Thermodynamic optimization for crystallization process of gas hydrate

INTERNATIONAL JOURNAL OF ENERGY RESEARCH

SHORT COMMUNICATION

Thermodynamic optimization for crystallization processof gas hydrate

Yuehong Bi1,2, Lingen Chen2,�,y and Fengrui Sun2

1Institute of Civil & Architectural Engineering, Beijing University of Technology, Beijing 100124, People’s Republic of China2Postgraduate School, Naval University of Engineering, Wuhan 430033, People’s Republic of China

SUMMARY

In this paper, thermodynamic optimization is applied to analyze the crystallization process of the gas hydraterelated to the gas hydrate cool storage system. Thermodynamic optimization model of the gas hydratecrystallization process is established. By taking the entropy generation minimization as the optimization objective,both the optimal control strategy and the optimal cooling rate of the gas hydrate crystallization process aredetermined. The minimum entropy generation corresponding to the optimal cooling rate decreases by 7.8%compared with normal situation. The results presented in this paper can provide important guidelines for optimaldesign and operation of the gas hydrate crystallization process. Copyright r 2010 John Wiley & Sons, Ltd.

KEY WORDS

crystallization process; gas hydrate; entropy generation minimization; thermodynamic optimization; cooling rate

Correspondence

*Lingen Chen, Postgraduate School, Naval University of Engineering, Wuhan 430033, People’s Republic of China.yE-mail: [email protected]

Received 11 April 2010; Revised 22 June 2010; Accepted 3 August 2010

1. INTRODUCTION

Gas hydrates have been paid more and more attentionsin many applications. The fields include energy,

environment, climate, natural gas industry [1–4] andcool thermal storage in air conditioning [5–8], etc. Coolstorage technology is gradually developed with the

appearance of air conditioning system using mechan-ical refrigeration. The development history of coolstorage air conditioning technology is the developmenthistory of cool storage materials [9–16]. The use of

gas hydrates as new phase change cool storage mediain air conditioning is attractive in recent years [5–8].The applications of gas hydrates involve complex

thermodynamic and kinetic problems, which need toestablish the relationship between the formation rateand thermodynamic variables [17]. Vysniauskas and

Bishnoi [18] took the lead in experimental study of thekinetics of gas hydrates formation above freezingpoint, and a consistent semi-empirical model was

formulated to correlate the experimental kinetic data.Englezos [19,20] and Bishnoi and Natarajan [21]established a gas hydrates intrinsic kinetic model basedon the crystallization theory coupled with the two-film

theory. The model did not contain any adjustable

parameters. This study revealed that the formationrate was proportional to the difference in the fugacityof the dissolved gas and the three-phase equilibrium

fugacity at the experimental temperature and was animportant landmark for the study of gas hydrates.Skovborg and Rasmussen [22] analyzed Englezos’

model over again and brought forward a masstransport limited extended Englezos’ model. Dholab-hai et al. [23,24] and Bishnoi and Dholabhai [25]studied on carbon dioxide and propane hydrate

formation kinetics in aqueous electrolyte solutions.Kalogerakis et al. [26] researched on the effects ofanionic, cationic and non-ionic surfactants on hydrate

formation kinetics. In recent years, thermodynamicoptimization theory has been applied for optimizingperformance of thermodynamic cycles and devices,

and lots of achievements have been made [8,27–49].However, so far there are only a few related references[33,50,51] that study on the thermodynamic optimiza-

tion for solution crystallization process. In thispaper, the thermodynamic optimization will be per-formed for the crystallization process of refrigerantgas hydrate.

Copyright r 2010 John Wiley & Sons, Ltd.

Published online 19 October 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.1779

Int. J. Energy Res. 2012; 36:269–276

269

2. PHYSICAL MODEL

The formation process of refrigerant gas hydrate canbe divided into two periods: nucleation and growth.The hydration reaction occurs when the hydration

medium (refrigerant and water) is cooled to subcoolingstate by coolant (t5 0), the temperature and pressureof the hydration medium keep unchangeable in the

period of nucleation, i.e. Tb0, Pb0. When t5 t1 issatisfied, the temperature rises in the period of growth,the fugacity of refrigerant at the liquid phase is bigger

than that at the interface of hydrate, the driving forcecaused by the fugacity difference makes the hydratecontinuously grow till t5 t2, and the period of grow is

end when the temperature falls down, i.e. the hydrationreaction finishes.When the temperature is constant, according to the

thermodynamic theory [52], one has:

dm ¼ RTdðln fÞ ð1Þ

where m is the chemical potential (Jmol�1), f is thefugacity of the hydration medium (Pa), R is the uni-

versal gas constant (8.314 Jmol�1K�1) and T is thetemperature (K).Supposing the crystallization process is isenthalpic,

then the entropy generation rate (s1) of the crystal-

lization process of gas hydrate is

s1 ¼ �U2ðtÞDm1=T ð2Þ

where U2(t) is the gas hydrate crystallization rate

(mol s�1), Dm1 is the chemical potential difference ofthe hydrate medium between the end and the start ofthe crystallization process (Jmol�1).

Substituting Equation (1) to Equation (2) yields:

s1 ¼ U2ðtÞR lnðfb=feqÞ ð3Þ

where fb is the fugacity of the hydrate medium in theliquid phase (Pa) and feq is the mean fugacity of the

hydrate medium at the interface of hydrate (Pa).Englezos [19,20] established a gas hydrates intrinsic

kinetic model. From which one can obtain the gas

hydrate crystallization rate

U2ðtÞ ¼ FpPK�cðfb � feqÞ; K��1c ¼ k��1d 1k��1r ð4Þ

where K�c is the crystallization rate constant of gashydrate (molm�2 s�1 Pa�1), K�d is the mass transfer

coefficient (molm�2 s�1 Pa�1), K�r is the surface reac-tion rate coefficient (molm�2 s�1 Pa�1), and FpS is thesum of the gas hydrate crystals surface area (m2).

The sum of the gas hydrate crystals surface area(FpS) is related to the size distribution of the gas hy-drate crystals. The surface of each crystal is propor-

tional to the power of 2/3 over the mole of each crystal.By consulting with Reference [33], FpS is substitutedwith �Fp� ¼ Daðn�Þ

2=3, where nS is the total mole of thegas hydrate crystals, Da takes into account the law of

mass transfer and the shapes of the crystals and Da canbe determined experimentally. �Fp� is less than the value

of the gas hydrate crystals surface area calculated byassuming that the moles of all the crystals are the sameand equal to the average crystal mole. As entropy

generation rate increases with the increase in thesurface of the crystals, the use of �Fp� gives a lowerbound on the entropy generation rate. Therefore, the

gas hydrate crystallization rate of Equation (4) is

U2ðtÞ ¼ dn1=dt ¼ DaK�cn

2=31 ðfb � feqÞ ð5Þ

where n1 is the total mole of the generated gas hydrate

at different time (mol).According to the ideas from References [33,50,51],

i.e. Equation (5) is expanded to Taylor series and only

the first item of Taylor series is reserved, the solution ofthe problem can be found much more easily if thecrystallization rate can be approximated as

U2ðtÞ ¼ dn1=dt ¼Y

n2=31 R lnðfb=feqÞ ð6Þ

that is, U2(t) is proportional to the difference of che-mical potentials and a phenomenological coefficient

P(mol4/3K J�1 s�1). Then the problem of minimizingentropy generation rate of the crystallization process ofgas hydrate can be written as

minð �s1Þ ¼ min1

t2

Z t2

0

Yn2=31 ½R lnðfb=feqÞ�2 dt

� �ð7Þ

Z t2

0

YRn

2=31 lnðfb=feqÞ dt ¼ �n ð8Þ

Nucleation period:

0tot1; n1ð0Þ ¼ 0; fbðtÞ ¼ fb0

Growth period:

t1tot2; n1ðt1Þ ¼ n0; n1ðt2Þ ¼ �n

tgr ¼ t2 � t1

where fb0 is the fugacity of the hydration medium inliquid phase in the nucleation period (Pa), n0 is thetotal mole of the critical crystals generated in the

period of nucleation (mol) and �n is the total mole ofthe gas hydrate generated in the time of t2 (mol).

3. THERMODYNAMIC OPTIMAL SOLUTION

The problem of minimizing entropy generation com-

bining Equations (6) and (7) with Equation (8) can besolved by establishing a Lagrange function. TheLagrange function is

L ¼ ½R lnðfb=feqÞ�1lY

n2=31 R lnðfb=feqÞ

h i�1ð9Þ

where l is the Lagrange multiplier.Taking the partial derivative of L with respect to fb

and setting it equal to zero (@L/@fb 5 0) yields

lnðfb=feqÞ ¼ffiffiffil

p= R

ffiffiffiffiffiffiffiYq� n1=31

� �for t1ptpt2 ð10Þ

Thermodynamic optimization for crystallizationY. Bi, L. Chen and F. Sun

270DOI: 10.1002/er

Int. J. Energy Res. 2012; 36:269–276 r 2010 John Wiley & Sons, Ltd.

From Equations (6) and (8), one hasffiffiffiffiffiffiffiffiffiffilYq¼ 3=2ðtgrÞ

�1ð �n2=3 � n2=3

0 Þ ð11Þ

Neglecting the total mole of critical crystals gener-

ated in the period of nucleation, i.e. supposing n0 5 0 issatisfied, Equation (11) can be changed asffiffiffiffiffiffiffiffiffiffi

lYq¼ 3=2ðtgrÞ

�1 �n2=3 ð12Þ

From Equations (8) and (11), one can obtain theoptimal dependence of the total gas hydrate mole ontime

n�ðtÞ ¼2

3

ffiffiffiffiffiffiffiffiffiffilYq� t

� �3=2¼ �nðt=tgrÞ

3=2

for t1ptpt2

ð13Þ

The optimal dependence of the gas hydrate crystal-

lization rate on time is

U�2ðtÞ ¼dn�1ðtÞdt¼

3

2�n

1

tgr

� �3=2 ffiffit

p¼

3

2

�n

tgr

ffiffiffiffiffit

tgr

s

¼3

2�U�2

ffiffiffiffiffit

tgr

s

for t1ptpt2

ð14Þ

where �U�2 is the average crystallization rate of the gashydrate in the interval of tgr(mol s�1).

The optimal dependence of the fugacity of thehydrate medium on time is

f�bðtÞ ¼ feq expffiffiffiffiffiffiffiffiffiffilYq Y

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffilQp

3t

s0@

1A�1

264

375

¼ feq exp3 �n1=3

2Q

Rffiffiffiffiffiffiffitgrt

p !

for t1ptpt2

ð15Þ

The fugacity model for refrigerant gas hydratedeveloped in Reference [53] has been proved that it canbe applied for gas hydrate phase equilibrium thermo-dynamics and engineering calculations in the design of

cool storage system. Therefore, the fugacity modeldeveloped in Reference [53] is used in this paper. It isexpressed as

f ¼ feq exp½vHðP� PeqÞ=RT� ð16Þ

where vH is the specific volume of refrigerant in gas

hydrate phase (m3mol�1), T is the temperature (K), Pis the pressure (Pa) and Peq is the equilibrium pressureof refrigerant at temperature T (Pa).From Equations (15) and (16), one can obtain the

optimal pressure

P� ¼ 3 �n1=3= 2Y

vHffiffiffiffiffiffiffitgrt

p T�1Peq

for t1ptpt2

ð17Þ

To Freon, which is commonly used as the gas hy-drate cool storage medium, the most simple line modelcan be applied to calculate the three-phase equilibrium,

i.e. the empirical equation is as follows [54]:

lnPeq ¼ aT1b ð18Þ

where a and b are the constants, which can be

determined experimentally.Supposing the pressure and the temperature at

critical decomposition point are Pm and Tm, respec-

tively, one can obtain

Peq ¼ Pm exp ½aðT� TmÞ� ð19Þ

Substituting Equations (16) and (19) into Equation

(15) yields

P� ¼ 3 �n1=3= 2Y

vHffiffiffiffiffiffiffitgrt

p T�1Pm exp½aðT�

� TmÞ�for t1ptpt2

ð20Þ

and t ¼ t1; P�ðt1Þ ¼ Pb0.where Pb0 is the pressure of refrigerant in the nucleation

period, i.e. the initial pressure at the growth stage (Pa).Letting A1 ¼ 3 �n1=3=ð2

QvH

ffiffiffiffiffitgr

pÞ. A1 is a coefficient

related to the characteristics of the crystallizationprocess of the hydrate medium (J s1/2K�1m�3).

Equation (20) can be simplified as

P� ¼ A1T�t�1=21Pm exp ½aðT� � TmÞ� ð21Þ

The corresponding temperature can be determinedby the energy balance equation, i.e.

U2Hr ¼ KfFDTf ð22Þ

DTf ¼ ðTc2 � Tc1Þ= ln½ðT� Tc1Þ=ðT� Tc2Þ�

where Hr is the heat of hydrate reaction, i.e. the latentheat of liquid–solid phase change (Jmol�1), Kf is thetotal heat transfer coefficient of heat exchanger in the

cold storage tank in the cool storage process(Wm�2K�1), F is the area of heat exchanger in thecold storage tank (m2), DTf is the logarithmic mean

temperature difference (K) and Tc1 and Tc2 are theinlet and outlet temperatures of the cooling medium,respectively (K).When the temperature difference between the inlet

and outlet of the cooling medium is relatively small,one can have DTf 5T–Tc, and then Equation (22) canbe changed to [55]

U2Hr ¼ KfFðT� TcÞ ð23Þ

where Tc is the mean temperature of the coolingmedium (K).

Substituting Equation (1) into Equation (23) yieldsthe optimal temperature

T� ¼ 3 �nHrð2KfFt3=2gr Þ

�1ffiffit

p1Tc for t1ptpt2 ð24Þ

and t ¼ t1; T�ðt1Þ ¼ Tb0.where Tb0 is the temperature of refrigerant in the nu-

cleation period, i.e. the initial temperature at thegrowth stage (K).

Thermodynamic optimization for crystallization Y. Bi, L. Chen and F. Sun

271, .

DOI: 10.1002/er


Letting B1 ¼ 3 �nHr=ð2KfFt3=2gr Þ. B1 is a coefficient

related to the characteristics of the heat transfer pro-cess of the hydrate medium (K s�1/2). Equation (24)

can be simplified as

T� ¼ B1

ffiffit

p1Tc for t1ptpt2 ð25Þ

Substituting Equation (25) into Equation (21) yields

P� ¼ A1Tct�1=21Pm expfa½B1

ffiffit

p1Tc

� Tm�g1A1B1 ð26Þ

At the temperature range of refrigerant gas hydrate

cool storage, Peq can be approximately substituted byPm. Using Equations (17) and (24), Equation (26) canbe simplified as

P� ¼ A1Tct�1=21A1B11Pm for t1ptpt2 ð27Þ

and t ¼ t1; P�ðt1Þ ¼ Pb0.Equations (24), (25) and (27) are the optimal

control strategies of the thermodynamic optimization

of process.The relation between induction time t1 and degree of

subcooling DTsc can be expressed as [55]:

t1 ¼ c0 � expa1

DTa2sc

� �� 1

� �ð28Þ

DTsc ¼ Tm � Tb0

where t1 is the induction time, c0 is the combinedcoefficient of the historical form of water in hydrate

medium, surfactant and agitation, etc., a1 and a2 arethe kinetic constants of gas hydrate formation andDTsc is the degree of subcooling.

4. NUMERICAL EXAMPLES ANDDISCUSSIONS

The cooling rate of the cooling medium acts as the

external control way in the crystallization processof gas hydrate. In fact, the thermodynamic optimiza-tion of gas hydrate crystallization process can be

realized if the cooling rate of the cooling medium meetsthe optimal control strategy, i.e. Equations (24), (25)and (27).The energy equation of the cooling medium acting

on the cool storage medium is given by the followingequation

U2ðtÞHr ¼ Cf _mcðTc2 � Tc1Þ ¼ _Qc ð29Þ

where Cf is the specific heat (J kg�1K�1), _mcis the mass

flow rate of the cooling medium (kg s�1) and Qc is thecooling rate of the cooling medium (W).Substituting Equation (14) into Equation (29) yields

_Qc ¼ 32�U�2Hr

ffiffiffiffiffiffiffiffiffit=tgr

pfor t1ptpt2 ð30Þ

_mcðTc2 � Tc1Þ ¼ 32�U�2HrC

�1f

ffiffiffiffiffiffiffiffiffit=tgr

pfor t1ptpt2

ð31Þ

Using Equation (30) to regulate the cooling rate ofthe cooling medium can meet the optimal controlstrategy equation. Actually, if the inlet temperatures

of cooling media (Tc1) holds a constant, usingEquation (31) to regulate the mass flow rate of thecooling media ( _mc) by real-time monitoring the outlet

temperature of the cooling medium (Tc2) can reach theaim of optimal control of the crystallization process.If the optimal control of gas hydrate crystallizationprocess is realized, the pressure and the temperature of

the hydrate medium are shown in Figures 1 and 2,respectively.In the calculations of Figures 1 and 2, the data for

the decomposition temperature and pressure of R141bcome from Reference [56], i.e. Tm 5 8.441C andPm 5 0.043MPa. Tc 5 41C and t2 5 8 h are experi-

mental measured data. B1 5 0.026 is calculated by thecorresponding experimental data. The supposed data isA1 5 50. Obviously, with the optimal cooling rate, thepressure precipitates and approximates rapidly to the

decomposition pressure Pm when entering the growthperiod, and the temperature rapidly increases from thesubcooling temperature of the nucleation period up to

the decomposition temperature Tm, and then thegrowth period is end.

Figure 1. Pressure versus time for optimal cooling rate.

Figure 2. Temperature versus time for optimal cooling rate.


272DOI: 10.1002/er


Equations (24), (25) and (27) include some kineticand heat transfer parameters, e.g. P, Kf, F, etc., so onecan see that the crystallization process of gas hydrate is

controlled by ‘kinetics control’ and ‘heat transfercontrol’. The coefficients A1 ¼ 3 �n1=3=ð2

QvH

ffiffiffiffiffitgr

pÞ1

and B1 ¼ 3 �nHr=ð2KfFt3=2gr Þ reflect the characteristics of

the crystallization process and the heat transfer processof the gas hydrate medium, respectively. The influencesof A1 and B1 on the pressure and the temperature areillustrated in Figures 3–5. In the calculations, the data

for the decomposition temperature and pressure ofR141b come from Reference [56], i.e. Tm 5 8.441C andPm 5 0.043MPa, and the experimental measured mean

temperature of the cooling medium is Tc 5 41C. Thevalues of t2 are experimental measured data. Thevalues of coefficient B1 are calculated by the corre-

sponding experimental data. The values of coefficientA1 are supposed.Figures 3 and 4 show that the pressure is influenced

mainly by A1 and the influence of B1 on the pressure is

very small with the optimal cooling rate. The increasein coefficient A1 means the increase in the hydratereaction speed. With the increase in A1, the speed of

pressure decrease slows up. The greater A1, the higherpressure of the hydrate reaction, and the faster growthspeed of crystal is. Therefore, the crystallization pro-

cess of gas hydrate is controlled mainly by ‘kineticscontrol’.The temperature is influenced only by B1 and A1

has no influence on the temperature for the optimalcooling rate. The increase in coefficient B1 meansthe increase in the heat transfer rate. Figure 5 showsthat the rate of temperature runs up with the increase

in B1. With A1 5 500 J s1/2K�1m�3, B1 5 0.026K s�1/2,t2 5 8.0 h, the same initial condition and the same totalmole of gas hydrate generated, the minimum entropy

generation rate corresponding to the optimal coolingrate decreases by 7.8% compared with that corre-sponding to the normal situation (no regulation).

5. CONCLUSIONS

Entropy generation minimization is performed forthermodynamic optimization model of the gas hydrate

crystallization process established in this paper. Theoptimal control strategy for the gas hydrate crystal-lization process is obtained. The cooling rate of coolingmedium often acts as the external control way in the

practical engineering. Using Equations (30) and (31) toregulate, the cooling rate of the cooling medium canmeet the optimal control strategy and make the

thermodynamic optimization of gas hydrate crystal-lization process in reality. The minimum entropygeneration rate corresponding to the optimal cooling

rate decreases by 7.8% compared with that corre-sponding to normal situation (no regulation). There-fore, the effect of thermodynamic optimization is

notable. During the deduction of the relationshipsand dependencies as described in this paper, thefugacity model developed in Reference [53], which issuitable only for simple refrigerant gas hydrates, is

used. Therefore, the relationships obtained in thispaper are only useful to develop a strategy on how to

Figure 3. Influence of A1 on the pressure versus time for

optimal cooling rate.

Figure 4. Influence of B1 on the pressure versus time for


Figure 5. Influence of B1on the temperature versus time for



273, .

DOI: 10.1002/er


run the crystallization process for simple refrigerantgas hydrates. To other gas hydrates, one could use thesame way to deduce the corresponding relationships.

NOMENCLATURE

a 5 constanta1, a2 5 kinetic constants of gas hydrate

formationA1 5 coefficient related to the characteristics

of the crystallization process of thehydrate medium (J s1/2K�1m�3)

b 5 constant

B1 5 coefficient related to the characteristicsof the heat transfer process of thehydrate medium (K s�1/2)

c0 5 combined coefficientC 5 thermal capacitance rate (J kg�1K�1)f 5 fugacity of the hydration medium (Pa)

FpS 5 sum of the gas hydrate crystals surfacearea (m2)

F 5 area of heat exchanger (m2)Hr 5 heat of hydrate reaction, i.e. the latent

heat of liquid-solid phase change(Jmol�1)

k�d 5mass transfer coefficient

(molm–2 s–1 Pa–1)k�r 5 surface reaction rate coefficient

(molm�2 s�1 Pa�1)

K* 5 crystallization rate constant of gashydrate (molm�2 s�1Pa�1)

Kf 5 heat transfer coefficient (Wm�2K�1)L 5Lagrange function_m 5mass flow rate (kg s�1)n1 5 total mole of the generated gas hydrate

at different time (mol)�n 5 total mole of the gas hydrate (mol)n* 5 optimal dependence of the total gas

hydrate mole on time (mol)

n0 5 total mole of the critical crystalsgenerated in the period ofnucleation (mol)

P 5 pressure (Pa)Pm 5 pressure at critical decomposition

point (Pa)P* 5 optimal pressure (Pa)_Q 5 rate of heat transfer (W)R 5 universal gas constant

(8.314 Jmol�1K�1)

t 5 time (s)T 5 temperature (K)T* 5 optimal temperature (K)

Tm 5 temperature at critical decompositionpoint (K)

U2 5 gas hydrate crystallization rate

(mol s�1)

U�2 5 optimal dependence of the gas hydratecrystallization rate(mol s�1)

Greek symbols

l 5Lagrange multiplierm 5 chemical potential (Jmol�1)

P 5 phenomenological coefficient(mol4/3K J�1 s�1)

s1 5 entropy generation rate of the

crystallization process of gashydrate (WK�1)

Subscripts

b 5 cool storage mediumc 5 cool storage charging process

eq 5 equilibriumf 5 heat transfer mediumgr 5 growthp 5 hydrate crystal

sc 5 subcooling

ACKNOWLEDGEMENTS

This paper is supported by Scientific ResearchCommon Program of Beijing Municipal Commissionof Education (Project No: KM200710005034), BeijingMunicipality Key Lab of Heating, Gas Supply,Ventilating and Air Conditioning Engineering andthe National Natural Science Foundation of People’sRepublic of China (Project No: 59836232). Theauthors wish to thank the reviewers for their careful,unbiased and constructive suggestions, which led tothis revised manuscript.

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Thermodynamic optimization for crystallization process of gas hydrate