Problem 1 Known: Dimensions and materials of the composition wall, 10 studs each with 2.5m high
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3 Prblem Knwn: Dimensins and materials f the cmpsitin wall, 0 studs each with.5m high Unknwn:. Thermal resistance assciate with wall when surfaces nrmal t the directin f heat flw are isthermal. Thermal resistance assciate with wall when surfaces parallel t the directin f heat flw are adiabatic Schematic: Assumptins:. Steady state cnditins. Cnstant prperties 3. Negligible cntact resistance. Temperature f cmpsite depends nly n x (surfaces nrmal t x are isthermal Prperties: k A 0.09W/m.K k B 0.6W/m.K k C 0.038W/m.K k D 0.7W/m.K These values can be fund frm table A3 Analysis: a Surfaces nrmal t the directin f heat flw are isthermal Thermal circuit f a single unit f the wall: element A D / B / C / 8.5 /.3.85k /W A B C D Where: LA A 0.05k k A 0.09*.5*0.65 A A
4 LB 0.3 B 8.5k kb AB 0.6*.5*0.0 LC 0.3 C.3k kc AC 0.038*.5*0.6 LD 0.0 D 0.03k k A 0.7*.5*0.65 D D Thermal circuit f the entire wall Where: 0.85k/W 0 / i 0 /.85 ttal i ttal 0.85k/W 0 b Surfaces parallel t the directin f heat flw are adiabatic Thermal circuit f a single element Where: A 0.85k 0.09 *.5* A 0.056k 0.09 *.5* D 0.706k 0.7 *.5* D 0.06k 0.7 *.5* 0.6 A A B C D D element A B D A C k /W D ttal 0/ element 0.888k/W Cmments: Frcing -D cnductin int -D cnductin prblem, which is the case fr bth thermal circuits, verestimates the heat flw. Thus underestimates the resistance. This means that the arrangement yielding the larger resistance is mre realistic. Since tt fr (b 0.888K/W > tt fr (a 0.85K/W, the apprximatin by the thermal circuit (b, which assumes surfaces parallel t the directin f heat flw are adiabatic, is mre realistic.
5 Prblem Knwn: Diameter, wall thickness thermal cnductivity f steel tubes Temperature f stream flwing thrugh the tubes Thermal cnductivity f insulatin and emissivity f aluminum sheath Temperature f ambient air and surrundings Cnvectin cefficient at uter surface and maximum allwable surface temperature Unknwn:. Minimum required insulatin thickness and crrespnding heat lss per unit length. Effect f insulatin thickness n uter surface temperature and heat lss Schematic: Assumptins:. Steady state. One-dimensinal radial cnductin 3. Negligible cntact resistances at the material interfaces. Negligible steam side cnvectin resistance ( T, i T s, i 5. Negligible cnductin resistance fr aluminum sheath 6. Cnstant prperties 7. Small bject in large surrundings Analysis: a Applying energy balance at the uter surface fr a unit length f pipe: q q cnv q ( (Here, q is heat flw per unit length, W/m, rad Where: q π r h ( T q cnv, 3 s, T, rad q π r εσ ( T T 3 s, i cnd, st cnd, st cnd, ins T s, s, T cnd, ins sur ln( r / r / πk st ln( r3 / r / πk ins
6 Substituting abve int equatin ( Ts, i Ts, πr3 h ln( r / r ln( r3 / r πk πk [ ( T T εσ ( T T ] ( s,, s, sur st ins π (88 33 πr3 ln(0.8 / 0.5 ln( r3 / [ 6 * ( *5.67 *0 ( ] Slving fr r 3 r m Then thickness f insulatin t ins r 3 -r 0.m The heat rate is then π (88 33 q 0W / m ln(0.8 / 0.5 ln(0.39 / b The effects f r 3 n T s, and q are shwn belw.
7 Beynd r 3 0.0m, there are rapidly diminishing benefits assciated with increasing the insulatin thickness.
8 Prblem 3 Knwn: Thermal cnductivity and thickness f tw insulating materials Temperature drp acrss the insulating materials Unknwn:. Outer diameter f the reactrs. Inner and uter surface temperatures f the brsilicate insulatin Schematic: T, 35 C T, t h c t D T i, 85 C D T i, T 5 C Silica ( k.38w/m.k Brsilicate ( k.09w/m.k T T T i,, 55 C Assumptins:. Steady state cnditins. -D cnductin (in radial directin 3. Cnstant thermal cnductivities. Same external envirnment fr the insulated reactrs 5. Same heat lses q q 6. The same utside heat transfer cefficients, h h h c Analysis: a The heat rate acrss spherical shell is given by: k( T T q π ( r r Due t the same heat lss f bth reactrs πk T πk T q q r r t r r t (
9 earrange (, we have tt ( k T k T r k Tt k Tt 0.* 0.0(.03* 55.38*50.38*50* *55*0. 0.5m D r 0.30m b At steady state, q q, cnv, cnv hc π ( r t ( T, T hc π ( r t ( T, T Bth side f abve equatin can be divided by h c and the nly unknwn is T,, s, (35 5 ( ( T 5 (, T, T 36.8 C T T C i,,
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