Analytical Model for Compressed Air System Analysis

This paper presents a simple analytical model for a compressed air system (CAS) supply side. The supply side contains components responsible for production, treatment and storage of compressed air such as a compressor, cooler and a storage tank. Simulation of system performance with different storage tank size and system pressure set-point were performed. Results showed that a properly sized tank volume reduces energy consumption while maintaining good system pressure stability. Moreover, results also showed that reducing system pressure reduced energy consumption, however a more detailed model that considers end-user equipment is required to study effect of pressure set-point on energy consumption. Future work will focus on developing a supply-demand side coupled model and on utilizing model in developing new control strategies for improved energy performance.

Maxwell and Rivera simulated in [6] the effect of different CAS control strategies on energy performance. Kleiser and Rauth studied in [7] system performance with different storage tank volumes was studied. In [8], Anglani et al. introduced a new tool that allowed a user to simulate different CAS components, configurations and settings.
This paper investigates a simplified model that can serve as a first tool for CAS design or retrofit evaluation. In Sect. 13.2, the mathematical models for the compressor, cooler and storage tank are presented. Section 13.3 evaluates, two different system modifications using the model: varying storage tank volume and decreasing system pressure. Finally, some conclusions and future work are presented.

Compressor
Assuming air behaved like an ideal gas, the work required (W comp ) to compress a volume (V 1 ) of air from air inlet pressure (P 1 ) to discharge pressure (P 2 ) was calculated using Eq. (13.1) [8].
where (n) is the polytropic compression exponent. The process was assumed to be isentropic and n = 1.4. To calculate the power, volume flow rate per unit time was used instead of volume. To estimate the power (W sup ) supplied to the compressor, efficiencies of the drive system (η ds ) and the compressor (η c ) were assumed constant at 90% and 80% respectively.

Air Cooler
In this study a counter flow air to air heat exchanger with effectiveness (ε) was assumed. The temperature of air leaving the cooler (T 3 ) was obtained using Eq. (13.2) [9].
T amb is the temperature of ambient air, which was assumed to be the cooling fluid. T 2 is temperature of air exiting the compressor and was obtained from ideal gas law.

Storage Tank
The purpose of a storage tank in a CAS was to store compressed air for when it was needed. Often, the pressure of air in storage was used as a control variable for the compressor. From the law of mass conservation, the mass of air in the storage tank was obtained with Eq. (13.3).
m in and m out were the air mass flowing in and out of the tank. m 0 is the mass of air in the tank at time t = 0. Assuming the temperature of air in the tank was equal to temperature of air leaving the cooler (T 3 ), the pressure of air in a tank (P tank ) of volume (V tank ) was obtained with Eq. (13.4) [6], where (R = 287 J/kg·K) is the gas constant of Air.

Simulation and Results
The mathematical model of the components presented in Sect. 13.2 were implemented in MATLAB. The compressor had a load/unload control, so the compressor would run at partial capacity (20%) for a period of time before shutting off. The system was assumed to leak 7% of its rated air capacity. System performance with no compressed air consumption, apart from leaks in the system, was simulated. Storage tank pressure is shown in Fig. 13.1. The load/unload control settings caused the tank pressure to cycled between the upper (9 bar) and lower (5 bar) pressure limits. In this case, consumption was only due to assumed leaks in the system.

Storage Tank Size
The impact of changing tank volume was studied. The compressed air consumption profile shown in Fig. 13.2 was assumed. Three different simulations with tank storage volumes of 100, 330 and 930 l were performed. The results compressor energy consumption and tank pressure are shown in Table 13.1 and Fig. 13.3 respectively. Results showed that a larger tank volume had a higher pressure stability, however this stability was not always justified in terms of energy consumption. The highest    energy consumption was for the 930 l tank, while the lowest was for the 330 l. Further analysis is required to study energy consumption for different compressed air consumption profile (Fig. 13.4).

System Pressure
Reducing system upper pressure limit from 9 to 8 bar was simulated assuming a tank volume of 330 l and the air consumption profile shown in Fig. 13.2. Energy consumption at different pressure levels is shown in Table 13.2. Results show that reducing system pressure led to 7% reduction in energy consumption. It should be noted that this model assumed constant compressor efficiency at different discharge pressures. In reality, efficiency varies with air discharge pressure. Moreover, leakage rate, which was assumed constant, changes proportionally with system pressure.
Decreasing system pressure removes artificial demand in pneumatic tools. This could be better analysed through modelling the demand side of the system.

Conclusion
A simplified CAS model was presented. Mathematical expressions that describe compressor, cooler and storage tank were implemented in MATLAB. Different air storage tank volumes and different system pressure levels were simulated. Results for tank volume show that too large or too small a tank led to excessive energy consumption. An adequate tank volume reduced energy consumption while maintaining system pressure stability. Moreover, simulation results showed that reducing system pressure reduced energy consumption, however a more detailed model that considers demand side is required to properly analyse the effect of system pressure on energy consumption. Future work will investigate model validation and developing a supply-demand coupled model. Moreover, development of novel control strategies to reduce energy consumption will be studied.
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