Question

12. The temperature at the inner and outer surfaces of a hollow cylindrical pipe with wall thickness L are held at constant values of T and T, respectively, where T, <To. The wall material has thermal conductivity that varies linearly according to k = k,(1 +BT), where k, and ß are constants. Draw a schematic of the system. Define and label a coordinate system. State assumptions and boundary conditions. b. Derive the governing differential equation for the heat transport and solve for the temperature profile by applying the boundary conditions. c. At what point will the actual temperature differ most from that which would exist is the case of constant thermal conductivity?

Answer 1

Engineering Chemical (1) Given that Given that Inner Temp. Iomer Temp. = Tisti Outer Temp. = To Tz (here To > T:) Wall thickness L K = Ro (1+BT) Assumptions (2) Steady State heat fronsher No head generection One Dirvensional heat transfer tion inside the cylinder (3) part (a) To-Tz To-T2 LIT Chor HdT Tif da Sed=L Top view Side view Boundary conditions The temp. is a Teat, cig at der (2) at SaSa The terp is & To-T2

Pool (b) We know that for a l-D, Steady state, without heat genered for tion and Variable thermal conductivity system. The general Eqni is ter (krd T ) = So a foro (1+BT) dT' dr Ko (1+ BT) dT =aldo ; Ko (1+BT) dT - G dar- > 2 at rer ; T = T KT- BT = clonata o and ser; To Tz putting in egn- o Ko (T, + BT ²) = C, log, & a Ko (Tz + B 72 ) = C, los + - Ean - En Ko (T. -T2) + ß (T? - T2 ² ) = C ln (87 Kot $ CT1+T3)] Ic = (T-T2] so; a = 2 (T-T2) [ko en (2) the c, in egn- "(T-T2) [kot & (T1+73)] ens = putting Ko (Tit BT 2. 2 lm (3782)

putting back the values of c, and Ca im coon. V ** T+?). (T-1[1* £. C7,+8) ber 82 tko (Tit T + BT 1c7.-T){**£ (741) bro this will be the temperature profile for the given Conditions. Part (C) for Constant Thermal Conductivity The governing Temprocature profile will be given by (7. - 12) en ( 2 ) + ln (1) - 0 Т. Now, we can compare egn 6 and 0 to find the point where the actual temperature aiffess most from the Corse of constent K.