Question

Question/12_ The room shown in the figure has a heater with heat flow rate input of go. The thermal capacitances of the heater and the room air are C, and C2, respectively. The thermal resistances of the heater-air interface and the room wall-ambient air interface are R, and R2, respectively. The temperatures of the heater and the room air are T, and T2, respectively. The temperature outside the room is T. which is assumed to be constant. 20 Room air T, C a. Derive the differential equations relating the temperatures T.T2, the input 90, and the outside temperature To. b. Using the differential equations obtained in Part (a), determine the state-space form of the system. Assume the temperatures T, and T, as the outputs. TC Heater

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Answer 1

Answer: Given Data, → The heater with heat flow rate input = 90 → The thermal capacitances at room temperature = 6, WC, → The thermal resistances are interface with room =R, & R₂ wall - ambient → The temperature of heater & room = T,&Te - To → The temperature outside of the room Roon air Tz C TC Heater

@ - The differential equatione relating the temperature T, AT " where, wo - heat stored theat going out = C, . dTAT - + dt consider the control volume, 7 I Iata LOV TR

3 En 0:0 in matrin form, raoror IR YR, Local? J* LYR, (%.* Yea)]l". 0 0 6 The temperatuvex T, and T, as the outputs :- State space consider, y = 4, : V a = T2 U2 = To .. we express the terms as 1 . Ax+ Bu where yer 7,2 17 J where 1 = (3:]-C53. ^[] of to] quo T 1 T. + B from the eq Os® we can express eqo Tico tempo o

1 A1 R, C, KC :. Arra eta bene last time Blo Y. [5] г. се ра (5). «(0)+of] 20-0 He: [..] Cat Du