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

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Bob on 29 Mar 2021
Answered: Star Strider on 29 Mar 2021
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."
Bob on 29 Mar 2021
Could you be more elaborate please?

Star Strider on 29 Mar 2021
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))