Question

Problem 1. (25%) The temperature on the inside surface of a CYLINDRICAL tank is T.(11) = 100°C under steady state conditions. The temperature outside the tank, T2(2), needs to be determined. The inner radius is 1 m and the outer radius is 1.3 m, and the tank material has a conductivity of k = 15 W/m. K. ONLY CONVECTION on the outer surface has to be considered. The ambient temperature is To = 25°C and the convection heat transfer coefficient is h = 20 W/m²K. Additionally, the external surface, with an absorptivity of a = 0.85, is receiving solar radiation at a rate of solar = 700 W/m2. NOTE: while radiation emitted from the outer surface to the surroundings is negligible in this case, you must consider the solar radiation absorbed by the outer surface. T() = T1 = 100°C T(12) = T2 =? To = 25°C h = 20 W/m²K = 1 m 15 = 1.3 m k = 15 W/mK solar = 700 W/m2 a=0.85 a. (22%) Starting with the appropriate GENERAL heat conduction equation, find an equation for the variation of the temperature with the radius, T(r), for the conditions shown above (find a particular solution). b. (3%) Compute the temperature at the outer surface of the tank, 7(ru) = 72. Is the heat going into or out of the cylinder? T= °C

Answer 1

The heat will flow outwards and temperature of outer surafce is 41.55 oC

We can е electrical analogy as shown below 1. T, nu • 70 Ri= ln holri 27K R2= hx972 General cheat conduction equation for steady state with no treat generatiis ЖА(*» 2) - 20 (к» 2) - دل кя да С. 2 дя КӘ7 3. C.. Я дл

KT + C₂ C. lnr GENERAL EQUATION Using Joundary conditions At r=r 7-7, 0 K7 = Ci ln rit C2 St r = 2 7= 72 © K 72= C. ln rr + C₂ K (72-7.) CI- k(72 - 71) - Ci Ion (34) ► eneris) K 71 + C2 Putting K(T2-11) Inri Inchi ualues constants in GENERAL EOS. K7 K (72-7, lnr, In/h of KT K (72-71) lnr + Impre IT - 71 K (72-71) en ( In/ha Isi (ri

Heat flow out by conduction - 7,- Tos In (8215) 2 TK I hx qah2 LOO 25 In (1.3/1) 1 20x93x1.3 22 x 15 8422.1 257 W Heat reciéned at autes surface Esolar 2 nr2 = 700 X 22 X1.3 571 7.698 W Net heat going out at outer sufae .-5717.6 9 8 + 842 2.1 257 2 704, 427 W 2704, 427 4 X 22X1.3 ( 72- 25) 72 = 41.55°C