Chapter 3  Boiling Heat Transfer

1. Pool Boiling

: Boiling inside the stagnant liquid through the heated surface

(1) Characteristic Boiling Curve

① Temperature controlled experiment

- In partially nucleate boiling region, a small number of nucleation center

come into play and only a few bubble exits(ONB)

These bubble rise quickly and are collapsed.  Thus the slight agitation

causes q'' to increase more rapidly with (Tw-Tsat)

∴ The heat transfer is due to  ● natural convection

● bubble agitation

- condensing vapor as heat source to prevent the wall failure

② Heat flux controlled experiment

; nuclear fuel element or electrically heated wire

o In this case, it more difficult            to observe TB than the tempe-

rature controlled experiment in

which the film at the bottom

proceeds more slowly.

(2) Natural convection region

[eqn: rm N_u ~=~ 0.55`` [Gr ``Pr ] ^{1over4}]    where,    [eqn: rm N_u = hD over k ~ , ~~~~~~P_r = {C_p μ} over k]

∴[eqn: rm H_1φ ~∝~ (`` T_w ~-~ T_sat ``) ^{1 over 4}]         [eqn: G_r = {ρ_l ^2 `g `β ``[ `T_v `-` T_sat `] ``D^3 } over {μ_l ^2}]

Thus, [eqn: rm q'' ~~∝~~ (``T_w ~-~ T_sat`` ) ^{5over4}]             [eqn: ~∝~~ T_v ~-~ T_sat]

or for turbulent natural convection from a horizontal flat surface.

[eqn: rm N u ~=~ 0.14`` [Gr ``Pr ] ^{1over3}]                                      (3.1.1)

(3) Nucleate Boiling

: Heat transfer is enhanced and thus a relatively large change in surface heat flux

for very slow increase in the surface temperature.

[eqn: rm T_w ~=~ T_sat ~+~ C``` (q'' ) ^n] , n = 0.25 ～ 0.5

And the heat transfer coefficient will be expressed as,

[eqn: rm N u ~=~ C ` ```R_e ^a ` ``P_r ^b]                               (3.1.2)

By Rohsenon at 1952 [ Trans. ASME Vol. 74, pp. 967-976 ]

[eqn: rm LEFT [ {C_pf ```△T_sat } over h_fg RIGHT ] ~~=~~ C_sf ~LEFT [ q'' over {μ_f h_fg } ``` LEFT ( σ over {g (ρ_f - ρ_g ) } RIGHT ) ^{1over2} RIGHT ]^{n} ~LEFT [ {C_p μ } over k RIGHT ] _f ^m+1]    (3.1.3)

Csf	fluid / surface
0.014	water / stainless-steel (m=1.0)
0.013	water / copper (m=1.0)
0.006	water / nickel (m=1.0)

By Foster and Zuber

: By using their Bubble radius and Bubble growth velocity,

[eqn: rm q'' over { ρ_g ``h_fg } ```left ( pi over α_f right ) ^{1over2} ```left [ {ρ_f ``r^*3 } over {2σ } right ] ^ {1over 4}~=~ 0.0015 ~left [``ρ_f over μ_f left ( { △T_sat ``k_f } over {ρ_g ``h_fg } right ) ^{1over2} pi over α_f left]^{5over8} P_rf ^{1over3}]

for water, 1 < P < 50 bar

Now since [eqn: rm H_2φ ~=~ q'' over { (`T_w ~-~ T_sat `)}] for water , m = 0

[eqn: rm {C_pf ^3```(T_w ``- ``T_sat )^3 } over h_fg ^3 ~=~ C_sf ^3~~ LEFT [ q'' over {μ_f ``h_fg } ``` LEFT ( {g_c `σ} over {g (ρ_f - ρ_g ) } RIGHT ) ^{1over2} RIGHT ] ~Pr_f ^3]     (3.1.4)

∴ [eqn: rm q'' ~∝~ (`` T_w ~-~ T_sat ``)]

and

[eqn: rm H_2φ ~=~ 1 over {C_f ^3 } ~ left { ~{ (` T_w ``-`` T_sat `)^2 ``k_f ^3 } over { h_fg ^2 ```μ_f ^2 } ``root { {g (ρ_f - ρ_g )} over { g_c α } } ~right }]         (3.1.5)

☞ Precise physical properties are required.

☞ complicated to calculate depending on the surface condition

Thus, simple dimensional correlation is

[eqn: rm H_2φ ~= ~A ^* ``(`q''` ) ^0.7 ```F``(``P``)]

where [eqn: rm A^* ```= ```0.1011 ```P_cr ^0.69],

[eqn: rm F``(``P``) ~=~ 1.8 ``Pr ^0.17 ~+~ 4 ``Pr ^1.2 ~+~ 10 ``Pr ^10]    by Mostinski

(3) Burnout in Pool Boiling

- Burnout occurs when the conditions are no longer such as to allow the           vapor generated in nucleate boiling to be removed from the vicinity of           the heated surface.

1) Mechanism of pool boiling CHF

① Hydrodynamic instability model [Zuber, 1958]

- Localized flooding

: occurs when [eqn: v_g] is sufficient to prevent an adequate amount of liquid             from rearching the heated surface

∵ when the bubble poplulation increases with heat flux, the outgoing               bubbles  jam the path of the incoming liquid.

☞ Helmholtz instability ; incipience of the boiling crisis

☞ Taylor instability : incipience of stable film boiling

★ Helmholtz Instability

o When two immiscible fluids flow relative to each other along an  interface of          separation, there's a maximum relative velocity above which a small                  disturbance of the interface will amplify and grow and thereby distort the             flow.

o  the incipience of the boiling crisis

- maximum vapor velocity for a stable vapor stream is given as,

[eqn: V_v ~=~ left[ {rho_l ``sigma ``2 pi g_c} over {rho_v ``lambda (rho_l + rho_v )} right]]                                   (3.1.6)

where [eqn: lambda] ; waver length of column

When [eqn: rm V_v] exceeds the above value, the stable surface can be no            longer sustained and a vapor blanket is formed, which prevents              bubbles from escaping on the heated surface.

★ Taylor Instability

o criterion for stable film boiling

o The stability of an interface of two fluids of different densities depends                  on the balance of the surface tension energy and the sum of the kinetic                 and potential energy of the wavy interface.

∴ If σ > K.E & P.E,

→ Stable film boiling cond. for a horizontal surface.

- condition of a stable wave for stable film boiling

[eqn: lambda ~〈~ 2pi sqrt{ {g_c sigma} over {g(rho_l - rho_v )}}]                                     (3.1.7)

② Microlayer dryout model [Haramura & Katto, 1983]

- exist the liquid layer of thickness, δ, between

vapor and heated surface

-  CHF when the layer is evaporated..

③ Bubble interaction model

- the critical heat flux is limited by the removal rate of bubbles which             carry away the heat in the form of evaporation

[eqn: {q''}_c ~=~ h_fg rho_v pi over 6 D_b^3 n f]

Rohsenow and Griffith adopted a correcting coefficient Cq to account             for conductive and convection composite of heat transfer, and assumed

[eqn: nD_b^2 ~=~ C_vb]

Then,   [eqn: {q''}_c over h_fg rho_v ~=~ pi over 6 C_q C_vb (fD_b )]

With empirical data,

[eqn: {q_c}'' = 143h_fg ρ_g [(ρ_l -ρ_g )/ρ_g ]^0.6 [g/gc]^0.25]

2) Critical vapor volumetric flux for pool boiling CHF

From Eq. (3.1.6),

[eqn: rm (j_g )_crit ~=~ left [ {rho_f ``σ`` 2pi ``g_c } over {rho_g ````λ_H ``` ( ``rho_f ``+`` rho_g ``) } right ] ^1/2]

Assume  [eqn: lambda_H ~=~lambda_crit] , then

[eqn: (j_g )_c ~=~ K left[ {sigma g g_c (rho_f - rho_g )} over rho_g^2 right]^1/4 left( rho_f over {rho_f + rho_g} right)^1/2]

For [eqn: rho_f ~》~rho_g]

[eqn: (j_g )_c ~=~ K left[ {sigma g g_c (rho_f - rho_g )} over rho_g^2 right]^1/4]   ; by Kutateladze

Or, from factional analysis (Droplet Fluidization)

[eqn: {rho_g over g_c j_g^2 (¼piD^2 )C_D} over {1 over 6 piD^3 (rho_l - rho_g ) g over g_c} ~X~ {rho_g over g_c j_g^2 (¼piD^2 )C_D} over {pi D sigma}]    = [eqn: K_g^4]

where,  ∙ drag force term :[eqn: rm { rho_g over g_c ``` j_g ^2 ``left [ 1 over 4 ``pi ```D_d ^2 ``right ]`` C_D }]

∙ buoyancy force term :[eqn: rm { 1 over 6 ``pi ``D_d ^3 ``( rho_l ``-`` rho_g ) ``g over g_c }]

∙ surface tension term :[eqn: rm { pi ```D_d ``σ }]

Then,

[eqn: K_g ~=~ {j_g rho_g^1/2} over {[ sigma g g_c (rho_l -rho_g )]^1/4}]           (3.1.8)

The volumetric flowrate of vapor formation is

[eqn: Q_g ~=~ q over {h_fg ρ_g} [ft^3 / hr]]

[eqn: ∴~~ j_g ~=~ Q_g over A_HT ~=~ q/A_HT over {h_fg ρ_g} ~=~ q'' over {h_fg ρ_g} [ft/sec]]

Thus,      [eqn: q''_c ~ =~ h_fg ``ρ_g (j_g )_c]

From Eq.(3.1.8),

[eqn: {q''}_c ~=~ K_g h_fg rho_g^1/2 left[ sigma g g_c (rho_f - rho_g )} right]^1/4]                           (3.1.9)

where,   Kg = 0.16  by Kutateladge

= 0.13      Zuber

= 0.18      Rohsenow

= 0.2    for droplet fluidization

3) Factors affecting on Pool Boiling

① Effect of subcooling

; increases CHF than that in the saturated liquid due to vapor                    condensation and sensible heat on the subcooled liquid

q"c,sub / q"c,sat = 1 + B(Tsat - Tbulk)                    (3.1.10)

㉠ Zuber's correlation for ΔTsub

[eqn: rm {q''}_{c,s ub} ~=~{q''}_c,sat left[1+{2k(T_sat - T_bulk )} over sqrt{pi alpha_f ``tau} right]]

where  τ ; period of growth and departure for bubble

[eqn: rm ≈ ~λ_m over v_crit ~=~ ( 3π over2 )^1/2 A ( ρ_g^2 over gσΔρ )^1/4 ~~~ and ~~ A = [ σ over gΔρ ]^1/2]

㉡ Ivey and Morris

[eqn: rm {q''}_{c, s ub} ~=~{q''}_c,sat left[1+0.1 left( rho_g over rho_f right)^1/4 Ja right]~,~~Ja ~=~ {rho_f c_pf TRIANGLE T_{s ub}} over {rho_g h_fg }]

☞ As [eqn: {dot q}''_CHF(sat) ~and/or ~ TRIANGLE T_{s ub}~i ncreases,~ {dot q}''_{CHF(s ub)}~i ncreases.]

☞ generally not well established, but depends on the geometry.

② Pressure effect

o Lower ΔT requires to activate cavities of a given size as the pressure           increases

o CHF increases as the pressure increases until a certain pressure.                However, above the pressure CHF decreases as the pressure increases            due to  the change of liquid properties.

③ Effect of Surface Finish or Roughness

o The boiling data including ONB for the rough surfaces lie to the left,            at lower wall superheat.

o At low heat fluxes, the direction of surface finish orientation affects             on boiling.

o Horizontal scratches require lower superheat to produce the same heat             flux.

o The influence of surface roughness on CHF is negligible.

④ Existance of Dissolved Gas

o The dissolved gases from the hot surface tend to move the q-ΔT               curve to the left and reduce the magnitude of CHF.

o generates unstable boiling and bumping with changing back and forth            between nucleate boiling and natural convection.

o The effect of noncondensible gases on CHF is most significant at               high subcooling.

⑤ Effect of Acceleration

; cause the fluctuations of void, and the variance of gravity in Eq.(3.1.9)           as,

[eqn: {q''}_crit ~SIM~ (a/g)^0.25]

By Adams(1962) with distilled water and graphite heaters,

[eqn: {q''}_crit ~SIM~ (a/g)^0.15]   for 0 < a/g < 10

[eqn: {q''}_crit ~SIM~ (a/g)^0.25]   for 10 <a/g < 100

Note that the exponent may vary with surface characteristics as well            as hydrodynamic condition such as pressure.

(4) Leidenforst point (MFB)

★ dancing of a liquid drop

; lower limit of stable film boiling corresponding to the breakdown of the            continuous insulating film and the onset of liquid-solid contact.

[eqn: {q''}_CHF over {rho_g h_fg} ~=~alpha_g v_g]

What is vmin for the Taylor instability(i.e terminal velocity) ?

In the hydrodynamic stability theory, the force balance gives

[eqn: { C_D rho_f v^2 } over 2 {pi D_l^2 } over 4 ~=~ (rho_f - rho_g ) g pi over 6 D_l^3]

∴ [eqn: v_min ~=~ left( 4 over 3 {g D_l triangle rho } over {C_D rho_f} right)^1/2]

Let, [eqn: D_b ~sim~ lambda_m over 2],    where the most dagerous wavelenth ;[eqn: lambda _m ~=~ 2 pi root3 A]         and  [eqn: A~=~ sigma over sqrt{g(rho_l -rho_g )}],

then, [eqn: q''_min ~ =~ C rho_g h_fg A^1/2 left( {g triangle rho} over {rho_g +rho_f } right)^1/2]

[eqn: = C rho_g h_fg left[ {sigma g (rho_l - rho_g )} over { (rho_g +rho_f )^2} right]^1/4]

where,   [eqn: c~=~0.13 ~~~~{ b y ~Z u b e r ~(1958) }]

[eqn: #&=~0.177 ~~~{ b y ~Z u b e r ~(1959) } #&=~0.09 ~~~~{ b y ~B e r n e n s o n ~(1961) }]

(5) Film Boiling

; the boiling mode in which a continuous blanket of vapor is generated            and maintained to isolate the liquid from the heating surface

1) vertical surface

o Analytical treatment of film boiling on verticla surface [Bromley, 1950]

The vapor rise by a gravitational buoyancy is retarded by fluid shear

stresses.

[eqn: g(rho_l - rho_g ) ~=~ -mu {d^2 T} over dy^2]

B.C ;    At y = 0,   u = 0

At y = δ,  u = 0    (for stagnant interface)

[eqn: du over dy] = 0   (for moving-up surface with vapor)

Then,

[eqn: u ~=~ {g TRIANGLE rho} over mu_v left( C {delta y} over 2 - y^2 over 2 right)]

where C=2 for the dynamic interface  and C=1 for the stagnant interface

If the heat transport is assumed to be by molecular conduction only, the

heat balance is

[eqn: h_fg dw over dx ~=~ {h_fg rho_v g TRIANGLE rho } over 2mu_v ( C over 2 - 1 over 3 ) 3 delta^2 {d delta} over dx ~=~ {k triangle T} over delta]        (3.1.11)

B.C  ;  δ=0  at x=0

From obtaining δby integrating Eq.(3.1.11), the local heat transfer                coefficient is,

[eqn: h ~=~ k over delta ~=~ left[ 8 over {3(C/2-1/3 )} right]^-1/4 left( {h_fg rho_v g TRIANGLE rho k^3 } over {triangle T x mu_v} right)^1/4]

The mean heat transfer coefficient is,

[eqn: bar{h} `=~ {5} over {4} LEFT [ {3(C/2-1/3)} over {8} {h _{fg} rho _{v} g TRIANGLE rho k ^{3}} over {TRIANGLE TL mu _{v}} RIGHT ] ^{1/4}]

[eqn: =~C _{1} LEFT ( {h _{fg} rho _{v} g TRIANGLE rho k ^{3}} over {TRIANGLE TL mu _{v}} RIGHT ) ^{1/4}]

where  C = 0.883 for the dynamic interface,

C = 0.625 for the stagnant interface

o By Nygel Bailey

[eqn: H_film &= q^'' over {(T_w -T_sat )} #&=0.4[{{k_g}^3 g` rho_g (rho_f - rho_g ) {h_fg}^' }over { mu _g `R (T_w -T_sat ) }]^1/4]               rod    vapor

R

[eqn: where,~ {h_fg }^' =h_fg [ 1+ 0.68 {C_pg (T_w - T_sat )} over {h_fg } ]]                        liquid

2) horizontal plate (or rod)  (Berenson) [RELAP]

- apply Taylor instability

[eqn: H_film &= q^'' over {(T_w -T_sat )} =0.425 left[{{k_g}^3 g` rho_g (rho_f - rho_g ) {h_fg}^' }over { mu _g ({ lambda _c} over {2pi}) (T_w -T_sat ) } right]^1/4]

where Tayler wavelength is [eqn: { lambda _c} over {2pi} =[{g_c sigma} over {g(rho_f -rho_g )}]^{{1} over {2}}]

■ to take account on the radiation

[eqn: h=h_c +0.75h_r]

where [eqn: h_r = sigma epsilon {T_w ^4 -T_sat ^4} over {T_w -T_sat}]

★ Effect of Parameters Affecting

Film Boiling

1) Effect of acceleartion

- ∝ 1/4 power

- reduced for a small cylinder                where σ is more important

2) velocity and subcooling effect

- suppression of film boiling when the relative motion of a heat                    surface to the Leidenfrost drop

- the film boiling cannot sustain when the subcooling is very high,                such that

[eqn: {T_s -T_b} over {T_w - T_s} > {h_radiation + 1.27 h_sat.FB} over h_{turb. conv.}]