Department f Cvl Engneerng & Appled Mechancs McGll Unversty, Mntreal, Quebec Canada CIVE 90 THEMODYNAMICS & HEAT TANSFE Assgnment #6 SOUTIONS. Cnsder a.-m hgh and -m-wde duble-pane wndw cnsstng f tw 3-mmthck layers f glass (k 0.78 W/m C separated by a -mm-wde stagnant ar space (k 0.06 W/m C. Determne the steady rate f heat transfer thrugh ths dublepane wndw and the temperature f ts nner surface fr a day durng whch the rm s mantaned at 4 C whle the temperature f the utdrs s -5 C. Take the cnvectn heat transfer ceffcents n the nner and uter surfaces f the wndw t be h 0 W/m C and h 5 W/m C, and dsregard any heat transfer by radatn. A duble-pane wndw cnssts f tw layers f glass separated by a stagnant ar space. Fr specfed ndrs and utdrs temperatures, the rate f heat lss thrugh the wndw and the nner surface temperature f the wndw are t be determned. Assumptns Heat transfer thrugh the wndw s steady snce the ndr and utdr temperatures reman cnstant at the specfed values. Heat transfer s ne-dmensnal snce any sgnfcant temperature gradents wll exst n the drectn frm the ndrs t the utdrs. 3 Thermal cnductvtes f the glass and ar are cnstant. 4 Heat transfer by radatn s neglgble. Prpertes The thermal cnductvty f the glass and ar are gven t be k glass 0.78 W/m C and k ar 0.06 W/m C. Analyss The area f the wndw and the ndvdual resstances are A (. m ( m 4. m ttal 3 ar, 0.047 h A (0 W/m. C(.4 m 0.003 m glass 0.006 ka (0.78 W/m. C(.4 m 0.0 m 0.93 ka (0.06 W/m. C(.4 m, 0.067 h A (5 W/m. C(.4 m + + + 0.047 + (0.006 + 0.93+ 0.067 cnv cnv cnv, cnv, 0.539 The steady rate f heat transfer thrugh wndw glass then becmes 3 T T [4 ( 5] C Q & 4 W T ttal 0.539 The nner surface temperature f the wndw glass can be determned frm T T Q & T T Q & cnv, 4 C (4 W(0.047 9. C cnv, Ar T
. A s cnstructed f tw layers f 0.5-n-thck sheetrck (k 0.0 Btu/h ft F, whch s a plasterbard made f tw layers f heavy paper separated by a layer f gypsum, placed 5 n apart. The space between the sheetrcks s flled wth fbreglass nsulatn (k 0.00 Btu/h ft F. Determne a the thermal resstance f the, and b ts -value f nsulatn n Englsh unts. A s cnstructed f tw layers f sheetrck wth fberglass nsulatn n between. The thermal resstance f the and ts -value f nsulatn are t be determned. Assumptns Heat transfer thrugh the s ne-dmensnal. Thermal cnductvtes are cnstant. Prpertes The thermal cnductvtes are gven t be k sheetrck 0.0 Btu/h ft F and k nsulatn 0.00 Btu/h ft F. Analyss (a The surface area f the s nt gven and thus we cnsder a unt surface area (A ft. Then the -value f nsulatn f the becmes equvalent t ts thermal resstance, whch s determned frm. sheetrck fberglass ttal k 3 + k 0.5 / ft 0.47 ft (0.0 Btu/h.ft. F 5 / ft 0.83 ft (0.00 Btu/h.ft. F 0.47 + 0.83.66 ft. F.h/Btu. F.h/Btu. F.h/Btu (b Therefre, ths s apprxmately a - n Englsh unts. 3 3 3. Cnsder a huse whse s are ft hgh and 40 ft lng. Tw f the s f the huse have n wndws, whle each f the ther tw s has fur wndws made f 0.5-n-thck glass (k 0.45 Btu/h ft F that are 3ft x 5 ft n sze. The s are certfed t have an -value f 9 (.e., an /k value f 9 h ft. Dsregardng any drect radatn gan r lss thrugh the wndws and takng the heat transfer ceffcents at the nner and uter surfaces f the huse t be and 4 Btu/h ft F, respectvely, determne the rat f heat transfer thrugh the s wth and wthut wndws. Tw f the s f a huse have n wndws whle the ther tw s have 4 wndws each. The rat f heat transfer thrugh the s wth and wthut wndws s t be determned. Assumptns Heat transfer thrugh the s and the wndws s steady and ne-dmensnal. Thermal cnductvtes are cnstant. 3 Any drect radatn gan r lss thrugh the wndws s neglgble. 4 Heat transfer ceffcents are cnstant and unfrm ver the entre surface. Prpertes The thermal cnductvty f the glass s gven t be k glass 0.45 Btu/h.ft F. The -value f the s gven t be 9 h.ft. Analyss The thermal resstances thrugh the wthut wndws are Wall A ( ft(40 ft 480 ft 0.00047 h ( Btu/h.ft. F(480 ft ttal, ka 9 hft 0.03958 h (480 m 0.0005 h (4 Btu/h.ft. F(480 ft + + 0.00047 + 0.03958 + 0.0005 0.0447 h T Q &
The thermal resstances thrugh the wth wndws are A wndws A 4 eqv ttal, 4(3 5 60 ft A ttal glass glass + A 0.5 / ft 0.000776 h ka (0.45 Btu/h.ft. F(60 ft + eqv wndws ka 480 60 40 ft 9 h.ft. 0.0454 h (40 ft + 0.000776 + 0.0454 0.00076 h 0.00047 + 0.00076 + 0.0005 0.0037 h Then the rat f the heat transfer thrugh the s wth and wthut wndws becmes ttal, T /,,. Δ ttal ttal 0 0447 ΔT / 0. 0037 7.7 ttal, ttal, ttal, eqv glass 4. Cnsder a -m-hgh electrc ht-water heater that has a dameter f 40 cm and mantans the ht water at 55 C. The tank s lcated n a small rm whse average temperature s 7 C, and the heat transfer ceffcent n the uter surface f the heater s W/m C. The tank s placed n anther 46-cm-dameter sheet metal tank f neglgble thckness, and the space between the tw tanks s flled wth fam nsulatn (k 0.03 W/m C. The thermal resstances f the water tank and the uter thn sheet metal shell are very small and can be neglected. The prce f electrcty s $0.08/kWh, and the hmewner pays $80 a year fr water heatng. Determne the fractn f the ht-water energy cst f ths husehld that s due t the heat lss frm the tank. Ht-water nsulatn kts cnsstng f 3-cm-thck fbreglass nsulatn (k 0.035 W/m C large enugh t wrap the entre tank are avalable fr abut $30. If such an nsulatn s nstalled n ths water tank by the hmewner hmself, hw lng wll t take fr ths addtnal nsulatn t pay fr tself? An electrc ht water tank s made f tw cncentrc cylndrcal metal sheets wth fam nsulatn n between. The fractn f the ht water cst that s due t the heat lss frm the tank and the payback perd f the d-t-yurself nsulatn kt are t be determned. Assumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvtes are cnstant. 4 The thermal resstances f the water tank and the uter thn sheet metal shell are neglgble. 5 Heat lss frm the tp and bttm surfaces s neglgble. Prpertes The thermal cnductvtes are gven t be k 0.03 W/m C fr fam nsulatn and k 0.035 W/m C fr fber glass nsulatn Analyss We cnsder nly the sde surfaces f the water heater fr smplcty, and dsregard the tp and bttm surfaces (t wll make dfference f abut 0 percent. The ndvdual thermal resstances are A πd π(. 046m ( m 89. m 0.09 ( W/m. C(.89 m fam T w T 3
fam ttal ln( r / r ln(3 / 0 0.37 π k π (0.03 W/m. C( m + fam 0.09 + 0.37 0.40 The rate f heat lss frm the ht water tank s & (55 Q T w T 7 C 70 W ttal 0.40 C / W The amunt and cst f heat lss per year are Q Q & Δt (. 007 kw(365 4 h / yr 63. kwh / yr Cst f Energy (Amunt f energy(unt cst (63. kwh($0.08 / kwh $49.056 $49.056 f 0.75 7.5% $80 If 3 cm thck fber glass nsulatn s used t wrap the entre tank, the ndvdual resstances becmes A πd π ( 0.5 m( m 3.67 m 0.06 fam fberglass ( W/m. C(3.67 m T w T fam fberglass ttal ln( r / r ln(3 / 0 0.37 πk π(0.03 W/m. C( m ln( r3 / r ln(6 / 3 0.79 πk π(0.035 W/m. C( m + fam + fberglass 0.06 + 0.37+ 0.79 0.676 The rate f heat lss frm the ht water heater n ths case s Tw T (55 7 C Q & 4.4 W ttal 0.676 The energy savng s savng 70-4.4 8.58 W The tme necessary fr ths addtnal nsulatn t pay fr ts cst f $30 s then determned t be Cst (0.0858 kw(tme perd($0.08 / kwh $30 Then, Tme perd 3, hurs 547 days.5 years 5. Steam at 450 F s flwng thrugh a steel ppe (k 8.7 Btu/h ft F whse nner and uter dameters are 3.5 n and 4 n, respectvely, n an envrnment at 55 F. The ppe s nsulated wth -n-thck fbreglass nsulatn (k 0.00 Btu/h ft F. If the heat transfer ceffcents n the nsde and the utsde f the ppe are 30 and 5 Btu/h ft F, respectvely, determne the rate f heat lss frm the steam per ft length f the ppe. What s the errr nvlved n neglectng the thermal resstance f the steel ppe n calculatns? A steam ppe cvered wth -n thck fberglass nsulatn s subjected t cnvectn n ts surfaces. The rate f heat lss frm the steam per unt length and the errr nvlved n neglectng the thermal resstance f the steel ppe n calculatns are t be determned. Assumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvtes are cnstant. 4 The thermal cntact resstance at the nterface s neglgble. Prpertes The thermal cnductvtes are gven t be k 8.7 Btu/h ft F fr steel and k 0.00 Btu/h ft F fr fberglass nsulatn. 4
Analyss The nner and uter surface areas f the nsulated ppe are A πd π(./ 35 ft ( ft 0. 96 ft A πd π(8 / ft ( ft. 094 ft ppe nsulatn T T The ndvdual resstances are ttal nsulatn ppe + (30 Btu/h.ft. F(0.96 ft 0.036 h ln( r / r ln( /.75 0.00 h πk π (8.7 Btu/h.ft. F(ft ln( r3 / r ln(4 / 5.56 h πk π (0.00 Btu/h.ft. F(ft 0.096 h (5 Btu/h.ft. F(.094 ft + + 0.036 + 0.00 + 5.56 + 0.096 5.65 h Then the steady rate f heat lss frm the steam per ft. ppe length becmes T T (450 55 F Q & 69.9Btu/h ttal 5.65 h If the thermal resstance f the steel ppe s neglected, the new value f ttal thermal resstance wll be ttal + + 0.036 + 5.56 + 0.096 5.648 h Then the percentage errr nvlved n calculatns becmes errr% (5. 65 5. 648 h 00 0.035% 565. h whch s nsgnfcant. 6. The thermal cntact cnductance at the nterface f tw -cm-thck cpper plates s measured t be 8,000 W/m OC. Determne the thckness f the cpper plate whse thermal resstance s equal t the thermal resstance f the nterface between the plates. The thckness f cpper plate whse thermal resstance s equal t the thermal cntact resstance s t be determned. Prpertes The thermal cnductvty f cpper s k 386 W/m C (Table A-5. Analyss Ntng that thermal cntact resstance s the nverse f thermal cntact cnductance, the thermal cntact resstance s determned t be c h 5 c 5.556 0 8,000 W/m. C m. Fr a unt surface area, the thermal resstance f a flat plate s defned as where s the thckness f k the plate and k s the thermal cnductvty. Settng c, the equvalent thckness s determned frm the relatn abve t be 5 k k c ( 386 W / m. C(5.556 0 m. C / W 0. 04 m.4 cm Therefre, the nterface between the tw plates ffers as much resstance t heat transfer as a.4 cm thck cpper. Nte that the thermal cntact resstance n ths case s greater than the sum f the thermal resstances f bth plates. 5
7. A 4-m-hgh and 6-m-wde cnssts f a lng 8-cm x 30-cm crss-sectn f hrzntal brcks (k 0.7 W/m OC separated by.5-cm thck plaster layers (k 0. W/m OC. There are als -cm-thck plaster layers n each sde f the, and a -cm-thck rgd fam (k 0.06 W/m OC n the nner sde f the. The ndr and utdr temperatures are O C and -4 O C, and the cnvectve heat transfer ceffcents n the nner and uter sdes are h 0 W/m OC and h 0 W/m OC, respectvely. Assumng -D heat transfer and gnrng radatn, determne the rate f heat transfer thrugh the. A cnssts f hrzntal brcks separated by plaster layers. There are als plaster layers n each sde f the, and a rgd fam n the nner sde f the. The rate f heat transfer thrugh the s t be determned. Assumptns Heat transfer s steady snce there s n ndcatn f change wth tme. Heat transfer thrugh the s ne-dmensnal. 3 Thermal cnductvtes are cnstant. 4 Heat transfer by radatn s dsregarded. Prpertes The thermal cnductvtes are gven t be k 0.7 W/m C fr brcks, k 0. W/m C fr plaster layers, and k 0.06 W/m C fr the rgd fam. Analyss We cnsder m deep and 0.33 m hgh prtn f whch s representatve f the entre. The thermal resstance netwrk and ndvdual resstances are T 3 4 cnv, 0.303 h A (0 W/m. C(0.33 m 0.0 m fam.33 ka (0.06 W/m. C(0.33 m 0.0 m 6 plaster 0.303 sde ka (0. W/m. C(0.33 m 0.8 m 3 5 plaster 54.55 center ka (0. W/m. C(0.05 m 0.8 m 4 brck 0.833 ka (0.7 W/m. C(0.30 m cnv, 0.5 h A (0 W/m. C(0.33 m + + + + md 0.8 md 3 4 5 54.55 0.833 54.55 ttal + + + md + 0.303 +.33 + (0.303 + 0.8 + 0.5 4.0 The steady rate f heat transfer thrugh the per 033. m s & [( ( ] Q T T 4 C 69. W ttal 40. Then steady rate f heat transfer thrugh the entre becmes & ( 4 6 m Q ttal (. 69 W 450 W 033. m 5 6 T 6
8. Cnsder a cld alumnum canned drnk that s ntally at a unfrm temperature f 3 0 C. The can s.5 cm hgh and has a dameter f 6 cm. If the cnvectn heat transfer ceffcent between the can and the surrundng ar at 5 O C s 0 W/m OC, determne hw lng t wll take fr the average temperature f the drnk t rse t 0 O C. In an effrt t slw dwn the warmng f the cld drnk, a persn puts the can n a perfectly fttng -cm thck cylndrcal rubber nsulatr (k 0.3 W/m OC. Nw hw lng wll t take fr the average temperature f the drnk t rse t 0 O C? Assume the tp f the can s nt cvered. A cld alumnum canned drnk that s ntally at a unfrm temperature f 3 C s brught nt a rm ar at 5 C. The tme t wll take fr the average temperature f the drnk t rse t 0 C wth and wthut rubber nsulatn s t be determned. Assumptns The drnk s at a unfrm temperature at all tmes. The thermal resstance f the can and the nternal cnvectn resstance are neglgble s that the can s at the same temperature as the drnk nsde. 3 Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 4 Thermal prpertes are cnstant. 5 The thermal cntact resstance at the nterface s neglgble. Prpertes The thermal cnductvty f rubber nsulatn s gven t be k 0.3 W/m C. Fr the drnk, we use the prpertes f water at rm temperature, ρ 000 kg/m 3 and C p 480 J/kg. C. Analyss Ths s a transent heat cnductn, and the rate f heat transfer wll decrease as the drnk warms up and the temperature dfference between the drnk and the surrundngs decreases. Hwever, we can slve ths prblem apprxmately by assumng a cnstant average temperature f (3+0/ 6.5 C durng the prcess. Then the average rate f heat transfer nt the drnk s πd π(. 006m A πd+ π(. 006m (. 05m + 009. m 4 4 Q & bare, ave h A( Tar Tcan, ave (0 W/m. C(0.09 m (5 6.5 C 5.40 W The amunt f heat that must be suppled t the drnk t rase ts temperature t 0 C s 3 m ρv ρπr ( 000 kg / m π( 0. 03 m ( 05. m 0. 353 kg Q mcpδt (. 0 353 kg(480 J / kg(0-3 C 0, 39 J Then the tme requred fr ths much heat transfer t take place s Q 0, 39 J Δt 9 s 3.9 mn 5.4 J / s We nw repeat calculatns after wrappng the can wth -cm thck rubber nsulatn, except the tp surface. The rate f heat transfer frm the tp surface s & Qtp, ave tp ( Tar Tcan, ave ( 0 W / m. C[ π ( 0. 03 m ]( 5 65. C 0.5 W Heat transfer thrugh the nsulated sde surface s A πd π(. 0 08 m (. 05 m 0. 034 m Q & nsulatn, sde ttal T 3.83 (0 W/m. C(0.034 m ln( r / r ln(4 / 3.88 πk π (0.3 W/m. C(0.5 m T + nsulatn 3.83 +.88 6.00 (5 6.5 C 6.00 ar can,ave sde ttal 3.08 W T can nsulatn T ar 7
The rat f bttm t the sde surface areas s ( πr /( πr r/( 3 /(. 5 0.. Therefre, the effect f heat transfer thrugh the bttm surface can be accunted fr apprxmately by ncreasng the heat transfer frm the sde surface by %. Then, & & nsulated Qsde+ bttm + Qtp. 3. 08 + 05. 3. 97 W Then the tme f heatng becmes Q 0, 39 J Δt 60 s 43.4 mn 3.97 J / s 9. A -mm-dameter and 0-m-lng electrc wre s tghtly wrapped wth a -mm-thck plastc cver whse thermal cnductvty s k 0.5 W/m OC. Measurements ndcate that heat s generated n the wre at a rate f 80 W. If the nsulated wre s expsed t ar at T 30 O C wth a heat transfer ceffcent f h 4 W/m OC, determne the temperature at the nterface f the wre and the plastc cver n steady peratn. Als determne f dublng the thckness f the plastc cver wll ncrease r decrease ths nterface temperature. An electrc wre s tghtly wrapped wth a -mm thck plastc cver. The nterface temperature and the effect f dublng the thckness f the plastc cver n the nterface temperature are t be determned. Assumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 3 Thermal prpertes are cnstant. 4 The thermal cntact resstance at the nterface s neglgble. 5 Heat transfer ceffcent accunts fr the radatn effects, f any. Prpertes The thermal cnductvty f plastc cver s gven t be k 0.5 W/m C. Analyss In steady peratn, the rate f heat transfer frm the wre s equal t the heat generated wthn the wre, W& e VI ( 8 V(0 A 80 W plastc cnv T The ttal thermal resstance s T cnv 0.336 (4 W/m. C[ π (0.004 m(0 m] plastc ttal ln( r / r ln( / 0.0735 πk π (0.5 W/m. C(0 m cnv + plastc 0.336 + 0.0735 0.405 Then the nterface temperature becmes T T Q & T T + Q & ttal 30 C + (80 W(0.405 6.4 C ttal The crtcal radus f plastc nsulatn s k 0.5 W/m. C r cr 0.0065 m 6.5 mm h 4 W/m. C Dublng the thckness f the plastc cver wll ncrease the uter radus f the wre t 3 mm, whch s less than the crtcal radus f nsulatn. Therefre, dublng the thckness f plastc cver wll ncrease the rate f heat lss and decrease the nterface temperature. 8