THERMAL TO MECHANICAL ENERGY CONVERSION: ENGINES AND REQUIREMENTS Vol. I - Fundamentals of the Heat Transfer Theory - B.M.

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1 FUNDAMENTALS OF THE HEAT TRANSFER THEORY B.M.Galtseysky Department of the Avaton Space Thermotechncs, Moscow Avaton Insttute, Russa Keywords: Heat transfer, conducton, convecton, radaton Contents 1. Types of heat transfer.. Investgaton method of heat transfer. 3. Dfferental equatons and unqueness condtons. 4. Smplfed equatons. 5. Transton from lamnar to turbulent flow. 6. Heat transfer coeffcent and frcton resstance. 7. Smlarty and modelng of heat transfer processes. 8. Crteral equatons for convectve heat transfer n the boundary layer. 9. Crteral equatons for convectve heat transfer n channels 10. Heat conducton process. 11. Radatve heat transfer. Bblography Bographcal Sketch Summary In desgnng heat engnes of flyng vehcles and a number of other facltes t s necessary to take nto account heat transfer processes. In some cases these processes become determnng ones n choosng a desgn, for example, the makng of thermal sheldng of combuston chambers of gas turbnes, nozzles of jet engnes, etc. 1. Types of Heat Transfer By convecton the complex heat transfer process s dvded nto a number of smpler processes: heat conducton, convecton and radaton. Each of these heat transfer processes obeys ts laws. Heat conducton s the process of molecular heat transfer by mcropartcles (molecules, atoms, ons, etc.) n a medum wth a non-unform temperature dstrbuton. Convecton s the process of heat transfer by dsplacng the macroscopc elements of a medum (molar volumes). Radaton s the process of heat transfer from one body to another by electromagnetc waves (or quanta). In technologcal facltes heat s as a rule transferred by two or three ways at a tme. Such a combned process s referred to as heat transfer. The heat transfer process condtoned by the smultaneous acton of convecton and heat conducton s called

2 convectve heat transfer. The partcular case of ths process s heat transfer representng convectve heat exchange between a movng medum and ts nterface wth another medum, sold body, lqud or gas.. Investgaton Method of Heat Transfer The phenomenologcal method s manly used for studyng the heat transfer processes n power plants. It s based on applyng the basc laws of physcs and some addtonal hypotheses on the course of thermal gas-dynamc processes. As a result of the use of ths method dfferental or ntegral heat conducton equatons are obtaned. In the smple cases, they are beng solved analytcally or numercally. In more complex cases, the method of smlarty or dmensons s used to obtan smlarty numbers, a relatonshp between whch s establshed as a result of the expermental study of the process. The phenomenologcal method of nvestgatng the heat transfer processes s based on the followng prncples: - the substance that takes part n heat transfer s consdered as a contnuous medum. Its molecular structure and also the mcroscopc mechansm of heat transfer are not vewed but are taken nto consderaton by ntroducng the quanttes that are responsble for the physcal propertes of substances (heat conducton, vscosty, heat capacty, densty, etc.); - to construct a mathematcal descrpton of the heat transfer process the frst law of thermodynamcs, the conservaton law of substance and the conservaton law of momentum are used. Fourer s law and Fck s law are adopted to set up a closed system of dfferental equatons. In dervng the energy equaton Fourer s law s used: the vector of the heat flux densty due to heat conducton at a gven pont and at a gven tme moment s drectly proportonal to that of the temperature gradent at ths very pont and at ths very tme q= λgradt where q s the heat flux densty determned as the amount of heat transferred per unt tme from unt surface; gradt = dt dn s the rate of threshold temperature varaton to an sothermal surface at a gven body pont and at a gven tme moment; λ s the thermal conductvty of substance servng as ts physcal characterstc; T - temperature; n normal to surface. Fourer s law s the partcular case of the common law of energy flux transfer and s rgorous only f the substance s homogeneous n every respect except temperature. In an nhomogeneous medum the heat transfer process can be condtoned not only by molecular heat conducton but also by dffuson of substance. The substance dffuson n the nhomogeneous medum can arse under the acton of the concentraton gradent of substances (concentraton dffuson), pressure gradent (barodffuson) and temperature gradent (thermodffuson). The nfluence of baro-and thermodffuson on heat transfer s small as aganst that of concentraton dffuson and can be neglected n the majorty of cases. In ths case, for a bnary gas mxture we have: λgrad ( ) q = T + g where 1 1 the subscrpts 1 and refer to the correspondng components of the bnary mxture, s the enthalpy, g s the densty of the dffuson flow of the mass of the component of the

3 mxture representng the amount of the substance of the mxture component transferred per unt tme from unt surface. The second term n ths equaton s responsble for heat transfer wth dffusng substance. When the gradents of other potentals are absent, the dffuson flow of substance n the bnary mxture s determned accordng to Fck s law: g = ρdgradc where С = ρ / ρ s the concentraton of the -th component, ρ s the densty of the -th component, n ρ = ρ s the mxture densty, D s the dffuson = 1 coeffcent. Ths relaton can be used for the mxtures composed of a small number of components, whose bnary dffuson coeffcents for all pars of substances are the same. Knowng the composton of the gas mxture t s possble to determne ts physcal propertes such as the mxture enthalpy n = = 1 C, the mxture mean heat n n n 1 capacty Cp = CC p, the thermal conductvty λ = 0.5 x λ + x / λ. = 1 = 1 = 1 In dervng the moton equaton Newton s law s used: shear stress of frcton between two layers of a lnearly movng lqud s drectly proportonal to the velocty gradent normal to the moton drecton. For example, n the case of the plane movng wth a velocty n parallel to the plane OXZ that s parallel to the OX-axs, the frcton shear u stress s equal to τ xz = μ, where μ s the dynamc vscosty coeffcent; along wth ths coeffcent use s made of the knematc vscosty coeffcent that s determned as the relaton ν = μ / ρ. Newton s law for three-dmensonal flows assumes a lnear dependence of the lqud stran velocty and s mathematcally wrtten as: j τ + j = μ for j j 3 τ = + + j μ μ for =j. 3 j j 3 Here the coordnates (x, y, z) are desgnated through x (=1,, 3), u s the projecton of the lqud velocty onto the x -axs. 3. Dfferental Equatons and Unqueness Condtons The theoretcal study of heat transfer amounts s frst of all assocated wth determnng a temperature feld T (t, x, y, z), a velocty feld u ( t, x, y, z), a pressure feld p (t, x, y, z), and a concentraton feld of substance (t, x, y, z). Knowng these quanttes and also the temperature and pressure dependences of the physcal propertes of substance, t s

4 possble to determne all quanttes that characterze heat transfer (heat flux, hydraulc resstance, etc.). In order to determne the above-mentoned nfluences the followng equatons are used. d dp = dv gradt + dv D: gradc + + q v (1) dt dt Energy equaton ρ ( λ ) ( ρ ) du = + + () dt 3 1 Moton equaton ρ R gradp μ u μgrad( dvu) u dv D: gradc + w = 0 (3) dt Dffuson equaton ρ ( ρ ) Contnuty equaton ρ + t ( ρu) ( ρv) ( ρw) + + z Here = c s the enthalpy of the mxture, q v s the volume densty of nternal heat sources, u s the velocty vector, F s the stress vector of the volume force, w s the velocty of flowrate (or of formaton) of the mass of the -th substance per unt volume of the chemcally reactng medum. The total dervatves n the above equatons for a movng medum are equal to: d = + u + v + w where u, v, w are the velocty projectons of the movng dt t z medum onto the x, y, z. -axes, respectvely. In the moton equaton F = gβ ΔT s the lftng force (Archmedean buoyancy force) where g s the gravtatonal acceleraton, β s the volume expanson coeffcent, Δ T = T T w s the temperature drop wth respect to the surface temperature T w. For the thermal and hydrodynamc processes to be analyzed, the above equatons must be supplemented by the unqueness condtons nvolvng the physcal, geometrcal, boundary and ntal condtons. The physcal condtons specfy the values of the = 0 physcal propertes of lqud (densty, vscosty, heat capacty, thermal conductvty, dffuson coeffcent, etc.) and ther temperature and densty dependences. Among the geometrcal condtons s the shape and szes of a regon where heat transfer s realzed, for example, the shape and szes of a body n the gas flow, the shape and szes of channels where heat carrer moves. The ntal condtons characterze the temperature, velocty and pressure dstrbutons of a medum at the ntal tme moment. If the heat transfer process s steady-state, then the ntal condtons drop out of consderaton. The boundary condtons characterze the laws of thermal and hydrodynamc nteracton of the surface of a consdered body wth the surroundngs. The medum velocty over the body surface s taken equal to zero as the movng medum adheres to the surface. When the njecton or sucton of lqud (coolant) s present over the surface of the consdered body, t s necessary to assgn the velocty of ts njecton or sucton. For the energy equaton the boundary condtons can be prescrbed n the form of the temperature (4)

5 dstrbuton over the body surface (boundary condtons of the frst knd) or n the form of the heat flux dstrbuton over the body surface (boundary condtons of the second knd). 4. Smplfed Equatons The boundary layer method s the most effcent procedure to solve the abovementoned system of equatons. The essence of the boundary layer method s as follows. In external flow past the body surface, by conventon t s possble to select two regons: the regon near the surface n whch the acton of vscosty and heat conducton (boundary layer) much manfests tself and the external flow regon (far from the surface) n whch t may be wth a suffcent accuracy consdered that the lqud s deal (non-vscous and non-heat conductng). The dynamc, thermal and dffuson boundary layers are dstngushed and are not equal n the general case (Fgure 1) In the case of flow past a flat surface, when the gravtatonal forces and chemcal reactons are absent, the system of the equatons for a steady-state flat boundary layer s wrtten n the form: T T T energy equaton ρ uc p + ρvcp = λ + qv (5) P moton equaton ρ u + ρv = μ (6) ρu ρv contnuty equaton + = 0 (7) Fgure 1: Dynamc (а) and shadow (b) boundary layers near the plate surface n the lqud flow.

6 At large veloctes (at the Mach numbers М>0.7) t s necessary to take account of the heat release n the boundary layer due to the knetc energy dsspaton of the gas flow μ ( ) and the work of pressure forces u( p ) P,.e., q v = μ + u. The energy equaton for the boundary layer at large veloctes of flow past the surface wth respect to a stagnaton temperature T0 = T + ( u C p ) s wrtten n the followng form: T 0 T0 T0 1 u ρ uc + p + ρvcp = λ μ 1 (8) Pr In the chemcally reactng boundary layer t s necessary to taken nto consderaton heat transfer by heat dffuson and release (or absorpton) due to chemcal transformatons. The energy equaton n ths case can be presented as: I 0 I0 λ J 0 1 u ρ u + ρv = + μ 1. (9) cp Pr u Here I 0 = I + s the total stagnaton enthalpy, I = ΣC J s the total enthalpy of a mxture, J = + h s the total enthalpy of the -th component ncorporatng the formaton heat of the -th component h due to chemcal transformatons. In these equatons Pr = μc p λ s the Prandtl number, Le = λ ρdcp s the Lews number. From the above-mentoned equatons t follows that at Pr=1 and Le=1 the energy equatons are wrtten n the same form as n the cases of no chemcal reactons. Hence, at small rates the heat transfer process s determned by the statc temperature feld; at large rates, by the temperature feld and n the presence of chemcal reactons, by the total enthalpy feld. 5. Transton from Lamnar to Turbulent Flow In external lqud or gas flow past a body both lamnar and turbulent boundary layers can develop on ts surface. In lamnar flow the lqud partcles proceed quet certan trajectores, all tme keepng ther moton n the drecton of the mean velocty vector. As the lqud flow velocty ncreases, the lamnar flow dsntegrates and there appear velocty, temperature, pressure pulsatons. Separate small volumes of lqud (moles) start movng across the flow and even n the opposte drecton relatve to the averaged moton. The transton from the lamnar flow to the turbulent one occurs not at a pont but over some secton (Fgure ). Flow over ths secton s unstable and s called the transent one characterzed by a perodc change of lamnar and turbulent states. The flow shape n the boundary layer s judged from the value of the crtcal Reynolds numbers Recr = u f xcr ν where х cr s the coordnate along surface and s reckoned from ts leadng edge The value of the crtcal Reynolds number n the general case depends

7 on many factors: the heat transfer ntensty, the mean velocty along the surface, ts roughness, the shape of the body leadng edge, the turbulence degree of the external flow, etc. Fgure Development of the turbulent boundary layer on the flat plate: 1 lamnar flow regon, transent flow regon, 3 turbulent flow regon, 4 vscous sublayer. In the case of lqud or gas flow past a smooth plate at М 0 the lower lmt of the crtcal Reynolds numbers (onset of dsntegraton of the lamnar boundary layer) s 5 Re = u x ν = and the upper lmt of the crtcal Reynolds numbers (steady cr1 f cr1 10 turbulent flow formaton) s Recr = uf xcr ν 10. When the lqud moves n cylndrcal channels, the channel dameter s used as a characterstc sze, therefore, Re cr1 300, and Re cr =10 4. To calculate the crtcal value of the Reynolds number Re cr n the case of flow past turbne and compressor grds the emprcal relaton s proposed: ( )( ) Tw Reсr1 = A + M0 + M Tt.3 Here М 0 s the Mach number at the pont of the mnmum of the pressure on the profle loop, the coeffcent A s determned dependng on the turbulence degree of the external 6 flow. A = for ε 0.1%; A = 0, ε 0,7 for ε = 0,1 1% A = 0,71 10 ε for ε = 1 3%. 6 (10) The length of the transent flow regon Δ l = x cr õ n the zero-pressure gradent flow cr1 uf l Recr can be estmated accordng to the relaton: Re cr = =. ms ν

8 Under the acton of turbulent pulsatons heat and momentum transfer wthn the turbulent flow regme ncreases as compared to the lamnar one. In the steady-state turbulent moton of lqud (or gas) ts mean parameters for some tme nterval reman constant. True or current parameters (velocty, temperature, pressure) of the flow contnuously dffer from ths mean value both n quantty and n drecton. Owng to ths, the current velocty, temperature, pressure can be presented as a sum of the mean values of these quanttes and pulsaton components. Averagng the equatons of energy and moton over tme yelds the Reynolds equaton for the character of the averaged turbulent lqud flow. The averaged Reynolds equatons enable one to present the turbulent flow equatons n the form of the equatons for lamnar flow n whch by thermal conductvty and vscosty coeffcent s understood a sum of molecular and turbulent components of vscosty and heat conducton,.e., λσ = μ + μt ; μσ = μ + μt. For the turbulent thermal conductvty λ Т and the vscosty coeffcent μ Т to be determned, both algebrac and one-and two-parameter dfferental sem-emprcal turbulence models can be adopted. These problems are detaled n (Avduevsky, V.S. (199)) Bblography TO ACCESS ALL THE 4 PAGES OF THIS CHAPTER, Vst: Cebec T, Bradshaw P. (1984) Physcal and Computatonal Aspects of Convectve Heat Transfer. Sprnger Verhag. New York, Berln, Tokyo. 590p. Eckert E.R.G. Drake R.M. (1977). Analyss of Heat and Mass Transfer McGraw New-York. Fundamentals of heat transfer n avaton and rocket-space technologes. / Avduevsky, V.S., Galtseysky, B.M., Globov, G.A., et al., Eds. V.S. Avduevsky and V.K. Koshkn. (199) Handbook. Second edton. Mashnostroyenye, Moscow, 58 p. Galtseysky B.M., Sovershenny V.D., Formalev V.E., Cherny M.S. (1996). Thermal protecton of turbne blades. Moscow, MAI. 356p. Bographcal Sketch B.M. Galtseysky: Professor of the Department of Avaton-Space Thermal Technques of Moscow Avaton Insttute MAI. 4. Volokolamskoe shosse, Moscow 15993, Russa. Date of brth Engneer (Moscow Avaton Insttute) Doctor of Phlosophy (Moscow Avaton Insttute) Doctor of Techncal Scences (Moscow Avaton Insttute) Ttled Professor (Moscow Avaton Insttute) , Russan Federaton State Prze Laureate 1990, MAI Prze Laureate 1989, 000, Honored Scentst of Russan Federaton 1998 He s a well-known specalst n the heat-mass transfer and space thermo technques. He conducted the nvestgatons of a heat transfer n the oscllatng flows, porous systems, jet systems, system coolng of power plant. He s specalzed n effectve methods of the heat transfer ntensfcaton and hghly productve methods of systems coolng calculaton.

9 He s author of more then 50 publshed works, concludng monograph: Heat Transfer n the Power Installatons of Spacecrafts (1975), Heat and Hydrodynamc Processes n Oscllatng Flows (1977). Heat transfer n avaton engnes (1985). Fundamentals of Heat Transfer n Avaton and Rocket-space Techncs (199, n cooperaton wth other authors). Thermal protecton of turbne blade (1996).