Analysis The characteristic length of the junction and the Biot number are

Transcription

1 -4 4 The temerature f a gas stream s t be measured by a thermule. The tme t taes t regster 99 erent f the ntal ΔT s t be determned. Assumtns The juntn s sheral n shae wth a dameter f D 0.00 m. The thermal rertes f the juntn are nstant. The heat transfer effent s nstant and unfrm ver the entre surfae. 4 Radatn effets are neglgble. 5 The Bt number s B < 0. s that the lumed system analyss s alable (ths assumtn wll be verfed. Prertes The rertes f the juntn are gven t be 5 W/m. C, ρ 8500 g/m, and 0 J/g. C. Analyss The haraterst length f the juntn and the Bt number are V πd / 6 D 0.00 m m A surfae πd 6 6 h (90 W/m. C(0.000 m B < 0. (5 W/m. C Sne B < 0., the lumed system analyss s alable. Then the tme erd fr the thermule t read 99% f the ntal temerature dfferene s determned frm T 0.0 ha h b ρ V ρ (8500 g/m Gas 90 W/m. C s (0 J/g. C(0.000 m h, T Juntn D T(t T e 0.0 e (0.654 s t t 7.8 s PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

2 -9-0 The heatng tmes f a shere, a ube, and a retangular rsm wth smlar dmensns are t be determned. Assumtns The thermal rertes f the gemetres are nstant. The heat transfer effent s nstant and unfrm ver the entre surfae. Prertes The rertes f slver are gven t be 49 W/m ºC, ρ 0,500 g/m, and 0.5 J/g ºC. Analyss Fr shere, the haraterst length and the Bt number are V πd / 6 D 0.05 m m 5 m Ar Asurfae πd 6 6 h, T h ( W/m. C(0.008 m B < 0. (49 W/m. C Sne B < 0., the lumed system analyss s alable. Then the tme erd fr the shere temerature t reah t 5ºC s determned frm b ha ρ V ρ h (0,500 g/m W/m. C s (5 J/g. C(0.008 m Cube: V A h B b T surfae ha ρ 6 e 5 e 0 ( s t 0.05 m m 6 6 ( W/m. C(0.008 m < 0. (49 W/m. C V ρ h (0,500 g/m t 48 s 40.5 mn 5 m 5 m 5 m W/m. C s (5 J/g. C(0.008 m Ar h, T T e Retangular rsm: V A h B b surfae (0,500 g/m ha ρ 5 e 0 ( s t ( W/m. C( m < 0. (49 W/m. C V ρ t 48 s 40.5 mn (0.04 m(0.05 m(0.06 m (0.04 m(0.05 m + (0.04 m(0.06 m + (0.05 m(0.06 m h W/m. C s (5 J/g. C( m 4 m 5 m 6 m m Ar h, T T e 5 e 0 ( s t t 6s 9.4 mn The heatng tmes are same fr the shere and ube whle t s smaller n retangular rsm. PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

3 9-6 Tmates are laed nt ld water t l them. The heat transfer effent and the amunt f heat transfer are t be determned. Assumtns The tmates are sheral n shae. Heat ndutn n the tmates s ne-dmensnal beause f symmetry abut the mdnt. The thermal rertes f the tmates are nstant. 4 The heat transfer effent s nstant and unfrm ver the entre surfae. 5 The Furer number s τ > 0. s that the ne-term arxmate slutns (r the transent temerature harts are alable (ths assumtn wll be verfed. Prertes The rertes f the tmates are gven t be 0.59 W/m. C, α m /s, ρ 999 g/m and.99 J/g. C. Analyss The Furer number s αt τ r 6 (0.4 0 m /s( 600 s 0.65 (0.04 m whh s greater than 0.. Therefre ne-term slutn s alable. The rat f the dmensnless temeratures at the surfae and enter f the tmates are θ θ T sn( s τ Ae s, sh T Ts sn( 0,sh T 0 τ A e T Substtutng, 7. 7 sn( Frm Table -, the rresndng Bt number and the heat transfer effent are B. hr B h B r (0.59 W/m. C(. 459 W/m. C (0.04 m The mum amunt f heat transfer s m 8ρV 8ρπD / 6 8(999 g/m [ π (0.08 m / 6].4 g m [ T ] (.4 g(.99 J/g. C(0 7 C 96.6 J Then the atual amunt f heat transfer bemes yl T (96.6 J 88 J sn s 0 7 sn(.040 (.040 s( (.040 Water 7 C Tmat T 0 C PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

4 -5-4 A lng ylndral shaft at 400 C s allwed t l slwly. The enter temerature and the heat transfer er unt length f the ylnder are t be determned. Assumtns Heat ndutn n the shaft s ne-dmensnal sne t s lng and t has thermal symmetry abut the enter lne. The thermal rertes f the shaft are nstant. The heat transfer effent s nstant and unfrm ver the entre surfae. 4 The Furer number s τ > 0. s that the ne-term arxmate slutns (r the transent temerature harts are alable (ths assumtn wll be verfed. Prertes The rertes f stanless steel 04 at rm temerature are gven t be 4.9 W/m. C, ρ 7900 g/m, 477 J/g. C, α m /s Analyss Frst the Bt number s alulated t be hr B (60 W/m. C(0.75 m (4.9 W/m. C The nstants and A rresndng t ths Bt number are, frm Table -,.0904 and A.548 The Furer number s 6 αt (.95 0 m /s(0 60 s τ (0.75 m whh s very lse t the value f 0.. Therefre, the ne-term arxmate slutn (r the transent temerature harts an stll be used, wth the understandng that the errr nvlved wll be a lttle mre than erent. Then the temerature at the enter f the shaft bemes τ (.0904 (0.548 θ 0, yl Ae (.548 e T C The mum heat an be transferred frm the ylnder er meter f ts length s m ρv ρπr (7900 g/m [ π (0.75 m (m] 760. g m [ T ] (760. g(0.477 J/g. C( C 90,640 J One the nstant J s determned frm Table - rresndng t the nstant.0904, the atual heat transfer bemes yl T T J( (90,640 J 5,960 J Ar T 50 C Steel shaft T 400 C PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

5 -0-46 A rb s rasted n an ven. The heat transfer effent at the surfae f the rb, the temerature f the uter surfae f the rb and the amunt f heat transfer when t s rare dne are t be determned. The tme t wll tae t rast ths rb t medum level s als t be determned. Assumtns The rb s a hmgeneus sheral bjet. Heat ndutn n the rb s ne-dmensnal beause f symmetry abut the mdnt. The thermal rertes f the rb are nstant. 4 The heat transfer effent s nstant and unfrm ver the entre surfae. 5 The Furer number s τ > 0. s that the ne-term arxmate slutns (r the transent temerature harts are alable (ths assumtn wll be verfed. Prertes The rertes f the rb are gven t be 0.45 W/m. C, ρ 00 g/m, 4. J/g. C, and α m /s. Analyss (a The radus f the rast s determned t be m. g m ρv V m ρ 00 g/m 4 V ( m V πr r m 4π 4π The Furer number s 7 αt (0.9 0 m /s( s τ 0.7 ( m r whh s smewhat belw the value f 0.. Therefre, the ne-term arxmate slutn (r the transent temerature harts an stll be used, wth the understandng that the errr nvlved wll be a lttle mre than erent. Then the ne-term slutn an be wrtten n the frm τ 60 6 (0.7 θ 0, sh Ae Ae T It s determned frm Table - by tral and errr that ths equatn s satsfed when B 0, whh rresnds t.07 and A Then the heat transfer effent an be determned frm hr B h B r (0.45 W/m. C( W/m. C ( m Ths value seems t be larger than exeted fr rblems f ths nd. Ths s rbably due t the Furer number beng less than 0.. (b The temerature at the surfae f the rb s T ( r, sn( / t T sn(.07 rad τ r r (.07 (0.7 θ ( r sh Ae (.9898 e T r / r.07 T ( r T ( r 59.5 C ( The mum ssble heat transfer s m ( T (. g(4. J/g. C(6 4.5 C 080 J Then the atual amunt f heat transfer bemes sn( s( sn(.07 (.07 s(.07 θ, sh ( ( (0.78(080 J 69 J (d The ng tme fr medum-dne rb s determned t be 7 6 τ θ 0, sh Ae (.9898 e T τr (0.6( m t α 7 (0.9 0 m /s (.07 τ 0,866 s 8 mn hr Oven 6 C τ 0.6 Rb 4.5 C Ths result s lse t the lsted value f hurs and 0 mnutes. The dfferene between the tw results s due t the Furer number beng less than 0. and thus the errr n the ne-term arxmatn. Dsussn The temerature f the uter arts f the rb s greater than that f the nner arts f the rb after t s taen ut f the ven. Therefre, there wll be a heat transfer frm uter arts f the rb t the nner arts as a result f ths temerature dfferene. The remmendatn s lgal. PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

6 -7-5 A ersn uts ales nt the freezer t l them quly. The enter and surfae temeratures f the ales, and the amunt f heat transfer frm eah ale n h are t be determned. Assumtns The ales are sheral n shae wth a dameter f 9 m. Heat ndutn n the ales s ne-dmensnal beause f symmetry abut the mdnt. The thermal rertes f the ales are nstant. 4 The heat transfer effent s nstant and unfrm ver the entre surfae. 5 The Furer number s τ > 0. s that the ne-term arxmate slutns (r the transent temerature harts are alable (ths assumtn wll be verfed. Prertes The rertes f the ales are gven t be 0.48 W/m. C, ρ 840 g/m,.8 J/g. C, and α. 0-7 m /s. Analyss The Bt number s Ar hr (8 W/m. C(0.045 m B 0.86 T 5 C (0.48 W/m. C The nstants and A rresndng t ths Ale Bt number are, frm Table -, T 0 C.476 and A.90 The Furer number s 7 αt (. 0 m /s(h 600 s/h τ 0. > 0. r (0.045 m Then the temerature at the enter f the ales bemes 0 ( 5 τ T (.476 (0. θ 0, sh Ae (.9 e C T 0 ( 5 The temerature at the surfae f the ales s θ ( r sh T( r T A e T ( r ( T ( r 0 ( 5 τ The mum ssble heat transfer s 4 m ρv ρ πr m ( T sn( r r / r.7 C / r (.9 e (.476 (0. 4. (840 g/m π (0.045 m 0.06 g (0.06 g(.8 J/g. C T Then the atual amunt f heat transfer bemes θ 0.40, sh (0.40(4.75 J 7. J sn(.476 rad.476 [ 0 ( 5 ] C 4.75 J sn( s( sn(.476 rad (.476 s(.476 rad ( (.476 PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.

7 Internal mbustn engne valves are quenhed n a large l bath. The tme t taes fr the valve temerature t dr t sefed temeratures and the mum heat transfer are t be determned. Assumtns The thermal rertes f the valves are nstant. The heat transfer effent s nstant and unfrm ver the entre surfae. Deendng n the sze f the l bath, the l bath temerature wll nrease durng quenhng. Hwever, an average anstant temerature as sefed n the rblem wll be used. 4 The Bt number s B < 0. s that the lumed system analyss s alable (ths assumtn wll be verfed. Prertes The thermal ndutvty, densty, and sef heat f the balls are gven t be 48 W/m. C, ρ 7840 g/m, and 440 J/g. C. Analyss (a The haraterst length f the balls and the Bt number are V.8( πd / 4.8D.8(0.008 m m A πd 8 8 s h (800 W/m. C(0.008 m B 0.0 < W/m. C Therefre, we an use lumed system analyss. Then the tme fr a fnal valve temerature f 400 C bemes has 8h 8(800 W/m. C b 0.88 s ρ V.8ρ D.8(7840 g/m (440 J/g. C(0.008 m (0.88 s t e e t 5.9 s T (b The tme fr a fnal valve temerature f 00 C s (0.88 s e e T ( The tme fr a fnal valve temerature f 5 C s 5 50 (0.88 s e e T t t t.5 s t 5.4 s (d The mum amunt f heat transfer frm a sngle valve s determned frm.8πd.8π (0.008 m (0.0 m m ρv ρ (7840 g/m g 4 4 m [ T ] ( g(440 J/g. C( C,400 J.4 J (er valve f Ol T 50 C Engne valve T 800 C PROPRIETARY MATERIA. 008 The MGraw-Hll Cmanes, In. mted dstrbutn ermtted nly t teahers and eduatrs fr urse rearatn. If yu are a student usng ths Manual, yu are usng t wthut ermssn.