability requirement 2018, 10, x FOR is satisfied. PEER REVIEW 9 of 12

40 V = 17.49 mm3 (d = 2.2 mm, L = 4.6 mm) 40 3 h , h V = 17.49 mm (dh = 2.2 mm Lh = 4.6 mm) )

-1 30 )

-1 30 min ⋅ min ⋅ 20 (atm c 20

Φ (atm filling c Φ 10 leakingfilling 10 leaking

0 0 50 100 150 50V (mm 1003) 150 V (mm3)

Figure 9. Effect of the hohlraum volume on the critical rate of pressure variation during the gas filling Figure 9. Effect of the hohlraum volume on the critical rate of pressure variation during the gas filling andFigure leaking 9. Effect processes. of the hohlraum volume on the critical rate of pressure variation during the gas filling andand leaking leaking processes. processes. It is well-known that ICF experiments will need to be performed at 10–60 Hz to be of interest for It is well-known that ICF experiments will need to be performed at 10–60 Hz to be of interest for real applicationIt is well-known of inertial that ICF confinement experiments fusion will need energy. to be Theperformed hohlraums at 10–60 will Hz be to filled be of interestwithin anfor realextremelyreal application application short of inertialamountof inertial confinement of timeconfinement before fusion the fusion energy.next energy.on Thee will hohlraums The be hohlraumsfilled. will In beparticular, will filled be within filledin fact, anwithin extremelyfor the an shortpreconceivedextremely amount short of application time amount before of theinertialtime next before confinement one the will next be filled.fuonsione will energy, In particular,be filled. the settingIn in particular, fact, filling for thepressurein fact, preconceived forof thethe applicationhohlraumpreconceived ofis inertialmuch application less confinement than of 15inertial atm fusion investigatedconfinement energy, the infu settingthesion current energy, filling work the pressure setting[29,42]. of filling Therefore, the hohlraum pressure Figure isof muchthe10 lesscompareshohlraum than 15 atmtheis much gas investigated fill less time than of in 15 the the atm hohlraum current investigated work vers [usin29 ,thethe42]. currentpressure Therefore, work variation Figure [29,42]. 10during Therefore, compares the gas Figure the filling gas 10 fill timeprocessescompares of the hohlraumunder the gas different fill versus time setting theof the pressurefilling hohlraum pressure variation vers of theus during thehohlraum. pressure the gas As variation fillingshown, processes there during is a positivethe under gas different linearfilling settingrelationshipprocesses filling under pressurebetween different ofthe the pressure setting hohlraum. filling variation Aspressure shown, during of therethe the hohlraum. gas is afilling positive As processes shown, linear relationshipthereand the is agas positive fill between time linear of the pressuretherelationship hohlraum. variation between That during is tothe thesay, pressuregas in order filling variation to processes get a duringshor andt gas the the fill gas gastime filling fill of timethe processes hohlraum, of the hohlraum.and we the should gas That fill increase time is to of say, inpressure orderthe hohlraum. to getvariation a shortThat rate. is gas to fillIfsay, the time in setting order of the tofilling hohlraum, get a pressureshort wegas shouldisfill 0.1 time atm, increase of thewhich hohlraum, pressure is lower we variation than should the rate. increasecritical If the pressure variation rate. If the setting filling pressure is 0.1 atm, which is lower than the critical settingpressure filling difference pressure ΔP isc = 0.1 0.5 atm, atm, whichthe gas is fill lower time of than thethe hohlraum critical is pressure much smaller difference than that∆Pc of= 1 0.5 atm atm, pressure difference ΔPc = 0.5 atm, the gas fill time of the hohlraum is much smaller than that of 1 atm theand gas 15 fill atm. time In ofthis the condition, hohlraum the is gas much fill time smaller of the than hohlraum that of may 1 atm be and reduced 15 atm. to 0.1 In s this as pressure condition, and 15 atm. In this condition, the gas fill time of the hohlraum may be reduced to 0.1 s as pressure thevariation gas fill time rate Φ of is the 68.2 hohlraum atm/min. may be reduced to 0.1 s as pressure variation rate Φ is 68.2 atm/min. variation rate Φ is 68.2 atm/min.

1000 Po|t→∞ = 15atm

(s) Φ t 10 c

(s) Φ t c 1 1 0.1 0.1 t < 0.1s t < 0.1s 1 10 100 1Φ (atm 10⋅min-1) 100 Φ (atm⋅min-1)

FigureFigure 10. 10.Effect Effect ofof the pressure variation variation during during the the gas gas filling ﬁlling processes processes on the on gas the fill gas time ﬁll of time the of thehohlraum.Figure hohlraum. 10. Effect of the pressure variation during the gas filling processes on the gas fill time of the hohlraum. 4. Conclusions 4. Conclusions Based on unified flow theory, a theoretical model of the gas flow behaviors during the gas filling and leakingBased on processes unified flow of the theory, ICF ahohlraum theoretical is model developed of the togas evaluate flow behaviors and analyze during the the gas fillingmass flowrateand leaking and rateprocesses of pressure of the variation. ICF hohlraum The role is of developed the fill tube to size evaluate and the and filling analyze pressure the on gas the mass gas fillingflowrate and and leaking rate of performancepressure variation. are examined The role ofand the analyzed. fill tube size The and guidelines the filling for pressure safe filling on the and gas leakingfilling andof the leaking ICF hohlraum performance are provided. are examined The conclusion and analyzed. are summarized The guidelines as follows: for safe filling and leaking of the ICF hohlraum are provided. The conclusion are summarized as follows:

Sustainability 2018, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/sustainability Sustainability 2018, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/sustainability Sustainability 2018, 10, 3763 10 of 12

4. Conclusions Based on unified flow theory, a theoretical model of the gas flow behaviors during the gas filling and leaking processes of the ICF hohlraum is developed to evaluate and analyze the gas mass flowrate and rate of pressure variation. The role of the fill tube size and the filling pressure on the gas filling and leaking performance are examined and analyzed. The guidelines for safe filling and leaking of the ICF hohlraum are provided. The conclusion are summarized as follows:

1. An increase in the variation rate of the filling/leaking pressure leads to a larger maximum pressure difference between the inside and outside of the ICF hohlraum during the filling/leaking process, and the critical pressure difference of the gas filling process is nearly equal to that of the gas leaking process. 2. Increase in fill tube diameter and decrease in its length both lead to a lower probability of the rupture of polymeric films at two ends of the ICF hohlraum, and thus increases the security of the hohlraum. 3. A departure in cross sectional shape of fill tube from circle to rectangle triggers an increase in pressure difference between the inside and outside of the ICF hohlraum, which raises the risk of polymeric films rupture and decreases the security of the hohlraum structure.

This investigation not only provides a deep understanding of gas filling and leaking processes of the ICF hohlraum with multiscale gas flow problems but also contributes to precisely controlling the gas mass flow rate and pressure variation rate via the optimized fill tube structure and hohlraum volume, which is of significance for the design of the ICF hohlraum with target microgripper. Since the gas flow in ICF hohlraum is a complex multiscale flow problem, the future research work could focus on the hydrodynamics of gas filling and retention via the multiscale three-dimensional unsteady numerical simulation.

Author Contributions: W.Y. provided the guidance and supervision. C.Y. and S.W. implemented the main research, discussed the results, and wrote the paper. All authors read and approved the final manuscript. Funding: This research was funded by National Natural Science Foundation of China (Nos. 51776037 and 51706194). Conflicts of Interest: The authors declare no conflict of interest.

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