2. Condensation Heat Transfer

; The removal of heat from a system in such a manner that vapor is             converted into liquid.

o homeogenous condensation

o heterogeneous condensation - •drop-wise condensation ↔ nucleate boiling

•film-wise condensation ↔ film boiling

(1) Drop-wise condensation

Drop-wise condensation occurs when the vapor is cooled sufficiently           below the saturation temperature to induce the nucleation of droplets.

o by contamination of non-wettable surfaces

o incidental reduction of interfacial tension between condensate and

surface

o occur homogeneously within vapor or heterogeneously on the entrained

particular matter

A reliable mechanism of drop-wise condensation have not yet been              established. However, the heat transfer by the drop-wise condensation is        several times larger than that of the film-wise condensarion.

Using △p([eqn: p_g ~-~ p_f ~=~ 2sigma over R]) for vapor formation, and Clausius-Clapeyron        equation, Hill, Witting and Demetri derived the rate of nucleation as,

[eqn: rm dn over dt ~=~ N v_f over v_g lambda exp[-4pi`` r^*^2 / 3kT_g ]]                            (5.2.1)

where [eqn: lambda ~=~( 2sigma/pi``m)^1/2] and N is the number of vapor mole per unit volume.

Also, Oswatitsch derived the droplet growth rate as,

[eqn: rm dr over dt ~=~ 3p_∞ over {8h_fg rho_f } sqrt{ 3R over {T_g M} (T_∞ -T_g )}]                           (5.2.2)

◆ Two fundamentally different model to describe the dropwise condensation

1)  By Jakob (1936)

; In the beginning, the condensation starts as a thin unstable film on              whole or a part of surface. When the film has a critical thickness to            break up, the liquid forms a droplet due to the  surface tension.

In 1964, Silver has quantitatively analyze the drop-wise condensation based       on the radial inward drainage of condensate.

[eqn: G_D over G_f ~=~ left[ {rho_f^2 D^2 g } over {24.2 mu_f G_f } right]^{1 over 9}]         D ; tube diameter           (5.2.3)

여기서  [eqn: G_D] ; the drop-wise condensation mass flux

[eqn: G_f] ;  the film-wise condensation mass flux for ideal condition

For steam at 1atm for [eqn: G_F ~=~ 1.36 times 10^-2 k_g /m^2 s] ,

[eqn: G_O / G_F ~=~6.5]   and   mean thickness of film [eqn: ～ ~2 mu m]

★ This model is proper when the temperature difference is large.

2) heterogeneous nucleation process  ― By Eucken (1937)

; In 1963, McCormick and Baer suggested that a lot of submicroscopic             droplets were randomly nucleated at active sites like minute groove on          the surface and collapsed to make big drop which can fall down from           the surface.

For hemispherical droplet, the growth rate is assumed to be only dependent       on the heat condensation through the drop as,

[eqn: LEFT ( {dr} over {dt} RIGHT ) ^{2} `=~4.15~~ {k _{f} (T _{gi} -T _{w} )} over {h _{fg} rho _{f}}]                      (5.2.4)

where [eqn: rm T_gi] ; interfacial gas temperature

By Gose, Mucciardi and Baer (1969),

minimum nucleation site density = [eqn: 5 times 10^4 sites/cm^2] ,

diameter of drop to be chiefly collapsed > 0.01 cm

★ This model is proper when the temperature difference is small.

☞ For small temperature difference, the drop-wise heat transfer coefficient          a little increases as the temperature increases, but significantly                  decreases in the case of large temperature difference due to change in           mechanism and presence of non-condensible gas in vapor phase.

☞ influence of pressure

☞ influence of vapor velocity

; maximum at 2m/sec

(2) Film-wise Condensation

- The condensate wets the surface and forms a continuous liquid film.

- The film behaves according to the fluid dynamic law which considers the

gravity, the vapor shear, the surface tension force and so on.

★ Various resistance to heat transfer during condensation

■ The mechanism of condensation at a plane liquid-vapor interface

Consider the interface mass transfer by kinetic theory

arrival rate; [eqn: G_+]     [eqn: G_-] ; departure rate

[eqn: T_g , ~p_g]

[eqn: T_f]            [eqn: G=G_+ - G_-]

From kinetic theory,

[eqn: vert G vert``=``left ( M over {2 pi R} right )^{1 over 2} ``P over {T^{1/2}}]	(5.2.5)

[eqn: G``=`G_+ ``-``G_-]	(5.2.6)

Since the interface is actually not in the static thermal equilibrium, it is         meaningless to use the thermostatic pressure or the temperature in each         side. Thus, we have to solve Boltzman transport equation with proper           boundary conditions and the asymptote of the thermal equilibrium in few         mfp. But for the purpose of engineering, a simplified kinetic theory              technique and proper correction factor may adopted.

A. Crude theory

For appropriate temperature and pressure of liquid and vapor, from Eqs.         (5.2.5) and (5.2.6).

[eqn: G``=``left ( M over {2 pi R} right )^{1 over 2} ``left[ {p_g} over {T_g}^1/2 ``-``{p_f} over {T_f}^1/2 right]]                              (5.2.7)

For small difference in pressure and temperature, by differentiating,

[eqn: G``=``left ( M over {2 pi RT} right )^{1 over 2} ``p```left[ {Delta p} over p ``-``{Delta T} over 2T right]]                          (5.2.8)

Assume that there is no temperature jump across the interface

[eqn: therefore ````G ``=``left ( M over {2 pi R T} right )^{1 over 2} Delta p]	(5.2.9)

This means that the condensation occurs when [eqn: p_g ``>`` p_f] ( or [eqn: Delta p``>``0])

Interfacial heat transfer coefficient

[eqn: H_i ``=``q'' / Delta T]	(5.2.10)

and

[eqn: q''``=``h_fg ``G]	(5.2.11)

From Eqs. (5.2.8), (5.2.9) and (5.2.10),

[eqn: H_i ``=``h_fg ``left [ M over {2 pi RT} right ]^{1 over 2} {Delta p} over DeltaT]	(5.2.12)

By Clausius - Clapeyron equation,

[eqn: H_i ``=`` left [ M over {2 pi R T} right ]^{1 over 2} {J ``h_fg ^2} over {T ``v_fg} ``left [1-{p``v_fg}over{2``h_fg} right ]]	(5.2.13)

B. Modification of the crude theory

By Silver, Eq. (5.2.13) is multiplied by a factor, f, called the "Molecular         exchange condensation factor"

f = 0.036 for water under vacuum condenser conditions.

Or, by Schrage, Eq. (5.2.7) can be rewritten as,

[eqn: G ``=`` left ( M over {2 pi R} right )^{1 over 2} left[ Gamma ``sigma_c `` P_g over T_g ^{1/2} ``-``sigma_l `` P_f over T_f ^{1/2} right ]]	(5.2.14)

where [eqn: sigma_c] , [eqn: sigma_l] : condensation and evaporation coefficient(usually same, σ)

Γ represents net motion of vapor toward surface and is a function

of a

a is the ratio of gas overall speed [eqn: u ```left (``=`` G / rho_g right )] to characteristic molecular       velocity, (2RT/M)1/2  as,

[eqn: a``=``u over {(2RT/M`)}^1/2]                                  (5.2.15)

For very large a, [eqn: Gamma(a)``->``2`a`pi^{1over2}]

[eqn: therefore```G``=``left (M over {2 pi R} right)^{1 over 2} ``left[``2 a pi^{1over2}``sigma``rho_g over {T_g^1/2}``-``sigma``rho_f over {T_f^1/2}right]]                 (5.2.16)

여기서 [eqn: sigma ``=``1] 이면(즉 surface에 닿는 모든 molecule 은 condensing.),             evaporation은 condensation에 비해 무시할 수 있다는 것이다. [eqn: ->] black body

For very small a (즉 low condensation rate), [eqn: Gamma(a) ``approx ``1``+``a` pi^{1over2}]

[eqn: Therefore````G``=``left(M over {2 pi R} right )^{1 over 2} left[ p_g over T_g^1/2 ``-``p_f over T_f^1/2 ``right] ``left(2sigma over {2-sigma} right)]                   (5.2.17)

[eqn: ->] Kucheror - Rikenglaz equation

☞ Above kinetic theory neglects the non-equilibrium interaction and,                moreover, is hard to consider the effect of the non-condensable gas.

C. Film condensation on a planar surface

○ Nusselt Equation for a laminar film

Assumptions

ㄱ. the flow of condensate in the film is laminar

ㄴ. constant fluid properties

ㄷ. negligible subcooling of the condensate

ㄹ. negligible momentum changes through the film

ㅁ. vapor is stationary and exerts no drag on the downward motion of the condensate

그림 5.10 laminar flow over     a inclined surface(Nusselt)

ㅂ. heat transfer is by conduction only

Velocity distribution in a liquid film

From force balance

[eqn: (``delta``-``y``)``dz``(rho_f ``-``rho_g ``)``g``sin theta]

[eqn: ``=``mu_f ``left( {d u_y} over {d y} right )`` dz]       (5.2.18)

Integrate with B.C ; [eqn: u_y ``=``0] at [eqn: y``=``0]

[eqn: therefore ```u_y ``=``{(rho_f ``-``rho_y )``g``sin theta} over {mu_f} ``left[y``delta``-`` y^2 over 2 right ]]

And the mass flow rate per unit width will be

[eqn: Gamma``=``rho_f Int_0 ^delta ``u_y ``d y ``=``{rho_f ``(rho_f ``-``rho_g )``g``sin theta ``delta^3} over {3``mu_f}]	(5.2.19)

Thus, the rate of increase of film flow rate with [eqn: delta] is

[eqn: { d Gamma} over {d delta}``=``{rho_f ``(rho_f ``-``rho_g ) ``g``sintheta ``delta^2} over {mu_f}]	(5.2.20)

If film surface temperature is [eqn: T_gi] and wall temperature is [eqn: T_w], the heat

transferred by conduction to an element of dz

[eqn: dq ``=``k_f over delta (T_gi -T_w `)`` dz``=``h_fg ``d Gamma]

[eqn: therefore] mass rate of condensation is

[eqn: d Gamma ``=``k_f over {delta ``h_fg} (T_gi ``-``T_w ) ``dz]	(5.2.21)

Inserting Eq. (5.2.20) into (5.2.21) and separating variables and integrating        with [eqn: delta``=``0] at [eqn: z``=``0],

[eqn: mu_f ``k_f ``(T_gi ``-``T_w )``z``=``rho_f ``(rho_f ``-``rho_g )``g``sin theta``h_fg ``left(delta^4 over 4 right)]

[eqn: therefore ```delta``=``left[ {4``mu_f ``k_f ``z``(T_gi ``-``T_w `)} over {g``sin theta ``h_fg ``rho_f ``(rho_f ``-``rho_g )} right]^{1 over 4}]	(5.2.22)

or [eqn: H_f ``(z)] will be

[eqn: H_f ``(z)``=``k_f over delta ``=``left[ {rho_f ``(rho_f ``-``rho_g ``)``g``sin theta ``h_fg ``k_f^3} over {4`` mu_f ``z ``(T_gi ``-``T_w ``)} right]^{1 over 4}]              (5.2.23)

and

[eqn: "Nu" ``=``{H_f ``z} over {k_f} ``=``left[ {rho_f ``(rho_f ``-``rho_g ``)``g``sin theta `` h_fg ``z^3} over {4 mu_f ``k_f ``(T_gi ``-``T_w ``)} right]^{1 over 4}]               (5.2.24)

The mean value of heat transfer coefficient over the whole surface is	[eqn: bar H_f ``=``1 over z ``int_0 ^z ``H_f ``(z)``d z ``=``0.943 left[ {rho_f ``(rho_f ``-``rho_g ``)``g``sin theta ``h_fg ``k_f^3} over {mu_f ``z``(T_gi ``-``T_w ``)} right]^{1 over 4}]    (5.2.25)

or

[eqn: bar H_f ``=``{Gamma_z ``h_fg} over {z``(T_gi ``-``T_w ``)}]	(5.2.26)

where [eqn: Gamma_z] is condensate flow at z from top.

Now, from Eqs. (5.2.21) and (5.2.26) ,

[eqn: delta ``=``{k_f ``Gamma_z ``dz} over {bar H_f ``z``d Gamma}]	(5.2.27)

And from Eqs. (5.2.19) and (5.2.27)

[eqn: k_f ``left[ ``{rho_f ``(``rho_f ``-``rho_g )``g``sin theta} over {3``mu_f} right]^{1 over 3} {dz} over z ``=``{bar H_f ``Gamma^{``1over3} ``d Gamma} over {Gamma_z}]

Integrate over z

[eqn: bar H_f ``=``0.925 ``left[ {rho_f ``(rho_f ``-``rho_g ``)``g``sin theta ``k_f^3} over {mu_f ``Gamma_z} right]^{1 over 3}]	(5.2.28)

Let [eqn: Re_Gamma ``=`` 4 Gamma_z ``/ mu_f] and [eqn: sin theta ``=``1]

[eqn: Bar H_f over k_f ``left[ {mu_f^2} over {rho_f ``(rho_f ``-``rho_g ``)``g} right]^{1 over 3} ``=``1.47``Re_Gamma^{- 1 over 3}], [eqn: Re_Gamma ``=`` {4 Gamma_z} over mu_f]              (5.2.29)

D. Condensation on horizontal tube

By Nusselt

[eqn: H_f ``(alpha) ``=``0.693 left [ {rho_f ``(rho_f ``-``rho_g ``)``g``sin alpha ``k_f^3 } over {Gamma_alpha ' ``mu_f} right]^{1 over 3}]                        (5.2.30)

where [eqn: alpha]  ; angle between horizontal tube and the vertical

[eqn: Gamma_alpha '] ; the local mass flow rate per unit length

[eqn: therefore ``bar H_f ``=``0.725 ``left[ {rho_f ``(rho_f ``-``rho_g )``g``h_fg ' k_f^3} over {D``mu_f ``(T_gi ``-``T_w ``)} right]^{1 over 4}]                     (5.2.31)

where D is outer diameter of tube,

[eqn: h_fg '] : modified [eqn: h_fg] to improve the original Nusselt theory	by Rohsenow

≜ [eqn: h_fg ``left [ 1``+``0.68 ``left ( { c_pf ``Delta T_f } over {h_fg} right) right]]                         (5.2.32)

or	[eqn: bar H_f over h_f ``left[ mu_f^2 over {rho_f ``(rho_f ``-``rho_g ) ``g} right]^{1 over 3} ``=``1.51 Re_Gamma ^{- 1 over 3}]                    (5.2.33)

- thicker condensate layer on subsequent tube

[eqn: therefore] HT coefficient will be reduced.

▶ Condensation within a horizontal tube

① Stratified flow

- small vapor velocity and low interfacial

shear forces

- modify Nusselt equation on the outside of a horizontal

tube as,

[eqn: ``bar H_f ``=``F``left[ {rho_f ``(rho_f ``-``rho_g )``g``h_fg ' k_f^3} over {D``mu_f ``(T_gi ``-``T_w ``)} right]^{1 over 4}]      (5.2.34)

where F is a factor which depends on the angle subtended from the

tube center to the chord forming the liquid level.

② Slug, plug and wavy flow

- significant vapor shear force

- different coefficient from top to bottom of the tube

☞ Top  ;   [eqn: F ~=~ 0.31 ``left[ {G_g D} over mu_g right] ^0.12]

☞ Bottom ;  [eqn: left[ {H_f (z) D} over k_f right] ~=~ {phi_lo^2 `` sqrt{8(G_f D /mu_f )} } over {5``[1+(1/Pr_f ) `` ln``(1+5``Pr_f )]}] (5.2.35)

E. The influence of various parameters on the  film condensation

① Turbulence

- Experiments shows that the wavy surface

has 40 ～ 50% higher HT coefficient than

theoretical value

∴ [eqn: H _{f} ^{0.85} `=~1.2H _{f} `(of`Eq.``(5.2.28))]  (5.2.36)

By Colburn, the heat transfer coefficient for

the case of turbulent is

[eqn: {h_f (z)} over k_f left[ mu_f^2 over {rho_f (rho_f - rho_g ) g } right]^5 ~=~ 0.056 ~left(4GAMMA_z over mu_f right)^0.2 ~left({c_pf ``mu_f } over k_f right)^{1 over 3}]

(5.2.37)

② Influence of interfacial shear. (That is, vapor velocity exists)

For inclined plane surface,  Eq.(5.2.18) will be

[eqn: ( delta -y)~ dz ~left(rho_f ``g~ sin theta - left(dP over dz right)right) ~+~ tau _i ``dz ~=~ mu`` left(du_y over du right) dz]       (5.2.38)

Define fictitious vapor density, [eqn: rho_g^*] , as

[eqn: dP over dz ~=~ rho_g^* g ~sin theta]             (5.2.39)

If the only pressure gradient is due to vapor head,

then [eqn: rho_g^* ~=~ rho_g]

Inserting Eq. (5.2.39) into (5.2.38) and integrating

[eqn: u_y ~=~ {(rho_f - rho_g^* ) g ~sin theta } over mu_f ~left[ y ~delta - y^2 over 2 right] ~+~ {tau_i ``y } over mu_f]            (5.2.40)

Then,  [eqn: GAMMA ~=~ rho_f INT _{o }^{delta } u_y ~dy~=~ left[{rho_f (rho_f - rho_g^* ) g ~sin theta ~delta^3 } over 3mu_y right ] ~+~ {tau_i ~rho_f ~delta^2 } over 2mu_f]

∴ [eqn: {d GAMMA} over {d delta} ~=~ left[ {rho_f (rho_f - rho_g^* ) g ~sin theta ~delta^2 } over mu_f right] ~+~ {tau_i ~rho_g ~delta } over mu_f]           (5.2.41)

With same procedure as that from Eq. (5.2.20) to (5.2.22),

[eqn: rm left[{4mu_f ~k_f ~z (T_gi - T_w ) } over {g ~sin theta ~h_{fg}^prime `` rho_f (rho_f -rho_g^* ) } right] ~=~ delta^4 ~+~ 4 over 3 left[{tau_i ~theta^3 } over {(rho_f - rho_g^* ) g ~sin theta } right ]]   (5.2.42)

③ non-condensable gas effect.

o The noncondensable gas near the     condensate film protects the          diffusion of the vapor from the       bulk mixture to the film and thus    reduce the mass and energy          transfer.

그림 1. Boundary Layer Temperature and Pressure Distributions

- simultaneously solve conservation

equations for both condensate film     and vapor-gas layer

o In nuclear plants, N2  from

accumulator or FP during LOCA

o hamper the performance of the

pressurizer.

1) For stagnant vapor-gas mixture

☞ By Sparrow and Lim, for free convection arising from density                   differences, the  condensing rate is proportional to

● bulk gas mass fraction [eqn: (rho_ell sigma _ell /rho_v sigma_v )^{1 over 2 }]

● vapor-gas mixture Schmidt #[eqn: (c_{p ell} (T_I - T_w ) / P_r_g ~rho_fg })]

그림  The Influence of Noncondensable Air in Steam

∴ non-condensable gas effect

increase with increase of

√  Sc

√ [eqn: (rho_ell sigma_ell / rho_v sigma_v )^{1 over 2}]

2) Moving vapor with noncondensable

gas

In this case, the non-condensable

gas effect is smaller than the case        of stationary vapor-gas  mixture.

That is, the gas concentration at          the interface is low relatively due to

the “sweep effect" of the moving          vapor-gas mixture.

3) Pressure

그림  Effect of Pressure on Condensation of Steam-Air Systems

- increasing pressure (or saturation

temperature) lower the effect of

noncondensables due to the less

effect on vapor saturation

temperature at higher pressure

- the heat transfer rate must be

smaller at lower pressure than at

higher pressure

4) Superheat in vapor

- larger density difference between

bulk mixture and mixture near the

interface with noncondensables than in pure vapor for larger superheat

-> larger buoyancy effects make much stronger convective flow to                  sweep past the cooled surface, and thus reduce the interfacial                   concentration & the thickness of noncondensable film

∴ The heat transfer rate increases.

5) RELAP5/MOD3 condensation model with noncondensible gas

- noncondensible component is assumed to be in thermal and mechanical

equilibrium

- three correlations for condensation heat transfer calculations

1) laminar film condensation on an inclined plane

; standard Nusselt film condensation correlation

2) laminar film condensation inside a horizontal tube with a stratified

liquid surface

; modification of original theory

3) turbulent film condensation inside a vertical tube

; Carpenter and Colburn correlation

- the reduction multiplier, FNC, for the presence of noncondensible gas

[eqn: FNC ~ =~ {(p_s - p_min )} over {p * f(Re_g )} * F] ,        for 0 < Reg < 20,000         (5.2.43)

where   [eqn: f(Re_g ) = 5 over {1+0.0001 Re_g }]

[eqn: F = 1+f(Re_g )*exp[-5( P_a over p ) ]]

and ps, pa, p are partial pressures of steam and air, and total pressure of         mixture, repectively. pmin is minimum pressure in the steam table.

F. Waviness effect

- The rate of mass transfer in the       liquid phase is enhanced by a         factor of 2-3 in the presence of       waves.

- The entire process of transfer in       the liquid is controlled by the         large scale wave structure.

o The rate of transfer is high          just after the lump passes and        decays with time until next lump arrives & a fresh film is generated;           renewal process & unsteady mass transfer

- The transfer in the gas phase is controlled by the small wave                   structure - the small waves act as a roughness- but, the roughness is of        unique shape.