On the fully developed turbulent compressible flow in an MHD generator Citation for published version (APA): Merck, W. F. H. (1971). On the fully developed turbulent compressible flow in an MHD generator. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR108911 DOI: 10.6100/IR108911 Document status and date: Published: 01/01/1971 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 07. Oct. 2021 ON THE FULL Y DEVELOPED TlTRBULENT COMPRESSIBLE PLOWIN ANMHD GENERATOR PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EIN OHOVEN OP GEZAGVAN DE RECTOR MAGNIFICUS VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG 23 NOVEMBER 1971 DES NA- MIDDAGS TE 4 UUR. DOOR WILLEM F.H. MERCK CEBOREN TT AMSTERDAM OP 29 JULI I 935 Ci.U VI· OFFSI'f '\.V. J·.I'JDIIOVJ:"J 1'171 DIT PROEFSCHRIFT IS GOI::DGEKElJRD DOOR DE PROMOTOR PROF DR. L.H.TH. RIETJENS Aan mijn moeder en Riet, die zoveel geduld met mij hebben gehad This work was performed as a part of the research.program of the group Direct Energy Conversion of the Eindhoven University of Technology, the Netherlands. The experimental part was realized thanks to the financlal support of the Eindhovens Hogeschoolfonds and the hospitality of the Max-Planck-Institut fÜr Plasmaphysik, Garching, Federal Republic of Germany. tot stand komen van een werk gewoonte V«rJ de schrijver zijn over weg die is afgelegd aldvorens 'Ome Frans' wekte in mijn prille jeugd reeds de techniek. moeder bracht mij van het ge cec;enne·ca daarmee m&Jn en verdiepend. Zij bracht nieuwe methode van elektrische door middel van Aan Technische Hogeschool te Eindhoven in dit nieuwe te verdiepen. Daar werd het onderzoek dat tot dit werk leidde, in promotor Prof.Dr. L.H.Th. wijze aanzette discussies met hem en onmisbare schakel in het en experimente Ze fei ·ten. zeer erkentelijk belangriJke ma·te die aan de Mijn vorming. 'Max-Planck-Institut für het bijzonder heer Hartmann, dank onmisbaar voor de voltooid dank zij en de heer volhouden van R1:an van Amelsvoort VPiend­ bijstand van Jas O&J het op steUen mijn grote epkenteliJkheid. bescheidenheid van m&,7'1 vrouw iJaren deze Eindhoven, september 1971 ivillem F'.H. Merck. 1. page I. Contents 7 2. Abstract 9 3. Nomenclature 10 4. Introduetion 15 4.1. General 4.2. HHD boundary layer and duet flows 4.2.1. Incompressible laminar MHD flow 16 4.2 •• Compressible laminar MHD flow 19 . 3. Incompressible turbulent HHD flow J 9 L.2.4. Compressible turbulent è·lliD flow 20 4.3. Entrance flow in a rectangular èffiD duet 22 4.4. Hartmann flow with variabie electrical conductivity 23 5. Turbulent MHD flow of perfect gases 27 5. I. Assumptions 27 5.2. Rednetion of the general momenturn equation 5.3. ~lliD equations for fully established two-dimensional turbulent flmv 30 5.4. Turbulence damping 37 5 .• Derivatives with respect to x 39 6. Numerical solutions 42 6.1. Solution the momenturn equation 42 6.2. Velocity profiles 6.3. Salution of the energy equation 6.4. Temperature profiles 50 i. Experiments and diagnostics 53 7. I. Velocity profile measurements 56 7.1. l. Pitot tube theory 58 7.2. Thermocouple measurements 7. 2. 1. Thermocouple t"":eory 61 7.3. Errordiscussion 67 7 page 7.3.1. Errors due to the closed loop system 67 7.3.2. Errors in the pitot tube measurements 68 7.3.3. Errors in the thermocouple measurements 70 8. Experimental results and discussion 74 8.1. Evaluation of the measurements 74 8.2. Camparisou of theory and experiment 75 9. Conclusions 82 10. References 84 I I . Appendices 89 Appendix I 89 Appendix II. Expressions for the x derivatives 89 Appendix III. Non-dimensionalisation and discretisation of the momenturn equation 93 Appendix IV. Non-dimensionalisation and discretisation of the energy equation 98 8 This work contains a theoretica! and experimental investigation of the gasdynamic properties of the turbulent subsonic flow of a per­ fect gas at the insulator walls in a magnetohydrodynamic duet. Using the ordinary aerodynamic turbulent flow equations, Maxwell's equations and Ohm's law a set of equations for the magneto­ hydrodynamic turbulent compressible steady flow is derived, excluding the Hall effect. These equations are applied to the flow between the insuiator walls of the MHD duet. Solutions of these equations are found for the case of a fully developed flow in a rectangular duet with constant cross section for various Hartmann and Reynolds numbers. Experiments were carried out in a segmented Faraday generator, which was part of a closed cycle MHD system with potassium seeded argon as a werking fluid and a 200 kW are burner as a plasma source. The gas pressure in the duet was slightly above I atm, the temperature about 1900 K and the gas velocity varied between 200 and 450 m/s. The diagnostics used, were a WRe-thermocouple to measure the temperature profiles and pitot tubes of various kinds to measure the total pres­ sure profiles. A computer program simultaneously calculated the gas temperature and velocity profiles from a given set of measurements. The shape of the measured temperature and velocity profiles is in agreement with the theoretically predicted flattening of the profiles. 9 3. NOMENCLATURE a transmission coefficient of optical system (7.11) A constant in the turbulence suppression term (5.29) A turbulent heat transfer quantity (5.19) q A =turbulent moment transfer quantity (5.19) T B = magnetic induction B = applied magnetic induction 0 c velocity of sound ltS c measure for sublayer thickness in turbulence damping term (5.26) 2 = friction coefficient =, 2Tw/p u 5 8 specific heat at constant pressure = specific heat at constant volume • diameter of thermocouple wire d electrical conductivity boundary layer thickness (4.9) D •• deformation tensor components Jl e electrica:l load factor • - E/u B s 0 E = electric field -E* electric field in moving coordinate system = Ê +;x B F. = general body force component 1 h interval length in finite difference equations Ha = Hartmann number • Bl(%}! (4.4) I = total electric current total electric current due to Hall effect = current density current density vector of Hall current Boltzmann's constant 1 duet width, transverse characteristic length L total duet length, axial characteristic length 10 m mass flow rate mass of electron .:f!_!) 2I M Mach number = u/ ( W M localMach number =u'/ L Nu Nusselt number = 2 q 1/À(T - T) w w s Nu local ~usselt number x p pressure = dynamic pressure = ptot dynamic pressure error static pressure = total pressure or total head Pr Prandtl number = C ~/WÀ p turbulent Prandtl number = A /A T q electron charge radiation heat flux, 7.2.1 .c turbulent heat flux (5.19) = universal gas constant Re Reynolds number = pul/~ s radiation constant of Stefan-Boltzmann (7.4) t time t error in estimated temperature profile (6.20) T absolute temperature fiT error in temperature measurement black body radiation temperature (7.12) local gas temperature, Fig. 7.7 thermocouple junction temperature, Fig. 7.7 insulator wall temperature, Fig. 7.7 T mean value of temperature profile s T thermocouple temperature at holder tip, Fig. 7.7 0 IJ u velocity component in x-direction = u 1 u velocity as particular salution of the Hartmann equation p (4.4) u mean value of velocity profile s _,. u general velocity vector ui, u. general velocity components, with i,j I, 2, 3 J V velocity component in y-direction u2 w velocity component in z-direction u~ w atomie or molecular weight of the gas x, y, z Cartesian coordinates x. Cartesian coordinates for i,j I, 2, 3 respectively xi' J a heat transfer coefficient, 7.2.1 .b B Hall parameter = qeB/meve (4.6) effective Hall parameter 8eff y specific heat ratio = C /C p V yl' y2, y3 turbulence damping factors, (5.26, 27, 29) E emission coefficient, 7.2.1.c 8 dimensionless temperature (T - Tw)/(Ts - Tw) 8 dilatation (5.1) K universa! turbulence constant in Prandtl mixing length equation (5.25) À heat conduction coefficient (7.5) ÀA heat conduction coefficient of argon (7.6) A turbulence damping factor = Ha (~) ~ (5.