Measurement and Analysis of Effective Thermal Conductivity of

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Measurement and Analysis of Effective Thermal Conductivity of MmNi4.5Al0.5 Hydride Bed E. Anil Kumar,† M. Prakash Maiya,‡ and S. Srinivasa Murthy‡,* † ‡

Discipline of Mechanical Engineering, Indian Institute of Technology Indore, Indore-452017 Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600036 ABSTRACT: The effective thermal conductivity (ke) of a MmNi4.5Al0.5 hydride powder bed was measured using an onedimensional steady-state axial heat transfer comparative method. The experiments showed that ke lies between 0.1 and 1.2 W/m K in the pressure and the temperature ranges of 0
50 bar and 0
100 °C, respectively. While hydrogen pressure and concentration showed strong influence on ke over the complete range, the temperature effects were minimal. Thermal conductivity of the solid materials, MmNi4.5Al0.5 alloy and its hydrides, was also back calculated from the experimentally measured ke values.

1. INTRODUCTION Hydrogenation and dehydrogenation reactions of metal hydrides evolve and absorb large amounts of heat. The reaction rates depend upon the temperature of the hydride bed. It is essential to design metal hydride reactors to exhibit heat and mass transfer characteristics needed for specific applications. Metal hydride beds can be considered as hydride particles in a discrete phase within hydrogen gas as a continuous phase. Knowledge of the effective thermal conductivity (ke) of the metal hydride bed is essential for the design of a hydrogen storage device. Different experimental methods reported to estimate the effective thermal conductivity of metal hydride beds can broadly be classified under steady-state and transient methods. In the former1
8 the metal hydride is filled in a container of simple geometry, required boundary conditions are imposed, and unidirectional heat flux is achieved. At steady state, temperature profiles and heat flux are measured. These data are compared with the respective solution of the Laplace or the Fouriers equation to estimate the ke of metal hydride bed. Steady-state methods can be further classified as comparative and absolute methods. In the former case,1
7 a reference material is placed in a series with the metal hydride bed, whereas in the later case,8 ke is calculated from experimental results without any additional information. Table 1 lists the ke measurements by steady-state methods reported in literature. In transient methods9
14 the thermal response of a probe is monitored when a sudden change of boundary conditions are applied. The effective thermal conductivity of metal hydride bed is calculated by correlating the measured temperature profile with the solution of Fourier equation. Transient methods used for ke measurement by different authors are listed in Table 2. The steady-state methods are more reliable and accurate than transient methods but are time consuming. The experimental values of ke from literature are summarized in Table 3. Several researchers have theoretically estimated the ke of hydride beds. Sun and Deng15 developed a model based on the concept conceived by Masmune and Smith16 for calculating ke. They considered a unit cell in which heat transfer occurs through three parallel mechanisms. Equating the sum of their separate contributions to the total heat transfer, an expression was derived r 2011 American Chemical Society

for computing ke. Pons and Dantzer17 and Kallweit and Hahne18 used the Zehner, Bauer, and Schlunder model to study the basic mechanism of thermal conductivity of metal hydride beds. Asakuma et al.19 developed a homogenization method for calculating the ke of metal hydride bed, in which a periodic composite consisting of two domains, solid and gas phases, is considered at microscopic scale. However all these equations/models need many factors which can be obtained only through experiments and hence cannot be directly used to calculate ke. Nevertheless these models give a qualitative understanding of the ke of metal hydride beds and are useful for analyzing the experimental results. Studies which highlight the effects of hydrogen pressure, bed temperature, and concentration on the effective thermal conductivity are scarce. Hence, in the work reported in this paper, using a steady-state measurement setup constructed according to ASTM standard E-1225,20 the ke of MmNi4.5Al0.5 hydride was measured during absorption and desorption at various operating conditions.

2. EXPERIMENTAL DETAILS 2.1. Description of the Thermal Conductivity Cell. The schematic diagram of ke measurement cell is shown in Figure 1a. The cell consists of the metal hydride bed, thermal conductivity of which is to be measured, the reference material, hot plate with disk heater, cold plate with coolant circulation arrangement, insulation and outer container with guard heaters. Metal hydride powder and reference material are placed in series inside a stainless steel (SS) 316 cylindrical container of 85 mm inner and 95 mm outer diameters and 52 mm height. Predetermined amount of metal hydride powder is filled into the top part of the container such that it also assumes a disk shape Special Issue: Ananth Issue Received: January 18, 2011 Accepted: May 19, 2011 Revised: May 10, 2011 Published: May 19, 2011 12990

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Table 1. Details of Steady-State Methods Used for ke Measurement S. no.

hydride

reference material

thermal conductivity of reference material W/m K

accuracy

1

TiMn1.5 hydride1

combination of ethyl alcohol and lead shots

1.64
0.0019ts (0 °C < ts < 60 °C)

2

Mg2Ni hydride6

stainless steel

14
16

(3%

3

MmNi4Fe hydride6

PTFE

0.21

4

MlNi4.5Mn0.5 hydride7

PTFE







5

TiFe0.85Mn0.15 hydride8







15%

15%

Table 2. Details of Transient Methods Used for ke Measurement S. no.

material

method/principle

10

1

LaNi4.7Al0.3 hydride

2

Ti0.98Zr0.02V0.43 Fe0.09Cr0.05Mn1.5 hydride10

3

Mg and 2% Ni hydride12

accuracy

A line source in contact with metal hydride bed is subjected to sudden heating.

10%

The time temperature response at some point in the bed is recorded. From the solution of transient heat conduction equation, ke is estimated. Radial temperature profile is studied by applying a constant heat flux at the boundary,

12%

ke is calculated using Fourier’s equation 4

MlNi4.5Mn0.5 hydride13

Metal hydride bed is subjected to oscillation heating, ke is calculated

12%

by tailoring the problem-specific mathematical result to the experimentally recorded temperature
time function

Table 3. Values of ke Obtained from Experiments by Various Authors operating variables S. no.

material

pressure bar

temperature K

ke value W/m K

1

TiMn1.5 hydride

0.1
50




0.2
1.2

2

Mg hydride9

40

313

1.3

3

Mg2Ni hydride9

45

373

0.8

4

Mg2Ni hydride6

45

373

0.83

5

MmNi4Fe hydride6

45

273

1.05

6

TiFe0.85Mn0.15 hydride8

10
8
55

273
423

0.1
1.5

7 8

MlNi4.5Mn0.5 hydride7 LaNi5 hydride11

30 0.01
10

300


1.3 0.1
1.5

9

LaNi4.7Al0.3 hydride10

10
6
60

193
413

0.02
1.2

1

10

Ti0.98Zr0.02V0.43 Fe0.09Cr0.05Mn1.5hydride

10
6
60

193
413

0.05
1.2

11

Mg and 2% Ni hydride12

0.1
50

573
673

4
9

12

Mg
Mg hydride13

0.1
50

523
653

2
8

10

with the same dimensions of 85 mm diameter and 20 mm thick, which is partitioned at the middle by a SS 316 disk of 3 mm thickness. The disk is welded to the inner surface of the container to withstand high pressure on metal hydride side. Seven “K”-type thermocouples (0.122 mm) are inserted into the metal hydride bed. All the thermocouples are silver brazed to the tube. Location of the thermocouples and their arrangement are shown in Figure 1b. The main intension of temperature measurement was to estimate the temperature gradient in the axial direction. Additionally the measurements ensure temperature uniformity in the radial direction. Thermocouples are intentionally staggered at a distance from the central axis in order to avoid disturbance to the bed. Dummy runs have been performed to ensure temperature uniformity on the planes perpendicular to the axis. A 6 mm diameter SS tube welded at a distance of 10 mm from the top supplies hydrogen to metal hydride. Poly(tetrafluoroethylene) (PTFE) is used as the reference material. Four “T”-type thermocouples of 0.122 mm diameter are attached in the groove provided

on the PTFE disk, two on the top and two on the bottom surfaces. The top surface of metal hydride bed is roofed with a hot plate, and the bottom surface of PTFE disk is covered with a cold plate. The hot plate is heated with a 250 W disk heater. The cooling chamber is made with SS 304 and has a 120 mm inner diameter, 3 mm thickness, and 80 mm height. The other end of the cooling chamber is closed with a SS 304 disk of 3 mm thickness and welded around the circumference. The coolant is pumped from a thermostatic bath, and the inlet and outlet temperatures are measured with two T-type thermocouples. The cell is enclosed in an outer cover, and the annular space is packed with glass wool insulation. A band heater is wound around the circumference of the covering cylinder at the bottom edge (which is concentrically placed with the metal hydride bed and reference material) covering the metal hydride bed and the reference material. The band heater acts as a guard heater and eliminates radial heat loss from metal hydride bed. The entire outer surface of covering cylinder is insulated with 50 mm thick glass wool. 12991

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Figure 1. (a) Schematic diagram of effective thermal conductivity (ke) measurement cell. (b) Metal hydride bed temperature measurement locations.

2.2. Thermal Conductivity of the PTFE Reference Disk. Since the effective thermal conductivity of metal hydride bed is less than unity, PTFE which has thermal conductivity similar to that of metal hydrides is considered as the reference material. The conductivity of PTFE is measured using a square guarded hot plate apparatus built based on the ASTM C177 standard. It was calibrated using a pair of polymethyl-methacrylate (Perspex) thermal conductivity reference material of lateral dimensions of 305  305 mm2 and a thickness of 31 mm supplied by the National Physics Laboratory, U.K. The details of the experimental setup and the procedure were described by Karithikeyan and Reddy.21 The measured values of thermal conductivity at 50, 75, and 100 °C are 0.23, 0.28, and 0.31 W/mK, respectively. The maximum error in the measurements is (5.75%. Variation of thermal conductivity of PTFE with temperature is a linear plot, as shown in Figure 2, given by the correlation:

kT ¼ 0:0016  T þ 0:1507

ð1Þ

The required disk of 85 mm diameter was machined from the PTFE block. 2.3. Description of the Experimental Setup. The experimental setup is shown schematically in Figure 3. It is a modified static PCI measurement setup which is discussed in the authors’ recent paper.22 The setup can be used within the pressure and average temperature limits of 0.1
50 bar and 30
160 °C, respectively. As shown in Figure 3, it was divided into eight zones. The volumes of all the zones are calculated. The thermal conductivity cell was charged with 530 g of activated metal hydride in a glovebox. Good contact was ensured between the surface of the metal powder and the heating plate by tightening the top flanges. The charged thermal conductivity cell was connected to the hydrogen measuring setup. Packing density and porosity of the bed are measured as 4670 kg/m3 and 0.43, respectively. 2.4. Experimental Procedure. The entire setup is first evacuated to 10
3 mbar by opening all valves except v9, v10, and v11. A constant power is applied to the main heater, and the 12992

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temperature of the coolant fluid is set. The heater power input and the coolant temperature combination required to achieve an average predetermined bed temperature is found by trial and error. All the thermocouples in the metal hydride bed and the reference material are monitored at 15 min intervals. The guard heater temperature is adjusted to the average temperature of the bed. After attaining steady state, the temperatures T1
T11 are recorded, as seen in Figure 4, for a typical case. Then the temperature gradient in the reference material is estimated as
dT


dx


ref

ðT8 þ T9 Þ ðT10 þ T11 Þ
2 2 ¼ Δx

ð2Þ

where T8 and T9 are hot surface temperatures, and T10 and T11 are cold surface temperatures of the reference material. Temperature gradient in the metal hydride bed dT/dx|bed is estimated by plotting the temperature recorded in the metal hydride (T1
T7) at different locations against the distance from the top end (distance in the direction of heat flow) as shown in Figure 1.

Figure 2. Variation of thermal conductivity of PTFE disk with temperature.

The ke of the bed at vacuum is estimated as   dT kref  dx   ref ke ¼ dT dx bed

ð3Þ

Then valves v7 and v12 are closed, and the hydrogen is supplied to volumes V1
V8 to a predetermined incremental pressure by opening valve v9. After reaching a steady state, valve v9 is closed, and initial pressure in the volumes V1
V3 and V6
V8 is recorded. Then valve v7 is opened, and the hydrogen is allowed to the measurement cell. After attaining a steady state, the final pressure is recorded, and the amount of hydrogen absorbed is calculated from the mass balance. Boundary conditions to the conductivity cell are then applied, and the effective thermal conductivity is estimated at new pressures and concentrations. The maximum error in ke estimation is calculated to be 5.32% as detailed in the Appendix. As observed in Figure 4, temperatures vary linearly along the height, and the slope decreases with an increase in hydrogen pressure, indicating a higher ke at higher pressures.

Figure 4. Temperature variation along the bed at various pressures.

Figure 3. Schematic diagram of effective thermal conductivity measurement setup. 12993

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Figure 5. Variation of effective thermal conductivity with pressure for different filling gases.

3. RESULTS AND DISCUSSION Effective thermal conductivity of MmNi4.5Al0.5 bed is measured at pressures, temperatures, and hydrogen concentrations in the range of 0
50 bar, 0
100 °C, and 0.1
1.2 wt %, respectively. The influence of each parameter on the effective thermal conductivity is discussed in the following sections. 3.1. Effect of Pressure. At a particular average temperature, as hydrogen pressure changes, the concentration of the hydride bed also changes. Therefore it is practically impossible to study the effect of gas pressure alone while using hydrogen as a filling medium. Hence in the present study, inert gases, namely argon, helium, and nitrogen, are also used as filling gases to study the effect of gas pressure. Figure 5 shows the variation of effective thermal conductivity (ke) of the MmNi4.5Al0.5 for different filling gases. As expected for packed beds, ke increases with increasing pressure in the form of the tilted “S” shaped curve. Similar variation was observed in the experimental results reported in literature.1,2,6,7,9 This variation may be divided into three regions. In the low pressure region up to about 1 bar, ke of the MmNi4.5Al0.5 bed is independent of pressure. In this region the density of the gas is low in the voids of the bed. Therefore the mean free path of the gas molecule is large compared to the void size, which is assumed to be equivalent to the average particle size of the bed. However, with the increase in gas pressure, the density of the gas in the pores increases due to which the mean free path decreases. When it approaches the size of the voids, collision of gas molecules is high resulting in higher gas conductivity. In this region of 1
15 bar, gas conductivity increases with pressure and thereby ke also increases. With a further increase of gas pressure, the mean free path of the gas molecules becomes much lower than the effective size of the voids. In this region, the process of heat transfer by the gas is similar to that in an infinite space and does not influence the conductivity of the gas. Thus for gas pressures above 15 bar, ke remains nearly constant. This pressure is called as breakaway pressure and is given by23 Pb ¼ 1770  10
24

ðT þ 491:67Þ s2 ls

ð4Þ

From the above equation the mean diameter ls of hydride particle with hydrogen as filling medium is calculated to lie between 10 and 15 μm.

Figure 6. Variation of effective thermal conductivity with pressure at different temperatures.

Figure 7. Static PCI curves for MmNi4.5Al0.5.24

For most of the metal hydrides, the middle region (where the influence of pressure on ke is felt) lies in a plateau region of hydriding or dehydriding process. Hence, in addition to the effect of hydrogen pressure, the effect of concentration should also be considered. However, with inert gases as the filling medium, the effect of gas pressure alone is seen. Effective thermal conductivity of the bed is found to be proportional to the thermal conductivity of the filling gas. Thermal conductivity of hydrogen is the highest of all the gases. The thermal conductivity of helium is nearer to that of hydrogen (kHe = kH2*0.8). At pressures near to 0.1 bar, ke is independent of the filling gas, because heat conduction at these low pressures is due to conduction in the particle grid. The effect of gas conductivity on ke increases with pressure. The difference between values of ke obtained under hydrogen and helium atmospheres increases with pressure as the concentration increases and stabilizes at the maximum concentration. In the pressure-independent region, the difference between ke values obtained from hydrogen and helium atmospheres approximately represents the increase in ke due to the hydrogen concentration. Figure 6 shows the variation of ke with pressure at different temperatures. The tilted ‘S’ shaped curve shifts toward right with an increase in temperature. At a particular pressure, effective thermal conductivity increases with a decrease in temperature. 12994

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Figure 8. Variation of effective thermal conductivity with pressure during absorption and desorption.

This can be explained with the help of PCI diagram, as shown in Figure 7.24 At a given pressure, the hydrogen concentration is higher at a lower temperature. Particle size increases with hydrogen concentration. In the case of metal hydride beds with fixed volume, expansion of particle causes compression of the bed due to which the contact area between the particles increases thereby increasing the solid to solid conductivity. The gas thermal conductivity may decrease slightly with temperature; however, the increase in solid to solid conductivity is expected to dominate. Hence effective thermal conductivity increases with a decrease in temperature. For the same reason, ke values during desorption are marginally higher than those during absorption, as shown in Figure 8. 3.2. Effect of Concentration. The experimental results1,6,7 have shown that the variation of ke with hydrogen concentration is linear. Suda et al.1 observed that, while the slope of the line was 0.46 regardless of pressure, ke was larger during desorption than that during absorption. However, Hahne and Kallweit18 observed that the variation of ke with concentration followed a curve similar to pressure concentration isotherm. Contrary to the above observations, the experimental studies on Mg/2 wt % Ni hydride by Kapischke and Hapke12 showed that the ke decreased with an increase in concentration. The same authors13 in their experiments on magnesium hydride observed that ke increased with concentration up to a particular concentration and decreased thereafter. Unlike the above observations Kempf and Martin8 found no effect of hydrogen concentration on ke for TiFe0.85Mn0.15 hydrides. To study the effect of concentration on ke, both pressure and temperature need to be kept constant. However due to a plateau slope, this cannot be achieved in the case of metal hydrides. Hence, in the present work the effect of hydrogen concentration is studied at a constant temperature allowing for the variation of hydrogen pressure. The variation of ke and hydrogen pressure with concentration is shown in Figure 9. Increase in concentration increases ke, which also includes the effect of increasing hydrogen pressure. The hydrogen concentration in the solid hydride particle influences the conductivity of the particle as well as the porosity of the bed. As the particle adsorbs hydrogen, it expands leading to a decrease in porosity because the total bed volume is constant. In the present case, the maximum expansion is about 8%. Due to this, the particle to particle contact area increases, and the contact resistance decreases, thereby improving the particle to particle

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Figure 9. Variation of effective thermal conductivity with concentration during absorption.

Figure 10. Variation of effective thermal conductivity with concentration during absorption and desorption.

heat transfer. However, embedding of hydrogen atoms in the metal alloy particles disturbs the electronic structure due to which the individual solid particle conductivity decreases. The value of ke is influenced by the net effect of these issues. Figure 10 shows the variation of ke with hydrogen concentration during absorption and desorption. 3.3. Effect of Temperature. With an increase in temperature radiation, heat transfer comes into play. In addition, the thermal conductivities of gas and solid also change. The total effect of temperature is the combination of these effects. The experimental studies on Mg, Mg/10 wt % Ni, and Mg2Ni9 revealed that for the first two hydrides ke increases with temperature at all hydrogen concentrations. However in the case of Mg2Ni a reverse trend was observed at higher concentrations. This may be due to the decrease in solid thermal conductivity with rise in temperature being dominant over increase in gas conductivity. For Mg2NiH4, it has been reported6 that ke decreases with increase in temperature. On the contrary, Sun and Deng7 observed that ke of MmNi4.5Mn0.5 hydride increases with temperature. Likewise, measurements on TiFe0.85Mn0.158 hydride have also shown an increase in ke with temperature. The effect was stronger at high pressures and negligible under vacuum. Thus, experimental results indicate that the dependence of ke on temperature is not unique. This may be 12995

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Figure 11. Variation of effective thermal conductivity with temperature with argon as filling gas.

Figure 13. Variation of effective thermal conductivity with temperature at different concentrations.

Figure 12. Variation of effective thermal conductivity with temperature at different pressures.

3.4. Theoretical Analysis. The experimental results on ke are analyzed using the Yagi and Kunii25 model for a hydride bed containing stagnant fluid assuming heat transfer mechanisms as: • through the void fraction by conduction and radiation. • through a series path consisting of an effective solid-path length and gas length. • through the solid phase, the energy flowing from one particle to the next through the area of contact. In the case of fine particles and motionless gas, the effective thermal conductivity is given by 0 1
ε @ke A ¼ β0 1 1 ð5Þ  kg kg @ A þ φ ks

due to the complexity in maintaining both pressure and hydrogen concentration constant as temperature varies. Also variation of solid thermal conductivity with temperature is dependent on the material and its hydrogen concentration. Therefore in the present work helium is used as the filling gas to study the effect of temperature on isobaric variation of ke, as shown in Figure 11. The ke is nearly constant at low pressures and increases with temperature at moderate and high pressures. The variation of ke is plotted along with the changes in concentration at constant pressures (Figure 12) and the pressure at constant concentrations (Figure 13) to bring out the effect of absorbed hydrogen in metal lattice. Figure 12 shows that at low pressure, the effect of temperature is negligible, and the concentration also remains constant. At a particular pressure, with increase in temperature the corresponding hydrogen concentration decreases and thereby the effective thermal conductivity decreases. The radiation effect is not significant at low temperature and small particle size. Figure 13 shows the variation of effective thermal conductivity with temperature at different concentrations. At a particular hydrogen concentration, the hydrogen gas pressure increases with an increase in temperature and thereby the effective thermal conductivity also increases.

Gas conductivity at different pressures and temperatures was calculated using REFPROP.26 The term j is empirically estimated as 0.078 in the porosity range of 0.3
0.5, while β is assumed to be unity.25 The effective thermal conductivity of the gas in the pore is calculated using the formula proposed by Chapman and Cowlings:2   kg 2σ 2
1 ð6Þ  ¼ 1þ Xe γ kg Pore diameter, which is assumed to be equivalent to the mean diameter of the particle is calculated from the experimentally found break even pressure using eq 4. The value of γ is taken as 0.15. The mean free path is a function of the density of the gas in the pore. In turn, it is a function of pressure and temperature,2 hence σ ¼

1:748  10
24  ðT þ 273Þ d2 P

ð7Þ

Thus effective thermal conductivity of the gas in the pore needs to be calculated as a function of pressure and temperature. 12996

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made for the analysis, the porosity variation is also shown in Figure 14. The thermal conductivity data for the pure MmNi4.5Al0.5 alloy is not available in the literature, and it has been back calculated using the data for the bed with argon, nitrogen, and helium, whose value is found to vary between 0.15 to 0.17 W/m K. The variation of effective thermal conductivity with solid thermal conductivity at different porosities is shown in Figure 15. The effective thermal conductivity increases with decrease in porosity, and from the trend, it appears that at a particular porosity, the effective thermal conductivity saturates beyond a certain value of solid thermal conductivity.

Figure 14. Variation of solid thermal conductivity with hydrogen concentration.

Figure 15. Variation of effective thermal conductivity with solid thermal conductivity at different porosities.

4. CONCLUSIONS Effective thermal conductivity of MmNi4.5Al0.5 hydride is measured by an one-dimensional comparative steady-state axial heat transfer method. The experiments yield ke values between 0.1 to 1.2 W/m K in the pressure and temperature ranges of 0
50 bar and 0
100 °C, respectively. Hydrogen pressure exerts a dual effect on ke. While the thermal conductivity contribution of the gas phase depends on pressure, hydrogen concentration in the hydride also increases with pressure. While ke increases with pressure in the form of tilted “S” curve, the effect of temperature is seen to be minimal. The shape of the “S” curve and the middle region, where the influence of pressure on ke is felt, lies between 1 and 15 bar for the MmNi4.5Al0.5 hydride at an average temperature of 60 °C. Since, for most of the metal hydrides the middle region lies in the plateau region of hydriding or dehydriding process, in addition to the effect of hydrogen pressure, the effect of concentration should also be considered. Alternately, to study the influence of concentration, the effect of pressure should be considered. This is more significant in the case of metal hydrides (especially the AB5-type material studied here) which exhibit a large plateau slope because there is significant variation of pressure with concentration in the plateau region. The solid conductivity of the hydride was back calculated from the ke data and was found to increase with hydrogen concentration. ’ APPENDIX

Porosity of the metal hydride bed depends on hydrogen concentration. Assuming linear variation, the porosity is given as18 ε ¼ εo
ð1
εo Þ  ðwt%=wtmax %Þ  f VE

ð8Þ

Maximum expansion factor is calculated as f VE ¼

ΔV  N A  ðH=MÞmax  Fs  2  10
3 M s  M H2

ð9Þ

The solid thermal conductivity is an important but unknown parameter. It is back calculated using the eqs 5
9 and the experimental results as a function of concentration. Then, the effective thermal conductivity of the bed at any particular pressure, temperature, and hydrogen concentration can be calculated using the same equations. Figure 14 shows that the solid conductivity increases with an increase in concentration similar to the variation of pressure with concentration for a PCI. The increase in heat conduction due to the increase in contact area and reduction of contact resistance dominates over the decrease in particle conductivity. Hence solid thermal conductivity increases as the alloy transforms into metal hydride. While it is rather difficult to track the variation of porosity, based on the linear variation assumption

Estimation of Uncertainty. Estimation of the effective thermal conductivity is dependent on the thermal conductivity of a PTFE disk and the temperature gradients across the reference material and the metal hydride bed. The maximum uncertainty is estimated from the minimum values of the measured quantities, and the accuracies of the measuring systems which are listed in Table A1. The temperature gradient and the effective thermal conductivity are calculated using eqs 2 and 3, respectively. Uncertainty in the temperature gradient across the reference material is computed as follows:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δT ¼ ðδT8 Þ2 þ ðδT9 Þ2 þ ðδT10 Þ2 þ ðδT11 Þ2 ¼ 0:2 ðA:1Þ
dT

δ
dx


ref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s   ffi 0:2 2 15:8  0:0001 2 ¼ þ
¼ 10:75 0:02 0:02  0:02 ðA:2Þ

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Table A1. Uncertainties of Various Measured Quantities S. no.

measured quantity

measuring device

accuracy

minimum value

maximum uncertainty

1

bed temperature

K-type thermocouple

(0.1 K

273 K

(0.13%

2

pressure of hydrogen gas

piezoresistive-type transducer

(0.01 bar

1 bar

(0.01%

3

mass of metal hydride

digital weighing balance

(1 g

530 g

(0.25%

4

coolant inlet and out let temperatures

T-type thermocouple

(1.0 K

293 K

(0.34%

5

thermal conductivity of reference, PTFE disk

SGHP apparatus

(0.01 W/m K

0.2 W/m K

(5%

6

length

vernier calipers

(0.1 mm

4 mm

(2.5%

Uncertainty in the effective thermal conductivity estimation is found as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 12 0 u0 u u
C
C uB  2 B C C B uB dT

C Dk dT

C B Dk
B Dk
C B B δke ¼ u δ þ δk þ δ

C ref uB

C B dx
A Dkref dx
C u@ dT
@D dT
ref bed A t D

dx
dx
ref

bed

ðA:3Þ Dke
dT

D
dx


¼ ref

kref
dT


dx


¼

0:21 ¼ 7:8  10
5 1537:5

ðA:4Þ

Dke
dT

D
dx


bed

δke ¼

ðA:5Þ

ðA:6Þ

bed

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7:8  10
5  10:75Þ2 þ ð0:51  0:01Þ2 þ ð7:01  10
5  36:65Þ2

ðA:7Þ

Minimum value of ke measured is 0.107 W/m K: δke 0:0057  100 ¼ 5:32%  100 ¼ 0:107 ke

β ε εo j

kref 

¼ 0:0057

thermal conductivity of gas in infinite space, W/m K thermal conductivity of gas in the pores, W/m K thermal conductivity of solid, W/m K mean diameter of the hydride particle, m molecular weight of alloy molecular weight of hydrogen Avogadro’s constant, 6.0221415  1023 pressure, N/m2 breakaway pressure, N/m2 diameter of hydrogen atom determined on the basis of viscosity,m temperature, °C volume expansion of the hydride, 2.9  10
30 m3/ H-atom effective pore length, m distance from the top of the cell, m

Greek Symbols

bed


dT


dx
0:21  790 ¼
0
1ref2 ¼
¼ 7:01  10
5 ð1537:5Þ2
@dT

A dx


T ΔV Xe x

bed


dT


dx
Dke 790 ¼ 0:51 ¼

ref ¼
1537:5 Dkref dT

dx


kg k*g k*s ls Ms MH2 NA P Pb s

ðA:8Þ

The maximum uncertainty in estimating ke is 5.32%.

’ AUTHOR INFORMATION

σ γ Fs

ratio of the average length between the centers of two neighboring solids in the direction of heat flow to the mean diameter of packing porosity of the bed initial porosity of the bed (without hydrogen) effective thickness of fluid in void in relation to the thermal conduction/mean diameter of the solid (lv/Dp) mean free path of gas molecules, m accommodation coefficient alloy density, kg/m3

Subscripts

1
11 b bed e g H2 o ref s T

thermocouple locations break away hydride bed effective gas hydrogen initial reference material solid temperature

Corresponding Author

*E-mail: [email protected]. Telephone: þ91-44-22574680.

’ NOMENCLATURE d molecular diameter of gaseous hydrogen, m maximum expansion factor fVE H/Mmax weight of hydrogen absorbed per mol of metal, g of H2/mol metal effective thermal conductivity of metal hydride bed, ke W/m K

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dx.doi.org/10.1021/ie200116d |Ind. Eng. Chem. Res. 2011, 50, 12990–12999

Industrial & Engineering Chemistry Research

ARTICLE

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dx.doi.org/10.1021/ie200116d |Ind. Eng. Chem. Res. 2011, 50, 12990–12999