Chapter 8 Sections 8.4 through 8.6 Internal Flow: Heat Transfer Correlations. In fully-developed region. Neglect axial conduction

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1 Chapter 8 Sectin 8.4 thrugh 8.6 Internal Flw: Heat Tranfer Crrelatin T v cu p cp ( rt) k r T T k x r r r r r x In fully-develped regin Neglect axial cnductin u ( rt) r x r r r r r x T v T T T T T u r x r r r x T T u r x r r r Unifr Surface Heat Flux Integrating twice 1 T u dt r r 1 r r r dx r u dt r r Trx (, ) Clnr C 4 dx 4 16r 1 The teperature reain finite at r 0 iplie C 1 0.

2 C u dt 3r T ( x) dx 16 Subtituting int T u r u r 4u T ( x) u r ur T ( x) T ( x) fr Eq. () 4 ur dt 3 1 r 1 r Trx (, ) T () x dx r0 4 r0 r u 1 r 8 ur T ( x) r 8 ur T ( x) r r 0 r u( x, r) T ( x, r) rdr dt dx T ( x, r) rdr ur dt dx dt dx ur dt dx dt dx dt dx 8 r r 3 1 r r r r r r r r 1 r 4 r 3 r 16 r 1 r 4 r r 16 r r dr 1 r 4 r q P c p q P u A c c p q u c 11 ur dt T( x) T( x) 48 dx p / u /4 4q 48 uc 11 q 48 c 11 q 48 k p p dr

3 11 ur dt T( x) T( x) 48 dx 11 q 48 k Since q h( T T ) r 48 k h 11 h 4.36 fr q cntant k independent f Re, Pr and axial lcatin.

4 1. Fully evelped Flw Lainar Flw in a Circular Tube: The lcal elt nuber i a cntant thrughut the fully develped regin, but it value depend n the urface theral cnditin. Unifr Surface Heat Flux ( q ): h k 4.36 Unifr Surface Teperature ( T ): h 3.66 k Turbulent Flw in a Circular Tube: Fr a th urface and fully turbulent cnditin ( Re 10, 000 ), the ittu Belter equatin ay be ued a a firt apprxiatin: n 0.03Re Pr 0.3 T n T n0.4 T T Pr 160 Re 10,000 L 10 The effect f wall rughne and tranitinal flw cnditin Re 3000 crrelatin: ay be cnidered by uing the Gnielinki Sth urface: f 8 Re 1000 Pr Pr f

5 6 f (0.790ln Re 1.64) 3000 Re 510 Surface f rughne >0 : f Figure8.3 Nncircular Tube: Ue f hydraulic diaeter a characteritic length: h 4A c P Since the lcal cnvectin cefficient varie arund the periphery f a tube, appraching zer at it crner, crrelatin fr the fully develped regin are aciated with cnvectin cefficient averaged ver the periphery f the tube. Lainar Flw: The lcal elt nuber i a cntant whe value (Table 8.1) depend n the urface theral cnditin ( T r q ) and the duct apect rati. Turbulent Flw: A a firt apprxiatin, the ittu-belter r Gnielinki crrelatin ay be ued with the hydraulic diaeter, irrepective f the urface theral cnditin.. Effect f the Entry Regin The anner in which the elt decay fr inlet t fully develped cnditin fr lainar flw depend n the nature f theral and velcity bundary layer develpent in the entry regin, a well a the urface theral cnditin. Lainar flw in a circular tube

6 Cbined Entry Length: Theral and velcity bundary layer develp cncurrently fr unifr prfile at the inlet. Theral Entry Length: Velcity prfile i fully develped at the inlet, and bundary layer develpent in the entry regin i retricted t theral effect. Such a cnditin ay al be aued t be a gd apprxiatin fr a unifr inlet velcity prfile if Pr 1.Why? Graetz Prble Average elt ber fr Lainar Flw in a Circular Tube with Unifr Surface Teperature: Cbined Entry Length: RePr L :

7 13 Re Pr 1.86 L RePr L : 3.66 Theral Entry Length: LRe Pr L Re Pr Average elt ber fr Turbulent Flw in a Circular Tube : Effect f entry and urface theral cnditin are le prnunced fr turbulent flw and can be neglected. Fr lng tube L 60 : 3, fd Fr hrt tubel 60 :, fd C C L Nncircular Tube: Lainar Flw: h depend trngly n apect rati, a well a entry regin and urface theral cnditin. See reference 11 and 1. Turbulent Flw: A a firt apprxiatin, crrelatin fr a circular tube ay be ued with replaced by h.

8 When deterining fr any tube geetry r flw cnditin, all prpertie are t be evaluated at T T T i,, Why d lutin t internal flw prble ften require iteratin? 3. The Cncentric Tube Annulu Fluid flw thrugh regin fred by cncentric tube. Cnvectin heat tranfer ay be fr r t inner urface f uter tube and uter urface f inner tube. Surface theral cnditin ay be characterized by unifr teperaturet, T r unifr heat flux q, q. i, i, Cnvectin cefficient are aciated with each urface, where q h T T i i, i q h T T, h h i k k i h h h i

9 Fully evelped Lainar Flw elt nuber depend n i / and urface theral cnditin (Table 8., 8.3) Fully evelped Turbulent Flw Crrelatin fr a circular tube ay be ued with replaced by h.

10 Prble 8.43 Fr an air paage ued t cl a ga turbine vane, calculate the air utlet teperature and heat reved fr the vane. KNOWN: iaeter and length f cpper tubing. Teperature f cllectr plate t which tubing i ldered. Water inlet teperature and flw rate. FIN: (a) Outlet teperature f the air clant fr the precribed cnditin and (b) Cpute and plt the air utlet teperature T, a a functin f flw rate, kg/h. Cpare thi reult with the fr vane having paage diaeter f and 4. ASSUMPTIONS: (1) Steady-tate cnditin, () Ideal ga with negligible vicu diipatin and preure variatin. PROPERTIES: Table A.4, Air (aue T = 780 K, 1 at): c p = 1094 J/kgK, k = W/K, = N/, Pr = 0.706; (T = 650C = 93 K, 1 at): = N/. ANALYSIS: (a) Fr cntant wall teperature heating, fr Eq. 8.41b, T T, PLh exp T T,i cp where P =. Fr flw in a circular paage, (1)

11 kg h h Re N The flw i lainar, and ince L/ = 75 /3 = 5, the Sieder-Tate crrelatin including cbined entry length yield 1/ h Re Pr 1.86 k L 1/ W K h W K Hence, the air utlet teperature i 650 T, W K exp C kg 1094J kg K T, 578 C (b) Uing the IHT Crrelatin Tl, Internal Flw, fr Lainar Flw with cbined entry length, alng with the energy balance and rate equatin abve, the utlet teperature T, wa calculated a a functin f flw rate fr diaeter f =, 3 and 4. The plt belw hw that T, decreae nearly linearly with increaing flw rate, but i independent f paage diaeter. () (3) 650 Outlet teperature, T (C) Flw rate, dt (kg/h)

12 Baed upn the calculatin fr T, = 578 C, T = 775 K which i in gd agreeent with ur auptin t evaluate the therphyical prpertie. Why i T, independent f? Fr Eq. (3), nte that h i inverely prprtinal t, h ~ -1. Fr Eq. (1), nte that n the right-hand ide the prduct P h will be independent f. Hence, T, will depend nly n. Thi i, f cure, a cnequence f the lainar flw cnditin and will nt be the ae fr turbulent flw.

13 Prble 8.5 eterine effect f abient air teperature and wind velcity n teperature at which ven gae are dicharged fr a tack. KNOWN: Thin-walled, tall tack dicharging exhaut gae fr an ven int the envirnent. FIN: (a) Outlet ga and tack urface teperature, T, and T,, fr V=5 / and T 4 C; (b) Effect f wind teperature and velcity n T,. ASSUMPTIONS: (1) Steady-tate cnditin, () Negligible wall theral reitance, (3) Exhaut ga prpertie apprxiately the f atpheric air, (4) Negligible radiative exchange with urrunding, (5) Ideal ga with negligible vicu diipatin and preure variatin, (6) Fully develped

14 flw, (7) Cntant prpertie. PROPERTIES: Table A.4, air (aue T, = 773 K, T = 83 K, 1 at): c p = 1104 J/kgK, = N/, k = W/K, Pr = 0.71; Table A.4, air (aue T = 53 K, T = 4C = 77 K, T f = 400 K, 1 at): = /, k = W/ K, Pr = ANALYSIS: (a) Fr Eq. 8.45a, PL T, T T T,i exp U c p U1 1 1 hi h where h and h are average cefficient fr internal and external flw, repectively. i Internal flw: With a Reynld nuber f kg Re 33,87 i N The flw i turbulent, and auing fully develped cnditin thrughut the tack, the ittu-belter crrelatin ay be ued t deterine h i. h i 4/ Re Pr k i W K 4/5 0.3 h i , W K 0.5 External flw: Wrking with the Churchill/Berntein crrelatin and Hence, V Re 94, /5 0.6Re 1/ 1/3 5/8 Pr Re /3 1/4 8, Pr

15 h W K W K The utlet ga teperature i then T, 4 C C exp W K 543 C 0.5 kg 1104 J kg K The utlet tack urface teperature can be deterined fr a lcal urface energy balance f the fr, which yield, T, h T, T h T i ht i, ht W T, 3 C h i h W K b) The effect f the air teperature and velcity are a fllw 560 Ga utlet teperature, T(C) Freetrea velcity, V(/) Tinf = 35 C Tinf = 5 C Tinf = -5C ue t the elevated teperature f the ga, the variatin in abient teperature ha nly a all effect n the ga exit teperature. Hwever, the effect f the freetrea velcity i re prnunced. icharge teperature f apprxiately 530 and 560C wuld be repreentative f cld/windy and war/till atpheric cnditin, repectively.

16 COMMENTS: If there are cntituent in the ga dicharge that cndene r precipitate ut at teperature belw T,, related perating cnditin huld be avided.