Solved: (Heating and cooling of a building) Heating and

Chapter 1, Problem 1.1.130

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(Heating and cooling of a building) Heating and cooling of a building can be modeled by the ODE

\(T^{\prime}=k_{1}\left(T-T_{a}\right)+k_{2}\left(T-T_{w p}\right)+P\)

where T = T(t) is the temperature in the building at time t, \(T_{a}\) the outside temperature, \(T_{w}\) the temperature wanted in the building. and P the rate of increase of T due to machines and people in the building, and \(k_{1}\) and \(k_{2}\) are (negative) constants. Solve this ODE, assuming P = const, \(T_{w}=\text { const }\), and \(T_{a}\) varying sinusoidally over 24 hours, say, \(T_{a}=A-C \cos (2 \pi / 24) t\). Discuss the effect of each tenn of the equation on the solution.

Text Transcription:

T^prime = k_1(T - T_a) + k_2(T - T_w) + P

T_a

T_w

k_1

k_2

T_w = const

T_a = A - C cos (2pi/24)t