INTRODUCTION
Today’s hydrogen delivery choices are energetically and economically expensive
(Department of Energy, 2005). As a cryogenic LH_{2}
is compact, but evaporates very rapidly, requiring high performance insulation
and care during transfers. In addition there are essentially electricity requirements
for liquefaction and conversion to para hydrogen (Peschka,
1992). At ambient, gas hydrogen occupies substantial volume even at high
pressures, requiring larger size and relatively heavy containers and therefore
low capacity delivery trailers (Yang and Ogden, 2004).
Absorbents (hydrides structures) and adsorbents (carbon structures) would reduce
the pressure and volume of a delivery trailer. But trade the weight of absorbent
materials and thermal management (e.g., heat exchangers) for the weight of pressure
vessel (Ahluwalia et al., 2009). Chemical storage
media often require 2way transportation by truck, filling and refilling. Reprocessing
is expensive and energy intensity and chemical carriers are often toxic, polluting,
heavy, or require high temperatures for dehydrogenation (Ahluwalia
et al., 2009).
Substantial reduction in delivery cost and energy appears to be possible with
development of advanced pressure vessels and a broadened range of thermodynamic
conditions under which hydrogen is trucked and delivered. Commercial hydrogen
delivery, occupy the extremes of this phase diagram Fig. 1.
Hydrogen is delivered as a compressed gas (red dot) at 22°C temperature
(horizontal axis), high pressure (dotted lines)and relatively low density (vertical
axis). Hydrogen can deliver at much higher density as a cryogenic liquid (blue
dot) with higher energetic cost (solid lines indicate the theoretical minimum
work, also known as thermomechanical exergy necessary to densify hydrogen).
The entire phase diagram analysis offers the possibility of finding operating
conditions (such as 70°C and 70 MPa) that may offer a favorable tradeoff
between the high transport cost of compressed hydrogen and the highenergy cost
of hydrogen liquefaction. The challenge is keeping capital costs under control
when operating in this region (Department of Energy, 2005).
Storing the gas in normal 20 MPa steel vessels offers a considerable reduction of storage volume (by a factor of 180 compared to atmospheric pressure), but the packaging in steel results in a dead weight a 100 times the mass of the hydrogen. Construction of pressure vessels by composite materials, with a metallic or ceramic liner and a wrapping of fibers embedded in resin constitute a considerable progress towards lightweight design with the worthwhile option of higher pressure up to 70 MPa Fig. 2. Much higher pressures delivery are not desirable because of the increasing technical effort for infrastructure and high pressure accessories as well as due to the "filling factor" of hydrogen decreasing progressively with rising pressure (difference of real gas from ideal gas law). Functionally Graded Materials (FGM) is interested primarily as heatshielding materials. The possibility of tailoring the desired thermal properties holds enormous application potential for FGMs.

Fig. 2: 
Cost of hydrogen delivery for metallic tube trailers, for
carbon composite tanks and for cold glass fiber tanks (70°C and 70
MPa), as a function of refueling demand (Department of
Energy, 2005) 
Aside from the thermomechanical barrier coatings, some of the potential applications
of FGMs include their use as interfacial zones to improve the bonding strength
and to reduce residual stresses in bonded dissimilar materials and as wearresistant
layers such as gears, cams, ball and roller bearings and machine tools (Erdogan,
1995). Most of the researches that conducted on FGMs were confined to the
analysis of thermal stress and deformation (Wetherhold
et al., 1996; Takezono et al., 1996;
Zhang et al., 1994; Obata
and Noda, 1994).
The works that describe the inhomogeneous properties and concerning the stress
analysis of cylindrical and spherical structural elements involve finite elements
and other numerical techniques due to the nature of functions made by Fukui
and Yamanaka (1992), Loy et al. (1999) and
Salzar (1995). Developing methodologies for solving
specific boundary value problems in solid mechanics involving inhomogeneous
media has always been difficult. Because of this difficulty, all the treatments
dealing with the mechanics of inhomogeneous solids are based on a simple function
representing material inhomogenity. For example, the elasticity problems considered
by Kassir and Chauprasert (1974) and Kassir
(1972), it is assumed that the shear modulus is a power function of the
depth coordinate of the form μ (y) = μ^{0}_{ym} and
the Poisson’s ratio is constant. Developing sufficiently general methods
for solving density and stiffness by the same powerlaw are proposed by Bert
and Niedenfuhr (1963), Reddy and Srinath (1974)
and Gurushankar (1975). The functionally gradient material
considered by Loy et al. (1999) is composed of
metallic materials where the volume fractions follow a powerlaw distribution.
Closedform solutions are obtained by Tutuncu and Ozturk
(2001) for cylindrical and spherical vessels with variable elastic properties
obeying a simple power law through the wall thickness which resulted in simple
EulerCauchy equations whose solutions were readily available. A similar work
was also published by Horgan and Chan (1999) where it
was noted that increasing the positive exponent of the radial coordinate provided
a stress shielding effect whereas decreasing it created stress amplification.
TherrDmodeling of FGM plates are obtained numerically by Reddy
and Cheng (2001) using the transfer matrix method. The overall mechanical
properties were calculated from the constituent properties by the wellknown
MoriTanaka method.
The aim of the present study is to model and simulate heterogeneous elastic
cylindrical vessel for Hydrogen transportation. A simple and convenient method
is developed to simulate Ncomposite layered cylindrical vessel subjected to
uniform internal pressures and external varying temperature. Two types of composed
materials are tested and the results are compared referring to previous numerical
reported researches. The extrusion stress between the vessel layers can be simply
obtained based on Lame’s solution, which is very useful in the design and
analysis for composites reinforced by unidirectional fiber layers. In order
to consider the graded property.
MATERIALS AND METHODS
The model can be simplified as a cylindrical vessel with continuously graded
properties as shown in Fig. 3. The pressure vessel is submitted
to uniform internal pressures and external varying temperature from ambient
to 400°C on the outer surfaces, without consideration of body force.

Fig. 3: 
Model of the graded vessel 
The model is composite of three layers.
Geometry selection: The model selected for the present analysis is a 2.5 mm cylindrical vessel with 223.5 mm internal diameter and 226 mm outer diameter. The vessel length is 1 m. The selected heads type is hemispherical Fig. 3.
Composite material properties: The model is composite of three layers. Two different materials have been investigated, Glass/Epoxy and Graphite/Epoxy. The material properties fed to the simulation are shown in Table 1.
Boundary conditions: The vessel internal pressure is applied uniformly on the internal surface at 70 MPa, while the external pressure is atmospheric. The initial temperatures are selected based on the normal storage temperature of the hydrogen, 70°C. The simulation is investigating the vessel material behavior during fast refueling, where the temperature is experienced to reduce to 130°C.
MATHEMATICAL MODELING
Thermodynamic analysis: This section describes a thermodynamic simulation
of a pressure vessel. The following assumptions are used in the analysis:
• 
Potential and kinetic energy of the hydrogen flowing out of
the vessel are neglected 
• 
Thermal conductivity of the FGM cylindrical wall is considered to be independent
of internal and external temperature 
• 
No conversion between phases (para and ortho) of hydrogen is considered 
• 
The pressure vessel is subjected to uniform internal pressures and external
varying temperature from 22 to 400 °C on the outer surfaces, without
consideration of body force 
From the first law of thermodynamics for a pressure vessel (Ahluwalia
et al., 2009):
In this equation, M is the total mass of LH2 stored in the vessel, u is the
specific internal energy of the hydrogen, t is time, M_{t} and c_{p,t
}are the mass and specific heat of the vessel within the insulation, T
is vessel temperature, Q is heat transfer rate from the environment into the
vessel, h is the enthalpy of the gaseous hydrogen, and is the mass flow rate
of hydrogen extracted from the vessel. Use of the identities h = u+ P/ρ
and reduces Eq. 1 to:
where, P is the vessel pressure and ñ is the density of the hydrogen
leaving the vessel. The lefthand side in Eq. 2 is positive
when the temperature increases as a function of time. Heat transfer into the
pressure vessel (Q in the equation) is positive and as expected, tends to increase
the temperature of the vessel. However, mass flow out of the pressure vessel
has a negative contribution to the change in temperature in the vessel. Considering
that the density of LH2 is very low, this term may reduce the heating effect
of Q, or it may even result in a reduction in temperature for the vessel, depending
on the rate of mass extraction. The last term in Eq. 2 is
commonly known as the flow work, since it is the work that the hydrogen stored
in the vessel has to do to push out the hydrogen being extracted. The required
property values are obtained from McCarty (1975). The
problem that appears more alternative for finding pressure and temperature when
internal energy and density are known. The specific heat of the vessel materials,
c_{p,t} is obtained as a function of temperature from correlations given
in the literature (Scott, 1967). For vessels made of Composite
material (i.e., Epoxy/Glasses fiber), the second term in the lefthand side
of Eq. 2 is written as a sum of two terms, each taking into
account the thermal mass of each material.
FGM analysis: A simple power law distribution is adopted and then the
V_{f} of the composite materials are expressed as (Rahimi
and Nejad, 2008):
where, V is volume fraction, k is volume fraction index, subscripts o, i represent
the volume fraction for composite material in the outer and inner layer, h =
thickness of the wall. The properties of the composite material for FG wall
can be obtained by linear rule of mixture (Wetherhold et
al, 1996) as:
where, P_{eff}, P_{o} and P_{i} are the effective material, outer and inner composite properties, respectively. Composite material properties of constituents must be dependent on temperature, since the FGMs are usually used in hightemperature environments. The properties P (T), of the composite material in (outer and inner) surface as FGMs can be expressed as
where; P_{0}, P_{1}, P_{1}, P_{2} and P_{3}
are the effective composite material properties and are the coefficients of
temperature. Using Eq. 35, the modulus
of elasticity E (z, T), the coefficient of thermal expansion α (z, T) and
the density ρ (z) are written as (Tutuncu and Ozturk,
2001):
RESULTS AND DISCUSSION
Today’s hydrogen delivery technologies (liquefied) are restricted to single points at extremes of the hydrogen phase diagram. Hydrogen delivery cost can be minimized by exploring the entire phase diagram and finding pressures and temperatures that result in high storage density without the heavy thermodynamic penalty of hydrogen liquefaction "cold highpressure hydrogen (70°C and up to 70 MPa)". Composite pressure vessels operating at low temperature (from 130 to 70°C) and high pressure (From 20 to 70MPa) minimize delivery cost through a synergistic optimization of hydrogen properties, fiber characteristics, pressure and temperature:
Pressure vessel operation at low temperature and highpressure has been simulated
Fig. 5, which explain the behavior of the FGM to reduce the
thermal conductivities across the wall thickness Fig. 4. As
well as Insulation performance is less critical when delivering 70°C and
highpressure hydrogen. This approach may also enable high speed refueling and
high capacity delivery in trucks, leading to affordable delivery.

Fig. 4: 
Thermal analysis across wall thickness for vessel subjected
to 400°C in the external surface 

Fig. 5: 
The difference between the external and internal temperature 
The approach is benign "environmentally", with low energy input and no need
for expensive 2way transportation or reprocessing. Delivering 70°C hydrogen
increases the density by 35% at a low energy cost and may enable fast refueling
without vessel overheating, over pressurization and stresses damage.
Use of inexpensive commercial glass fiber: Glass fiber is considered an inexpensive low performance alternative to carbon fiber. However, glass fiber is synergistic at cooling operation, potentially strengthening ~50% as it is cooled down up to 130°C when the internal temperature varying from 70 to 130°C at fast refueling (Fig. 6) which it appear in stress simulation (Fig. 7) as well as the strain (Fig. 8).
Proof of concept of pressure vessel deformation which it appear clearly in Fig. 9, show promise for engineering the strengthening of glass fiber composites up to 70% when operating at low temperature(130°C). Further experimentation at cooling operation will determine the extent of the strengthening as a function of temperature and pressure.

Fig. 6: 
Variation of stresses in the vessel wall with respect to internal
temp. and pressure 

Fig. 7: 
Stresses analysis when the internal temp. varying from 130
to 70°C, at fast refueling 
Cryogenic capable pressure vessels can operate at cold temperature with no
structural damage. In the current work we tested the pressure vessels of FGM
at cold temperature (down to 70°C) and high pressure multiple times (thousands)
without any damage to the vessel (Fig. 10).

Fig. 8: 
Strain analysis when the internal temp. varying from 130
to 70°C at fast refueling 

Fig. 9: 
Varying of deformation in the vessel wall when the internal
temperature varying from 130 to 70°C at fast refueling 

Fig. 10: 
Total deformation across wall thickness when the internal
temperature varying from 130 to 70°C at fast refueling 

Fig. 11: 
Varying of stresses in the vessel wall with respect to varying
of internal pressure 
Finite element analysis by ANSYS package indicates that no significant plastic
deformation occurs after the first few cycles and therefore vessels can be expected
to have a long life even at cryogenic conditions (Fig. 11).
CONCLUSIONS
When hydrogen cooled to 130°C, it densifies by 45% at low energetic cost as well as with low energy input and no need for expensive 2way transportation or reprocessing. It may enable fast refueling without vessel overheating, over pressurization and stresses damage. The inexpensive glass fiber is strengthen by ~70% when the internal temperature varying from 70 to 130°C, at fast refueling, the graphite fiber weaker in coldness. The concept of FG composite materials represents a good thermal controlling (i.e., the difference between inner and outer temperature is up to 60°C) for safely transportation of hydrogen for industry. Cryocompressed vessels have considerably larger thermal endurance (~10x) than liquid hydrogen tanks. Dispensing of cold (70°C) hydrogen reduces automobile vessel cost by about 25%.