How to find the TF of a House Heating System. (New and I need help)

I would appreciate if somebody could explain me, how can I find the Transfer Function from the Heating System below?

I don't know how to start it, I don't know how to start thinking of starting this.

"In the House Heating System below, we insert a Step change from 0 to a Maximum Heating Power of 100kW.

At the same time, we monitor the increase in Temperature.

The Initial Temperature was 283K and the Final Temperature is 303K.

It took 3.33 hours to reach the Final Temperature.

The Thermal Capacity of the house is C = 15*10^6.

Find the Transfer Function and confirm it with Step in Matlab."

[darova]

Looks like ordinary diff equation. What about ode45?

[Star Strider]

Since it appears to be a linear differential equation with constant coefficients, calculate the Laplace transform of the differential equaiton, and solve thr the output divided by the input to get the transfer function.

[Bob]

L [ C * dTin/dt ] = L [ Win - k*(Tin - Tout) ]

[Bob]

Could you be more elaborate please?

Answer 1

[Star Strider]

This is actually easier to do by hand:

syms C Tin(t) Win k Tout(t) s H(s) 
Eqn = C*diff(Tin) == Win - k*(Tin - Tout)
LEqn = laplace(Eqn)
LEqn = subs(LEqn, {laplace(Tin(t), t, s), laplace(Tout(t), t, s)}, {Tin(s),Tout(s)})
LEqn = expand(LEqn/Tin(s))
LEqn = subs(LEqn, {Tout(s)/Tin(s)}, {H(s)})
LEqn = isolate(LEqn, H(s))