2.3.3.3 Other Underground Storage Methods
One of the options available is by converting the abandoned mines to gas storage reservoirs due to certain circumstances. However, due to uncertain evidence of tightness, such mines are of no practical value as hydrogen storage facility.
The other option available are using the lined rock cavern which built by existing mining technology. Stainless steel is used to lined the cavern. The surrounding walls of cavern were sealed off by the liner. There is one such facility currently available. It is the Swedish LRC Demonstration Project for storage of natural gas (Tengborg, et al., 2014 cited in Bünger, et al., 2016, p.47).
As for rock cavern for hydrogen gas storage, an investigation on the storage of hydrogen gas in line rock cavern has been conducted by Johansson, et al. (2018).
The lined rock cavern presents a reasonable route. A large-scale natural gas underground storage was constructed in Sweden in 2002 and has been used safely.
The investigation looks for further research required in order to use the facility to store hydrogen gas. The report enlists out several potential researches for further study. In additions, the report also lists out, as a critical research issue, the potential effect of hydrogen embrittlement on the steel lining. The concern being to ensure the safe storage of hydrogen gas in such lined rock caverns. The report concluded that the concept of line rock cavern seems a reasonable way forward.
There are a few analytical methods available for investigating heat conduction problem, with the differential equations and partial differential equations are the principal ones (Zalba, et al., 2003). To assess these equations, separation of variable solution and series method are generally used (Jabbari, Sohrabpour and Eslami, 2003).
2.4.1 System of Coordinate for Cylindrical Wellbore
In the studies and assessments of a physical shape, there exist three principle systems of coordinate that may be applied for dealing with different geometry. According to Cengel, et al. (2008), they are rectangular Cartesian coordinate, cylindrical coordinate and spherical coordinate. As the current study is about cylindrical wellbore of hollow cylindrical in nature, the cylindrical coordinate system will be elaborate more herein.
To study the heat transfer of a cylindrical wellbore or a hollow cylinder, as described by Incropera, et al. (2013), it is proposed to initially look into a small part of the cylinder medium by considering a differential control volume with cylindrical coordinates as shown in Figure 2.10.
An infinitesimally small differential control volume is firstly defined.
Consider the heat transfer and distribution process that taking place, then the relevant energy transfer processes and corresponding conduction heat flows are then identified onto that small differential control volume accordingly.
Figure 2.10: A Differential Control Volume in Cylindrical Coordinates (Incropera, et al., 2013).
To investigate the heat conduction problem, Qiu and Tien (1993) apply the theory of conservation of energy onto the control volume. The equation of conservation of energy is as below
̇ ̇ ̇ ̇ (2.1)
where
̇ inflow energy, J/s ̇ energy generation, J/s ̇ outflow energy, J/s ̇ storage energy, J/s
Next, for that control volume, the heat flux components in axial, circumferential and radial directions can be determined by using Fourier‟s law as well as heat flux vector (Incropera, et al., 2013). The corresponding heat flux components in axial , circumferential , and radial which have shown respectively in the Figure 2.10, are expressed in following equations
(2.2)
(2.3)
(2.4)
where
axial heat flux, W/m2
circumferential heat flux, W/m2 radial heat flux, W/m2
radius, m
thermal conductivity, W/m K temperature, K
temperature gradient along respective radial, circumferential and axial direction, K/m
Finally, from the theory of conservation of energy, heat conduction equation in cylindrical coordinates can be arranged in the general form as follows
( )
( )
(
)
(2.5)
where
thermal conductivity, W/m K heat generation, W/m3
density, kg/m3
specific heat, J/kg K
2.4.2 Single Layer Cylindrical Wellbore Model
In the investigation of heat conduction problems involving structures of cylindrical shape, such as solid cylinder structure or hollow cylinder structure, the system of coordinate adopted is cylindrical coordinate system (Hetnarski, Eslami and Gladwell, 2009). Although the mechanism of heat and temperature distribution is similar for both structures, Acharya and Dash (2017) pointed out that hollow cylinder requires more considerations and because it has more boundary conditions as compare to solid cylinder.
For this section, a single layer cylindrical wellbore is under investigation. A cross section of such cylindrical wellbore with inner radius and outer radius is shown in the following Figure 2.11. In the diagram, the boundary conditions for a heat conduction problem are shown. Temperature on the inner surface is denoted ( ) and temperature on the outer surface is denoted ( ).
Figure 2.11: Cross Section of Single Layer Cylindrical Wellbore
where
inner radius, m outer radius, m
( ) internal surface temperature at radius , K ( ) external surface temperature at radius , K
2.4.3 Steady-State One Dimension Radial Flow Heat Conduction
Investigation of heat conduction problem in normal real situations usually involved considerations in all three geometrical dimensions. The external environment and various other variables that affect the steady state and transient response of the system under investigation will make the response of the system complex to be estimated (Wrobel and Brebbia, 1979). Due the above complexities, some reliable assumptions are to be made so that response of the system could be evaluated easier by remove away those variables that are not related and not relevant. Generally, the exclusion of the variables is depending on the application or purpose of investigation.
From these considerations, Cengel and Ghajar (2011) simplify the heat conduction equations by proposing two reasonable assumptions. The system is assumed to be in steady state and one flow dimension. The heat conduction equation in cylindrical coordinates, Eq. (2.5), is then simplified as follow
(
) (2.6)
The above equation shows the heat conduction of the single layer cylindrical wellbore in the radial flow consideration.
𝑇(𝑎) a
b
𝑇(𝑏)
2.4.4 Non-heat-generation Heat Conduction
There is term notated as R in Eq. (2.6) which represents heat generation. Heat generation in a body, as explained by Incropera, et al. (2013), is a presentation of certain energy conversion process. Such process usually involved thermal energy and other energy, for example, chemical, nuclear and electrical energy. Schiffer, Linzen and Sauer (2006) pointed out that if a medium is having a negative heat generation, then that medium or component is a heat sink. The medium or component is utilising or consuming thermal energy. On the other hand, if a medium is having a positive heat generation, then that medium is termed as heat source. The medium is generating or producing heat and dissipates the heat in other form of energy.
In the current investigation, the cylindrical wellbore to be studied is not a heat source. It is not generating any heat energy. It is also not converting other form of energy like electrical or chemical energy to thermal energy. In addition, the heat consumed or utilised by the cylindrical wellbore is never to be significant.
Hence, the heat generation could be excluded in the current study as its effect and influence throughout the analysis shall not be significant. The heat generation term, R in the Eq. (2.6) shall be ignored and neglected. The heat conduction equation, Eq. (2.6) could then be further simplified into following form
(
) (2.7)
Referring back to Figure 2.1, the boundary conditions of the single layer cylindrical wellbore are as follows
( ) (2.8)
( ) (2.9)
where
inner radius, m outer radius, m
inner surface temperature at radius , K outer surface temperature at radius , K
Integrating the further-simplified heat conduction equation of Eq.(2.7) gives
(2.10)
where A and B are the constants of integration.
By applying boundary conditions of Eq. (2.8) and Eq.(2.9), constant A and B can be calculated. Substituting the computed constants A and B into Eq. (2.10) gives temperature distribution equation ,
⁄ ( ) (2.11)
Eq. (2.11) enable the thermal analysis of a single layer cylindrical wellbore with no heat generation, in one dimensional radial flow, steady state heat conduction problem, to be computed. The boundary conditions of the cylindrical wellbore are inner radius of , outer radius of , temperature at inner surface and temperature at outer surface . A closer look into Eq. (2.11) reveals that the distribution of temperature is a function of inner and outer radius.