Dimensionless lumped formulation for performance assessment of adsorbed natural gas storage

Applied Energy 87 (2010) 1572–1580

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Dimensionless lumped formulation for performance assessment of adsorbed natural gas storage M.J.M. da Silva, L.A. Sphaier * Laboratory of Theoretical and Applied Mechanics — LMTA/PGMEC, Department of Mechanical Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, bloco E, sala 216, Niterói, RJ 24210-240, Brazil

a r t i c l e

i n f o

Article history: Received 19 June 2009 Received in revised form 11 September 2009 Accepted 15 September 2009 Available online 9 October 2009 Keywords: Natural gas Adsorption Dimensionless groups Lumped analysis

a b s t r a c t Adsorbed natural gas (ANG) has been emerging as an attractive alternative to compressed natural gas or liquefied natural gas, on various circumstances. However, in spite of the advantages associated with ANG over other storage modes, there are some issues that need be properly addressed in order to ensure a viable employment of such alternative. One major problem is that the thermal effects associated with the sorption phenomena tend to diminish the storage capacity, thereby resulting in poorer performance. Hence, in order to design commercially viable storage vessels, the heat and mass transfer mechanisms that occur in these devices must be carefully understood and controlled. With the purpose of improving the understanding of mass and energy transport within ANG vessels, dimensionless groups associated with this problem have been developed in this study, resulting in an innovation to the ANG literature. Along with the dimensionless groups, a lumped-capacitance formulation has been also proposed. Although this type of formulation is limited compared to the multi-dimensional formulations present in the literature, its computational solution is remarkably faster. Numerical solution results using the proposed lumped formulation are compared with those of a previous study, suggesting that the simpler model can be applied to larger process times. The process of charging and discharging ANG vessels was then simulated employing the proposed formulation for different combinations of the developed dimensionless groups. In order to properly assess charge and discharge processes, a performance coefficient was employed. The results show that increasing the heat capacity ratio and dimensionless heat transfer coefficient tend to augment the performance coefficient, whereas an increase in the dimensionless heat of sorption worsens performance. The proposed normalization scheme is applicable to both multi-dimensional and spatially-lumped formulations, thereby facilitating the analysis of heat transfer enhancement in these storage vessels. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The problem of natural gas storage and transportation has been shown to have a significant importance in several areas. The current storage and transportation methods for natural gas are compressed natural gas (CNG) and liquefied natural gas (LNG). Compressed gas has the disadvantage of working at high pressures, which require heavy reservoirs for transportation and high compression costs. On the other hand, liquefied gas needs cryogenic temperatures and specialized equipment for re-gasification. Adsorbed natural gas (ANG) comes as promising technology for storage and transportation since lower pressures are employed and there is no need for extreme temperature reductions. Compared to CNG this means that equivalent storage capacities are feasible at lower pressures, thereby reducing compression costs. Wegrzyn * Corresponding author. Tel.: +55 21 2629 5576; fax: +55 21 2629 5588. E-mail address: [email protected] (L.A. Sphaier). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.09.011

and Gurevich [1] compared adsorbed natural gas storage with CNG and LNG, providing estimates of costs associated to each technology. The main problem with adsorbed natural gas is that the sorption heating (and cooling) effect degrades the storage capability; hence, improving heat transfer within ANG reservoirs becomes extremely important. In this context, a couple of solutions to remedy the problem have been proposed. Vasiliev et al. [2] numerically simulated the use of a heat pipe to reduce the unwanted thermal effects, and Yang et al. [3] performed an experimental analysis, proposing a solution which consisted of using a u-shaped heat exchanging pipe to supply heat to the central region of the storage vessel for reducing the unwanted thermal effects during discharge. Regardless of the of proposed solution strategy, a proper understanding of the heat transfer and mass mechanisms that occur within ANG reservoirs is required. Because of this, a number of studies that consider the physical and mathematical modeling of the problem have been carried-out. Barbosa Mota et al. [4,5]

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Nomenclature cp CR C i isor H m _ m Mg Ml p u t T V

constant pressure specific heat performance coefficient thermal capacity ratio specific enthalpy heat of sorption dimensionless heat transfer coefficient mass mass flow rate maximum compressed gas fraction maximum adsorbed gas fraction pressure specific internal energy time temperature reservoir volume

Greek symbols specific mass or concentration  porosity

Superscripts  dimensionless quantity Subscripts e effective or apparent g gas phase i inter-particle voids l adsorbed (liquid) phase max maximum min minimum s solid phase p particle w reservoir wall l micro-pores k meso-pores j macro-pores T isothermal process

q

provided a formulation for evaluating charge and discharge dynamics of adsorbed methane; Ergun’s equation was used to relate the gas-velocity with the pressure gradient and a Langmuir isotherm was employed for calculating the adsorbed amount. Study [4] was specifically focused on examining the effects of intra-particle diffusion. Vasiliev et al. [2] employed a model that takes into account temperature variations in radial and axial directions, but gas concentration variations were considered only in the radial direction; methane storage was considered and a Dubinin– Radushkevich isotherm was adopted. Bastos-Neto et al. [6] considered a model similar to the one used in [5]; however, no intra-particle diffusion effects were considered; Darcy’s equation was used for the velocity–pressure relation and a virial relationship was used as the adsorption isotherm. This study also performed an experimental analysis, the results of which were compared with those obtained from the solution of the formulation. Another similar heat and mass transfer model was employed by Basumatary et al. [7]; however, in that study, the thermal conductivity of the adsorbent medium was allowed to vary with temperature. Different from the above mentioned studies, Walton and LeVan [8] presented a one-dimensional mathematical model for examining the influence of non-isothermal effects and impurities contained in natural gas during storage-cycles, and Zhou [9] employed a simple lumped formulation for natural gas storage. In a recent study, Hirata et al. [10] presented a one-dimensional model for gas discharge and employed a hybrid analytical–numerical methodology for the solution of the governing equations. Although heat and mass transfer modeling and simulation is an important feature in the development of ANG technology, the majority of studies are focused in the development and assessment of new adsorbents for natural gas storage. Monge et al. [11] experimentally investigated the problem of methane storage in two series of activated carbon fibers. Biloe et al. [12] performed a characterization of adsorbent composite blocks for methane storage, and a Dubinin–Astakhov isotherm relation was employed by [13] for determining the influence of microporous characteristics of activated carbon on both charge and discharge of an ANG system. Lozano-Castello et al. [14] analyzed the influence of pore size distribution on methane storage in order to determine optimum pore size for activated carbons. Pupier et al. [15] experimentally studied the effect of cycling operations on adsorbed natural gas storage. Experimental comparisons of methane storage using dry and wet

active carbons were performed by Perrin et al. [16,17] and Zhou et al. [18,19]. Esteves et al. [20] presented experimental results of adsorption equilibria of several natural gas compounds on an activated carbon. Carvalho et al. [21] investigated the use of activated carbons from cork waste for adsorbing natural gas components. Bastos-Neto et al. [22] and Rios et al. [23] performed studies regarding methane adsorption using sorbents obtained from coconut shells. The applicability of new materials for methane storage was studied by Duren et al. [24] using molecular simulations. Muto et al. [25] investigated adsorption characteristics of activated carbon monoliths made from phenol resin and carbonized cotton fiber, whereas Kowalczyk et al. [26] studied the storage of methane on nanoscale tubular vessels. The adsorption of methane on different synthesized and commercial activated carbons was experimentally studied by Balathanigaimani et al. [27]. A calculation of methane adsorption storage capacity on different materials [28] showed that there is a linear relationship between methane uptake and specific surface area; this study also suggested that different materials and new approaches might be required for natural gas storage. Recently, novel materials for natural gas storage have been emerging; among these, Metal Organic Frameworks (MOFs) have been the subject of a number of investigations [24,29–31]. Although a significant amount of work has been done towards characterizing adsorption properties of natural gas and its components, studies focused on the adsorption of other gases indicate a potential for adsorption storage of other substances, such as CO2 [32]. In spite of the relevance of the previously proposed mathematical formulations, it seems that all of them are limited with respect to the normalization of the problem. As a matter of fact, apparently, no previous study encountered in the literature presents a discussion on the dimensionless groups relevant to heat and mass transfer in adsorbed natural gas storage. In this context, the purpose of this paper is to provide a meaningful presentation of dimensionless parameters associated to the problem of heat and mass transfer in adsorbed natural gas storage. In addition, a simple lumped-capacitance formulation is provided for for evaluating heat and mass transfers in the adsorbed natural gas reservoir. This model is simpler than previous multi-dimensional models, which take into account temperature and pressure gradients within the reservoir. Although, its range of applicability is narrower than multi-dimensional models, results from this formulation are compared with those from a previous study, showing that it could be applied

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to certain cases. The proposed model is normalized using the developed dimensionless groups, and numerical solutions of different test-cases are carried-out. The results provide dimensionless temperature, pressure, and concentration histories for the different analyzed cases. Also, a performance coefficient is employed, and its value is calculated for different situations. With these simulation results the impact of the dimensionless parameters on charge and discharge operations is illustrated. 2. Mathematical model The problem of natural gas (NG) storage using adsorption, involves the transport of heat and mass within a reservoir which is filled with a porous adsorbent. The resulting porous material that fills the reservoir may composed of adsorbent particles or appear in a monolithic form. In either way, the reservoir will end up having an effective or total porosity. The reservoir volume V can be subdivided, in a general form, in:

V ¼ Vs þ Vd þ Vi þ Vp ¼ Vp þ Vi ;

ð1Þ

where Vi þ Vp is the total usable void volume (the fraction of V that could be filled with natural gas). The dead-end pores have no effect on mass transfer and hence the effectively useful (interconnected) void volume excludes Vd . The volume Vp is the particle volume and the inter-particle void space Vi is null for a monolithic material. Assuming that void spaces are uniformly distributed within the porous medium, the total (usable) porosity, the bed porosity (assuming that the medium is composed of porous particles) and the particle (or monolithic) porosity are defined as:

¼

Vi þ Vp ; V

b ¼

Vi ; V

p ¼

Vp ; Vp

ð2Þ

dmg ¼ qg dVg ¼ qg dV;

ð9Þ

dml ¼ ql dV:

ð10Þ

The concentrations qg and ql can be expressed in terms of temperature and pressure by introducing a equation of state and an adsorption isotherm (or equilibrium relation). The specific mass of the gaseous phase is written as:

qg ¼ qg ðT; pÞ ¼

p ; RT

ð11Þ

where the second equality stands for ideal gas behavior. The specific mass of the adsorbed phase is written in terms of an isotherm relation:

ql ¼ ql ðT; pÞ;

ð12Þ

in which the functional form can depend on the selected adsorbent. Following [4,5], a Langmuir adsorption isotherm is adopted:

ql bp ¼ qm ; 1 þ bp qb

ð13Þ

in which

qm ¼ 55920 T 2:3 ;

b ¼ 1:0863  107 expð806=TÞ:

ð14Þ

2.1. Mass and energy balances: lumped analysis In order to obtain a simple model for describing heat and mass transfer within the reservoir, a lumped capacitance analysis is performed. Denoting space-averaged quantities with bar over-scripts, a lumped mathematical model for describing reservoir temperature and pressure is now derived. A global mass conservation balance for the natural gas can be written as:

where b ¼ 0 for a monolithic material. The total porosity is related to the bed and particle porosities as:

dmg dml _ g; þ ¼m dt dt

ð1  Þ ¼ ð1  b Þð1  p Þ:

_ g represents the net mass flow rate into the reservoir. In where m terms of the concentrations in gaseous and adsorbed phases, the following equation is obtained:

ð3Þ

The skeletal volume of the particle (or a monolithic material) is defined as the volume fraction occupied by the voidless solid and closed-pores; hence the skeletal fraction can be written as:

Vs þ Vd ¼ 1  p : Vp

ð4Þ



ð15Þ

_g  g dq l m dq þ ¼ : dt dt V

ð16Þ

A lumped energy balance for the natural gas reservoir can be written as:

Assuming that the mass of solid is uniformly distributed within the medium, the skeletal density (also called true density), which includes the volume of voidless solid and closed-pores, is defined as:



ms qs ¼ : Vs þ Vd

Combining the above equation with the mass conservation formula (16), one finds:

ð5Þ

The bulk volume of the porous solid material is defined in terms of the total volume:

qb ¼

ms ¼ qs ð1  b Þð1  p Þ ¼ qs ð1  Þ: V

ð6Þ

The concentration of the gaseous phase is written in terms of void spaces excluding closed-pores:

qg ¼

dmg ; dVg

ð7Þ

whereas the adsorbed phase concentration is defined in terms of the total volume:

ql ¼

dml : dV

ð8Þ

Hence, infinitesimal masses of gas and liquid (adsorbate) can be written as:

  d d d du g q l q  s qb Þ V þ mw w  g Þ þ ðu  l Þ þ ðu ðu dt dt dt dt _ g Þj ; ¼ Q_ þ ðig m



in

q g

ð17Þ

!    dig dp di dq di du  l l þ ðil  ig Þ l þ qb s V þ mw w þq  dt dt dt dt dt dt

_ g: ¼ Q_ þ ðig jin  ig Þm

ð18Þ

Introducing the definition of the differential heat of sorption [33] and specific heats relations

isor ¼ ig  il ; 1 ; dig ¼ cp g dT þ ð1  bTÞ dp q g  w ¼ cw dT; du dis ¼ cs dT;

ð19Þ dil ¼ cl dT;

ð20Þ ð21Þ

one arrives at:

qe ce

  dq l dT Q_ dp _ g; ¼ þ bT þ isor þ Dig m dt V dt dt

ð22Þ

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where qe ce is the effective (or apparent) thermal capacity:

m c qe ce ¼ ðqb cs þ ql cl Þ þ qg cp g  þ w w : V

ð23Þ

The total heat transfer at the storage vessel outer surface (exchanged with the surroundings) is given by:

Q_ ¼ 

Z

A particular situation occurs if inlet flow rates are independent of operating conditions (i.e. pressure and temperature), in which the previous quantities can be written in terms of the isothermal process time and a critical time, t c :

Z

Dmreal ¼

tc

_ g jdt þ jm

0

hðT  T 0 ÞdA  hðT  T 0 ÞAe ;

ð24Þ

Ae

Dmmax ¼

Z

Z

tf

_ g;cr jdt; jm

ð31Þ

tc tf

_ g jdt; jm

ð32Þ

0

where h is an overall heat transfer coefficient that accounts for thermal resistances within the reservoir and at the outer surface. Then, the energy equation is finally written as:

  dT A dp dq qe ce ¼ hðT  T 0 Þ e þ bT þ isor l þ Dig m_ g : dt dt V dt

ð25Þ

The term Dig will depend on an approximate relation between the averaged enthalpy in the reservoir and at its inlet, being given by:

Dig ¼ ig jin  ig :

ð26Þ

Nevertheless, due to the lumped approximation these enthalpies are approximately equal, and hence Dig  0. The initial conditions for the following problem are given in terms of pressure and temperature:

Tðt ¼ 0Þ ¼ T 0 ;

ð27Þ

in which p0 ¼ pmax for discharge and p0 ¼ pmin for charge. Since this mathematical formulation is based on a lumped analysis, it can only provide accurate results for cases in which small temperature and pressure gradients are present. This is expected to occur in relatively slow charge and discharge process. Although high-gradient situations can frequently occur in these types of problems (especially during charge), there are situations under which small gradients can occur. In this cases, the presented formulation would be better suited for calculations due to the extremely low required computational effort. 3. Performance assessment In order to assess the performance of a process (either charge or discharge), the parameter CR, introduced by Barbosa Mota et al. [5], is adopted:

CR 

Dmreal ; Dmmax

ð28Þ

representing a ratio between the actual amount of gas delivered during discharge (or stored, during charge), Dmreal , and the maximum amount that can be delivered (or stored), which occurs for isothermal operation. The actual (real) and the ideal (isothermal) masses of NG stored can be written as:

Dmreal ¼

Z

tf

_ g jdt; jm

Dmmax ¼

0

Z 0

for charge;

ð33Þ

pðt c Þ ¼ pmin ;

for discharge:

ð34Þ

_ g is a function of t only and m _ g;cr is Under this special condition m the modified mass flow rate (dependent on operating conditions) due to the fact that the critical pressure has been reached. This flow _ g , yielding CR values smaller than rate is clearly smaller than m unity. 4. Development of dimensionless groups Assuming that both charge and discharge processes start from an initial state that is in equilibrium with the surroundings, the initial temperature for both cases is given by the surroundings temperature itself, T 0 . The minimum reservoir pressure (for the gas phase) occurs in the beginning of the charge cycle (or in the end of the discharge cycle). The maximum pressure occurs at the end of the charge cycle (or at the beginning of the discharge cycle). Using Eq. (11) the minimum and maximum values of gas concentration can be written:

qg;max ¼

pmax ; RT 0

qg;min ¼

pmin ; RT 0

ð35Þ

where qg;max corresponds to the gas concentration in the end of the charge cycle, or in the beginning of the discharge cycle for isothermal operation. For non-isothermal operation, this maximum value will not be reached, and hence less gas will be stored in the reservoir. In a similar fashion, qg;min represents the gas concentration in the beginning of the charge cycle, or in the end of the discharge cycle for isothermal operation. Again, for non-isothermal operation the value qg;min cannot be attained during discharge, resulting a limited gas recovery. Analogous to gas phase concentrations, minimum and maximum values of adsorbed phase concentrations are defined in terms of pmin ; pmax and T 0 :

ql;max ¼ ql;max ðpmax ; T 0 Þ;

ql;min ¼ ql;min ðpmin ; T 0 Þ:

ð36Þ

0.90 0.85

tf

_ g jT dt; jm

ð29Þ

in which t f is the process (charge or discharge) time and the T subscript indicates an isothermal process. Naturally, for isothermal operation tf will be the time taken for the reservoir pressure reach the critical pressure (pmax for charge and pmin for discharge). Based on the minimum and maximum pressures one defines the maximum storage capacity:

Dmmax ¼ ðDqg þ Dql ÞV:

pðt c Þ ¼ pmax ;

ð30Þ

where Dqg and Dql represent the maximum possible variations in qg and ql , respectively.

0.80 CR

ðt ¼ 0Þ ¼ p0 ; p

in which the parameter t c represents the instant at which the critical reservoir pressure is reached. In summary:

0.75

Current Literature

0.70 0

1

2 τ

3

4

Fig. 1. Validation results: current formulation versus previous literature results (from [5]).

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Once more, some of these values can only be achieved under isothermal operation, since the effects of heating (during charge) and cooling (during discharge) worsens gas storage and recovery. Based on the minimum and maximum values for qg and ql , one defines the following differences:

Dqg  qg;max  qg;min ;

Dql  ql;max  ql;min ;

ð37Þ

representing the maximum changes in these concentrations. The first dimensionless groupings represent the fractions of gas that are stored in gaseous form and in adsorbed form, being defined as:

M g ¼

Dqg V ; Dmmax

Ml ¼

Dql V : Dmmax

ð38Þ

Naturally, these parameters must always satisfy Mg þ Ml ¼ 1. The next parameters comprise thermal capacity ratios between different capacities and the thermal capacity of the amount of gas charged (or discharged) under isothermal conditions. In this context, the thermal capacity ratios of the adsorbent material, the reservoir wall and the minimum gas contained in the reservoir are respectively defined as:

C s ¼

cs qb V ; cp g Dmmax

C min ¼

C w ¼

cw mw ; cp g Dmmax

ðcp g qg;min þ cl ql;min ÞV cp g Dmmax

:

ð39Þ

However, this will be a relevant parameters if the storage of different type of gases are considered since it depends only on the physical properties of the stored substance. For natural gas this parameter is approximately unity ðcp  1Þ. A last, and probably one of the most relevant parameters, is the dimensionless heat transfer coefficient. It is defined in a form that resembles the number of transfer units in heat exchangers; however, the fluid heat capacity rate is the average heat capacity rate associated with the amount charged (or discharged) for an isothermal process:

H ¼

hAe tf : cp g Dmmax

ð42Þ

With the introduction of dimensionless groups, the number of parameters in which the reservoir temperature and pressure can depend is reduced to a minimum. This greatly facilitates the analysis of the heat and mass transfer that occur within these types of problems. In addition, all of the presented parameters have an actual physical interpretation, which allows for a better understanding of the transport mechanisms involved. At this point it is also worth noting that although dimensionless groups for a lumped formulation are being defined, the same parameters will appear in multi-dimensional models, with the addition of extra parameters to account for gradients within the material.

ð40Þ

The ratio between the specific heats of NG in the gaseous and adsorbed phases is defined as:

4.1. Dimensionless governing equations

ð41Þ

The dependent and independent variables are normalized by introducing the following dimensionless forms:

Fig. 2. Charge process: pressure, temperature, and concentration histories for H ¼ 1.

Fig. 3. Charge process: pressure, temperature, and concentration histories for H ¼ 10.

cp ¼

cl : cp g

M.J.M. da Silva, L.A. Sphaier / Applied Energy 87 (2010) 1572–1580

t ¼

t ; tf

qg ¼

T ¼

T ; T0

qg  qg;min ; Dqg

p  pmin ; Dp q  ql;min ql ¼ l : Dql p ¼

_ g  m

ð44Þ

the dimensionless thermal expansion coefficient:

where Dp ¼ pmax  pmin . Once the previous definitions are introduced in Eqs. (16) and (25), a dimensionless version of the problem is obtained:

dqg dq _ g ; M g  þ M l l ¼ m dt dt   dT  dp ¼ H ð1  T  Þ þ Mg xb T   M g qg þ cp M l ql þ C  dt dt    dql þ M l isor  : dt

ð45Þ

ð46Þ

b  bT 0 ¼

ð47Þ

The terms on the right hand side of Eq. (46) represent a competition between different effects: heat transfer to the surroundings, heating/cooling to to compression and heating/cooling due to sorption. The combination of these parameters will result in a temperature variation, which can be diminished or augmented according to the value of the thermal capacity ratio C  . The remaining parameters are the dimensionless heat of sorption: 

isor 

isor ; T 0 cp g

ð48Þ

the dimensionless mass inflow (negative for outflow):

Fig. 4. Charge process: pressure, temperature, and concentration histories for H ¼ 100.

1 ; T

ð49Þ

ð50Þ

and a thermodynamic ratio:

x

Dp j1 ¼ ; c p g Dq g T 0 j

ð51Þ

where the second equalities in Eqs. (50) and (51) reflect the fact that ideal gas behavior is assumed ðj ¼ cp g =cv g Þ. Finally, a dimensionless version of the total mass in the reservoir is defined:

where C  is the added contributions of the thermal capacity ratios:

C  ¼ C min þ C s þ C w :

_ g tf m ; Dmmax

ð43Þ

1577

m 

mreal : Dmmax

ð52Þ

Using the mass conservation equation, it is easily shown that:

m ¼ m ðp ; T  Þ ¼ M g qg þ Ml ql :

ð53Þ

where the functional dependence of m comes from the fact that qg and ql depend on the dimensionless pressure and temperature histories. By observing the obtained dimensionless equations, one sees that the temperature and pressure evolution will depend on four characteristic parameters: H , representing the heat transfer rate to the surroundings taking into account the process time; C  , rep resenting the relative thermal capacity; isor , representing the sorp tion heating; and M l , representing the fraction of gas that is stored by adsorption (the remaining fraction is stored by compression).

Fig. 5. Discharge process: pressure, temperature, and concentration histories for H ¼ 1.

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in which sðt  Þ is a shut-off function which becomes zero if p reaches a critical value. In other words:

4.2. Performance assessment Using the normalized variables, the performance coefficient CR is written as

CR ¼

ð54Þ

where

Z

1

0

_ g jdt ; jm

Dmmax ¼

Z 0

_ g jT dt  ; jm

CR ¼









ð55Þ



m ðp ; T Þjt ¼1  m ðp ; T Þjt ¼0 ; m ðpT ; 1Þjt ¼1  m ðpT ; 1Þjt ¼0

ð56Þ

where pT corresponds to the pressure history for an isothermal process (with T  ¼ 1).

5. Results and discussion Simulation results for the current model are numerically calculated for two different processes: discharge with a constant mass outflow rate, and charge with a constant inlet pressure. These test-cases reflect real situations for consumption and fueling, respectively. Both cases must take into account the fact that, once the critical pressure has been reached, the mass flow rate must be _ g is given by: interrupted. In order to achieve this condition m

_ g ¼ sðp Þ/ðp Þ; m

0 < p < 1;

for

ð57Þ

Fig. 6. Discharge process: pressure, temperature, and concentration histories for H ¼ 10.

ð58Þ

otherwise;

ð59Þ

and the function /ðp Þ will have different forms depending on the type of process:

sðp Þ ¼ a ðpin  p Þ;

1

Using the mass conservation (Eq. (45)) it can be easily shown that: 



sðp Þ ¼ 0;

Dmreal ; Dmmax

Dmreal ¼

sðp Þ ¼ 1;



sðp Þ ¼ 1;

for charge;

ð60Þ

for discharge;

ð61Þ

where is the dimensionless inlet pressure and a is a dimensionless parameter that accounts for the flow resistance, depending on the permeability of the reservoir and other parameters. pin



5.1. Validation In order to validate the prosed formulation simulation results for a discharge case presented by Mota et al. [5] is performed and compared with the results presented in that study. Using the dimensional data provided in [5] the following values were obtained for the dimensionless parameters:

Ml ¼ 0:6830;  isor

¼ 1:575;

C  ¼ 6:589; H ¼ /ðsÞs;

ð62Þ ð63Þ

where s is the dimensional discharge time in hours. The function / varies with s due to variations on the convective heat transfer coefficient on the outer surface, having a maximum value of 0.43 at s ¼ 4. Since the model used for discharge in [5] was one-dimensional, considering only radial variations and heat losses only on

Fig. 7. Discharge process: pressure, temperature, and concentration histories for H ¼ 100.

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1579

the curved portions of the storage vessel, the dimensionless parameters were also calculated this way. Fig. 1 presents the calculated values of the performance coefficient CR using the presented lumped formulation, compared with those presented in [5]. As can be seen, for large discharge time (2.8–4 h) the current formulation is in very good agreement with the previously published results. As expected, for shorter discharge times, the current formulation cannot match the results calculated in [5], since that simulation is based on a formulation that can account for temperatures and pressure gradients within the reservoir. Naturally, the maximum error occurs for very small discharge time (s ¼ 0:02 h), in which the current formulation overestimates the performance coefficient by almost 23%. According to the lumped analysis hypothesis, the validity of the employed formulation requires small temperature and pressure gradients. These gradients arise from the combination of three effects: internal heating/cooling due to sorption (as well as compression), heat transfer to/from the environment, and internal reservoir characteristics (e.g. thermal capacity and heat transfer resistances). In traditional lumped analysis, the relation between heat transfer interactions with the surroundings and material properties (represented by a Biot number) is what usually dictates the validity of the formulation. Nevertheless, for the problem herein considered, the internal heating effect plays a significant role on the magnitude of temperature and pressure gradients, such that a Biot number alone is insufficient for determining whether small gradients will in fact occur. Although a thorough analysis is required for precisely determining the range of applicability of the lumped formulation, one can expect that relatively small internal heating/cooling rates as well as small rates of heat transfer to/from the surrounding will lead to smaller gradients. Hence, processes involving small charge

Now that the proposed formulation and numerical solution have been validated, simulation results are carried-out for investigating the effect of varying the dimensionless parameters on charge and discharge processes. Initially, the evolution of dimensionless pressure, temperature, concentration, and the amount of stored gas are evaluated for different situations. For the charge process, the value of pin was selected as 1.1, and the value of a was calculated so that at t ¼ 1 the isothermal solution reaches full capacity, i.e. p ¼ 1. Figs. 2–4 display the results  for different values of H considering Ml ¼ 0:9; isor ¼ 1:5; C  ¼ 5, which are typical values for this type of problem. As can be seen, for smaller values of H the temperature rise is much more pronounced and consequently the maximum pressure is reached at a much earlier stage than that of the isothermal solution. As a result, for these cases, a smaller amount of gas is charged, which will lead to a worse CR. The results for H ¼ 100 are almost isothermal, which will lead to higher CR values. The next set of Figs. 5–7 presents the discharge results, considering a constant mass outflow rate, for the same values of the dimensionless parameters used for simulating charge processes. As one can observe, there is a significant cooling effect for smaller values of H , which reduces the discharge capacity, as clearly seen in the concentration results. Naturally, these cases will result in a poorer performance. One point that is worth mentioning is that two distinct regimes can be seen in these results. The first is governed by pressure; until

Fig. 8. Performance coefficient for different charge processes.

Fig. 9. Performance coefficient for different discharge processes.

and discharge rates will be better suited for the application of the lumped formulation. 5.2. Charge and discharge results

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the critical pressure is reached p and m equal the isothermal solution. The second regime is governed by temperature, commencing after the critical pressure is reached; at this stage mass is admitted into the reservoir (at a lower rate) as heat is lost to the surroundings. If an adiabatic reservoir were considered, there would be no mass inflow in this second stage. A similar observation could be done for the charge solution; however the distinction between the two stages is not equally clear. The effect of varying the dimensionless parameters on the performance coefficient CR is presented in Fig. 8 for charge processes and in Fig. 9 for discharge processes. As can be seen, the higher values of H and C  yield the better performance. In addition, for high H values, the effect of C  on performance becomes negligible, and vice-versa. The larger value of C  may seem as an exaggerated situation, in which the thermal capacity of the reservoir is almost two orders or magnitude greater than that of the total stored gas; nevertheless, it is included to show how increasing the reservoir thermal capacity over a certain value, results in negligible change in performance. Besides analyzing the effect of varying C  and H , the presented results also show the impact of varying the heat of  sorption on the storage performance. As can be seen, as isor is increased, the coefficient CR is decreased, demonstrating how increasing the heat of sorption worsens charge and discharge performance. 6. Conclusions This paper was aimed at developing dimensionless groups for heat and mass transfer in adsorbed natural gas storage. Four dimensionless parameters associated with different heat and mass transfer mechanisms resulted from this analysis: the thermal capacity ratio ðC  Þ, the adsorbed gas fraction ðMl Þ, the dimension less heat of sorption ðisor Þ, and the dimensionless heat transfer coefficient ðH Þ. Other parameters were also seen in the analysis, such as the gas–liquid specific heat ratio ðcp Þ and the dimensionless thermal expansion coefficient ðb Þ; however, these depend only on the properties of the stored substance and have a fixed value when natural gas storage is considered. A lumped capacitance heat and mass transfer formulation was employed and normalized using the developed dimensionless groups. Simulation results from this model were compared with those of a previous study, suggesting that the lumped analysis is valid for longer process times. Then, in order to illustrate the influence of the dimensionless parameters on the behavior of charge and discharge operations, simulation results for different test-cases were carried-out. The results demonstrated that higher values of C  and H lead to better performance, and that the heat of sorption has a degrading effect on performance. The obtained dimensionless groups can improve the analysis of heat and mass transfer in natural gas storage via adsorption, thus allowing better designs of ANG systems to be achieved. At the current stage, the dimensionless groups were employed to a lumped formulation; nonetheless they can be easily employed to multidimensional formulations. Once this is done, a comparative analysis of different models can be carried-out, and one can precisely determine the validity of lumped capacity approaches, based on the values of the dimensionless groups. As a final remark, it should also be mentioned that although the current formulation was developed for natural gas storage, it could similarly be employed for the adsorption storage of other gases. Acknowledgement The authors would like to acknowledge the financial support provided by, CNPq (Brazil), FAPERJ (Brazil) and Universidade Federal Fluminense (Brazil).

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