Appendix A: Lumped Parameter Assumption Using Biot Number

Transcription

1 Appendix A: umped Prmeter Assumption Using Biot Number e dimensionless group known s te Biot number, Bi, is te rtio of conductive nd convective resistnces to et trnsfer [1]. rge vlues of Bi imply tt te limiting et trnsfer mecnism is conduction becuse te internl, or conductive, resistnce is muc greter tn te surfce, or convective, resistnce to et trnsfer. e opposite is true for smll vlues of Bi indicte tt te limiting et trnsfer mecnism is convection. e ltter cse indictes te pplicbility of lumped prmeter nlysis [1]. If constnt temperture in given direction is ssumed, ten Biot number, Bi, less tn 0.1 indictes tt lumped prmeter nlysis results in less tn 5% error [1]. erefore, bsed on te vlue of te Bi in te rdil direction, te vlidity of te constnt rdil temperture ssumption used to form Eqution 11, or te differentil eqution, cn be evluted. Eqution A1 defines Bi in terms of, k, te re of et trnsfer, A (m nd te volume of et trnsfer, V (m 3 [1]. V A Bi (A1 k For cylindricl element, V nd A re defined by Equtions A nd A3, were d (m is te dimeter, nd (m is te lengt of te element [1]. A πd (A V d π (A3 4 ubstituting Equtions A nd A3 into Eqution A1 yields te Biot number for cylindricl element, s sown by Eqution A4. d π Bi 4 d cyl k πd 4 k (A4 ble A1 sows te clcultion of te Bi number for te steel nd luminum rods using expected nd experimentl vlues of nd k from Appendix F. As seen in tese tbles, te Biot number is less tn 0.1 for ll conditions, ing te ssumptions used to derive Eqution 11 vlid. 0

2 ble A1- Biot Number Clcultion for te teel nd Aluminum Rods for Different Conditions teel Rod (d m Aluminum Rod (d0.019m Conditions (W/(m K k (W/(mK Bi (W/(m K k (W/(mK Bi iterture E E-04 Air-cooled wit "ermocouple" boundry conditions Wter-cooled wit "ermocouple" boundry conditions E E E E-04 Air-cooled wit "End of Rod" boundry conditions E E-04 Wter-cooled wit "End of Rod" boundry conditions E E-04 1

3 Appendix B: Confirmtion of θ olutions is ppendix will sow tt te specific boundry conditions for te ir nd wter exposed metl rods will stisfy te solutions of te energy blnce differentil equtions. e differentil eqution is: d m ese solutions for ir nd wter re sown below: Air Cooled: ( 0 cos[ m( + sin[ m( θ cos[ m] + sin[ m] (B1 (B Wter Cooled: sin[ mx] sin[ m( x ] θ θ (B3 sin[ m] sin[ m] were θ (B4 nd 4 m (B5 kd e boundry conditions, BC#, for te two situtions re te following: Air Cooled: BC1 At x 0 BC At x d k ( Wter Cooled: BC1 w At x 0 BC w At x Air Cooled Metl: ubstitution nd mnipultion of te equtions will sow tt Eqution B is stisfied by te first boundry condition nd is solution of te differentil eqution. Eqution B nd B4 must be equted to prove tt BC1 stisfies Eqution B. ubstitution of BC1 into te bove eqution gives te following: cos[ m( + sin[ m( θ cos[ m] + sin[ m] (B6

4 cos[ m( 0] + sin[ m( 0] 1 θ 1 x 0 cos[ m] + sin[ m] Eqution B7 sows tt bot sides of te eqution equls one fter te vlues re cnceled. e second boundry condition lso works wit Eqution B. Rerrnging Eqution B6 nd differentition wit respect to x gives: k ( cos[ m( + sin[ m( + cos[ m] + sin[ m] d d ( k( msin[ m( cos[ m( k cos[ m] + sin[ m] msin[ m( cos[ m( k cos[ m] + sin[ m] (B7 (B8 (B8b (B9 were d [sin( x] cos[ x] nd d [cos ( x] sin[ x] ubstituting BC into Eqution B9 gives te following: k( m sin[ m( ] cos[ m( ] k( m sin[0] cos[0] d k k ( k x cos[ m] + sin[ m] cos[ m] + sin[ m] ( ( ( cos[ m] + sin[ m] ( cos[ m] + sin[ m] (B10 were sin[0] 0 nd cos[0] 1 Eqution B10 is te sme s te rerrnged eqution for θ, s sown below. θ x cos[ m( ] + sin[ m( ] 1 ( cos[ m] + sin[ m] cos[ m] + sin[ m] ( cos[ m] + sin[ m] 3

5 Eqution B stisfies te differentil eqution, Eqution B1, by te following process. Eqution B8 is differentited nd rerrnged. ( d m ( m m cos[ m( + sin[ m( k cos[ m] + sin[ m] cos[ m( + cos[ m] + sin[ m] sin[ m( d + m m ( cos[ m( + d m + m cos[ m] + sin[ m] sin[ m( e vlue inside te lrge brckets is equl to from Eqution B8. us, Eqution B11 simplifies to (B11 d m m ( m (B1 e bove eqution is te rerrnged form of Eqution B1, wic sows tt te solution does stisfy te differentil eqution. Wter Cooled Metl: e sme process of boundry condition substitution into te solutions employed for te ir cooled metl is used to prove tt te wter cooled solution stisfies te boundry conditions. is process of substitution nd rerrngement is sown in te following steps for bot wter boundry conditions: For BC1 w θ sin[ mx] sin[ m( x ] sin[ m] sin[ m] sin[0] sin[ m( ] sin[ m( ] 1 1 x 0 sin[ m] sin[ m] sin[ m] θ were sin[-m] - sin[m] For BC w θ sin[ m] sin[ m( ] x sin[ m] sin[ m] Using te metod from te ir cooled sitution, Eqution B3 stisfies te differentil eqution, Eqution B1. Eqution B3 must be differentited: 4

6 sin[ mx] sin[ m( x ] ( θ + (B13 sin[ m] sin[ m] 1 [( θ sin[ mx] sin[ m( x ]] + sin[ m] d 1 [( θ m cos[ mx] m cos[ m( x ]] sin[ m] d 1 [( θ m sin[ m] sin[ mx] m sin[ m( x ]] d sin[ mx] sin[ m( x ] m ( θ + sin[ m] sin[ m] m ( e lst substitution ws possible becuse of Eqution B13. e bove eqution is te rerrnged form of Eqution B1, wic sows tt te solution does stisfy te differentil eqution. 5

7 Appendix C: ermocouple eory A termocouple is used to mesure te totl temperture of system. ese devices work becuse dissimilr metls produce voltge tt is proportionl to te temperture difference t te contct junction, wic is bsed on te terml properties of te conductor metl [4]. Becuse voltge is produced, n externl power source is not necessry, nd te termocouple is n ctive sensor [5]. Furtermore, te current will continue until te tempertures t te ot nd cold junctions re te sme. is cretion of voltge nd current bsed on temperture differences is known s te eebeck Effect. A scemtic of typicl termocouple circuit is sown in Figure A1 [4]. Figure C1 ypicl ermocouple Circuit ree lws necessry for obtining ccurte temperture redings ve been developed bsed on reserc on termocouple circuits. ese re te lws of omogenous circuits, intermedite metl sttes, nd successor or intermedite temperture sttes. e lw of omogenous circuits declres tt circuit of te sme metl cnnot generte current from temperture differences regrdless of te condition nd type of te metl wires. us, if current is detected from circuit, ten te wires must be different metls. e lw of intermedite metls clims tt te sum of te terml voltges is zero if te temperture trougout multi-metl circuit is constnt. Consequently, te ccurcy of termocouple will not be ffected if ny mesuring device of different metl is dded to te circuit nd te tempertures of te device connection junctions re te sme s te circuit tempertures. e lw of te successive temperture llows for te use of known, reference junction temperture to determine n unknown, different junction temperture. If voltge, V 1 is produced for temperture difference of Δ 1 nd noter voltge V is produced for temperture difference of Δ, ten te voltge produced from temperture difference of Δ 1 + Δ is equl to V 1 + V. ome metods used to obtin te constnt reference temperture include using eiter n ice bt or temperture controlled oven [4]. In conclusion, te ppliction of te tree lws of termoelectricity permits te determintion of ccurte temperture mesurements. e mesuring junction is criticl in te mesurement of tempertures. wo suc junctions re te grounded nd ungrounded junctions. Bot junctions re sown in Figure. [6]. 6

8 Figure C Grounded nd ungrounded termocouple digrm were te grounded termocouple s extr insultion to prevent contct wit te set. For te grounded termocouple, te mesuring junction is normlly in contct wit te conductive, metl set in wic it is enclosed. is contct llows for quick response times nd te protective set protects te junction from corrosive environments to provide long product lifetime. e ungrounded junction is enclosed in, but isolted from conductive set to prevent contct. is loss of contct between te set nd te junction gives slower, but more ccurte response time. is junction lso s long service time under corrosive nd sock conditions [4]. us, te mesuring device ffects mny importnt crcteristics of temperture mesurements including te speed nd ccurcy of te response. 7

9 Appendix D: ermocouple Positions of Experimentl etup teel Rod, d 1.90 cm, ir-cooled: end 46.0 Aluminum Rod, d 1.31 cm, wter-cooled: end 53.0 Aluminum Rod, d 1.90 cm, ir-cooled: end 45.8 teel Rod, d 1.31 cm, wter-cooled: end 53.0 e difference in dimeters is ttributed to mesuring errors, nd constnt dimeter of 1.3 cm ws used in ll of te clcultions. 8

10 Appendix E: Rw Dt All te tempertures re in C, were ir is te ir temperture, wter is te wter temperture, nd stem is te stem temperture. Furtermore, zone represents termocouple. ble E Air Exposed teel Rw Dt Initil tem trted ime Air Wter tem :51: :51: :51: :51: :51: :51: :5: :5: :5: :53: :53: :54: :54: :54: :54: :55: :55: :55: :55: :56: :56: :56: :56: :57: :57: :57: :57: :58: :58: :58: :58: :59: :59: :59: :59: :00: :00: :00: :00: :01: :01:

11 ime Air Wter tem :01: :01: :0: :0: :0: :0: :03: :03: :03: :03: :04: :04: :04: :04: :05: :05: :05: :05: :06: :06: :06: :06: :07: :07: :07: :07: :08: :08: :08: :08: :09: :09: :09: :09: :10: :10: :10: :10: :11: :11: :11: :11: :1: :1: :1: :1: :13: :13: :13:

12 tedy tte ime Air Wter tem :13: :14: :14: :14: :14: :15: :15: :15: :15: :16: :16: :16: :16: :17: :17: :17: :17: :18: :18: :0: :0: :1: :: :3: :4: :5: :6: :7: ble E3 Air-Exposed Aluminum - Alredy t tedy tte ime Air Wter tem :38: :39: :40: :41: :4: :43: :44: ble E4 Wter-Exposed Aluminum Initil Vlues ime Air Wter tem :50: :50: :53: :54:

13 Wter On tedy tte ime Air Wter tem :55: :56: :57: :58: :59: :00: :01: :0: :03: :04: :05: :06: :07: :08: :09: :10: :11: ime Air Wter tem ble E5 Wter-Exposed teel - Alredy t tedy tte :18: :19: :0: :1: :: :3:

14 Appendix F: Representtive Clcultions for te Air-Exposed teel Becuse te equtions were pplied in te sme mnner, te only clcultions sown re for te irexposed steel. ese clcultions will be ssumed to be representtive of te oter clcultions. e djusted distnce of te stem ws clculted by subtrcting te distnce of te first termocouple from ll of te oter distnces. is sifted te origin to te position of te first termocouple. For te ir-exposed mesurements, te stedy stte vlues were used in Equtions 14, 18, nd 1 to obtin te vlues for θ s function of for te steel nd s function of k Al for te luminum. P π d 4 (14 m ka d kd kπ 4 θ (18 s cos θ cos [ m( x ] + sin[ m( x ] [ m] + sin[ m] (1 For te wter exposed sets of dt, Eqution 17 replced Eqution 1, wile te oters remined te sme. ( sin[mx] sin[m x ] θ θ (17 sin[m] sin[m] ble F1 Clcultions for Air-Exposed teel θ Mesurement (- Distnce from tem (m Adj Distnce from tem (m Averge ('C tndrd Devition ( C Clculted (Eqution 1 Experimentl (Eqution 18 Difference Air E Wter E tem E E E E E E E E E E E E E E E-01.97E E E E E E E E E E E E E-0.938E E E E-0.48E E E E-0.476E-04 Wit k steel 16 W/(m K for ll mesurements, nd using solver to minimize te squre distnce, te vlues of nd k could be found using difference reference tempertures nd distnces. For exmple, using te tempertures for te 33

15 first nd lst termocouple of te ir-exposed steel s te temperture of te stem nd rod end respectively, or using te termocouple metod, gve 18.8 W/(m K. Becuse ws ssumed to be independent of te mteril, tis ws lso used for te ir-exposed luminum. Performing te sme set of clcultions on te ir-cooled luminum mesurements, but solving for k Al sowed tt k Al W/(m K. e end rod metod of clcultion used te stem temperture for to give 18.8 W/(m K nd k Al W/(m K. e sme procedure ws repeted for te wter cooled dt, except Eqution 17 ws used to find te Clculted θ insted of Eqution 1. Furtermore, for te end rod clcultion metod, te wter temperture ws used s t n estimted position,, to find θ becuse of te boundry condition of Eqution 16. For tis metod, te stem temperture ws used s. Using te termocouple metod, te first nd lst termocouple tempertures were te initil nd finl tempertures in te clcultion of te experimentl θ nd θ. All of te results re summrized in ble F. ble F ummry of Clcultions of Coefficiencts itution Clcultion Metod First nd st ermocouple Ends of Rods 1 (W/m K k 1 (W/(m K (W/m K k (W/(m K Air Exposed Wter Exposed Percent Difference 69.1% 43.1% -53.% -38.9% 34

16 Appendix G: Error Anlysis e uncertinties in te mesurements re sown in ble G1. e lengt uncertinty ws estimted by looking t te divisions, or tick mrks, on te mesuring device. e temperture uncertinty ws obtined from te stndrd devition clcultion in Appendix F. e dimeter uncertinty ws estimted from te number of decimls on te clipers. e conductivity of steel ws obtined from tble [], wic sowed little devition of te et conductivity constnt wit respect to temperture. ble G1 Uncertinties in Mesurements Mesurement (Units Vlue Uncertinty emperture ( C Vried Vried: ~ 0.1 engt (m Vried E-3 Dimeter (m ~0.013 ~0 teel erml Conductivity (W/m s 16 ~ 0 θ (None Clculted Vried ~ 0 e eqution for dimensionless temperture, θ, is Differentiting in terms of, s nd : θ θ θ ( ( s Finding te totl error: θ 1 ( + ( ` θ θ θ ( θ + + (G1 s e dt for te ir-exposed steel ws substituted into te bove equtions. is is sown in ble G. ble G Error Clcultions of Air Exposed teel Mesurement (- Distnce from tem (m Adj Distnce from tem (m Averge ( C tndrd Devition ( C dθ/d dθ/d dθ/d otl dθ Air E Wter E tem E E E E E-03.58E E E E-03 1.E E E E E E E E E E-04.36E E-03 35

17 Mesurement (- Distnce from tem (m Adj Distnce from tem (m Averge ( C tndrd Devition ( C dθ/d dθ/d dθ/d otl dθ E E E E E E E E E E E E E E E E E E E E E E E E E-03 is ws te error used for ll of te θ results becuse it ws ssumed tt it wouldn t vry significntly between te different situtions. Also, te ir-exposed steel tempertures fluctuted between te mximum nd minimum tempertures observed. 36