Makeup gas for thermal swing systems

In any process where the temperature varies cyclically, such as packed bed thermal energy storage or temperature swing adsorption, makeup gas is a subtle but important concept. To demonstrate the need for makeup gas, let's take the example of a packed bed at 10 bar and 25 °C. Now, say we flow a hot gas through the packed bed at 300 °C and stop when the outlet temperature reaches a pre-defined temperature, e.g., 50 °C. Although there has been no mass transfer, the total number of moles in the gas phase within the bed at the end of charge will be less than the start.

This is due to fluid compressibility and described by the equation of state which relates the gas molar density to the state variables, pressure, and temperature. For an ideal gas:

$$C = \frac{P}{RT}$$

The molar concentration in [mol/m3] decreases as the temperature [K] increases. To calculate the number of moles within the bed, we must integrate along the length due to the varying temperature.

$$n_{bed} = A \epsilon_b \int_{z=0}^{z=L} C dz$$

The bed void fraction is included as a fraction of the total volume occupied by the solid. In the above equation, $\epsilon_b$ is assumed to be constant throughout the bed; however, if this varies, then it should be included in the integral. This is the case for multi-layered packed beds where each layer may have a different void fraction. Additionally, for porous materials, the total void fraction should be used instead of the bed void fraction.

The difference in the number of moles between the start and end is:

$$n_{makeup} = A \epsilon_b \int_{z=0}^{z=L}(C^{t=t_{end}} - C^{t=t_{start}})dz$$

In the case of a hot temperature thermal store, makeup gas is required during discharge of the tank.

During idle periods, the hot bed will slowly cool, and as a result, the density of the gas will increase. If no gas enters or leaves the system, this will result in a decrease in pressure. Therefore, to maintain pressure, additional make-up gas is required.

The opposite is true for low-temperature thermal stores. When the cold temperature is below ambient, the vessel will increase in temperature during idle periods. That will consequently increase the pressure, and excess gas will need to be vented to avoid overpressure. This is a very similar concept to boil-off in cryogenic liquid storage tanks.