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Heat and mass transfer in a concentric-annular-tube bed packed with ZrV1.9Fe0.1 particles
Journal of Alloys and Compounds 375 (2004) 305–312
Heat and mass transfer in a concentric-annular-tube bed packed with ZrV1.9Fe0.1 particles S. Fukada∗ , S. Morimitsu, N. Shimoozaki Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan Received 6 October 2003; received in revised form 8 December 2003; accepted 8 December 2003
Abstract Pressure–composition isotherms of the ZrV1.9 Fe0.1 –hydrogen system were determined by a constant volume method and were correlated to an empirical van’t Hoff equation. The enthalpy change of hydrogen absorption was correlated to a function of the hydrogen-to-alloy atomic molar ratio. The entropy change was independent of it. Heat and mass transfer process in a concentric-annular-tube bed packed with the ZrV1.9 Fe0.1 particles was studied experimentally and numerically as a basic study of hydrogen storage at elevated temperature. A one-dimensional concentric-annular-tube bed was charged with hydrogen. The experimental profiles of hydrogen partial pressure and temperature were compared with numerical simulation calculated by the pressure–composition isotherm, a hydrogenating rate equation and two balance equations of hydrogen and heat. Another simplified analysis using three dimensionless lump parameters succeeded in estimating temperature profiles in the hydrogen-absorbing-alloy bed more easily. The simulation was able to make it better to understand contributions of experimental parameters such as the flow rate, surrounding temperature and other physical properties. © 2004 Published by Elsevier B.V. Keywords: Hydrogen storage materials; Gas–solid reactions; Heat conduction; Thermal analysis
1. Introduction Hydrogen-absorbing alloys have been studied for various purposes such as hydrogen storage [1], chemical heat pumps [2], refrigerators [3], purification of hydrogen or inert gas [4,5], hydrogen isotope separation [6–8] and so on. Heat and mass transfer in particle beds of hydrogen-absorbing alloys such as LaNi5 was well studied experimentally and analytically [9–12]. One-dimensional [9], two-dimensional [10,11] and even three-dimensional analysis [12] were carried out. Quantitative agreement using LaNi5 that has high hydrogen absorption capacity around room temperature was obtained between the numerical results and measured temperatures and amounts of absorbed hydrogen. However, since there are still many parameters to be determined from fitting curves to experimental data, we need further experimental and numerical efforts to elucidate heat and mass transfer processes in hydrogen-absorbing beds.
∗ Corresponding author. Tel.: +81-92-642-4140; fax: +81-92-642-3800. E-mail address: [email protected] (S. Fukada).
0925-8388/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.jallcom.2003.12.008
Hydrogen is produced at higher temperature in some industrial processes. When hydrogen is produced by steam-reforming of natural gas around 600–800 ◦ C, it seems to be more advantageous to store high-temperature hydrogen tentatively on its way to the subsequent line. Unfortunately, less studies were carried out on hydrogen absorption under such a high-temperature condition previously except for our experiment using a long-tube bed of a Zr–V–Fe alloy [13]. In the present study, we selected ZrV1.9 Fe0.1 alloy as a hydrogen-absorbing material holding sufficient hydrogen absorption capacity even at elevated temperature. However, there are no available data on pressure–composition isotherm. As the first part of our experiment, therefore, pressure–composition isotherms of the ZrV1.9 Fe0.1 – hydrogen system were determined and were correlated to an empirical van’t Hoff equation. In the next place, profiles of hydrogen partial pressure and temperature in a ZrV1.9 Fe0.1 bed were experimentally determined using a concentric-annular-tube vessel packed with the alloy particles. Hydrogen diffused there approximately in the onedimensional radial direction. As the last thing, a simplified analysis was applied to the heat and mass transfer process in the one-dimensional hydrogen-absorbing bed. Such an
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Nomenclature cp,g cS D E
heat capacity of gas (J/kg K) heat capacity of alloy particle (J/kg K) diffusion coefficient (m2 /s) activation energy of reaction rate constant (J/mol) hT heat-transfer coefficient between particles and wall (W/m2 K) H enthalpy change of hydrogen absorption (J/mol) k absorption rate constant (mol/m3 Pa0.5 s) kL first-order reaction-rate constant (s−1 ) Ke Darcy’s permeability (m2 ) p hydrogen pressure (Pa) peq equilibrium pressure (Pa) p0 atmospheric pressure, 1.0133 × 105 (Pa) q hydrogen absorption amount (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) qm averaged hydrogen absorption amount (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) saturated value of hydrogen absorption amount q0 (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) r radial distance from the center (m) rin inner radius of concentric-annular-tube bed (m) rout outer radius of concentric-annular-tube bed (m) rp particle radius (m) Rg gas law constant (J/mol K) S entropy change of hydrogen absorption (J/mol K) t time (s) T temperature (K) Tm temperature averaged over the whole bed (K) surrounding temperature (K) T0 u gaseous velocity (m/s) V total volume of packed bed (m3 ) W volumetric hydrogen flow rate (m3 /s) x dimensionless absorption amount defined as x = qm /q0 z longitudinal distance from the top of bed (m) Greek letters α dimensionless parameter defined as α = (−∆H)γq0 /(1 − ε)ρS cS T0 β dimensionless parameter defined as β = hT A/kL,0 V(1 − ε)ρS cS δ dimensionless parameter defined as δ = E/Rg T0 γ packed density of alloy (mol/m3 ) ε void ratio ρg gaseous hydrogen density (kg/m3 ) ρS solid density (kg/m3 ) λe effective thermal conductivity in bed (W/mK) µ gaseous viscosity (Pas) θ dimensionless temperature defined as θ = (Tm − T0 )/T0
analytical solution will make it clear how various physical parameters contributed to the profiles of temperature and hydrogen pressure in the bed. The analytical results were compared with experimental data. Hydrogen was supplied from the top of cylindrical beds of hydrogen-absorbing alloys in many previous experiments [9–13]. Then, hydrogen diffused non-uniformly in either radial or longitudinal direction [10,11]. Therefore, the profiles of hydrogen absorption amount and temperature in the bed distributed differently in the two directions. Such non-uniformity seemed to deteriorate hydrogen-absorption performance. In the present system, hydrogen-absorbing-alloy particles were packed between two concentric-annular tubes. Hydrogen was supplied through an inner cylindrical tube walls with many pores. The hydrogen concentration was almost uniform in the longitudinal direction. Consequently, the hydrogen reaction zone and the temperature profile proceeded across the concentric-annular-tube bed with a uniform pattern as shown later. It will be proved in the present study that such change in the configuration makes the heat and mass transfer process simpler and more effective.
2. Pressure–composition isotherm of ZrV1.9 Fe0.1 –H2 system A Zr–V–Fe alloy was manufactured from a mixture of particles of each element metal (Zr, V, and Fe) by an argon-arc-melting method. Particles of Zr, V and Fe adjusted to be ZrV1.9 Fe0.1 were melted several times. The alloy was crashed and was screened between 12 and 32 mesh (0.495–1.397 mm) in an argon–atmosphere glove box. Uniformity of the alloy particles was checked by a X-ray diffraction (XRD) method. The XRD pattern was presented previously [13]. The hydrogen absorption capacity was determined by a constant volume method. A hydrogen storage vessel and a reaction vessel were made of 316 stainless-steel. The alloy of 1.00 g was put in the reaction vessel. Pressure in the storage vessel before and after hydrogen absorption was measured using two MKS Balatron absolute-pressure gages with the accuracy of 0.5%. Before the measurement, the alloy was activated at 800 ◦ C under vacuum of around 10−3 Pa for 1 h. The pressure–composition isotherm of the ZrV1.9 Fe0.1 – hydrogen system is shown in Fig. 1. There was no plateau region in the ZrV1.9 Fe0.1 alloy. The no-plateau tendency was consistent with that of ZrV2 [14]. Displacement of a part of V with Fe was found to enhance hydrogen absorption rates without loosing hydrogen absorption capacity [15]. All of the pressure–composition isotherms were correlated to the following empirical van’t Hoff equation: peq Rg T ln = H(q) − TS(q) (1) p0
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alloy. This was because there were many kinds of tetrahedral sites in the cubic C15 Laves phase.
105
peq [Pa]
104
3. Heat and mass transfer experiment using concentric-annular-tube bed
103 Key Temp. 302K 371K 423K 530K 631K 734K
102 101
Estimation
100 0.0
0.5
1.0
1.5
2.0
2.5
q [H/ZrV1.9Fe0.1] Fig. 1. Pressure–composition isotherm of the ZrV1.9 Fe0.1 –hydrogen system.
Fig. 3 shows a schematic illustration of the concentricannular-tube bed for hydrogen absorption of the present experiment. Fig. 4 shows a schematic diagram of the experimental system. The particle size of the ZrV1.9 Fe0.1 alloy ranged from 12 to 32 mesh (0.495–1.397 mm). The packed amount was 5015.12 g. The bed volume was 1137 cm3 . The inlet hydrogen flow rate was in the range from 1.5 to 10.0 l(NTP)/min. Hydrogen was purified by a liquid-nitrogen cold trap. The inlet gas temperature ranged
The pressure–composition isotherm shown in Fig. 1 was compared with Eq. (1). So that, the S values were found to be constant independent of not only temperature, T, but also the hydrogen-to-alloy atomic molar ratio, q. They were correlated to the following simple relation: S = −11.9Rg
(2)
The independence of S from q and T means that hydrogen atoms dissolved in the solid phase lost almost all degrees of freedom that had in the gaseous phase. Fig. 2 shows variations of the enthalpy change of hydrogen absorption as a function of the atomic molar ratio of H/ZrV1.9 Fe0.1 . Thus, H increased with q independent of T. The H values were correlated to the following polynomial equation: H = −122.7 + 45.96q + 10.28q2 − 5.708q3
(3)
The solid line in the figure was one calculated from Eq. (3). The variations of H with q were considered to come from different energy levels of hydrogen atoms dissolved in the
RgTln(peq/p0)+T∆S [kJ/mol-H]
0 -20
∆S/Rg=-11.93 (2) ∆H(q)=-122.7+45.96q+10.28q2-5.708q3
Fig. 3. Concentric-annular-tube ZrV1.9 Fe0.1 bed.
(3)
-40 -60
Key Temp. 302K 371K 423K 530K 631K 734K Eq.(3)
-80 -100 -120 0.0
0.5
1.0
1.5
2.0
q [H/ZrV1.9Fe0.1] Fig. 2. Enthalpy changes of hydrogen absorption.
Exhaust H2 gas cylinder Cold trap Mass-flow controller Pressure sensor Absorbing reactor
Internal heater Thermocouples Diffusion pump Rotary pump
Fig. 4. A schematic diagram of the experimental apparatus.
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cp,g ∂ ∂ ∂T (ρg T) + (1 − ε)cS ρS + (ruρg T) ∂t ∂t r ∂r ∂T 1 ∂ ∂qm rλ = + γ(−H) r ∂r ∂r ∂t
from 294 to 835 K. Seven Chromel-Almel thermocouples were inserted in the concentric-annular-tube bed. They were located at different positions of r = 28, 38, 48, 58 and 70 mm in the radial direction at the same height of z = 100 mm and at z = 50, 100 and 150 mm in the longitudinal direction at the same radial position of r = 38 mm. Before the absorption experiment, the bed was activated by heating at 800 ◦ C under vacuum lower than 10−3 Pa for 5 h. Then electricity of the internal heater was cut off, and the bed was cooled down slowly. When temperature reached a desired one, a constant flow rate of purified hydrogen was introduced into the bed through an inner cylindrical porous tube. Electricity of the internal heater was not supplied during the hydrogen supply.
εcp,g
4. Analysis
γ
4.1. Numerical analysis using partial differential equations Three equations for a hydrogen-absorbing rate, a hydrogen material balance and a heat balance as well as the pressure–composition isotherm of Eqs. (1)–(3) are simultaneously solved numerically. We set the following assumptions and conditions for the three equations: (i) temperature of the gaseous phase in the bed is the same as that in the particle phase; (ii) there is no hydrogen absorbed in the bed at the initial stage. The bed temperature at start is equal to the surrounding temperature, T0 ; (iii) the particle bed is a concentric-annular-tube type. Hydrogen gas is supplied through the inner porous tube under a constant flow rate and a constant temperature of T0 until the upstream pressure reached a specified one. After the upstream pressure has reached it, the pressure is maintained there and the flow rate starts decreasing quickly; (iv) hydrogen diffuses in the radial direction. The hydrogen flow velocity inside the bed is described by the Darcy’s law; (v) part of heat generated in the bed is consumed to elevate temperature of alloy particles and gas. The rest is transferred in the radial direction and is released outside through the outer tube; (vi) thermal conduction in the longitudinal direction of the bed is neglected. So, the system is of one-dimensional; (vii) hydrogen permeation through the outer tube is negligibly small. Under the above assumptions and conditions, the equations of material balance and heat balance were described as follows: ∂ρg MH2 γ ∂qm 1 ∂ ε (ruρg ) = − (4) + r ∂r 2 ∂t ∂t
ρg =
MH2 p Rg T
(5) (6)
Here, the gaseous velocity in the radial direction, u, was related with the pressure difference by the Darcy’s law in the same way to the previous study [9] as follows: u=−
Ke ∂p µ ∂r
(7)
Hydrogen absorption rate was described in a similar way to the previous work [10] as follows: p0.5 − p0.5 ∂qm eq =k ∂t (1 − (qm /q0 ))−1/3 − 1
(8)
Here, p and peq are bulk hydrogen pressure and pressure in equilibrium with qm , respectively. The latter was calculated from Eqs. (1)–(3) as a function of qm . The initial and boundary conditions were as follows: t = 0, r = rin , r = rout ,
T = T0 , T = T0 , −λ
qm = 0,
p=0 Ke ∂p u0 = − µ ∂r r=rin
∂T = hT (T − T0 ), ∂r
∂p = 0. ∂r
(9) (10) (11)
Eqs. (4)–(8) under the conditions of Eqs. (9)–(11) with the equilibrium isotherm of Eqs. (1)–(3) were solved numerically by the finite difference method. 4.2. Simplified analysis The above simulation was expected to give detailed profiles of temperature and hydrogen absorption amount as a function of time and the radial distance in a similar way to LaNi5 studied previously [9]. It will be shown later that the simulation is valid also in the ZrV1.9 Fe0.1 bed. However, Eqs. (4)–(8) included many parameters to be determined from comparing the analysis with experimental data. Part of these parameters were unique to the ZrV1.9 Fe0.1 bed. There are no reliable data for the thermal conductivity in the particle bed and the hydrogenating rate constant of ZrV1.9 Fe0.1 beforehand. Therefore, there were many difficulties in comparing the numerical simulation with experimental profiles. On the other hand, simplified analysis based on reasonable consideration may become more useful if obtained. This was because several parameters appearing in the original equations were lumped. Therefore, the number of parameters to be determined decreased. In order to simplify the governing equations and so to obtain an ordinary differential equation that can be easily understood, we set the following additional three assumptions.
S. Fukada et al. / Journal of Alloys and Compounds 375 (2004) 305–312
(viii) We focus on temperature averaged over the bed instead of local temperature. Then, Eq. (5) was integrated over the whole bed. The manipulation resulted in the following macroscopic heat balance equation: ∂Tm ∂Tm + (1 − ε)ρS cS V + ρg cp,g WT0 ∂t ∂t ∂qm = γ(−H)V − hT A(Tm − T0 ) (12) ∂t
ερg cp,g V
(ix) The hydrogenating equation of Eq. (7) is replaced by a rate equation using linear-driving-force approximation. The rate equation was simplified to the following one by the linear-driving-force approximation: ∂qm E (13) = kL,0 exp − (q∗ − qm ), ∂t Rg T The linear-driving-force approximation was believed to be valid under usual conditions of adsorption or hydrogenation [16]. The rate constant kL,0 exp(−E/Rg T) in Eq. (13) is related with the hydrogen diffusivity in solid by 15D/rp 2 . A comparison was given to prove it in the previous reference [17]. (x) Thermal capacity of the solid phase is much greater than that of the gaseous phase. With the assumption (x), the first and third terms on the left-hand side of Eq. (12) were neglected, because of less contribution to the whole heat balance compared to those of the other terms. With the use of the dimensionless independent parameters, x and θ, that are defined in the nomenclature, we could reduce Eq. (12) to the following dimensionless ordinary differential equation that was simplified more than the original partial differential equation. dθ βθ δ =α− exp (14) dx 1−x 1+θ
5. Results and discussion 5.1. Temperature curves Fig. 5 shows examples of variations of experimental temperature with time at different radial or longitudinal locations in the ZrV1.9 Fe0.1 bed. All temperature curves except for the wall one showed similar time variation regardless of the different thermocouple locations of No.(1)–(7). The curves were characterized by steep temperature rise and then gradual drop. The tendency was held also under different conditions of the H2 flow rate, W0 , and the surrounding temperature, T0 . Time when temperature started rising was the earliest at the inner location depicted by No.(1) in Fig. 5. That in the outer location depicted by No.(6) was the latest. The time was almost the same regardless of different heights at the same radial position depicted by Nos.(2), (4) and (5). All those things mean that a cylindrical reaction zone moved across the ZrV1.9 Fe0.1 bed uniformly in the longitudinal direction. The reaction zone moved very quickly with a uniform pattern. Temperature differences among the thermocouple positions, No.(1)–(7), were small compared with variations with time. Therefore, we focused on variations of temperature with time as shown below. Fig. 6 is examples of comparison of temperature variations with time between experiment and numerical simulation using Eqs. (4)–(11) for different T0 conditions. The position of thermocouples was r = 38 mm and z = 100 mm. The numerical simulation showed similar characteristics of steep temperature rise and gradual drop. Good agreement was obtained between the experiment and the numerical simulation. This was valid also under other W0 conditions. The value of the effective thermal conductivity in the Zr–V–Fe alloy bed used for the numerical simulation was λe = 0.14 W/mK, that of the Darcy’s permeability parameter Ke = 8.9 × 10−8 m2 , and that of the reaction rate constant kL = 1.0 × 800 (4)
The boundary conditions are as follows: at x = 0 and x = 1
(1)
700
Line z [mm] (1) 100 (2) 100 (3) 100 (4) 50 (5) 150 (6) 100
(2)
(5)
(15)
The three dimensionless lump parameters, α, β and δ, in Eq. (14) are defined in the nomenclature. Thus all experimental parameters were correlated to the three dimensionless lump parameters. Integration curves of Eq. (14) started from θ = 0 at x = 1 and ended at θ = 0 at x = 0. The dimensionless maximum temperature rise, θ max , could be expressed as follows. βθmax δ xmax = 1 − (16) exp α 1 + θmax The value of x is in proportion to time under the constant H2 flow condition.
Temperature [K]
θ=0
309
600
r [mm] 28 38 48 38 38 wall
(3) Tmax-T0
500
(6) 400
T0=298K W=10L/min
tmax 300
0
1000
2000
3000
4000
5000
6000
Time [s]
Fig. 5. Variations of experimental temperature in ZrV1.9 Fe0.1 bed with time.
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800
400 300
experiment simulation
Temperature [C]
600
T0=200C T0=30C
400
200 W0=5L(NTP)/min 0 0
2000
4000 6000 Time [s]
8000
0 [C]
100 β
10 Values of parameters α, β andδ [-]
T0=400C
200
10000
Flow rate 5 L/min 7.5L/min 10 L/min
5
α
β
δ
α
0
δ -5 1.5
2.0
2.5
3.0
3.5
-1
Fig. 6. Comparison of temperature profiles between experiment and analytical equations.
10−3 exp(−9.98/Rg T). This numerical simulation wasted a lot of time due to solve the nonlinear governing equations, especially Eqs. (4) and (5), simultaneously. In addition to that, the values of physical parameters used for the simulation were still uncertain. This was because there was possibility in that the parameters may depend on temperature and hydrogen pressure. Therefore, their dependence was investigated by the simplified analysis using lump parameters shown below. Average temperature was evaluated before comparison between experiment and the simplified analysis. So that, it was found that variations of the average temperature with time are almost coincident with those of the local temperature at the location of r = 38 mm and z = 100 mm (No. (2)). Therefore, the comparison was made between the simplified analysis and the local temperature at No. (2) for simplicity. Fig. 7 shows a comparison of the temperature profile
2.0 experiment calculation
T0=294K
W=7.5L(NTP)/min z=100mm, r=38mm
1.5
1000/T [K ]
Fig. 8. Values of parameters α, β, and δ.
between the experimental variation and the analytical equation of Eq. (14). As seen in the figure, close agreement was also obtained between them regardless of different T0 conditions. This situation was held also in curves of any other W0 conditions. The values of α, β, and δ were determined from fitting curve as a function of T0 and W0 . Fig. 8 shows the values of α, β, and δ determined. The values of α decreased with elevation of T0 regardless of different W0 values. Therefore, the value of (−H)γ/(1−ε)ρS cS except for q0 /T0 in α was considered constant independent of temperature and hydrogen pressure. On the other hand, the values of δ defined by E/Rg T0 increased with elevation of T0 . From the slope of the δ line, the activation energy of 10 kJ/mol was obtained. The value was coincident with that of the activation energy of the rate constant k used for the numerical simulation. The values of β defined by hT A/kL,0 V(1−ε)ρS cS were constant independent of T0 . Therefore, the values of hT A/kL,0 V(1−ε)ρS cS were considered constant independent of temperature and hydrogen pressure. Consequently, the value of the heat-transfer coefficient between the wall and the particle bed was hT = 20 W/m2 K, and the value was independent of T0 and W0 .
(T-T0)/T0 [-]
5.2. Hydrogen absorption T0=374K
1.0
0.5
T0=523K T0=621K
0.0 0.0
0.2
0.4
0.6
0.8
1.0
qm/q0 [-]
Fig. 7. Comparison of temperature profiles between experiment and simplified analysis.
Fig. 9 shows variations of the total amount of hydrogen absorbed in the ZrV1.9 Fe0.1 bed with time. The experimental variation with time was almost linear at the early stage of hydrogen absorption. The slope was in proportion to W0 . After certain time depending T0 and W0 , the curve started decreasing. Thus, the present hydrogen-absorbing alloy bed showed a good absorption performance. Comparatively, good agreement was obtained between the experiment and the numerical simulation using Eqs. (4)–(11). Profiles of hydrogen absorption amount in the bed were almost flat across the bed, although not shown in the figure. The absorption performance did not change regardless of repeated
S. Fukada et al. / Journal of Alloys and Compounds 375 (2004) 305–312
500
2.5 experiment simulation
30
1.5
20
1.0
Key
400
2.0 W0=5L(NTP)/min T0=300K
Tmax-T0 [K]
40
H/ZrV1.9Fe0.1 [-]
T0 303K 473K 680K 835K
300 200 100 r=48 mm z=100 mm
10
0.5
0 0
0 0
2000
4000 6000 Time [s]
8000
0.0 10000
Fig. 9. Changes of total amount of hydrogen absorbed in ZrV1.9 Fe0.1 bed.
absorption–desorption cycles until about 50 of the maximum experimental run number in a similar way to Pd [18]. 5.3. Maximum temperature rise Fig. 10 shows a correlation of the results of the experimental temperature-rising rate defined by (Tmax − T0 )/tmax . The rising rates were found to be in proportion to W0 , independent of T0 . Thus, the rising rate in the early absorption time was controlled only by W0 , consequently the hydrogen absorption rate. It was found that the temperature rising rate is in proportion to the hydrogen absorption rate, if we assume that the two contributions of the second term on the left-hand side of Eq. (5) and the second term on the right-hand side are dominant in the earlier time of hydrogen absorption. Since the hydrogen absorption rate was in proportion to W0 at the
2 4 6 8 10 12 H2 Flow rate, W0 [L(NTP)/min]
Fig. 11. Relation between maximum temperature rise and flow rate.
earlier time, the following equation would be obtained from Eq. (5). T ∼ (−H)W0 = t (1 − ε)cS ρS V
(17)
Here, V is a volume where temperature varied more heavily. The right-hand side of Eq. (17) was considered in proportion to W0 and was independent of T0 . This thing was coincident with the result of Fig. 10. Then, the contribution of thermal conduction in the radial direction in the temperature rising region was found to be smaller than that of the heat generation. The contribution was considered to increase with an increase in W0 . Fig. 11 shows a plot of the maximum temperature rise defined by Tmax − T0 as a function of W0 and T0 . As seen in the figure, the plot can be divided into two regions. At the lower W0 region, Tmax − T0 depended on W0 . Therefore, the values of Tmax − T0 were controlled by a balance between the hydrogen absorption rate, consequently heat generation rate, and the heat conduction. 500
Key Flow rate 1.5L/min 3.5L/min 5 L/min 7.5L/min 10 L/min q0 (from PCT curve)
25 Key T0 296K 473K 680K 835K
400
Tmax-T0 [-]
(Tmax-T0)/tmax [K/min]
20
15
14
300
2.5
2.0
1.5
200
1.0
100
0.5
10
q0 [H/ZrV1.9Fe0.1]
Total amount of hydrogen absorbed [mol]
50
311
5 r=48 mm z=100 mm
0 300
0 0
2
4 6 8 H2 flow rate [L(NTP)/min]
10
12
Fig. 10. Experimental values of temperature rising rates.
400
500
600
700
800
0.0 900
T0 [K]
Fig. 12. Relation between maximum temperature rise and surrounding temperature.
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On the other hand, Tmax −T0 was independent of W0 in the higher W0 region. In order to make it clear what parameter controlled Tmax − T0 in the higher W0 region, another plot was tested. Fig. 12 shows experimental values of Tmax − T0 as a function of T0 and W0 . As seen in the figure, Tmax − T0 at higher W0 values than W0 = 5 l(NTP)/min showed almost the same temperature dependence with that of the hydrogen absorption capacity, q0 . The temperature profile could be correlated by Eq. (16) successfully. Judging from xmin almost independent of T0 , the dependence of Tmax − T0 on T0 should be coincident with the dependence of q0 . In other words, Tmax − T0 in this system was limited by q0 . All those things could be explained by the simplified analysis of Eq. (16). 6. Conclusions Pressure–composition isotherms of the ZrV1.9 Fe0.1 –H2 system were correlated to an empirical van’t Hoff equation. The entropy change was independent of the hydrogen absorption amount and temperature. The enthalpy change was a unique function of the hydrogen absorption amount independent of temperature. Experimental profiles of temperature and hydrogen absorption in the concentric-annular-tube bed packed with the ZrV1.9 Fe0.1 particles were analyzed with two types of simulation. The concentric-annular-tube bed was found to be efficient for hydrogen absorption. Hydrogen reaction zone proceeded uniformly in the longitudinal direction. The microscopic balance equations of Eqs. (4)–(8) could simulate local temperature profiles in the ZrV1.9 Fe0.1 bed accurately. We determined experimental parameters of λe = 0.14 W/mK, Ke = 8.9 × 10−8 m2 , kL = 1.0 × 10−3 exp(−9.98/Rg T), and hT = 20 W/m2 K for the ZrV1.9 Fe0.1 bed. These parameters were found to be independent of the surrounding temperature and hydrogen flow rate. Another analysis using three lump parameters of
α, β, and δ was found to be efficient to observe how all physical parameters affected the hydrogen absorption and the temperature distribution in the bed. The behavior of the experimental absorption and temperature variations could be understood by the analytical simulation and numerical simulation.
References [1] Y. Osumi, Hydrogen-absorbing-alloys, their properties and applications, Agune Inc., Tokyo, 1993 (in Japanese). [2] M. Balakumar, S.S. Murthy, K.M. Murthy, C. Sastri, Heat Recovery Syst. 5 (1985) 527–534. [3] T. Nishizaki, K. Miyamoto, K. Yoshida, J. Less-Comm. Met. 89 (1983) 559–566. [4] S. Fukada, Y. Toyoshima, M. Nishikawa, Fusion Eng. Des. 49–50 (2000) 805–809. [5] S. Fukada, Y. Toyoshima, J. Alloys Compd. 289 (1999) 306– 310. [6] S. Fukada, H. Fujiwara, J. Chromatogr. A 898 (2000) 125–131. [7] H. Fujiwara, S. Fukada, M. Samsun-Baharin, M. Nishikawa, J. Nucl. Sci. Technol. 37 (2000) 724–726. [8] H. Fujiwara, S. Fukada, M. Nishikawa, Sep. Sci. Technol. 36 (2001) 337–348. [9] H. Choi, A.F. Mills, Int. J. Heat Mass Transfer 33 (1990) 1281–1288. [10] A. Isselhorst, J. Alloys Compd. 231 (1995) 871–879. [11] A. Jemni, S.B. Nasrallah, J. Lamloumi, Int. J. Hydrogen Energy 24 (1999) 631–644. [12] K. Aldas, M.D. Mat, Y. Kaplan, Int. J. Hydrogen Energy 27 (2002) 1049–1056. [13] S. Fukada, S. Morimitsu, N. Shimoozaki, K. Onoda, J. Alloys Compd., in press. [14] A. Pebler, E.A. Gulbransen, Trans. Metall. Soc., AIME 239 (1967) 1593–1600. [15] D. Shaltiel, I. Jacob, D. Davidov, J. Less-Comm. Met. 53 (1977) 117–131. [16] S. Fukada, T. Nakahara, N. Mitsuishi, J. Nucl. Mater. 171 (1990) 399–407. [17] D.M. Ruthven, Principles of adsorption and adsorption processes, Wiley, New York, 1984 pp. 258–262. [18] S. Fukada, M. Samsun-Baharin, H. Fujiwara, Int. J. Hydrogen Energy 27 (2002) 177–181.
Heat and mass transfer in a concentric-annular-tube bed packed with ZrV1.9Fe0.1 particles S. Fukada∗ , S. Morimitsu, N. Shimoozaki Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan Received 6 October 2003; received in revised form 8 December 2003; accepted 8 December 2003
Abstract Pressure–composition isotherms of the ZrV1.9 Fe0.1 –hydrogen system were determined by a constant volume method and were correlated to an empirical van’t Hoff equation. The enthalpy change of hydrogen absorption was correlated to a function of the hydrogen-to-alloy atomic molar ratio. The entropy change was independent of it. Heat and mass transfer process in a concentric-annular-tube bed packed with the ZrV1.9 Fe0.1 particles was studied experimentally and numerically as a basic study of hydrogen storage at elevated temperature. A one-dimensional concentric-annular-tube bed was charged with hydrogen. The experimental profiles of hydrogen partial pressure and temperature were compared with numerical simulation calculated by the pressure–composition isotherm, a hydrogenating rate equation and two balance equations of hydrogen and heat. Another simplified analysis using three dimensionless lump parameters succeeded in estimating temperature profiles in the hydrogen-absorbing-alloy bed more easily. The simulation was able to make it better to understand contributions of experimental parameters such as the flow rate, surrounding temperature and other physical properties. © 2004 Published by Elsevier B.V. Keywords: Hydrogen storage materials; Gas–solid reactions; Heat conduction; Thermal analysis
1. Introduction Hydrogen-absorbing alloys have been studied for various purposes such as hydrogen storage [1], chemical heat pumps [2], refrigerators [3], purification of hydrogen or inert gas [4,5], hydrogen isotope separation [6–8] and so on. Heat and mass transfer in particle beds of hydrogen-absorbing alloys such as LaNi5 was well studied experimentally and analytically [9–12]. One-dimensional [9], two-dimensional [10,11] and even three-dimensional analysis [12] were carried out. Quantitative agreement using LaNi5 that has high hydrogen absorption capacity around room temperature was obtained between the numerical results and measured temperatures and amounts of absorbed hydrogen. However, since there are still many parameters to be determined from fitting curves to experimental data, we need further experimental and numerical efforts to elucidate heat and mass transfer processes in hydrogen-absorbing beds.
∗ Corresponding author. Tel.: +81-92-642-4140; fax: +81-92-642-3800. E-mail address: [email protected] (S. Fukada).
0925-8388/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.jallcom.2003.12.008
Hydrogen is produced at higher temperature in some industrial processes. When hydrogen is produced by steam-reforming of natural gas around 600–800 ◦ C, it seems to be more advantageous to store high-temperature hydrogen tentatively on its way to the subsequent line. Unfortunately, less studies were carried out on hydrogen absorption under such a high-temperature condition previously except for our experiment using a long-tube bed of a Zr–V–Fe alloy [13]. In the present study, we selected ZrV1.9 Fe0.1 alloy as a hydrogen-absorbing material holding sufficient hydrogen absorption capacity even at elevated temperature. However, there are no available data on pressure–composition isotherm. As the first part of our experiment, therefore, pressure–composition isotherms of the ZrV1.9 Fe0.1 – hydrogen system were determined and were correlated to an empirical van’t Hoff equation. In the next place, profiles of hydrogen partial pressure and temperature in a ZrV1.9 Fe0.1 bed were experimentally determined using a concentric-annular-tube vessel packed with the alloy particles. Hydrogen diffused there approximately in the onedimensional radial direction. As the last thing, a simplified analysis was applied to the heat and mass transfer process in the one-dimensional hydrogen-absorbing bed. Such an
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Nomenclature cp,g cS D E
heat capacity of gas (J/kg K) heat capacity of alloy particle (J/kg K) diffusion coefficient (m2 /s) activation energy of reaction rate constant (J/mol) hT heat-transfer coefficient between particles and wall (W/m2 K) H enthalpy change of hydrogen absorption (J/mol) k absorption rate constant (mol/m3 Pa0.5 s) kL first-order reaction-rate constant (s−1 ) Ke Darcy’s permeability (m2 ) p hydrogen pressure (Pa) peq equilibrium pressure (Pa) p0 atmospheric pressure, 1.0133 × 105 (Pa) q hydrogen absorption amount (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) qm averaged hydrogen absorption amount (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) saturated value of hydrogen absorption amount q0 (H-to-alloy atomic molar ratio) (H/ZrV1.9 Fe0.1 ) r radial distance from the center (m) rin inner radius of concentric-annular-tube bed (m) rout outer radius of concentric-annular-tube bed (m) rp particle radius (m) Rg gas law constant (J/mol K) S entropy change of hydrogen absorption (J/mol K) t time (s) T temperature (K) Tm temperature averaged over the whole bed (K) surrounding temperature (K) T0 u gaseous velocity (m/s) V total volume of packed bed (m3 ) W volumetric hydrogen flow rate (m3 /s) x dimensionless absorption amount defined as x = qm /q0 z longitudinal distance from the top of bed (m) Greek letters α dimensionless parameter defined as α = (−∆H)γq0 /(1 − ε)ρS cS T0 β dimensionless parameter defined as β = hT A/kL,0 V(1 − ε)ρS cS δ dimensionless parameter defined as δ = E/Rg T0 γ packed density of alloy (mol/m3 ) ε void ratio ρg gaseous hydrogen density (kg/m3 ) ρS solid density (kg/m3 ) λe effective thermal conductivity in bed (W/mK) µ gaseous viscosity (Pas) θ dimensionless temperature defined as θ = (Tm − T0 )/T0
analytical solution will make it clear how various physical parameters contributed to the profiles of temperature and hydrogen pressure in the bed. The analytical results were compared with experimental data. Hydrogen was supplied from the top of cylindrical beds of hydrogen-absorbing alloys in many previous experiments [9–13]. Then, hydrogen diffused non-uniformly in either radial or longitudinal direction [10,11]. Therefore, the profiles of hydrogen absorption amount and temperature in the bed distributed differently in the two directions. Such non-uniformity seemed to deteriorate hydrogen-absorption performance. In the present system, hydrogen-absorbing-alloy particles were packed between two concentric-annular tubes. Hydrogen was supplied through an inner cylindrical tube walls with many pores. The hydrogen concentration was almost uniform in the longitudinal direction. Consequently, the hydrogen reaction zone and the temperature profile proceeded across the concentric-annular-tube bed with a uniform pattern as shown later. It will be proved in the present study that such change in the configuration makes the heat and mass transfer process simpler and more effective.
2. Pressure–composition isotherm of ZrV1.9 Fe0.1 –H2 system A Zr–V–Fe alloy was manufactured from a mixture of particles of each element metal (Zr, V, and Fe) by an argon-arc-melting method. Particles of Zr, V and Fe adjusted to be ZrV1.9 Fe0.1 were melted several times. The alloy was crashed and was screened between 12 and 32 mesh (0.495–1.397 mm) in an argon–atmosphere glove box. Uniformity of the alloy particles was checked by a X-ray diffraction (XRD) method. The XRD pattern was presented previously [13]. The hydrogen absorption capacity was determined by a constant volume method. A hydrogen storage vessel and a reaction vessel were made of 316 stainless-steel. The alloy of 1.00 g was put in the reaction vessel. Pressure in the storage vessel before and after hydrogen absorption was measured using two MKS Balatron absolute-pressure gages with the accuracy of 0.5%. Before the measurement, the alloy was activated at 800 ◦ C under vacuum of around 10−3 Pa for 1 h. The pressure–composition isotherm of the ZrV1.9 Fe0.1 – hydrogen system is shown in Fig. 1. There was no plateau region in the ZrV1.9 Fe0.1 alloy. The no-plateau tendency was consistent with that of ZrV2 [14]. Displacement of a part of V with Fe was found to enhance hydrogen absorption rates without loosing hydrogen absorption capacity [15]. All of the pressure–composition isotherms were correlated to the following empirical van’t Hoff equation: peq Rg T ln = H(q) − TS(q) (1) p0
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307
alloy. This was because there were many kinds of tetrahedral sites in the cubic C15 Laves phase.
105
peq [Pa]
104
3. Heat and mass transfer experiment using concentric-annular-tube bed
103 Key Temp. 302K 371K 423K 530K 631K 734K
102 101
Estimation
100 0.0
0.5
1.0
1.5
2.0
2.5
q [H/ZrV1.9Fe0.1] Fig. 1. Pressure–composition isotherm of the ZrV1.9 Fe0.1 –hydrogen system.
Fig. 3 shows a schematic illustration of the concentricannular-tube bed for hydrogen absorption of the present experiment. Fig. 4 shows a schematic diagram of the experimental system. The particle size of the ZrV1.9 Fe0.1 alloy ranged from 12 to 32 mesh (0.495–1.397 mm). The packed amount was 5015.12 g. The bed volume was 1137 cm3 . The inlet hydrogen flow rate was in the range from 1.5 to 10.0 l(NTP)/min. Hydrogen was purified by a liquid-nitrogen cold trap. The inlet gas temperature ranged
The pressure–composition isotherm shown in Fig. 1 was compared with Eq. (1). So that, the S values were found to be constant independent of not only temperature, T, but also the hydrogen-to-alloy atomic molar ratio, q. They were correlated to the following simple relation: S = −11.9Rg
(2)
The independence of S from q and T means that hydrogen atoms dissolved in the solid phase lost almost all degrees of freedom that had in the gaseous phase. Fig. 2 shows variations of the enthalpy change of hydrogen absorption as a function of the atomic molar ratio of H/ZrV1.9 Fe0.1 . Thus, H increased with q independent of T. The H values were correlated to the following polynomial equation: H = −122.7 + 45.96q + 10.28q2 − 5.708q3
(3)
The solid line in the figure was one calculated from Eq. (3). The variations of H with q were considered to come from different energy levels of hydrogen atoms dissolved in the
RgTln(peq/p0)+T∆S [kJ/mol-H]
0 -20
∆S/Rg=-11.93 (2) ∆H(q)=-122.7+45.96q+10.28q2-5.708q3
Fig. 3. Concentric-annular-tube ZrV1.9 Fe0.1 bed.
(3)
-40 -60
Key Temp. 302K 371K 423K 530K 631K 734K Eq.(3)
-80 -100 -120 0.0
0.5
1.0
1.5
2.0
q [H/ZrV1.9Fe0.1] Fig. 2. Enthalpy changes of hydrogen absorption.
Exhaust H2 gas cylinder Cold trap Mass-flow controller Pressure sensor Absorbing reactor
Internal heater Thermocouples Diffusion pump Rotary pump
Fig. 4. A schematic diagram of the experimental apparatus.
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cp,g ∂ ∂ ∂T (ρg T) + (1 − ε)cS ρS + (ruρg T) ∂t ∂t r ∂r ∂T 1 ∂ ∂qm rλ = + γ(−H) r ∂r ∂r ∂t
from 294 to 835 K. Seven Chromel-Almel thermocouples were inserted in the concentric-annular-tube bed. They were located at different positions of r = 28, 38, 48, 58 and 70 mm in the radial direction at the same height of z = 100 mm and at z = 50, 100 and 150 mm in the longitudinal direction at the same radial position of r = 38 mm. Before the absorption experiment, the bed was activated by heating at 800 ◦ C under vacuum lower than 10−3 Pa for 5 h. Then electricity of the internal heater was cut off, and the bed was cooled down slowly. When temperature reached a desired one, a constant flow rate of purified hydrogen was introduced into the bed through an inner cylindrical porous tube. Electricity of the internal heater was not supplied during the hydrogen supply.
εcp,g
4. Analysis
γ
4.1. Numerical analysis using partial differential equations Three equations for a hydrogen-absorbing rate, a hydrogen material balance and a heat balance as well as the pressure–composition isotherm of Eqs. (1)–(3) are simultaneously solved numerically. We set the following assumptions and conditions for the three equations: (i) temperature of the gaseous phase in the bed is the same as that in the particle phase; (ii) there is no hydrogen absorbed in the bed at the initial stage. The bed temperature at start is equal to the surrounding temperature, T0 ; (iii) the particle bed is a concentric-annular-tube type. Hydrogen gas is supplied through the inner porous tube under a constant flow rate and a constant temperature of T0 until the upstream pressure reached a specified one. After the upstream pressure has reached it, the pressure is maintained there and the flow rate starts decreasing quickly; (iv) hydrogen diffuses in the radial direction. The hydrogen flow velocity inside the bed is described by the Darcy’s law; (v) part of heat generated in the bed is consumed to elevate temperature of alloy particles and gas. The rest is transferred in the radial direction and is released outside through the outer tube; (vi) thermal conduction in the longitudinal direction of the bed is neglected. So, the system is of one-dimensional; (vii) hydrogen permeation through the outer tube is negligibly small. Under the above assumptions and conditions, the equations of material balance and heat balance were described as follows: ∂ρg MH2 γ ∂qm 1 ∂ ε (ruρg ) = − (4) + r ∂r 2 ∂t ∂t
ρg =
MH2 p Rg T
(5) (6)
Here, the gaseous velocity in the radial direction, u, was related with the pressure difference by the Darcy’s law in the same way to the previous study [9] as follows: u=−
Ke ∂p µ ∂r
(7)
Hydrogen absorption rate was described in a similar way to the previous work [10] as follows: p0.5 − p0.5 ∂qm eq =k ∂t (1 − (qm /q0 ))−1/3 − 1
(8)
Here, p and peq are bulk hydrogen pressure and pressure in equilibrium with qm , respectively. The latter was calculated from Eqs. (1)–(3) as a function of qm . The initial and boundary conditions were as follows: t = 0, r = rin , r = rout ,
T = T0 , T = T0 , −λ
qm = 0,
p=0 Ke ∂p u0 = − µ ∂r r=rin
∂T = hT (T − T0 ), ∂r
∂p = 0. ∂r
(9) (10) (11)
Eqs. (4)–(8) under the conditions of Eqs. (9)–(11) with the equilibrium isotherm of Eqs. (1)–(3) were solved numerically by the finite difference method. 4.2. Simplified analysis The above simulation was expected to give detailed profiles of temperature and hydrogen absorption amount as a function of time and the radial distance in a similar way to LaNi5 studied previously [9]. It will be shown later that the simulation is valid also in the ZrV1.9 Fe0.1 bed. However, Eqs. (4)–(8) included many parameters to be determined from comparing the analysis with experimental data. Part of these parameters were unique to the ZrV1.9 Fe0.1 bed. There are no reliable data for the thermal conductivity in the particle bed and the hydrogenating rate constant of ZrV1.9 Fe0.1 beforehand. Therefore, there were many difficulties in comparing the numerical simulation with experimental profiles. On the other hand, simplified analysis based on reasonable consideration may become more useful if obtained. This was because several parameters appearing in the original equations were lumped. Therefore, the number of parameters to be determined decreased. In order to simplify the governing equations and so to obtain an ordinary differential equation that can be easily understood, we set the following additional three assumptions.
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(viii) We focus on temperature averaged over the bed instead of local temperature. Then, Eq. (5) was integrated over the whole bed. The manipulation resulted in the following macroscopic heat balance equation: ∂Tm ∂Tm + (1 − ε)ρS cS V + ρg cp,g WT0 ∂t ∂t ∂qm = γ(−H)V − hT A(Tm − T0 ) (12) ∂t
ερg cp,g V
(ix) The hydrogenating equation of Eq. (7) is replaced by a rate equation using linear-driving-force approximation. The rate equation was simplified to the following one by the linear-driving-force approximation: ∂qm E (13) = kL,0 exp − (q∗ − qm ), ∂t Rg T The linear-driving-force approximation was believed to be valid under usual conditions of adsorption or hydrogenation [16]. The rate constant kL,0 exp(−E/Rg T) in Eq. (13) is related with the hydrogen diffusivity in solid by 15D/rp 2 . A comparison was given to prove it in the previous reference [17]. (x) Thermal capacity of the solid phase is much greater than that of the gaseous phase. With the assumption (x), the first and third terms on the left-hand side of Eq. (12) were neglected, because of less contribution to the whole heat balance compared to those of the other terms. With the use of the dimensionless independent parameters, x and θ, that are defined in the nomenclature, we could reduce Eq. (12) to the following dimensionless ordinary differential equation that was simplified more than the original partial differential equation. dθ βθ δ =α− exp (14) dx 1−x 1+θ
5. Results and discussion 5.1. Temperature curves Fig. 5 shows examples of variations of experimental temperature with time at different radial or longitudinal locations in the ZrV1.9 Fe0.1 bed. All temperature curves except for the wall one showed similar time variation regardless of the different thermocouple locations of No.(1)–(7). The curves were characterized by steep temperature rise and then gradual drop. The tendency was held also under different conditions of the H2 flow rate, W0 , and the surrounding temperature, T0 . Time when temperature started rising was the earliest at the inner location depicted by No.(1) in Fig. 5. That in the outer location depicted by No.(6) was the latest. The time was almost the same regardless of different heights at the same radial position depicted by Nos.(2), (4) and (5). All those things mean that a cylindrical reaction zone moved across the ZrV1.9 Fe0.1 bed uniformly in the longitudinal direction. The reaction zone moved very quickly with a uniform pattern. Temperature differences among the thermocouple positions, No.(1)–(7), were small compared with variations with time. Therefore, we focused on variations of temperature with time as shown below. Fig. 6 is examples of comparison of temperature variations with time between experiment and numerical simulation using Eqs. (4)–(11) for different T0 conditions. The position of thermocouples was r = 38 mm and z = 100 mm. The numerical simulation showed similar characteristics of steep temperature rise and gradual drop. Good agreement was obtained between the experiment and the numerical simulation. This was valid also under other W0 conditions. The value of the effective thermal conductivity in the Zr–V–Fe alloy bed used for the numerical simulation was λe = 0.14 W/mK, that of the Darcy’s permeability parameter Ke = 8.9 × 10−8 m2 , and that of the reaction rate constant kL = 1.0 × 800 (4)
The boundary conditions are as follows: at x = 0 and x = 1
(1)
700
Line z [mm] (1) 100 (2) 100 (3) 100 (4) 50 (5) 150 (6) 100
(2)
(5)
(15)
The three dimensionless lump parameters, α, β and δ, in Eq. (14) are defined in the nomenclature. Thus all experimental parameters were correlated to the three dimensionless lump parameters. Integration curves of Eq. (14) started from θ = 0 at x = 1 and ended at θ = 0 at x = 0. The dimensionless maximum temperature rise, θ max , could be expressed as follows. βθmax δ xmax = 1 − (16) exp α 1 + θmax The value of x is in proportion to time under the constant H2 flow condition.
Temperature [K]
θ=0
309
600
r [mm] 28 38 48 38 38 wall
(3) Tmax-T0
500
(6) 400
T0=298K W=10L/min
tmax 300
0
1000
2000
3000
4000
5000
6000
Time [s]
Fig. 5. Variations of experimental temperature in ZrV1.9 Fe0.1 bed with time.
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800
400 300
experiment simulation
Temperature [C]
600
T0=200C T0=30C
400
200 W0=5L(NTP)/min 0 0
2000
4000 6000 Time [s]
8000
0 [C]
100 β
10 Values of parameters α, β andδ [-]
T0=400C
200
10000
Flow rate 5 L/min 7.5L/min 10 L/min
5
α
β
δ
α
0
δ -5 1.5
2.0
2.5
3.0
3.5
-1
Fig. 6. Comparison of temperature profiles between experiment and analytical equations.
10−3 exp(−9.98/Rg T). This numerical simulation wasted a lot of time due to solve the nonlinear governing equations, especially Eqs. (4) and (5), simultaneously. In addition to that, the values of physical parameters used for the simulation were still uncertain. This was because there was possibility in that the parameters may depend on temperature and hydrogen pressure. Therefore, their dependence was investigated by the simplified analysis using lump parameters shown below. Average temperature was evaluated before comparison between experiment and the simplified analysis. So that, it was found that variations of the average temperature with time are almost coincident with those of the local temperature at the location of r = 38 mm and z = 100 mm (No. (2)). Therefore, the comparison was made between the simplified analysis and the local temperature at No. (2) for simplicity. Fig. 7 shows a comparison of the temperature profile
2.0 experiment calculation
T0=294K
W=7.5L(NTP)/min z=100mm, r=38mm
1.5
1000/T [K ]
Fig. 8. Values of parameters α, β, and δ.
between the experimental variation and the analytical equation of Eq. (14). As seen in the figure, close agreement was also obtained between them regardless of different T0 conditions. This situation was held also in curves of any other W0 conditions. The values of α, β, and δ were determined from fitting curve as a function of T0 and W0 . Fig. 8 shows the values of α, β, and δ determined. The values of α decreased with elevation of T0 regardless of different W0 values. Therefore, the value of (−H)γ/(1−ε)ρS cS except for q0 /T0 in α was considered constant independent of temperature and hydrogen pressure. On the other hand, the values of δ defined by E/Rg T0 increased with elevation of T0 . From the slope of the δ line, the activation energy of 10 kJ/mol was obtained. The value was coincident with that of the activation energy of the rate constant k used for the numerical simulation. The values of β defined by hT A/kL,0 V(1−ε)ρS cS were constant independent of T0 . Therefore, the values of hT A/kL,0 V(1−ε)ρS cS were considered constant independent of temperature and hydrogen pressure. Consequently, the value of the heat-transfer coefficient between the wall and the particle bed was hT = 20 W/m2 K, and the value was independent of T0 and W0 .
(T-T0)/T0 [-]
5.2. Hydrogen absorption T0=374K
1.0
0.5
T0=523K T0=621K
0.0 0.0
0.2
0.4
0.6
0.8
1.0
qm/q0 [-]
Fig. 7. Comparison of temperature profiles between experiment and simplified analysis.
Fig. 9 shows variations of the total amount of hydrogen absorbed in the ZrV1.9 Fe0.1 bed with time. The experimental variation with time was almost linear at the early stage of hydrogen absorption. The slope was in proportion to W0 . After certain time depending T0 and W0 , the curve started decreasing. Thus, the present hydrogen-absorbing alloy bed showed a good absorption performance. Comparatively, good agreement was obtained between the experiment and the numerical simulation using Eqs. (4)–(11). Profiles of hydrogen absorption amount in the bed were almost flat across the bed, although not shown in the figure. The absorption performance did not change regardless of repeated
S. Fukada et al. / Journal of Alloys and Compounds 375 (2004) 305–312
500
2.5 experiment simulation
30
1.5
20
1.0
Key
400
2.0 W0=5L(NTP)/min T0=300K
Tmax-T0 [K]
40
H/ZrV1.9Fe0.1 [-]
T0 303K 473K 680K 835K
300 200 100 r=48 mm z=100 mm
10
0.5
0 0
0 0
2000
4000 6000 Time [s]
8000
0.0 10000
Fig. 9. Changes of total amount of hydrogen absorbed in ZrV1.9 Fe0.1 bed.
absorption–desorption cycles until about 50 of the maximum experimental run number in a similar way to Pd [18]. 5.3. Maximum temperature rise Fig. 10 shows a correlation of the results of the experimental temperature-rising rate defined by (Tmax − T0 )/tmax . The rising rates were found to be in proportion to W0 , independent of T0 . Thus, the rising rate in the early absorption time was controlled only by W0 , consequently the hydrogen absorption rate. It was found that the temperature rising rate is in proportion to the hydrogen absorption rate, if we assume that the two contributions of the second term on the left-hand side of Eq. (5) and the second term on the right-hand side are dominant in the earlier time of hydrogen absorption. Since the hydrogen absorption rate was in proportion to W0 at the
2 4 6 8 10 12 H2 Flow rate, W0 [L(NTP)/min]
Fig. 11. Relation between maximum temperature rise and flow rate.
earlier time, the following equation would be obtained from Eq. (5). T ∼ (−H)W0 = t (1 − ε)cS ρS V
(17)
Here, V is a volume where temperature varied more heavily. The right-hand side of Eq. (17) was considered in proportion to W0 and was independent of T0 . This thing was coincident with the result of Fig. 10. Then, the contribution of thermal conduction in the radial direction in the temperature rising region was found to be smaller than that of the heat generation. The contribution was considered to increase with an increase in W0 . Fig. 11 shows a plot of the maximum temperature rise defined by Tmax − T0 as a function of W0 and T0 . As seen in the figure, the plot can be divided into two regions. At the lower W0 region, Tmax − T0 depended on W0 . Therefore, the values of Tmax − T0 were controlled by a balance between the hydrogen absorption rate, consequently heat generation rate, and the heat conduction. 500
Key Flow rate 1.5L/min 3.5L/min 5 L/min 7.5L/min 10 L/min q0 (from PCT curve)
25 Key T0 296K 473K 680K 835K
400
Tmax-T0 [-]
(Tmax-T0)/tmax [K/min]
20
15
14
300
2.5
2.0
1.5
200
1.0
100
0.5
10
q0 [H/ZrV1.9Fe0.1]
Total amount of hydrogen absorbed [mol]
50
311
5 r=48 mm z=100 mm
0 300
0 0
2
4 6 8 H2 flow rate [L(NTP)/min]
10
12
Fig. 10. Experimental values of temperature rising rates.
400
500
600
700
800
0.0 900
T0 [K]
Fig. 12. Relation between maximum temperature rise and surrounding temperature.
312
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On the other hand, Tmax −T0 was independent of W0 in the higher W0 region. In order to make it clear what parameter controlled Tmax − T0 in the higher W0 region, another plot was tested. Fig. 12 shows experimental values of Tmax − T0 as a function of T0 and W0 . As seen in the figure, Tmax − T0 at higher W0 values than W0 = 5 l(NTP)/min showed almost the same temperature dependence with that of the hydrogen absorption capacity, q0 . The temperature profile could be correlated by Eq. (16) successfully. Judging from xmin almost independent of T0 , the dependence of Tmax − T0 on T0 should be coincident with the dependence of q0 . In other words, Tmax − T0 in this system was limited by q0 . All those things could be explained by the simplified analysis of Eq. (16). 6. Conclusions Pressure–composition isotherms of the ZrV1.9 Fe0.1 –H2 system were correlated to an empirical van’t Hoff equation. The entropy change was independent of the hydrogen absorption amount and temperature. The enthalpy change was a unique function of the hydrogen absorption amount independent of temperature. Experimental profiles of temperature and hydrogen absorption in the concentric-annular-tube bed packed with the ZrV1.9 Fe0.1 particles were analyzed with two types of simulation. The concentric-annular-tube bed was found to be efficient for hydrogen absorption. Hydrogen reaction zone proceeded uniformly in the longitudinal direction. The microscopic balance equations of Eqs. (4)–(8) could simulate local temperature profiles in the ZrV1.9 Fe0.1 bed accurately. We determined experimental parameters of λe = 0.14 W/mK, Ke = 8.9 × 10−8 m2 , kL = 1.0 × 10−3 exp(−9.98/Rg T), and hT = 20 W/m2 K for the ZrV1.9 Fe0.1 bed. These parameters were found to be independent of the surrounding temperature and hydrogen flow rate. Another analysis using three lump parameters of
α, β, and δ was found to be efficient to observe how all physical parameters affected the hydrogen absorption and the temperature distribution in the bed. The behavior of the experimental absorption and temperature variations could be understood by the analytical simulation and numerical simulation.
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