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2012

An experimental and theoretical study of heat and mass transfer An experimental and theoretical study of heat and mass transfer

during the venting of gas from pressure vessels during the venting of gas from pressure vessels

W. S. Winters Sandia National Laboratories, Livermore, CA

G. H. Evans Sandia National Laboratories, Livermore, CA

S. F. Rice Sandia National Laboratories, Livermore, CA

R. Greif University of California, Berkeley, CA

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Winters, W. S.; Evans, G. H.; Rice, S. F.; and Greif, R., "An experimental and theoretical study of heat and mass transfer during the venting of gas from pressure vessels" (2012). US Department of Energy Publications. 133. https://digitalcommons.unl.edu/usdoepub/133

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An experimental and theoretical study of heat and mass transfer during the ventingof gas from pressure vessels

W.S. Winters a,⇑, G.H. Evans a, S.F. Rice a, R. Greif b

a Sandia National Laboratories, Livermore, CA 94551, United Statesb Department of Mechanical Engineering, University of California, Berkeley, CA 94720, United States

a r t i c l e i n f o

Article history:Received 10 December 2010Received in revised form 15 August 2011Accepted 15 August 2011Available online 19 September 2011

Keywords:Compressible flowHeat transferDepressurizationGas venting

a b s t r a c t

Non condensing gas flow and heat transfer during venting of vessels are studied using experiments andanalysis. A high pressure helium supply vessel is connected to a low pressure receiver via orifice and tub-ing. A single control volume analysis and a multi-dimensional analysis are used to predict pressure andmass-averaged temperature in the supply. Experiments utilizing transient PVT methods are conducted toobtain transient pressure and mass-averaged temperature data for validating the analysis. Measuredtransient pressures and mass-averaged temperatures in the supply are reproduced by analysis. Heattransfer is due to natural convection except for the early part of transfer.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Pressurization and depressurization of gas vessels occur innumerous industrial processes and gas storage applications andare also of fundamental interest. Time scales for filling or emptyingvessels are often sufficiently long and enable significant heat trans-fer to occur between the resident gas and the interior vessel wall. Inthese situations the determination of the heat transfer is importantin determining the mass of gas in the vessel as a function of time.

The majority of recent literature on vessel heat transfer has fo-cused on the filling of high pressure reservoirs, the primary appli-cation being high pressure storage of gaseous hydrogen for theemerging hydrogen economy. Ranong et al. [1] determined heattransfer coefficients for filling pressure vessels. A number of exper-iments have been conducted using thermocouples to measuretransient temperature distributions in tanks and cylinders duringhydrogen filling (see, e.g. [2–4]). Recent literature on dischargingvessels is surprisingly sparse. In the present work our analysesand experimental validation will focus mainly on the heat andmass transfer for pressure vessels undergoing depressurization.In addition to overall or global considerations the spatial variationof the heat transfer is also studied which assists in the elucidationof the basic transport mechanisms.

For vessels undergoing rapid depressurization, gas velocities areextremely low except near the exit hole where velocities can ap-proach sonic values. (Here we assume the exit hole diameter to be

orders of magnitude smaller than the vessel diameter.) As a resultthe static pressure of the gas is essentially uniform throughout thevessel. For rapid depressurization, the interior vessel wall tempera-ture remains relatively constant. As depressurization begins, the gastemperature decreases rapidly creating large temperature excur-sions and heat transfer from the interior vessel wall to the gas. Inthe first stage of this heat transfer, referred to here as ‘‘the earlystage,’’ the heat transfer is characterized by heat conduction andconvection due to gas expansion effects. Thermal instabilitiesquickly form and the heat transfer transitions into the second stagein which the principal mode of heat transfer is by natural convectionand may be either laminar or turbulent depending on the magni-tude of the depressurization, vessel size, and properties of the gas.

A number of studies have been conducted to understand earlystage heat transfer during depressurization. Landram [5] developedan approximate solution in which the time-dependent near wallheat conduction layer thickness in the gas was obtained using anintegral method. Johnston and Dwyer [6] utilized Schlieren cinema-tography to study the flow structure in a discharging gas reservoir.They developed numerical solutions that generally agreed withtheir experimental data. In each of these studies the influence ofconvective heat transfer was neglected. Paolucci [7] developed ananalytical solution that included the convective effects due to gasexpansion. His results compared favorably to available data for veryearly times. Greif et al. [8] developed a similar solution for heattransfer to the sidewalls of a channel in which gas is compressedby a piston. Lin and Armfield [9] utilized direct numerical simula-tion to study unsteady heat transfer in closed rectangular and cylin-drical containers. They identified three main stages of flow

0017-9310/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2011.08.023

⇑ Corresponding author. Tel.: +1 925 830 1320.E-mail address: billwinters@comcast.net (W.S. Winters).

International Journal of Heat and Mass Transfer 55 (2012) 8–18

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International Journal of Heat and Mass Transfer

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evolution and quantified differences in these regimes depending onwhether the container geometry was rectangular or cylindrical.

An early study of heat transfer to or from gases in vessels under-going depressurization and pressurization was conducted byReynolds and Kays [10]. Their analysis utilized a correlationsuggested by McAdams [11] for natural convection, i.e.,

Nu ¼ cRan; ð1Þ

where Nu is the Nusselt number, Ra is the Rayleigh number and theconstants c and n are 0.13 and 0.33 respectively. An experimentalvalidation of the analysis of Reynolds and Kays was performed byLyons [12]. Means and Ulrich [13] conducted a series of experi-ments to quantify transient convective heat transfer during andafter gas injection into a vessel. They utilized the correlation givenby Eq. (1) to calculate the natural convection heat transfer for a per-iod of time after gas injection. The following values for c and n werefound to represent their data:

laminar flow : ðRa � 108Þ : c ¼ 0:53; n ¼ 0:25; ð2Þ

turbulent flow :ðRa � 108Þ : c ¼ 0:12; n ¼ 0:33: ð3Þ

Clark [14] conducted a series of experiments for the rapid dischargeof helium and nitrogen from a vessel and determined the followingvalues for c and n:

laminar flow : ðRa � 1:24� 108Þ : c ¼ 1:15; n ¼ :22; ð4Þ

turbulent flow :ðRa � 1:24� 108Þ : c ¼ 0:14; n ¼ :333: ð5Þ

Charton et al. [15] modeled the discharge of helium and deuteriumfrom a vessel to a vacuum chamber via a long tube of small diam-eter. They also utilized Eq. (1) to describe the heat transfer in thedischarge vessel. The following parameters were used to provide afit to their data:

laminar flow : ðRa � 108Þ : c ¼ 0:47� 0:49 and; n ¼ :25; ð6Þ

turbulent flow :ðRa � 108Þ : c ¼ 0:1; n ¼ :33; ð7Þ

where in the laminar flow correlation, c = 0.47 for cylindrical vesselsand c = 0.49 for spherical vessels.

More recently, Woodfield et al. [16] studied charging and dis-charging of cylinders with hydrogen, nitrogen and argon gas. Theyspatially averaged transient thermocouple and heat flux measure-ments to determine heat transfer to and from the discharging andcharging gas. For discharging hydrogen gas they were able to correlatetheir measurements using Eq. (1) by using c = 0.104 and n = 0.352 forthe latter part of the discharge. Early in the discharge they measuredhigher heat transfer rates which they attributed to transient behaviorcorresponding to the startup of convection currents.

2. Problem statement

In this work we focus on the gas transfer apparatus shown inFig. 1a. A high pressure supply vessel is connected to a low pressurereceiver vessel via a flow path that includes a flow restricting orifice,a valve and several short lengths of relatively large diameter tubing.The pressure drop across the valve and tubing is usually small com-pared to the pressure drop across the orifice. Initially the gas andinterior vessel walls are at the ambient temperature. The vessel walltemperature is considered to be constant throughout the transfer.At time zero the valve is opened and gas flows from the supply tothe receiver. In all cases considered here the supply-to-receiverpressure ratio is large enough to choke the flow at the orifice forsome part of the gas transfer. The pressure drop across the orificeeventually becomes small enough that the flow unchokes afterwhich pressure driven flow continues until the supply and receiverpressures are nearly equal. At this point in time the temperature inthe supply is lower-than-ambient temperature and the temperaturein the receiver is higher-than ambient temperature. Heat transfer tothe supply gas from the supply vessel wall and from the receiver gasto the receiver vessel wall causes additional mass transfer until thetemperature and pressure in both vessels have equilibrated. Weanalyze the heat and mass transfer in the supply vessel during theentire time of transfer. In the calculations and the experiments(which are discussed later) the gas is helium.

3. Single control volume analysis with heat transfer correlation

The gas system in Fig. 1a was analyzed using the network flowanalysis code NETFLOW [17]. NETFLOW calculates compressible

Nomenclature

Ae exit areaAs surface area, area for heat transferb Abel-Noble co-volume constantc constant in natural convection heat transfer correlationCp specific heat at constant pressureCv specific heat at constant volumeD characteristic spherical diameterh heat transfer coefficientk thermal conductivitym mass_me mass flow rate at exit

n exponent constant in natural convection heat transfercorrelation

P pressureP�S supply pressure at time t⁄

P�R receiver pressure at time t⁄

Pe pressure at exitq heat fluxR gas constantt timet⁄ time at which transient PVT data is obtained

T temperatureTave mass-averaged gas temperatureT�S supply mass-averaged temperature at time t⁄

T�R receiver mass-averaged temperature at time t⁄

Tw wall temperatureu internal energy per unit massV volume

Dimensionless groupsGr Grashof number = gb(Tw � T)q2D3/l2

Nu Nusselt number = hD/kPr Prandtl number = Cpl/kRa Rayleigh number = Gr Pr

Greek lettersb volume expansivityc ratio of specific heatsl dynamic viscosityq densityq�S supply mass-averaged density at time t⁄

q�R receiver mass-averaged density at time t⁄

W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18 9

flow in networks consisting of vessels, tubes, orifices, valves andflow branches. A vessel is assumed to behave as a single well-mixed control volume at uniform pressure and temperature. Heattransfer correlations are used to determine the heat transfer be-tween the gas in the vessel and the interior vessel wall. Flow intubes is determined using one-dimensional-transient flow conser-vation equations. The frictional pressure drop in tubes is accountedfor using quasi-steady pressure drop formulations (see, e.g. Moody

[18]). The heat transfer is accounted for using appropriate heattransfer correlations for tube flow (see, e.g. Dittus and Boelter [19]).

The NETFLOW formulation for the supply vessel in Fig. 1a issummarized here. The continuity equation for the supply is givenby

dmdt¼ � _me; ð8Þ

where m is the mass of gas in the supply and _me is the exit massflow rate.The energy equation for the supply is given by

dðmuÞdt

¼ � _meðuþPqÞ þ hAsðTw � TÞ; ð9Þ

where u, P, q and T are the gas internal energy per unit mass, staticpressure, density and temperature, respectively, and, As, Tw and hare the interior vessel surface area, vessel wall temperature andaverage heat transfer coefficient, respectively.

Since some of the initial supply pressures studied here exceedideal gas conditions, a real gas equation-of-state (EOS) is used torelate gas density, pressure and temperature. We use the Abel-No-ble form of the van der Waals EOS in which the molecular attrac-tion constant is neglected and the co-volume constant is slightlychanged (Chenoweth [20]):

P ¼ qRT1� bq

; ð10Þ

where R is the gas constant and b is the adjusted co-volume con-stant. The co-volume constant determined by Chenoweth [20] forhelium (2:673� 10�3m3=kg) is used in the present work. Internalenergy is related to temperature using a constant specific heat atconstant volume:

u ¼ CvT: ð11Þ

The NETFLOW formulation utilizes an analysis developed by Birdet al. [21] and others to describe the rate at which mass exits thesupply vessel. The analysis assumes that flow very near the exithole accelerates isentropically from its stagnation value to the exitplane. The solution for mass flow rate has two branches dependingon whether the flow is choked at the exit hole. For unchoked flowthe exit mass flow rate is given by:

_me ¼ Ae

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Pq

cc� 1

� �Pe

P

� �2=c

� Pe

P

� �ðcþ1c Þ

" #vuut ; ð12Þ

where c is the ratio of specific heats, Ae is the flow area of the exithole, and Pe is the time varying pressure downstream of the exithole.

For choked flow the exit mass flow rate is independent of Pe andgiven by

_me ¼ Ae

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficPq

2cþ 1

� �cþ1c�1

vuut: ð13Þ

The choked flow branch is valid when the following condition ismet:

Pe

P� 2

cþ 1

� � cc�1

: ð14Þ

Eqs. (12)–(14) were developed for an ideal gas. Exit mass flow rela-tionships for an Abel-Noble gas are considerably more complex andcannot be expressed in closed form. Our work shows that the idealgas expressions presented here, while not being thermodynamicallyprecise, are a sufficiently accurate approximation for the exit flow ofa real gas provided the Abel-Noble EOS is used to relate pressure,

Fig. 1. High pressure gas system: (a) simple system; (b) schematic of experimentalapparatus.

10 W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18

density and gas temperature within the vessel. As a result Eqs. (12)–(14) are used to express the exit flow in NETFLOW.

For the case where h = 0 in Eq. (9) and b = 0 in Eq. (10), NET-FLOW reproduces the isentropic ideal gas vessel depressurizationsolution of Bird et al. [21]. For non adiabatic solutions, the heattransfer in the supply is determined from a free convection heattransfer correlation having the form of Eq. (1). The Rayleigh num-ber in (1) is the product of the Grashof and Prandtl numbers and isgiven by:

Ra ¼ GrPr ¼ gbðTw � TÞq2D3

l2

Cplk

ð15Þ

and the Nusselt number is related to the heat transfer coefficient inEq. (9) by

Nu ¼ hDk; ð16Þ

where g is the gravitational constant, D is the characteristic diame-ter of the supply vessel and Cp;b;l; and k are the gas specific heat atconstant pressure, volume expansivity, dynamic viscosity and ther-mal conductivity, respectively.

4. Multi-dimensional analysis

Flow and heat transfer in the supply vessel during depressuriza-tion are both multi-dimensional and transient. Utilizing the singlecontrol volume analysis and heat transfer correlations in NETFLOWmay not sufficiently address the multi-dimensional nature of theproblem. A multi-dimensional analysis may be necessary whenstudying conditions that exceed the parameter space addressedby the heat transfer correlations (e.g., a new geometrical configura-tion, a different range of initial conditions, etc.).

A complete multi-dimensional analysis for the system shown inFig. 1a is difficult since the initial supply-to-receiver pressure ra-tios often lead to choked flow in the connecting piping for a signif-icant fraction of the gas transfer time. During this time largepressure gradients can result in highly under-expanded jets thatcreate complex shock structures in the receiver. Resolving timescales related to this supersonic flow while simultaneously resolv-ing the longer time scales associated with vessel heat transfer (tensof seconds) will inevitably lead to excessively long computationaltimes even with modern numerical algorithms and massively par-allel computing.

In order to overcome these difficulties a code-coupling ap-proach was used in which the multi-dimensional code FUEGO[22] was coupled to NETFLOW at the supply exit hole. FUEGO, acompressible, unstructured, control volume/finite element methodanalysis code was used to solve the multi-dimensional flow andheat transfer problem in the supply while NETFLOW was used tosolve the flow and heat transfer in the interconnecting tubingand receiver. This coupling method was successful in isolatingthe FUEGO computational space from supersonic flow since flowchoking and unchoking in the tubing and at the receiver inletwas accounted for by NETFLOW. Code coupling made it possiblefor NETFLOW to provide the time-dependent mass flow boundarycondition for the FUEGO computational space.

For supply pressures greater than 2.07 MPa (3000 PSIA) theideal gas EOS is not accurate; the previously discussed Abel-NobleEOS was implemented into FUEGO using user subroutines. Initialconditions for FUEGO were assumed to be uniform temperatureand pressure; vessel wall boundary conditions were assumed tobe no-slip with a constant wall temperature.

5. Experiments

The transient PVT method (Clement and Desormes [23]) wasutilized to measure the supply and receiver time varying pressureand gas mass-averaged temperature during a number of heliumtransfer experiments. The mass-averaged temperature representsthe spatially uniform temperature that characterizes the thermalenergy content in each vessel. Since significant thermal gradientsexist in the supply and receiver during the transfer, the mass-averaged temperature is a far more meaningful parameter thanthermocouple or other temperature measurements made at oneor more discrete locations in the vessel. The mass-averaged gastemperature is expressed as

Tave ¼R

V TqdVm

: ð17Þ

The mass-averaged temperature is analogous to the temperature, Tin Eqs. (9)–(11) and Eq. (15) which are used for the single controlvolume analysis of vessel discharge. Measurements of mass-aver-aged temperature are also useful for validating multi-dimensionalanalysis methods since predicted non uniform temperature fieldscan be integrated using Eq. (17) and compared directly to the mea-sured values of mass-averaged temperature.

The transient PVT method developed by Clement and Desormes[23] was also used by Johnston and Dwyer [24] and more recentlyby Clark [14]. The method requires a fast acting valve, a time accu-rate pressure transient measurement and thermocouple measure-ments inside each vessel. The thermocouple measurements are notused to measure temperature transients.

The procedure outlined below describes the transient PVTmethod for determining the pressure and mass-averaged temper-ature in the supply and receiver at a time equal to t�.

1. Charge supply and receiver to desired initial conditions andwait until the temperature in each vessel is uniform.

2. At t = 0, open the valve between the supply and the receiver andallow gas to flow.

3. At t ¼ t�, close the valve and record P�S and P�R, the supply andreceiver pressures at t ¼ t�.

4. Wait for the temperature and pressure in the supply and recei-ver to become uniform in space and use these measured tem-peratures and pressures together with the equation of state tocompute q�S and q�R, the supply and receiver gas densities att ¼ t�.

5. Use P�S and P�R with q�S and q�R in the equation of state (10) todetermine T�S and T�R, the mass-averaged supply and receivertemperatures at t ¼ t�.

6. Repeat steps 1–5 for other values of t� until sufficient data arecollected to determine the transient pressure and mass-aver-aged temperature histories in the supply and receiver overthe time period of interest.

A schematic of the experimental apparatus is shown in Fig. 1b.A solenoid-actuated ball valve was used to control the flow be-tween the supply and receiver. Pressures were measured usingTeledyne Taber dynamic pressure transducers with varying modelnumbers and ranges depending on the magnitude of the pressureexcursion. Temperatures were measured at two locations in eachvessel, one near the wall and one near the center of the vessel;Omega 0.00076 m (0.030 inches) Type K thermocouples were usedfor the measurements. Temperatures in each vessel were assumeduniform when the two thermocouples recorded the same temper-ature. All data were recorded on Nicolet Odyssey data recorder.

Six transient PVT experiments were conducted using a190 � 10�6 m3 supply and various sized receivers. Supply pres-sures varied from 2.17 MPa to 41.51 MPa covering an ideal to real

W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18 11

gas range. Except for one experiment in which the receiver wasevacuated initially, the initial receiver pressure was 1 atmosphere.All experiments were conducted at ambient temperature (295 Kfor the initial condition). Considerable effort was expended to in-sure that each series of transient PVT tests began at nearly identicalinitial conditions. Most masses (supply plus receiver gas mass)computed from measured pressures and mass-averaged tempera-tures were within 1% of total system mass over the entire durationof the transfer. The parameters for each transient PVT test seriesare summarized in Table 1.

Results in the next section will frequently be referenced by testname consisting of three numbers separated by dashes. The se-quence of numbers represents the supply size, the receiver sizeand the nominal initial supply pressure in PSIA. For all tests the ori-fice diameter downstream of the supply was 0.000508 m.

6. Results

In this section results from the single control volume analysisand the multi-dimensional analysis of the supply vessel are pre-sented. Comparisons are made with data obtained from the tran-sient PVT experiments. In both the single control volume andmulti-dimensional analyses, the pressure downstream of the ori-fice, PeðtÞ was computed from a NETFLOW simulation of flowthrough tubing to the receiver. Fig. 2 compares the NETFLOW-pre-dicted receiver pressure to the measured receiver pressure for test190-658-3000PSI. Similar agreement was obtained for all tests. TheNETFLOW-predicted supply pressure is compared to the measuredsupply pressure for test 190-658-300PSI in Fig. 3a. Agreement be-tween measured and predicted pressure in Fig. 3a is typical of theagreement for all tests.

6.1. Single control volume analysis with heat transfer correlation

For the present study, the following constants for the naturalconvection heat transfer correlation (Eq. (1)) were found to providethe best fit to the data:

laminar flow : ðRa � 1:24� 108Þ : c ¼ :933and; n ¼ :25; ð18Þ

turbulent flow : ðRa � 1:24� 108Þ : c ¼ 0:168; n ¼ :33: ð19Þ

The exponents for the Rayleigh number were selected to correspondto traditional values for laminar and turbulent free convection fromvertical surfaces (see, e.g. [25]). These constants may not providethe best fit for heat transfer in much larger vessels.

Figs. 3b–d show measured and predicted supply temperaturesfor three tests in which the supply and receiver volumes were190 � 10�6 m3 and 658 � 10�6 m3 respectively. The initial supplypressures varied between 2.17 MPa and 41.51 MPa. The measure-ments show that higher initial supply pressures result in largertransient temperature excursions. This expected and observedtrend is well replicated by the NETFLOW predictions. The timingand magnitude of the minimum supply temperature and the

recovery back to ambient temperature are accurately reproducedby the calculations.

Fig. 4 shows measured and predicted supply temperatures fortwo tests in which the supply and receiver volumes were190 � 10�6 m3 and 83 � 10�6 m3 respectively. The initial supplypressure in Fig. 4a (Tests 190-83-3000PSI) was 20.82 MPa. InFig. 4b (Test 190-83-6000PSI) the initial supply pressure was41.51 MPa. The reduction in receiver size from 658 � 10�6 to83 � 10�6 m3 results in a more rapid transfer to pressure equilib-rium. Correspondingly, the minimum supply temperature occursearlier in time. This trend is captured by the calculations althoughpredicted minimum temperatures are several degrees less thanmeasured values. The rate of temperature decrease and the recov-ery back to ambient temperature are well-reproduced by thecalculations.

Fig. 5 shows measured and predicted supply temperatures forTEST 190-12909-6000PSI. In this test helium initially at41.37 MPa in the 190 � 10�6 m3 supply is transferred to a rela-tively large 12909 � 10�6 m3 receiver which is initially evacuated.When compared to other tests with smaller receivers and similarinitial supply pressures (i.e., TEST 190-658-6000PSI in Fig. 3d andTEST 190-83-6000PSI in Fig. 4b), it can be seen that the initialrecovery to ambient temperature occurs more slowly. This trendand the time and magnitude of the minimum supply temperatureare well-reproduced by the calculations.

The influence of heat transfer on gas flow from a high pressuresupply is illustrated in Fig. 6. The figure shows the temperature(Fig. 6a) and mass inventory (Fig. 6b) in the supply for three differ-ent heat transfer conditions. The geometry and initial conditionsfor TEST 190-658-3000PSI are used in the calculations. The three

Table 1Test matrix.

Test name Nominal supply volume (m3) Nominal receiver volume (m3) Nominal initial supply pressure (MPa) Nominal initial receiver pressure (MPa)

190-658-300PSI 190 � 10�6 658 � 10�6 2.17 0.1190-658-3000PSI 190 � 10�6 658 � 10�6 20.79 0.1190-658-6000PSI 190 � 10�6 658 � 10�6 41.51 0.1190-83-3000PSI 190 � 10�6 83 � 10�6 20.82 0.1190-83-6000PSI 190 � 10�6 83 � 10�6 41.51 0.1190-12909-

6000PSI190 � 10�6 12909 � 10�6 41.37 Vacuum

Fig. 2. Measured and predicted receiver pressure for Test 190-658-3000PSI.

12 W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18

curves shown are for the following three heat transfer conditions:(1) heat transfer using the natural convection correlation definedby Eqs. (1), (18), and (19); (2) adiabatic (zero heat transfer); and(3) isothermal (nearly infinite heat transfer rate). Experimental re-sults are also included in Fig. 6. The experimental mass valuesshown in Fig. 6b were computed using the helium EOS and themeasured supply pressure, measured mass-averaged temperature,and supply volume. Neglecting heat transfer entirely (the adiabaticassumption) produces significant and lasting deviations from themeasured temperature and mass after one second of transfer. Asexpected, the isothermal assumption produces good estimates fortemperature and mass inventory at late times as the gas in the sup-ply approaches ambient temperature. The isothermal assumptionproduces poor results during the earlier part of the transfer evenafter pressure equilibrium is essentially achieved at approximately2 s. If the transfer is interrupted with a valve closure prior toachieving thermal equilibrium, estimates of the mass remainingin the supply vessel could be in error by as much as 9%.

6.2. Multi-dimensional analysis

In this section results are presented of the coupled calculationsof Test 190-658-300PSI. For these coupled calculations the multi-dimensional FUEGO code is used to simulate the supply vessel fluidflow, heat, and mass transfer during depressurization, and the NET-FLOW code is used to simulate the transport processes in the pipenetwork and receiving vessel. The initial supply pressure is2.17 MPa (315 PSIA), and the initial receiver pressure and pipe net-work pressure downstream of the valve are 0.1 MPa. The initialtemperature is uniform throughout the system at 293.05 K. Thesupply vessel interior volume is 1:9� 10�4 m3; the outlet openingdiameter is 0.000508 m; the orientation of the spherical supplyvessel is such that the outlet opening is pointed downward (inthe direction of gravity as shown in Figs. 1a, b).

Due to the relatively uniform pressure within the supply vesselduring depressurization, the velocity is small except in the imme-diate vicinity of the outlet. Furthermore, as shown in Fig. 7, the

Fig. 3. Measured and predicted supply pressure and temperature: (a) pressure and (b) temperature for TEST 190-658-300PSI; (c) temperature for TEST 190-658-3000PSI; (d)temperature for TEST 190-658-6000PSI.

W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18 13

Rayleigh number based on the diameter of the supply vessel andthe difference between the vessel wall temperature and themass-averaged gas temperature, Eq. (15), is well below the valuefor transition to turbulence. A value of 1.977 � 10�5 kg/m-s forthe viscosity of helium and ideal gas values for density and thermalexpansion coefficient were used in the calculation of the Rayleighnumber in Fig. 7.

Thus the results from the FUEGO calculations that are shownhere are for laminar flow conditions. The initial time step was10�7 s; the maximum allowed time step was 10�4 s; discretizationwas first order in time and space; the three dimensional continuity,momentum, and energy equations were solved sequentially withno relaxation and 6 nonlinear iterations were taken at each timestep. All variables are collocated on the unstructured mesh withpressure stabilization used to avoid oscillations. The results pre-sented here are for a hexagonal mesh of 72,600 elements, distrib-uted non-uniformly as shown in Fig. 8 to resolve the gradients near

the vessel wall and outlet. A circular outlet is located at the bottom.Sensitivity of the results to mesh size, time step, and numericalparameters was determined; the results presented here are suffi-ciently independent of those quantities.

The time-dependence of the pressure at the center of the supplyvessel and the mass-averaged temperature calculated using thethree dimensional FUEGO results are compared with the experi-mental data and the NETFLOW results in Figs. 9a and b. The pres-sure is seen to be slightly over-predicted whereas the temperatureis in excellent agreement with the experimental data and the net-work flow result.

The evolution of the heat transfer predicted by the FUEGO sim-ulation is shown in Fig. 10a–c. In this figure the temperature fieldon a central cut plane and the local heat transfer around a line onthe vessel surface defined by the intersection of a central cut planewith the spherical vessel surface are shown at three times duringdepressurization. Fig. 10a shows that at 0.1 s there is a thin andapproximately uniform thermal boundary layer adjacent to the in-ner surface of the vessel and the heat transfer is large and, with theexception of the region adjacent to the outlet, relatively uniform.At this time there is little natural convection of gas in the vessel;but there is convection normal to the vessel surface due to thegas expansion. This is the ‘‘early stage’’ heat transfer regime de-scribed in the introduction. In Fig. 10b at 0.6 s natural convectionhas started as evidenced by the non-uniform thermal boundarylayer and Nusselt number distribution. The largest Nusselt numberis near the outlet at the bottom of the vessel and the smallest Nus-selt number is at the top of the vessel, consistent with the temper-ature field shown in Fig. 10b. The average Nusselt number at thistime is significantly smaller than at 0.1 s (cf. Fig. 10a). In Fig. 10cat 1.26 s natural convection has resulted in a strong recirculationpattern in the upper half of the spherical supply vessel and theNusselt number is highly non-uniform. The average Nusselt num-ber is only slightly smaller at this time compared with its valueat 0.6 s (cf. Fig. 10b).

It is of interest to compare the average supply vessel Nusseltnumbers during depressurization computed by (a) the correlationof Eq. (1) using the constants of Eq. (18) and (b) the local heat fluxat the vessel wall from the three-dimensional simulation, averagedover the surface area of the vessel. The latter is determined as fol-lows. From Eq. (16)

Fig. 4. Measured and predicted supply temperature for (a) TEST 190-83-3000PSIand (b) TEST 190-83-6000PSI.

Fig. 5. Measured and predicted supply temperature for TEST 190-12909-6000PSI.

14 W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18

Nu ¼ hDk¼ qD

kðTw � TÞ ¼DR

AsqlocaldAs

kðTw � TÞAs¼�D

RA krT � n̂dAs

kðTw � TÞAs

�R

AsrT � n̂dAs

pDðTw � TÞ ; ð20Þ

where As ¼ pD2 is the surface area of the spherical supply vessel.The results are shown in Fig. 11 where the heat transfer determined

Fig. 6. Influence of heat transfer on predicted supply temperature (a) and mass (b).

Fig. 7. Supply vessel Rayleigh number, calculated from the FUEGO solution.

Fig. 8. Mesh of fluid region on a central plane through the spherical supply vessel.

Fig. 9. Predicted and measured histories of pressure (a) and mass-averagedtemperature (b) for Test 190-658-300PSI.

W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18 15

from the NETFLOW model using the natural convection correlationis seen to be comparable to that determined from the multi-dimen-sional model for t > 0.5 s. For 0 < t < 0.5 s, the multi-dimensionalmodel predicts a large Nusselt number which rapidly diminisheswith time whereas the NETFLOW flow model predicts a Nusselt

number that rises to a maximum. Also shown in Fig. 11 is the Nus-selt number (green curve) from the analysis of Paolucci [7] for the‘‘early stage’’ heat transfer from a vessel to a discharging pressur-ized gas (i.e., Eq. (46) in [7] for the dimensionless heat transfer).The Paolucci analysis includes conduction and convection normal

Fig. 10. Computed temperature field and heat transfer: (a) 0.1 s; (b) 0.6 s; (c) 1.26 s.

16 W.S. Winters et al. / International Journal of Heat and Mass Transfer 55 (2012) 8–18

to the vessel wall but not free convection. There is very good agree-ment between the FUEGO multi-dimensional result and the ‘‘earlystage’’ result for t < 0.25 s which corresponds to the time periodprior to the onset of free convection as seen in Fig. 10a–b. We notethat although the free convection Nusselt number in the NETFLOWmodel deviates significantly from the multi-dimensional and ‘‘earlystage’’ Nusselt numbers for t < 0.25 s, the effect of this difference onthe prediction of the mass-averaged temperature in the supply ves-sel is negligible. However, it seems clear from the multi-dimen-sional calculations and Paolucci’s analysis that the assumption ofnatural convection during the early stages of depressurization isfundamentally incorrect.

In all cases studied here, the discharge hole was extremely smallcompared to the vessel volume. As a result the pressure in the vesselwas uniform during discharge and pressure driven velocities are ex-tremely low except for the region of flow very near the exit hole. Be-cause of this, both the early stages (conduction and convectionnormal to the wall) and later stages (natural convection) of the heattransfer are unaffected by the orientation of the exit hole.

7. Conclusions

The transient PVT method is useful in determining mass-aver-aged temperature in vessels undergoing rapid discharge and filling.The transient mass-averaged temperature is a measure of the totalthermal energy in the gas. In the present work, transient PVT wasused to obtain validation data to support calculations that predicthelium gas flow and heat transfer in vessels undergoing rapiddepressurization. Calculations for high pressure discharge fromvessels were performed using two methods, (1) a single controlvolume method with a network flow analysis code and (2) a mul-ti-dimensional method with a control volume/finite element code.

When the exit diameter is small relative to the vessel diameter,pressures in the discharging vessel are nearly uniform in spaceduring the entire discharge transient. As a result, the process lendsitself to a single control volume analysis. Since the heat exchangebetween the resident gas and the vessel walls is inherently mul-ti-dimensional, the single control volume method relies on a heattransfer correlation to account for these multi-dimensional effects.Over the range of initial conditions and geometries considered inthis study, the single control volume analysis and natural convec-tion heat transfer correlation were shown to predict both pressureand mass-averaged temperature in vessels undergoing rapiddepressurization.

Multi-dimensional calculations were performed to simulate oneof the transient PVT experiments. In these calculations the

multi-dimensional nature of heat transfer was captured with directnumerical simulation rather than relying on a heat transfer corre-lation. Multi-dimensional calculations have the advantage of beingable to predict heat transfer and gas flow for geometries and initialconditions that are beyond the range of specific heat transfer cor-relations. However, for the geometries and the initial conditionsconsidered here, multi-dimensional calculations for the dischargevessel, interconnecting tubing and the receiver are costly to per-form since supersonic flow exists for much of the transfer. In thepresent work these difficulties were overcome by coupling multi-dimensional calculations for flow and heat transfer in the dis-charge vessel with network flow calculations for the downstreamtubing and receiver.

The coupled multi-dimensional calculation for heat transferreproduced the measured transient pressure and mass-averagedtemperature in the discharging vessel. Close examination of thesolution showed that the early part of the vessel heat transfer pro-cess is characterized by convection and conduction that is normalto the wall. Away from the exit hole the heat transfer boundarylayer is relatively uniform. Only at later times does free convectionbegin to dominate leading to large variations in heat transfer alongthe vessel wall.

Acknowledgements

The authors wish to acknowledge Sandia National Laboratories.Sandia National Laboratories is a multi-program laboratory oper-ated by Sandia Corporation, a wholly owned subsidiary of Lock-heed Martin Corporation, for the United States Department ofEnergy’s National Nuclear Security Administration under ContractDE-AC04-94AL85000.

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