VVT35 - Transient temperature with a PID controller

Description

The purpose of this verification test is to verify how a PID controller maintains a set point temperature over time by adjusting control inputs.

Simulation model

The model consists of one 2D element generated using a thin shell with a size of 0.13 mm and thickness of 100 mm.

The meshed elements have the following material and physical properties:

• Shell material for the plate with:
  • Mass density: ρ = 1000 kg/m³
  • Thermal conductivity k = 1 W/(m·K)
  • Specific heat: c_p = 2.5 J/(kg·K)
• Environmental fluid material: Air
  • Mass density: ρ = 1.2041 kg/m³
  • Specific heat at a constant temperature: C_p = 1004.5 J/kg·K

The following boundary conditions are applied:

• The Convection to Environment constraint to convect from the top and both sides with a convection coefficient h_i = 50 W/m^2◦C and a fluid ambient temperature.
• The Heat Load type of Thermal Loads with a heat load of 100 W with the defined active heater controller.
• The Active heater Controller modeling object with the following options:
  • Type = PID Controller
  • Set Point Temperature = 100 °C
  • Gain = 25 °C
  • Integral Term Constant = 1
  • Derivative Term Constant =0
  • Bias = 0

This model uses the Simcenter 3D Space Systems Thermal environment.

The following solution options are set:

• Solution Type = Transient
• Fluid Temperature = 20 °C
• Integration Method = Implicit, because the model has an active heater controller that oscillate fast and have large time steps
• Time Step = 1
• End Time = 500 s

The default solver parameters are selected.

Theory

The energy balance on the element for a temperature control process involving a PID controller is given by:

where:

• C is the heat capacity of the system.
• G is the thermal conductance between the system and the environment.
• T_enc is the ambient temperature.
• Q_in is the heat input, which is controlled by the PID controller.

The PID controller inputs the load according to the following relationship.

where:

• Set Point Temperature is the target temperature T_s that you define. This temperature is used to calculate the error e(t)=T_s-T_element(t), where T_element(t) is the temperature of a given element at time t.

• K_p is the proportional gain parameter.

• K_I is the integral gain parameter.

• K_e is the derivative gain parameter.

• e(t) is the error function over time.

• Offset is the bias.

Results

The following graphs compare results predicted by the thermal solver with the calculated theoretical results. The simulation results are in agreement with the theoretical calculations.