CAES System Dynamic Modelling:

The CAES system can be described as an unsteady open system due to the significant variations in air temperature, pressure and mass during both charging and discharging processes (Grazzini and Milazzo 2012). In this study, small D-CAES based on solar PV as the energy source is proposed. The D-CAES cycle is implemented with TES for storing thermal energy generated during the compression phase and produce adiabatic D-CAES cycle. Figure 3.4 shows the proposed cycle configuration which consists of:

 Solar PV to generate the electricity needed to drive the air compressor.

 Air compressor for air compression to charge small vessel.

 Small high pressure cylinder to store the energy in the form of compressed air.

 Micro turbine in which the energy can be extracted via air expansion to produce power.

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Figure 3.4 Proposed advanced D-CAES based on solar PV.

In the proposed cycle, the solar PV can be used to run the compressor to store the energy in the form of compressed air and the stored energy can be recovered to generate electricity by air expansion through micro turbine. The air entering the turbine can be heated up using TES.

3.4.1 Compression phase:

In the compression phase, the atmospheric air is compressed to the desired pressure. The outlet pressure and temperature of the air leaving the compressor can be calculated using:

𝑝𝑐,𝑜𝑢𝑡 = 𝑝𝑎𝑚𝑏∗ 𝜋𝑐 (3.8)

𝑇_{𝑐,𝑜𝑢𝑡} = 𝑇_𝑎𝑚𝑏 ∗ (𝜋_𝑐)𝑛𝑐−1𝑛𝑐 (3.9)

Where:

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𝑝𝑎𝑚𝑏, 𝑇𝑎𝑚𝑏 are the atmospheric pressure and temperature. 𝜋𝑐 is the compressor pressure ratio.

𝑛_𝑐 is the polytropic index for the compressor. The compressor power input can be calculated as:

𝑊_𝑐 = 1

𝜂_𝑐𝑚̇𝑐,𝑎𝐶𝑃𝑇𝑎𝑚𝑏[𝜋𝑐 𝑛𝑐−1

𝑛𝑐 − 1] (3.10)

For high output pressure, a multi stage of compression processes are used.

3.4.2 Thermal Energy Storage:

To recover the thermal energy generated during compression processes to be used for reheating the air entering the turbine, TES was investigated using both sensible heat and phase change materials (PCM) in order to select the most effective thermal storage option for D- CAES cycle. For the sensible heat storage technology implementation in the proposed distributed CAES configuration, a concrete storage media was used to store the heat produced during the compression process. A cylindrical thermal insulated concrete with heat exchanger coils as shown in Figure 3.5 is studied.

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The heat balance in the concrete TES includes two heat exchange processes. In the first process the concrete absorbs the heat produced during compression and in the second process the concrete heats the air before entering the micro turbine during discharging/expansion stage. The heat balance can be expressed as following:

mTESCP(TES) dT_TES

dt = 𝑞̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛− 𝑞̇𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛− 𝑞̇𝑙𝑜𝑠𝑠

(3.11)

ρ_TES V_TESC_P(TES)dTTES

dt = 𝑞̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛− 𝑞̇𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛− 𝑞̇𝑙𝑜𝑠𝑠

(3.12)

Where the ρTES is the density of concrete TES (2750 kg/m3), CP(TES) is the specific heat of concrete TES (916 J/kg.K), 𝑞̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 is the heat generated during compression, 𝑞̇𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 is the heat required to reheat the air entering the turbine during expansion stage, and 𝑞̇𝑙𝑜𝑠𝑠 is the heat lost to the surrounding.

The change in TES temperature can be calculated by applying heat exchange theory (Sukhatme and Sukhatme 1996) as following:

𝑞̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 = 𝑚̇𝑐𝑜𝑚CP(TC−out− TCAES−in) (3.13) The temperature of the air entering the CAES can be calculated as:

TC−out− TCAES−in

T_C−out− T_TES = 1 − e

[−(UA)TES_{𝑚̇𝑐𝑜𝑚CP}] (3.14)

Where U is the overall heat transfer coefficient of TES, A is the heat exchange area, T_C−out is the temperature of the air at compressor exit, T_CAES−inis the temperature of the air entering the CAES, TCAES−in is the temperature of the air at CAES inlet, and TTES is TES temperature.

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𝑞̇_{𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛} = 𝑚̇_𝑡C_P(T_TES− T_TIT)(1 − e[−(UA)TES𝑚̇𝑡CP ]₎ (3.16)

𝑞̇_{𝑙𝑜𝑠𝑠} = (UA)_TES(T_TES− T_∞) (3.17)

3.4.3 Air Storage Tank:

For the storage tank and assuming the tank is adiabatic with constant volume, both the charging and discharging processes can be described using ideal gas laws as:

𝑑𝑝 𝑑𝑡 = 𝑑 𝑑𝑡( 𝑚𝑅𝑇 𝑉 ) = 𝑅 𝑉 𝑑 𝑑𝑡(𝑚𝑇) (3.18)

For the ideal gas:

𝑇𝛾−1𝛾

𝑝 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

(3.19)

Equation (19) can be written in a derivational form with respect to the time as: 𝑑𝑇 𝑑𝑡 = 𝑇 𝑝[1 − 1 𝛾] [ 𝑑𝑝 𝑑𝑡] (3.20)

Using equations (18) and (20) the rate of temperature change can be determined as: [𝑑𝑇 𝑑𝑡]_{𝑡𝑎𝑛𝑘}= 1 𝑚_{𝑎𝑡𝑎𝑛𝑘}(1 − 1 𝛾) [𝑚̇𝑎𝑖𝑟𝑖𝑛 𝑇𝑎𝑖𝑟𝑖𝑛 − 𝑚̇𝑎𝑖𝑟𝑜𝑢𝑡𝑇𝑎𝑖𝑟𝑜𝑢𝑡] (3.21)

Where 𝑚𝑎𝑡𝑎𝑛𝑘 is the instantaneous air mas in the tank which can be expressed as:

𝑚_{𝑎𝑡𝑎𝑛𝑘} = ∫ [𝑚̇_𝑎𝑖𝑟𝑖𝑛 _{− 𝑚̇} 𝑎𝑖𝑟 𝑜𝑢𝑡_] 𝑡 0 𝑑𝑡 (3.22)

3.4.4 Expansion Phase:

In this phase, the compressed air is expanded through a turbine to extract the stored energy. The air entering the turbine is taken from the storage tank at nearly ambient temperature and

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passed through the TES to recover stored thermal energy and to increase its temperature before entering the turbine. The output power of the turbine can be determined as:

𝑊𝑡 = 𝜂𝑡𝑚̇𝑡𝐶𝑃𝑎𝑇𝑡,𝑎𝑖𝑟

𝑖𝑛 _{[1 − (𝜋} 𝑡)

𝑛𝑡−1

𝑛𝑡 _] (3.23)

Where 𝑚̇_𝑡 is discharge mass flow rate and 𝑇_{𝑡,𝑎𝑖𝑟}𝑖𝑛 _{is the temperature of the air leaving TES and} entering the turbine.