International Journal of Heat and Mass Transfer 54 (2011) 4856–4863

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Release of stored thermochemical energy from dehydrated salts

Mehdi Ghommem a, Ganesh Balasubramanian a, Muhammad R. Hajj a, William P. Wong b,Jennifer A. Tomlin c, Ishwar K. Puri a,⇑a Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, USAb Science Applications International Corporation, Ottawa, ON, Canada K1P 5Y7c Science Applications International Corporation, Blacksburg, VA 24060, USA

a r t i c l e i n f o

Article history:Received 2 May 2011Received in revised form 23 June 2011Accepted 23 June 2011Available online 27 July 2011

Keywords:Thermochemical reactionEnergy storageSalt hydratesSensitivity analysisPolynomial chaos expansion

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.06.041

⇑ Corresponding author. Address: 223A Norris Hal24061, USA. Tel.: +1 540 231 3243; fax: +1 540 231 4

E-mail address: ikpuri@vt.edu (I.K. Puri).

a b s t r a c t

Thermochemical materials, particularly salt hydrates, have significant potential for use in thermal energystorage applications. When a salt hydrate is heated to a threshold temperature, a chemical reaction is ini-tiated to dissociate it into its anhydrous form and water vapor. The anhydrous salt stores the sensibleenergy that was supplied for dehydration, which can be later extracted by allowing cooler water or watervapor to flow through the salt, transforming the stored energy into sensible heat. We model the heatrelease that occurs during a thermochemical hydration reaction using relations for mass and energy con-servation, and for chemical kinetics and stoichiometry. A set of physically significant dimensionlessparameters reduces the number of design variables. Through a robust sensitivity analysis, we identifythose parameters from this group that more significantly influence the performance of the heat releaseprocess, namely a modified Damköhler number, the thermochemical heat capacity, and the heat fluxand flowrate. There is a strong nonlinear relationship between these parameters and the process effi-ciency. The optimization of the efficiency with respect to the parameters provides guidance for designingengineering solutions in terms of material selection and system properties.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Long-term energy storage and release can be facilitated by theuse of salt hydrates [1]. Upon heating, these materials release thewater coordinated to the solid crystal through a thermochemicalreaction. This results in an anhydrous form of the salt that alsochemically stores the energy supplied for the thermochemicalreaction. Anhydrous salts are typically hygroscopic and thus ableto absorb or adsorb atmospheric water vapor even at room temper-atures [2]. The reverse mechanism of salt hydration releases thestored energy. Although recrystallization and nucleation effectsduring hydration raise important questions about their durablecyclical use over time, thermochemical salt hydrates have severaladvantages over phase change materials [3–5] and latent heat stor-age devices [6,7] for long-term energy storage [8], transportationand release [9,10]. For instance, salt hydrates exposed to solar en-ergy during summer months will result in the corresponding dehy-drated salts and water vapor. These products can be separatelystored until the winter when they can be recombined through salthydration to release the stored heat. This process, which can be

ll rights reserved.

l (MC 0219), Blacksburg, VA574.

used to warm buildings, has been demonstrated for centralized so-lar seasonal energy storage [3,11–13].

We have previously described the fundamentals associated withthe energy storage in a salt hydrate [1]. Here, we provide a similaranalysis of the reverse heat release process from these salts. Sincethe performance of the thermal storage and release can be signifi-cantly improved by exploiting specific material behaviors, our anal-ysis provides additional information by which to evaluate theimpact of various system parameters on the recovery of the thermo-chemically stored energy. Unlike a typical parametric analysis basedon the classical sweep approach that requires several simulations,we use uncertainty quantification tools [14–16] that provide a ro-bust sensitivity analysis [17,18]. Thus, we are able to quantitativelydetermine the role of the key thermodynamic process parameters[1] that influence the performance of thermochemical storage-release systems. Magnesium sulfate heptahydrate (MgSO4 � 7H2O)is chosen as the model salt due to its large volumetric energy storagecapability [19]. (The words ‘anhydrous’ and ‘dehydrated’ are usedinterchangeably hereafter.)

2. Modeling the energy release process

Experiments designed to measure the thermochemical energyrelease process comprise of an insulated container filled with thedehydrated salt. Cooler water vapor in excess of that required for

Nomenclature

A frequency factor in Arrhenius’ equation (s�1)C specific heat capacity (J/kg K)Dm modified Damköhler numberE activation energy in Arrhenius’ equation (kJ/mol)DH enthalpy of hydration (J/kg)K thermal conductivity (W/m K)L length of the simulation domain (m)M molar mass (gm/mol)N concentration (mol/m3)Q energy released per unit volume (J/m3)R Universal gas constant (J/(K mol))T temperature (K)n unit normal vectorq heat flux (W/m)n number of random variablesp order of the polynomial chaosr rate of reaction (s�1)t time (s)

Greek symbolsC simulation domain boundaryP process efficiencya⁄ generic variable

b mass fractionc rate of water vapor supplied to the anhydrous salt (s�1)g dimensionless heat capacity per unit volumeq density (kg/m3)v dimensionless thermochemical heat capacityf number of molesk uncertain parameterl mean valuer standard deviationn random variable

Subscriptsd dimensional parametere end of overall heat transfer processh salt hydrateg water vaporr initiation of thermochemical reactionr, e end of thermochemical reactions anhydrous saltx axis along the horizontal directiony axis along the vertical direction

M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863 4857

complete hydration is passed over the salt. On coordination withwater, the anhydrous salt converts to the hydrated form, simulta-neously releasing energy that warms the excess water vapor flow-ing over the hydrated salt [12,13]. We, instead, employ an imposedheat flux that is extracted continuously from our model system asan analogy to the energy carried out by the warmer water vapor inthe experiments. While our model system is simple and con-structed to understand the fundamentals of the process, the studyof a detailed geometric configuration to mimic the experiment isone of our future goals.

Fig. 1 contains a schematic of the simulation configuration. Weconsider a square Lx � Ly two-dimensional box (Lx = Ly = L = 0.01 m)that is filled with the dehydrated MgSO4. This anhydrous salt,which has a higher internal energy than its hydrated counterpartdue to the thermochemical energy stored in it, is also at a highertemperature than its surroundings that are at room temperature.

Fig. 1. Schematic of the h

Thermal energy is extracted from the system at a constant rateqd from its right boundary. The enclosing walls on the top, leftand bottom, shown in Fig. 1, are insulated and consideredadiabatic.

The thermochemical process is initiated when the local temper-ature decreases to the reaction temperature Tr = 473 K. This thresh-old triggers the interaction (hydration) of the anhydrous salt withthe cooler water vapor that is introduced into the system. Dehy-drated salts are porous media. As an increasing fraction of the por-ous volume is filled by the supplied water vapor, we assume thatthe rate of vapor flow into the system decreases, since there isnow a greater resistance to the vapor inflow as well as relativelyless void spaces for the water molecules to diffuse into. Therefore,we introduce water vapor into the system using a time dependentfunction Ng = Ng,i(arctan(cdt)/(p/2)) until all the vacant spaces inthe porous medium are filled. Here, Ng,i denotes the concentration

eat release process.

4858 M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863

of water vapor required for complete hydration of the entire anhy-drous salt in the simulation box and cd a porosity-dependent ratefor filling the domain with water vapor. A salt with a larger voidvolume has a correspondingly larger cd which facilitates a rela-tively more rapid flow of water vapor. Similarly, cd is lower for aless porous salt. Such a model can be physically thought of as rep-resenting the action of a model pump. As the anhydrous MgSO4 iscooled, it experiences a relatively large temperature decrease whenthe thermochemical reaction occurs and the enthalpy of hydrationis released. This process is represented by

MgSO4|fflfflfflffl{zfflfflfflffl}Anhydrous salt

þ 7H2O|fflffl{zfflffl}Water vapor

!Removal of stored heatMgSO4 � 7H2O|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

Salt hydrate

: ð1Þ

2.1. Mathematical model

The heat release is modeled and numerically solved using rela-tions for energy (Eq. (2)) and mass (Eq. (3)) conservation, chemicalkinetics (Eq. (4)) and stoichiometry (Eq. (5)) [1,20]. First, the anhy-drous salt gradually cools due to the passage of water vapor over it.The consequent extraction of sensible energy reduces the salt tem-perature until the thermochemical reaction temperature Tr isreached. At this temperature, hydration is initiated due to whichthe salt combines with water vapor to produce the salt hydrate.The energy required for the process depends upon the reaction rater and the enthalpy of hydration DH of the hydrate, i.e.,

@

@tððMhNhCh þMsNsCs þMgNgCgÞTÞ ¼ rðKrTÞ � rMsNsDH; ð2Þ

where r = Aexp(�(E/RT)). The thermal conductivity K = bhKh +bsKs + bgKg, where bj = (MjNj/qh) for j 2 {h,s,g}. We approximate DHas a Gaussian function of temperature [1,19].

Since the system is closed and isolated, there is no mass exchangewith its surroundings. Thus, a decrease in the concentrations of theanhydrous salt and water vapor increases the concentration of thesalt hydrate. Hence, to satisfy the conservation of mass

Mh@Nh

@tþMs

@Ns

@tþMg

@Ng

@t¼ 0: ð3Þ

The changes in the masses of the anhydrous salt and water va-por are governed by the stoichiometry as well as the thermochem-ical kinetics, which determine the sorption rate. The rate ofdecrease in the concentration of MgSO4 due to its hydration(chemical reaction) into MgSO4 � 7H2O can be described in anArrhenius form as

@Ns

@t¼ �rNs: ð4Þ

Since the hydration of f moles of MgSO4 produces f moles ofMgSO4 � 7H2O, the rates of change of the concentrations of theanhydrous salt and its hydrated counterpart are identical,

@Nh

@t¼ � @Ns

@t: ð5Þ

Using the dimensionless parameters x ¼ x=L; y ¼ y=L; bT ¼T=Tr ; bE ¼ E=ðRTrÞ; t ¼ ðt=ðL2qhCh=KhÞÞ; bK ¼ K=Kh; c ¼ cd=ðKh=ðL2

qhChÞÞ;Dm¼ðL2qhCh=KhÞ=ð1=AÞ;gj¼ðMj NjCj=ðqhChÞÞ for j 2 {h,s, g},

Table 1Nondimensional numbers relevant to the modeling.

Symbol Description

Dm Modified Damkohlv Dimensionless The

q Dimensionless Heac Dimensionless Flow

and v = (DH/(ChTr)), the corresponding governing equations are(where the hats are removed in Eqs. (6)–(9) and thereafter fornotational convenience),

@

@tððgh þ gs þ ggÞTÞ ¼ Kr2T � Dmghv expð� expð�E=TÞÞ; ð6Þ

@gh

@tþ Ch

Cs

@gs

@tþ Ch

Cg

@gg

@t¼ 0; ð7Þ

@gs

@t¼ �Dmgs expð�E=TÞ; and ð8Þ

@gh

@t¼ �MhCh

MsCs

@gs

@t: ð9Þ

Since energy is extracted from one of the system boundaries,the boundary conditions are,

KrT � n ¼ þq on C1; andKrT � n ¼ 0 on Ci ðinsulatedÞ; for i 2 f2;3;4g; ð10Þ

where q = (qdL)/(KhTr), n denotes the unit normal to the surface. Weuse a heat flux of qd = 1000 W/m (i.e., q = 0.0431) per unit depth torepresent a baseline case but subsequently employ a range of valuesto perform the sensitivity analyses, which is described later.

From Eqs. (6)–(10), the significant dimensionless parameters(shown in Table 1) for the process are Dm, which describes howrapidly the thermochemical energy transfer occurs in comparisonto heat conduction through the material, v that conveys the mag-nitude of the dehydration enthalpy relative to the total heat capac-ity of the material, q, which represents the ratio of the extractedflux to that diffusing through the material [1] and c that accountsfor the porosity of the salt, and hence the heat and mass transferwithin it. The properties of MgSO4 considered herein are presentedin Table 2. The simulation technique is similar to that used by uspreviously [1].

2.2. Results and discussion

Fig. 2(a) presents the temperature variation at (X,Y) = (9Lx/10,Ly/2), which is a location close to C1. The temperature of theanhydrous salt decreases from Ti (=1.06) due to heat conductionuntil the reaction temperature T = Tr = 1 is reached after whichthere is an abrupt transition to a lower temperature Tr,e as the saltis hydrated. Thereafter, the temperature again decreases due to re-moval of sensible energy by heat conduction from the hydratedsalt through C1. Thus, the sharp temperature change betweent = 0.6 and t = 0.8 occurs due to the release of the enthalpy ofhydration during the thermochemical reaction in addition to theextracted energy q while the temperature distribution during othertimes is simply the result of heat extraction alone.

Fig. 2(b) presents the temporal variations in the concentrationsof the chemical species. The water vapor concentration increasesas it fills the system until the required value to transform all thedehydrated salt contained in the system is attained. Both the dehy-drated salt gs and water vapor gg concentrations decay to zero oncethe thermochemical reaction is complete, i.e., all of the salt is

er number Number ¼ Rate of thermochemical energy transferRate of heat diffusion

rmochemical Heat Capacity ¼ Enthalpy of dehydration of MgSO4 �7H2OHeat capacity of MgSO4 �7H2O per unit mass

t Flux ¼ Input heat fluxDiffusive heat flux

Rate ¼ Rate of water vapor flow to fill the simulation boxRate of heat diffusion

Table 2Material properties for salt obtained from the literature [19,27,1].

Symbol Description Value

Mh Molecular mass of MgSO4 � 7H2O (gm/mol) 246Ms Molecular mass of MgSO4 (gm/mol) 120Mg Molecular mass of H2O (vapor) (gm/mol) 18qh Density of MgSO4 � 7H2O (kg/m3) 1680qs Density of MgSO4 (kg/m3) 2660qg Density of H2O (vapor) (kg/m3) 0.46645Ch Specific heat of MgSO4 � 7H2O (J/kg K) 1546Cs Specific heat of MgSO4 (J/kg K) 800Cg Specific heat of H2O (vapor) (J/kg K) 1975DH Enthalpy of hydration (J/kg) 8536.18Kh Thermal conductivity of MgSO4 � 7H2O (W/mK) 0.48Ks Thermal conductivity of MgSO4(W/mK) 0.48Kg Thermal conductivity of H2O (vapor) (W/mK) 0.026A Frequency factor in Arrhenius’ equation (s�1) 1.67 � 105

E Activation energy in Arrhenius’ equation (kJ/mol) 55Tr Hydration temperature (K) 473R Universal gas constant (J/(Kmol)) 8.314

M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863 4859

hydrated. The concentration of the salt hydrate gh correspondinglyincreases to a value that satisfies the stoichiometry, in accord withthe chemical kinetics [1].

Next, we define a thermodynamic process efficiency parameter,

P ¼ 1� ðQ ex=Q reÞ; ð11Þ

where the energy extracted during the entirety of the process

Q ex ¼ qttot ;

and the cumulative heat release

Q re ¼Z Tr

Ti

gs dT þ ghvþZ Te

Tr;e

gh dT:

Here, ttot denotes the total simulation time, and Tr,e and Te the tem-peratures at the end of the thermochemical reaction and the overallheat transfer process, respectively. The model system is analogousto a heat engine, where Qre represents the cumulative energy re-leased from the salt while Qex is the output for external use. Heattransfer to the cooler water vapor that is supplied to initiate thereaction in the system leads to a loss of usable energy. The valuesof the simulation parameters for the baseline case are presentedin Table 3.

The temporal evolution of Qre is presented in Fig. 3(a). Beforethe thermochemical reaction is initiated and afterwards, heat lossthrough C1 occurs at a constant rate q. During the thermochemicalreaction, additional heat release, represented through v, enhancesthe extracted energy. Thus, materials with larger enthalpies alsoincrease the energy released. Fig. 3(b) presents the normalizedcumulative heat release _Qre from the salt, which is obtained by nor-malizing Qre with respect to the initial heat extraction rate from thedehydrated salt. Before the reaction occurs and afterwards _Qre isconstant. During the thermochemical reaction _Qre first rapidly in-creases and then declines when the concentration of the anhy-drous salt becomes lower than of its hydrated counterpart,eventually becoming zero when all of it is transformed toMgSO4 � 7H2O. The peak value represents the significant differencebetween sensible energy extracted and that due to the thermo-chemical reaction. Use of a material with a larger enthalpy ofhydration will increase the value of this peak.

3. Sensitivity analysis

Combinations of different system parameters can have a non-linear influence on the process, making an assessment of therelative importance of a single parameter difficult. Therefore,

we conduct a sensitivity analysis based on a nonintrusive polyno-mial chaos expansion [18,21,22]. Here, an uncertainty (or smallchange) in the input parameters is introduced and propagatedthrough the model equations to determine the rates of variationof the output parameters that result from the forced inputvariations.

3.1. Uncertainty propagation using the nonintrusive polynomial chaosexpansion

A classical approach to determine the effect of parameter ran-domness or uncertainties on a system’s response is to use MonteCarlo simulations (MC) to solve deterministic problems whenparameter values are randomly varied. Since the relatively lowconvergence rate associated with this approach results in a largerequired number of samples, it can become impractical due to highcomputational costs for complex problems. More sophisticatedsampling strategies such as Latin Hypercube [23] can somewhatimprove the efficiency of MC simulations.

The polynomial chaos expansion (PCE) is a spectral method inprobability space that approximates the model output by apolynomial expansion of uncertain parameters. The spectral rep-resentation of uncertainty is based on the decomposition of arandom function (or variable) into separable deterministic andstochastic components [24]. Thus, any generic variable a⁄ isexpressed as

a�ðnÞ ¼XP

m¼0

amWmðnÞ: ð12Þ

Here, am denotes the deterministic component, which is the ampli-tude of mth fluctuation, n the random variable vector (uncertain in-put parameter), and Wm(n) the random basis functioncorresponding to the mth mode. P + 1 = (n + p)!/n! is the numberof output modes, which is a function of the order of the polynomialchaos p and the number of random variables n. Many choices arepossible for the basis functions depending on the type of the prob-ability distribution selected for the uncertainty of the random var-iable vector n [25]. For variables with probability distributionsthat are Gaussian, Hermite polynomials are used because they forman orthogonal set of basis functions [26]. The Hermite polynomial oforder n is [26]

Hqðni1 ; . . . ; nin Þ ¼ ð�1Þq @q

@ðni1 Þc1 . . . @ðnin Þcne�

12n

T n; ð13Þ

wherePn

k¼1ck ¼ q and q is the degree of the polynomial.The Hermite polynomials form a complete orthogonal set of

basis functions in random space. If n is a standard Gaussian distrib-uted variable, then the inner product takes the form

hWiðnÞ;WjðnÞi ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffið2pÞn

q�� �Z 1

�1� � �Z 1

�1WiðnÞWjðnÞeðn

T n=2Þdn;

ð14Þ

with the density function of the n variate standard Gaussian distri-bution as a weighting function. We note that the inner product ofthe basis functions with respect to each other is zero, i.e.,

hWi;Wji ¼ hW2i idij; ð15Þ

which yields

ai ¼ha�ðnÞ;WiðnÞihW2

i ðnÞi: ð16Þ

An efficient and accurate alternative to obtain the coefficients ofthe polynomial expansion ai is based on solving the following lin-ear system [18]:

Table 3Parameter values for the baseline case. gh�init is the amount of anhydrous salt presentinitially in the simulation box, and gg,i is the quantity of water vapor to completelytransform it into the corresponding salt hydrate [1].

4860 M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863

a�1...

a�N

0BB@

1CCA ¼

W0ðn1Þ . . . WPðn1Þ... . .

. ...

W0ðnNÞ . . . WPðnNÞ

0BB@

1CCA

a0

..

.

aP

0BB@

1CCA; ð17Þ

gh�init gg, i c q ttot Ti Tr,s Tr, e Te P

0.23 0.6 400 0.0431 1.25 1.039 1.0 0.9 0.87 0.91

which represents the discretized form of Eq. (12). If the number ofsamples N is set equal to the number of polynomials in the expan-sion P + 1, the matrix in Eq. (17) is square. It can then be inverted toobtain the expansion coefficients from the outputs a⁄s. For the casewhere N is larger than P + 1, the system is solved by minimizing theerrors in the least square sense. This approach more accuratelycomputes the PCE coefficients [18]. It leads to the so-called linearregression method which, in comparison with other methods likequadrature-based methods, is more flexible in the choice of sam-pling points, and can make efficient use of a strategy such as LatinHypercube sampling.

Here, we consider uncertainties in the dimensionless parame-ters Dm, q, v, and c and investigate their effects on the performance

0 0.2 0.4 0.60.85

0.9

0.95

1

1.05

T

Thermochemicalreaction

(a)

(b)

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1.2

η

Introduce watervapor

to the system

Thermochemicalreaction

Dehydrate salt

Water vapor

Fig. 2. Transient evolution of (a) temperature and (

indices, which are the time taken to initiate the reaction tr and theprocess efficiency P. Using PCE, variations in the input parametersare introduced and propagated through the numerical model todetermine the rates of variations of the output parameters. We de-note the random vector composed of the normalized uncertainparameters as n ¼ fnDm ; nq; nv; ncg, where nk = (k � lk)/rk and lk

and rk are the mean value and standard deviation of k, respec-tively, and k 2 {Dm,q,v,c}. Then, we approximate the outputs tr

and P by polynomial expansions as given by Eq. (12). The PCE coef-ficients are determined from a nonintrusive PCE procedure, i.e., by:

0.8 1 1.2 1.4t

Significant temperaturedrop

due to the thermochemicalreaction

0.8 1 1.2 1.4t

ηsηhηg

Hydrate salt

b) concentrations of all three chemical species.

Table 4Sensitivity of performance indices to variations in dimensionless parameters.

Uncertain parameter (nk) nDm nq nv nc

@tr/@nk 0.0004 �0.1214 �0.00001 �0.0001@P/@nk 0.0012 0.0215 0.0026 0.0016

0 0.2 0.4 0.6 0.8 1 1.20

0.02

0.04

0.06

0.08

0.1

0.12

t

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

t

(a)

(b)

Fig. 3. Transient evolution of (a) the cumulative heat release and (b) the normalizedrate of cumulative heat release.

M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863 4861

1. Generating samples of the uncertain parameters using LatinHypercube sampling [23]. To generate samples, variations ofthe uncertain parameters are obtained by assuming Gaussianprobability distributions. Their mean values are based on thematerial properties specified in Table 2 (baseline case) andstandard deviations are set equal to 15% of the mean values.

2. Numerically solving the governing equations of the heat release.3. Using all samples to evaluate the PCE coefficients based on lin-

ear regression method, as described above.

3.2. Analysis

To analyze the statistics of the MC and nonintrusive PCE, weconsider the impact of the process parameters on tr and P.Fig. 4(a) and (b) present the empirical density functions of theseparameters. The density functions are determined from 200 real-izations based on both the MC simulations and those determinedfrom nonintrusive PCE. The results from the MC and third-orderPCE for tr are in excellent agreement. The need to include quadraticand cubic terms in the PCE representation to match the MC resultsindicates that the relationship between the input parameters and tr

is nonlinear. Analogous results for the stochastic efficiency P are

shown in Fig. 4(b). A seventh-order PCE is required to reproducethe MC results. This represents a strong nonlinear relationship be-tween P and the input parameters.

Thus, although it is computationally simpler, PCE provides anaccurate representation of the relationship between the inputparameters and performance indices. It incorporates the physicsof the system and identifies the parameters that improve processperformance. The approach also indicates that the coupling be-tween the process parameters is highly nonlinear. The sensitivityof the output a⁄ to changes of the ith input parameter nk is

@a�ðnÞ@nk

¼XP

m¼0

am@WmðnÞ@nk

: ð18Þ

These sensitivities for Dm, q, v, and c, which are presented in Ta-ble 4, help to reduce the number of design variables, thus facilitat-ing the search for conditions that enhance process performance.Based on the magnitudes of the sensitivities in the table, fractionalvariations in q are much more influential on process performancethan are similar variations in Dm, v and c. Increasing Dm increasesboth tr and P. Dm increases, for instance, when the thermal conduc-tivity of the hydrated salt is lowered (e.g., through a poorer packingof the salt within the box) or by increasing its specific heat (e.g., byincluding additives). As expected, increasing q (i.e., by increasingthe heat flux through C1 or decreasing Tr) reduces tr and increasesP. v does not have as significant an influence on tr althoughincreasing it yields a higher process efficiency. This implies that amaterial with a larger enthalpy of (de)hydration also enhancesthe thermochemical energy storage and release process. Increasingc (i.e., increasing the rate at which the system is filled with watervapor) leads to increases in P and decreases in tr. Thus, as intui-tively expected, faster filling of the system with vapor acceleratesthe thermochemical reaction, improving system performance.

Considering, for instance, the influence of both the extracted heatqd and the hydrate thermal conductivity Kh, the ratio of the sensitiv-ities of the performance measurea are related to their variations, i.e.,

@a=@qd

@a=@Kh¼ ð@a=@nqÞ � ð@nq=@qÞ � ð@q=@qdÞð@a=@nDm Þ � ð@nDm=@DmÞ � ð@Dm=@KhÞ

¼ � 1Khqd

� @a=@nq

@a=@nDm

� �; ð19Þ

where a 2 {tr,P}. Based on the values in Table 4 and the propertiesof (MgSO4 � 7H2O) [1], decreasing qd by 100 W/m (per unit depth)must be compensated by an �0.03 W/m K increase in Kh to main-tain the same value of tr. Furthermore, for the same reduction inqd, Kh should be reduced by �0.0018 W/m K to obtain an identicalefficiency P. Similarly, either the density qh must be decreased by�30 kg/m3 or the specific heat Ch of the hydrate lowered by�27.7 J/kg K to compensate for a 100 W/m (per unit depth) reduc-tion in qd while maintaining P at a constant value. Thus, combiningthe heat release model with PCE identifies those parameters thathave a greater impact, which can then be tuned to yield improve-ments in the process efficiency. This should assist in process designand to identify suitable materials for targeted applications.

4. Summary

The relations for mass and energy conservation, and chemicalkinetics and stoichiometry are applied to model the thermochem-ical energy release from dehydrated salts. The model system allowsfor constant heat extraction from a boundary. Water vapor is intro-duced into it to initiate a thermochemical reaction that releases theenthalpy of dehydration. A robust sensitivity analysis shows thatthere is a strong nonlinear relationship between the process effi-ciency and the dimensionless parameters Dm, q, v and c. The

0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

ρ

tr

1st order PCE2nd order PCE3rd order PCEMC

0.84 0.86 0.88 0.9 0.92 0.94 0.960

5

10

15

20

25

30

35

40

45

50

ρ

Π

1st order PCE2nd order PCE7th order PCEMC

(a)

(b)

Fig. 4. Probability distribution functions of (a) the time taken to initiate the reaction (tr) and (b) the process efficiency (P) obtained with Monte Carlo simulations and PCE ofdifferent orders.

4862 M. Ghommem et al. / International Journal of Heat and Mass Transfer 54 (2011) 4856–4863

efficiency is most influenced by the energy extracted from the sys-tem. As q, c or v increase, the time required to initiate the thermo-chemical reaction decreases and the system efficiency increases,whereas changing the thermal conductivity or reaction rate to alterDm has a weaker influence on system performance. These observa-tions provide guidance for the development of thermochemicalmaterials for engineering applications.

References

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