THERMAL EFFECT ON MASS FLOW-RATE OF SONIC NOZZLE

Transcription

1 THERMAL SCIENCE: Year 08, Vol., No. A, pp Introuction THERMAL EFFECT ON MASS FLOW-RATE OF SONIC NOZZLE by Hong-Bing DING, Chao WANG *, an Gang WANG Tianjin Key Laboratory of Proce Meaurement an Control, School of Electrical Engineering an Automation, Tianjin Univerity, Tianjin, China Original cientific paper Sonic nozzle are wiely ue a flow meaurement an tranfer tanar. The thermal effect of onic nozzle i ignificant at low Reynol number. It inclue two correction factor, C T for the thermal bounary-layer an C α for contraine thermal eformation of throat area. Firtly, uing the imilarity olution, the formula for correction factor C T over wall temperature range from 0.8T 0 to.t 0 wa obtaine. For γ =.33, C T = Re / T/T 0 ; for γ =.4, C T = Re / T/T 0 ; for γ =.67, C T = 4.00 Re / T/T 0. Seconly, thermal an tre moel for partially contraine expanion were built. Unlike the free expanion, truth lope of C α for three nozzle are , an , repectively. Latly, the experimental ata of Cu nozzle wa ue to valiate preent reult. It reveale that moifie experimental value are in goo agreement with the preent reult. Key wor: onic nozzle, thermal effect, thermal bounary-layer, contraine thermal eformation Backgroun Sonic nozzle are applie in accuracy meaurement an control of ga-flow ue to it tability, imple tructure, an goo repeatability. Over 70% onic nozzle were ue a a tanar ga meter to calibrate other type meter []. Uner critical flow conition, real ma flow-rate of onic nozzle i calculate by []: CCA * nt p0 qm = Cq mi = () T R m 0 where C i the icharge coefficient for aiabatic wall, C * the critical flow factor, A nt = π /4, R m = R/M, p 0 an T 0 are inlet tagnation preure an temperature, repectively. The ga expan an accelerate in onic nozzle an it temperature will rop greatly. Then, the nozzle boy will be coole by force convection heat tranfer between the wall urface an the flui [3]. Subequently, thermal bounary-layer an throat area will be change which i calle thermal effect. The thermal effect i ignificant at low Reynol number [4]. For the lat ecae, thermal effect of onic nozzle ha been invetigate by Bignell an Choi [5], Li an Mickan [6], Hu et al. [7], Wright et al. [3], an Unal et al. [4]. * Correponing author, wangchao@tju.eu.cn

2 48 THERMAL SCIENCE: Year 08, Vol., No. A, pp Coniering thermal effect, ma flow rate eq. () i rewritten: C* ( Cα Ant ) p0 qmt, = [ C w + ( CT )] () R T where C T i the correction factor for the thermal bounary-layer, C α = A/A ref the correction factor for the thermal eformation of the throat area. Thermal bounary-layer During the lat few ecae, many invetigator, uch a, Illingworth [8], Li an Nagamatu [9], Cohen an Rehotko [0], Ball [], Back [], Aziz [3], an Kenouh [4], mae a great eal of reearche about the effect of wall temperature on laminar bounary-layer. For the compreible flow, Illingworth [8], an Li an Nagamatu [9] put forwar imilarity metho for calculating the propertie of laminar compreible bounary-layer in an axial preure graient with heat tranfer. However, the reult are only uitable for planar flow an the main-tream velocitie mut atify pecial-relationhip which iffer from flow characteritic of onic nozzle. Subequently, Cohen an Rehotko [0], an Ball [] preente an approximate metho for the calculation of the compreible laminar bounary-layer with heat tranfer an arbitrary preure graient, bae on Stewarton tranformation an Thwaite correlation concept. Back [] propoe a imilarity olution of the laminar bounary-layer equation for a large range of flow acceleration, urface cooling, an flow pee in uperonic nozzle. However, in their work, the target variable i local Reynol number, Re x, rather than throat Reynol number. Thee reult can not be irectly applicable to flow meter. For ma flow-rate of onic nozzle, Tang [5], Geropp [6], an Ihibahi an Takamoto [7] preente the analytical imilarity olution of bounary-layer of onic nozzle by ome remarkable an praieworthy tranform. Unfortunately, their moel bae on aiabatic wall bounary were not able to analyze the thermal effect on ma flow rate of onic nozzle. Johnon et al. [8] preente CFD reult howing a ecreae in icharge coefficient for a nozzle warmer than the aiabatic boy temperature ue to a thermal bounary-layer an pointe out that the magnitue of the thermal bounary-layer effect i proportional to Re / an pecific heat ratio, γ. Thermal expanion of the nozzle throat area For the thermal expanion on material, it i neceary to conier whether the boy i free to expan or i contraine. If the boy i free to expan, the thermal eformation or train i imply calculate by the thermal expanion coefficient which prouce no tre. If the boy i partially contraine, then internal tre an thermal eformation i more complicate [9]. Thoma et al. [0] an Park et al. [] analyze the thermal an mechanical behavior of contraine copper mol. The comparion of tre-total train between uncontraine an practical contraine conition wa obtaine. Cragun an Howell [] preente the contraine thermal expanion micro-actuator. Ifahani et al. [3] imulate thermal tre an cooling eformation of the mol which i contraine by the ie. Anola et al. [4] mentione Chevron an Guckel micro-actuator will eform an prouce lateral bening ue to contraine thermal expanion. In 03, Stavely [5] invetigate the eformation occur in contact-aie compliant mechanim with temperature change. The reult inicate that the contraine iplacement i normal to the tructural connection joint. All reearche can help u unertan thermal effect on onic nozzle. m 0

3 THERMAL SCIENCE: Year 08, Vol., No. A, pp Preent work In thi tuy, the following pecific tak were mae. To obtain correction factor, C T, for the thermal bounary-layer. A imilarity olution for it thermal bounary-layer with wall heat tranfer wa preente. By efinition of C T, it formula wa obtaine. To obtain correction factor C α for contraine thermal eformation of throat area. The thermal an tre moel for partially contraine expanion were built. Thermal tre an eformation were illutrate to analyze the proce of partially contraine expanion. The truth formula of C α for ifferent nozzle were obtaine. Latly, the experimental ata of Cu nozzle wa ue to valiate the formula. After moifie, the experiment agree well with preent formula. Similarity olution for C T Similarity equation an reucibility For a curvilinear ytem of co-orinate, fig., x i wall urface an y i the co-orinate at perpenicular angle to the urface. The boy raii r w (x) i perpenicular to axi. The velocity component parallel an normal to the wall will be enote by u an v, repectively. The continuity, momentum an energy equation of compreible laminar bounary-layer are expree a [6]: Figure. Sonic nozzle an curvilinear ytem of co-orinate ( rurw) ( rvrw) + = 0 x y u u p u p r u + v = + µ, 0= x y x y y y ( ct p ) ( ct p ) p u T r u + v = u + µ + k x y x y y y where = 0 for planar an = for axiymmetric. It i aume the flui i perfect ga. Aitionally, the effect of vicoity an heat conuction in the main-tream of nozzle can be neglecte. Further aume that: 0 (3) µ 0 µ = T (4) T 0 = k = µ cp cp T (5) T0 Pr = In orer to reuce PDE (3) to ODE, we efine tream function y a: µ

4 50 THERMAL SCIENCE: Year 08, Vol., No. A, pp an introuce a co-orinate tranformation: Then, further aume that: y uxy (, ) y ( xy, ) = y (6) Txy (, ) 0 y u ( x) η = (7) 0 y N( xt ) ( x, y) y( xy, ) = NxK ( ) ( η) (8) uxy (, ) = u( xfη ) ( ) (9) exy (, ) = e( xgη ) ( ) (0) where η i the imilar variable. K(η), F(η), an G(η) are imenionle tream function, velocity, an total energy, repectively. Combining eq. (6), (8), an (9), we get: uf TNK η = y where K = /η. Compare with eq. (7), it i implie that F = K. Next, ome aumption houl be put forwar to reuce bounary-layer equation to ODE. Becaue Geropp aumption i not uitable to non-aiabatic wall, the follow equation were introuce: c p T rw N u γ + + = M m + () R T x r x N x u x m w R m u u γ = m + M m 0 T0 N p u x where η(x, y) i erive in Appenix A.. The nozzle parameter m i a function of nozzle geometry an ientropic exponent which i erive in Appenix A.. Finally, the bounary-layer equation can be rewritten a: where the bounary conition are: ( ) () (3) KK + K = K G (4) m G + KG = 0 (5) T w = 0 : K = K = 0, G = Gw = (6) T0 η η =, K =, G = (7) where it i aume that η = 0 or when y = 0 or, repectively.

5 THERMAL SCIENCE: Year 08, Vol., No. A, pp Next, to thermal effect on it ma flow rate, the thermal bounary-layer profile an iplacement thickne were invetigate in etail. Bounary-layer profile A tuy of the imilarity olution of bounary-layer can provie the unertaning of the effect of wall heat tranfer. Taking R = an γ =.4 a an example, we get m = Uing eq. (4)-(7), imilarity olution of K an G for m = an ifferent T w /T 0 were plotte in fig., where the expreion T w /T 0 0 mean T w ten to abolute zero. Beie, it i foun that: γ λ T γ + = G+ ( G K ) (8) T γ λ γ + Hence, the imenionle ma flux, J, at the throat i calculate by: J uρ u u = = = u T ρ T Figure. The imilarity olution of K an G for m = (R = an γ =.4) an ifferent T w /T 0 K γ G+ G K ( ) The imilarity olution of T/T an J at nozzle throat are hown in fig. 3 an 4. (9) Figure 3. Similarity olution of T/T for m = (R = an γ =.4) an ifferent T w /T 0 at throat Figure 4. Similarity olution of J for m = (R = an γ =.4) an ifferent T w /T 0 at throat Accoring to fig. an 3, it i foun that the imenionle parameter u(x, y)/u (x) an T(x, y)/t (x) at the throat will rop with the ecreae of T w /T 0.

6 5 THERMAL SCIENCE: Year 08, Vol., No. A, pp Aitionally, eq. (9) how that ma flux J(x, y) i proportional to u/u an inverely proportional to T/T. A hown in fig. an 3, the ecline rate of T/T i larger than that of u/u. Hence, J(x, y) at nozzle throat will graually rie with the ecreae of T w /T 0, a hown in fig. 4. Correction factor C T Accoring to eq. (A.8), we get: m ( m+ ) γ γ m rw 0 rnt y= Nc u Tη Hence, the iplacement thickne of bounary-layer i calculate by: (0) ρ u T u = y = y = ( J) y = ρ u Tu m ( m+ ) γ ( ) ( ) γ + γ γ γ γ m T u 0 γ + T 0 u = N λ λ c u T η () Subtituting eq. (A.6) an (8) into eq. (), give: m ( γ ) γ + γ λ γ + = G ( G K ) K + η γ 0 m λ + m γ γ γ + Reλ λ γ + where the throat Reynol number Re = r cr c cr /µ 0. At the throat, eq. () i reuce to: () ( γ ) γ + = + Re γ G ( G K ) K η (3) 0 The itribution of iplacement thickne of onic nozzle for variou T w /T 0 an γ are hown in fig. 5 an 6, where, X = 0 at the throat. Figure 5 an 6 how that the iplacement thickne become thicker with the increae of γ an T w /T 0, an the ecreae of throat Reynol number. Aitionally, eq. () euce that = 0 ( J)y, an fig. 4 how that all value of ma flux J are greater than when T w /T 0 ten to 0. Thu, i below 0 while T w /T 0 ten to zero which mean the actual flow-rate for non-aiabatic wall i larger than ieal flow-rate for aiabatic wall. For aiabatic wall, icharge coefficient, C, i ivie into three part [7], namely vicou icharge coefficient, C, affecte by ga vicoity, invici icharge coefficient, C, inuce by multi-imenional flow [8], an virial icharge coefficient, C 3, affecte by phyical propertie of real ga. Becaue C 3 coul be negligible when the ga preure i low, the icharge coefficient C jut conier firt two item an i ecribe by [9]:

7 THERMAL SCIENCE: Year 08, Vol., No. A, pp q 4 m C = = CC = C qmi where,nt the iplacement thickne at nozzle throat.,nt (4) Figure 5. Ditribution of iplacement thickne of Barchorff nozzle for R =, T w /T 0 = an ifferent γ Figure 6. The itribution of iplacement thickne of Barchorff nozzle for R =, γ =.4, an ifferent T w /T 0 (where X = 0 at the throat),beie, accoring to Hall theory [8], C can be calculate by: γ + 8γ + 754γ + 97γ C = + R 96 R 4608 R For R = an γ =.4. The C remain contant Combining eq. (3), the icharge coefficient at T w = T 0 i expree a eq. (6) which i baically accor with empirical equation (accuracy: 0.%) of ISO 9300 []. By efinition, the correction factor C T for aiabatic wall: 0.5 C = Re (6) For non-aiabatic wall, uing the imilarity olution, the correction factor C T can be approximate to eq. (7) over the wall temperature range from 0.8T 0 to.t 0 : 0.5 T Re for γ =.33 T0 0.5 T CT = Re for γ =.4 (7) T0 0.5 T 4.00 Re for γ =.67 T0 where T i equal to T w T 0. Next, CFD imulation are conucte to compare with eq. (7) by imilarity olution. The experimental valiation will be preente after invetigating the characteritic of thermal eformation. (5)

8 54 THERMAL SCIENCE: Year 08, Vol., No. A, pp Software FLUENT i ue to imulate the flow. The axiymmetric wirl laminar moel an tructure qua-map meh are aopte [30, 3]. Dicretization of the governing equation employ econ-orer upwin cheme, an the olution i obtaine by a enity-bae approach. Flui i perfect ga. The thermal conuctivity an ynamic vicoity are function of local temperature. The throat Reynol number i proportional to inlet preure. Inlet an outlet bounary conition are both preure. The back preure ratio i fixe at 0. to enure to avoi the hock at the ivergent ection. The inlet tagnation temperature T 0 i fixe at 300 K. The wall temperature T w = 85, 90, 95, 300, an 305 K. A gri ize of Figure 7. Comparion of C T between eq. (7) an CFD for T 0 = 300 K an variou γ, T w wa performe to guarantee a gri inepenent olution. All reult are hown in fig. 7. The tagnation (reference) temperature T 0 remain contant 300 K. The flui inclue air (γ =.4), nitrogen (γ =.4), an argon (γ =.67). It i obviou that eq. (7) agree well with imulate. Partially contraine thermal expanion for C α Nozzle geometry, fixture, an loa Three onic nozzle reporte by Wright et al. [3] were ue to analye the thermal eformation of nozzle throat. A i hown in fig. 8, the nozzle throat iameter at the reference temperature 98 K are 3. mm,. mm, an 0.65 mm, repectively. The material of nozzle i Cu whoe thermal conuctivity i 380 W/mK. A heathe platinum reitance thermometer (RTD) wa ue a a PID controlle heater to maintain the CFV at the eire T w et point value of 98 K, 303 K, 308 K, an 33 K. Firtly, the geometry of nozzle boy i plotte in fig. 8(b). Thee nozzle have the ame external profile, thu the mall iameter nozzle ha the larger wall thickne. Seconly, each nozzle wa intalle between inlet an outlet pipe mae of fibergla fille PTFE with O-ring eal, a hown in fig. 8(a). The PTFE material reuce conuctive heat tranfer between the heate nozzle boy an the tainle teel pipe. The thermal conuctivity of PTFE i 0.6 W/mK which i about /400 th that of Cu. It mean that the temperature of PTEE an pipeline are almot unaffecte by heate nozzle. Beie, the left an right ege of nozzle boy on O-ring eal are contraine to have no iplacement in axi irection (x-irection). Latly, the boy temperature itribution i nearly uniform an equal to the external controlle temperature, ince the thermal conuctivity of Cu i 380 W/mK an the Biot number i lower. Thu, thermal loa are that boy temperature et to a contant precribe temperature. Thermal an tre moel The governing equation an bounary conition controlling the performance of thermal an tructure of nozzle boy were ecribe.

9 THERMAL SCIENCE: Year 08, Vol., No. A, pp Figure 8. Experiment etail reporte by Wright et al. [3]; (a) experimental arrangement, (b) the geometry of onic nozzle It i aume that the material of nozzle boy ha contant thermal conuctivity, ince the boy temperature range i relatively mall. The tranient equilibrium equation without internal heat reource can be written: Tw Tw = 0 (8) a t where thermal iffuivity a = k/(rc p ). In thi tuy, the boy temperature i nearly uniform an teay, hence eq. (8) i reuce to T w cont. Then, it i aume the thermal eformation i mall an reverible, an the elatic behavior i linear where the train i proportional to the tre. Beie, the irreverible platic an creep eformation were not coniere. Hence, the contitutive equation for a linear iotropic thermo-elatic continuum i [3]: + υ υ eij = σij σii α( Tw Tref ) δi, j E E (9) where e an σ are train an tre. The bounary conition in accorance with bounary loa in fig. 8 are applie to the element noe: Mechanical Noe on O-ring eal of left an right ege are contraine to have no iplacement in the X-irection. Noe on other urface are un-contraine.

10 56 THERMAL SCIENCE: Year 08, Vol., No. A, pp Thermal All noe have the ame temperature T w. No internal heat i generate. Thermal eformation an correction factor C α If the nozzle boy i allowe to expan freely, the thermal eformation of the raiu of flow channel i r /r w = α T. Thu, C α = A/A ref = + α T. However, the reult will be ifferent when ome part or ege are contraine. Next, the partially contraine thermal eformation of nozzle boy wa invetigate in etail. For Cu, Young' moulu an yiel Figure 0. The raiu variation of flow channel for 3. mm nozzle trength are. 0 Pa an Pa. Thermal conuctivity k = 380 W/mK, thermal expanion coefficient α = an Poion' ratio υ = 0.3. Beie, the reference temperature T ref = T 0 = 98 K. For 3. mm nozzle, the preicte evolution of thermal eformation at 33 K ( T = 5 K) are hown in fig. 9. In cae of free boy thermal expanion, if the material i allowe to expan or contract freely, there are no tree in the boy. However, for partially contraine expanion, the eformation an tre become more complicate. fig. 9(a) how thermal tre in X-Y plane i not equal to zero. The maximum of thermal tre i near the O-ring eal (contraine part) an i about Pa which i le than yiel trength of Cu. Thu, the irreverible platic an creep eformation o not exit. fig. 9(b) illutrate that Y- iplacement in X-Y plane i obviouly ifferent from free expanion. (a) Figure 9. Preicte evolution of thermal eformation at 33 K for 3. mm nozzle; (a) thermal tre in X-Y plane, (b) Y-iplacement in X-Y plane (for color image ee journal web ite) The raiu variation r of flow channel for 3. mm nozzle are hown in fig. 0 an tab.. At T = 5 K, for free expanion, r i alway greater than 0 an for contraine expanion, r near the entrance i negative. Beie, although r i poitive at throat, it value mm i le than mm of free expanion. (b)

11 THERMAL SCIENCE: Year 08, Vol., No. A, pp For. mm nozzle, the thermal tre an eformation at 33 K ( T = 5 K) are plotte in fig.. Similarly, the thermal tre in thi cae i not equal to zero. The maximum of thermal tre i alo le than yiel trength of Cu. The raiu variation r for. mm nozzle are hown in fig. an tab.. At T = 5 K, r uptream of throat i alway negative. r at the throat i mm i ignificant le than mm of free expanion. It mean that the ifference between contraine an uncontraine expanion get bigger with the ecreae of throat iameter. Thi might be becaue the mall iameter nozzle ha the larger wall thickne. Table. Thermal eformation of contraine thermal expanion (T 0 = T ref = 98 K) Throat Diameter, Boy temperature T w Raiu change r at the throat A(T w )/A(T ref ) C α 3. mm 308 K mm C α = T 33 K. 0 5 mm K mm mm 308 K mm C α = T 33 K mm K mm mm 308 K mm C α = T 33 K mm K mm (a) (b) Figure. Preicte evolution of thermal eformation at 33 K for. mm nozzle; (a) thermal tre in X-Y plane, (b) Y-iplacement in X-Y plane (for color image ee journal web ite) Beie, fig. 0 an alo how the raiu change for. mm nozzle at ifferent T = 5, 0, an 5 K. The reult how the r at ifferent temperature have a ame tenency an it magnitue increae with the increae of T. Table lit the reult of correction factor C α. The lope of C α for 3. mm,. mm, an 0.56 mm nozzle are , , an , repectively, while the lope of C α i a contant for free expanion.

12 58 THERMAL SCIENCE: Year 08, Vol., No. A, pp Figure. The raiu change of flow channel for. mm nozzle Figure 3. The C T v. Re / T/T 0 for Cu nozzle, comparing the imilarity olution with experimental ata by Wright et al. [3] Experimental valiation The experimental ata of Cu nozzle by Wright et al. [3] wa ue to valiate the reult of thi tuy. The inlet tagnant temperature of ry air i approximate to 98 K. Each nozzle wa calibrate with ry air at ix preure etpoint (00~700 kpa). The uncertainty of ma flow rate i about 0.06%. For ry air, the lope of C T of Wright et al. [3] an eq. (7) of thi work are 7.05 an 3.845, repectively. The reaon for thi icrepancy wa foun out. In Wright tuy, the imple correction factor C α = + α T for free expanion wa ue irectly whoe lope i greater than truth value. Hence, by efinition of eq. (), the calculate lope of C T will be le than truth value. Now, the truth value of C α in tab. for ifferent nozzle were ue to calculate new moifie experimental value of C T. The reult are plotte in fig. 3. It reveal that the experimental value of C T are in goo agreement with eq. (7). Concluion To tuy thermal effect on ma flow rate of onic nozzle, both correction factor C T for the thermal bounary-layer an correction factor C α for contraine thermal eformation of the throat area were propoe. The reult are outline a follow. A imilarity olution for it thermal bounary-layer with wall heat tranfer wa preente. For non-aiabatic wall, the formula for correction factor C T over the wall temperature range from 0.8T 0 to.t 0 wa obtaine. For γ =.33, C T = 3.800Re / T/T 0 ; for γ =.4, C T = 3.845Re / T/T 0 ; for γ =.67, C T = 4.00Re / T/T 0. Thermal an tre moel for partially contraine thermal expanion of nozzle boy were built. The truth lope of C α for 3. mm,. mm, an 0.56 mm nozzle are , an , repectively, which are ifferent from a contant value of free expanion The experimental ata of Cu nozzle wa ue to valiate the reult of thi tuy. It howe that moifie experimental value of C T are in goo agreement with eq. (7).

13 THERMAL SCIENCE: Year 08, Vol., No. A, pp Acknowlegment Thi work i upporte by National Natural Science Founation of China uner Grant No , Natural Science Founation of Tianjin uner Grant 6JCQNJC03700 an No. 5JCYBJC900, Reearch Fun of Tianjin Key Laboratory of Proce Meaurement an Control uner Grant No. TKLPMC-06, an Program for New Century Excellent Talent in Univerity uner Grant No. NCET Nomenclature A area, [m ] a thermal iffuivity, a = k/(rc p ), [m ] c p iobaric heat capacity, [Jkg K ] C icharge coefficient for the aiabatic wall, [ ] C T, C α correction factor, [ ] C * critical flow factor, [ ] c oun pee, [m ] throat iameter of nozzle, [mm] E Young'/elatic moulu, [Pa] e total energy (c p T + u /), [Jkg ] F, G, K, N efine in eq. (8)-(0), [ ] J imenionle ma flux, [ ] k thermal conuctivity, [Wm K ] m the nozzle parameter in eq. () M Mach number, [ ] N 0 integration contant, [ ] Pr Prantl number, [ ] p preure, [Pa] q m, q mi real/ieal ma flow-rate, [kg ] R raiu of curvature, [m] Re x local Reynol number (= r u x/µ 0 ), [ ] Re throat Reynol number (= r cr c cr /µ 0 ), [ ] R m pecific ga contant, [Jkg K ] for plannar = 0; for axiimetric = T T t temperature, [K] temperature ifference (=T w T 0 ), [K] time, [] u, v velocity in the x-, y-irection, [m ] X, Y, Z Carteian co-orinate, [m] x, y, r w curvilinear ytem of co-orinate, [m] Greek ymbol α thermal expanion coefficient, [K ] γ ientropic exponent, [ ] iplacement thickne, [mm] i, j Kronecker elta, [ ] r raiu change of flow channel, [mm] e train, [ ] η imilar variable, [ ] θ iffuer angle, [ ] λ imenionle velocity, [ ] µ ynamic vicoity, [Pa ] n kinematic vicoity, [m ] r enity, [kgm 3 ] σ tre, [Pa] υ Poion' ratio, [ ] y tream function, [m K ] Subcript 0 at tagnation conition main-tream cr critical point nt at nozzle throat ref reference temperature w nozzle boy Appenix A. Sufficient conition of reucibility to ODE Equation () can be tranforme into: N m u ( m+ ) γ m c r = N x u x c x x Integrating eq. (A.), we get: γ rw m ( m+ ) γ m γ rw 0 rnt N= Nu c w (A.30) (A.3) where N 0 i an integration contant. Li an Nagamatu [9] irectly aume that N 0 = Rµ 0 /T 0. However, for onic nozzle, eq. (3) i rewritten:

14 60 THERMAL SCIENCE: Year 08, Vol., No. A, pp m- ( m+ ) γ - m- γ - rw u m0 rnt x r T m r N0 0T 0 r0 T0 γ - + M u ( c ) = Eq. (A.3) i reuce to: m+ ( m+ ) γ m 5γ γ γ m rw γ + n 0 γ λ λ λ = c 0 x γ + r γ + mt N Let: nt (5 ) ( ) 0 0 (A.3) (A.33) T N γ + = (A.34) ν c γ γ where i throat iameter. Thu, integration contant N 0 can be expree: At the moment, eq. (A.4) i tranforme into: λ N 0 0 0,5 (5 ) ( ) γ γ ν 0c0 γ + = (A.35) T m+ ( m+ ) γ γ w rnt γ m r x m λ λ λ = (A.36) γ + γ + λ m Beie, accoring to eq. (7) an eq. (A.), η(x, y) i expree: A. The parameter m for the nozzle flow y γ ( m+ ) m γ w 0 0 nt m r y η( xy, ) = c u N r Txy (, ) (A.37) To olve the bounary-layer of onic nozzle, the value of parameter m for the nozzle houl be etermine. Auming the main-tream flow i -D ientropic, the cro-ection area can be calculate by [33]: + rw = rnt γ + γ γ λ λ γ + Subtituting eq. (A.9) into eq. (A.7), give: γ (A.38) λ m m + γ γ γ m + x λ λ λ = m (A.39) γ + γ + λa where = 0 or. Accoring to the wall co-orinate near the nozzle throat, it i implie that:

15 THERMAL SCIENCE: Year 08, Vol., No. A, pp ( rw rnt ) 4 ( x ) R = (A.40) ( rw rnt ) ( x) Aitionally, to take the erivative of λ to eq. (A.0), we get: m λ γ + γ γ x = λ m γ + ( m+ ) γ (m + ) λ At the nozzle throat, accoring to eq. (A.9), (A.), an (A.): (A.4) 3γ γ γ + = m R (A.4) The expreion of eq. (A.3) i the ame a that of Ihibahi an Takamoto [7]. It i obviou that the value of parameter m for both planar an axiymmetric nozzle flow are the ame. If R =, when γ =.33,.40, an.67, m = , 0.483, an 0.49 repectively. Reference [] Yin Z. Q., et al., Dicharge Coefficient of Small Sonic Nozzle, Thermal Science, 8 (04), 5, pp [] *** ISO 9300, Meaurement of Ga Flow by Mean of Critical Flow Venturi Nozzle, Britih Stanar, October, 005 [3] Wright, J. D., et al., Thermal Effect on Critical Flow Venturi, Proceeing, 9 th ISSFM, Arlington, Virg., USA, 05 [4] Unal, B., et al., Numerical Aement of Dicharge Coefficient an Wall Temperature Depenence of Dicharge Coefficient for Critical-Flow Venturi Nozzle, Proceeing, 9 th ISSFM, Arlington, Virg., USA, 05 [5] Bignell, N., Choi, Y. M., Thermal Effect in Small Sonic Nozzle, Flow Meaurement an Intrumentation, 3 (00),, pp.7- [6] Li, C. H., Mickam, B., Flow Characteritic an Entrance Length Effect for MEMS Nozzle, Flow Meaurement an Intrumentation, 33 (03), Oct., pp. -7 [7] Hu, C. C., et al., Dicharge Characteritic of Small Sonic Nozzle in the Shape of Pyramial Convergent an Conical Divergent, Flow Meaurement an Intrumentation, 5 (0), June, pp. 6-3 [8] Illingworth, C. R., The Laminar Bounary-layer Aociate with Retare Flow of a Compreible Flui, ARC RM 590, 946 [9] Li, T. Y., Nagamatu, H. T., Similar Solution of Compreible Bounary-Layer Equation, Journal of the Aeronautical Science, 0 (955), 9, pp [0] Cohen, C. B., Rehotko, E., The Compreible Laminar Bounary-layer with Heat Tranfer an Arbitrary Preure Graient, NACA Report 94, Wahington DC, 956 [] Ball, K. O. W., Similarity Solution for the Compreible laminar Bounary-layer with Heat an Ma Tranfer, Phyic of Flui, 0 (967), 8, pp [] Back, L. H., Acceleration an Cooling Effect in Laminar Bounary-layer-Subonic, Tranonic, an Superonic Spee, AIAA Journal, 8 (970), 4, pp [3] Aziz, A., A Similarity Solution for Laminar Thermal Bounary-layer over a Flat Plate with a Convective Surface Bounary Conition, Communication in Nonlinear Science an Numerical Simulation, 4 (009), 4, pp [4] Kenouh, A. A., Theoretical Analyi of Heat an Ma Tranfer to Flui Flowing Acro a Flat Plate, International Journal of Thermal Science, 48 (009),, pp

16 6 THERMAL SCIENCE: Year 08, Vol., No. A, pp [5] Tang, S. P., Theoretical Determination of the Dicharge Coefficient of Axiymmetric Nozzle uner Critical Flow, Project SQUID Technical Report, PR-8-PU, 969 [6] Geropp, D., Laminar Bounary-Layer in Flat an Rotation-Symmetric Laval Nozzle (in German), Deutche Luft- un Raumfahrt Forchungbericht, 97, pp [7] Ihibahi, M., Takamoto, M., Theoretical Dicharge Coefficient of a Critical Circular-Arc Nozzle with Laminar bounary-layer an It Verification by Meaurement uing Super-Accurate Nozzle, Flow Meaurement an Intrumentation, (000), 4, pp [8] Johnon, A. N., et al., Numerical Characterization of the Dicharge Coefficient in Critical Nozzle, Proceeing, NCSL Workhop an Sympoium, Albuquerque, N. Mex., USA, 998, pp [9] Teoorecu, P. P., Introuction to Thermoelectricity, Treatie on Claical Elaticity, Springer, Amteram, The Netherlan, 03, pp [0] Thoma, B. G., et al., Analyi of Thermal an Mechanical Behaviour of Copper Mol During Continuou Cating of Steel Slab, Iron an Steelmaker, 5 (998), 0, pp [] Park, J. K., et al., Analyi of Thermal an Mechanical Behaviour of Copper Moul During Thin Slab Cating, Proceeing, 83 r Steelmaking Conference, Pittburgh, Penn., 000, Vol. 83, pp. 9- [] Cragun, R., Howell, L. L., A Contraine Thermal Expanion Micro-Actuator, Micro - Electro - Mechanical Sytem (MEMS), 66, (998), pp [3] Ifahani, A. H. G., et al., Simulating Thermal Stree an Cooling Deformation, Die Cating Engineer, (0), Mar., pp [4] Anola, R., et al., Evolutionary Optimization of Compliant Mechanim Subjecte to Non-Uniform Thermal Effect, Finite Element in Analyi an Deign, 57 (0), Sept., pp. -4 [5] Stavely, R. L., Deign of Contact-Aie Compliant Cellular Mechanim for Ue a Paive Variable Thermal Conuctivity Structure, Ph. D. thei, The Pennylvania State Univerity, Pittburgh, Penn., USA, 03 [6] Schlichting, H., et al., Bounary-layer theory, 8 th e., Springer, New York, USA, 000 [7] Johnon, A., Numerical Characterization of the Dicharge Coefficient in Critical Nozzle, Ph. D. thei, The Pennylvania State Univerity, Pittburgh, Penn., USA, 000 [8] Hall, I. M., Tranonic Flow in Two-Dimenional an Axially-Symmetric Nozzle, Quarterly Journal of Mechanic an Applie Mathematic, 5 (96), 4, pp [9] Wang, C., et al.. Influence of Wall Roughne on Dicharge Coefficient of Sonic Nozzle, Flow Meaurement an Intrumentation, 35 (04), Mar., pp. 5-6 [30] *** Fluent Inc., Fluent Uer Guie, Fluent Inc.; 003 [3] Verteeg, H. K., et al., An Introuction to Computational Flui Dynamic: the Finite Volume Metho, Wiley Pre, New York, USA, 995 [3] Srihar, M. R., et al., Review of Elatic an Platic Contact Conuctance Moel-Comparion with Experiment, Journal of Thermophyic an Heat Tranfer, 8 (994), 4, pp [33] Ding, H. B., et al., An Analytical Metho for Wilon Point in Nozzle Flow with Homogeneou Nucleating, International Journal of Heat an Ma Tranfer, 73 (04), June, pp Paper ubmitte: November 4, 05 Paper revie: June 4, 06 Paper accepte: June 3, Society of Thermal Engineer of Serbia. Publihe by the Vinča Intitute of Nuclear Science, Belgrae, Serbia. Thi i an open acce article itribute uner the CC BY-NC-ND 4.0 term an conition.