Comparative analysis of proportional, proportional–integral, and proportional–integral–derivative controllers for thermal regulation in a cylindrical cooling system with multiple O-Ring heat sources
Authors: Chanda Shorup , Ratul Md. Moyeenul Hossain , Tasnim Most. Nishat , Moni Meril , Suvro Sourav Das , Saha Sumon 
Published: August 2024

This study is intended to determine the thermal management capability of different closed-loop controllers inside a cylindrical three-dimensional cooling system with multiple O-Ring type heat sources. The motivation for this research stems from the need for efficient thermal regulation in advanced cooling systems, which is critical for applications ranging from electronics cooling to industrial processes. The uniqueness of this study lies in its comprehensive evaluation of different controllers such as proportional (P), proportional–integral (PI), and proportional–integral–derivative within a geometrically complex cooling environment using a novel trial-and-error approach for tuning controller parameters. The observational domain is a hollow cylinder, and the four discrete heat sources are placed at regular intervals along the cylinder's longitudinal axis, with all the remaining walls insulated. At the center of the system is a temperature probe that measures and provides feedback to the control module to continuously compare it to a pre-specified setpoint temperature. The flowing fluid, air, enters through the semi-circular inlet at one end, whose velocity is controlled by a controller response, and is discharged through the semi-circular outlet at the other end at atmospheric conditions. This study uses the Galerkin finite element method, using proper initial and boundary conditions to solve the governing equations, namely, the Navier–Stokes and heat energy equations. We analyze the time-dependent behaviors of the cooling system by measuring controller responses such as overshoot, rise time, oscillation, steady-state error, and settling time. Additionally, the impacts of the Reynolds number (Re), Richardson number (Ri), and Grashof number (Gr) on the overall mean Nusselt number over time are observed to understand the influence of flow regulation due to the controllers' actions. This comprehensive analysis provides insights into the controllers' inlet flow control based on the set temperature at the probe's center location. A unique trial-and-error approach for selecting Kp and Ki values, which is crucial for controller analysis, is also presented. Based on the results, it can be inferred that increasing the Kp value from 0.003 to 0.010 ms−1 K−1 reduces the steady-state error from 40.8% to 13.97%. For a Ki value of 0.006 ms−2 K−1, the PI controller achieves a zero steady-state error with faster settling at 3.29 s, along with an overshoot of approximately 80.51%. Conversely, a lower Kd value of about 0.0001 mK−1 results in a reduction in the settling time and overshoot compared to the PI controller, while a higher Kd value ensures optimum stability with a higher settling time and a stable Nusselt number of 1.93. This trial-and-error approach to parameter tuning provides valuable insights into the design of controlled environments and the effective management of thermal conditions in various thermo-fluidic applications.