A SHORT INTRODUCTION TO TWO-PHASE FLOWS Condensation and boiling - PowerPoint PPT Presentation

A SHORT INTRODUCTION TO TWO-PHASE FLOWS Condensation and boiling heat transfer Herv´ e Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40, herve.lemonnier@cea.fr herve.lemonnier.sci.free.fr/TPF/TPF.htm ECP, 2011-2012

HEAT TRANSFER MECHANISMS • Condensation heat transfer: – drop condensation – film condensation • Boiling heat transfer: – Pool boiling, natural convection, ´ ebullition en vase – Convective boiling, forced convection, • Only for pure fluids. For mixtures see specific studies. Usually in a mixture, h � � x i h i and possibly ≪ h i . • Many definitions of heat transfer coefficient, q Nu = hL h [W/m 2 /K] = ∆ T , k , k ( T ?) Condensation and boiling heat transfer 1/42

CONDENSATION OF PURE VAPOR • Flow patterns – Liquid film flowing. – Drops, static, hydrophobic wall ( θ ≈ π ). Clean wall, better htc. • Fluid mixture non-condensible gases: – Incondensible accumulation at cold places. – Diffusion resistance. – Heat transfer deteriorates. – Traces may alter significantly h Condensation and boiling heat transfer 2/42

FILM CONDENSATION • Thermodynamic equilibrium at the interface, T i = T sat ( p ∞ ) • Local heat transfer coefficient, q q h ( z ) � = T i − T p T sat − T p • Averaged heat transfer coefficient, � L h ( L ) � 1 h ( z )d z L 0 • NB: Binary mixtures T i ( x α , p ) and p α ( x α , p ). Approximate equilibrium condi- tions, – For non condensible gases in vapor, p V = xP sat ( T i ), Raoult relation – For dissolved gases in water, p G = Hx G , Henry’s relation Condensation and boiling heat transfer 3/42

CONTROLLING MECHANISMS • Slow film, little convective effect, conduction through the film (main thermal resistance) • Heat transfer controlled by film characteristics, thickness, waves, turbulence. • Heat transfer regimes, Γ � M L Re F � 4Γ P , µ L – Smooth, laminar, Re F < 30, – Wavy laminar, 30 < Re F < 1600 – Wavy turbulent, Re F > 1600 Condensation and boiling heat transfer 4/42

CONDENSATION OF SATURATED STEAM • Simplest situation, only a single heat source: interface, stagnant vapor, • Laminar film (Nusselt, 1916, Rohsenow, 1956), correction 10 to 15%, � 1 � k 3 L ρ L g ( ρ L − ρ V )( h LV +0 , 68 C P L [ T sat − T P ]) 4 h ( z ) = 4 µ L ( T sat − T P ) z • Averaged heat transfer coefficient ( T W = cst) : h ( z ) ∝ z − 1 4 , h ( L ) = 4 3 h ( L ) • Condensate film flow rate, energy balance at the interface, Γ( L ) = h ( L )( T sat − T P ) L h LV • Heat transfer coefficient-flow rate relation, � 1 � ¯ µ 2 h ( L ) 3 − 1 L = 1 , 47 Re 3 F k L ρ L ( ρ L − ρ V ) • h LV and ρ V at saturation. k L , ρ L at the film temperature T F � 1 2 ( T W + T i ), • µ = 1 4 (3 µ L ( T P ) + µ L ( T i )), exact when 1 /µ L linear with T . Condensation and boiling heat transfer 5/42

SUPERHEATED VAPOR • Two heat sources: vapor ( T V > T i ) and interface. • Increase of heat transfer wrt to saturated conditions, empirical correction, � 1 � 1 + C P V ( T V − T sat ) 4 h S ( L ) = ¯ ¯ h ( L ) h LV • Energy balance at the interface, film flow rate, ¯ h S ( L )( T W − T sat ) L Γ( L ) = h LV + C P V ( T V − T sat ) Condensation and boiling heat transfer 6/42

FILM FLOW RATE-HEAT TRANSFER COEFFICIENT • Laminar, � 1 � ¯ µ 2 h ( L ) 3 − 1 L = 1 , 47 Re 3 F k L ρ L ( ρ L − ρ V ) • Wavy laminar and previous regime (Kutateladze, 1963), h ( z ) ∝ Re − 0 , 22 ), F � 1 � ¯ µ 2 h ( L ) Re F 3 L = 1 , 08Re 1 , 22 k L ρ L ( ρ L − ρ V ) − 5 , 2 F • Turbulent and previous regimes (Labuntsov, 1975), h ( z ) ∝ Re 0 , 25 , F � 1 � ¯ µ 2 h ( L ) Re F 3 L = 8750 + 58Pr − 0 , 5 (Re 0 , 75 k L ρ L ( ρ L − ρ V ) − 253) F F • NB: Implicit relation, Re F depends on h ( L ) through Γ. Condensation and boiling heat transfer 7/42

OTHER MISCELLANEOUS EFFECTS • Steam velocity, v V , when dominant effect, • V v descending flow, vapor shear added to gravity, • Decreases fil thickness, • Delays transition to turbulence turbulence, 1 h ∝ τ 2 i • See for example Delhaye (2008, Ch. 9, p. 370) • When 2 effects are comparable, h 1 stagnant, h 2 with dominant shear , 1 h = ( h 2 1 + h 2 2 ) 2 Condensation and boiling heat transfer 8/42

CONDENSATION ON HORIZONTAL TUBES • Heat transfer coefficient definition, � π h = 1 ¯ h ( u )d u π 0 • Stagnant vapor conditions, laminar film, Nusselt (1916) � 1 � k 3 0 . 728 L ρ L ( ρ L − ρ V ) gh LV 4 ¯ h = µ L ( T sat − T p ) D (0 . 70) • 0.728, imposed temperature, 0.70, im- posed heat flux. • Γ, film flow rate per unit length of tube. Condensation and boiling heat transfer 9/42

• Film flow rate- heat transfer coefficient, energy balance, � 1 � ¯ µ 2 1 . 51 h 3 − 1 L = Re 3 F k L ρ L ( ρ L − ρ V ) (1 . 47) • Vapor superheat and transport proprieties, same as vertical wall • Effect of steam velocity (Fujii), � u 2 � 0 . 05 ¯ ¯ h V ( T sat − T P ) k L h = 1 . 4 1 < < 1 . 7 , h 0 gDh LV µ L h 0 • Tube number effect in bundles, (Kern, 1958), h (1 , N ) = N − 1 / 6 h 1 Condensation and boiling heat transfer 10/42

DROP CONDENSATION • Mechanisms, – Nucleation at the wall, – Drop growth, – Coalescence, – Dripping down (non wetting wall) • Technological perspective, – Wall doping or coating – Clean walls required, fragile – Surface energy gradient walls. Self- draining Condensation and boiling heat transfer 11/42

• heat transfer coefficient, h = 1 1 + 1 + 1 + 1 h G h d h i h co • G : non-condensible gas, d : drop, i : phase change, co coating thickness. • Non-condensible gases effect, ω i ≈ 0 , 02 ⇒ h → h/ 5 • Example, steam on copper, T sat > 22 o C, h in W/cm 2 / o C, h d = min(0 , 5 + 0 , 2 T sat , 25) Condensation and boiling heat transfer 12/42

� � � POOL BOILING • Nukiyama (1934) • Only one heat sink, stagnant saturated water, • Wire NiCr and Pt, – Diameter: ≈ 50 µ m, – Length: l – Imposed power heating: P � � � Condensation and boiling heat transfer 13/42

� � BOILING CURVE • Imposed heat flux, � � � � � � � � � P = qπDl = UI • Wall and wire temperature are equal, � � � D → 0 R ( T ) = U I , < | T> | 3 ≈ T W • Wall super-heat: ∆ T = T W − T sat • Heat transfer coefficient, q � � � � � � � � � � h � � � � � � T W − T sat Condensation and boiling heat transfer 14/42

� � � � � � �� � � BOILING CURVE � � � � � � � � � � � � � � � � � � � � � � http://www-heat.uta.edu , Next Condensation and boiling heat transfer 15/42

� � � � � � HEAT TRANSFER REGIMES � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • OA: Natural convection • DH: Transition boiling • AD: Nucleate boiling • HG: Film boiling Condensation and boiling heat transfer 16/42

� � � � � � TRANSITION BOILING STABILITY • Wire energy balance, � � � � � � � � � � � � � � � � � d T � � � � � � � � � � � � � � � � MC v d t = P − qS � � � � � � • Linearize at ∆ T 0 , q 0 , T = T 0 + T 1 , � � � � � � � � � � � � d T 1 − S ∂q � � � � � � � � � MC v d t = P − q 0 S ∂ ∆ T T 1 � �� � � � � � � =0 • Solution, linear ODE, � ∂q � S T 1 = T 10 exp( − αt ) , α = MC v ∂ ∆ T T 0 • 2 stable solutions, one unstable (DH), ∂q ∂ ∆ T < 0 • Transition boiling, imposed temperature experiments (Drew et M¨ uller, 1937). Condensation and boiling heat transfer 17/42

NATURAL CONVECTION • Wire diameter D , natural convection Nu = hD q = h ( T F − T sat ) , k Ra = gβ ( T F − T sat ) D 3 Pr = ν L , α L ν L α L • Nusselt number is the non-dimensional heat transfer coefficient ( h ). • k L , α L , ν L at the film temperature 1 2 ( T F + T sat ), β ` a T sat . • Churchill & Chu (1975), 10 − 5 < Ra < 10 12 ,   2 0 , 387 Ra 1 / 6   Nu =  0 , 60 +  � 9 / 16 � 8 / 27 � � 0 , 559 1 + Pr Condensation and boiling heat transfer 18/42

NATURAL CONVECTION ON A FLAT PLATE • Scales A , P , plate area and perimeter. Length scale, L = A P . Ra = gβ ( T P − T ∞ ) L 3 Nu = hL qL k = k L ( T P − T ∞ ) , ν L α L • Two regimes,  0 , 560 Ra 1 / 4  1 < Ra < 10 7    � 1 + (0 , 492Pr) 9 / 16 � 4 / 9 Nu = � 1 + 0 , 0107Pr �  0 , 14 Ra 1 / 3 Ra < 2 10 11  0 , 024 � Pr � 2000 ,   1 + 0 , 01Pr • Thermodynamic and transport properties Raithby & Hollands (1998). For liquids: all at T F = 1 2 ( T P + T ∞ ) Condensation and boiling heat transfer 19/42