Transformation Analysis of Thermochemical Reactor Based on

Ind. Eng. Chem. Res. 2000, 39, 4127-4139

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Transformation Analysis of Thermochemical Reactor Based on Thermophysical Properties of Graphite-MnCl2 Complex Jong Hun Han,† Kun-Hong Lee,*,† Dong Hyun Kim,‡ and Hwayong Kim‡ Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja-Dong, Nam-Ku, Pohang, KyungBuk, South Korea 790-784 and Department of Chemical Engineering, Seoul National University, San 56-1, Sinlim-Dong, Kwanak-Ku, Seoul, South Korea 151-742

Transformation of the graphite-MnCl2 complex/ammonia system was investigated in a thermochemical reactor through numerical simulation. Thermophysical properties such as effective thermal conductivity and gas permeability were measured for the reactive medium with a bulk density of 100-250 kg m-3, and the values obtained were in the range of 14.0-25.6 W m-1 K-1 and (8.0 × 10-15)-10-12 m2, respectively. A model for the thermochemical reactor was developed on a macroscopic scale, and then the measured thermophysical properties were used in the model equations. The results of the simulation showed that, when the operating pressure was 200 kg m-3, the rate of global conversion was significantly impeded by the mass-transfer limitation. The absence of a distinctive heat front in the reactor shows that heat transfer is not a limiting factor with the graphite-MnCl2 complex. Introduction Sorption processes based on reversible solid-gas reactions have been developed for heat and cold production or thermal storage.1-10 In particular, these processes are favorable because natural refrigerants such as ammonia, water, methanol, and methylamine, which do not affect the global environment in terms of ozone depletion or global warming, can be used as working fluids. The basic system of the process is composed of a thermochemical reactor connected to an evaporator/ condenser in which the phase change of the working fluid takes place. A number of solid-gas couples can be used to produce thermal energy over a wide range of temperature (-50 to 350 °C) and pressure (0.1-50 bar).3 The solid-gas reactions used are of the following type:

MS‚nG + mG S MS(n + m)G + m∆Hr

(1)

where MS is the reactive solid, typically alkaline earth or transition metal halogenides, and G is the working fluid, typically NH3 or its derivative. These reversible reactions take place in a fixed-bed reactor under nonequilibrium conditions imposed by a double constraint of temperature and pressure.11 Various models for the transformation of the thermochemical reactor have been developed on the basis of the conservation laws of mass and energy coupled with chemical kinetics. Modeling approaches used in the study of the sorption processes have been classified into two main categories.12 The local model,11,13-15 which considers local and uniform variables on a small representative volume, allows simulation of the dynamic response and optimization of a thermochemical reactor. Studies of this type can be performed by solving a set * Corresponding author. Tel.: +82-54-279-2271. Fax: +8254-279-8298. E-mail: [email protected]. † Pohang University of Science and Technology. ‡ Seoul National University.

of governing equations with partial derivatives that need discretization in space and time. The local model can be used for the identification of kinetic and thermal parameters of the reactive media.11,13 On the other hand, global models11,16-20 consider uniform variables i.e., equivalent temperature, averaged in a representative volume with the scale of the reactor. The global model describes the complicated phenomena through the heat or gas coupling between the thermochemical reactor and the auxiliary components such as the evaporator, condenser, heat source, and heat sink in the complete unit.12 The time transients of all of these components can be achieved through the global model, which makes the study of the optimal control, predimensioning, or sizing of a complete unit easier by taking into consideration the exergetic efficiency and the coefficient of performance (COP). Table 1 summarizes previous efforts on the modeling of sorption heat pumps. In this paper, local modeling of a thermochemical reactor was performed on the macroscopic level (or on the pellet level) so as to observe the dynamic behavior of the reactor with knowledge of the heat and mass transfer through the reactive medium. For the observation of the transformation of the reactor, it is important to have accurate identification of the thermophysical properties as well as accurate modeling and simulation. Our modeling work was concentrated on a graphitemetallic salt complex, which is the consolidated composite formed by expanded graphite powders and the reactive solid, as a reactive medium in a sorption heat pump. Graphite-metallic salt complexes, which were developed by Spinner et al., simultaneously solve several lasting problems related to heat and mass transfer within the reactive medium. Graphite-metallic salt complexes have been known to be superior to the composite of the porous metallic foam such as aluminum, nickel, and copper.21,22 Special concern has been placed on the possibility of manufacturing sorption heat pumps using a graphite-metallic salt complex. Among many different technologies, STELF (Syste´mes Thermochemiques Energetiques aliant ELF) technology,

10.1021/ie9904394 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/04/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

Table 1. Previous Modeling Approaches of the Sorption Heat Pump author

work

findings

Lebrun and Spinner (1990)

Simulation for the development of solid-gas chemical heat pump pilot plants

Mazet et al. (1991)

Analysis of transformation and heat transfer in a nonisothermal solid-gas reacting medium Numerical modeling of the operation of chemical heat pump with model parameters that were obtained in specific lab-scale experiments Unreacted-core model for solid-gas reaction submitted to Tc and Pc. Simulation of the rate of reaction and identification of the thermal parameters in reactor Modeling and experimental investigation of thermochemical transformer based on the coupling of two solid-gas reactions Modeling of gas-solid reaction through coupling of heat and mass transfer with chemical reaction

Dimensioning method by the dynamic simulation allowed us to calculate an installation to produce aimed performance. Optimization of discontinuous and pseudo-continuous operating cycles could be carried out by phase chaining rules The kinetic and thermal parameters were identified separately through the preliminary experimental study and modeling for unconsolidated reactive media. The validity of this method was demonstrated for a reactor with scale of 25 kW by the excellent correlation between experimental and simulated results.

Lebrun and Neveu (1992) Goetz and Marty (1992) Lepinasse et al. (1994) Lu et al. (1996)

Stitou et al. (1997) Stitou et al. (1997) Neveu and CastaingLasvignottes (1997) CastaingLasvignottes and Neveu (1997)

Analytical modeling based on average functioning characteristics of a solid-gas thermochemical reactor during an interval of time Dimensioning nomograms for the design of fixed-bed thermochemical reactors with various geometrical configurations Impact of the microscopic process on the dynamic behavior of a solid-gas reactor Influence of external couplings on the dynamic behavior of a thermochemical reactor

The concept of grain-pellet could be used for identification of parameters of chemical kinetics, mass transfer at the grain level, and heat transfer at the pellet level. Strong dependence of reaction rate on ∆Peq and ∆Teq was shown. The transformation of the reactive medium was mainly limited by conductive heat transfer. The feasibility of a thermochemical transformer was demonstrated by a 1 kW laboratory plant based on the coupling via the gaseous phase. This model could be used for the simulation and optimization of the sequence of different phases during a working cycle. This model indicated the significant influence of mass transfer in the reactive media at low pressure and low permeability via the knowledge of the transfer mechanism. When a reactive medium was given, this model could determine the pressure range in which this medium could function correctly and efficiently. The analytical model allowed simple calculations to be made for the predimensioning of thermochemical machines in the decision-making process without any need for numerical simulation. No mass-transfer limitation was assumed in this work. The nomograms enabled us to do rapid dimensioning of solid-gas reactor according to working conditions, geometrical configurations, and characterization of the reactive medium. No mass-transfer limitation was assumed in this work. In terms of global approach for the sizing of thermochemical transformer, the kinetic expression defined by the Carnot temperature (θ) and global exchange coefficient (Usw) could be advantageous in comparison with other kinetic laws. In this model, a single parameter (Usw) that characterized the heat exchange between the reactive medium was evaluated for different configurations of a laboratory-scale model. Usw depended only on the geometry of the reactor and physical and thermal properties of the reactive medium. The mass-transfer problems appearing at low pressure were not taken into account in this work.

which was also developed by Spinner et al., has been actively investigated.23-32 Efforts to develop commercial machines for such diverse application areas as air conditioners, refrigerators, and recovery of industrial waste heat are being made by several countries, including South Korea under a contract with ELF. The heat- and mass-transfer characteristics of the reactive medium are closely associated with the thermophysical properties such as thermal conductivity and gas permeability, which were measured on a preliminary level by Mauran and L’Haridon.33,34 Although numerous local models of the thermochemical reactor have been developed, most of them have considered heat transfer only on the macroscopic scale of reactive media with unlimited mass transfer. However, Lu et al. reported that the pressure gradient, as well as the pressure perturbation, in the reactive medium could become a considerable factor in the global transformation of the reactor.15 They also reported that ignoring the dominant influence of mass transfer at low pressure for a reactive medium of low permeability, especially for a reactive medium of high density, could lead to erroneous results. Poyelle et al. reported that consolidated adsorption beds, which are made of NAX zeolite, binder, and expanded graphite powders, have very different behavior unlike that of unconsolidated beds because of mass-transfer resistance.35 Lu et al. have developed a coupled heat- and mass-transfer model and

investigated the dynamic behavior of the reactor through the sensitivity analysis of parameters such as gas permeability and operating pressure. However, their simulations were carried out with fixed thermophysical properties although such properties vary during the transformation of a real reactor. Moreover, they could not provide specifications for a reactive medium for practical applications such as refrigeration or air conditioning because of a lack of thermophysical properties. In the present work, the effective thermal conductivity and gas permeability of the graphite-MnCl2 complex were measured and directly used in model equations. The transformation of a reactor was investigated through numerical simulation of the model on the macroscopic scale of the reactive medium that takes into consideration heat and mass transfer coupled with chemical kinetics. The transformation of the reactor was then analyzed at various operating pressures and temperatures for a reactive medium with a specific bulk density and weight fraction of graphite. Experimental Section Preparation of the Graphite-MnCl2 Complex. Graphite hydrosulfate powders in flake form were dried in a circulation oven at 60 °C for 10 h to remove moisture before heat treatment. A steel crucible containing ca. 5 g of dried expandable graphite powder was

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rapidly inserted into a furnace (Fisher, Coal Analyzer 490) and kept at a constant temperature of 700 °C for 2 min. Expanded graphite powder was obtained by the heat treatment via expansion or exfoliation of the graphite complex. The expanded graphite powder was poured into a rectangular mold of hardened steel and then pressed by a single acting platen to obtain a porous graphite matrix with a preferred bulk density and dimension. The graphite-MnCl2 complex was prepared using the impregnation method. A 25 wt % salt solution was prepared by dissolving MnCl2 powder (Shinyo Co., 98.5%) in distilled water. The porous graphite matrix was placed in a container filled with the aqueous MnCl2 solution. Vacuum pressure of ca. 0.018 bar was applied to the container for several minutes to soak the porous graphite complex with the MnCl2 solution. The soaked graphite complex was dried through two steps. In the first step, the complex was placed in a circulation oven at 60-80 °C to remove water in the free state. In the second step, the complex was vacuum dried (Precision Scientific Inc., 5851) at 210 °C for 3 h to remove the bonded water molecules from MnCl2‚4H2O. The material aspects of this graphite-metallic salt complex have been reported elsewhere in detail by Han et al. 22 Heat- and Mass-Transfer Parameters. The thermal conductivity of the graphite-MnCl2 complex was measured with the transient one-dimensional heat flow technique. A solution to the heat-transfer equation was given by Carslaw and Jaeger for the case of heat conduction in a slab initially at constant temperature, To, that is subjected to a constant heat flux, Fo, at the surface x ) L. 36 The temperature-time relationship at any surface x within the material is given by

T(t) - To )

Fot FcpL

+

[

FoL 3x2 - L2 ke

-

2

∞

∑

(-1)n

exp 6L2 π2n)1 n2 -n2π2Ret nπx cos (2) L L2

(

)

( )]

where t is the time, ke the effective thermal conductivity, L the length of the specimen, and Re the thermal diffusivity. The temperature distribution at x ) L, with consideration of a finite temperature drop, ∆Tc, due to incomplete contact between the heat source and the surface of the specimen, can be given as

T(t) - To ) ∆Tc + f(w) ) w +

1 3

-

2 π

FoL f(w) ke

∞

∑ n-2 exp(-n-2π2w) 2 n)1

w)

(3a)

Ret L

2

)

t Θ

(3b)

(3c)

where Θ is the characteristic time. The effective thermal conductivity can be determined from the slope of a leastsquares fit of the temperature variation, T(t) versus f(w). Figure 1 shows the schematic diagram of the apparatus for thermal conductivity measurements under a controlled atmosphere. The principle of this technique and the experimental procedure were described in detail by Han et al.37,38 Thermal conductivity was measured with

Figure 1. Schematic diagram of the apparatus for thermal conductivity measurements: (A) measuring cell and (B) electrical circuit.

Figure 2. Schematic diagram of the apparatus for gas permeability measurements.

the reactive complex in the direction perpendicular to the compression force in the molding of the porous graphite matrix, which corresponds to the radial direction of the reactive complex of cylindrical shape. The gas permeability of the graphite-MnCl2 complex was measured using Darcy’s law as the driving force is the pressure gradient in the gas phase.14 Assuming that ammonia is an ideal gas under isothermal viscous flow at a low flow rate, the equation of gas permeation can be derived39

K)

(

P2 2QµL A P2 - P2 1 2

)

(4)

where K is the coefficient of permeability, Q the gas flow rate, µ the gas viscosity, L the thickness of specimen, P1 the upstream pressure, and P2 the downstream pressure. Figure 2 shows the schematic diagram of the apparatus for gas permeability measurements. Gas permeability was measured for the radial direction of the reactive complex with dimensions 65 × 65 × 11 mm3. Before the absorption reaction occurs, a vacuum of 10-4 Torr was applied at 150 °C to remove the air and to make partially hydrated MnCl2 into the completely anhydrous form inside the permeation cell. The cell was heated to a fixed temperature by an electric coil that can be controlled by PID. Preheated ammonia gas was allowed to flow into the bottom of the cell and then to flow axially into the reactive medium. An ammonia atmosphere was maintained for 4-6 h, which is sufficient to complete the absorption reaction in the following chemical reaction:

MnCl2‚2NH3 + 4NH3 S MnCl2‚6NH3 ( ∆Hr (5)

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Figure 3. Geometric configuration of the reactor used in this simulation. The heat-transfer parameters (ke, hw, hf) controlling the thermal process in the reactor were described with the temperature profile.

A pressure gradient between the two ends of the permeation cell was developed by opening the metering valve slowly so that the chemical reaction would not be induced by an abrupt change in pressure. The pressure was controlled by the back pressure controller (Tescom, 100 psig) and measured by a pressure transducer (Valcom, VPRN-5K) at both ends of the cell. The volumetric flow rate of gas was measured with a flowmeter of float type (Gilmont Instruments,150 mm). In this case, the gas permeability of the reactive medium containing MnCl2‚6NH3 was measured. After the completion of this test, the cell was heated under isobaric conditions to bring about the desorption reaction in eq 5. The gas permeability of the reactive medium with the ammoniated state of MnCl2‚2NH3 was measured at this time. The pressure during the experiment was in the range of 1.09-1.30 bar, and the pressure gradient, ∆P ) P1 - P2 was between 0.01 and 0.20 bar. Model Description Geometry of the Reactor. Figure 3 shows the geometric configuration of the reactor used in this simulation. The fixed-bed reactor, which has an external heat exchanger at the outer periphery and a gas diffuser at the center, has frequently been adapted in the sorption machine. The outer and inner radii were 150 and 10 mm, respectively. Thermodynamic Equilibrium of MnCl2-NH3. Sorption heat pumps are based on thermal effects of reversible solid-gas chemical reactions that take place in a fixed-bed reactor connected by pipes with either a condenser or an evaporator according to the corresponding cycle phase. These reversible chemical reactions can be expressed as in eq 1, where the equilibrium is monovariant. Because the liquid-vapor equilibrium is also monovariant, the complete cycle can easily be represented with a Clausius-Clapeyron diagram in which the solid-gas equilibrium is given by the following equation:40

∆Hrxn ∆Srxn ln(Peq) ) + RT R

(6)

Figure 4. Basic unit of sorption heat pump system and its corresponding Clausius-Clapeyron diagram. Two equilibrium drops of ∆Teq and ∆Peq were designated for an absorption reaction with the constraint conditions (Tc, Pc) on the diagram.

where Peq is the equilibrium pressure in the MnCl2NH3 system, ∆Hrxn the enthalpy change of the reaction, and ∆Srxn the entropy change of the reaction. In case of liquid-vapor equilibrium, ∆Hrxn and ∆Srxn were replaced by the enthalpy and entropy changes, respectively, for the phase change. Figure 4 shows the basic unit of a sorption heat pump and the ClausiusClapeyron diagram for the MnCl2-NH3 system. Using a differential scanning calorimeter (TA Instruments, PDSC), the values of ∆Hrxn and ∆Srxn were evaluated as 48 903 J mol-1 and 232 J mol-1 K-1, respectively. For comparison, Lepinasse et al. reported that their values were 47 416 J mol-1 and 228.07 J mol-1 K-1, respectively, which differ slightly from our results.41 The solid-gas chemical reaction takes place when the operating conditions deviate from the thermodynamic equilibrium line. The difference between the operating conditions of the reactive solid and the corresponding state of thermodynamic equilibrium is known as the equilibrium drop.12,15,19,42 Two equilibrium drops can be expressed by ∆Teq ) |Tc - Teq(Pc)| and ∆Peq ) |Pc - Peq(Tc)|. These equilibrium drops have a direct influence on the reaction rate and, accordingly, the performance of a sorption machine. Thus, anincrease in ∆Teq or ∆Peq leads to an increase in the rate of the chemical reaction and, accordingly, to an increase in the power of sorption machine. Chemical Kinetics. A number of kinetic laws have been developed using various models to describe the transformation of the solid-gas reaction.11,12,14,15 In particular, Goetz et al. presented the unreacted core model using the grain-pellet concept with which they could give the structural changes of the reactive solid in a phenomenological view.14 However, this approach was somewhat tedious as seven parameters, including the hydraulic radius of the pore, the internal porosity of the grain, the kinetic absorption/desorption parameters, and the radius of the grain, should be identified through a procedure involving the minimization of the difference between the simulated and experimental instantaneous rates. In this work, overall and semiem-


pirical kinetic models developed by Mazet et al. were selected, and the reaction rate was then expressed as a function of local conversion (x), T, and P in the following form:

dx/dt ) f(x) k(P,T)

(7)

For the f(x) term, expressions of the form (1 - x)n for the absorption reaction and xn for the desorption reaction can be used. The reaction pseudo-order n is a parameter to be identified, and its value is in the range of 0-2. The k(P,T) term, which is similar to the specific rate in homogeneous kinetics, is an increasing function of temperature. However, for reversible heterogeneous reactions, k(P,T) should be expressed by two factors considering the Arrhenius term and the thermodynamic equilibrium drop as follows:

k(P,T) ) s exp(-Ea/RT) g(P,T) ) Ar g(P,T)

(8)

where s is the preexponential factor, Ea the activation energy, and Ar the Arrhenius term. Mazet et al. reported that the g(P,T) term can be dominant over the Arrhenius term, particularly in the case of the absorption reaction. As the temperature decreases, a strong increase in the reaction rate was observed by Mazet et al., although the Arrhenius term was decreasing. There have been various expressions for g(P,T),12 i.e., ln[Pc/ Peq(Tc)] and [(Pc - Peq(Tc))/Pc]n, and then Mazet et al. expressed it as [Pc - Peq(Tc)]/Pc.11,13 Thus, for the absorption reaction of MnCl2-NH3, the overall kinetic model was given by

Pc - Peq(Tc) dx ) Ar (1 - x)n dt Pc

(9)

The two parameters Ar and n should be identified with the experimental results. The values of Ar and n were empirically determined by Choi as 1.018 7 × 10-3 s-1 and 1.185, respectively, under the conditions Pc ) 1.0 bar and Tc ) 288 K.43 Although the values of Ar and n vary slightly with Pc, Tc, and the state of the reactive medium, the overall kinetics of eq 9 give satisfactory results in this work. Specific Heat and Porosity. Two parameters used to identify the graphite-MnCl2 complex are given as follows:

Fb )

mg mg , fg ) Vb ms + mg

(10)

where mg is the mass of graphite, Vb the volume of the reactive complex, and ms the mass of salt. The reactive complex in the reactor is made up of various components such as expanded graphite powders, MnCl2‚2NH3, MnCl2‚ 6NH3, and ammonia gas. Thus, the specific heat of the reactive complex should be defined according to the relative proportion of each component. Because the equation of heat transfer was developed in the representative volume, specific heat was expressed on the basis of unit volume as follows:

Cp(x) ) (Vc/Vb)Cp,C + (v2m/Vb)(ms/MW2)(1 - x)Cp2 + (v6m/Vb)(ms/MW2)xCp6 + (x)Cp,NH3 (11) where Vc is the volume occupied by graphite, Cp,C the specific heat of graphite, v2m the molar volume of MnCl2‚

2NH3, MW2 the molecular weight of MnCl2‚2NH3, Cp2 the specific heat of MnCl2‚2NH3, v6m the volume occupied by MnCl2‚6NH3, Cp6 the specific heat of MnCl2‚ 6NH3, total porosity, and Cp,NH3 the specific heat of ammonia gas. Total porosity can be expressed on the basis of unit volume in the following form:

(x) ) 1 - Vc/Vb - Vs/Vb ) 1 - Fb/Fg - [(1 - fg)/fg][v2m + (v6m - v2m)x] (Fb/MW2) (12) where Vs is the volume occupied by the reactive solid and Fg is the true density of graphite (2260 kg m-3). The values of the parameters used for the evaluation of specific heat and total porosity are given as

Cp2 ) 233.4 J kg-1 K-1, Cp6 ) 554.48 J kg-1 K-1 v2m ) 82.27 × 10-6 m3 mol-1, v6m ) 162.27 × 10-6 m3 mol-1 Equation of Heat Transfer. The most interesting characteristic of the graphite-MnCl2 complex is the anisotropy in the thermophysical properties. The effective thermal conductivity in the radial direction is larger than that in axial direction depending upon the bulk density of the graphite matrix. Such an anisotropy of thermal conductivity is attributable to the alignment of the basal planes in the graphite crystals during the compression molding of expanded graphite powders.22 Spinner et al. reported that the anisotropy ratio was in the range of 1.5-3.5 for reactive complexes with bulk densities of 50-200 kg m-3, which was also confirmed by Han et al.37,38 The preferred design of reactors in sorption machines frequently takes a cylindrical shape with the gas diffuser in the center and the heat exchanger at the wall. The reactor is insulated at two ends in the axial direction. From these facts, a strong preference for heat transfer exists in the radial direction. Thus, assuming that heat transfer takes place only in the radial direction and neglecting convective heat transfer between ammonia and the reactive complex, the following equation of heat transfer can be derived:

∂T ) ∇‚(ke3T) + So∆Hrxn ∂t

Cp[x(r)]

(13)

where So is a source term related to the mass flow involved in the chemical reaction. This term can be evaluated by the overall kinetics and molar concentration of salt

So ) γNsalt

dx dt

(14)

where Nsalt is the number of moles per unit volume of reactive complex and γ is the stoichiometric ratio of the chemical reaction that is associated with the number of ammonia molecules generated during the absorption or desorption reaction. Nsalt can be expressed as follows:

Nsalt ) (ms/MW2)/Vb ) [(1 - fg)/fg](Fb/MW2) (15) Equation of Mass Transfer. The pressure in the gas diffuser was asummed to be uniform along the axial direction as the solid-gas reaction proceeds. Spinner

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reported that, in the case of gas permeability, the anisotropy ratio was in the range of 1.1-2.5 for reactive complexes with bulk densities of 100-200 kg m-3.44 Their results were confirmed at the preliminary level for porous graphite matrix by Han et al.45 Thus, assuming that the mass transfer proceeds only in the radial direction, one can derive the equation of mass transfer to be one-dimensional as follows:

( ) (

)

Dp ∂ P 3P - So ) ∇‚ ∂t RT RT

(16)

where Dp is the diffusivity coefficient and is defined as KP/µ. Model Equations. The heat transfer equation in eq 13 could be represented as follows for discretization in the numerical analysis.

Cp

∂T 1 ∂ ∂T rke + So∆H ) ∂t r ∂r ∂t

(

)

(17)

The effective thermal conductivity, ke, is mainly a function of the bulk density (Fb), the weight fraction of graphite (w), and the local conversion (x). However, ke can become a function of local conversion only if Fb and fg are given for a reactive complex, and a linearized function of local conversion was used in this simulation. Under the conditions of continuity of heat conduction at the wall and no heat flow at the inner radius, the following boundary and initial conditions were used.

∂T ke ) hsw(Tc - T) at r ) rw ∂t

(18a)

∂T ) 0 at r ) ri ∂r

(18b)

T ) Tc at t ) 0

(18c)

where hsw is the heat transfer coefficient, which is strongly influenced by the degree of contact between the wall and the reactive complex. The value of hsw depends on the local conversion as well. For two extreme cases of reactive complexes with MnCl2‚6NH3 or MnCl2‚2NH3, values of 550 and 150 W m-2 K-1, respectively, were used. Thus, in this simulation, a linearized function of local conversion was also used for a given reactive complex. The variables rw and ri indicate the positions of outer radius and inner radius for the reactive complex, respectively. The mass transfer equation in eq 16 could be also represented as follows for discretization in the numerical analysis:

(

)

RT ∂ Dp ∂P ∂P P ∂T ) + r - RTSo ∂t T ∂t r(x) ∂r RT ∂r

(19)

The diffusivity coefficient, Dp is influenced by the gas permeability, operating pressure, and temperature. As a result, Dp can be significantly varied by the pressure distribution or disturbance, temperature distribution, and local conversion inside the reactive complex as the solid-gas reaction proceeds. The gas permeability, K, is mainly a function of Fb, fg, and x. However, a linearized function of x was used for the gas permeability in this simulation because Fb and fg were fixed for a specific reactive complex. Because there was no outlet for the gas at the wall and the pressure in the gas diffuser was assumed to be constant, the following

Table 2. Effective Thermal Conductivity of the Graphite-MnCl2 Complex Used in the Heat Transfer Equation Fb (kg m-3)

fg (wt %)

states of metallic salt

ke (W m-1 K-1)

100

50

150

50

200

50

250

50

100

30

150

30

MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3

14.8 ( 0.15 13.9 ( 0.20 16.1 ( 0.40 15.5 ( 0.12 21.9 ( 0.25 20.8 ( 0.20 25.6 ( 0.31 24.7 ( 0.23 11.7 ( 0.21 10.9 ( 0.14 14.0 ( 0.21 12.8 ( 0.16

boundary and initial conditions were used:

∂P at r ) rw ∂r

(20a)

P ) Pc at r ) ri

(20b)

P ) Peq(Tc) at t ) 0

(20c)

The chemical kinetics in eq 9 were coupled through the source term of eq 14 with the equations of heat and mass transfer in eqs 17 and 19, respectively. The finite difference method was used as a tool for the numerical analysis. The equations of heat and mass transfer were discretized according to an implicit scheme. The overall numerical code is composed of four subroutines that run independently or in combination with other subroutines. The simulation was carried out with a relative numerical error of less than 10-3, which was checked by the other standardized module. The space was divided into 10 lattices with the time interval of 1.0 s to observe the overall behavior of the reactor, which was sufficient for the purposes of this work. Results and Discussions Thermophysical Properties. Table 2 shows the effective thermal conductivity of the graphite-MnCl2 complex. The correlation coefficients for temperature change versus f(w) in eq 3a were over 0.9995.38 As shown in Table 2, the thermal conductivity is in the range of 14.0-25.6 W m-1 K-1, which is much higher than the value for powder beds of 0.1-0.5 W m-1 K-1. Thus, we can expect that the limitation of heat transfer in the powder bed can be remarkably diminished using the graphite-MnCl2 complex in the fixed-bed reactor. A significant enhancement in heat transfer results in a much shorter cycle time, which is desirable in refrigeration and the air conditioning operations. We should note that the change in thermal conductivity is small for the graphite-MnCl2‚nNH3 (n ) 6, 2) complexes as the value of n varies. However, for a certain series of reactive complexes, i.e., the graphite-CaCl2‚nNH3 (n ) 8, 4, 2) complexes, the change in thermal conductivity is fairly large as the value of n varies,38 so that the assumption of constant thermal conductivity would lead to erroneous results in the simulation. Because the mean pore radius of the graphitemetallic salt complex is mostly on the micrometer scale,45 Darcy’s law can be applied to the reactive complex when the gas flow rate is maintained low enough that inertial effects during experiments can be


Figure 5. Influence of the bulk densities of the graphite-MnCl2 complex on the global conversion: (a) Fb ) 100 kg m-3, (b) Fb ) 150 kg m-3, (c) Fb ) 200 kg m-3, and (d) Fb ) 250 kg m-3. Table 3. Gas Permeability of the Graphite-MnCl2 Complex Used in the Mass-Transfer Equation Fb (kg m-3)

fg (wt %)

states of metallic salt

K (m2)

100

50

150

50

200

50

250

50

100

30

150

30

MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3 MnCl2‚6NH3 MnCl2‚2NH3

2.5 × 10-13 8.6 × 10-13 4.0 × 10-14 1.2 × 10-13 1.3 × 10-14 3.0 × 10-14 8.1 × 10-15 1.8 × 10-14 8.0 × 10-14 2.5 × 10-13 1.5 × 10-14 4.3 × 10-14

neglected. For a granular or unconsolidated bed, the limitation by mass transfer has not been considered as the gas permeability is as high as 10-9-10-11 m2. It was even assumed to be unlimited mass transfer inside the reaction media for a consolidated bed.20,42 However, Lu et al. reported that the gas permeability of the reactive complex on the macroscopic scale might have a significant influence on the global kinetics when the operating pressure is less than 5.0 bar.15 Table 3 shows the gas permeability of the graphite-MnCl2 complex. The gas permeability is in the range of 10-12-(8.0 × 10-15) m2 depending on the bulk density, weight fraction of graphite, and ammoniated state of MnCl2. We can observe that the gas permeability becomes small as the solid-gas reaction progresses as the effective open porosity decreases because of the change in molar volume of MnCl2‚nNH3 from 82.3 × 10-6 m3 mol-1 (n ) 2) to 162.3 × 10-6 m3 mol-1 (n ) 6). Global Conversion. The parameters that can influence global conversion are the effective thermal conductivity ke, the gas permeability K, the heat transfer coefficient at the wall hw, and the operating conditions, i.e., pressure and temperature. Because the parameters such as ke, K, and hw are known for an envisaged graphite-MnCl2 complex, the influence of the operating conditions on the global conversion is studied using simulation. The operating conditions are closely related to the local reaction rate and the mass transfer. At first, we investigated the transformation of the reactor in the case of fg ) 50%. Figure 5 shows the influence of the bulk densities of the graphite-MnCl2 complex on the global conversion. The global conversion is a very sensitive function of the bulk density of the reactive

Figure 6. Influence of the operating pressure on the global conversion under the ∆Teq,max ) 36 K: (a) Pc ) 1.0 bar, (b) Pc ) 2.0 bar, (c) Pc ) 3.5 bar, and (d) Pc ) 5.0 bar.

complex at 1.0 bar, which is caused by the mass-transfer limitation. As the operating pressure increases, however, the global conversion rapidly increases and become almost the same irrespective of the bulk density at 5 bar. The high operating pressure increases the gas diffusivity, Dp () KPc/µ) through the porous reactive complex. Thus, the reduction in mass-transfer resistance and the increase in reaction rate rapidly increase the global conversion, and it becomes almost the same for all reactive complexes at Pc ) 5 bar. However, the increase in Pc also increases the equilibrium temperature drop, ∆Teq [) |Tc - Teq(Pc)|]. The maximum equi-

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Figure 7. Local conversions with bulk densities of the graphite-MnCl2 complex when Pc ) 1.0 bar and Tc ) 353 K: (a) Fb ) 100 kg m-3, (b) Fb ) 150 kg m-3, (c) Fb ) 200 kg m-3, and (d) Fb ) 250 kg m-3. s, X ) 0.1; - -, X ) 0.3; ‚‚‚‚, X ) 0.5; -‚-‚-, X ) 0.7; and -‚‚-‚‚, X ) 0.9.

librium temperature drop was 5.3 K at Pc ) 1.0 bar, whereas it was 45.6 K at Pc ) 5.0 bar. In the complete cycle of the sorption heat pump, the increase of ∆Teq,max with Pc means that the temperature of cold production increases and the theoretical coefficient of performance (COPt) decreases,40 because of the increases in the evaporation temperature and sensible heat of reaction complex, respectively. Thus, an increase in Pc is not always favorable in the practical operation of a sorption machine. Figure 6 shows the influence of the operating pressure on the global conversion with the adjustment of ∆Teq,max to 36.0 K for all cases through variations in the operating temperature, Tc. The same ∆Teq,max value means a similar driving force to bring about the chemical reaction, so that we can observe the effect of operation pressure on the global conversion independently. The global conversion increases rapidly with reaction time in case of Fb ) 150 kg m-3. It also increases with Pc even though the effect of the operating pressure becomes negligible at high Pc. On the other hand, there is a distinctive difference in global conversion in case of Fb ) 250 kg m-3 depending on Pc, as shown in Figure 6. Considering that the effective thermal conductivity, gas permeability, and chemical kinetics are the same in this case, it comes from the difference in mass diffusivity. It is very important to elevate the reaction rate at low operating pressure as the sorption machines frequently operate at the pressure less than 1.0 bar in low-temperature applications such as refrigeration and air conditioning.

Local Conversion. Figure 7 shows the profiles of local conversions when Pc ) 1.0 bar. Lu et al. reported that the profile for local conversion could show the existence of reactive fronts in the progress of the reaction.15 If there is no mass-transfer limitation in the reactor, a heat front progresses from the external heat exchanger to the gas diffuser. However, if there is masstransfer limitation, a mass front moves in the opposite direction from the gas diffuser to the heat exchanger. As shown in Figure 7, we can observe that a mass front is gradually developed as the bulk density increases. This implies that the limitation by mass transfer exists in the reactive complex with high bulk density, i.e., >200 kg m-3, which corresponds to the above statements with regard to global conversion. In the case of Fb ) 150 kg m-3, the transformation of the reactor can be controlled by two fronts, which are the mass front from the gas diffuser and the heat front from the heat exchanger. From the profile of local conversion, we can observe that there is a retarding zone of reaction, which is a local zone of low reaction rate, depending on the heatand mass-transfer characteristics of the reactive complex. The minimum point in this zone is called the retarding point. The position of the retarding point also allows for a determination of which phenomenon is the limiting one between heat and mass transfer. A retarding point is close to the gas diffuser in the case of Fb ) 100 kg m-3, indicating that heat transfer is a limiting factor, but it appears not to be very large because of the smoothness of the retarding zone. It also implies


Figure 8. Local temperatures with bulk densities of the graphite-MnCl2 complex when Pc ) 1.0 bar and Tc ) 353 K: (a) Fb ) 100 kg m-3, (b) Fb ) 150 kg m-3, (c) Fb ) 200 kg m-3, and (d) Fb ) 250 kg m-3. s, X ) 0.1; - -, X ) 0.3; ‚‚‚‚, X ) 0.5; -‚-‚-, X ) 0.7; and -‚‚-‚‚, X ) 0.9.

that heat transfer does not limit the transformation of reactor even though the reactive complex is of low density. This is logical because the thermal conductivity of the reactive complex is larger than 14.0 W m-1 K-1 in comparison with unconsolidated beds for which the thermal conductivity is 0.1-0.5 W m-1 K-1.13,14 It is also confirmed from Figure 8 that there is no significant temperature gradient inside the reaction complex. In the case of Fb ) 150 kg m-3, a retarding point exists in the middle of the gas diffuser and the heat exchanger, indicating that heat and mass transfer are equally important. In the cases of Fb ) 200 and 250 kg m-3, the retarding point is closer to the heat exchanger. It can be explained that MnCl2-NH3 reaction occurs more slowly near the heat exchanger because of hindered mass transfer through the reactive complex. As Pc increases to 5.0 bar, the retarding zone becomes wider, and the retarding point moves closer to the gas diffuser, although these results are not shown here. Local Temperature and Pressure. The profiles of local conversion, temperature, and pressure inside the reactor are closely associated with one another. The measurement of local conversion inside the reactor is practically very difficult. Through the measurement of local temperature and pressure, however, we can investigate the evolution of the local conversion. Figures 8 and 9 show the profiles of local temperature and pressure, respectively. From these figures, we can observe that the values of Tlocal and Plocal become reduced as the bulk density increases in the process of the solid-

gas reaction, which makes a retarding reaction zone form inside the reactor. The reaction rate mainly depends on two factors: one is (1 - xlocal) and the other is |Tlocal - Teq(Plocal)|. In the short initial stage of the solid-gas reaction, the operating conditions of Tc and Pc are imposed upon the thermochemical reactor. Under these conditions, if there are no transfer limitations within the reactor, the reaction rate is maximal state as the values of (1 - xlocal) and |Tc - Teq(Pc)| are maximum. Actually, the temperature of the reactor increases instantaneously toward only the maximum local temperature that is adapted by the coupling of the chemical kinetics and the heat and mass transfer for the reactive complex, as shown in Figure 8. Because the reactor is cooled by an external heat exchanger, the reactor approaches the constraint temperature in the long run, at which point the reaction rate becomes almost zero. A slow decline in the reaction rate is also observed in the slope of global conversion in Figure 5. In the case of Fb ) 100 kg m-3 of Figure 8a, the local temperature increases to near Teq(Pc) and then decrease gradually toward Tc as the reaction progresses, which implies that there are essentially no mass transfer limitations. However, as the bulk density increases, the local temperature increases only slightly, as shown in Figure 8d. As a result, the available thermodynamic equilibrium drop is just Teq(Plocal) - Tc, which is too small to bring about the chemical reaction and results in a much longer cycle time for the sorption machine. As Pc increases to 5.0 bar, the local temperature also

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Figure 9. Local pressures with bulk densities of the graphite-MnCl2 complex when Pc ) 1.0 bar and Tc ) 353 K: (a) Fb ) 100 kg m-3, (b) Fb ) 150 kg m-3, (c) Fb ) 200 kg m-3, and (d) Fb ) 250 kg m-3. s, X ) 0.1; - -, X ) 0.3; ‚‚‚‚, X ) 0.5; -‚-‚-, X ) 0.7; and -‚‚-‚‚, X ) 0.9.

increases to near Teq(Pc) because of better mass transfer and more rapid kinetics. Figure 9 shows the profiles of local pressure when Pc ) 1.0 bar. Before the solid-gas reaction occurs, the initial pressure in the reactor is the equilibrium pressure that can be determined by the constraint temperature. The pressure then drops to the local values because of the hindrance of mass transfer by the decrease in gas permeability during the absorption reaction and then approaches Pc in the end. When the bulk density is low, the pressure gradient is small, as shown in Figure 9a. However, as the bulk density increases, its gradient becomes larger, causing the driving force for chemical reaction to reduce significantly. Weight Fraction of Graphite. The thermophysical properties of the graphite-MnCl2 complex vary noticeably depending on the weight fraction of graphite. These effects have been reported by Han et al. for the reactive complexes of graphite-MnCl2‚nNH3 (n ) 6, 2), CaCl2‚ nNH3 (n ) 8, 4, 2), and BaCl2‚8NH3.38 As shown in Tables 2 and 3, the thermal conductivity and the gas permeability significantly decrease as the weight fraction of graphite decreases. Figure 10 shows the change in global conversion with the weight fraction of graphite when Pc ) 1.0 bar. The decrease in weight fraction of graphite leads to the reduction of global conversion because of poor heat- and mass-transfer characteristics. Furthermore, its degree of reduction becomes more significant as the bulk density increases. For the case of Fb ) 150 kg m-3 and fg ) 30%, ∆t0.5, which is the

total reaction time for the global conversion to reach 0.5, is over 15 h. This result indicates that this specific reactive complex cannot function correctly and efficiently and is thus unlikely to be used in practical applications. Figure 11 shows the change in global conversion with the weight fraction of graphite when Pc ) 5.0 bar. The global conversion increases in comparison with when Pc ) 1.0 bar because of the better mass diffusivity and chemical kinetics, as can be expected. The decrease in the weight fraction of graphite means an increase in the cooling or heating capacity per unit volume because of the relative increase in reactive solid. However, in term of power density, it depends on both the rate of global conversion and the number of moles of reactive solid, which can be represented by the following equation:

P hv )

( )(

)

∆X v∆HNsalt ∆t Mw

(21)

Thus, in the practical design of a sorption machine, the cooling capacity and powder density should be simultaneously considered to fit the design conditions such as cold production temperature, cycle time for the cooling load, and COP. Conclusion A transformation analysis of a thermochemical reactor was carried out using the graphite-MnCl2 complex/


Figure 10. Change of the global conversion with the weight fraction of graphite when Pc ) 1.0 bar and Tc ) 353 K: (a) Fb ) 100 kg m-3 and (b) Fb ) 150 kg m-3.

NH3 pair through numerical simulation with various operating pressures and temperatures. The absence of a distinctive heat front in the reactor shows that heat transfer is not a limiting factor with the graphiteMnCl2 complex. In real operations of a sorption machine, the suitable operating pressure is in the range of 0.5-2.0 bar for cold production below 10 °C if we assume that the heat sink temperature is 35 °C. Therefore, the graphite-MnCl2 complex with a low bulk density can be used for lowtemperature applications such as refrigeration and air conditioning. However, a sorption machine may not function correctly or may result in a much longer cycle time at low pressure with the graphite-MnCl2 complex over F ) 200 kg m-3 and the fg ) 50% as mass transfer is significantly limited. In this case, applications in a thermal storage should be considered as such applications do not require short cycle times. It is important to keep the pressure gradient inside the reactive complex as small as possible at operating pressures less than 1.0 bar. This condition can be achieved by changing the geometry of the reactive complex or the reactor configuration. The thickness of the reactive complex should also be optimized considering the thermal mass of heat exchanger so as to decrease the mass-transfer limitation by reducing the diffusion path. A parallel-plate reactor or a shell-and-tube heat exchanger with the reactive complex in the shell side might be useful for this purpose.

Figure 11. Change of the global conversion with the weight fraction of graphite when Pc ) 5.0 bar and Tc ) 353 K: (a) Fb ) 100 kg m-3 and (b) Fb ) 150 kg m-3.

Acknowledgment The authors are pleased to acknowledge the financial assistance of the Korea Science and Engineering Foundation (Project No. 95-0502-04-01-3) and the Pohang Iron and Steel Co., Ltd. (Project No. 1-UD-97012-01) to this work. Nomenclature A ) area of specimen (m2) Cp,NH3 ) heat capacity or NH3 (J kg-1 K-1) Cp ) heat capacity or specific heat (J kg-1 K-1) Cp,C ) specific heat of graphite (J kg-1 K-1) Cp,2 ) specific heat of MnCl2‚2NH3 (J kg-1 K-1) Cp,6 ) specific heat of MnCl2‚6NH3 (J kg-1 K-1) Dp ) mass diffusivity of gas (m2 s-1) Ea ) activation energy (J mol-1) Fo ) constant heat flux by a heat source (W m-2) fg ) weight fraction of graphite G ) reactive gas or working fluid hsw ) heat transfer coefficient at the wall (W m-2 K-1) K ) coefficient of permeability or gas permeability (m2) ke ) effective thermal conductivity (W m-1 K-1) L ) length of specimen (m) MS ) reactive solid Mw ) molecular weight of reactive solid MW2 ) molecular weight of MnCl2‚2NH3 mg ) mass of graphite (kg) ms ) mass of reactive solid (kg) Nsalt ) number of moles per unit volume of reactive solid (mol m-3) n ) pseudo-order of solid-gas reaction

4138


P ) pressure (bar) P1 ) upstream pressure (Pa) P2 ) downstream pressure (Pa) Pc ) constraint pressure (bar) Peq ) thermodynamic equilibrium pressure (Pa) P h v ) average power per unit volume or power density (W m-3) Q ) gas flow rate (m3 s-1) R ) gas constant (J K-1 mol-1) ri ) position at inner radius for reactive complex (m) rw ) position at inner radius for reactive complex (m) So ) source term, molar flow rate generated by chemical reaction (mol s-1 m-3) s ) preexponential factor (s-1) T ) temperature of specimen (°C) t ) time (s or h) Tc ) constraint temperature (°C) To ) initial temperature of specimen (°C) Vb ) volume of reactive complex (m3) Vc ) volume occupied by graphite (m3) Vs ) volume occupied by reactive solid (m3) v2m ) molar volume of MnCl2‚2NH3 (m3 mol-1) v6m ) molar volume of MnCl2‚6NH3 (m3 mol-1) w ) dimensionless number X ) global conversion of solid-gas reaction x ) local conversion of solid-gas reaction Greek Letters Re ) thermal diffusivity of specimen (m2 s-1) ) total porosity of reactive complex Fb ) bulk density of reactive complex (kg m-3) Θ ) characteristic time (s) µ ) viscosity of gas (Pa s) γ ) stoichiometric ratio of the chemical reaction ∆Hrxn ) reaction enthalpy (J mol-1) ∆P ) pressure gradient (Pa) ∆Srxn) reaction entropy (J K-1 mol-1) ∆Tc ) finite temperature drop by incomplete contact (°C)

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Received for review June 17, 1999 Revised manuscript received May 15, 2000 Accepted August 10, 2000 IE9904394

Transformation Analysis of Thermochemical Reactor Based on

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