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MHMT11 Momentum Heat Mass Transfer D source Dt Heat transferconvection Thermal boundary layer. Forced convection (pipe, plate, sphere). Natural convection. Rohsenow W.M., Hartnett J.P., Cho Y.I.: Handbook of Heat Transfer. McGraw Hill, 3rd Edition, 1998 Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010 MHMT11 Heat transfer - convection Heat flux from a surface to fluid is evaluated using heat transfer coefficient , assuming linear relationship between heat flux and driving force, which is the temperature difference between solid surface and temperature of fluid far from surface (outside a thermal boundary layer) q (Tw T f ) Heat transfer coefficient is related to thickness of thermal boundary layer T y T T Tw x MHMT11 Thermal boundary layer Integral boundary layer theorem can be derived in the same way like for the momentum boundary layer H H 2 T T T Upper bound H ( u u ) dy a 0 x x y y 0 y 2 dy H ( 0 H 0 H 0 H 0 u xT u u yT u T T x T x )dy a[ ] yy 0H x x y x y u xT T dy [u yT ] yy 0H a | y 0 x y u xT dy T x H 0 u x T dy a | y 0 x y T (u x (T T ))dy a | y 0 x y d dx H ux (T T )dy ux (T T ) | H 0 d H T a | y 0 dx y this is zero at boundary MHMT11 Thermal boundary layer Final form of integral boundary equation u x (T T ) qw d [(Tw ( x) T )U ( x) dy ] dx U ( x)(Tw ( x) T ) c p 0 T thermal energy thickness qw heat flux This equation can be applied as soon as the momentum boundary layer was identified (velocity profile inside the thermal boundary layer must be known). This equation is used for prediction of heat transfer at flows along plates, pipes, cylinder, spheres… However, in the following slides we shall analyze heat transfer in a pipe and along a plate in a simpler way, using the penetration theory (instead of integral theorem). The penetration theory predicts not so accurate but still qualitatively correct results. MHMT11 Heat transfer – pipe Forced convection: Reynolds number is given, Nusselt number is to be calculated. Nu 3 Re Pr D / L Duchamp MHMT11 Heat transfer – pipe (laminar) Convective heat transfer in a pipe for the laminar flow and fully developed velocity profile (therefore it is not necessary to solve the problem of penetration momentum boundary layer). theory Velocity profile u( y) u 8y D 2 4ax axD u ( ) 2u axD 3 2u parabolic velocity profile D slope of the velocity profile behind the boundary layer is the inlet temperature T0 Re Pr Re Pr Pe Leveque formula Tw x uD 2 uD D ax a x Gz Graetz number T0 y D D 3 2uD 2 Nu ax 2 D uD Nux c3 c 3 Gz x ax Remark: This correlation predicts local value of (x), which is stressed by lower index in Nusselt and Graetz number MHMT11 Heat transfer – pipe (laminar) Anyway, qualitatively the same result can be obtained using integral theorem qw Tw T0 d T u ( T T ) dy a x 0 dx 0 cp T assuming linear temperature and velocity profile in the thermal boundary layer y 8y T Tw (Tw T0 ) u( y) u T D Tw T0 d T 8y y u ( T T )(1 ) dy a w 0 dx 0 D T T d 8 T2 aD ( ) dx 6 T u T3 9 aDx 8 u …this results differs only by a constant (9/8) from the previous result (1/2) but this constant is anyway usually modified using experiments or more accurate assumptions. MHMT11 Heat transfer – pipe (laminar) Local value (x) increases to infinity with decreasing distance from the tube inlet. From practical point of view a more important is the average value of at pipe of the length L . 1 c 3 D 3c 3 D 3 ln dx Re Pr dx Re Pr 1.6 Gz L0 D L 0 x D 2 L D L L index ln is used because this ln is related to the mean logarithmic temperature difference used in the heat exchanger design Correlations used in practice for heat transfer prediction in laminar flows in pipes ln D Léveque formula for Gz>50 (short pipes) 1.67 3 Gz D 0.0668Gz Hausen formula for arbitrary long pipes Nu ln 3.66 2/3 1 0.04Gz Nu Note the fact that at very long tubes the Nu (and ) is constant 3.66, and that the Hausen correlation reduces to Leveque for Gz MHMT11 Heat transfer – pipe (turbulent) Turbulent flow is characterised by the energy transport by turbulent eddies which is more intensive than the molecular transport in laminar flows. Heat transfer coefficient and the Nusselt number is therefore greater in turbulent flows. Basic differences between laminar and turbulent flows are: Nu is proportional to 3 0.8 u in laminar flow, and u in turbulent flow. Nu doesn’t depend upon the length of pipe in turbulent flows significantly (unlike the case of laminar flows characterized by rapid decrease of Nu with the length L) Nu doesn’t depend upon the shape of cross section in the turbulent flow regime (it is possible to use the same correlations for elliptical, rectangular…cross sections using the concept of equivalent diameter – this cannot be done in laminar flows) The simplest correlation for hydraulically smooth pipe designed by Dittus Boelter is frequently used (and should be memorized) Nu 0.023Re0.8 Pr m m=0.4 for heating m=0.3 for cooling MHMT11 Heat transfer - plate Analysis of heat transfer at external flows (around a plate, cylinder, sphere…) differs from internal flows (for example heat transfer in pipe) by the fact that velocity profile at surface is not known in advance and therefore not only the thermal boundary layer but also the momentum boundary must be solved Nux Re x 3 Pr Duchamp MHMT11 Heat transfer - plate Heat transfer in parallel flow along a plate is characterised by simultaneous development of thermal and momentum boundary layers. It will be assumed, that the thickness of momentum boundary layer H is greater than thermal boundary layer T y x (this is previously obtained Linear velocity profile u ( y ) U u ( T ) U T result for thickness of the H 4.6 H H U momentum boundary layer) Thermal boundary layer thickness H T T2 T Tw x Nu 4ax 4ax 4ax x H 4.6 u (T ) U T U T U x U H 4.6 y T x loca value of heat transfer 4ax u ( T) x T (18.4a)1/3 ( )1/6 U Note the fact, that the ratio of thermal and momentum boundary layer thickness is independent of x and velocity U ( xU )1/2 1/6 ( ) T (18.4a)1/3 x 1 18.41/3 this constant should be replaced by 0.332 in accurate solution ( xU Re )1/2 ( 1/3 ) a Pr MHMT11 Heat transfer - plate Mean value of heat transfer is obtained from the previous formula by integration along the length of plate L H y x U T H / Pr1/3 x Laminar flow regime L Nu 0.664 Re L Pr1/3 Turbulent flow regime (Re> 500000) 1/3 Nu 0.0365Re0.8 Pr L Compare these results with the heat transfer in pipe Pipe: Laminar flow NuRe1/3 turbulent flow NuRe0.8 Plate: Laminar flow NuRe1/2 turbulent flow NuRe0.8 MHMT11 Heat transfer – sphere… Flow around a sphere (Whitaker) Front side Important for heat transfer from droplets… Rear side (wake) See next slide Nu 2 (0.4 Re1/2 0.06 Re2/3 ) Pr 0,4 0,71Pr380 3,5Re7,6.104 . Cross flow around a cylinder (Sparrow 2004) Important for shell&tube and fin-tube heat exchangers Nu 0.25 (0.4 Re1/2 0.06 Re2/3 ) Pr 0.37 0,67Pr300 1Re105 . Cross flow around a plate (Sparrow 2004) Nu front 0.6 Re Nurear 0.17 Re 2/3 Heat transfer – sphere MHMT11 Conduction outside a sphere, see 1D solution of temperature profile Tw D 1 d 2 dT (r )0 2 r dr dr T r T (r ) T D (T Tw ) 2r q (T Tw ) T 2 |r D /2 (T Tw ) r D D Nu 2 MHMT11 Natural convection Velocity and Reynolds number is not known in advance. Flow is induced only by buoyancy. Instead of Reynolds it is necessary to use the Rayleigh number. Duchamp MHMT11 Natural convection Vertical wall. Wall temperature Tw>Tf. Heated fluid flows upward along the plate due to buoyant forces. Forces equilibrium (wall shear stress=buoyant force) u m g T w g um Penetration depth at distance x x 4ax 2 4a um 2 um x Substituting um into force balance 4ax 4 ax 3 g T 4 y g T x x g Tx3 g Tx3 Nu 4 c4 c 4 Ra 4 a a where Ra is Rayleigh number (x represents a characteristic dimension in the direction of gravity, e.g. height of wall) MHMT11 Natural convection Turbulent flow regime occurs at very high Ra>109 and instead the 4th root of Ra the 3rd root is used in correlations Nu cturb 3 Ra Note the fact, that at turbulent flow regime the heat transfer coefficient is almost independent of x (size of wall) x g Tx3 g T 3 3 cturb cturb a a MHMT11 EXAM Forced convection Natural convection MHMT11 What is important (at least for exam) What is it heat transfer coefficient and how is related to thermal boundary layer thickness Forced convection Nu(Re,Pr) q (Tw T f ) T ln D 1.6 3 Gz 1.6 3 Re Pr D / L Heat transfer in a pipe (laminar) Nu Heat transfer a plate (laminar) Nu 0.664 Re L Pr1/3 Heat transfer turbulent Nu Re0.8 Pr1/3 MHMT11 What is important (at least for exam) Free convection Nu(Ra) Laminar flow (Re<1010) Turbulent flow x g Tx3 Nu c4 c 4 Ra a x g Tx3 Nu ct 3 ct 3 Ra a