MODULE 2: Worked-out Problems

Transcription

1 MODUE : Worked-out Problems Problem : he steady-state temperature dstrbuton n a one dmensonal wall of thermal conductvty 5W/m and thckness 5 mm s observed to be ( C) abx, where a C, B- c/ m, and x n meters (a) What s the heat generaton rate n the wall? (b) Determne the heat fluxes at the two wall faces In what manner are these heat fluxes related to the heat generaton rate? nown: emperature dstrbuton n a one dmensonal wall wth prescrbed thckness and thermal conductvty Fnd: (a) the heat generaton rate, n the wall, (b) heat fluxes at the wall faces and relaton to Schematc: ssumptons: () steady-state condtons, () one dmensonal heat flow, (3) constant propertes nalyss: (a) the approprate form of heat euaton for steady state, one dmensonal condton wth constant propertes s d d dx dx k d dx d dx(a bx ( CC / m k ) d [bx] dx bk ) 5W / m W / m 5 3 (b) he heat fluxes at the wall faces can be evaluated from Fourer s law,

2 x (x) d k dx x Usng the temperature dstrbuton (x) to evaluate the gradent, fnd d x (x) k [abx ] -kbx dx he flux at the face, s then x x atx, x () x (l) kb 5W / m( C / m (),W / m ) 5m Comments: from an overall energy balance on the wall, t follows that E n x E out Eg () x () x x () (),w / m 5m 5 W / m 3

3 Problem : salt gradent solar pond s a shallow body of water that conssts of three dstnct flud layers and s used to collect solar energy he upper- and lower most layers are well mxed and serve to mantan the upper and lower surfaces of the central layer at unform temperature and, where > lthough there s bulk flud moton n the mxed layers, there s no such moton n the central layer Consder condtons for whch solar radaton absorpton n the central layer provdes non unform heat generaton of the form e -ax, and the temperature dstrbuton n the central layer s ax (x) e bx c ka he uanttes (W/m3), a (/m), B (/m) and C() are known constants havng the prescrbed unts, and k s the thermal conductvty, whch s also constant (a) Obtan expressons for the rate at whch heat s transferred per unt area from the lower mxed layer to the central layer and from central layer to the upper mxed layer (b) Determne whether condtons are steady or transent (c) Obtan an expresson for the rate at whch thermal energy s generated n the entre central layer, per unt surface area nown: emperature dstrbuton and dstrbuton of heat generaton n central layer of a solar pond Fnd: (a) heat fluxes at lower and upper surfaces of the central layer, (b) whether condtons are steady or transent (c) rate of thermal energy generaton for the entre central layer Schematc: ssumptons: () central layer s stagnant, () one-dmensonal conducton, (3)constant propertes nalyss () the desred fluxes correspond to conducton fluxes n the central layer at the lower and upper surfaces general form for the conducton flux s

4 cond k Hence ka e ax B al l cond(x k e B u cond(x ) k B ka ka (b) Condtons are steady f / t pplyng the heat euaton, t k α t k e ax k e ax Τ α t Hence condtons are steady snce (for all <x<) t For the central layer, the energy generaton s E E g g e a dx ax e axdx a (e a ) ( e a a ) lternatvely, from an overall energy balance, E g Eg - (- cond(x))-( cond(x) ) E g k ka B ka e a B ( e a a ) Comments: Conducton s the negatve x-drecton, necesstatng use of mnus sgns n the above energy balance

5 Problem 3: he steady state temperatures dstrbuton n a one-dmensonal wall of thermal conductvty and thckness s of the form ax 3 bx cxd derve expressons for the heat generaton rate per unt volume n the wall and heat fluxes at the two wall faces(x, ) nown: steady-state temperature dstrbuton n one-dmensonal wall of thermal conductvty, (x)x 3 Bx CXd Fnd: expressons for the heat generaton rate n the wall and the heat fluxes at the two wall faces(x, ) ssumptons: () steady state condtons, () one-dmensonal heat flow, (3) homogeneous medum nalyss: the approprate form of the heat dffuson euaton for these condtons s d Or dx k Hence, the generaton rate s d dx d k dx k[6x B] d [3x dx d k dx Bx C ] whch s lnear wth the coordnate x he heat fluxes at the wall faces can be evaluated from Fourer s law, d x k k[3x Bx C] dx Usng the expresson for the temperature gradent derved above Hence, the heat fluxes are: Surface x; x ()-kc Surface x; () - [3 BC] x COMMENS: () from an over all energy balance on the wall, fnd E E n x g E out () E x 3k () ( kc) ( )[3 g Bk B C] E g From ntegraton of the volumetrc heat rate, we can also fnd E E g (x)dx g 3k Bk k[6x B]dx k[3x Bx]

6 Problem 4: he one dmensonal system of mass M wth constant propertes and no nternal heat generaton shown n fg are ntally at a unform temperature he electrcal heater s suddenly energzed provdng a unform heat flux o at the surface x the boundares at x and else where are perfectly nsulated (a) Wrte the dfferental euaton and dentfy the boundary and ntal condtons that could be used to determne the temperature as a functon of poston and tme n the system (b) On -x coordnates, sketch the temperature dstrbutons for the ntal condton (t<) and for several tmes after the heater s energzed Wll a steady-state temperature dstrbuton ever be reached? (c) On x -t coordnates, sketch the heat flux x (x,t) at the planes x, x/, and x as a functon of tme (d) fter a perod of tme t e has elapsed, the heater power s swtched off ssumng that the nsulaton s perfect, the system wll eventually reach fnal unform temperature f Derve an expresson that can be used to determne f a functon of the parameters o,t e,, and the system characterstcs M,c p, and (the heater surface area) nown: one dmensonal system, ntally at a unform temperature, s suddenly exposed to a unform heat flux at one boundary whle the other boundary s nsulated Fnd: (a) proper form of heat dffuson euaton; dentfy boundary and ntal condtons, (b) sketch temperature dstrbutons for followng condtons: ntal condton (t<), several tmes after heater s energzed ;wll a steady-state condton be reached?, (c) sketch heat flux for x, /, as a functon of tme, (d) expresson for unform temperature, f, reached after heater has been swtched off the followng an elapsed tme, te, wth the heater on] Schematc: ssumptons: () one dmensonal conducton, () no nternal heat generaton, (3) constant propertes nalyss: (a) the approprate form of the heat euaton follows lso the approprate boundary and ntal condtons are: x α t Intal condton: (x, ) unform temperature

7 o > Boundary condtons: x k / x) t x / x) Insulated (b) he temperature dstrbutons are as follows: No steady-state condton wll be reached snce E E n out and E n s constant (c) he heat flux as a functon of tme for postons x, / and appears as: ( d) If the heater s energzed untl tt o and then swtched off, the system wll eventually reach a unform temperature, f Perform an energy balance on the system, for an nterval of tme Δtt e, t e E n Est En Qn sdt ost e E Mc( ) ost It follows that ost e Mc(f ) OR f Mc st e f

8 Problem 5: m-long steel plate (k5w/m) s well nsulated on ts sdes, whle the top surface s at C and the bottom surface s convectvely cooled by a flud at C Under steady state condtons wth no generaton, a thermocouple at the mdpont of the plate reveals a temperature of 85 C What s the value of the convecton heat transfer coeffcent at the bottom surface? nown: length, surfacethermal condtons, and thermal conductvty of a Plate Plate mdpont temperature Fnd: surface convecton coeffcent Schematc: ssumptons: () one-dmensonal, steady conducton wth no generaton, () Constant propertes nalyss: for prescrbed condtons, s constant Hence, 5 C cond 5W / m / 5m / 5W / mk 3 C 5W / m ( / k) (/ h) ( / h)m / W h 3W / m Comments: he contrbutons of conducton and convecton to the thermal resstance are R t,cond m / W R t,cond 33m / W h

9 Problem 6: he wall of a buldng s a composte consstng of a -mm layer of common brck, a -mm layer of glass fber(paper faced 8kg/m ), a -mm layer of gypsum plaster (vermculte), and a 6-mm layer of pne panel If the nsde convecton coeffcent s W/m and the outsde convecton coeffcent s 7W/m, what are the total resstance and the overall coeffcent for heat transfer? nown: Materal thckness n a composte wall consstng of brck, glass fber, and vermculte and pne panel Inner and outer convecton coeffcents Fnd: otal thermal resstance and overall heat transfer coeffcent Schematc: Brck glass Gypsum Pne panel, p h 7W/m b gl hw/m mm mm 6mm h b b k g l gl k gy gy p p h ssumptons: () one dmensonal conducton, () constant propertes, (3) neglgble contact resstance Propertes: 3: Brck, k b 3 W/m: Glass fber (8kg/m 3 ), k g 38W/m: gypsum, k gy 7W/m: pne panel, k p W/m nalyss: consderng a unt surface rea, the total thermal resstance

10 R R R R tot tot tot tot h 7 k 3 38 ( )m / W B B g g 93m / W k gy gy 7 p p h 6 m W he overall heat transfer coeffcent s U R R U tot 34W / m tot (93m / W) Comments: an antcpated, the domnant contrbuton to the total resstance s made by the nsulaton

11 B Problem 7: he composte wall of an oven conssts of three materals, two of whch are known thermal conductvty, k W/m and k C 5W/m, and known thckness, 3m and C 5m he thrd materal, B, whch s sandwched between materals and C, s of known thckness, B 5m, but unknown thermal conductvty kb Under steady-state operatng condtons, measurements reveal an outer surface temperature of s, C, an nner surface temperature of s, 6 C and an oven ar temperature of 8 C he nsde convecton coeffcent h s known to be 5W/m What s the value of kbb? nown: hckness of three materal whch form a composte wall and thermal conductvtes of two of the materals Inner and outer surface temperatures of the compostes; also, temperature and convecton coeffcent assocated wth adjonng gas Fnd: value of unknown thermal conductvty, k B Schematc: s, C s, 6 C 8 C h 5W/m B C 3m B C 5m k W/m k C 5W/m B C ssumptons: () steady state condtons, () one-dmensonal conducton, (3) constant propertes, (4) neglgble contact resstance, (5) neglgble radaton effects nalyss: Referrng to the thermal crcut, the heat flux may be expressed as

12 s, s, B B / B C C W / m 3m 8 (6 ) C 5m 5m 5W / m B he heat flux can be obtaned from h( s, 5W / m ) 5W / m (8 6) C Substtutng for heat flux, 5 B B W / m Comments: radaton effects are lkely to have a sgnfcant nfluence on the net heat flux at the nner surface of the oven

13 Problem 8: steam ppe of m outsde dameter s nsulated wth a -mm-thck layer of calcum slcate If the nner and outer surfaces of the nsulaton are at temperatures of s, 8 and s, 49, respectvely, what s the heat loss per unt length of the ppe? nown: hckness and surface temperature of calcum slcate nsulaton on a steam ppe Fnd: heat loss per unt ppe length Schematc: D 6m s, 8 Steam Calcum slcate nsulaton D m s, 49 ssumptons: (steady state condtons, () one-dmensonal conducton, (3) constant propertes Propertes: calcum slcate (645): k89w/m nalyss: he heat per unt length s r r π(s, ) r s, ln(d / D) π(89w / m)(8 49) ln(6m / m) r 63W / m Comments: heat transferred to the outer surface s dsspated to the surroundngs by convecton and radaton

14 Problem 9: long cylndrcal rod of cm conssts of a nuclear reactng materal (kw/m) generatng 4,W/m 3 unformly throughout ts volume hs rod s encapsulated wthn another cylnder havng an outer radus of cm and a thermal conductvty of 4W/m the outer surface s surrounded by a flud at C, and the convecton coeffcent between the surface and the flud s W/m Fnd the temperatures at the nterface between the two cylnders and at the outer surface nown: cylndrcal rod wth heat generaton s cladded wth another cylnder whose outer surface s subjected to a convecton process Fnd: the temperature at the nner surfaces,, and at the outer surface, c Schematc: ssumptons: () steady-state condtons, () one-dmensonal radal conducton, (3), neglgble contact resstance between the cylnders nalyss: he thermal crcut for the outer cylnder subjected to the convecton process s

15 R R ln ro / r πk hπr o Usng the energy conservaton reurement, on the nner cylnder, E E out g Fnd that πr he heat rate euaton has the form Δ / R, hence (R R )and Δ / R Numercal values: R R ln / / / W / m 4,W / m π 4W / m π m 3 π () m 76m / W 398m / W 754W / m Hence C C 754W / m (76 cccc)m / W C C 754W / m 398m / W 3 3 C Comments: knowledge of nner cylnder thermal conductvty s not needed

16 Problem : n electrcal current of 7 flows through a stanless steel cable havng a dameter of 5mm and an electrcal resstance of 6* -4 /m (e permeter of cable length) he cable s n an envronment havng temperature of 3C, and the total coeffcent assocated wth convecton and radaton between the cable and the envronment s approxmately 5W/m (a) If the cable s bar, what s ts surface temperature? (b) If a very thn coatng of electrcal nsulaton s appled to the cable, wth a contact resstance of m /W, what are the nsulaton and cable surface temperatures? (c) here s some concern about the ablty of the nsulaton to wthstand elevated temperatures What thckness of ths nsulaton (k5w/m) wll yelds the lowest value of the maxmum nsulaton temperature? What s the value of the maxmum temperature when the thckness s used? nown: electrc current flow, resstance, dameter and envronmental condtons assocated wth a cable Fnd: (a) surface temperature of bare cable, (b) cable surface and nsulaton temperatures for a thn coatng of nsulaton, (c) nsulaton thckness whch provdes the lowest value of the maxmum nsulaton temperature Correspondng value of ths temperature Schematc: s E g ssumptons: () steady-state condtons, () one-dmensonal conducton n r, (3) constant propertes

17 nalyss: (a) the rate at whch heat s transferred to the surroundngs s fxed by the rate of heat generaton n the cable Performng an energy balance for a control surface about the cable, t follows that e I R h( πd )( It follows that s )wth I R e (7) E g 4 (6 or, for the bare cable, Ω / m) 94W / m s s 7787 hπd C 3 C (5W / m 94W / m ) π(5m) (b) Wth thn coatng of nsulaton, there exsts contact and convecton resstances to heat transfer from the cable he heat transfer rate s determned by heatng wthn the cable, however, and therefore remans the same, s s R t,c R t,c hπd πd hπd πd (s ) R t,c h nd solvng for the surface temperature, fnd s s R t,c πd h 53 C 94W / m π(5m) m m 4 3 W W C he nsulaton temperature s then obtaned from s R t,e Or

18 s R 7787 t,c C R t,c 53 C πd W m C m W π(5m) (c) he maxmum nsulaton temperature could be reduced by reducng the resstance to heat transfer from the outer surface of the nsulaton Such a reducton s possble D <D cr r k 5W / m h 5W / m cr m Hence, D cr 4m> D 5m o mnmze the maxmum temperature, whch exsts at the nner surface of the nsulaton, add nsulaton n the amount D D t t 75m D cr D (4 5) m he cable surface temperature may then be obtaned from ` s s 3 C R t,c ln(dc,r/d ) m /W ln(4/5) πd ππ hππ π(5m) ππ(5w/ ) W c,r 5 π(4m) m hence, W 94 m s 695 recognzng C R 38 C s s 3 C s 3 C (7 66 3)m/W 5m/W t,c, that, ( ) W m R 94 t,c 695 C m W s πd π(5m) s /R t,c, Comments: use of the crtcal nsulaton n leu of a thn coatng has the effect of reducng the maxmum nsulaton temperature from 7787 C to 38 C Use of the crtcal nsulaton thckness also reduces the cable surface temperatures to 695 C from 7787 C wth no nsulaton or fro 53 C wth a thn coatng

19 Problem : he steady state temperature dstrbuton n a complete plane wall of three dfferent methods, each of constant thermal conductvty, s shown below (a) On the relatve magntudes of and 3 and of 3 and 4 (b) Comment on the relatve magntudes of k and k B B and ok k and k BB C (c) Plot the heat flux as a functon of x nown: emperature dstrbuton n a composte wall Fnd: (a) relatve magntudes of nterfacal heat fluxes, (b) relatve magntudes of thermal conductvtes, and (c) heat fluxes as a functon of dstance x Schematc: 3 4 B C ssumptons: () steady-state condtons, () one-dmensonal conducton, (3) constant propertes nalyss: (a) for the prescrbed condtons (one-dmensonal, steady state, constant k), the parabolc temperature dstrbuton n C mples the exstence of heat generaton Hence, snce d/dx ncreases wth decreasng x, the heat flux n C ncreases wth decreasng Hence, > 3 4 However, the lnear temperatures dstrbutons n and B ndcate no generaton, n whch case 3

20 (b) Snce conservaton of energy reures that 3, B 3, C and d/dx/ B <d/dx) C, t follows from Fourer s law that > C Smlarly, snce,,b and d/dx) > d/dx) B, t follows that < B (d) It follows that the flux dstrbuton appears as shown below x O x x 3 x Comments: Note that, wth d/dx) 4,C, the nterface at 4 s adabatc

21 Problem : When passng an electrcal current I, a copper bus bar of rectangular cross secton (6mm*5mm) experences unform heat generaton at a rate al, whereα 5W / m If the bar s n ambent ar wth h5w/m and ts maxmum temperature must not exceed that of the ar by more than 3C, what s the allowable current capacty for the bar? nown: Energy generaton, (I), n a rectangular bus bar Fnd: maxmum permssble current Schematc: ssumptons: () one-dmensonal conducton n x (W>>), () steady-state condtons, (3) constant propertes, (4) neglgble radaton effects Propertes: copper: k 4W/m nalyss: the maxmum md plane temperature s s Or substtutng the energy balance results,

22 , / h s I m W m W m m m W C I h k I hence h k I h k o o 86 / 5 / 8 3 ) 3 / ( 5 3 ) / / ( 5, 5 max 3 max