Energy Fuels 2010, 24, 2596–2606 Published on Web 03/11/2010
: DOI:10.1021/ef901430a
Thermal Stability of Ammonia Borane: A Case Study for Exothermic Hydrogen Storage Materials Scot D. Rassat,* Christopher L. Aardahl, Tom Autrey, and R. Scott Smith Pacific Northwest National Laboratory, P.O. Box 999, 902 Battelle Boulevard, Richland, Washington 99352 Received November 23, 2009. Revised Manuscript Received February 3, 2010
A model to determine the thermal stability of chemical hydrogen storage materials was developed to predict the stability of ammonia borane at 50-60 °C, the extreme range of environmental temperatures for hydrogen storage materials in PEM fuel cell applications. Experimental data from differential scanning calorimetry measurements between 70 and 85 °C were used to develop isothermal and adiabatic reactivity models that could be extrapolated down to temperatures of interest. Results of the analysis show that solid ammonia borane is metastable in the 50-60 °C range, having stability against appreciable reaction on the order of a week at 60 °C and months at 50 °C. An adiabatic analysis is the most extreme case, and stability of many days under such extreme conditions indicates that solid ammonia borane may be suited to most hydrogen storage applications. This type of analysis is also applicable to other candidate hydrogen storage materials.
system targets, which include both hydrogen storage material and storage system components as a basis.8,9 Furthermore, there has been significant progress to address other important targets: enhanced rates of hydrogen release,10-14 purity of hydrogen gas,4,15-17 and more efficient synthesis procedures to decrease the cost of AB.18 On the other hand, there is little information regarding the thermal stability of AB under extreme operating conditions. This target is critical for a material that releases hydrogen at low temperatures7,16,19 in an exothermic reaction,16,19 from a safety and engineering perspective. The US DOE has established 60 °C as an upper operating temperature limit for hydrogen storage on vehicles. The literature provides some evidence that solid ammonia borane is stable at or below this temperature. Mayer reported 0.6 mL H2 released from 1.3 mmol of solid ammonia borane when stored 24 h at 60 °C.20 Although the gas volume measurement conditions were not specified, it can be conservatively estimated that ∼0.02 mol of H2 per mole of ammonia
Introduction As evidence builds for global climate change, strong motivation has developed for energy sources with a low carbon footprint. While centralized power production using coal or natural gas allows for large-scale CO2 sequestration to limit carbon emissions, distributed and mobile power applications are being addressed by another potential approach. That is, employing hydrogen-based fuel cell technology, which could allow zero emissions at the point of energy usage. Such approaches have historically suffered from low energy density or expensive means to contain the hydrogen until needed to produce power. High density hydrogen storage via solid chemical fuels is one approach being examined to allow carbon-free approaches to applications such as vehicles, back-up power, and power sources for small instruments such as consumer electronics and sensors.1,2 Ammonia borane (AB), NH3BH3, is a particularly interesting hydrogen storage material due to an inherently large gravimetric and volumetric hydrogen density.3 For example, solid AB releases between ∼2 and 2.5 mol equiv of hydrogen,4-7 corresponding to a material gravimetric capacity of 13 to 16 wt % H2. The large gravimetric and volumetric capacities for neat ammonia borane provide some margin to meet or exceed the US Department of Energy (DOE) storage
(8) U.S. Department of Energy; 2009, http://www1.eere.energy.gov/ hydrogenandfuelcells/storage/pdfs/targets_onboard_hydro_storage.pdf (9) Aardahl, C. L.; Rassat, S. D. Int. J. Hydrogen Energy 2009, 34, 6676–6683. (10) Bluhm, M. E.; Bradley, M. G.; Butterick, R., III; Kusari, U.; Sneddon, L. G. J. Am. Chem. Soc. 2006, 128, 7748–7749. (11) Keaton, R. J.; Blacquiere, J. M.; Baker, R. T. J. Am. Chem. Soc. 2007, 129, 1844–1845. (12) Blaquiere, N.; Diallo-Garcia, S.; Gorelsky, S. I.; Black, D. A.; Fagnou, K. J. Am. Chem. Soc. 2008, 130, 14034–14035. (13) Denney, M. C.; Pons, V.; Hebden, T. J.; Heinekey, D. M.; Goldberg, K. I. J. Am. Chem. Soc. 2006, 128, 12048–12049. (14) Neiner, D.; Karkamkar, A.; Linehan, J. C.; Arey, B.; Autrey, T.; Kauzlarich, S. M. J. Phys. Chem. C 2009, 113, 1098–1103. (15) Baumann, J.; Baitalow, F.; Wolf, G. Thermochim. Acta 2005, 430, 9–14. (16) Gutowska, A.; Li, L.; Shin, Y.; Wang, C. M.; Li, X. S.; Linehan, J. C.; Smith, R. S.; Kay, B. D.; Schmid, B.; Shaw, W.; Gutowski, M.; Autrey, T. Angew. Chem., Int. Ed. 2005, 44, 3578–3582. (17) Palumbo, O.; Paolone, A.; Rispoli, P.; Cantelli, C.; Autrey, T. J. Power Sources 2010, 195, 1615–1618. (18) Heldebrant, D. J.; Karkamkar, A.; Linehan, J. C.; Autrey, T. Energy Environ. Sci. 2008, 1, 156–160. (19) Wolf, G.; Baumann, J.; Baitalow, F.; Hoffmann, F. P. Thermochim. Acta 2000, 343, 19–25. (20) Mayer, E. Inorg. Chem. 1972, 11, 866–869.
*To whom correspondence should be addressed. E-mail: sd.rassat@ pnl.gov. (1) Crabtree, G.; Dresselhaus, M.; Buchanan, M. Phys. Today 2004, 57 (12), 39–44. (2) Orimo, S.; Nakamori, Y.; Eliseo, J.; Zuttel, A.; Jensen, C. Chem. Rev. 2007, 107, 4111–4132. (3) Hamilton, C. W.; Baker, R. T.; Staubitz, A.; Manners, I. Chem. Soc. Rev. 2009, 38, 279–293. (4) Hu, M. G.; Geanangel, R. A.; Wendlandt, W. W. Thermochim. Acta 1978, 23, 249–255. (5) Baitalow, F.; Baumann, J.; Wolf, G.; Jaenicke-R€ oßler, K.; Leitner, G. Thermochim. Acta 2002, 391, 159–168. (6) Storozhenko, P. A.; Svitsyn, R. A.; Ketsko, V. A.; Buryak, A. K.; Ul’yanov, A. V. Russ. J. Inorg. Chem. 2005, 50, 980–985. (7) Zheng, F.; Rassat, S. D.; Heldebrant, D. J.; Caldwell, D. D.; Aardahl, C. L.; Autrey, T.; Linehan, J. C.; Rappe, K. G. Rev. Sci. Instrum. 2008, 79, 084103. r 2010 American Chemical Society
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borane was released assuming standard temperature and pressure conditions (e.g., 0 °C and 1 atm). Mayer also reported that ammonia borane produced by the described method yielded a product that was stable at room temperature for at least 2 months. Storozhenko et al. investigated the physicochemical properties of crystalline ammonia borane and solutions.6 They observed that no gas was evolved from the pure solid when stored for 4 months under vacuum at room temperature, and the composition of dilute ether solutions of ammonia borane were stable for at least 40 h at 50 °C. However, Shaw and co-workers showed that concentrated solutions of AB are not stable, that is, a 1 M solution of AB in THF has a half-life of hours at 50 °C.21 This current work generally defines a framework to judge thermal stability of all hydrogen storage materials when stored for long periods at the US DOE targets of 50 to 60 °C, and, as a specific example, seeks to extend the limited understanding of ammonia borane thermal stability. The thermal stability modeling approach described here, an expansion of our previous analysis,22 is complementary to direct experimental measurements. Modeling can be used to more quickly screen the expected stability of possible storage materials under a wide range of conditions provided kinetic data at higher temperatures are available and the reaction mechanism is the same at low temperature. Further, there are practical experimental limitations in long-duration storage stability tests such as complications from thermal fluctuations, power outages, or gas leaks that will affect the uncertainty of experimental measurements. This paper describes the development of a kinetic model to predict the thermal stability of a chemical hydrogen storage material at temperatures too low to determine conveniently by experimental means, but at temperatures that are of upmost importance from safety and engineering implications. By example we investigate the thermal stability of ammonia borane under isothermal and adiabatic storage conditions. The adiabatic model, which accounts for the exothermic heat of reaction retained in the fuel bed, gives a more conservative, limiting estimate of AB thermal stability. While this paper specifically addresses neat solid ammonia borane as an example, the thermal stability considerations and aspects of the modeling approach are more generally applicable to any exothermic hydrogen storage material. Additionally, the stability analysis in this paper was completed for a representative source (supplier) of ammonia borane; other sources of AB may present different stability characteristics,22 and these could be assessed by applying the methods developed here directly. Thermal Stability Criteria. What constitutes a thermally stable hydrogen storage material is subject to some interpretation, as there are no clear-cut guidelines. In all cases, it is reasonable to define “thermal stability” in terms of a shelf life: the duration that the material can be stored in a stable configuration. Differences arise in the definition of a stable system. Possible differentiators include absolute limits on the loss of hydrogen storage capacity, criteria for an acceptable rate of loss of storage capacity, and system physical limitations (e.g., maximum design storage pressure). Such factors are used here to establish thermal stability targets.
The US DOE provides current targets for on-board storage systems for light-duty vehicles through their Web site. These targets are progressively more challenging with goals established for 2010, 2015, and ultimate performance. Included is the maximum operating temperature limit for systems in full sun exposure (“full-solar load”), which is defined as 50 °C in 2010 and 60 °C in 2015 and beyond. While the target limiting ambient operating conditions are defined, the criteria for acceptable performance degradation in these extremes have yet to be determined. In lieu of specific guidelines, thermal stability criteria are defined in this paper in terms of other DOE targets such as acceptable “loss of useable H2” due to leaks or venting: in 2010, 0.10 (g/h)/kg H2 stored; and in and after 2015, 0.05 (g/h)/kg H2 stored.8 Other practical considerations include the maximum storage system pressure. A benefit of ammonia borane and other chemical hydrogen storage materials is that they typically require no hydrogen overpressure. Therefore, storage vessels can be simplified and the weight reduced (e.g., compared to liquid hydrogen and compressed hydrogen gas tanks). However, this general benefit means that H2 generated unintentionally during storage will need to be safely vented from the storage tank at much lower pressure. A practical storage tank for a chemical hydrogen fuel will likely be rated to handle at least 8 atm (absolute) pressure, consistent with the initial DOE target minimum delivery pressure to a fuel cell (4 atm in 2010) and a 2-fold safety factor. For H2 internal combustion engines (ICE), the specified minimum delivery pressure is 35 atm, and in the extreme, the DOE target for the maximum delivery pressure from the storage system regardless of power plant is 100 atm. For the purposes of this paper, a simple physical model for a solid ammonia borane storage system is used. One possible solid ammonia borane storage configuration is a packed bed of powder or spherical pellets (0.74 kg/L intrinsic density23,24) having a minimum close-packed porosity of 30 vol%.9 Assuming that 1000 mol of ammonia borane (30.9 kg) in this form is enclosed by walls, with no additional gas headspace volume, the total internal storage system volume is 60 L, the gas pore volume is 18 L, and the volumetric density of the stored material ranges from ∼0.068 to 0.084 kg H2/L, depending on the number of moles of H2 released per mole of AB. 2.0-2.5 mol of H2 are likely to be targeted for release from each mole of ammonia borane in a practical reaction system, which is connected to but operated independently of the fuel storage unit.9 This corresponds to 4.0-5.0 kg of H2 stored in 1 kmol of ammonia borane. Conservatively, the lesser of these values is assumed in subsequent calculations. The pore volume of the storage system is also assumed to be initially filled with H2 gas at 1 atm, and the vessel walls are designed to contain H2 up to the established pressure limit (e.g., 8 atm in a fuel cell vehicle). Combining the DOE H2 loss targets, the storage pressure limits, and the features of the representative solid ammonia borane storage system, Figure 1 provides a storage time referenced thermal stability map based on 2010 criteria. It is assumed that any H2 released from ammonia borane in storage is retained within the pore volume of the system up to the defined storage system pressure limit (e.g., 1, 8, 35, or 100 atm). At 50-60 °C, this equates to ∼0.005, ∼0.022, and
(21) Shaw, W. J.; Linehan, J. C.; Szymczak, N. K.; Heldebrant, D. J.; Yonker, C.; Camaioni, D. M.; Baker, R. T.; Autrey, T. Angew. Chem., Int. Ed. 2008, 47, 7493–7496. (22) Rassat, S. D.; Smith, R. S.; Autrey, T.; Aardahl, C. L.; Chin, A. A.; Magee, J. W.; VanSciver, G. R.; Lipiecki, F. J. Prepr. Pap.-Am. Chem. Soc., Div. Fuel Chem. 2006, 51, 517–519.
(23) Hughes, E. W. J. Am. Chem. Soc. 1956, 78, 502–503. (24) Lippert, E. L.; Lipscomb, W. N. J. Am. Chem. Soc. 1956, 78, 503– 504.
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was supported on a silica scaffold framework. Gutowska et al. further related the exothermic heat released from neat AB measured by DSC to the extent of ammonia borane conversion (= integrated heat released up to elapsed time (t) per total heat released in complete reaction). This normalization resulted in a characteristic sigmoid-shaped fractional conversion curve for AB, similar to the results of evolved H2 gas volume measurements.7,19,25 An inherent assumption of this earlier work was that exactly one (1.0) equivalent of hydrogen was ultimately released in the reaction, consistent with the observations of Wolf et al.19 The Wolf et al. DSC data19 and the Gutowska et al. reaction extent normalization approach16 were used in our earlier AB thermal stability modeling studies.22 Gutowska and co-workers also completed isothermal DSC tests for neat AB in the range of 70-85 °C and reported the time to half conversion (t1/2) as a function of temperature. These DSC data are reassessed in the current paper, as described below, to estimate the thermal stability of ammonia borane during storage. The kinetics of H2 release from AB under isothermal conditions, as inferred from DSC data, is typified by sigmoidal release curves following a delay or induction period. The induction period increases in duration with decreasing reaction temperature.16,19 Similar kinetics for the thermal decomposition of neat AB have been reported for direct measurements of the evolved H2 gas volume as a function of time using, for example, isothermal gas buret systems.7,25 These direct measurements also indicate nominally one equivalent of H2 released for temperatures 1 is possible in systems where the rate of nucleation accelerates during the growth phase,30,31 therefore, n > 4 is realistic (e.g., β = 3 and λ > 1). Brown et al. define the logarithmic form of eq 1 as the Avrami-Erofeev equation,29 giving credit to two of the independent developers. In this paper it is simply referred to as the Avrami equation, which is common.30,31 In the literature,30,31 the Avrami eq 1 is often expressed with τ set equal to zero. The more restrictive form of the equation is applicable to the portion of a solids transformation curve starting from the point where stable nuclei have been formed
Eτ τ ¼ Aτ exp RT
ð3Þ
where R is the ideal gas constant; Ak and Aτ are the preexponential factors for k and τ, respectively; and Ek and Eτ are the apparent release reaction and induction process activation energies, respectively. The least-squares best fit values of the pre-exponential factors and apparent activation energies were subsequently applied in eqs 2 and 3 to estimate k and τ at the temperatures of interest. Applying the temperature-dependent k and τ values from eqs 2 and 3 in the Avrami eq 1 gives a “global” isothermal kinetics model. It was used to evaluate the reaction conversion as a function of time at a temperature of interest, which may be interpolated between or extrapolated outside the range of the temperatures of the “individual” isothermal Avrami curve fits used to develop the model. Rearranging eq 1, the global kinetics model can be used to estimate the duration t to a specific conversion fraction χ " #1=n 1 1 þτ t ¼ ln k 1 -χ 1 ¼ ½ -lnð1 -χÞ1=n þ τ; 0eχe1:0 k
(28) Wolf, G.; van Miltenburg, J. C.; Wolf, U. Thermochim. Acta 1998, 317, 111–116. (29) Brown, W. E.; Dollimore, D.; Galwey, A. K. In Comprehensive Chemical Kinetics - Volume 22, Reactions in the Solid State; Bamford, C. H.; Tipper, C. F. H. Eds; Elsevier Scientific Publishing Company: Amsterdam, 1980; pp 41-113. (30) Doremus, R. H. Rates of Phase Transformations; Academic Press, Inc.: Orlando, 1985; pp 24-26. (31) Rao, C. N. R.; Rao, K. J. Phase Transitions in Solids, An Approach to the Study of the Chemistry and Physics of Solids; McGrawHill, Inc.: New York, 1978; pp 81-95.
ð4Þ
Additionally, the time-derivative of the Avrami equation gives the reaction rate at a specified duration at isothermal temperature dχ ¼ nkn ðt -τÞn -1 exp½ -fkðt -τÞgn ; 0 for teτ ð5Þ dt Although DSC experimental results are analyzed in the framework of the Avrami equation in the current investigation, 2599
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step at T3 results in an overall conversion of χ = 0.39 mol equiv at storage time t4 (∼12 units). The greater overall conversion in the adiabatic process compared to the isothermal process (χ=0.39 vs χ=0.28) is due to acceleration in the reaction rate dχ/dt with increasing fuel temperature. The adiabatic reaction scheme depicted in Figure 2 is descriptive of a stepwise calculation process, but it is not quantitatively accurate. In particular, the time steps were made large to make the changes in the extent of reaction and differences in the rate of reaction readily observable, and the temperature change per unit conversion was manipulated to control the spread of the family of release curves. Representative calculation results are obtained by simultaneously solving the differential forms of the mass and energy balances for the adiabatic system and integrating over infinitely small step sizes. The following development is derived from standard treatments of heat transport phenomena in the chemical engineering literature (e.g., Bird et al.).32 It is developed in some detail here to emphasize the conservative assumptions used in the ammonia borane case study and to note opportunities to use the calculation framework to design systems to enhance thermal stability (e.g., by integration of heat exchangers). For a closed storage system in which no mass (e.g., hydrogen or ammonia borane) is transferred across the system boundary, no mechanical work is done on or by the system, and changes in kinetic and potential energy in the system are negligible (e.g., a fixed storage bed with no flow of species and no elevation change), the time rate of change of total internal energy in the system (dUtot/dt) is equal to the net rate that heat (Q) is added to the system
Figure 2. Schematic representation of an adiabatic, exothermic reaction process resulting in increasing reactant temperature (T3 > T2 > T1) and acceleration in the reaction rate compared to an isothermal process at T1.
the limited data are not used to gain further insight into the details of the reaction mechanism. Rather, eq 1 is used empirically as a suitable means to model the sigmoid ammonia borane conversion versus time data. The isothermal kinetic model was developed from and for the first equivalent of H2 released from ammonia borane, therefore it is not applicable above ∼120 °C when the second equivalent is released. Further, since the primary goal of this work is to develop an understanding of thermal stability at 50-60 °C, the model was tuned to favor extrapolation to lower temperatures. Energy Balance and Adiabatic Model. Figure 2 is a schematic comparison of reaction progress for isothermal and adiabatic storage systems during thermal dehydrogenation of an exothermic fuel such as ammonia borane. The individual isothermal conversion curves in the figure are representative of Avrami kinetics, eq 1. The picture shows the state of the fuel bed after some decomposition at temperature T1, giving a conversion fraction χ = 0.055 at elapsed time t1 (∼10.5 units). Following the isothermal release curve T1, the fractional conversion increases to χ = 0.10 at t2 after a finite time step Δt of 0.5 units, and after three equal time steps, a final conversion χ = 0.28 is reached at t4 (∼12 units). The equivalent adiabatic process in three equal time steps is depicted in Figure 2 by a series of filled circles, and lines and arrows connecting the points, on the three release curves corresponding to increasing storage bed and reaction temperature (T3 > T2 > T1). Like the isothermal process, the first time step at T1 of the adiabatic process increases the conversion fraction to χ = 0.10 at t2. However, in the adiabatic system, the heat released in the first time step causes the fuel bed temperature to increase from T1 to T2. The hydrogen release kinetics of the second time step shown on the conversion curve at T2 are given by eq 1 using n, k, and τ evaluated at T2 [e.g., per eqs 2 and 3]. Although the clock time (or actual storage time) at the start of the second step is t2, the time required to reach a conversion fraction χ=0.10 if the system were isothermal at T2 is the effective time t2,eff (∼8.8 units), as determined from eq 4 for T2. After a 0.5 unit time step at T2, the fractional conversion increases to χ = 0.19. The average reaction rate, Δχ/Δt, in this second step is 0.18 mol equiv per time unit [= (0.19-0.10)/0.5 units], twice the rate of the first step at T1. In turn, the heat released during the second step results in a further increase in the bed temperature to T3. After determining the effective time at the start of the third step (t3,eff = ∼6 units) from eq 4 using χ = 0.19 and the kinetic parameters for T3, the last 0.5 unit time
Xd X dUtot Ui ¼ Qacc ðtÞ ¼ ¼ Qj dt dt j i
ð6Þ
Subscript i denotes components of the storage system, and subscript j denotes various heat transfer pathways. Equation 6 is an overall energy balance expressing the rate of heat accumulation in the system, Qacc, in terms of the rate of change of internal energy in the storage system components. The change in internal energy for component i in the storage system is given by the product of the component mass (mi), the difference in temperature at any time, T(t), from a reference state (or initial) temperature, T0, and the constant volume mass specific heat capacity of the component (Cv,i) Ui ¼ mi Cv, i ½TðtÞ -T0
ð7Þ
The rate of change of internal energy for component i is the time derivative of Ui d d d Ui ¼ Cv, i ðTÞmi ðtÞ TðtÞ þ Cv, i ðTÞ½TðtÞ -T0 mi ðtÞ dt dt dt ð8Þ In this equation, the time and temperature dependence of the quantities is noted explicitly. For components of constant mass (e.g., the storage system structure), the rightmost term is zero and the change in internal energy is strictly a function of the rate of temperature change, dT(t)/dt. (32) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons, Inc.: New York, 1960; pp 456-491.
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any endothermic reactions or phase transformations in the fuel, the rate at which heat is lost at the storage vessel surface to the surroundings, the rate at which heat is conducted away from the storage system, and the rate at which heat is removed by heat exchanger elements (e.g., cool fluid in a radiator loop). The net heat transfer rate of exothermic and endothermic reaction processes occurring within the storage system is Qrxn, and the net external heat transfer rate due to storage system surface heat transfer processes, conduction, and auxiliary heat exchange components is Qext. The net rate that heat is added to the storage system is then X Qj ¼ Qin ðtÞ þ Qout ðtÞ ¼ Qrxn ðtÞ þ Qext ðtÞ ð11Þ
For incompressible fluids and solids, such as the storage system structure and the crystal-dense ammonia borane fuel pellets contained in it, the constant volume specific heat capacity is approximated by the mass specific heat capacity at constant pressure, Cp,i. Noting this, substituting eq 8 into eq 6, and expanding the internal energy summation quantity of eq 6 in terms of the individual storage system components gives the time-dependent rate of heat accumulation in the storage system Xd ð9Þ Ui ¼ ::: Qacc ðtÞ ¼ dt i -½mAB ðtÞCp, AB ðTÞ þ mAB -H2 ðtÞCp, AB -H2 ðTÞ
j
dT þ mH2 ðtÞCv, H2 ðTÞ þ mstr Cp, str ðTÞ ::: dt Z T Z T dmAB dmAB - H2 -½ -½ Cp, AB ðTÞdT Cp, AB -H2 ðTÞdT dt dt T0 T0 Z T dmH2 -½ Cv, H2 ðTÞdT dt T0
By convention, net heat addition terms are negative in sign whereas net heat loss terms are positive. In the most conservative thermal stability analysis for a fuel such as ammonia borane that releases H2 exothermically, the storage system is assumed to be adiabatic and there is no heat lost from the system by any means (i.e., Qext = 0). In this scenario, it is also assumed that once the environment has provided the heat to raise the fuel bed temperature to the initial storage temperature (T0) at time zero, the reaction heat is the only heat source. The net heat input rate is then given in terms of the reaction enthalpy
The storage system components are unreacted ammonia borane (subscript AB), ammonia borane that has reacted to liberate one molar equivalent of hydrogen gas (subscript AB-H2), pre-existing and liberated hydrogen gas (subscript H2), and the storage structure including outer walls and any internal elements (subscript str). The dmj/dt terms of eq 9 account for the rate of heat accumulated in components of changing mass referenced to T0 (i.e., the internal energy is zero at T0). An equivalent total heat accumulation result is obtained if the mass specific heat capacities are replaced with their molar specific equivalents (C0v,i and C0p,i) while the masses of components are replaced with the component molar quantities (ni). The total moles of remaining fresh (unreacted) ammonia borane [nAB(t)] and reacted ammonia borane [nAB-H2(t)] is equal to the moles of ammonia borane initially present in the storage system, and the change in the number of moles of fresh ammonia borane is equal in magnitude but opposite in sign to the change in the number of moles of reacted ammonia borane (i.e., dnAB = -dnAB-H2). Making molarquantity substitutions for fresh and reacted ammonia borane and hydrogen gas and assuming that the molar heat capacity of unreacted and reacted ammonia borane is the same, eq 9 simplifies to
X j
dχðtÞ nAB, 0 dt
ð12Þ
where nAB,0 is the total number of moles of ammonia borane fuel stored. Substituting eqs 10 and 12 into eq 6 subject to the conservative adiabatic model assumptions noted above, the overall energy balance becomes dT -½nAB, 0 C p0 , AB ðTÞ þ nH2 ðtÞC v0 , H2 ðTÞ dt dnH2 dχðtÞ - C v0 , H2 ½TðtÞ - T0 nAB, 0 ¼ ΔHrxn ð13Þ dt dt In this equation, the storage system structure mass has also been conservatively neglected as a heat sink. This is equivalent to assuming an infinite fuel bed, including any gas volume (e.g., pore space) and a mass-less wall to retain any evolved H2. It results in the most rapid rate of temperature increase in the fuel bed if all exothermic reaction heat is retained (i.e., adiabatic assumption). Assuming that all hydrogen gas released from the fuel is retained within the storage system, the total quantity of H2 in the system at t is a function of the molar equivalents of ammonia borane converted in the reaction from t = 0 to t, χ(t)
Qacc ðtÞ ¼ -½nAB, 0 C p0 , AB ðTÞ þ nH2 ðtÞC v0 , H2 ðTÞ dT dnH2 þ mstr Cp, str ðTÞ - C v0 , H2 ½TðtÞ - T0 dt dt
Qj ¼ Qrxn ðtÞ ¼ Qexo ðtÞ ¼ ΔHrxn
ð10Þ
nH2 ðtÞ ¼ nH2 , 0 þ nAB, 0 ½χðtÞ - χ0
In the rightmost term, the integral form of the hydrogen heat capacity has been replaced by the approximate form based on the mean molar heat capacity of hydrogen referenced to T0 (C 0v,H2), which is nominally constant in the temperature range of interest. The rate of heat input to the system, Qin, is the sum of the heating rate due to exothermic reactions (Qexo), the rate at which heat is added at the outer storage vessel surface (e.g., air convection and solar thermal radiation), the rate at which heat is conducted into the system through structural components, and the rate at which heat is added by auxiliary heaters or heat exchanger elements. Likewise, the rate of heat loss from the system, Qout, is the sum of the cooling rate due to
ð14Þ
Here, nH2,0 is the initial number of moles of hydrogen gas in the system, and χ0 is the initial mole fraction of ammonia borane that is reacted (= moles H2 released per mole of ammonia borane stored
