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THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y. 10017 S The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Authorization to photocopy material for internal or personal use under circumstance not falling within the fair use provisions of the Copyright Act is granted by ASME to libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service provided that the base fee of $0.30 per page is paid directly to the CCC, 27 Congress Street, Salem MA 01970. Requests for special permission or bulk reproduction should be addressed to the ASME Technical Publishing Department. Copyright®1995 byASME All Rights Reserved 95-GT-17 Printed in U.S.A. A NEW ANALOGY FUNCTION FOR THE NAPHTHALENE SUBLIMATION TECHNIQUE TO MEASURE HEAT TRANSFER COEFFICIENTS ON TURBINE AIRFOILS M. Haring Swiss Federal Institute of Technology E PF L-LTT Lausanne, Switzerland° B. Weigand ABB Power Generation Ltd. Gas Turbine Development Baden, Switzerland Abstract The naphthalene sublimation technique is based on the analogy between mass and heat transfer. This analogy is only fully valid for incompressible flow and if the Prandtl and Schmidt number are equal. In the present investigation the energy- and mass transfer equations were solved simultaneously to establish an analogy function which allows the calculation of the Nusselt number from the Sherwood number in function of the Mach, the Prandtl and the Schmidt number. For a laminar flow this new analogy function is based on similarity solutions of the conservation equations for high Mach number flows. Also a numerical investigation was conducted to study the influence of the pressure gradient and the Soret effect as well as varying fluid properties. For a turbulent flow, a flat plate solution was established for Pr=1. Energy and mass transfer equations were additionally solved for a two dimensional duct flow to study the influence of the Prandtl number on the analogy function independently. The resulting analytical and numerical solutions are shown for various pressure gradients, Prandtl and Mach numbers. In addition, approximations for the analogy function are derived. The influence of the present theory on heat transfer measurements on a turbine airfoil is shown. The theory is validated against experimental results in Haring et. al. (1995) showing a good agreement between the heat transfer coefficients calculated with the new analogy function and measurements of actual heat transfer. Nomenclature Symbol Unit Description c cp cf C C, mass concentration (naphthalene) specific heat at constant pressure skin friction coefficient Chapman Rubesin Parameter see Eq. (1) and (2) [kgN/kg] [J/(kg K)] [ [ [ - ] - ] - ] dh D Ec f, G fn,gn g [m] [m2/s] [ -] [ -] _] [-] [ h h H K L I [m] [m] m [kg/(m2s)] [J] [m] [J] 1-1 M Nu n Pr Prt p Re Sc Sc t Sh t T [ -] -] -] [-] [ [ 1-1 [Pa] -] -] -] _] [ ] [ [ [ [ T05 [K] [K] Tr u,v [K] [m/s] up w [ x [m/s] yZ -] [m] _] x [ a [W/m2K] hydraulic diameter mass diffusion coefficient Eckert number non-dimensional stream function see eq. (39) non-dimensional stagnation enthalpy enthalpy duct hight stagnation enthalpy see eq. (19) thermal entrance length mixing length mass flux Mach number Nusselt number exponent in the analogy function (eq. 3) Prandtl number turbulent Prandtl number pressure Reynolds number Schmidt number turbulent Schmidt number Sherwood number integration variable (eq. 23) temperature total temperatur (free stream) recovery temperature velocity components in the x, directions, respectively mean velocity in the duct Sutherland exponent (eq. 18) coordinate axes y x =x/ h/(hRe Pr) heat transfer coefficient Presented at the International Gas Turbine and Aeroengine Congress and Exposition Houston, Texas - June 5-8, 1995 Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab aT[kg/(ms)] thermal diffusion factor see eq. (31) [-1 a1, 3 1 mass transfer coefficient [m/s] R [-] wedge angle (eq. 27) boundary layer thickness [m] 5 eddy diffusivity for momentum Em [-] Illingworth variable (x-coordinate) [ ] Illingworth variable (y-coordinate) rl [ ] stream function yr [ ] coefficient of heat conductivity [W/mK] [-] eigenvalues dynamic viscosity [Pa s] kinematic viscosity [m2/s] v p [kg/m3] density PN 8 flow over a flat plate Gnielinski (1975) established an empirical correlation in function of the Prandtl, the Schmidt and the Reynolds number, which gives values for n between n=0.45 (ReL=3.e5) and n=0.74 (ReL=1.e7, Sc=2.5, Pr=0.7). All these studies were conducted at low Mach numbers (incompressible flow). The objective of the present investigation is to develop an analogy function, that is also valid for the compressible flow range and for different Schmidt and Prandtl numbers. Analysis Laminar flow with zero pressure gradient To establish an analogy function the continuity-, the [kg/m3] partial density naphthalene momentum-, the energy- and the mass transfer 1-1 non-dimensional enthalpy equations have to be solved for the boundary layer flow. Continuity equation Subscripts: h i N p w S homogeneous solution refers to initial conditions refers to naphthalene particular solution refers to wall conditions refers to free stream conditions ax (Pu) + ya (pv) = 0(4) Momentum equation in x direction p u au + v a u _ dp + a µau ay) = dx ay ay ax Introduction The naphthalene sublimation technique has been used for many years to study heat transfer phenomena. The () 5 Momentum equation in y direction d Y method is based on the analogy between the energy equation and the mass transfer equation and is only fully (6) = 0 Energy equation valid for the incompressible flow range and if the Schmidt number is equal to the Prandtl number. In most aHaH p u ax + p v ay = PrPr 1 tay ( ^ u `aYI ) } + Pr Lay \µ ay/J applications this is not the case. Sublimates such as Naphthalene, Thymol, Jod, Pardibrombenzol, etc. have a Schmidt number between Sc=2 and 2.5, whereas the Prandtl number of air is Pr=0.7. Empirical correlations are commonly applied to correlate the heat and mass transfer coefficients when the Schmidt number and the (7) Mass transfer equation Prandtl number are not equal. Often power law relations are used for the Nusselt and the Sherwood number Nu = C, Rem Prn \ Sh = C, Rem Scn (8) (2) For a "similar" laminar boundary layer flow this set of equations can be reduced to ordinary differential For the same Reynolds number the analogy function becomes Nu =Sh (SOn J pl uax + v ay =ay (pia ) (1) equations by introducing the Illingworth Stewardson variables y (3) rl (x,Y) _ P S u Sp ^s where the exponent n is an empirical constant. Chen 0 ps dy(9) x and Goldstein (1992) applied a numerical value of n=1/3, the same accounts for Berg (1991), Brakel and =^(x)= f Ps us µs d x Cowel (1986), and Kestin and Wood (1970) whereas (10) Presser (1968) gives values of n=0.33 for laminar and n=0.44 for turbulent flow conditions. For the turbulent 2 Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab 00 which allow a separation of the stream function yr, the velocity u and the stagnation enthalpy H in a product of "size" times "shape" iV(x,Y) = G(4) f(ii) (11) u(x,Y) = uS(4) f'(il) (12) H(x,y) =H8(^) g(TI)• (13) To normalize the boundary conditions the nondimensional variables O = (g - g w )/ (1 - g w ) for the +K$ t(' (f„)Pr del 00(20) (f „ ) Pr r (f..) Pr(f,f„),drIdt .1 r (f„)Pr del 0J and the concentration distribution in the boundary layer is given by: 00 enthalpy and c = (c-c w )/(cg-c w ) for the concentration shall be introduced. c = 1 - (f(f„)Sc dTl / J (f„)Sc del (21) Introducing (9) and (10) into (5) to (8) leads to momentum-, energy and mass transfer equations in the following form: Once the enthalpy- and the concentration distribution in the boundary layer are known, the Nusselt and (Cf")'+ff"+mo u d s P5 - f' 2 =0 (14) K(Cf'f")'+(CO')'+PrfEY=0 (15) (Cc')'+fScc'=0 (16) us dp with the associated boundary conditions 1=0: f=f'=0 f'=1 i 0=0 0=1 c=0 c=1 Sherwood number can be calculated. By introducing the Illingworth Stewardson variables ri and E the analogy function becomes: Nu To s TW e , Sh - T r- TW c' 1=0 where O' and c' can be derived from eqs. (20) - (21). (17) C is the Chapman Rubesin Parameter, which takes the variation of the fluid properties in account based on the Sutherland law (0.5 <w < 1): 0 Pr Pr r (f „ ) -Pr O'I = f )— 1+K j(f') (f'f")'dildt 1=00 0J (f „ ) Pr f 0 P µ s _ s\w-1 _( p µ C JI w-1 T' 8-) c , (18) and the parameter K is defined as ,0=0 (23) dn 00 = (f„)Sc/ j(f')Scdi(24) The Prandtl and Schmidt number dependency functions can be approximated numerically, leading to K _ Pr - 1) ( U52 l H I- the analogy function (19) 1 Nu T - T w Pr 3 (Sc) (1+K{0.249 0.15644log, o (Pr)]) Sh - T r - T W The set of eqs. 14-17 can be solved analytically for the simple case of zero pressure gradient and C=1. The enthalpy distribution in the boundary layer becomes: 00 r (f „ ) Pr d _ (22) (25) K can be expressed in function of the Mach number, the Prandtl number and the wall to gas temperature ratio: 0 „ ) Pr ( (f „ ) 0=1- ^J +K J^ (f Pr (f, f')' dry dt OJ $ (f,,)Pr dTl K = 2(Pr-1) I1+0.5(K-1) Mg 2 ] (K -1) M5 2 I (26) 1 + 0.5(K -1) Mgt - TS^ [ (K -1) M3 2 +2] The result obtained (eqs. (25)-(26)) shows that the transfer function for Nu/Sh is reduced to the classical solution Nu/Sh=(Pr/Sc) 113 for the case of Pr=1 for all 3 Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab 0.76 Mach numbers. For Pr#1 the classical solution is Nu obtained for M5- 0 because in this case To is equal T r . Sh For the case of very high Mach numbers eq. (26) shows the interesting result of K=2(Pr-1). lical solution 0.74 auskas (1985) 0.72 Influence of variable fluid properties 0.70 Nu Sogin (1961) M=0.2 Sh M=0.4 0.65 0.68 0.0 0.4 p 0.6 0.8 1.0 Influence of the Soret effect Mass diffusion is a function not only of concentration M = 0.6 gradients but also of other potential gradients acting on the system. At higher Mach numbers the primary concern is the coupling effects of mass concentration and temperature gradients. Although pressure gradients M = 0.8 M = 1.0 0.50 0.5 0.2 Fig. 2 Influence of the pressure gradient (Pr=1, Sc=2.5, C=1) 0.60. 0.55 ant solution 0.70 M = 0.05 Eckerl(1942) 0.6 0.7 0.8 0.9 and body forces, among others, can also induce mass diffusion, they are not generally important in the type of application considered. If the thermo-diffusion or Soret effect is taken into account eq. (8) becomes: 1.0 w Fig. 1 Influence of variable fluid properties (Pr=O. 7, Sc=2.5, dp/dx=0, T /T8=0.9) p To study the influence of variable fluid properties, eqs. (14) to (17) were solved numerically using a shooting method (Press et. al (1989)). For a zero pressure ( u ac ax D ^c + a p Dc (1-c) aT ai ac' ay I ay) ay (P ay ) ayT (29) At room temperature the concentration of naphthalene at the surface is approximately c W , N = 3.e-4 kg N /kg, gradient and a Chapman Rubesin Parameter of C=1, eq. (25) was verified. Fig. 1 shows that the influence of the exponent w in the Sutherland law (eq. 18) on the therefor c(1-c)can be simplified to c. This reduces analogy function can be neglected for the temperature equation (29) to an ordinary differential equation after applying the Illingworth Stewardson transformation. variations presented here (T/T5=0.8-1.2). With the temperature distribution in the boundary layer approximated by the Crocco Buseman integral (Schlichting (1965)) eq. (29) becomes: Influence of the pressure gradient The above shown results were based on the assumption of a zero pressure gradient. In the leading edge region the influence of pressure gradient has to be taken into account. Fig. 2 shows the numerical solution of eqs. (14) to (17) for different pressure gradients, presented by the c"+fScc'- u52a11c f'f„1. 2 =0. (30) cp (T6 + 2 cp [1 - f' 2 l) p parameter: Eq. (30) was solved numerically and the result is shown = ug du5 d^ (27) in Fig. 3. It can be seen that the influence of the Soret effect depends strongly on the thermo-diffusion factor aT. For a binary mixture aT can be calculated in function The results are compared with measurements conducted by Zukauskas (1985) and Sogin (1961) on a of the molecular weight, the thermal conductivity and the temperature (Hirschfelder (1954)). At room temperature the thermo-diffusion factor for the pressure gradient can be additionally taken into account naphthalene air mixture becomes aT=0.16, which shows that the Soret effect has to be taken into account cylinder in cross flow and a simplified solution established by Eckert (1942). The influence of the by modifying the analytical solution: Nu Nu Sh=oh S )analyt. (1-0.02[) for Mach numbers higher than M=1.5. (28) ll Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab component in y direction becomes zero. Introducing the turbulent Prandtl number Pr t and the non- 0.7 ShNu dimensional parameters 0.6 Hi _ Hw ;u=u ;Y =h;x -h;p=P (33) 0 the energy equation can be written as: 0.5 0 1 2 m e = R ax a { Pr + E Pr 3M 4 Y C Fig. 3 Influence of the Soret effect (Pr=0.7, Sc=2.5, tJ C=1, dp/dx=0) L ((1 - P) + C + y Turbulent flow When the boundary layer is turbulent the energy and r ` m+ aH } (1 Pr l)u ^U Ec \ t/ Y with the boundary conditions the mass transfer equations can not be solved in the same way as for laminar flow. The solution proposed by y=0 H'=0 the authors is based=0 on a simple model for the velocity H y= 1 distribution and on the von Karman analogy. Energy and mass transfer equations were additionally solved for (34) x=0 H= 1 and the Eckert number defined as Ec=u 0 2/(c p AT). The velocity distribution u of the fully developed duct flow, fully developed channel flow to study the influence of the Prandtl number. appearing in eq. (34), was calculated from the momentum equation using the mixing length model for the eddy viscosity E m +=1 2 jau/ayj/v. The well known mixing length distribution given by Nikuradse and Flat plate solution for Pr=1 Van Driest (1951) solved the momentum equation for a turbulent boundary layer flow on a flat plate. The integral method was used to obtain surface friction based on a modified by a Van Driest damping term near the wall was applied: velocity profile derived from the mixing length theory. To take the variation of fluid properties into account the =h[0.14-0.08(y/h) 2 -0.06(y/h) 4 ] (1 -e - Y 26 Crocco Buseman temperature distribution was assumed. The skin friction coefficient is given by an implicit relationship: ) (35) For more detailed information the reader is referred to Cebeci and Bradshaw (1988). A solution of eq. (34) can be derived by splitting the problem into a solution for the homogeneous equation 0.242 arc sinal + arc sin[31 a (cf T W /T5) 1i2 = 0.41 + log (Re x cf) - w log 10 (TS) 1Y and a particular integral. A particular solution can be found easily since the second term of the right side of eq. (34) is only a function of y. By introducing H P =H p (y) (31) into eq. (34) the following ordinary differential equation can be derived: where a1 and R1 are functions of the Mach number and the gas to wall temperature ratio (Jischa (1982)). Once the skin friction coefficient is known an analogy function 1H p between heat and mass transfer can be derived applying the von Karman analogy to obtain Nusselt and aY = Ec u ua 1 _ (1 + S m + ) Pr aY Sherwood numbers: Nu P1 + 5 ' Sh = Sc For air the Prandtl number is equal to Pr=0.7 and the turbulent Prandtl number is approximately Prt=0.9 2 { Sc - 1 + ln[(5 Sc + 1)/6]} leading to a ratio of Pr/Prt=1. This assumption permits the solution of eq. (36) analytically and results in the - 1+5' (36) (1 + ER,+ Pr Pr tJ particular solution 2 { Pr - 1 + ln[(5 Pr + 1)16]} 2 (32) H p =(1-Pr) Ec 2 . (37) Fully developed channel flow In order to study the effect of different Prandtl numbers on the analogy function a solution for fully developed channel flow was derived. In this case the velocity 5 Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab It should be noted that eq. (37) satisfies the two (0.6526+0.07065log, 0 M) boundary conditions for y=0 and y=1 for H. For values of the Prandtl number where Pr/Prt^1, eq. (36) has to be solved numerically. This analogy function was compared with the flat plate solution, and the results expressed in form of the exponent n (eq. 3) are shown in Fig. 4. A good The homogenous part of eq. (34) is given by Re aH = a 1 + E r"—+ agreement between the flat plate and the channel flow 1 a H solution for Pr=0.7 and Sc=2.5 can be seen. (38) Prt ay J y ax ayy ^(Vr 0.70 n which is known as the turbulent Graetz-Problem. A solution can be found by introducing 0.60 channel flow solution which reduces eq. (38) to the Sturm-Liouville Eigenvalue problem. This problem was solved numerically by using a forth-order Runge-Kutta method. The resulting eigenvalues and constants were compared with solutions given by Shah and London 0.55 (1978) and a good agreement was found. The general 0.50 solution of the energy equation (34) is the addition of the homogenous and the particular solution 0.1 00 cn 9n(y) e n+HP (40) n=1 seen that the effect of Prandtl number on the analogy X 2— cn 12 7 n x- 8 (1-Pr)Ec g (1) e 0.9 function becomes important for higher Prandtl numbers. - Nu(x) = M numerically. Fig. 5 shows the exponent n in the analogy function (3) for different Prandtl numbers and it can be Co e n 0.7 valid any longer and equation (36) has to be solved boundary conditions by using the orthogonality of the eigenfunctions. Finally the Nusselt and the Sherwood number are derived: cn g 0.5 For other fluids than air the simplification Pr/Prt=1 is not The integration constants c n can be found via the 4 0.3 Fig. 4 Exponent n for the turbulent boundary layer flow (Pr=O.7, Sc=2.5) a2 n=1 flat plate solution 0.65 (39) H = fn (X)'gn (y) H=Hh+Hp= (43) 1Sc^ 0.7 (41) n — Xn n=1 0.6 Channel flow solution cn gn'(1) e Sh(x) = 4 0.5 X n2 X Flat plate solution n 1(42) 2 1 C 2 gn'( 1 ) e n=1 0.4 0 2 4 6 8 10 Pr ^n Fig. 5 The exponent n in the analogy function for different Prandtl numbers (Sc=2.5, M=O. 1) With the Nusselt and Sherwood number being known an analogy function was established by integrating over the thermal entrance region (L=20 dh) The theory is validated against experimental results (Haring et. al (1995)) comparing the naphthalene sublimation (mass transfer data) with the liquid crystal technique (heat transfer data) in compressible flow. Both techniques were used on a cylinder, on a flat plate ti i Nu _ Nu(x)dx/ Sh(x)dx) Sh 0 0 L Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab and a turbine airfoil and show a good agreement between the two methods. exponent n in the analogy function becomes very important. Measurements on a turbine airfoil a [W/m 2 Kl 1200 The naphthalene sublimation technique was applied to measure the local heat transfer coefficient on a turbine airfoil. The measurements were conducted in a linear test facility consisting of five blades. The test facility and the measurement equipment are described in detail in Haring et al (1994). To obtain the Mach number Measurements TEXSTAN 800 distribution the airfoil surface pressures were measured at midspan of the airfoil surfaces (both on suction and pressure sides, Fig.6, right axis) forming the two center flow passages of the cascade. On the suction side the velocity rapidly increases. After the strong acceleration 400 the velocity increases only slightly to reach the maximum at s/L=0.8, from where a decrease can be observed. On the pressure side the acceleration is almost uniform. The sublimation losses were measured in 595 points (85 on the perimeter, 7 heights). -1.0 OC [W/m 2 K] 1200 -0.2 — a nbyEq.25and43 n=0.33 Pressure side 1.2 1000 800 0.6 1.0 Suction side 1.0 Fig. 7 Numerical comparison with TEXSTAN (M2=0. 7 Re2=1 '900'000, Tu=2%) 0.8 The experimental data is compared with the numerical ^' b b^ 600 0.2 s/L M 1400 . -0.6 results obtained with the boundary layer code TEXSTAN (Crawford (1986)), using the Lam Bremhorst 0.6 PTM k-c model (Schmidt and Patankar (1991)). With the 400 0.4 200 0.2 -1.0 -0.6 -0.2 0.2 0.6 exception of the stagnation point region a good agreement between predicted and measured values was found (Fig.7). Conclusions The naphthalene sublimation technique to measure heat transfer coefficients is based on the analogy between the energy and the mass transfer equation. This analogy is only fully valid in the incompressible flow range and it the Prandtl and Schmidt number are equal. 1.0 s/L Pressure side Suction side The theory described in this paper allows the application of the sublimation technique in the Fig. 6 Heat transfer coefficients on a turbine airfoil for different exponents n in the analogy function (Eq. 3) (M2=0.7 Re2=1'900'000) and Mach number distribution. compressible flow range and for different Prandtl and Schmidt numbers. • For laminar boundary layer flow the energy and the mass transfer equations were solved based on similarity solutions. A numerical investigation has been Fig. 6 (left axis) shows the measured heat transfer distribution on a turbine airfoil. The present analogy function for laminar and turbulent boundary layer flow was used to evaluate the measured mass transfer data and the result is compared with a constant exponent n=0.33 in equation (10). In the laminar flow region at low conducted to study the influence of the pressure gradient, variable fluid properties and the Soret effect. • The influence of the pressure gradient and variable fluid properties on the analogy function is small, whereas for Mach numbers higher then M=1.5 the Soret effect has to be taken into account. Mach number the differences between the two analogy • functions are very small whereas at higher Mach number For the turbulent boundary layer flow a flat plate solution (for Pr=1) established by van Driest was used and in the turbulent flow region the influence of the to obtain an analogy function. The energy and mass 7 Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab transfer equations were then solved for a fully developed channel flow to study the influence of the Prandtl number. For Pr=0.7 and Sc=2.5 a good agreement was found between the two solutions. • The theory is validated against experimental results (Haring et. at (1995)) comparing the naphthalene sublimation with the liquid crystal technique in compressible flow. Both techniques were used on a cylinder, on a flat plate and a turbine airfoil and show a good agreement between the two methods. • Measurements conducted with the naphthalene sublimation technique on a typical turbine airfoil (M2=0.7, Re2=1'900'000, Tu=2%) were compared to data obtained with a numerical boundary layer code (TEXSTAN) and good agreement was found except for the stagnation region. Acknowledgments The coating of the airfoil and the thickness measurements, to obtain the heat transfer measurements shown in Fig. 7 and 8 were conducted at the Technical University Darmstadt, Dept. of Flight Propulsion, Darmstadt, Germany, by Dipl.-Ing. J.Richter. The authors kindly acknowledge the encouragement and helpful discussions with Dr. Harasgama, ABB Gas Turbines Development, concerning the development of the above theory. Additionally the authors would like to thank ABB Power Generation Ltd. for permission to publish this paper. References Brakel, J.P., Cowell, T.A., 1986 An analysis of the uniform naphthalene layer mass transfer analogue methode applied to laminar flow over a flat plate J. of Heat Transfer, 1986, Vol. 2 pp. 495-499 Berg, P.H.,1991 Experimentelle Bestimmung des ortlichen inneren Warmeubergangs von Turbinenleit- and Laufschaufeln mit Hilfe der Analogie zwischen Warme- and Stoffubergang. Dissertation, T.H. Darmstadt, Institut fur Flugantriebe Cebeci, T., Bradshaw, P., 1988 Physical and Computational Aspects of Convective Heat Transfer. Springer Verlag, New York Chen, P.H., Goldstein, R.J. 1992 Convective Transport Phenomena on the Suction Surface of a Turbine Blade Including the Influence of Secondary Flows Near the Endwall. Journal of Turbomachinery, Vol 114, pp. 776-787 Crawford, M.E., 1986 Simulation codes for calculation of the heat transfer to convectively cooled turbine blades. VK/ - LS- 1986-06, Convective heat transfer & film cooling in turbomachinery Eckert, E., 1942 Die Berechnung des Warmeubergangs in der laminaren Grenzschicht umstromter Korper. VDI Forsch.- Heft, Nr. 416, Berlin Gnielinski, V., 1975 Berechnung mittlerer Warme- and Stoffubergangskoeffizienten an laminar and turbulent umstromten Einzelkorpern mit Hilfe einer einheitlichen Gleichung. Forsch. Ing. Wes. 41, Nr. 5, pp. 145-153 HAring, M., BSlcs, A., Harasgama, S.P., Richter, J., 1994a Heat transfer measurements on turbine airfoils using the naphthalene sublimation technique. ASME Paper 94GT-171 Haring, M., Hoffs, A., Bolcs, A., Weigand, B.,1995 An experimental study to compare the naphthalene sublimation with the liquid crystal technique in compressible flow. Submitted for publication at ASMEIGTI 95P. Hirschfelder, J.O., Curtiss, F.C., Bird, R.B., 1954 Moleculat Theory of Gases and Liquids. J. Wiley & Sons, New York Jischa, M., 1982 Konvektiver Impuls-, Warme- and Stoffaustausch Vieweg, Braunschweig, Weissbaden Kestin, J., Wood, R.T., 1970 The Influence of Turbulence on Mass Transfer from Cylinders. J. of Heat Transfer, pp.321 - 327 Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T, 1989 Numerical Recipices. Cambridge University Press, Cambridge Presser, J., 1968 Experimentelle Prufung der Analogie zwischen konvektiver Warme- and Stoffubertragung bei nicht abgelosster Stromung. Warme- and Stoffubertragung, Voll ,S225-236 Schlichting, H., 1965 Grenzschichttheorie G. Braun Verlag, Karlsruhe Schmidt & Patankar S.V., 1991 Simulating boundary layer transtition with low Reynolds number k-epsilion models: Part 2; an approach to improve the predictions. Trans. ASME J. of Turbomachinery, Vol. 113, pp 18-26 Shah, R.K., London, A.L., 1978 Laminar flow forced convection in ducts. Advances in Heat transfer, Academic Press, New York Sogin, H.H., Subramanian, V.S., 1961 Local mass transfer from circular cylinders in cross flow Journal of Heat tTransfer, Nov. 19961, S.483-493 Van Driest, E.R., 1959 Turbulent flows and Heat Transfer. High Speed Aerodynamics and Jet Propulsion, Vol. V, 1959, pp. 372-396 Zukauskas, A., Ziugzda, J., 1985 Heat Transfer on a Cylinder in Crossflow Springer Verlag, New York Downloaded From: https://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82298/ on 06/17/2017 Terms of Use: http://www.asme.org/ab