Physc 231 Lecture 33 Man pnts f tday s lecture: eat and heat capacty: Q cm Phase transtns and latent heat: Q Lm ( ) eat flw Q k 2 1 t L Examples f heat cnductvty, R values fr nsulatrs Cnvectn R L / k Radatn P σ e 4 where σ 5.67x10-8 W / ( K 4 )
eat and heat capacty We saw that fr atms r mlecules n an deal gas s <KE>f/2k B. In general, the atms r mlecules n matter ncrease n energy f the bject s heated. hs s heat energy. f s the number f degrees f freedm (3 fr atm). eat Q s the energy that flws frm a ht bject t a cld bject slely because f the dfference n. When the heat flw s suffcent, the tw bjects reach the same temperature and we say they are n thermal equlbrum at the same. Fr a mnatmc deal gas at cnstant vlume Q E th U U f U f 2 Nk b f f 2 Nk b f 2 Nk b Q f 2 Nk f b 2 nr nc c mlar,v f/2r s the heat capacty per mle at mlar,v cnstant vlume fr an deal gas Fr many ther materals, the relatnshp between heat transferred and temperature change s gven by: (usually transferred at cnstant V r P) Q cm c sthe specfc m s the mass and cm heat capacty per unt mass the ttal ( f ) heat capacty f the bject.
Example f equlbratn and energy cnservatn t a fabrcatn plant, a ht frgng has a mass f 75 kg and a specfc heat capacty f 430 J/(kg C). harden t, the frgng s quenched by mmersn n 710 kg f l that has a temperature f 32 C and a specfc heat capacty f 645 cal/(kg C). he fnal temperature f the l and frgng at thermal equlbrum s 47 C. ssumng that heat flws nly between the frgng and the l, determne the ntal temperature f the frgng. Energy cnservatn: Q c m Q c m Q + Q 0 l ( ) l l l ( f l ) 4.19J cal f cl 645cal/(kg C) 2700kJ/(kg C) ( ) ( ) c m c m f l l f l f + f ( ) c m l l f l c m ( ) c m l l f l c m Q Q l m m l c 75 kg 710 kg ( )( ) C) ( 75kg) 430J/kg/ C 2700J / (kg C) 710kg 47 32 C 47 C 939 C c l 645cal/kg/ C + 430J / (kg f l,0,0? 47 C 32 C
Quz 100 kg mass f water at a temperature f 30 C s drpped nt a thermally slated vessel cntanng 100 kg f water whch s a temperature f 10 C. fter system cmes t thermal equlbrum the fnal temperature s (useful nfrmatn: c water 4186 J/(kg C)) a) 10 C b) 15 C c) 20 C d) 25 C ( ) + c water 100kg( f 10 C) 0 c water 100kg f 30 C / / f 30 C + f 10 C 0 / / f ( 30 C +10 C) / 2 20 C
Questn 13.1 hermal Cntact I w bjects are made f the same materal, but have dfferent masses and temperatures. If the bjects are brught nt thermal cntact, whch ne wll have the temperature change f greater magntude? a) the ne wth the hgher ntal temperature b) the ne wth the lwer ntal temperature c) the ne wth the greater mass d) the ne wth the smaller mass e) the ne wth the hgher specfc heat Q heavy + Q lght 0 cm ( heavy f ) heavy + cm ( lght f ) lght 0 cm heavy ( f ) heavy cm ( lght f ) ( lght f ) lght m heavy m lght ( f ) heavy
Phase transtns he calrc curve that descrbes vs. Q. Regns wth a lnear ncrease crrespnd t a cnstant heat capacty where Q cm Q Q /(cm) he flat regns ccur when the system has a phase transtn. Fr water, there s ne where ce changes t water and ther where water changes t steam. If the pressure remans cnstant durng the phase change, the temperatures wll reman cnstant. he heat requred t change a mass m f the matter s gven by the latent heat L fr the phase change. Q Lm 4 5 Lfusn 33.5x10 J / kg Lvaprzat n 22.6x10 J / kg
Example When t rans, water vapr n the ar cndenses nt lqud water, and the energy s released. (a) w much energy s released when 0.0254 m (ne nch) f ran falls ver an area f 2.59x10 6 m 2 (ne square mle)? (b) If the average energy needed t heat ne hme fr a year s 1.5x10 11 J, hw many hmes can be heated wth the energy determned n part (a)? a) m water ρ water V water 1000kg / m 3 ( 2.59x10 6 m 2 )(.0254m) 6.58x10 7 kg Q released L vaprzatn m water ( 22.6x10 5 J / kg) ( 6.58x10 7 kg) 1.5x10 14 J b) n huses 1.5x10 11 J 1.5x10 14 J n huses 1000
eat flw here are three majr prcesses that transfer heat frm ne pnt t anther. Cnductn Cnvectn Radatn Cnvectn results frm the fact that ht bjects generally expand. hs decreases ther densty. If ths ccurs n a flud, the less dense ht flud rses and the clder denser flud falls.
Cnductn Cnductn cncerns the transfer f heat thrugh materals wthut cnvectn. Cnsder an cncrete wall f a heated garage. he utsde f the garage s at temperature 1 and the nterr f the garage s a temperature 2. he cnductve heat flw thrugh a prtn f the wall wth area s gven by: ( ) Q k 2 1 k t L L s the heat flw thrugh the wall, k s the thermal cnductvty f cncrete, and L s the thckness f the cncrete wall. Example: Calculate the heat flw thrugh a 2 m 2 sectn f a 20 cm thck cncrete wall when the utsde temperature s 0 C and the nsde temperature s 20 C. ssume the thermal cnductvty f the cncrete s 1.3 J/(s m C) Q t k L 1.3J / (s m C) 2m 2 0.2m ( )( 20 C) 260W
Quz w materals have the same nsulatng value f the same amunt f heat per secnd per square meter flws thrugh each due t the same temperature dfference. Ignrng ar cnvectn, what thckness f bdy fat s requred t gve the same nsulatng value as a 0.010 nch thckness f ar? (k fat 0.2 J/(sm C), k ar 0.023 J/(sm C)) a) 0.09 nches b) 0.7 nches c). 2.3 nches d) 4.2 nches want fat ar / / / / k fat L fat k ar L ar k fat L fat k ar L ar multply bth sdes f eq. by L fat L ar k fat L L fat ar L fat / / / k ar L fat L ar k fat k ar k ar L L fat ar L ar / k ar / / k ar ( 0.01nch) 0.2 0.023.09nch
Layered materals R values Cnsder the layered nsulatng structure at the rght. he area f each layer s the same. Each layer can have a dfferent thckness L and heat cnductvty k. ere the mprtant thng t remember s that we are cncerned wth a steady state slutn. here s n buld up f heat n any f these layers. Fr each layer ne has: L R k L L k t Q ere R t the R value f the th nsulatng layer. Large R value means better nsulatn and less heat lss n wnter. he ttal temperature dfference fr a layered structure s gven by the sum. R t Q R R.e.
Example f a layered structure w rds, ne f alumnum and the ther f cpper are jned end t end. he crss-sectnal area f each s 4.0x10-4 m 2, and the length f each s 0.04 m. he free end f the alumnum rd s kept at 302 C, whle the free end f the cpper rd s kept at 25 C. he lss f heat thrugh the sdes f the rds may be gnred. (a) What s the temperature at the alumnum-cpper nterface? (b) w much heat s cnducted thrugh the unt n 2.0 s? (k alum 238 J/(s m C), k Cu 397 J/(s m C)) a) alum R alum L alum k alum Cu L Cu k Cu alum Cu L alum k alum L Cu k Cu k Cu k alum 397 238 1.67 alum + Cu Cu ( alum Cu +1) 2.67 Cu Cu 2.67 302 C 25 C 2.67 103.7 C nterface 25 C +103.7 C 128.7 C b) k Cu Cu 397 4x10 4 103.7 W 412W L Cu.04
Example f a layered structure w rds, ne f alumnum and the ther f cpper are jned end t end. he crss-sectnal area f each s 4.0x10-4 m 2, and the length f each s 0.04 m. he free end f the alumnum rd s kept at 302 C, whle the free end f the cpper rd s kept at 25 C. he lss f heat thrugh the sdes f the rds may be gnred. What s the temperature at the alumnum-cpper nterface between Cpper and lumnum? Useful nfrmatn: k alum 238J/(sm C) and : k Cu 397J/(sm C). a) alum R alum Cu R Cu R alum L alum.04m k alum 238J / (s m C) 1.681x10 4 s m 2 C / J R Cu L Cu.04m k Cu 397J / (s m C) 1.008x10 4 s m 2 C / J tt R tt R tt R Cu + R lum 2.69x10 4 s m 2 C / J tt R tt nt erface 25 C + Cu 25 C + R Cu nt erface 25 C + tt R tt R Cu 25 C + (302 25) 1.008 2.69 129 C