Modeling a simple Water Heater System

Consider a mathematical model (differential equation) for the simple water heater system which describes its
dynamics.

The different signals and components needed to derive a mathematical model for the thermal system are

 V= Voltage applied to the heater (input),
T=Temperature of water, (output),
Ta= Ambient temperature,
Qh=Energy supplied by heater,
Qs=Energy stored by liquid,
QL=Energy lost to the surrounding environment by conduction.

The differential equation is obtained by applying heat balance equation.
Energy supplied by the heater (Qh) = Energy stored by water (Qs) + Energy lost tothe surrounding environment by conduction (QL) i.e. Qh = Qs + QL.

k*V = C*(dT/dt) + (T-Ta)/R

Where, k is a constant provided by the heater manufacturer ,C is a thermal capacitance of the liquid, R is the thermal resistance of the tank wall (heat conduction) and tis the time. Equation (1) is the differential equation that describes  the dynamics of the thermal system. Taking Laplace transform at both the side of equation (1) and rearranging in the transfer function form.

                                               k*V(s) = C*s*T(s) +1/R *(T(s))         ..... (2)
                                               T(s)/V(s) = k/(C*s+1/R)                      .... (3)
 

Equation (3) is a First order system with the time constant C*R. k*R can be treated as system static gain. if we add more complexity in the system by adding time delay in the system and substituting system parameters, its transfer function is derived , which is shown in following Figure 2.at the MATLAB command window.