Question

Differential Equations Help

PX 0.0.0 A warehouse is being built that will have neither heating nor cooling. Depending on the amount of insulation, the time constant for the building may range from 1 to 5 hr. To illustrate the effect insulation will have on the temperature inside the warehouse, assume the outside temperature varies as a sine wave, with a minimum of 14°C at 2:00 AM. and a maximum of 34°C at 2:00 P.M. Assuming the exponential term (which involves the initial temperature To) has died off, what is the lowest temperature inside the building if the time constant is 1 hr? If it is 5 hr? What is the highest temperature inside the building if the time constant is 1 hr? If it is 5hr? If the time constant is 1 hr, then the lowest temperature inside the building is about (Round to the nearest tenth as needed.) 'c.

Answer 1

The answer for above problem is explained below.

By using the equation, M = Mo - B cos (at) 0 Here C-= : Sine wave ovce a いし 24-ho period] And Mo-B = 14 - And Mo +B= 34 - And given, at Jam, t=0 And at apm, t =12 By Adding equations ① & ② a Mo = 48 es Mo=24] And Mo + B = 34 B = 34 -M. = 34-24 = 10 TB=10) The Forcing function is, QCH) = k C Mo - B cos(wt)) - Q Ct) = K (24 - 10 cos (wt)) And, Temperature, TCt) = B.- B F(t) + cē

Where, cos (Wt) + ( h) sincut) F(t) = It w2 K2 When K= 1 lhe get, -kt T(t) = 24 - 10 FCH) + cé (o Here, Ho=0, Bo= Mo] - I cos (wt) + sin(at) And F(t) = It wa Vito 2

FCE) = sin(wt + tawless ]] C: maximum value of sin és 1]

The exponential term died off, Trowest = 24-10 FCE) highest 2) Ilowest 24-10 - V 1 + 0 / 2 substitute ww = Tya and K=1 24 a 10 Telowest = 10 = 24- Florest - 14.306° ) with ] K = 1] With I K= 1 lowe Then, Teowest = 10 24- - 1st T1% x 25 124 i Teowest = 17.927°C] with (K = %5 2

minimum value of sin es - 24 + 10 highest for kal 10 = 24t highest : Thighest = 33.67600) for k=1] And for K= 75 Thighest = 24+ It T2825 144 in Thighest = (3000726°C) for [ k= %5]