hi
i figured out the equation describing the electrically heated tube which contains the streams going in and out, the temperature loss to the environment and the heat from the electric wire that is placed between the tube and the isolation.
energy budget: P_in-P_out=dE_th/dt
IN:
stream going into the tube: c_p*m_dot*T_in
electric heater: k_tube*A_contact*deltaT_middle1
OUT:
stream going out of the tube: c_p*m_dot*T_out
heat loss from tube to environment: 2*Pi*l*k_tube&isolation*deltaT_middle2
deltaT_middle1=((T_in-T_wire)-(T_out-T_wire))/ln((T_in-T_wire)/(T_out-T_wire))
deltaT_middle2=((T_in-T_environment)-(T_out-T_environment))/ln((T_in-T_environment)/(T_out-T_environment))
P_in=c_p*m_dot*T_in+k_tube*A_contact*deltaT_middle1
P_out=c_p*m_dot*T_out+2*Pi*l*k_tube&isolation*deltaT_middle2
final equation:
dT_out/dt=(c_p*m_dot*(T_in-T_out)+k_tube*A_contact*deltaT_middle1-2*Pi*l*k_tube&isolation*deltaT_middle2)/c_p*m_dot
the temperature of the stream needs to be constant. the temperature of the environment can be viewed as constant. i want to know the difference between the energy consumption of a pid, pi and a bang-bang controller. the controller shall vary the current of the electric heater.
