SKV_Cryo_ads_0D

7/31/2019 SKV_Cryo_ads_0D

1/11

This article appeared in a journal published by Elsevier. The attached

copy is furnished to the author for internal non-commercial research

and education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling or

licensing copies, or posting to personal, institutional or third partywebsites are prohibited.

In most cases authors are permitted to post their version of the

article (e.g. in Word or Tex form) to their personal website or

institutional repository. Authors requiring further information

regarding Elseviers archiving and manuscript policies are

encouraged to visit:

http://www.elsevier.com/copyright
http://www.elsevier.com/copyrighthttp://www.elsevier.com/copyright


2/11

Author's personal copy

A lumped-parameter model for cryo-adsorber

hydrogen storage tank

V. Senthil Kumara,*, K. Raghunathana, Sudarshan Kumarb

aIndia Science Lab, General Motors R & D, Creator Building, International Technology Park, Bangalore 560066, IndiabChemical and Environmental Sciences Lab, General Motors R & D, 30500 Mound Road, Warren, MI 48090, USA

a r t i c l e i n f o

Article history:

Received 24 March 2009

Received in revised form

30 April 2009

Accepted 1 May 2009

Available online 2 June 2009

Keywords:

Hydrogen storage modeling

Cryogenic adsorption

Metal-organic frameworks

a b s t r a c t

One of the primary requirements for commercialization of hydrogen fuel-cell vehicles is

the on-board storage of hydrogen in sufficient quantities. On-board storage of hydrogen by

adsorption on nano-porous adsorbents at around liquid nitrogen temperatures and

moderate pressures is considered viable and competitive with other storage technologies:

liquid hydrogen, compressed gas, and metallic or complex hydrides. The four cryo-

adsorber fuel tank processes occur over different time scales: refueling over a few minutes,

discharge over a few hours, dormancy over a few days, and venting over a few weeks. The

slower processes i.e. discharge, dormancy and venting are expected to have negligible

temperature gradients within the bed, and hence are amenable to a lumped-parameter

analysis. Here we report a quasi-static lumped-parameter model for the cryo-adsorber fuel

tank, and discuss the results for these slower processes. We also describe an alternativesolution method for dormancy and venting based on the thermodynamic state description.

2009 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights

reserved.

1. Introduction

One of the primary requirements for thecommercialization of

hydrogen fuel-cell vehicles is the on-board hydrogen storage

for a range of about 500 km, see Satyapal et al. [1]. On-board

storage of hydrogen by adsorption at low temperatures(around 77 K, liquid nitrogen temperature) and moderately

high pressures (less than 60 bar) is considered viable and

competitive with other storage technologies: liquid hydrogen,

compressed gas, and metallic or complex hydrides, see Zhou

[2]. Nano-porous metal-organic frameworks (MOFs) as adsor-

bents offer good gravimetric capacity and fast and reversible

kinetics. For example, MOF-5 has a reversible maximum

excess adsorption capacity of about 6 wt% at 77 K, see Zhou

et al. [3]. MOF-5 is the adsorbent considered in this report to

study the fuel tank performance.

In this work we assume that the operating pressure of the

cryo-adsorber fuel tank is 20 bar. We assume that liquid

nitrogen at 77 K is available as a coolant at the fuel station.

Hence, the lowest temperature of the cooled hydrogen feed

gas is about 80 K. The four processes occurring in a cryo-

adsorber fuel tank are:

Refueling A depleted fuel tank at low pressure (say 1.1 bar)

and higher temperature (say 120 K) is charged with

hydrogen. Heat released during adsorption is removed by

the recirculation of cool hydrogen gas entering at 80 K.

Discharge When the vehicle is in motion, the fuel-cell stack

demands are met by desorbing hydrogen from the cryo-

adsorber bed. Desorption is enhanced by therecirculation of

hot hydrogen gas. The tank pressure eventually drops to

1.1 bar.

* Corresponding author. Tel.: 91 80 41984567; fax: 91 80 41158262.E-mail addresses: [email protected], [email protected] (V. Senthil Kumar).

A v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m

j o u r n a l h o m e p a g e: w w w . e l s e v i e r . c om / l o c a t e / h e

0360-3199/$ see front matter 2009 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.ijhydene.2009.05.025

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 5 4 6 6 5 4 7 5


3/11


Dormancy & Venting While the vehicle is parked, its pres-

sure builds up to vent pressure, say 25 bar, due to contin-

uous heat leak into the fuel tank, and gas is vented to the

fuel cell or atmosphere. The process of venting is also called

boil-off, by analogy with liquid hydrogen storage systems.

The time period until the start of venting is called

dormancy.

These fuel tank processes occur over different time scales:

refueling over a few minutes, discharge over a few hours,

dormancy over a fewdays, andventing overa fewweeks. When

themoleculartransportprocessesare fast, slower processeslike

discharge, dormancy and venting are expected to have negli-

gible internal gradients and are generally amenable to a lum-

ped-parameter analysis. In thisreport we describe a quasi-static

lumped-parameter model for the cryo-adsorber fuel tank, and

present the results for these slower processes. The fastest fuel

tank process, refueling, requires higher dimensional models to

describe the temperature gradients occurring within the bed,

and these are being currently developed.

Section 2 describes the adsorption datafor MOF-5. Section 3describesa fuel tank designconsidered in this study. Section 4

identifiesthe assumptions involved in the model development

and Section 5 presents the quasi-static lumped-parameter

model for the cryo-adsorber. The simulation results for

dormancy, venting and discharge are presented in Sections 6

8, respectively. While some of the thermophysical properties

are directly taken fromexisting databases, someare computed

in this work; these details are given in the appendix.

2. Adsorption data

Zhou et al. [3] have reported the adsorption data for MOF-5

powder over a wide range of temperature and pressure. Over

the range of 60125 K temperature and 130 bar pressure, their

excess adsorption data canbe fitted to a Langmuir isotherm of

the following form:

qT; P qmbP

1 bP; (1)

b b0 exp

B

T

; (2)

qm qm0

fT; (3)

where we have assumed

fT 1 AT2: (4)

For the hydrogen adsorption data in AX-21, Benard and

Chahine [4] use

fT 1 AT: (5)

However, for the MOF-5 data in the considered range of

temperature and pressure, we find that equation (4) gives

a good fit. Fig. 1 shows the Langmuir fit for data in the range

130 bar and 60125 K, with the following parameters :b0 2.8164 10

3 bar1, B 332.0158 K, qm0 11.7134 102

kg H2/kg MOF and A 131.3231 106 K2. At 20 bar and 77 K,

these parameters give q* 0.0532 kg H2/kg MOF.

ForMOF-5 powderZhou et al. [3] reporta skeletal density of

about 1.78 g/cc and 66.2% porosity. Due to thelack of densified

MOF-5 adsorption data, we assume that MOF-5 pellets have

same density as the powder. Hence, solid density

rs 1780 kg/m3, pellet porosity 3p 0.662, and pellet density

rp rs

1 3p

601:64 kg=m3: (6)

Assuming a dense random packing of spheres gives a bed

porosity of3b 0.36, bed density

rb rp1 3b 385:05kg=m3 (7)

and total porosity

Nomenclature

T, P Temperature and pressure, K, bar

ms, Vb Skeletal mass and volume of the adsorbent bed,

kg, m

3

mw Mass of tank wall, typically steel, kg_mf; _m Inlet and outlet gas mass flow rates, kg/s

3p, 3b, 3t Pellet, bed and total porosities, 3t 3b (1 3b)3p,

m3/m3

rs, rp, rb Skeletal, pellet and bed densities, rp rs(1 3p),

rb rp(1 3b), kg/m3

rg, vg Gas density and specific volume vg 1/rg, kg/m3,

m3/kg

mg, kg, ks Gas viscosity, gas thermal conductivity and

adsorbent thermal conductivity, Pa s, W/m/K,

W/m/K

dp Pellet diameter, m

U Superficial bed velocity, m/shg Convective heat transfer coefficient on the surface

of a pellet, W/m2/K

Nu, Bi, Re, Pr Nusselt, Biot, Reynolds and Prandtl numbers

vw; vs Mean specific volumes of the steel and

adsorbent, m3/kg

aPg; kTg Isobaric thermal expansion coefficient and

isothermal compressibility, 1/K, 1/barHg, Hq, Hs, Hw Specific enthalpy of gas, adsorbate, adsorbent

(solid) and steel components (wall), J/kg

CPg, CPs, Cpw Specific heat capacity of gas, adsorbent and

steel, J/kg/K_Ql; _Qh Heat leak rate, heat addition rate into the

tank, W

q, q* Excess adsorbate concentration and its

equilibrium value, kg H2/kg adsorbent

qm, b The two Langmuir adsorption isotherm

parameters, kg/kg, 1/bar

DHa Heat of adsorption, J/kg H2 adsorbed

Hsys Enthalpy of the inner thermal masses, J

TN Ambient temperature, KReff Effective resistance for heat leak into the

inner thermal masses, K/W

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 5 4 6 6 5 4 7 5 5467


4/11


3t 3b 1 3b3p 0:7837: (8)

For MOF-5 Panella et al. [5] report a heat of adsorption of

3.8 0.8 kJ/mol, Dailly et al. [6] report 4.1 kJ/mol, Kaye and

Long [7] report 4.75.2 kJ/mol, and Zhou et al. [3] report an

initialheat of adsorption of about 4.8 kJ/mol. Here we assume

a constant average heat of adsorption DHa 4.0 kJ/mol.

3. Fuel tank design

A lumped-parameter model does not distinguish between

different packed bed designs. Hence for illustration we

assumea longitudinal packed bed as shown in Fig. 2. A header

distributes the hydrogen feed stream to a packed bed of cryo-

adsorbent pellets. The bed adsorbs hydrogen, the unadsorbed

gasflow gets collected andflows outof the tank. Hydrogen gas

flowing out of the tank can be cooled and recirculated for

further adsorption.

The adsorbent bed is designed for an assumed 5 kg total

hydrogen capacity at 20 bar and 77 K, with 20% excess bed

mass allowed to provide for operating temperatures higher

than 77 K. Total hydrogen in the fuel tank, at any temperature

and pressure, is the sum of hydrogen in the adsorbed and gas

(in bed voids and pellet pores) phases. Thus,

mH2 T; P msqT; P Vb3trgT; P: (9)

At 20 bar and 77 K, rg 6.4851 kg/m3. Using these values in

Equation (9) gives an adsorbent mass of 75.3 kg. Allowing

a 20% excess gives ms 90.37 kg and Vb 0.2347 m3 234.7 l.

The bed volume will be significantly lower for densified

adsorbent pellets.

4. Model assumptions

In the next section we develop a quasi-static lumped-param-eter model for the cryo-adsorber, andhere we justify or list the

assumptions of quasi-static behavior and lumping. Quasi-

static behavior implies local thermal and mass equilibrium at

any time i.e. the transient system passes through a series of

equilibrium states. The adsorbents typicallyoffer largespecific

internal surface area, leading to intimate gassolid contact.

Hence, at any location within the bed, there is negligible

temperature difference between the solid and the gas i.e.

Ts r!; tzTg r

!; thT r!; t; (10)

and the adsorbate loading is close to its equilibrium value i.e.

q r!

; tzq

T r!

; t; P r!

; t: (11)These near-equalities can be easily demonstrated using

heat and mass transfer correlations for packed beds, see for

example Wakao et al. [8] and Wakao and Funazkri [9]. These

quasi-static approximations are implicitly assumed for

example in the work of Mota et al. [10]. The advantages of

quasi-static approximation are the following: when local

thermal equilibrium prevails a single energy balance can be

used to describe both solid and gas phases; when local mass

equilibrium prevails, we can replace dq/dt (usually described

using a linear driving force model) by dq*/dt. When the fuel

tank processes are quasi-static, to design or simulate the fuel

tank, only adsorption isotherms and heat capacity of the

adsorbent need to be measured at different temperatures.A lumped-parameter model involves intra-pellet lumping

andacross-the-bedlumping of temperature, pressure andsolid-

phase concentration fields. The intra-pellet lumping of

temperature and concentration can be systematically justified

by a Biot number analysis. For example at 80 K and 20 bar, the

hydrogen gas thermal conductivity is 0.061663 W/m/K, NIST

web book [11]. Then, for a 0.5 mm diameter pellet, in the stag-

nant gas limit from Nu hgdp/kg 2 we get the convective heat

transfer coefficient as hg 246.65 W/m2/K. Huang et al. [12]

report an effective thermal conductivity ofksz 0.32 W/m/K for

MOF-5. Using all these data we compute the Biot number of

a spherical pellet as Bi hgdp/6ks 0.0642. Since the computed

Biot number is lower than 0.1, we can safely assume that the

pellets are thermally thin i.e. the intra-pellet temperature

gradients are negligible, see Incropera et al. [13].

Fig. 2 Fuel-tank details: longitudinal (left) and cross-

sectional (right) views.

Fig. 1 Langmuir fit (lines) for MOF-5 excess adsorption

data (symbols) of Zhou et al. [3].



5/11


The pressure drop across a bed of highly porous pellets is

often negligible, and it can be easily shown using an Ergun

equation analysis or see for example the results of Sankararao

and Gupta [15], justifying an across-the-bed pressure lumping.

For example at 80 K and 20 bar, the hydrogen gas density and

viscosity are 6.2069 kg/m3 and 3.7453 106 Pas, NIST web book[11]. We assume a particle diameter of dp 0.5 mm and a bed

porosity of 3b 0.36. For a particular design of 0.5467 m diam-

eter, a 60 g/s flow of 80 K 20 bar gas corresponds to a superficial

velocity of U 0.041181 m/s. Using these values in the Ergun

equation DP/L amgU brgU2, where a 1501 3b

2=33bd2

p and

b 1:751 3b=33bdp, Bird et al. [14], gives DP/L 0.013 bar/m,

which is negligible compared to the feed pressure of 20 bar.

Across-the-bed lumping of temperature and concentration

fields is intuitively assumed for the slower tank processes:

discharge, dormancy and venting. For a fast process like

refueling, however, we anticipate significant temperature

gradients within the bed for which higher dimensional

models are being currently developed.

5. Model development

The quasi-static lumped-parameter model contains transient

mass and energy balances, resulting in two equations. For

a given bed and feed flow (when present), there are three

unknowns: the tank temperature, pressure and mass outflow

rate. Closure is achieved by specifying one of these three

unknowns:

1. During discharge the mass outflow rate is specified by the

fuel-cell demand. Hence, the model is solved for tempera-ture and pressure.

2. During dormancy the outflow is zero by definition. Hence,

the model is solved for temperature and pressure.

3. The vent is typically operated by a pressure controller.

Therefore neglecting valve transients, venting can be

treated as an isobaric process. Hence, the model is solved

for mass outflow rate and temperature.

5.1. Transient mass balance

The rate of change of hydrogen content of the tank balances

the net flow into the tank. Hence the transient mass balance

for hydrogen is given by

dmH2dt

_mf _m: (12)

Using Equation (9) we get

msdqdt

Vb3tdrgdt

_mf _m: (13)

Now, expressions are developed for the two time deriva-

tives in the above equation. Assuming that the gas phase is in

equilibrium at the corresponding temperature and pressure

gives

rgt rgTt; Pt: (14)

Then,

drgdt

vrg

vT

P

dTdt

vrg

vP

T

dPdt

: (15)

The isobaric temperature derivative of density is related to

the isobaric thermal expansion coefficient aPg according to

aPg 1vg

vvgvT

P

1rg

vrg

vT

P

(16)

The isothermal pressure derivative of density is related to

the isothermal compressibility factor according to

kTg 1

vg

vvgvP

T

1rg

vrg

vP

T

; (17)

see Smith et al. [18], page 62. Using these results, the time

derivative of density is given by

drgdt

rgaPgdTdt

rgkTgdPdt

: (18)

The quasi-static approximation for the adsorbateconcentration

qt qTt; Pt (19)

gives

dqdt

dq

dt

vq

vT

P

dTdt

vq

vP

T

dPdt

: (20)

The temperature-dependence ofq* comes from that of the

two Langmuir parameters: b(T ) and qm(T ). Using Equations (2)

and (3), and simplifying give

vqvT

P q

f0TfT

B

1 bPT2!: (21)

The pressure dependence of q* is only due to the explicit P

terms. Hence, from Equation (1) we get

vq

vP

T

q

1 bPP: (22)

Using Equations (21) and (22) in (20) we get

dq

dt q

f0T

fT

B

1 bPT2

!

dTdt

q

1 bPP

!

dPdt

: (23)

Using equations (18) and (23) in equation (13) and rear-

ranging, the transient mass balance takes the form

a11dTdt

a12dPdt

b1; (24)

with

a11 msq

f0T

fT

B

1 bPT2

! Vb3trgaPg; (25)

a12 msq

1 bPP Vb3trgkTg; (26)

and

b1 _mf _m: (27)

Note that the dimensions of a11, a12and b1 are mass/

temperature, mass/pressure and mass/time, respectively.



6/11


5.2. Transient energy balance

The thermal masses associated with the fuel tank are the gas

phase, adsorbed phase, adsorbent, pressure vessel including

the bed restrainers and other bed internals, insulation layer,

outer shell and ambient, as shown in Fig. 2. The insulationlayer isolates the inner thermal masses (gas, adsorbed phase,

adsorbent, and pressure vessel) from the outer ones (shell and

ambient). When the fuel-tank processes are slow, a single

lumped temperature can be used for the inner thermal

masses. Sircar [16] shows that any adsorption measurement

directly gives only the Gibbsian surface excess adsorption and

not the absolute adsorption, and suggests using mass and

energy balances with excess adsorption to simulate practical

adsorption systems. The transient energy balance for the

inner thermal masses (system) is given by

dHsys

dt _mfHgTf; Pf _mHgT; P _Ql Vb3t

dP

dt: (28)

Computation of Hsys is elaborated in the appendix. Each

term in the above equation is analyzed below, and the time

derivatives of temperature and pressure are collected

explicitly.

The differential change in the enthalpy of inner thermal

masses is defined as

dHsysT;P dmwHw dmsHs d

msqHq

d

Vb3trgHg: (29)

All the steel components in the inner thermal masses

(distribution tubes, restrainers, holding plates and pressure

vessel) are accounted as a steel wall of mass mw. For

a particular design for example we had mwz 125 kg. Adsor-bate enthalpy is obtained from the definition of heat of

adsorption as

Hq Hg DHa: (30)

This study assumes a constant average heat of adsorption,

independent of temperature and pressure. Hence, dHq dHg,

and

dHsysT; P mwdHw msdHs

msq Vb3trg

dHg

ms

Hg DHadq Vb3tHgdrg: (31)

And its time derivative is

dHsysdt

mwdHw

dt ms

dHsdt

msq Vb3trg

dHgdt

ms

Hg DHadq

dt Vb3tHg

drgdt

: (32)

Substituting equation (32) in equation (28), and collecting

the terms containing Hg(T, P) on one side and the remaining

time derivatives on another side gives

mwdHw

dt ms

dHsdt

msq Vb3trg

dHgdt

msDHadqdt

Vb3tdPdt

_mfHg

Tf; Pf

_m Vb3t

drgdt

msdqdt

HgT; P _Ql:

(33)

From the mass balance equation (13), note that the coeffi-

cient ofHg(T, P) in the above equation is _mf. Then,

mwdHw

dt ms

dHsdt

msq Vb3trg

dHgdt

msDHadqdt

Vb3tdPdt

_mfHg

Tf; Pf

HgT; P

_Ql: 33

Note that the reference state used in computing the gas

phase enthalpy does not determine the system evolution,since the gas enthalpy appears as a difference between two

states on the right hand side of equation (34). Hence, as

expected, the energy balance and the results deriving from it

are independent of the reference state used in computing the

gas phase enthalpy.

Now, using quasi-static approximations, all the time

derivatives are expressed in terms of the time derivatives of

temperature and pressure. For the time derivatives ofrg and q

equations (18) and (23) are used. The time derivative ofHg(T, P)

is expanded as

dHgdt

vHgvT

P

dTdt

vHgvP

T

dPdt

; (35)

see for example Section 2.1 of Ahluwalia and Peng [17].

Using the thermodynamic identities

vHgvT

P

T

vSgvT

P

Cpg (36)

and

vHgvP

T

vg T

vSgvP

T

vg T

vvgvT

P

vg1 aPgT

; (37)

see Smith et al. [18], page 191 we have,

dHgdt

CPgdTdt

vg1 aPgTdPdt

: (38)

For the solid phases (pressure vessel and adsorbent),

neglecting the thermal expansion of the material, similar

equations are written as

dHwdt

CPwdTdt

vwdPdt

; (39)

and

dHsdt

CPsdTdt

vsdPdt

; (40)

Using equations (18), (23), (38)(40) in equation (34), and

rearranging the transient energy balance is in the form

a21dTdt

a22dPdt

b2; (41)

with

a21 mwCpw msCps

msq Vb3trg

CPg

msqDHa

f0T

fT

B

1 bPT2

!; (42)

a22 mwvw msvs

msqvg Vb3t

1 aPgT

msqDHa1 bPP

Vb3t; (43)

and

b2 _mfHg

Tf; Pf

HgT; P

_Ql: (44)



7/11


Note that the dimensions of a21, a22 and b2 are energy/

temperature and energy/pressure and energy/time

respectively.

We assume that the heat leak has the form

_Ql TN T=Reff (45)

We have assumed a typical value of Reff 74.0K/W, so that

the heat leak into the tank during typical dormancy condi-

tions is about 3 W. Engineering calculations show that with

a 1-inch multi layer vacuum super insulation and good

mechanical design such values ofReffare realistic.

5.3. Solution procedures

Solving equations (24) and (41) simultaneously, gives

dTdt

b1a22 b2a12

a11a22 a21a12 f1T; P; (46)

and

dPdt

b1a21 b2a11a11a22 a21a12

f2T; P: (47)

Among the coefficients appearing in these two equations

only b1 contains _m. Suppose _m is known (as in the case of

discharge from the fuel-cell operation requirement or during

dormancy where it is zero by definition), these two coupled

first order differential equations are solved with initial condi-

tions T(t 0) T0 and P(t 0) P0, to give the temperature and

pressure evolution T(t) and P(t). For the isobaric processes of

venting, equation (47) gives the isobaric criterion as

b1a21 b2a11: (48)

Using the expression for b1 in the isobaric criterion and

rearranging gives the isobaric mass outflow as

_m _mf a11a21

b2: (49)

Equation (46) is solved with the above isobaric flow rate to

get T(t). This formulation allows an elegant implementation

of isobaric or isothermal processes, by setting f1 or f2 to zero

respectively, although for an arbitrary process neither of

them is zero. For an isobaric process, instead of deriving an

expression for the time derivative of pressure and then

setting it to zero, one could directly get an expression for _mby setting the pressure derivative to zero during the deriva-

tion. Such a procedure is mathematically identical to the

above method. We prefer the model described by equations

(46) and (47) since it gives a unified formalism for all the fuel

tank processes.

6. Dormancy

In a parked vehicle, continuous heat leak into the fuel tank

causes the hydrogen pressure to build up and gas is vented to

the fuel cell. The time period until the start of venting is called

dormancy. Let T0 and P0 respectively be the temperature and

pressure of the tank at the beginning. Let the vent pressure be

Pvent 25 bar. During dormancy, there is no flow into or out of

thetank i.e. _mf _m 0. Hence, the model equationsreduce to

b1 0, b2 _Ql. Therefore,

dTdt

_Qla12

a11a22 a21a12(50)

and

dPdt

_Qla11

a11a22 a21a12: (51)

Note that both dT/dt, and dP/dt are proportional to _Ql. The

temperature and pressure evolution equations are solved

simultaneously with the initial conditions T0, P0 and the time

taken for the tank to reach the vent pressure gives the

dormancy. Figs. 3 and 4 show that a full tank (5 kg load),

starting at T0 85.6 K and P0 20 bar (for a given pressure and

hydrogen load the corresponding equilibrium temperature

can be got from equation (9)) reaches the vent pressure in 3.5

days, and the temperature at that time is 90.4 K.

An alternate, quick but approximate, method to estimate

dormancy is as follows: The hydrogen content of the tank at

the beginning of the dormant phase is mH2 T0; P0. Let the

temperature of the tank at the end of the dormant phase be

Tvent. By definition, hydrogen is not lost from the tank during

dormancy. Hence, Tvent is computed by solving the equation

mH2 T0; P0 mH2 Tvent; Pvent: (52)

Neglecting the temperature variation ofReff, the arithmetic

average heat leak during dormancy is

C _QlD TN T0 Tvent=2=Reff: (53)

For dormancy, setting_

mf _

m 0 in the transient energybalance equation (28), and integrating over the dormant

period gives

HsysTvent; Pvent HsysT0; P0 C _QlDDtdorm Vb3tPvent P0: (54)

The appendix details the computation of Hsys(T, P). Since

Tvent is already known, this equation is solved for Dtdorm. Note

that this equation is exact with a time-averaged heat leak. But

Fig. 3 Pressure evolution during full load dormancy, with

a 1-inch multi layer vacuum super insulation. The point

marks the end of dormancy.



8/11


9/11


10/11


11/11

Documents

SKV_Cryo_ads_0D