Nucleate Boiling Chapter 10: Boiling 10.3 Nucleate Boiling
10.3 Nucleate Boiling Chapter 10: Boiling 10.3 Nucleate Boiling
Three stages of bubble production, each of which will be discussed in the following subsections, are defined as follows: (1) initiation or nucleation, (2) growth, (3) detachment The section then closes with a summary of modeling approaches that have been developed to describe heat transfer in nucleate boiling.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 1 10.3 Nucleate Boiling Chapter 10: Boiling 10.3.1 Nucleation Nucleation, or bubble initiation in typical industrial applications is characterized by the cyclic formation of vapor bubbles at preferred sites on the solid heating surface of the system.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 2 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.4 Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity (Wang and Dhir, 1993a).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 3 10.3 Nucleate Boiling Chapter 10: Boiling Young’s equation σ + − = 2 (10.1) pg p v pl Rb Vapor pressure for a curved bubble interface − 2σ ρ p= p( T )exp v v v, sat ρ (10.2) pv, sat() T R b l Eq. (10.2) can be approximated as 2σ ρ pB p( T ) 1 − v v v, sat ρ (10.3) pv, sat() T R b l Combining eqs. (10.1) and (10.3) yields 2σ ρ (10.4) p+ p( T ) − p B 1 + v g v, sat l R ρ b l Clapeyron equation h dp = lv (10.5) ( ρ− ρ ) dT T 1/v 1/ l
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 4 10.3 Nucleate Boiling Chapter 10: Boiling Ideal gas law = ρ p Rg T (10.6) ρl>>ρv h dp = lv 2 dT (10.7) p Rg T Integrating eq. (10.7) p h T− T ln ∞ = lv sat (10.8) pl Rg T sat T Substituting eq. (10.4) into eq. (10.8) R T T2σ ρ p ∆TTT = − =g satln 1 + 1 +v − g (10.9) sat ρ hlv p l R b l p l Eq. (10.9) can be further simplified RTT σ ∆ = − =g sat 2 − (10.10) T T Tsat p g pl h lv R b
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 5 10.3 Nucleate Boiling Chapter 10: Boiling Rearranging eq. (10.10) 2σ T R = sat (10.11) b ρ ∆ hlv v T Nucleation occurs beyond the point where the curvature reaches its maximum; the corresponding superheat is 4σ T ∆ = − = sat TTTKsat max pv hl v D c (10.12) The average liquid film thickness in terms of bubble lift
time t0 and liquid kinematic viscosity by the following equation 8 δ= (3 ν t )1/ 2 7 l 0 (10.13)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 6 10.3 Nucleate Boiling Chapter 10: Boiling Example 10.1 A steam bubble with a radius of 5 μm is surrounded by liquid water at 120 ˚C. Will this bubble grow or collapse?
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 7 10.3 Nucleate Boiling Chapter 10: Boiling Solution: 5 The liquid pressure is p l = 1 atm = 1.013x10 Pa. The
saturation temperature at this pressure is Tsat = 100°C = 373.15 K. The properties of water at this temperature are -3 h and σ = 58.9x10 N/m, l v = 2251.2 kJ/kg, and σv = 0.5974 kg/m3. Therefore, the critical bubble radius can be determined2σ T from eq. (10.11), i.e., R = sat b ρ ∆ hlv v T − 3 2× 58.9 × 10 × 373.15 − = =1.63 × 106 m = 1.63 μm 2251.2× 103 × 0.5974 × (120 − 100) Since the radius of the bubble, 5 μm, is greater than the critical bubble radius, the bubble will grow.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 8 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.5 Variation of bubble radius as the bubble grows in and emerges from a cavity.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 9 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.6 Liquid microlayer under a vapor bubble at a nucleation site.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 10 10.3 Nucleate Boiling Chapter 10: Boiling 10.3.2 Bubble Dynamics and Detachment Several types of forces influence bubble growth, including the inertia of the surrounding liquid, shear forces at the interface, surface tension, and the pressure difference between the vapor and the liquid. The growth of a spherical bubble in an extensive and uniformly-superheated liquid is considered first.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 11 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.7 Vapor bubble in superheated liquid.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 12 10.3 Nucleate Boiling Chapter 10: Boiling Incompressible, radially-symmetric invscid flow, the continuity equation is 1∂ (r2 u ) = 0 r2 ∂ r (10.14) At the liquid-vapor interface, velocity is equal to growth dR rate u(,) R t = dt (10.15) Integrating eq. (10.14) over the interval (R,r) 2 dR R (10.16) u(,) r t = dt r Momentum equation for the surrounding liquid ∂u ∂ u ∂ p (10.17) ρ +u = − l ∂t ∂ r ∂ r Substituting eq. (10.16) into eq. (10.17) ρdR 2 d2 R 2 ρ R4 dR 2 ∂ p (10.18) l+2 − l = − 22RR 2 5 r dt dt r dt ∂ r
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 13 10.3 Nucleate Boiling Chapter 10: Boiling Integrating eq. (10.18) from the bubble surface (r=R) to infinity d2 R3 dR 2 1 R+ =[ p()() R − p ∞ ] (10.19) 2 ρ l l dt2 dt l Substituting eq. (10.1) into eq. (10.19) 2 d2 R3 dR 1 2σ (10.20) R+ = p − p () ∞ − 2 ρ v l dt2 dt l R The pressure difference can be evaluated using the Clausius- Clapeyron equation ρ h[ T− T[ p ( ∞ )]] p− p () ∞ = vl v∞ sat l (10.21) v l ∞ Tsat [ pl ( )] Substituting eq. (10.21) into eq. (10.20) and neglecting the surface tension in eq. (10.20) 2 2 d R3 dR ρ h T∞ − T[ p ( ∞ )] (10.22) R + = vl v sat l 2 ρ ∞ dt2 dt l Tsat [ p l ( )] Subject to the initial condition R=0, t = 0 (10.23)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 14 10.3 Nucleate Boiling Chapter 10: Boiling The instantaneous bubble radius can be obtained by 1 2 ρ h T− T[ p ( ∞ )] 2 (10.24) R() t= vl v∞ sat l t ρ ∞ 3lTsat [ p l ( )] ∂ ∂α ∂ ∂ TTTl l l2 l (10.25) +u = r ∂t ∂ rr 2 ∂ r ∂ r Initial and boundary conditions for eq. (10.25) = ∞ = ∞ = ∞ Tr( ,0) Tl ( ), TRtTpTtT ( , )sat ( v ), ( , ) l ( ) (10.26) Energy balance at the liquid-vapor interface ∂ T dR k= ρ h (10.27) l∂ v l v rr= R dt Instantaneous bubble radius = α (10.28) R( t ) 2 CR l t
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 15 10.3 Nucleate Boiling Chapter 10: Boiling
For large Jakob, CR can be obtained 3 C= Ja (10.29) R π Jakob number ρ c[ T()()∞ − T p ] Ja = lp l l sat v (10.30) ρ vhl v For a small Jakob number (corresponding to high pressure) = Ja CR 2 (10.31) Relationship between bubble radius and time 3 3 +2 + + = +2 − 2 − R( t 1) ( t ) 1 (10.32) 3 Where R+ and t+ are nondimensional radius and time 2 +RA + tA (10.33) R= t = BB2 2
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 16 1 10.3 Nucleate Boiling Chapter 10: Boiling b[ T(∞ ) − T ( p ( ∞ ))] h ρ 2 A = lsat l l v v ρ ∞ lTsat ( p l ( )) 1 (10.34) 12α 2 B= l Ja π (10.35) ρ c[ T (∞ ) − T ( p ( ∞ ))] Ja = lp l l sat l (10.36) ρ vhl v For saturation conditions, the maximum and minimum cavity sizes that can be active under a constant temperature are 1/ 2 δ 12σ T (10.37) (R )= 1 ± 1 − sat c max,min δ ρ − 3vhl v ( T w T sat ) Criterion for vapor bubble growth 4σ T TTR− >sat , > δ (10.38) w satρ δ c vhl v Temperature required for bubble growth 2σ T 1 (10.39) TTR− >sat , < δ w satρ− δ c vhl v R c1 R c /(2 )
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 17 10.3 Nucleate Boiling Chapter 10: Boiling Example 10.2 A vapor bubble is initiated and grows in liquid water at 120 ˚C. Find the time at which the bubble sizes predicted by the inertia-controlled model and the heat transfer- controlled model are the same. What is the bubble size obtained by both models?
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 18 10.3 Nucleate Boiling Chapter 10: Boiling Solution:
The saturation temperature at 1 atm is Tsat = 100°C = 373.15K. The -3 properties of water at this temperature are σ = 58.9x10 N/m, h l v = 2251.2 kJ/kg, ρ = 958.77 kg/m3, σ = 0.5974 kg/m3, c = 4.216 kJ/ l v plα= ρ kg=K, and =k l0.68 W/m-K The thermal diffusivity is l = k l/() l cp l 1.68x10-7 m2/s. The Jakob number is obtained from eq. (10.30), i.e., ρ c[ T()()∞ − T p ] 958.77× 4.216 × (120 − 100) Ja=lp l l sat v = = 60.25 ρ × vhl v 0.596 2251.2
and CR for the heat transfer-controlled model can be obtained from eq. (10.29): 3 3 C= Ja = ×60.25 = 58.9 R π π
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 19 10.3 Nucleate Boiling Chapter 10: Boiling The bubble sizes in the inertia-controlled model and heat transfer- controlled model can be obtained from eqs. (10.24) and (10.28), respectively. The time at which the bubble sizes obtained by both models are the same can be found by equalizing eqs. (10.24) and (10.28): 1/ 2 2 ρ h T− T[ p ( ∞ )] vl v∞ sat l t= 2 Cα t 3ρ T [ p (∞ )] R l i.e., lsat l 2 ρ h T− T[ p ( ∞ )] t= 4 C 2α vl v∞ sat l R l ρ ∞ 3lTsat [ p l ( )] 3 − 2 0.596× 2251.2 × 10 120 − 100 =4 × 58.92 × 1.68 × 10 7 × 3 958.77 373.15 =4.66 × 10− 5 s The bubble radius obtained is =α = × × ×−7 × × − 5 R2 CR l t 2 58.9 1.68 10 4.66 10 = 3.30 × 10− 4 m = 0.330 mm
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 20 10.3 Nucleate Boiling Chapter 10: Boiling
(a) (b)
Figure 10.8 Criteria for growth of hemispherical bubble nucleus
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 21 10.3 Nucleate Boiling Chapter 10: Boiling The following relation for bubble growth using microlayer theory is applicable to both inertia and heat transfer controlled region for pure liquids: R()() t R t R() t = 1 2 + (10.40) R1()() t R 2 t where ρ h( T− T )exp[ − ( t / t )1/ 2 ] R( t )= 0.8165 vl v w sat d t (10.41) 1 ρ lTsat
1/ 2 t TT− R( t )= 1.954 R* exp − + ∞ sat Ja(α t ) 1/ 2 2 − l td T w T sat (10.42) 1/ 2 1/ 2 − t +0.373Pr1/ 6 exp− Ja(α t ) 1/ 2 l l td
R() t − R*=1.39082 d − 0.1908Pr 1/ 6 α 1/ 2 l (10.43) Ja(ltd )
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 22 10.3 Nucleate Boiling Chapter 10: Boiling Liquid Bubbl Liquid Bubbl 2 e 2 e
S R δ S R δ Boundary Liquid Boundary Liquid Layer 1 Layer 1 Edge Edge
T T0 0 T δ
Tsat Tsat
R R+δ S R S R+δ (a) (b) Figure 10.9 Schematic illustration of bubble growth in a droplet suspended in an immiscible fluid. (a) Early stage where thermal boundary layer (δ) is within the droplet; (b) later stage when boundary layer extends into liquid 2 (Avedisian and Suresh, 1985).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 23 10.3 Nucleate Boiling Chapter 10: Boiling The radius of the final bubble is =( − ε ) − 1/3 RSf1 o (10.44) The velocity, pressure, and temperature in the two liquids are governed by the continuity, momentum, and energy equations for one-dimensional and unsteady conditions, given as 1 ∂ ( r2 v ) = 0 r 2 ∂ r i (10.45) ∂v ∂ v1 ∂ p 1 ∂ ∂ v 2 v i+ν i = − i + ν r 2 i − i ∂ti ∂ rρ ∂ r i r2 ∂ r ∂ r r 2 (10.46) ∂TTT ∂1 ∂ ∂ i+ν i = α r 2 i ∂ti ∂ r i r 2 ∂ r ∂ r (10.47) where i = 1 and 2, vi is the radial velocity, pi is the pressure within liquid
i, Ti is the temperature within liquid i, and αi is the thermal diffusivity of liquid i.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 24 10.3 Nucleate Boiling Chapter 10: Boiling The boundary and initial conditions are ( ) =( ) = T1 r,0 T 2 r ,0 To (10.48) ( ) = ( ) T1 R, t Tv t (10.49)
T( S,, t) = T( S t ) 1 2 (10.50) ∂TT ∂ k1= k 2 1∂ 2 ∂ (10.51) rS,, t r S t ( ∞) = T2 , t To (10.52) ( ) = RR0 o (10.53)
& = R ( 0) 0 (10.54)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 25 10.3 Nucleate Boiling Chapter 10: Boiling The interfacial energy balance around the bubble yields ∂ T k1 = ρ h R& 1 ∂ vl v r R, t (10.55) The velocity can be found by integrating eq. (10.45) and applying a mass balance around the bubble. 2 = ε R & vi R r 2 (10.56) The Rayleigh equation for this problem is obtained by integrating eq. (10.46) over r twice: from R to S and again form S to ∞. β ∂ ∂ ∂∂ ∂ vi+ v i = −1 p i + 1 2 v i − 2 v i ∫ vi v i r dr R ∂t ∂ rρ ∂ rr2 ∂ r ∂ r r 2 (10.57) where β = S or ∞. When r = R, 2σ ∂ v p− p =1 − 2µ 1 v R1 1 ∂ R r r= R
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 26 10.3 Nucleate Boiling Chapter 10: Boiling and when r = S 2σ ∂ v p− p =12 −2 ( µ − µ ) 1 (10.59) SS1 2 2 1 ∂ S r r= S The equation of motion for a bubble growing in the center of a droplet is found by combining eqs. (10.56) and (10.55), integrating twice, and substituting eqs. (10.58) and (10.59): &&2 4 3 &&+ & 2 −εRRRRR − ε − ε + − µ RR2 R 1 1 4 v1 1 SR 2 SS4 3 (10.60) p− p 2σ σ R =v o −11 + 12 ρ ε ε ρ σ 1 1RS 1
The analytical solution for boiling of a droplet in another immiscible 1 3 liquid is given as 1/ 2 2 − 1 9 S =( 1 −ε) 3 1 − ε ( 1 − ε) JaPe1/ 2 τ − 1 2π (10.61)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 27 10.3 Nucleate Boiling Chapter 10: Boiling
10-1 540
T0 15.0 Figure 10.10 Early time 10-2 520 variation of vapor pressure, p (T ) v 0 12.5 vapor temperature and T (K) 10-3 v radius of a bubble growing p (atm) 500 v in a superheated n-octane Tsat (p0) -4 10.0 10 Tv droplet. Initial conditions Evaporator Adiabatic Condenser correspond to the kinetic 480 limit of superheat of octane 10-5 p0 = pv(Tsat) p 7.5 v at the indicated pressure (Avedisian and Suresh 10-6 460 10-11 10-10 10-9 10-8 10-7 10-6 1985). τ
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 28 10.3 Pool Boiling Regimes Chapter 10: Boiling
10.3.3 Bubble Detachment Any heating method requires that the liquid wets a surface of characteristic length L, which is larger than the bubble or capillary scale
σ LL> = b ρ− ρ g()l g
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 29 10.3 Pool Boiling Regimes Chapter 10: Boiling
10.3.3 Bubble Detachment Any heating method requires that the liquid wets a surface of characteristic length L, which is larger than the bubble or capillary scale
σ LL> = b ρ− ρ g()l g
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 30 10.3 Nucleate Boiling Chapter 10: Boiling 10.3.3 Bubble Detachment Bubble departure diameter from a balance of buoyancy and surface tension 2σ D = 0.0208β (10.62) b ρ− ρ g()l v Bubble diameter departure 5 (10.63) − 2σ D=1.5 × 10 4( Ja * ) 4 b ρ− ρ g()l v 5 − 2σ (10.64) D=4.65 × 10 4( Ja * ) 4 b ρ− ρ g()l v where ρ c T Ja* = lp l sat (10.65) ρ vhl v
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 31 10.3 Nucleate Boiling Chapter 10: Boiling Product of bubble release frequency and bubble diameter at departure = Vb fb D b 1 (10.67) π 1 − + ρ ′′ 1Vb v hl v / q Where Vb is the bubble departure velocity D g()ρ− ρ 2σ V =bl v + b ρ+ ρ ρ + ρ (10.68) 2(lv )D b ( l v )
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 32 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.11 Forces acting on a vapor bubble growing on a heating surface (Eastman, 1984).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 33 2ρ dR Re = l R µ l dt 10.3 Nucleate Boiling Chapter 10: Boiling The bubble departure can be determined by a force balance: + = + + FFFFFd s i p B (10.69) The surface tension force is proportional to the fluid surface tension, σ, and the contact angle, θ, i.e. = π σ θ FRs2 b sin (10.70) The drag force can be calculated as: ρ dR 2 FCR= l π 2 d d 2 dt (10.71)
where drag coefficient Cd can be calculated by using the following experimental correlation 45 C = d Re (10.72) where 2ρ dR Re = l R µ (10.73) l dt
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 34 10.3 Nucleate Boiling Chapter 10: Boiling For a spherical bubble, which is submerged in a stagnant fluid, the buoyancy force is equal to the weight of the fluid displaced. So the buoyancy force is 4π R3 F=( ρ − ρ ) g (10.74) B3 l v According to Newton’s second law, the inertial force can be approximated as follows, based on the above argument: 11 d2 R FR= π3 ρ i 6 l dt 2 (10.75) The pressure force results from the contribution of the dynamic excess vapor pressure and capillary pressure. It can be written as follows: 2σ F= + p π R2 pR v b (10.76) where the vapor pressure is given below 2σ p− p = (10.77) v l R
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 35 10.3 Nucleate Boiling Chapter 10: Boiling 10
8
6 Internal Pressure 4 Buoyancy ) 2 Inertia N 5 5 10 ( .10 .11 .12 .13 .14 .15 .16 .17 .18 -2 Radius in cm Force -4 Total
-6
-8
-10 Surface Tension
Figure 10.13 Forces acting on a bubble in saturated water under 0.229-g acceleration (Eastman, 1984).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 36 10.3 Nucleate Boiling Chapter 10: Boiling
10
8
Buoyancy 6
4 Internal Pressure
5 2
Forc e (10 N) Inertia Figure 10.12 Forces acting on a .5 .6 .7 .8 .9 .10 .11 .12 .13 Radius in cm -2 bubble in saturated water with 1-g Total
-4 acceleration (Eastman, 1984).
Surface Tension -6
-8
-10
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 37 10.3 Nucleate Boiling Chapter 10: Boiling 10.3.4 Nucleate Site Density The number density of sites, or total number of active sites per unit area, is a function of contact angle, cavity half angle, and heat flux (or superheat) (Fig. 10.6), i.e., ′′ =θ φ ∆ Na f( , , T ,fluid properties) (10.78) Equation (10.11) indicated that for a given local heat flux or
superheat, a cavity will be active if Rmin is greater than Rb, 2σ T ≥ sat Rmin hρ ∆ T (10.79) ′′ lv v N a for water on a variety of surfaces and pressure ranges from 1 to 198 atm by 1/ 4.4 − 0.44 D NDF′′ = 2 c a d Dd (10.80)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 38 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.14 Number density of active sites for boiling on a copper surface (Lorenz et al., 1974).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 39 10.3 Nucleate Boiling Chapter 10: Boiling
− ρ− ρ 3.2 ρ − ρ 4.13 F =2.157 × 10− 7 lv 1 + 0.0049 l v (10.81) ρ ρ v v =σ[ + ρ ρ ] ⋅ − − Dc41(/)/l v p l{ exp h l v ( T v T sat )/( R g T v T sat )1 } (10.82)
σ ρ− ρ 0.9 D =0.0208θ ⋅ 0.0012l v (10.83) d ρ− ρ ρ g()l v v
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 40 10.3 Nucleate Boiling Chapter 10: Boiling 10.3.5 Bubble Growth and Merger
y=Y Macro region
Liquid Vapor
T = T g sat
θ y r Figure 10.15 Macro r=R wall and micro region in numerical simulation Micro region (Son et al., 2002) 0 δ h / 2 y δ
r r = R1 r = R0 wall
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 41 10.3 Nucleate Boiling Chapter 10: Boiling Conservation of mass in the microlayer ∂ δ q′′ =v − (10.84) ∂ l ρ tl h lv Liquid velocity normal to the surface, vl ∂ δ = − 1 (10.85) vl∫ ru l dy r∂ r 0 Momentum equation in the thin film ∂p ∂ 2 u l= µ l (10.86) ∂rl ∂ y2 Assuming conduction is the mechanism of heat transfer, heat flux is TT− q′′ = k w δ (10.87) l δ Evaporating heat flux can be written using Clausius-Clapyeron 1/ 2 equation 2 ρ h2 () p− p T q′′ = vl v T − T + l v v (10.88) π δ v ρ Rg T v T vl h l v
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 42 10.3 Nucleate Boiling Chapter 10: Boiling The pressures in the liquid and vapor are related by momentum balance at interface (Chapter 5) q′′ 2 p= p −σ K − p + l v d ρ 2 vhl v (10.89) where surface tension, σ, is a function of temperature, and K is curvature: 2 1 ∂ ∂δ ∂ δ K= r 1 + ∂ ∂ ∂ r r r r (10.90) The combination of eqs. (10.84) – (10.88) yields a fourth order ordinary differential equation in the following form: δ′′′′= δ δ ′ δ ′′ δ ′′′ f(,,,,,) r t (10.91) which is subject to the following boundary conditions: δ= δ δ′ = δ ′′′ = = 0; 0 at r R 0 (10.92) δ= δ′ = θ δ ′′ = = h/ 2; tan ; 0 at r R1 (10.93)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 43 10.3 Nucleate Boiling Chapter 10: Boiling Governing equations for the macro region are ∂ V ρ+VV ⋅ ∇ = ∇p + ρ g − ρ β () T − T ∂ t sat (10.94) −σKH ∇ + ∇ ⋅ µ ∇VV + ∇ ⋅ µ ∇ T ∂ ρT + ⋅ ∇ = ∇ ⋅ ∇ φ cpl V T k T for >0 (10.95) ∂ t = φ T Tsat( p v ) for =0 m′′ (10.96) ∇ ⋅V = ∇ρ + V& ρ 2 micro Step function H 0 φ ≤ -1.5h (10.97) H= 0.5+φ /(3h)+sin[2 π φ /(3h)]/(2 π ) φ ≤ 1.5 h 1 φ > 1.5h
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 44 10.3 Nucleate Boiling Chapter 10: Boiling The source term is obtained by conservations of mass and energy at the interface ∇ ′′ k T (10.98) m=ρ () Vδ − V = hlv Rate of volume of production form the microlayer R1 k() T− T & = l w δ (10.99) Vmicro ∫ rdr R0 ρ δ ∆ vhl v Vmicro Level set function Φ ∂ φ (10.100) = −Vδ ⋅ ∇ φ ∂ t Reinitialized by solving ∂ φ φ =0 (1 − ∇ φ ) (10.101) ∂ t φ 2+ 2 0 h
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 45 10.3 Nucleate Boiling Chapter 10: Boiling
Boundary conditions for the macro region
∂ φ u= v =0, T = T , = − cosϕ at y = 0 w ∂ y (10.102)
∂v ∂ T ∂ φ u= = = at r = 0, R ∂r ∂ r ∂ r (10.103)
∂u ∂ v ∂ φ = ==0, T = T , at y = Y ∂y ∂ y ∂ y sat (10.104)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 46 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.16 Bubble growth and demerger pattern for ∆T=10K and waiting time of 1.28 ms
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 47 10.3 Nucleate Boiling Chapter 10: Boiling
10.3.6 Heat Transfer in Nucleate Boiling
In general, heat flux and heat transfer coefficient during evaporation and nucleate boiling can be correlated with the driving temperature
difference (Tw – Tsat) according to the following equations: ′′ =( − ) m (10.105) q c1 Tw T sat and since q” = h(Tw – Tsat) one can get =( −) m− 1 =( − ) n (10.106) h c2 Tw T sat c 2 T w T sat m− 1 =′′m = ′′ P h c3 q c 3 q (10.107) h= c( T − T ) 1/ 4 2 w sat Laminar flow (10.108) =( − ) 1/3 h c2 Tw T sat Turbulent flow (10.109)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 48 10.3 Nucleate Boiling Chapter 10: Boiling the heat transfer coefficient is approximately proportional to the third power of this temperature difference, i.e., =( − ) 3 (10.110) h c2 Tw T sat Heat flux during partial nucleate boiling (10.111) KK2 2 π q′′ =1π( kcfDNT ρ )2 ∆ + 1 − 1 NDhTNDhT π 2 ∆ + 2 ∆ 2pbbal 2 abNC a 4 bEvp Contraction for nucleate boiling 1 − − Nu = Re1m Pr 1 n (10.112) Nusselt number l 1/ 2Cs,l 1/ 2 hσ q′′ σ Nu = = ρ− ρ − ρ − ρ (10.113) kl()()() lv g k l T w T sat l v g 1/ 2 Reynold’s number q′′ σ 1 Re = (10.114) ρ ρ− ρ ν lh lv () l v g l
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 49 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.17 Temperature profile above a heating surface during nucleate boiling (Stephan, 1992).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 50 10.3 Nucleate Boiling Chapter 10: Boiling
Figure 10.18 Evaporation and nucleate boiling of water at 100 °C on a heated surface (Stephan, 1992).
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 51 10.3 Nucleate Boiling Chapter 10: Boiling
Table 10.1 Heat transfer comparison of evaporation and nucleate pool boiling*
m q′′ = c( T − T ) =( − ) n h= c q′′ P 1 w sat h c2 Tw T sat 3
Evaporation Laminar m = 5/4 n = 1/4 P= 1/5 Turbulent m = 4/3 n = 1/3 P= 1/4
Nucleate boiling m = 4 n = 3 P= 3/4
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 52 10.3 Nucleate Boiling Chapter 10: Boiling Substituting eq. (10.113) and (10.114) into eq. (10.112) 1 3 g()ρ− ρ 2 c() T− T q′′ = µ h l v pl w sat (10.115) l lv σ n Cs,l h l v Pr l Rearranging eq. (10.115) if the heat flux1 is specified 1 3 ′′ σ 2 − = hlv n q TTCw satPrl s, l (10.116) cµ h g() ρ− ρ pl l l v l v Heat transfer coefficient for pool boiling − =0.12 0.4343ln Rp − −0.55 − 0.5′′ 0.67(10.117) h55 pr ( 0.4343ln p r ) M q
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 53 10.3 Nucleate Boiling Chapter 10: Boiling
Table 10.2 Values of C s l and n for various combinations of surfaces and fluids Surface Surface finish Fluid C n material sl Brass Water 0.0060 1.0 Chromium Benzene 0.0101 1.7 Chromium Ethyl alcohol 0.0027 1.7 Copper Carbon tetrachloride 0.0130 1.7 Copper Isopropanol 0.0130 1.7 Copper Lapped n-Pentane 0.0049 1.7 Copper Polished n-Pentane 0.0154 1.7 Copper Polished Water 0.0130 1.0 Copper Scored Water 0.0068 1.0 Nickel Water 0.0060 1.0 Platinum Water 0.0130 1.0 Stainless steel Chemically etched Water 0.0130 1.0 Stainless steel Ground and polished Water 0.0060 1.0 Stainless steel Mechanically polished Water 0.0130 1.0 Stainless steel Teflon pitted Water 0.0058 1.0
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 54 10.3 Nucleate Boiling Chapter 10: Boiling A fluid-specific correlation that includes the effect of the surface roughness: = ′′ ′′ nf 0.133 < h h0FPF( q / q 0 ) ( R p / R p 0 ) , 0.0005< p r 0.95 (10.118)
For the fluid that is not listed in Table 10.3, h0 can be obtained by the experimental results or other correlations. The pressure correction factor is 1.73p0.27+ [6.1 + 0.68/(1 − p )] p 2 , for water F = r r r PF 0.27 + + − 1.2pr 2.5 p r p r /(1 p r ), for all fluids except water and helium (10.119)
and the exponent in eq. (10.118), nf, decreases with increasing pr: 0.9− 0.3p0.3 , for water nf = r − 0.15 0.9 0.3pr , for all fluids except water and helium (10.120)
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 55