heat transfer process where a liquid undergoes a phase change into a vapor (gas)

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1 Two-Phase: Overview Two-Phase two-phase heat transfer describes phenomena where a change of phase (liquid/gas) occurs during and/or due to the heat transfer process two-phase heat transfer generally considers processes that occur at a solid/fluid interface and are therefore a sub-field of convection because of the change of phase, the latent heat (h fg ) of the fluid must be considered the surface tension (σ) is another parameter that plays an important role Boiling heat transfer process where a liquid undergoes a phase change into a vapor (gas) Condensation heat transfer process where a vapor (gas) liquid undergoes a phase change into a liquid

2 Boiling: Overview Boiling associated with transformation of liquid to vapor (phase change) at a solid/liquid interface due to convection heat transfer from the solid agitation of the fluid by buoyant vapor bubbles provides for large convection coefficients è large heat fluxes at low-to-moderate surface-to-fluid temperature differences Modified Newton s Law of Cooling q " s = h( T s T sat ) = hδt e T s surface temperature T sat saturation temperature of liquid ΔT e ( T s T sat ) excess temperature

3 Boiling: Overview Flow Cases Pool Boiling liquid motion is due to natural convection and bubble-induced mixing Forced Convection Boiling (Flow Boiling/2-Phase Flow) liquid motion is induced by external means and there is also bubble-induced mixing Temperature Cases Saturated Boiling liquid temperature is slightly higher than saturation temperature Subcooled Boiling liquid temperature is less than saturation temperature

4 Boiling: The Boiling Curve Boiling Curve identifies different regimes during saturated pool boiling inflection point nucleate boiling transition boiling film boiling free convection Leidenfrost point ΔT e ( T s T sat ) Water at Atmospheric Pressure

5 Boiling: Boiling Curve Free Convection Boiling (ΔT e < 5 C) little vapor formation liquid motion is primarily due to buoyancy effects Nucleate Boiling (5 C < ΔT e < 30 C) onset of nucleate boiling ΔT e ~ 5 C (ONB) isolated vapor bubbles (5 C < ΔT e < 10 C) liquid motion is strongly influenced by nucleation of bubbles on surface h and q s increase sharply with increasing ΔT e heat transfer is primarily due to contact of liquid with the surface (single-phase conduction) and not to vaporization jets and columns (10 C < ΔT e < 30 C) increasing number of nucleation sites causes bubble interactions and coalescence into jets and slugs liquid/surface contact is impaired by presence of vapor columns q s increases with increasing ΔT e h decreases with increasing ΔT e

6 Boiling: Boiling Curve Nucleate Boiling (5 C < ΔT e < 30 C) critical heat flux (CHF) (ΔT e ~ 30 C) maximum attainable heat flux in nucleate boiling water at atmospheric pressure CHF ~ MW/m 2 h max ~ W/m 2 -K Transition (30 C < ΔT e < 120 C) & Film Boiling (ΔT e > 120 C) heat transfer is by conduction and radiation across the vapor blanket liquid/surface contact is impaired by presence of vapor columns q s decreases with increasing ΔT e until the Leidenfrost point corresponding to the minimum heat flux for film boiling and then proceeds to increase a reduction in the surface heat flux below the minimum heat flux results in a abrupt reduction in surface temperature to the nucleate boiling regime Heat flux controlled heating: burnout potential if the heat flux at the surface is controlled it can potentially increase beyond the CHF this causes the surface to be blanketed by vapor and the surface temperature can spontaneously achieve a value that potentially exceeds its melting point (ΔT e > 1000 C) if the surface survives the temperature shock, conditions are characterized as film boiling

7 Boiling: Pool Boiling Correlations Due to complexity of fluid mechanics and phase-change thermodynamics, boiling heat transfer correlations are empirical Rohsenow Correlation: Nucleate Boiling note: can be as much as 100% inaccurate! ( ) σ & g ρ q " s = µ l h l ρ v fg ( ' Critical Heat Flux ) + * 1 2 & c p,l ) ( n + ' C s, f h fg Pr l * 3 ( ) 3 subscripts: ΔT e µ l viscosity; c p,l specific heat; g acceleration due to gravity σ surface tension; h fg latent heat of vaporization; ρ density l è saturated liquid state v è saturated vapor state ( ) ρ v 2 & σg ρ q " max = 0.149h fg ρ l ρ v v ( ' ) + * 1 4 correction factor required for surfaces with small characteristic lengths

8 Boiling: Pool Boiling Correlations Rohsenow Correlation ( ) σ & g ρ q " s = µ l h l ρ v fg ( ' ) + * 1 2 & c p,l ) ( n + ' C s, f h fg Pr l * 3 ( ) 3 ΔT e

9 Boiling: Pool Boiling Correlations Minimum Heat Flux Film Boiling correlation for spheres & cylinders NuD = h convd k v & σg ρ q " min = 0.09h fg ρ l ρ v v ( '( ( ρ l + ρ v ) 2 ( ) = C g ( ρ l ρ v ) h $ fg D 3 1 & ) cylinder ( + C =. ' ν v k v ( T s T sat ) * / 0.67 sphere ) + * Leidenfrost point reduced latent heat h " fg = h fg c p,v T s T sat ( ) total average heat transfer coefficient due to cumulative & coupled effects of convection (due to boiling) and radiation across the vapor layer h = h 3 conv + h rad h rad = εσ T 4 4 s T sat T s T sat ( )( h 1 3 ) h = h conv h rad h conv > h rad ( ) σ Stefan - Boltzmann constant

10 Condensation: Overview Condensation occurs when the surface temperature is less than the saturation temperature of an adjoining vapor heat is transferred from vapor the surface to the surface Film Condensation entire surface is covered by the condensate which flows continuously from the surface and presents a thermal resistance to heat transfer from the vapor to the surface typically due to clean, uncontaminated surfaces can be reduced by using short vertical surfaces & horizontal cylinders Dropwise Condensation surface is covered by drops ranging from a micron to large agglomerations thermal resistance is lower than that of film condensation surface coatings may inhibit wetting and stimulate dropwise condensation

11 Condensation: Film Condensation Vertical Plate thickness δ and flow rate m of condensate increase with increasing x generally, the vapor is superheated (T v, >T sat ) and may be part of a mixture that contains noncondensibles a shear stress at the liquid/vapor interface induces a velocity gradient in the vapor as well as the liquid Laminar Flow Analysis assume pure vapor assume negligible shear stress at liquid/vapor interface u = 0 y y=δ negligible advection in the film

12 Condensation: Film Condensation Vertical Plate: Laminar Flow Analysis film thickness % δ( x) = 4k lµ l ( T sat T s )x( ' * & gρ l ( ρ l ρ v )h fg ) flow rate per unit width Γ m b = gρ l condensation rate 1 4 ( ρ l ρ v )δ 3 3µ l average Nusselt number modified latent heat NuL = h LL = gρ l ( ρ l ρ v ) h $ fg L 3 1 % ( 4 h " fg = h fg ( Ja) ' * k l & k l µ l ( T sat T s ) ) Jakob number heat transfer rate Ja = c T T p ( s sat) q = h L A s ( T sat T s ) h fg m = q " h fg

13 Condensation: Film Condensation Vertical Plate: Turbulence transition may occur in the film and three flow regimes may be delineated Re δ = 4Γ = 4m µ l µ l b = 4ρ u l mδ wave-free laminar region (Re δ <30) h L ( ν 2 l g) =1.47Re δ k l µ l Re δ = 4gρ l ( ρ l ρ v )δ 3 3µ l 2 wavy laminar region (30<Re δ <1800) h L ( ν 2 l g) 1 3 Re = δ k l 1.08Re 1.22 δ 5.2 turbulent region (Re δ >1800) h L ( ν 2 l g) 1 3 = k l Re δ Pr 0.5 Re δ ( )

14 Condensation: Film Condensation Vertical Plate: Calculation Procedure assume a flow regime and use the corresponding equation for to determine Re δ if Re δ value is consistent with flow regime assumption, calculate total heat rate and mass flow rate if Re δ value is inconsistent with flow regime assumption, iterate on flow regime assumption until it is consistent h L

15 Condensation: Film Condensation Radial Systems: Single Tubes/Spheres ( ) $ % h D = C gρ 3 l ρ l ρ v h fg k l ( ' * & µ l ( T sat T s )D ) 1 4 Tube: C =0.729 Sphere: C=0.826

16 Condensation: Film Condensation Radial Systems: Vapor Flow in a Horizontal Tube if vapor flow rate is low, condensation in both circumferential and axial directions # Re v,i = ρ u D & v m,v % ( $ ' µ v ( ) $ % h D = gρ 3 l ρ l ρ v h fg k l ( ' * & µ l ( T sat T s )D ) i < 35,000 h " fg h fg ( T sat T s ) 1 4 for high flow rates, flow is two-phase annular flow

17 Condensation: Dropwise Condensation Dropwise Condensation heat transfer rates ~order of magnitude greater than film condensation heat transfer coefficients highly dependant on surface properties Steam on copper with surface coating h dc = T sat W m 2 K h dc = [ W m 2 K] T sat >100 C [ ] 22 C < T sat <100 C