Pressure variations in a cryogenic liquid storage tank subjected to periodic excitations

International Journal of Heat and Mass Transfer 66 (2013) 223–234

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Pressure variations in a cryogenic liquid storage tank subjected to periodic excitations C. Ludwig a, M.E. Dreyer b,⇑, E.J. Hopfinger c a

DLR, German Aerospace Center, Institute of Space Systems, Robert-Hooke-Strasse 7, 28359 Bremen, Germany ZARM, University of Bremen, 28359 Bremen, Germany c LEGI/CNRS/UJF, B.P. 53, 38041 Grenoble, France b

a r t i c l e

i n f o

Article history: Received 30 November 2012 Received in revised form 4 June 2013 Accepted 29 June 2013 Available online 2 August 2013 Keywords: Sloshing Cryogenics Pressure drop Propellant tanks Launcher upper stages

a b s t r a c t The pressure change in a partially filled liquid nitrogen tank, subjected to periodic lateral forces, has been investigated experimentally. The cylindrical tank has a radius of R = 0.148 m and is filled up to 69% of the total volume (43  103 m3). Six different sloshing conditions were considered, with the wave amplitude b of the first asymmetric mode ranging from b/R = 0.12 up to wave breaking conditions of b/R P 0.54. The tank was pressurized with nitrogen vapor and the pressure at sloshing initiation was in general pi  250 kPa. Pressure drops in the order of 100 kPa have been measured and the dependency of these pressure drops on wave amplitude has been determined. The integrated temperature sensors allowed to measure the vapor temperature change and, with a high resolution, the thermal boundary layer in the liquid. The effective diffusion coefficient model of Das and Hopfinger (2009) [2] has been extended and allows calculating the temperature distribution in the thermal boundary layer, as well as the pressure drop as a function of the effective diffusion coefficient. A novel result is that the calculated temperature distribution in the thermal boundary layer deviates from the measured one, relating to a stop in the pressure drop. The sloshing Nusselt number NuS = De/D0 is shown to correlate well with a Reynolds number that contains the wave amplitude. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In cryogenic propellant tanks of rockets, as well as in other liquefied gas storage tanks, large pressure changes can occur due to condensation or evaporation at the liquid–vapor interface. These pressure changes, and especially the rate of pressure change, can be considerably enhanced when the tank is subjected to excitations in a way that the liquid vapor interface is sloshing. It is of great practical and fundamental interest to be able to relate the pressure change to the sloshing conditions. Moran et al. [1] showed that in a partially filled liquid hydrogen (LH2) tank, pressurized to 250 kPa, a pressure change (pressure drop) of about 100 kPa can occur in less than 10 s. Das and Hopfinger [2] and Hopfinger and Das [3] conducted experiments with volatile liquids, storable at room temperature, aimed at understanding the physical processes and the dependency of the pressure change on the physical properties of the liquid and sloshing conditions. A condensation-evaporation model has been developed by them, giving the rate of pressure change as a function of the physical properties of the

⇑ Corresponding author. Tel.: +49 421 218-57866. E-mail address: [email protected] (M.E. Dreyer). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.06.072

liquid (expressed by the Jakob number), the temperature gradient near the liquid–vapor interface, and an effective thermal diffusivity depending on the sloshing conditions. This effective thermal diffusivity coefficient can be two orders of magnitudes larger than the molecular thermal diffusivity [2]. Arndt [4] showed that, as expected, the presence of a non-condensable gas decreases condensation and increases evaporation. The latter experiments were limited to constant sloshing conditions. Das and Hopfinger [2] used only axial excitations and storable liquids with different pressurization conditions. For a better understanding of the relation between the pressure drop and the sloshing conditions of cryogenic liquids, further experiments and analysis were clearly needed. The experiments, presented in this paper have been conducted with liquid nitrogen (LN2) for identical initial thermodynamic conditions but different sloshing conditions. These experiments were aimed at establishing a Nusselt number dependency on sloshing conditions. Special attention was given to adequately resolving the thermal boundary layer in the liquid. In the next section, an extended form of the mass transfer model of Das and Hopfinger [2] is presented, containing the thermal boundary layer development. The experimental setup and procedures are discussed in Section 3, together with a detailed presentation of the sloshing conditions with the measured and evaluated wave amplitudes. Section 4 pre-

224

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

Nomenclature

Roman letters Af forcing amplitude B, B1 prefactors b wave amplitude C constant or coefficient in (8) cp specific heat capacity D0 liquid thermal diffusivity De effective thermal diffusivity d tank wall thickness f frequency f1 natural frequency Hl liquid height Hv vapor height Dhv latent heat of evaporation/condensation K constant of order 1 m mass p pressure heat flux Q_ q heat flux per unit area r radial coordinate R tank radius Rs specific gas constant S free surface area SA cross-sectional wall area Se evaporation surface T temperature mean vapor temperature T^ v Tw wall temperature Tx oscillation period, Tx = 1/f t time td decay time after sloshing end Uj liquid velocity field V volume w velocity in z direction we evaporation velocity wc condensation velocity z vertical coordinate

damping coefficient mixing thickness thermal boundary layer thickness thermal boundary layer gradient thickness eigenvalue of first asymmetric mode frequency ratio characteristic temperature difference thermal conductivity dynamic viscosity kinematic viscosity density azimuthal coordinate damping rate angular wave frequency natural angular wave frequency

d dM dT dT,g

e1 g1 H k

l m q / X

x x1

Subscripts 0 start c critical exp experiment f end i initial j coordinates j = (r, u, z), z vertical l liquid m maximal S sloshing s saturation u ullage v vapor w wall C liquid surface Nondimensional numbers 3

Ga ¼ gR m2

Galilei number

qc H Ja ¼ q l p;lDhv v ;s

Jacob number

NuS ¼ DD0e sloshing Nusselt number Pr ¼

cp l k

Prandtl number 2

Greek letters bi bifurcation parameters of Miles [9] with i = 2, 3, 4 c amplitude–radius ratio

sents the experimental results of the pressure drops as well as the thermal boundary layer development, including a comparison to a theoretical model. From the measured pressure drops, the values of the effective diffusion have been determined. The sloshing Nusselt number NuS is presented in Section 5 as a function of a sloshing Reynolds number ReS. In Section 6, the conclusions drawn from this study are presented.

ReS ¼ xmb sloshing Reynolds number

on the sloshing conditions. For capillary waves, such a Nusselt number has been proposed by Hopfinger and Das [3] as a function of the wave amplitude. However, for gravity wave sloshing this was not possible mainly because of the lack of adequate experimental data. The thermal energy flux equation is

ql cp;l 2. Interfacial mass transfer model When the liquid surface (the liquid–vapor or gas interface) is motionless, the mass transfer depends on the thermodynamic conditions at the interface. We assume that the mass flux is related to the heat flux in the liquid. Additionally, it is assumed that the heat flux in the vapor is at least one order of magnitude lower and is therefore negligible. When the liquid is sloshing, the mass transfer is increased and Hopfinger and Das [3] and Das and Hopfinger [2] proposed to express this increased mass transfer by an effective diffusion coefficient. This effective diffusion coefficient with respect to the thermal diffusivity is a Nusselt number, that depends

  @T @ @2T þ TU j ¼ ql cp;l D0 2 @t @xj @z

ð1Þ

with the liquid density ql, the specific heat capacity of the liquid cp,l, the temperature T, the liquid velocity field Uj and the liquid’s thermal diffusivity D0. Incompressibility conditions were used. It is possible to decompose temperature and liquid velocity into a mean and  j þ U 0 . In cylindrical coorfluctuating part, i.e. T ¼ T þ T 0 and U j ¼ U j dinates, the index j is defined as j = (r, u, z), where z is vertical, r radial and u azimuthal, measured from the direction of container excitation x = (r, u = 0) (see Fig. 1).  j ¼ 0 and time averaging By setting the mean velocity field U over a few oscillation periods, eliminating horizontal gradients (i.e. in r and u direction), we can write

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

225

where Ja is the Jakob number Ja = [(qlcp,l)/(qv,sDhv)]H (according to [5]). In boiling studies, the Jakob number is commonly used without the density ratio. Here, the density ratio is included for compactness. The pressure change in the system is

dp Rs T^ v ;i dmv p dT^ v ¼ þ i ^ dt V u dt T v ;i dt

ð6Þ

where T^ v is the ullage volume averaged mean vapor temperature, T^ v ;i the initial mean vapor temperature, Vu the ullage volume, Rs the specific gas constant, mv the vapor mass and pi the initial pressure. The vapor is considered as an ideal gas. The change of vapor mass is dmv/dt = wcqv,sS + weqv,sSe with S as the total area of the free surface. We assume that the liquid in the vicinity of the wall is superheated due to heat flux along the container wall. The container walls are heated by conduction from the warm top to the cold bottom. Liquid evaporates on an annular surface Se, adjacent to the tank wall with the evaporating vapor velocity we. Substituting for dmv/dt in Eq. (6) we get:

wc ¼

Fig. 1. Test tank with locations of temperature sensors. The liquid–vapor interface is at z = Hl = 0.455 m and Hv = 0.195 m. Q_ w is the wall heat flow, the wall thickness is d and the tank radius is R = 0.148 m. Tank excitation is in the direction x = (r, u = 0) and is marked with the excitation amplitude Af.

ql cp;l

! @ T @T 0 w @ 2 T ¼ ql cp;l D0 2 þ @z @t @z

ð2Þ

where w is the vertical fluctuating velocity component. The time averaging, expressed by overbars, is possible due to the wave period being much shorter than the time in which significant mass transfer occurs. When expressing the temperature–velocity fluctuation cor relation in terms of a wave diffusion coefficient T 0 w ¼ D0e @ T=@z we obtain

ð3Þ

with the effective diffusion coefficient De = D0 e + D0 and D0 = kl/ (qlcp,l) with kl as the thermal conductivity of the liquid, and the bars have been dropped. The spatial temperature gradient is the mean gradient, averaged over several wave periods. Integration of Eq. (3) from z = 0 to the liquid height Hl gives

Z

Hl

Tdz ¼ De

0

 @T  H ¼ De @z z¼Hl dT;g

ð4Þ

where H ¼ T s  T l is the characteristic temperature difference, with Ts as the saturation temperature and Tl the bulk temperature of the liquid and dT,g is the thermal boundary layer gradient thickness, defined by (10). The heat flux at the interface per unit surface is qC ¼ ql cp;l De ð@T=@zÞjz¼Hl and the associated condensation heat flux is wcqv,sDhv = qC with the vapor density at saturation conditions qv,s and the latent heat of evaporation Dhv. The vapor condensation velocity normal to the mean interface wc is therefore given by

wc ¼ 

dT^ v v ;i dt

 T^pi

Vu Se  we S Rs T^ v ;i qv ;s S

ð7Þ

The surface area Se increases in time by conduction. In the presence of sloshing, it can be assumed to be equal to S because of the fluid surface motion caused by sloshing. Eqs. (5) and (7) can be used to determine De from measurement of dp/dt and dT^ v =dt, which is the objective of the present experiments. More specifically, the aim is to determine a Nusselt number NuS = De/D0 as a function of a parameter that contains the sloshing conditions and container size (see Section 5). Another important parameter is the thermal boundary layer thickness:

dT ¼ C

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De t þ ðd2T;i =C 2 Þ

ð8Þ

The initial thickness at onset of sloshing dT,i depends p on the ffiffiffiffiffiffiffiffiffiffiffiffi experimental conditions and is calculated as dT;i ¼ C D0 t S;0 . The time tS,0 is the experimental time at which sloshing is started. The constant factor C in Eq. (8) can be determined from the integration of the transient heat conduction equation (3) (assuming that De is constant) which gives



@T @2T ¼ De 2 @t @z

@ @t

dp dt

De ql cp;l H De ¼ Ja qv ;s Dhv dT;g dT;g

ð5Þ

T  T l ¼ ðT s  T l Þerfc

 Hl  z pffiffiffiffiffiffiffi ; 2 De t

ð9Þ

where erfc is the complementary error function which tends to zero pffiffiffiffiffiffiffi when ðHl  zÞ=2 De t ¼ 2. The thermal boundary layer thickness is pffiffiffiffiffiffiffi therefore dT ¼ 4 De t . Here we are primarily interested in the heat flux at the interface z = Hl. The general expression of the heat flux is q =  qlcp,lDe(@T/ @z). From Eq. (9) follows:

T s  T l ðHl zÞ2 q ¼ ql cp;l De pffiffiffiffiffiffiffiffiffiffiffi e 4De t pD e t

ð10Þ

At the interface, z = Hl and q = qC, giving pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ð@T=@zÞz¼Hl ¼ ðT s  T l Þ= pDe t . We call the thickness pDe t the therpffiffiffiffi mal boundary layer gradient thickness dT;g ðC ¼ pÞ. If the effective diffusion coefficient were known (a Nusselt number were known), the heat flux and hence the pressure drop as a function of time could be calculated from (12) and (13). As mentioned before, the objective here is to determine De and to establish such a Nusselt number in terms of the sloshing conditions. De is determined from the measured maximum rate of pressure change (dp/dt)m, using (7), (5) and (8), resulting in the following equation

226

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

Fig. 2. Typical tank pressure as a function of time (texp) with definition of the time scales. The values of the different times are here for experiment E2. The time DtS = (dT,g,i)2/pDe = tS,0D0/De takes into account the existence of a thermal boundary pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi layer of gradient thickness ðdT;g;i Þ ¼ pD0 t S;0 when sloshing starts at tS,0.

 De ¼ 

dp dt

m

dT^ v v ;i dt

 T^pi

Rs T^ v ;i qv ;s

Vu S

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pD0 tS;0 qv ;s Dhv Hql cp;l

ð11Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where pD0 t S;0 is the thermal boundary layer gradient thickness at start of sloshing, that is at tS = 0 as depicted in Fig. 2. For the calculation of De, the evaporating vapor velocity we is not known. At the start of sloshing, where De is determined, we is estimated to be less than 10% of wc. Its contribution is therefore neglected in (11). The pressure change as a function of time during sloshing is obtained from (7), (5) and (8) in the form

pffiffiffiffiffiffi   De Ja S dp Rs T^ v ;i qv ;s p dT^ v ¼ we Se  pffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i dt Vu p t S þ Dt S T^ v ;i dt

ð12Þ 3.2. Pressurization and relaxation

where De has been determined from Eq. (11). Integration of (12) from DtS to t 0S (see Fig. 2), keeping the physical properties constant, gives the following equation for the pressure variation as a function of time, which can be compared to the experimental pressure variation.

p ¼ pi

pffiffiffiffiffiffi qffiffiffiffi   pffiffiffiffiffiffiffi T^ v ðtÞ Rs T^ v ;i qv ;s 2 Ja S De pffiffiffiffi t0S  Dt S  we Se t S  Vu p T^ v ;i

During the experiments, the tank pressure and the fluid temperatures are measured. For the pressure measurement, one Sensortechnics CTE 9010 ANO pressure sensor is used. For the measurement of the fluid temperatures, sixteen LakeShore DT670 Band B silicon diodes, designed for cryogenic conditions are applied with two LakeShore 218 Temperature Monitor controllers. The data acquisition rate for the temperature sensors is 3 Hz and for the pressure sensor 10 Hz. Fig. 1 shows the positions of the temperature sensors. The exact locations of the temperature sensors are summarized in Table 1. The sensors are distributed in the tank as follows: a support rod is situated in the tank, on which seven temperature sensors are placed in the liquid phase and three in the vapor phase. One sensor is placed next to the diffuser in the vapor phase. Furthermore, four sensors are attached to the tank wall and one on the inner side of the lid. During the experiments, up to five temperature sensors can be located in the liquid’s thermal boundary layer. The supplier of the temperature sensors gives an accuracy of ±0.5 K. Internal evaluation showed that a higher accuracy of ±0.1 K can be achieved. The pressure sensor is located in the tank ullage to measure the tank pressure with an accuracy of ±7.4 kPa. For the pressurization of the LN2, GN2 is used as pressurant gas which is injected via the diffuser situated in the lid. The physical properties of LN2 at different tank pressures are summarized in Table 2 together with the data for LOX, LH2 and HFE-7000, required for the subsequent comparison of the experimental results. For the analysis presented in this paper, the following times are defined as depicted in Fig. 2. The experimental time texp starts with the beginning of the pressurization and continues to the end of the experiment. Important points in time are: tp,0 (at texp = 0 s) the beginning of the pressurization, tp,f the end of pressurization, tS,0 beginning of sloshing and tS;f þtd , a time after sloshing, when the sloshing motions have sufficiently decayed, in order to make the determination of the temperature profile feasible.

ð13Þ

The time tS is the sloshing time and t0S ¼ t S þ Dt S . The time DtS = (dT,g,i)2/(pDe) has to be considered in order to take into account, that at the initial state tS,0, already a thermal boundary layer exists. As mentioned above, we ffi 0.1wc and Se = S = const. Without sloshing Se is a function oftime. 3. Experimental conditions 3.1. Experimental setup In the following, the experimental setup is briefly introduced. The cylindrical test tank used in the experiments is shown in Fig. 1. The tank has an inner diameter of 2R = 0.296 m and a round shaped bottom. The total inner volume is 43  103 m3 . In the top third of the tank the inner wall is made of d ¼ 1:5  103 m stainless steel. The tank is embedded in a vacuum casing for better insulation. The test tank is partly filled with liquid nitrogen (LN2) up to a fill height of Hl = 0.455 m (±1  103 m), resulting in a height of the vapor phase of Hv = 0.195 m. The volumes of the liquid and vapor phases are V l ¼ 29:7  103 m3 and V u ¼ 13:3  103 m3 .

At the beginning of the experiments, the tank was filled with LN2, several hours before the start of the experiment. Evaporating nitrogen could leave through the tank outlet. When the liquid surface reached the defined position between temperature sensor T5 and T4, indicated by a characteristic thermal stratification, the vapor outlet was closed. In the next step, at experimental time texp = 0 s, the pressurization was started by injecting GN2 at a temperature of 294 K via the diffuser (see Fig. 1). Table 3 summarizes the conditions of the performed experiments. The pressure and temperature evolutions in the tank during

Table 1 Positions of the temperature sensors, x is along the sloshing axis x = (r, u = 0). Sensor

x [m]

z [m]

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16

0.0980 0.0980 0.0980 0.0980 0.0980 0.0980 0.0980 0.0980 0.1480 0.1480 0.1480 0.1480 0.0980 0.0063 0.0980 0.0980

0.330 0.430 0.440 0.450 0.460 0.510 0.610 0.110 0.450 0.460 0.625 0.630 0.650 0.644 0.435 0.445

227

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234 Table 2 Fluid properties at saturation conditions [6,2]. The Jakob number is Ja = [(qlcp,l)/(qv,sDhv)](Ts  Tl). In the LN2 experiments Tl = 77.6 K. p [kPa]

qv,s [kg/m3]

Ts [K]

ql,s [kg/m3]

cp,l  103 [J/(kg K)]

Rs [J/(kg K)]

Dhv  103 [J/kg]

Ja [–]

D0  108 [m2/s]

LN2 LN2 LN2 LN2

100 200 250 300

77.24 83.63 85.93 87.90

4.6 8.7 10.7 12.7

806.6 776.8 765.6 755.7

2.04 2.08 2.10 2.12

296.8 296.8 296.8 296.8

199.3 190.5 187.1 184.0

– 5.9 6.7 7.1

8.8 8.2 8.0 7.7

LOX LH2 HFE-7000

250 250 220

99.81 23.75 329.43

10.3 3.1 14.6

1091.9 66.4 1310

1.74 12.28 1.4

259.2 4157.2 41.6

202.8 420.1 120

8.8 2.4 38

7.2 12.9 3.8

Table 3 Conditions of the performed experiments. The experimental time texp and the tank pressures p are summarized for tp,0 (texp = 0 s) the beginning of the pressurization, tp,f the end of pressurization, tS,0 the beginning of sloshing and tS;f þtd a decay time after sloshing end (compare Figs. 3 and 6). Exp

Af  103 [m]

f [Hz]

x/x1

texp [s] at

p [kPa] at

tp,f

tS,0

t S;f þtd

tp,0

tp,f

tS,0

t S;f þtd

470.8 485.1 457.7 447.6 436.5 220.5 ðtp;f þtd Þ 1947.7

101.6 101.0 101.5 101.0 101.3 105.0

303.6 300.9 298.5 304.0 302.6 308.6

246.5 242.2 245.7 242.5 244.4 –

105.0

308.6

299.0

230.2 201.0 195.0 147.5 133.4 253.8 ðtp;f þtd Þ 296.6

E1 E2 E3 E4 E5 E0

5.1 5.1 5.1 5.1 5.1 0

1.40 1.49 1.52 1.65 1.81 0

0.796 0.848 0.865 0.939 1.030 0

55.5 52.7 54.7 55.4 54.8 64.3

207.7 203.8 206.7 207.4 207.9 –

E11

10.0

1.40

0.796

64.3

978.1

Fig. 3. (a) Tank pressure, (b) liquid temperatures, (c) vapor temperatures, (d) wall and lid temperatures during pressurization and relaxation of experiment E2 (see Table 3). T14 is the pressurant gas temperature at the diffuser. Pressurization starts at tp,0 (texp = 0 s) and ends at tp,f (texp = 53 s).

and after pressurization are shown in Fig. 3 for the experiment E2 (see Table 3). The tank pressure (Fig. 3(a)) increases during pressurization almost linearly up to the final tank pressure of 300 kPa (at tp,f). Subsequently, the inflow of the pressurant gas was stopped and the tank was closed. Consequently, relaxation takes place and the pressure curve asymptotically decreases to 240 kPa. For the liquid temperature (Fig. 3(b)), only the four uppermost temperature

sensors detected a change in temperature over the considered time frame. The influence of the warm pressurant gas on the vapor temperature during the pressurization process can be seen in the vapor temperature evolution (Fig. 3(c)). After the end of the pressurization, the temperature distribution reduces analogously to the pressure. The wall temperature distribution (Fig. 3(d)) is are also affected by the pressurization process, but much less than the va-

228

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

por temperatures due to the slow reaction of the wall material. It can be seen that the lid temperature (T13) is not changing over time. Fig. 4 shows the vertical temperature profile histories for experiment E2 of the liquid (Fig. 4(a)) and the vapor phase (Fig. 4(b)) before pressure ramping (tp,0), after ramping (tp,f) and after pressure relaxation (tS,0). (The profiles at tS,0 correspond to the temperatures at texp = 204 s, see Fig. 6.) At tp,0, there is no stratification in the liquid phase. After the end of the pressurization, at tp,f, a strong gradient in the liquid’s temperature stratification appeared and the thermal boundary layer extends from the free surface (z = 0.455 m) to just below sensor T16 (z = 0.445 m). After relaxation, at tS,0, the thickness of the thermal boundary layer is increased to a point between T3 (z = 0.440 m) and T15 (z = 0.435 m) and the temperature gradient is less sharp due to the decreasing pressure. In Fig. 4(a), the temperature sensors T1 and T8 are not shown because these sensors have the same temperature as T2 over all analyzed time steps. Saturation temperature at the relating tank pressure is assumed at the free surface (z = 0.455 m), which is calculated using the NIST database [6]. The evolution of the thermal boundary layer with time can be calculated with Eq. (9) and the effective thermal diffusivity, required for tS,0 is calculated using Eq. (11). These are plotted in Fig. 4 at times tp,f and tS,0 together with the measured profiles. It can be seen that the calculated profiles match the experimental data well but slightly underestimate the thickness of the thermal boundary layer. This might be due to the fact, that the theory disregards the heat conduction from the tank wall into the liquid. Fig. 4(b) shows the evolution of the thermal stratification in the vapor phase. At pressurization start (tp,0), the temperature in the ullage increases linearly from saturation temperature on the free surface to 280 K, which is the temperature at the inner side of the lid. At pressurization end (tp,f), the temperature sensor T14 (z = 0.644 m), next to the diffuser dominates the profile of the stratification, which shows now a more curved progression. This curvature decreases during relaxation. The wall temperature profile at texp = 200 s in Fig. 3(d) is used to evaluate the supply of heat to the surface layer of the liquid through the container wall (compare Q_ w in Fig. 1). The downward wall heat flow can be calculated with the following formula.

DT w SA Q_ w ¼ kw Dz

ð14Þ

From Fig. 3(d) we get DTw = T10  T9 = 11 K for a distance between the sensors of Dz = 0.011 m. This corresponds to a heat flow along

the wall of Q_ w ¼ 12:1 W, with kw = 8.62 W/(m K) (according to Barron [7]) for the thermal conductivity of the wall material (stainless steel) in the appearing temperature range. The cross-sectional wall area is SA ¼ 1:4  103 m2 . In preliminary experiments, an increase in pressure by 50 kPa in 570 s due to self-pressurization was measured. This results in a pressure increase of 0.09 kPa/s. However, the required heat flow for this pressure increase is only 4.5 W. The remaining 7.6 W would increase the average temperature of the total liquid volume by 0.33 K in 103 s. However, Fig. 3(b) shows that primarily the uppermost liquid layer is heated up. 3.3. Sloshing conditions For the sloshing excitation, the test tank is mounted on a movable platform, which is driven by a crank shaft to convert the rotation of an electrical motor into a linear sinusoidal oscillation of the form x = Af sin 2pft, where Af is the maximum tank displacement (excitation amplitude) and x = (r, u = 0). Sloshing was started at the time tS,0 after pressure relaxation, resulting in a tank pressure drop due to enhanced condensation (see Fig. 6(a)). After the tank pressure has reached a minimum, sloshing excitation was stopped. However, the tank pressure and the fluid temperatures were observed until tS;f þtd , when the liquid motion has sufficiently decayed. In order to determine the effect of sloshing on the interfacial mass transfer, five different sloshing conditions were considered in this study: three stable asymmetric sloshing modes, one chaotic sloshing mode and one swirl mode. The amplitude of the periodic container excitation set to Af/R = 0.0344 for the experiments E1 to E5. The angular frequency of the experiments was varied from 0.8 6 x/x1 6 1.03 (with x = 2pf and x1 as the natural angular wave frequency), which covers the range of stable planar wave motion to swirl waves [8,9]. Swirl waves are waves, moving in circumferential direction inside the tank. In experiment E11, the excitation amplitude has been increased by a factor of 2 to Af/R = 0.0675 and the frequency was chosen such to maintain stable sloshing f = 1.4 Hz and x/x1 = 0.8 (see Table 3 and Fig. 5). This supplied an additional data point in the Nusselt number relation. For the large liquid depth to radius ratio, appearing in the presented experiments, the natural frequency x1 offfi the first asympffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi metric mode was x1 ¼ e1 g=R ¼ 1:841g=R ¼ 11:04 1=s or f1 = 1.757 Hz (with e1 = 1.841 as the eigenvalue of the first asymmetric mode). According to Miles [9] weakly non-linear theory, stable planar asymmetric wave motion exists for Af/R = 0.0344, up to a value of x/x1 = 0.876, corresponding to f = 1.54 Hz. Chaotic

Fig. 4. (a) Liquid and (b) vapor temperature profiles of experiment E2 before pressure ramping (tp,0), at pressurization end (tp,f) and at start of sloshing (tS,0). Theoretical profile with Eq. (9) at end of pressurization tp,f with D0 = 7.7  108 m2/s and at start of sloshing with D0 = 8.0  108 m2/s (tS,0 with (9)). Dashed lines are only for better visualization. The saturation temperature at free surface (z = 0.455 m) is calculated from [6]. Temperatures measured by sensor T1 and T8 are not shown because these measured the same value as sensor T2, located at z = 0.430 m.

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

wave motion is observed in a range of 0.88 6 x/x1 6 0.97 and swirling in a range of 0.97 6 x/x1 6 1.05. The excitation frequencies of f = 1.65 Hz (x/x1 = 0.94) and f = 1.81 Hz (x/x1 = 1.03), imposed in the presented experiments E4 and E5 are therefore well situated within the chaotic and the swirl mode regime. For the stable planar wave motion experiments, the chosen frequencies are for E1 and E11 f = 1.4 Hz (x/x1 = 0.796), for E2 f = 1.49 Hz (x/ x1 = 0.848) and for E3 f = 1.52 Hz (x/x1 = 0.865). The phase diagram in Fig. 5 depicts the boundaries of the sloshing modes, as defined by Miles’ [9] theory for deep water conditions and low damping

½x=x1 i ¼

" #1=2  Af 2=3 1:684 bi þ 1 R

ð15Þ

where b2 = 0.36, b3 = 1.55 and b4 = 0.735 are specific values of the bifurcation parameter, giving the boundaries of the sloshing modes, determined by Miles [9]. The used indices i = 2, 3, 4 are the same as in Miles [9]. In order to be sure that the actual sloshing modes are met, preceding qualitative observations at ambient pressure were made by covering the container with a transparent lid. From these observations, the following mean amplitudes of the planar waves were estimated: 1.8  103 m, 2.5  103m and 3.5  103 m for respective frequencies f = 1.4 Hz, 1.49 Hz and 1.52 Hz. These wave amplitudes are close to those, determined from Eq. (16). It should be noted that, because of the interference of two frequencies, namely x and x1, the wave amplitudes of the planar waves periodically reach amplitudes of twice the given mean values with the variations decaying in time [8]. The frequency of 1.52 Hz corresponds to x/x1 = 0.865 and is close to the boundary of bifurcation to chaotic motion. The observations showed that an excitation with a frequency of f = 1.53 Hz (x/x1 = 0.870), caused a chaotic sloshing behavior when starting from rest. Miles [9] weakly non-linear theory gives a bifurcation boundary of x/ x1 = 0.876. The conditions of experiment E11 are also close to the bifurcation boundary, but the wave motion is planar and stable. The observed maximum rise of the chaotic fluid motion is approximately equal to R and its mean value is close to bc = g/x2 = 0.54R. The swirl wave has a maximum amplitude of

229

about 0.7R and steady swirling is established after about 10 s. In summary, according to the present observations and Royon-Lebeaud et al. [8], the sloshing characteristics are as follows: (a) Stable planarwaves: the maximum amplitudes b of the stable, planar wave motions can be estimated from:

b=Af  2½ðx=x1 Þ2 =ð1  ðx=x1 Þ2 Þ:

ð16Þ

For the excitation Af/R = 0.0344, Eq. (16) yields maximum amplitudes of b  1.77  103 m, b  2.6  103 m and b  3.1  103 m for the frequencies f = 1.40 Hz, f = 1.49 Hz and f = 1.52 Hzrespectively. This is close to the observations. As mentioned earlier, these steady, planar wave amplitudes are mean values and are reached only when the natural frequency component of f1 = 1.757 Hz has decayed. When the container is set into motion, the amplitude modulation varies by a factor of two due to frequency beating at (f1  f), with the natural frequency decaying at the rate of X ¼ dx1 [8]. The damping coefficient d = K(m2/R3g)1/4 can be written in terms of the Galilei number Ga as d = K(1/Ga)1/4  103. Therefore, it takes about 100 s or 200Tx (container oscillation periods Tx = 1/f) in the presented experiments for the amplitude of the natural frequency component to decay by 1/e of the initial amplitude [8]. For comparison, the maximum amplitude is already reached after approximately two to three oscillations. (b) Chaotic sloshing: in the chaotic sloshing regime, the wave amplitude is increasing until bc = g/x, or 0.54R, which is when the downward wave crest acceleration is equal to the gravitational acceleration. Subsequently, it collapses rapidly to small amplitude irregular motions before a new growth cycle begins. The wave motion is slightly asymmetric with a negative wave amplitude of bc and a positive amplitude of approximately b/R = 1. The growth is approximately linear:

Af t b p R R Tx

ð17Þ

Thus, it takes a few oscillations periods for the wave amplitude to break up. Observations under isothermal conditions showed that breaking conditions b/R P 0.54 are reached in about 5 s (9Tx) after initiation of tank movement. After that, violent breaking of the waves appeared until about 20Tx. Then slosh waves collapse rapidly and a new cycle starts. It should be noted that wave breaking gives rise to large surface perturbations with plunging jets that cause mixing in the surface layer. (c) Swirl wave: the swirl wave, once established, is very stable and has a mean wave amplitude of about 0.7R. After start of container excitation, here Af/R = 0.0344, the wave motion is planar and grows up to breaking conditions in about 9 s (15Tx). Then the swirl grows exponentially [8] and is stable after about 30Tx. 4. Sloshing effect on pressure drop

Fig. 5. Phase diagram of dimensionless forcing amplitude as a function of frequency ratio. The performed experiments are indicated by E1 to E11. The solid lines are the bifurcation boundaries b2, b3, b4 of the sloshing regimes determined by Miles [9].

The initial conditions of the sloshing experiments, except experiment E11, are those corresponding to texp = 204 s in Fig. 6, when the pressure is about 240 kPa and quasi-stable conditions are established. At this time, denoted as the sloshing time tS = 0, sloshing is initiated. In experiment E11 sloshing is initiated at texp = 978.1 s. The sloshing conditions of all performed experiments are summarized in Table 3. The pressure and temperature evolutions for stable sloshing are shown in Fig. 6 for x/x1  0.85 (experiment E2 in Table 3). Fig. 5(a) shows a typical trend of the pressure drop in a tank due to stable sloshing conditions. The pressure drop is Dp = 40 kPa in a time of 150 s. After a pressure minimum, the pressure slowly began to increase again due to heat supply caused by wall heat conduction. Most of the observed pressure drop is due to condensation of vapor at the liquid surface. However, the decrease in vapor

230

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

Fig. 6. (a) Tank pressure, (b) liquid temperatures, (c) vapor temperatures, (d) wall and lid temperatures during and after planar wave sloshing of experiment E2. Sloshing starts at tS,0 (texp = 204 s) and consists of a planar wave motion with mean wave amplitude of b/R = 0.18 (f = 1.49 Hz). Sloshing ends at texp = 421 s, and t S;f þtd is td = 64 s after end of sloshing (texp = 485 s).

(a)

(b)

Fig. 7. (a) Liquid and (b) vapor temperature profiles for experiment E2 at sloshing start (tS,0) and td = 64 s after end of sloshing ðtS;f þtd Þ. Theoretical profile after end of sloshing with Eq. (9) (tS;f þtd with (9)), using a value of De = 4.4  107 m2/s. The dashed lines are only for better visualization; saturation temperatures at free surface (z = 0.455 m) are calculated with [6]: for tS,0 Ts = 83.7 K and for tS;f þtd Ts = 85.6 K.

temperature as indicated in Fig. 6(c) must be taken into account when the condensation rate is determined from the measured pressure drop.

In Fig. 7(a), the experimental liquid temperature profiles at the start of sloshing (tS,0) and a decay time after sloshing ðtS;f þtd Þ are plotted for E2 together with the theoretical profile, calculated with

231

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

Fig. 9. Experimental tank pressure as a function of time and theoretical evolution, calculated with Eq. (13) for experiments E2 and E4. Sloshing starts at tS,0 (details see Table 3).

Fig. 8. Liquid temperature profiles for experiment E4 (see Table 3) at sloshing start (tS,0) and td = 64 s after end of sloshing ðtS;f þtd Þ. Theoretical profile after end of sloshing with Eq. (9) (tS;f þtd with (9)), using a value of De = 19.5  107 m2/s. Note, that at the position of sensor T8 the liquid temperature at tS;f þtd is the same temperature as the bulk temperature for tS,0. The dashed lines are only for better visualization; saturation temperatures at free surface (z = 0.455 m) are calculated with [6]: for tS,0 Ts = 85.8 K and for tS;f þtd Ts = 80.7 K.

Eq. (9). Fig. 8 depicts the same profiles for the wave breaking case E4. The temperature profile for the vapor phase for E2 (Fig. 7(b)) shows, that the temperature distribution stays almost linear between the saturation temperature and the temperature at the inner side of the lid. For E4, the vapor profile is not presented as the temperature sensors were wetted during sloshing and do not appear to represent vapor temperatures. The value of the decay time td approximately corresponds to the time when the sloshing motion has sufficiently decayed for making temperature measurements possible. This occurs at td = 64 s after the end of sloshing for E2 and E4. At ts;f þtd , the theoretical liquid temperature profile accord-

ing to formula (9), using the effective diffusion coefficient given in Table 4, differs from the measured profile. The temperature gradient in the vicinity of the liquid surface is considerably weaker in the experiment. The heat flux from the liquid surface is therefore less than the computed value. Consequently, the pressure drop follows this trend. This is apparent from Fig. 9, where the measured values are compared to results from Eq. (13) for the cases E2 and E4. The conditions and pressure changes of all experiments, including those without sloshing, are summarized in Table 4. The conpffiffiffiffi stant p in Eq. (13), which comes from the derivation of Eq. (10), could be replaced by a coefficient C(tS) to account for the deviation of the temperature gradient (due to mixing, see Fig. 10) from the calculated temperature gradient. However, this coefficient is not known. It is seen in Fig. 9, that for E2, at time t > 270 s, the calculated pressure continues to decrease, contrary to the measured pressure drop which approaches the final value. The explanation is schematically depicted in Fig. 10: at this time, the growth of the thermal boundary layer dT stops at dM and the temperature gradient near the liquid surface rapidly decreases, hence decreasing the heat flux and the related vapor condensation rate. At a greater

Table 4 Experimental conditions and results of the sloshing experiments. Sloshing timeline starts from tS = 0 s and tDp = Dpm/(dp/dt)m. The non-sloshing experiment is the limiting situation. Liquid temperature is Tl = 77.6 K, wc,m is determined from Eq. (7) using (dp/dt)m andDe from (11). For stable sloshing experiments, the maximal rate of pressure drop ðdp=dtÞm ¼ ð@p=@tÞm was determined over the time 1/j(f  f1)j, for E4 and E5, the time interval was 8 s and 5 s, representing the period of the maximum pressure. Exp

f [Hz]

pi [kPa]

tS;f þtd [s]

(dp/dt)m [kPa/s]

Dpm [kPa]

tDp [s]

D0  108 [m2/s]

E1 E2 E3 E4 E5 E0 E11

1.40 1.49 1.52 1.65 1.81 0 1.40

246.5 242.2 245.7 247.5 244.4 308.6 299.0

263 281 251 240 229 – 970

0.79 1.44 3.57 11.19 13.14 2.85 1.67

27.0 44.9 54.7 110.6 115.3 54.7 48.8

34.2 31.2 15.3 9.9 13.0 – 29.2

7.97 7.99 7.98 7.97 7.98 7.71 7.75

b/R

dT,i  103 [m]

T^ v ;i [K]

dT;ðS;f þtd Þ  103 [m]

@ T^ v =@t [K/s]

wc,m  104 [m/s]

De  108 [m2/s]

E1 E2 E3 E4 E5 E0

0.12 0.18 0.21 0.6 0.7 0

160 161 159 159 159 181

1.62 3.06 5.18 18.6 35.98 7.65

0.24

13.9 18.3 21.9 39.0 52.7 7.4 ðdp;f þtd Þ 38.7

0.24 0.44 1.46 4.13 2.62 0.13

E11

7.2 7.2 7.2 7.2 7.2 3.9 (dp,f) 15.4

0.85

1.95

17.0 32.1 54.3 194.8 379.8 7.9 (D0) 41.6

157

232

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

pressure decrease comes abruptly to a stop (see experimental curve of E4 in Fig. 9). 5. A sloshing Nusselt number The Nusselt number is defined as the vapor condensation velocity with sloshing motion to the vapor condensation velocity without sloshing. At the beginning of the sloshing, which is when De is determined from Eq. (11), the thermal boundary layer thicknesses are equal with and without sloshing, we get from (5):

NuS ¼

Fig. 10. Schematic presentation of stop of boundary layer growth and formation of a mixed layer dM. Curve (a) is the temperature profile calculated with (9) and (b) is the measured profile.

depth, the diffusivity drops back to D0 such that a stable density interface is established with a mixed layer dM above. The time when the calculated pressure drop deviates from the maximum pressure drop is about 285 s in Fig. 9, which corresponds to t0 s = (285–208) s + Dts where Dts = 50 s. The thermal boundary layer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thickness atffi time t0s ¼ 127 s is equal to dT ¼ 4 32:1  108  127 ¼ 2:6  102 m which corresponds to the wave amplitude b in experiment E2. This indicates that the thermal diffusivity equals De in a layer whose thickness is in the order of the wave amplitude. The maximum pressure drop occurs in a time (t0s  127 s), which is required for the thermal boundary layer to reach a thickness dT  b. Then, the temperature gradient at the interface decreases rapidly due to mixing, which is equivalent to a rapid increase in the thermal boundary layer gradient thickness pffiffiffiffiffiffiffi pffiffiffiffi dT;g ¼ Cðt S Þ De t with Cðts Þ > p. When wave breaking occurs, mixing takes place down to a depth dM  R in a considerably shorter time, than the time needed for the thermal boundary layer dT to grow to a depth of R. This can be explained by the jets formed during the violent breaking events (as mentioned in Section 3.3), which penetrate to the depth of R in the recoil motion of the wave. When the surface layer is mixed, the

De D0

ð18Þ

When De is replaced by NuSD0 in Eq. (5), the condensation velocity is given by

NuS D0 Ja wc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p NuS D0 ðts þ Dts Þ

ð19Þ

where Dts ¼ ðdT Þ2i =p NuS D0 . Therefore, if the Nusselt number can be established in terms of sloshing conditions, the pressure drop can be calculated from Eq. (13) by substituting De = NuS D0. All thermodynamic properties of the fluids are contained in Eq. (13). The heat transfer, hence the vapor condensation rate, depends on the effective diffusivity, which depends on the wave amplitude b, the fluid velocity xb and the surface layer instabilities. Viscosity will play a role in these instabilities. Thus, it is likely that the Nusselt number depends on the Reynolds number ReS = xb2/m in the following form

NuS ¼ BðReS Þn

ð20Þ

where B is a prefactor to be determined from the experimentally found NuS–ReS correlation. It is of interest to express ReS in terms of container geometry, here just R. For given excitation conditions, the wave amplitude b = cR with c being the amplitude-container radius ratio. The frequency is given by x = g1x1 with g1 as the frequency ratio. Therefore, ReS in Eq. (20) takes the following form

ReS ¼ g1

pffiffiffiffiffi

e1 c2

ðgR3 Þ

m

1=2

ð21Þ

Fig. 11. Sloshing Nusselt number as a function of sloshing Reynolds number for the presented experiments E1 to E11 as well as the experiments of Moran et al. [1] and Das and Hopfinger [2], as specified in Table 5.

233

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

Table 5 Given and estimated values of the experiments by Moran et al. [1] and Das and Hopfinger [2]. The value * is measured. Exp

pi [kPa]

R [m]

T^ v ;i [K]

(dp/dt)m [kPa/s]

dT^ v =dt [K/s]

tp,0  tp,f [s]

Moran: #870 LH2 Moran: #869 LH2 Das: HFE-7000

250 250 220

0.746 0.746 0.05

40 40 327

17.5 0.76 4.1

1.0 0.05 0.05

50 50 79

Exp

dT,i  103 [m]

wc,m  103 [m/s]

De  107 [m2/s]

b/R [–]

g1 [–]

ReS  105 [–]

NuS [–]

Moran: #870 LH2 Moran: #869 LH2 Das: HFE-7000

4.4 4.4 10⁄

8.1 0.32 1.1

148 5.8 2.9

0.6 0.06 0.6

1.3 1.64 1.3

59 0.66 0.53

117.5 4.6 7.7

with e1 being the eigenvalue of the first sloshing mode and m the kinematic viscosity of the liquid. Fig. 11 shows the sloshing Nusselt number NuS versus the sloshing Reynolds number ReS of the presented experimental data, completed by experimental data obtained by Moran et al. [1] and Das and Hopfinger [2]. The data of these experiments are summarized in Tables 2 and 5. The experiments of Moran et al. [1] were conducted with LH2, pressurized with GH2 in a very large tank and allow extending the Nusselt and Reynolds number ranges up to 3 decades. It is seen that the Nusselt number data collapse well onto a straight line of slope n = 0.69 in the log–log representation, determined by a least square fit. The straight line intersects NuS = 1 at ReS = 4.0  103. This Reynolds number value can be considered as the critical Reynolds number (ReS)c = 4.0  103 ± 20%, below which the sloshing motion does not noticeably increase the heat transfer and thus the pressure change. The free surface remains practically smooth. The Nusselt number is then as follows

NuS ¼



ReS ðReS Þc

0:69 ð22Þ

indicating that B ¼ ðReS Þ0:69 . c Generally, the Nusselt number is also a function of the Prandtl number Pr. The experimental determination of this dependency would be difficult since common cryogenic liquids have a Prandtl number of 1 < Pr < 2. For volatile, non-cryogenic liquids like HFE7000, the Prandtl number is also low (Pr  6). A formal, general expression for NuS including Pr would be:

NuS ¼ B1 ðRenS Prm Þ

ð23Þ

The Prandtl number will not change the exponent n = 0.69, and a common value of m in convective heat transfer is m = 1/3. The expression for NuS is then NuS ¼ B1 ðRe0:69 Pr1=3 Þ. NuS = 1 gives S 0:69 1=3 B1 ¼ ððReS Þ1 Pr Þ which is equal to B. The expression of NuS including Pr is then:

NuS ¼

Re0:69 Pr1=3 S ðReS Þ0:69 c

NuS = De/D0, can be correlated with a Reynolds number ReS based on wave amplitude and frequency. A critical Reynolds number emerges, below which sloshing does not affect the pressure drop. These results were enabled by the instrumentation, that allowed obtaining a high resolution of the thermal boundary layer in the liquid. Furthermore, the establishment of the Nusselt–Reynolds number relation was possible due to the wide range of sloshing conditions of the performed experiments and the precision of the pressure and temperature measurements in the liquid and vapor phases. The effective diffusion coefficient model of Das and Hopfinger [2] has been refined. Using this model, the effective diffusion coefficient De has been determined from the measured rate of pressure drop. The thermal boundary layer evolution as a function of time can be calculated from (9) and an explicit expression for the pressure change is given (Eq. (13)). The calculated pressure drop diverges from the measured pressure drop at a time when the temperature gradient near the liquid surface is reduced due to mixing. It is of interest to point out that without sloshing the pressure drop (e.g. after the pressurization end) also comes rapidly to a stop. But in this case, it is due to the heat supply from the tank wall, that causes evaporation which is first compensating the condensation, and then evaporation is the dominating effect and is, subsequently, increasing the tank pressure. From Eq. (12), the rate of pressure drop can be expressed in dimensionless form.

ðdT Þi V u dp pffiffiffiffi qv ;s  p Ja NuS D0 Spi dt ql

ð25Þ

This dimensionless expression of pressure change is of interest because it gives a synthetic view of how the different parameters affect the pressure drop. Furthermore, the Nusselt number is a function of Reynolds number (see Eq. (22)), containing the wave amplitude and tank dimensions (see Eq. (21)).

ð24Þ

6. Conclusions In this paper, a study of the pressure variation in a cryogenic tank in the presence of sloshing is presented. Experimental data are compared to analytical results of a revised interfacial mass transfer model. A sloshing Nusselt number is defined that correlates with a sloshing Reynolds number and the results are compared to the experiments and literature data. The two main and novel conclusions are as follows: (i) In the presence of sloshing, the pressure drop can be large but comes rapidly to a stop. The reason for this arrest in pressure drop is the formation of a mixed layer below the liquid surface of the thickness dM  b < R (Fig. 10). (ii) The sloshing Nusselt number, expressing the ratio of effective De to molecular thermal diffusivity D0,

Acknowledgements The authors acknowledge gratefully Peter Prengel and Frank Ciecior for their effort in preparing and performing the experiments. E.J. Hopfinger acknowledges financial support from the ZARM during his stay there. References [1] E.M. Moran, N.B. McNeils, M.T. Kudlac, M.S. Haberbusch, G.A. Satorino, Experimental results of hydrogen slosh in a 62 cubic foot (1750 liter) tank, in: 30th Joint Propulsion Conference, 1994, AIAA-94-3259. [2] S.P. Das, E.J. Hopfinger, Mass transfer enhancement by capillary waves at a liquid–vapour interface, Int. J. Heat Mass Transfer 52 (5) (2009) 1400–1411. [3] E.J. Hopfinger, S.P. Das, Mass transfer enhancement by capillary waves at a liquid–vapour interface, Exp. Fluids 46 (4) (2009) 597–605. [4] T. Arndt, Sloshing of Cryogenic Liquids in a Cylindrical Tank under normal Gravity Conditions, first ed., Cuvillier Verlag Göttingen, Göttingen, 2012. p. 102.

234

C. Ludwig et al. / International Journal of Heat and Mass Transfer 66 (2013) 223–234

[5] V.P. Carey, Liquid–Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, first ed., Taylor & Francis, 1992. p. 196. [6] E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, National Institute of Standards and Technology Standard Reference Data Program, 2010.

[7] R.F. Barron, Cryogenic Heat Transfer, third ed., Taylor & Francis, Philadelphia, 1999. p. 23. [8] A. Royon-Lebeaud, E.J. Hopfinger, A. Cartellier, Liquid sloshing and wave breaking in circular and square based cylindrical containers, J. Fluid Mech. 577 (2007) 476–494. [9] J.W. Miles, Resonantly forced surface waves in a circular cylinder, J. Fluid Mech. 194 (1984) 15–31.