Heat and Mass Transfer Models
3.5 Solution methodology .1 Lab-scale MHHSD
The governing equations are discretized by using the finite volume method (FVM). The solution methodology is similar to the one presented in Muthukumar and Ramana, 2009. An alternative direction implicit (ADI) scheme with tri-diagonal matrix algorithm (TDMA) is used for solving the governing equations. Velocity terms are controlled using staggered grids to catch the heat transfer across the control volume by convection. Boundary conditions are applied using half control volume method. Values of different thermo-physical parameters are taken from Muthukumar et al., 2003; Muthukumar and Ramana, 2009 and Chung and Lin, 2009, and the
from Askri et al., 2003. The radiative information has been calculated as per the FVM outlined in Kim, 2008. The thermo-physical properties of hydriding alloy and various constants used in this thermal model are listed in Table 3.2. For the dimensions of the reactor given in Table 3.2, the effect of sizes of control volumes
r z
on the average bed temperature with time is studied. There is no change in results beyond 64 90 control volumes. Hence, in the following, all analyses are carried out with 64×90 control volumes.3.5.2 MHHSD with ECT
The solution procedure and the assumptions to simplify the governing equations are similar to the one presented in Muthukumar et al. (2012). The thermo physical properties of LmNi4.91Sn0.15
and hydrogen used to compute the present problem are listed in Table. 3.3. The mesh of computational domain considered is free tetrahedral mesh and simulations are also performed using COMSOL Multiphysics 4.3. It effectively solves the energy equation (convection and conduction mode heat transfer) (Eq. 3.1a) and the hydrogen mass (diffusion) (Eq. 3.4) and momentum (Darcy’s law) (Eq. 3.6) transport equations in porous media module. User – defined functions are incorporated for calculating the specific parameters such as the rate of hydrogen absorption (Eq. 3.7), rate of hydrogen desorption (Eq. 3.8), hydride concentrations during hydriding and dehydriding processes and equilibrium pressure (Eq. 3.9).
The geometry of MHHSD with 60 ECT and mesh map related to the calculation tolerance are defined first. Fig. 3.4 shows the 3-D computational domains implemented. The first domain simulates the geometry of the above described basic 60 ECT. The second domain describes the MH reaction bed. The outer wall of ECT is introduced by fixing variable wall temperature
boundary condition. Two sub – domains considered are the porous MH bed and the ECT which is used as a heat exchanger for the flow of HTF (i.e. water, oil). To reduce the calculation time and also for symmetry reasons the computational domain simulates a half reactor.
After introducing the MH and gas properties, the boundary conditions and initial values are assigned. The convergence of the solutions determines temperature, pressure and MH density distributions.
Table 3.2 – Thermo – physical properties of Mg2Ni, hydrogen and constants used in the analysis (Muthukumar et al., 2003; Askri et al., 2003; Muthukumar and Ramana, 2009;
Chung and Lin, 2009) Reactor geometry
Length of the geometry : 450 mm
Inner radius of inner cylinder : 6 mm
Inner radius of outer cylinder : 13.5 mm
Thickness of cylinder wall : 3 mm
Properties of Mg2Ni
Density of metal : 3200 kg/m3
Specific heat of metal : 1414 J/kg·K
Effective thermal conductivity of metal (including
copper additive) : 1.4 W/m·K
Porosity : 0.5
Effective density of solid : 3200 kg/m3
Effective density of solid at saturation state : 3276 kg/m3
Activation energy Ea : 55,000 J/mol
Ed : 58,500 J/mol
Entropy of reaction ∆Sa : 124.5 J/mol·K
∆Sd : 131.5 J/mol·K
Enthalpy of reaction ∆Ha : 64,550 J/mol
∆Hd : 70,776 J/mol
Properties of hydrogen
Thermal conductivity of hydrogen : 0.1272 W/m·K
Specific heat hydrogen : 14,283 J/kg·K
Density of hydrogen : 0.0838 kg/m3
Constants used
Universal gas constant : 8.314 J/mol·K
Reaction constant Ca : 100 1/s
Cd 40 1/s
Slope factor : 0.35
Constant : 0.15
Hysteresis factor : 0.2
Scattering albedo, ω : 0.03
Extinction coefficient, β : 5.0
Table 3.3 – Thermo – physical properties of LmNi4.91Sn0.15, hydrogen and constants used in the analysis (Paya et al., 2009; Satheesh and Muthukumar, 2010)
Reactor geometry
Length of the reactor : 160 mm
Cooling tube outer diameter : 6.35 mm
Cylinder inner diameter : 103.4 mm
Thickness of cylinder wall : 6.05 mm
Properties of LmNi4.95Sn0.15
Density of metal : 8500 kg/m3
Specific heat of metal : 500 J/kg·K
Effective thermal conductivity of hydride : 0.2 W/m·K
Porosity : 0.5
Effective density of solid : 4250 kg/m3
Effective density of solid at saturation state : 4310 kg/m3
Activation energy Ea : 30,500 J/mol
Ed : 28,000 J/mol
Entropy of formation ∆Sa : 105.4 J/mol·K
∆Sd : 110.6 J/mol·K
Enthalpy of formation ∆Ha : 27,000 J/mol
∆Hd : 32,400 J/mol Properties of hydrogen
Thermal conductivity of hydrogen : 0.1272 W/m·K
Specific heat hydrogen : 14,283 J/kg·K
Density of hydrogen : 0.0838 kg/m3
Constants used
Universal gas constant : 8.314 J/mol·K
Reaction constant Ca : 80 1/s
Cd : 40 1/s
Slope factor : 0.35
Constant : 0.15
Hysteresis factor : 0.2
Mass flow rate of HTF : 3.2 l/min
3.6 Summary
In this chapter, the detailed physical models of lab-scale MHHSD and MHHSD with ECT and their respective thermal modeling are reported. The formulations of the thermal model for predicting the coupled heat and hydrogen transfer characteristics during hydriding and dehydriding processes, initial conditions and boundary conditions are presented separately for both lab-scale MHHSD and MHHSD with ECT models. For lab-scale MHHSD, the governing equations are numerically solved using FVM and the detailed solution methodology is also reported. For MHHSD with ECT, 2-D and 3-D mathematical models developed for predicting the hydrogen absorption/desorption characteristics using COMSOL Multiphysics 4.3 and the detailed solution methodology are reported. Finally, the thermo-physical properties of MH alloys, hydrogen and constants used in the numerical analysis are listed in Tables 3.2 and 3.3.
The detailed discretization of governing equations are reported in Appendix A.
Chapter 4
Experimental Set-up
In this chapter, schematic (assembled) and photographic views of the hydrogen storage reactors with 36 and 60 ECT, the detailed layout of the test set-up, and experimental procedure are discussed.