Thermal Science & Engineering Vol.14 No.1 (2006)

Reaction Rate of Ti0.18Zr0.84Cr1.0Fe0.7Mn0.3Cu0.057 to Use for the Heat Driven
Type Compact Metal Hydride Refrigerator*
Sang-chul BAE† and Masafumi KATSUTA‡
Abstract
Our final goal of this study is to develop the heat driven type compact metal hydride (MH) refrigeration system forthe vending machine and the show case, and to attain a refrigeration temperature of 243 K by using a heat source ofabout 423K. The reaction rate of the MH to use for the heat source, MH used for heat source is studied firstly because
the MH refrigeration system consists of two MHs, one is used for the heat source and the other is used for the coolingload extracting. As for the reaction rate in the hydriding process, initially, a rapid surface reaction, governed by the
relation 1 − (1 − F )1 3 = k h t . After the MH surface has been covered by hydride, the reaction becomes diffusion
controlled with the relation 1 − 3(1 − F ′) 2 3 + 2(1 − F ′) = k h′ t . The reaction rates, kh and k’h, are exponentially
proportional to the pressure difference and increase with temperature. And, as for the dehydriding process, it is foundout that the rate-controlling step is uniquely diffusion reaction. The dehydriding reaction rate is exponentially
proportional to the pressure difference and the initial reacted fraction, and increases with temperature. Finally, on thebasis of these experimental results, the brand new rate correlations are reasonably derived. The predicted results for thiscorrelation are in successfully agreement with the experimental ones.
Key Words： Reaction rate, Plateau pressure, Rate-controlling step, Reacted fraction
NomenclatureA: Surface area (m2)C
: Heat capacity (J/K)
c: Hydrogen concentration (wt %)Di: Diffusion reaction coefficient (cm2/s)
E: Activation energy (J/molH2)
F: Reacted fraction (c/cmax.)
dF/dt: Reaction ratek
: Reaction rate constantks: Surface reaction rate constant (cm/s)
P: Pressure (MPa)R: Gas constant (J/kgK)r: Radius (mm)x
: Mass(g)T: Temperature (T)t: Time (s)

di
d
h
high
nc
p
s
v
l
max.

: Diffusion reaction: Dehydriding reaction: Hydriding reaction: Used at a higher temperature for heat source: Temperature constant: Plateau: Surface: Volume: Initial: Maximum

1. Introduction
Recently, the cascade use of the energy, especiallylow-quality thermal energy, is desired all over the world,to achieve the reduction of CO2 discharge amountregulated by COP3 Kyoto Protocol and to conserve theenergy source. In the refrigeration field, the research
on a refrigerator using waste heat and natural energy
attracts attention from such demand. One of thepromising candidates is Metal Hydride (MH)refrigeration system [1, 2].
In order to design the heat exchanger to use for theheat driven type MH refrigerator, we must satisfy thenext heat flow equation [3].

Greek letters: Reaction enthalpy (J/molH2): Pressure difference (MPa): Effective thermal conductivity (W/mK): Thickness (mm): Packing density (kg/m3)

ΔH
ΔP
λ
δ
ρ

Subscripts∗ Received : September 26, 2005, Editor: Akira YABE†
Institute of Ocean Energy, Saga University (1 Honjo-machi, Saga, Saga, 840-8502, JAPAN)‡
Department of Mechanical Engineering, Waseda University (3-4-1 Okubo, Sinjuku-ku, Tokyo, 169-8555, JAPAN)-1 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

ΔH ⋅ cmax . ⋅ x dF
2.016 ⋅ 100 dt

=

λA
δ

(T − TS ) + C

dTexperiment. The two points of pressure in reservoir andreactor are measured with a strain gauge. The vacuum
level in the system is measured with the vacuum gauge.
The accuracy of the temperature and pressure
measurement were ±0.1 ℃ and ±25 Pa, respectively.
The sampling interval of measurements is about 0.17 s incase of integration time 16.7 msec. MH alloy is crushed
beforehand and the average initial diameter of MH using
our experiment is about 0.43 mm. In this research, thecapacity method (Sievert's method) is used as a methodof measuring the hydrogen storage capacity and reaction
rate of MH alloy. In the Sievert's method, the hydrogenstorage capacity of MH alloys is calculated from thepressure change before and after the equivalence state of
two containers of already-known capacity, reservoir andreactor. Therefore, the precise capacity measurement ofreservoir and reactor are important. The accuracy of
capacity measurement is ±0.1 cm3 .
PCT (Pressure-Concentration-Temperature) curvemeasurement based on the vacuum condition was used
as an origin point of the hydrogen storage capacity of theMH alloy. Moreover, details of these methods are basedon Japan Industrial Standards (JIS).

(1)

dt

where cmax. is the hydrogen concentration (metal hydride)of equilibrium pressure 1.5 MPa at every temperature,λA/ δ is the heat transfer factor (λ, A and δ are the MH
effective thermal conductivity, area and thickness
respectively), C is the heat capacity of a system anddF/dt is the reaction rate．
In this paper, our interests firstly focused to thedetermination of the reaction rate, namely, dF/dt in theheat flow equation [3]. Then, the dF/dt of MHHigh (MHused at a higher temperature for heat source) is studied.
2．Experimental apparatus and methods
2.1 Experimental apparatus
Figure 1 shows details of the experimental apparatus.
The pipeline of this apparatus is mainly composed ofstainless steel，6.35mm O.D. (1/4 in) tube. Three pointsof temperature, environment air, at temperature constantbath and reservoir are measured by using mineralinsulated thermo-couple (T type) of 1mm O.D. in theGas

Thermocouple
T

ReleaseVacuumTd：ValveReservoir
Tank

P
PIRANI
Vacuum：JointPd

One - way Valve

Strain Gage
Transducer
Pr

He

H2

HeliumGas

Hydrogen
Gas
CylinderSupplyCylinderGas

FilterMH
PowderTr
Dynamic
StrainAmplifier

ReactorConstant
Temperature
BathData Acquisition
Controller

Personal
ComputerFig. 1 Schematic Diagram of Experimental Facility-2-© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

φ 84
φ 64
φ 40
φ 30

O-ring(G25)

O-ring13
O-ring(G35)22
φ 25FilterMH bedFig. 2 Detail of Reaction Rate Measuring Reactor
2.1.1 Reactor for Reaction rate measurement
MH reactor was made of stainless steel. Moreover,to suppress the temperature gradient by thegeneration of reaction heat in the MH alloy bed, areactor internal surface area is widely designed asmuch as possible. In order to act as the heat sink, thethermal capacity of the reactor to the MH alloy isdesigned by 1000 times or more. Figure 2 shows thedetails of reactor.

experiment is expressed by dotted lines. Proceduresof hydriding reaction rate experiment are as follows:
1) Close the valve between reactor and reservoir. 2)
Supply hydrogen gas of certain pressure to reservoir.
3) Vacuum the reactor which has the activated metal
hydride inside, and immerse it into the thermal
constant bath. 4) Open the valve between reactor andreservoir. 5) Measure and record the pressure changeof the system. 6) Calculate the reaction rate.
For the dehydriding experiment, almost all theprocedures are the same as the hydriding test which
has listed above. Only hydrogen gas will be supplied
to the reactor and the reservoir will be vacuumed
every time.
2.2.2 Experimental conditions
Tables 1 and 2 show the pressure and thetemperature dependency of the hydriding reaction
rate experiment condition. Reservoir capacity in the
hydriding reaction rate experiment is 132.2 cm3 .
Table 1 Hydriding Experimental Conditions ofDependency on Pressure Difference
Temperature
30℃
Initial Hydrogenwt%
0Concentration
Pressure Difference
MPa 0.53, 0.64, 0.72,
0.82, 0.91

2.2 Experimental procedure and conditions
2.2.1 Experimental methods

Table 2 Hydriding Experimental Conditions ofDependency on Temperature
Pressure DifferenceMPa
0.63
Initial Hydrogenwt%
0Concentration
Temperature
30, 50, 70, 90℃

E q u i lib r u i m P r essu r e
Hy d r og en P r ess u r e

A bsorp tion
Hydriding
Process

Pressure

Pr ocess

Moreover, experimental conditions of dehydriding
reaction rate are shown in Tables 3 and 4. But, theinitial hydrogen concentration of MH in thedehydriding condition cannot be controlled. The
condition concerning the dependency of an initialhydrogen concentration of MH is included in these
conditions. Reservoir capacity in the dehydriding
reaction rate experiment is 80.7 cm3 .

D esorp tion

Dehydriding
ProcessP r ocess

HHydrogenyd r og en
C om p o wt%
sitio nConcentration
Table 3 Dehydriding Experimental Conditions ofDependency on Pressure Difference
Temperature
160℃
Initial Hydrogenwt%
about 0.834
Concentration(1.5 MPa)
PressureMPa
0.41, 0.50, 0.59,
Difference
0.68, 0.78

Fig.3 Experimental Procedure
The inner condition of a reactor and reservoir in
the reaction rate experiment can be shown by the
PCT curve of Fig. 3. It expresses the changing oftemperature, hydrogen concentration and pressure of
the reactor and the reservoir during the reaction rate
experiment. The system pressure change during-3 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)a minor role during this stage, as that part of thealloy not covered by hydride is always in contact
with hydrogen. The surface reaction is therefore
postulated as the rate-controlling step. After the alloy
particles have been completely covered by a hydride
layer, the transport of materials through the layer bydiffusion becomes rate-controlling step. At this stage,
the reaction rate decreases and becomes slow.

Table 4 Dehydriding Experimental Conditions ofDependency on Temperature and Initial Hydrogen
Concentration
Pressuredifference,
MPa
130℃
140℃
150℃
160℃

0.27, 0.35, 0.43, 0.58
0.30, 0.36, 0.41, 0.46
0.31, 0.34, 0.40, 0.46
0.38, 0.40, 0.46, 0.50

0.8
0.7
Reacted Fraction F

Initialhydrogenconcentration, wt%

0.39

Moreover, MH alloy bed thickness is about 0.9mm
in the experiment.
3. Experimental results
3.1 The radius and the packing density of MH
particles
The surface area radius rs and the volume radius rv
of MH particles after the activation are measured byusing the scanning electron microscope (SEM). To
determine the surface reaction rate Eq. (4) and the
diffusion rate Eq. (5), these measured rs and rv aresubstituted, respectively. In addition, the rs and rv
before activation are also measured. They are shownin the Table 5. As for MH particles size, it isunderstood that it is atomized down to about 1/40 by
the activation.

Surface Radius rs
Volume Radius

rv

After

Activation

Activation

0.215 mm0.00437 mm

0.219 mm

0.91MPa

0.4
303K

0.3

0.82MPa
0.72MPa

0.2

0.63MPa

0.1

0.54MPa

0
0

20
30
40
Time sec
Fig. 4 Hydriding reacted fraction vs. time10

1.2
ΔP=P-P p

303K

Mean Radius of MH before and after
Activation
Before

ΔP=P-P p

0.5

dF/dt

Table 50.6

1

0.91MPa

0.8

0.82MPa
0.72MPa

0.6

0.63MPa②

0.4

0.54MPa①

0.2

0.00530 mm

0The packing density is estimated using thefollowing assumptions. 1) MH particle is a globularform. 2) The Peret diameter is used as a particle
diameter. As a result, this value is 3.665 kg/m3 .

0

0.2
0.4
0.6
Reacted Fraction F
Fig. 5 Hydriding rate-controlling step

3.2 The reaction rate of hydriding process

0.8

After various trials, we attained the result that kh,
k’h of MHHigh are governed by the following
expressions [4, 5].

3.2.1 The pressure difference dependency
The reacted fraction F at constant temperature(303K), as a function of time t, is shown in Fig. 4.
Fig.5 shows the relations between dF/dt and thereacted fraction F. A relatively rapid reaction ratetakes place for values of the reacted fraction F up to
about 0.3. It is evident that hydrogen diffusion plays()

kh = k s g P , Pp h(T ) / Rρrs

k 'h = 6 Dih g ' ( P , Pp ) h ' ( T ) / R ρ rv

-4-

(2)
2

(3)

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

where ks is a rate constant in cm/s, Dih is a diffusion
1/ 3
coefficient in cm2/s. Plots of 1 − (1 − F )and
2/3
1 − 3(1 − F ′) + 2(1 − F ′) at constant temperature(303K), as a function of time, are shown in Fig.6. k’his given by the slope of the straight lines shown in
Fig. 6. As ΔP becomes larger, it is recognized that kh
and k’h are enhanced. The experimental data are
possible to be correlated by using the following
exponential approximation [6, 7] as function of ΔP
among the present approximations.

3.2.2 The temperature dependency
1/ 3
Fig. 7 shows the plots of 1 − (1 − F )and
2/3
1 − 3(1 − F ′) + 2(1 − F ′) as a function of time atconstant ΔP. kh and k’h are given by the slope of thestraight lines shown in Fig. 7. kh and k’h become
higher with increasing temperature. After various
trials, the temperature dependency of kh and k’h canbe represented by the Arrhenius expressions [8, 9]
and the activation energies are also obtained asshown in Eqs. (6) and (7).
Surface reaction;

Surface reaction;k h = 0.2662 ( P − Pp )0.58

k h = 0.2605 exp( −545.7 / RT ) ,(4)

Eh , s = 545.7 J/mol H2

Diffusion reaction;

k 'h = 0.01818 ( P − Pp )1.62

Diffusion reaction;(5)

k h′ = 0.7000 exp( −1.214 × 10 4 / RT ) ,0.12
ΔP=P-P p

0.1

Eh,di = 1.214 ×104 J/molH2

1-(1-F)

1/3

0.91MPa
0.08

0.72MPa

0.06

0.54MPa

(7)
−2

0.04

The constant,and thek s = 1.653 × 10
−6coefficient, Dih = 1.425 × 10 in the Eqs. (2) and(3) are estimated by using the above results.
Therefore, the brand new rate correlations of
hydriding process are expressed as follows.

khSurface reaction step ( F < 0.3)

0.02

1 − (1−F )

1/ 3

0
0

0.1

0.2
Time sec

0.3= kh t−20.4

k = (1.653 ×10
× ( P−PP )

0.25

0.58

/ Rρ r )

(8)

s

h

(a) Surface reaction

× exp(−545.7 / RT )Diffusion reaction step ( F > 0.3)

(

)

0.2

1 − 3 (1−F ')0.15

k 'h = (6 ×1.425 ×10

2/3+ 2 1 − F ' = k 'h t
−6

2/3

1-3(1-F') +2(1-F')

(6)

k'h

0.1

ΔP=P-P p

× ( P−PP )

1.62

2

/ Rρ r )ν

(9)4
× exp(−1.214 ×10 / RT )0.91MPa

where F ′ = ( F − F0 ) /(1 − F0 ), F0 = 0.3 and F0 is areacted fraction at the beginning of diffusion
reaction.

0.72MPa

0.05

0.54MPa00

1020
Time sec

30

3.3 The reaction rate of dehydriding process

3.3.1 The pressure difference dependency
The reacted fraction F at constant temperature (433K), as a function of time t, is shown in Fig. 8.

(b) Diffusion reaction
Fig. 6 f(F) as a function of time at 303 K-5 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

0.12

1
363 K
343 K
323 K
303 K

0.08

433 K

0.06

Reacted Fraction F

1-(1-F)

1/3

0.1

kh

0.04
0.02

0.8

0.41MPa
0.50MPa

0.6

0.59MPa
0.68MPa

0.4

0.78MPa

0
0

0.1

0.2
Time sec

0.3

0.2

0.4

0

10
15
20
25
Time sec
Fig. 8 Dehydriding reacted fraction vs. time(a) Surface reaction
0.25

5

0
363 K
343 K
323 K
303 K

0.2

k'h

ΔP=P p-P

-0.2
dF/dt

0.15

433 K

-0.1

2/3

1-3(1-F') +2(1-F')

ΔP=P p-P

0.1

0.41MPa-0.3

0.50MPa-0.4

0.59MPa
0.68MPa

-0.5

0.05

0.78MPa
-0.6

0

-0.7
0

5

10
15
Time sec

20

25

0

(b) Diffusion reaction
Fig. 7 f(F) as function of time at ΔP =0.63 MPa

0.2

0.4
0.6
0.8
Reacted Fraction F

1Fig. 9 Dehydriding rate-controlling step

Figure 9 shows the relations between the reaction
rate dF/dt and the reacted fraction F, and it is foundout that the dehydriding process is dominated by aunique rate-controlling step.
About the dehydriding process the relations of thereacted fraction F vs. time t are expressed with theEq. (9). But, the F’ of Eq. (9) is calculated as
F’=F1-F. F1 indicates an initial reacted fraction
during the dehydriding process. The initial conditionof dehydriding experiments coincide with the end of
hydride process so that it is difficult to keep theinitial reacted fraction F1 at every test, this effectshould be taken into account to the dehydriding
process. Therefore, as for the dehydriding process,we propose Eq. (10) by introducing the c1nc termwith respect to F1.

k 'd = (6Did / Rρ rv 2 ) g '( PP , P)h '(T )c1nc(10)

where c1 is the initial concentration of thedehydriding process in wt %.
2/3
Plots of 1 − 3(1 − F ′) + 2(1 − F ′) at constanttemperature (433K), as a function of time, are shownin Fig.10. The reaction rates k’d are given by theslopes of the straight lines shown in Fig.10. The
qualitative trend of k’d to ΔP is almost similar as
those of ks and k’s. The correlate of experimental
data is also possible by the exponential function [6,7] of the pressure difference ΔP such as Eq. (11).

k 'd = 0.085( Pp − P )1.8

-6-

(11)

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

5

0.12
Δ P =P p -P

Δ P= 0 .4 1 M Pa

4 .5

0.78M P a
0.68M P a4

0.59M P a

3 .5

nc

+2 (1 -F')

0.06

1 -3 (1 -F')

0.08

2/3

0.1

0.50M P a

0.04

k 'd

3

0.41M P a

2 .5

0.02

2

0
0

2

4

400

6

410

Time sec

420
430
Te m pe ratu re K

440

Fig. 12 nc vs. temperature

Fig. 10 f (F) as function of time at 433K

The Arrhenius plot [8, 9] for the apparent rate
constant defined as the k d′ / c lnc at the constant ΔP(0.41 MPa) is shown in Fig. 13. The activationenergy of dehydriding process is found to be− 7.1 × 10 4 J/molH2. From the above, the constant
Did of the Eq. (10) is decided as 62.07. Therefore, it
can get the brand new rate equation of thedehydriding process as follows;

3.3.2 The temperature and the initial reacted fraction
dependency
A plot of the reaction rate k’d at the constanttemperature (403K), and the constant ΔP (0.41MPa),
with respect to c1, is shown in Fig.11.
The tendency of these experimental data can be
related to the exponential function of the c1 as shown
Eq. (12). And, the relation of the nc vs. thetemperature at constant ΔP (0.41 MPa) is illustrated
in Fig.12.

1 − 3(1 − F ′) + 2(1 − F ′) = k d′ t
2

3

k d′ = (6 × 62.07 / R ρ rd2 ) × ( PP − P )1.8

k d′ ∝ c1nc = 0.00495c12,82

× exp( −7.1 × 10 4 / RT ) × c10.0066T − 23.58(12)= 0.00495c10.066T − 23.58

0 .0 0 1 2

0.1

0 .0 0 1

Δ P= 0 .4 1 M Pa

Δ P=0.41MPa

k'd/c１nc

0 .0 0 0 8
k' d

(13)

0 .0 0 0 6

0.01

0 .0 0 0 4
0 .0 0 0 2
0.001

0
0 .2

0 .3

0 .4
c 1 w t%

0 .5

2.3

0 .6

Fig. 11 Dependency of kd′ on initial concentration

2.35

2.4
1000/T

2.45

2.5

Fig. 13 Arrhenius plot of dehydriding process

at 403 K-7 -© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No.1 (2006)

0.025
+10%

0.26
0.22

±0%
-10%

0.18
0.14

Experimental Value k'hExperimental Value kh

0.3

0.1

0.02

+20%

±0%

0.015
-20%

0.01
0.005
0

0.1

0.15
0.2
0.25
Calculated Value kh

0.30(a) Surface reaction

0.005
0.01
0.015
Calculated Value k'h

0.02(b) Diffusion reaction

Fig.14 Calculated data vs. experimental data in hydriding process
shown in our experimental results, because thereaction of MHHigh becomes a saturated condition
within 20 seconds, it can be used for the heat driven
type MH refrigerator.

3.4 Comparison

3.4.1 Hydriding process.
Comparison between experimental reaction rates
and predictions is made and represents in Fig.14. As
for the surface reaction step, the experimental results
are expressed by the brand new rate Eq. (8) in theerror within ±10%. Additionally, as for diffusion
reaction step, the experimental data agrees withprediction using brand new rate correlation within±20 %.

Acknowledgements
This work has been partially funded by Waseda
Advanced Research Institute of Science andEngineering.
References[1] Takeda, H. , Kabutomori, T., Murai, M. and Jin,
T., JSW technical report, No.54, (1998), 39-46.
[2] Uehara, I., Sakai, T. and Ishikawa, H., Journal of
Alloys and Compounds, Vol. 253-254, (1997),
635-641.
[3] Rudman, P. S., Journal of the Less-Common
Metals, Vol. 89, (1983), 93-110.
[4] Stander, C. M., Physic. Chemie Neue Folge,(1977), 229-238.
[5] Levenspiel O., Chemical Reaction Engineering,John Wiley & Sons, Inc., (1972).
[6] Suda, S. and Komazaki, Y., Journal of the LessCommon Metals, Vol. 89, (1983), 127-132.
[7] Suda, S., Kobayashi, N. and Yoshida, K., ibid.,
Vol. 73, (1980), 119-126.
[8] Wang, X. L. and Suda, S., ibid., Vol. 159, (1990),
83-90.
[9] Han, S. C. and Lee, J. Y., ibid., Vol. 128, (1987),
155-165.

3.4.2 Dehydriding process
On the other hand, for the dehydriding process, it
is understood that the experimental data are
predicted by the brand new rate Eq. (13) in the error
within ±15% from the comparison.
4. Conclusion
In the initial stage of the hydriding process of
MHHigh (F<0.3) the rate-controlling step is a surface
reaction. And, at a later stage of the hydriding
process (F>0.3) the diffusion reaction is the rate
controlling step. As for the dehydriding process, it isunderstood that only diffusion reaction is therate-controlling step. The reaction rates are
exponentially proportional to the pressure difference
and increases with temperature. The experimental
reaction rates of MHHigh are represented by the brandnew rate correlations in the error within ±20 %. As-8-© 2006 The Heat Transfer Society of Japan