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A thermodynamic analysis for heterogeneous boiling nucleation on a superheated wall
International Journal of Heat and Mass Transfer 54 (2011) 4762–4769
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A thermodynamic analysis for heterogeneous boiling nucleation on a superheated wall Xiaojun Quan, Gang Chen, Ping Cheng ⇑ MOE Key Laboratory for Power Machinery and Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 January 2011 Received in revised form 20 May 2011 Accepted 20 May 2011 Available online 28 June 2011
A thermodynamic model based on Gibbs free energy and availability is developed for onset of heterogeneous nucleation on heated surfaces with different wettabilities in pool boiling. Different from classical nucleation theory, this model takes into consideration the temperature gradient in the superheated liquid layer adjacent to the wall as well as the contact angle between the liquid and the wall. Using Gibbs free energy equilibrium condition, a closed form solution is obtained on the critical radius for onset of heterogeneous boiling nucleation on walls with different wettabilities. Effects of contact angles and wall temperatures on the critical radius, the wall temperature gradient of the superheated liquid layer and the heat flux at onset of heterogeneous nucleate boiling are illustrated. These effects on the change of availability during the heterogeneous nucleation process, representing the energy barrier for the occurrence of the first-order phase transition, are also discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Heterogeneous nucleation Thermodynamic equilibrium Critical radius Wall temperature Contact angle
1. Introduction In the classical theory of homogeneous nucleation [1] in a superheated liquid at a uniform temperature Tl and pressure pl, the critical radius for onset of homogeneous nucleation is
r Homo ¼ 2r‘v T s =qv hfg DT c
ð1Þ
with Ts being the saturation temperatures corresponding to pl and DT = Tl Ts. The change in availability (defined as W = U + p0V T0S with p0 and T0 being the pressure and temperature at a reference condition) at equilibrium during homogeneous nucleation is
DWHomo ¼ WHomo ðT l ; pl Þ W0 ðT l ; pl Þ ¼ c c
4prlv r 2c : 3
ð2Þ
For heterogeneous nucleation on a perfectly smooth wall at a wall temperature Tw equal to the temperature of the superheated liquid at a uniform temperature of Tl, i.e., no temperature gradient, it is known [1,2] that the critical radius is also given by Eq. (1). The change in availability during heterogeneous nucleation at equilibrium is given by
DWc ¼
prlv r2c 3
ð2 þ 3 cos h cos3 hÞ:
ð3Þ
The above equation with h = 0 reduces to homogeneous nucleation given by Eq. (2). Comparing Eqs. (2) and (3), it is apparent that (i) the homogeneous nucleation would require more work to achieve ⇑ Corresponding author. Tel./fax: +86 21 34206337. E-mail address: [email protected] (P. Cheng). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.05.026
than heterogeneous nucleation in a superheated liquid at uniform temperature; (ii) the classical homogeneous nucleation theory given by Eqs. (1) and (2) is identical to heterogeneous nucleation theory when the wall temperature is equal to the uniform liquid temperature (Tw = Tl) and the contact angle is zero (h = 0), i.e., a perfectly hydrophilic wall. 2. Existing models for heterogeneous nucleation with wall temperature gradient When a wall exists in the superheated liquid, however, a wall temperature gradient k always exists in the superheated liquid during heterogeneous nucleation as shown in Fig. 1, where the solid curve is the temperature distribution in the thermal boundary layer dt which increases from a bulk temperature of T1 to Tw at the wall. The superheated liquid layer [3] has a thickness det (where det < dt for subcooled boiling with T 1 < T s as seen from Fig. 1) within which the fluid temperature is higher than the saturated temperature Ts corresponding to the pressure of the liquid. In the following, we will first review existing models for heterogeneous nucleation with liquid temperature gradient near the wall taken into consideration. 2.1. Hsu’s model Hsu [4] was apparently the first one to study the onset of heterogeneous nucleation in a cavity, having mouth radius of rca, on a superheated wall at Tw (where Tw > Ts(pl)) in pool boiling with liquid temperature gradient near the wall taken into consideration.
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Nomenclature cp g G h hfg k m p q r R s S T U V x
specific heat (J/(kg K)) specific Gibbs free energy (J/kg) Gibbs free energy (J) specific enthalpy (J/kg) latent heat (J/kg) temperature gradient (K/m) mass (kg) pressure (Pa) heat flux (W/m2) bubble radius (lm) gas constant (J/(kg K)) specific entropy (J/(kg K)) entropy (J/K) temperature (K) internal energy (J) volume (m3) projection height from the heated wall (m)
h
deg
H
H = U + pV TwS (J)
density (kg/m3) surface tension of liquid and vapor (N/m) availability (J)
q rlv W
Subscripts 0 reference condition, initial state b bubble c critical state ca cavity l liquid s saturated state v vapor w wall 1 bulk liquid Superscript Homo homogeneous nucleation
Greek symbols dt thermal boundary layer thickness (m) superheated liquid layer thickness (m) det k thermal conductivity (W/(m K)) l specific chemical potential (J/kg)
Acronyms ONB onset of nucleate boiling
and since Tl = Tv, we have
T s ðpv Þ ¼ T v ¼ T l ¼ T s ðpl Þ þ
Fig. 1. Formation of an embryo bubble in a superheated liquid layer with thickness det on a heated wall.
He assumed that a linear temperature distribution exists in the thermal boundary layer (with a thickness of dt) of the liquid adjacent to the wall as,
x ; T l ðxÞ T 1 ¼ ðT w T 1 Þ 1 dt
ð0 6 x 6 dt Þ:
ð4aÞ
or
T l ðxÞ ¼ T w kx with k ¼
ðT w T 1 Þ ; dt
ð4b; cÞ
where k is the temperature gradient of the liquid near the wall, x is the distance from the heated wall, T1 is the liquid temperature at the edge of the thermal boundary layer. Hsu [4] proposed that the onset of nucleation occurs when the temperature at tip of the bubble, Tl(xb), is at least equal to the saturation temperature corresponding to the pressure inside the bubble Ts(pv), i.e.,
T l ðxb Þ P T s ðpv Þ;
ð5Þ
where xb is the projection height of the vapor bubble formed on the heated wall (see Fig. 1) and Ts(pv) is the saturation temperature corresponding to the vapor pressure inside bubble. The value of Ts(pv) can be determined approximately as follows. The Clausius–Clapeyron equation gives:
T l ¼ T s ðpl Þ þ
2rlv T s ðpl Þ : rb qv hfg
ð6aÞ
Making the approximation
T s ðpv Þ ffi T v
ð6bÞ
2rlv T s ðpl Þ ; r b qv hfg
ð6cÞ
which is a constant independent of x. Fig. 2(a) is a sketch showing the temperature profiles in the bubble given by Eq. (6c) (represented by a vertical straight line) and in the thermal boundary layer given by Eq. (4a) or (4b) (represented by an inclined straight line) given by Hsu’s model [4]. Note that for a bubble embryo to grow, the condition of Tl P Tv (both of which are greater than saturation temperature of the liquid Ts(pl)) must be satisfied on the vapor/liquid interface. It can be seen from Fig. 2(a) that the condition of Tl > Tv on the vapor/liquid interface is always satisfied in Hsu’s model [4]. For a truncated bubble sitting on a cavity mouth with a radius of rca, Hsu [4] assumed that the height and radius of the bubble are related to the radius of the month rca by xb = 2rca and rb = 1.25rca. The values of rca for a given wall superheat DTw = Tw Ts can be obtained by imposing Eq. (5), i.e., letting Tl(xb) from Eq. (4a) equal to Ts(pv) from Eq. (6c), resulting in a second-order algebraic equation in terms of rca which has two real roots if
dt P
12:8rlv T s ðT w T 1 Þ
qv hfg ðDT w Þ2
ð7Þ
:
Assuming that heat is transferred by conduction in the superheated layer and substituting the above expression into the heat conduction equation gives the required heat flux for onset of nucleation 2
qONB ¼
kðT w T 1 Þ khfg qv ðT w T s ÞONB ¼ ; dt 12:8rlv T s
ð8Þ
which is the expression originally given in Hsu’s paper [4]. Note that for a truncated bubble sitting on a cavity mouth with radius of rca and assuming that the bubble is of spherical shape, the bubble height xb and bubble radius rb are related to rca and contact angle h by xb = rca(1 + cosh)/sinh and rb = rca/sinh. A comparison of these expressions with Hsu’s assumptions of xb = 2rca and rb = 1.25rca shows that sin h = 0.8 and cosh = 0.6, i.e., h = 53.1°. Thus, Hsu’s assumptions of xb = 2rca and rb = 1.25rca are equivalent to the case
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tribution of the thermal boundary layer given by Eq. (4b) is an inclined straight line which intercepts each other at x = xb/2, where Tl(xb/2) = Tv. It can be seen from this figure that Tv > Tl(x) in the upper half of the vapor bubble (xb =2 < x 6 xb ), and Tv < Tl(x) for the lower half of the vapor bubble (0 6 x < xb =2). It follows that the condensation takes place along the upper half of vapor/liquid interface while evaporation takes place along the lower half of the interface. Based on second law of thermodynamics and defining a function H = U + plV TwS, Wu et al. [8] on discussed the growth and collapse of a bubble based DH. In this paper, we study heterogeneous boiling nucleation taking into consideration the temperature gradient in the superheated liquid adjacent to the heated wall and contact angle of the wall based on consideration of the changes in Gibbs free energy (G) and availability function (W) after an embryo is formed. At thermodynamic equilibrium condition, a closed-form expression is obtained for embryo’s critical radius at onset of heterogeneous nucleation in a superheated liquid layer. The change in the availability function DW is used to investigate the growth and collapse of an embryo bubble.
3. Heterogeneous nucleation with wall temperature gradient In this section, we will develop a thermodynamic model for heterogeneous nucleation on a wall with wall temperature gradient and wettability taken into consideration from the view point of changes in Gibbs free energy and availability. 3.1. Change in Gibbs free energy
Fig. 2. Comparisons of previous models by Hsu [4] and Wu et al. [8] with the present model.
of a contact angle of 53.1°. Li and Cheng [5] extended Hsu’s theory to include the effects of contact angle and dissolved gas on heterogeneous nucleation in pool boiling, and they showed that the nucleation temperature decreased with the increase in the contact angle or amount of gas dissolved in water. Bergles and Rohsenow [6] as well as Basu et al. [7] have applied Hsu’s model for onset of nucleation in flow boiling. 2.2. Wu et al.’s model Most recently, Wu et al. [8] developed a thermodynamic model for heterogeneous bubble nucleation on a wall with temperature gradient taken into consideration. They assumed that the liquid temperature is linear within the thermal boundary layer which is also given by Eq. (4b). Based on the assumptions that (i) temperature inside the bubble Tv is uniform and (ii) the net heat transfer through the bubble interface is equal to zero, i.e.,
Z klv Alv
ðT l T v Þ dAlv ¼ 0: dlv
ð9Þ
Wu et al. [8] obtained the following expression for temperature of the vapor bubble
T v ¼ T w krb ð1 þ coshÞ=2 ¼ T w kxb =2;
During a constant temperature and pressure process, the second law of thermodynamics given by dS P 0 (where S is the entropy) leads to dG 6 0 where G is the Gibbs free energy defined as:
G ¼ U þ pV TS;
where U is the internal energy. For bubble nucleation which is a constant pressure and constant temperature process,
dG ¼ Gv Gl 6 0;
ð12Þ
where Gv(pv, Tv) = Uv + pvVv TvSv and Gl(pl, Tl) = Ul + plVl TlSl, respectively. It is noted that the chemical potential l is defined as
l ¼ h Ts:
ð13Þ
Therefore, the chemical potential on a mass basis is equal to the specific Gibbs function (l = G/m = g) for a pure substance. Since the temperature of superheated liquid Tl is equal to that of superheated vapor Tv at thermal equilibrium, the change in Gibbs free energy between vapor and liquid is:
Dg ¼ g v g l ¼ lv ll ¼ ðhv hl Þ T l ðsv sl Þ;
ð14Þ
where
hv hl ¼ hfg þ ðcpv cpl ÞðT l T s Þ;
ð15Þ
hfg Tl p sv sl ¼ R ln v : þ cpv cpl ln Ts Ts pl
ð16Þ
Substituting Eqs. (15) and (16) into Eq. (14) gives
Dg ¼ g v g l ¼ Dl ¼
ð10Þ
where Tw is the wall temperature and xb = rb(1 + cosh) is the bubble projection height from the wall with h being the contact angle. Fig. 2(b) is a sketch showing that the temperature profile in the bubble given by Eq. (10) is a vertical line and the temperature dis-
ð11Þ
hfg p DT þ RT l ln v ; Ts pl
ð17Þ
where DT = Tl Ts with Ts being the saturation temperature corresponding to the liquid pressure pl, and hfg is the latent heat; pv and pl are related by the Young-Laplace equation given by
pv ¼ pl þ 2rlv =r
ð18Þ
X. Quan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4762–4769
if the embryo bubble is in mechanical equilibrium with the surrounding superheated liquid. Note that if gv < gl, i.e., Dg < 0, then nucleation will take place. Conversely, if gv > gl, i.e., Dg > 0, then the evaporation is impossible and condensation will take place instead. Setting Dg = 0 in Eq. (17) gives the critical radius for homogeneous nucleation given by Eq. (1). To investigate the critical condition for heterogeneous bubble nucleation with k – 0 based on Gibbs free energy consideration, we now make the following assumptions: (i) A necessary condition for bubble nucleation is that the temperature at the tip of the bubble (at x = xb) is at least equal to the saturation temperature corresponding to the liquid pressure
T l ðxb Þ P T s ðpl Þ
ð19Þ
where xb is the projection height of bubble from the heated wall (see Fig. 1). (ii) The temperature of the superheated liquid Tl within the superheated layer is assumed to be linear [3] given by Eq. (4b) with k being the temperature gradient at the wall given by,
k ¼ ðT w T s Þ=det
ð20Þ
(iii) The temperature of vapor bubble Tv(x) in the differential control distance dx is assumed to be equal to the liquid phase outside of the bubble Tl(x) at the same distance x from the heated wall, i.e.,
T v ðxÞ ¼ T l ðxÞ ¼ T w kx:
ð21Þ
which is the ‘‘local thermal equilibrium’’ assumption along the vapor/liquid interface. Fig. 2(c) is a sketch showing the temperature profiles of the vapor and the superheated liquid in the present model where Tv(x) = Tl(x) = Tw kx is an inclined straight line with temperature of vapor bubble greater than the saturation temperature Ts(pl) everywhere along the vapor/liquid interface except at the tip of the bubble where Tv(xb) = Tl(xb) = Ts(pl). We can speculate the evaporation and condensation processes along the interface based on Eq. (17), where the second term is always positive as pv > pl. At the tip of the bubble x = xb where Tl = Ts (i.e., DT = 0), the first term in Eq. (17) vanishes. It follows that lv > ll near the tip which means that the interface of the embryo bubble near the bubble tip will be condensed due to the higher vapor chemical potential compared with that of the liquid. Away from the tip where DT > 0, the first term in Eq. (17) becomes more negative at smaller x. At some point at x where the first term is larger than the second term in Eq. (17) so that lv < ll and therefore evaporation will take place along the embryo bubble interface near the wall. At steady state, the total mass of condensation is equal to the total mass of evaporation along the interface so that the radius of the embryo bubble remains constant. Since the liquid temperature Tl(x) given by Eq. (21) is a function of x within the superheated liquid layer, the change in the Gibbs free energy before and after a bubble is formed on the heated surface can be obtained by integrating Eq. (17) with respect to x from x = 0 to x = xb to give:
Z
DG ¼ Gv ðT l ; pv Þ Gl ðT l ; pl Þ ¼ ½g v ðT l ; pv Þ g l ðT l ; pl Þdmv Z ¼ qv DgdV v h Z xb i h p ¼ qv fg ðT l T s Þ RT l ln v p r2 ðr cos h xÞ2 dx: Ts pl 0 ð22Þ
4765
Substituting Eqs. (18) and (21) to Eq. (22) and integrating with respect to x from x = 0 to x = xb give,
q pð1 þ cos hÞ2 r3 DG ¼ v 12 ½ð8 4 cos hÞðT w T s Þ krð1 þ cos hÞð3 cos hÞhfg =T s R lnðpv =pl Þ½ð8 4 cos hÞT w krð1 þ cos hÞð3 cos hÞÞ ð23Þ which has taken into consideration temperature variations in the embryo bubble and in the superheated liquid layer with a thickness of det . Note that if Gv < Gl or DG < 0, then the embryo bubble will grow. Conversely, if Gv > Gl or DG > 0, then the bubble will collapse. It is relevant to note that the assumption for onset of nucleation in the present model given by Eq. (19) differs from Hsu’s criteria given by Eq. (5). Since pv > pl, i.e. Ts(pv) > Ts(pl), therefore it can be seen from Eq. (17) that gl is always greater than or equal to gv at the liquid/vapor interface for Hsu’s criteria given by Eq. (5), which means that the nucleation condition along the vapor/liquid interface is always satisfied by Hsu’s criteria, i.e., DG < 0. While for present model, onset of nucleation take places when DG = 0. The above implies that the criteria for onset of nucleation given by Hsu [4] is more strict than that given in the present model. It has also been pointed out by Davis and Anderson [9] that Hsu’s criteria provides an upper limit to the wall superheat required to initiate nucleate boiling. In the present model, the critical radius can be obtained by setting Eq. (23) to zero (DG = 0) for the following two cases: (i) For heterogeneous nucleation at uniform liquid temperature (Tw = Tl = Tv), i.e., for the case of k = 0, the critical radius can be determined by setting Eq. (23) to zero, which reduces to Eq. (1) for homogenous nucleation. This means that the critical radius for heterogeneous nucleation without a temperature gradient (i.e., k = 0) is the same as that of homogeneous nucleation [1]. (ii) For the case of heterogeneous nucleation with a temperature gradient (k – 0) in the superheated liquid, the critical temperature gradient at the wall with temperature of Tw for the embryo bubble with radius r is given by Eq. (20) by substituting det ¼ xb to give
kc ðrÞ ¼ ðT w T s Þ=xb ¼ ðT w T s Þ=½rð1 þ coshÞ:
ð24Þ
Substituting Eq. (24) for the term k in Eq. (23) gives the change in Gibbs free energy during heterogeneous nucleation with k – 0: cos hÞ2 r 3 DG ¼ qv pð1þ 12 ð5 3 cos hÞhfg ðT w T s Þ=T s R lnðpv =pl Þ½ð5 3 cos hÞT w þ ð3 cos hÞT s g
ð25Þ
At equilibrium, the embryo’s Gibbs free energy must be equal to the liquid’s Gibbs free energy, i.e., Gv(Tv, pv) = Gl(Tl, pl). Thus, setting Eq. (25) to zero gives the critical radius for onset of nucleation for heterogeneous nucleation with k – 0:
rc ¼
2Rrlv T s ½ð5 3 cos hÞT w þ ð3 cos hÞT s pl hfg ðT w T s Þð5 3 cos hÞ
ð26Þ
The critical temperature gradient for critical radius rc for onset of heterogeneous nucleation with k – 0 can be obtained by substituting Eq. (26) into Eq. (24) to give,
kc ðr c Þ ¼
hfg pl ð5 3 cos hÞðT w T s Þ2 2Rrlv T s ð1 þ cos hÞ½ð5 3 cos hÞT w þ ð3 cos hÞT s
ð27Þ
Consequently, the heat flux at ONB for heterogeneous nucleation with k – 0 can be obtained as,
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qONB
¼ kkc ðr c Þ khfg pl ð5 3 cos hÞðT w T s Þ2ONB
¼ 2Rrlv T s ð1 þ cos hÞ ð5 3 cos hÞðT w ÞONB þ ð3 cos hÞT s ð28Þ
where k is the thermal conductivity of the superheated liquid.
Solving Tw from Eq. (24) in terms of kc and substituting it into Eq. (32) gives the change in availability in terms of (r, h, kc) for heterogeneous nucleation with k – 0 as, hÞ2 r 3 DWðr; h; K c Þ ¼ qv pð1þcos ð5 3 cos hÞkc rhfg ð1 þ cos hÞ=T s 12 R lnðpv =pl Þ½kc rð5 3 cos hÞð1 þ cos hÞ þ ð8 4 cos hÞT s g þprlv r2 ð2 þ 3 cos h cos3 hÞ=3
ð33Þ
3.2. Change in availability It should be noted that heating the liquid phase to a supersaturated state is only a necessary condition for the boiling nucleation to occur. However, there is no guarantee that phase transition from liquid to vapor will actually take place due to the fact that the metastable state is separated from the truly stable one by an energy barrier. The change of availability DW represents such an energy barrier for the occurrence of the first-order phase transition, and an increase of which will cause onset of nucleation more difficult. The change in availability for homogeneous nucleation is [1]:
4. Results and discussion 4.1. Critical radius for onset of heterogeneous nucleation
DWHomo ¼ W W0 ¼ Gv ðT v ; pv Þ Gl ðT l ; pl Þ þ ðpl pv ÞV v þ 4prlv r 2 :
ð29aÞ
Substituting Eqs. (18) and (17) (where the latter was obtained under the thermal equilibrium condition of Tl = Tv) into Eq. (29a) gives:
DWHomo ¼
Table 1 summarizes the results obtained for the critical radius and change in availability at equilibrium for the cases of heterogeneous nucleation with and without liquid temperature gradient and contact angle as well as homogeneous nucleation with liquid temperature equal to the wall temperature. Note that the change in availability for heterogeneous nucleation (at the same h) at equilibrium is the same for both k = 0 and k – 0 as mentioned earlier.
hfg 4pqv r 3 p 4prlv r 2 DT þ RT l ln v : 3 pl Ts 3
ð29bÞ
At equilibrium, the first term on the right-hand side of the above equation vanishes which reduces to the expression given by Eq. (2). On the other hand, the change in availability for heterogeneous nucleation for any k is [1]:
DW ¼ W W0 ¼ Gv ðT v ; pv Þ Gl ðT l ; pl Þ þ ðpl pv ÞV v 1 1 þ 4pr 2 rlv ð1 þ cos hÞ þ cos hð1 cos2 hÞ ; 2 4
Fig. 3 shows the variation of the critical radius for onset of heterogeneous nucleation of water on a wall at different wall temperatures with different contact angles, which was computed from Eq. (26) for water at atmospheric pressure (pl = 101,325 Pa, Ts = 373 K). It can be seen that the critical embryo bubble radius rc of water decreases with increase of Tw and the contact angle h (i.e., a more hydrophobic surface). Consequently, the value of rc for heterogeneous nucleation with temperature gradient (k – 0), depends on the contact angle, which is different from the results of heterogeneous nucleation without temperature gradient (i.e. k = 0), where rc is independent of contact angle for the same working fluid [1]. At a given contact angle, rc decreases sharply at first with increasing Tw and decreases gradually with further increase in Tw. The predicted values of rc shown in Fig. 3 are in fair agreement with
ð30Þ
where the first two terms on the right-hand side of Eq. (30) depending on k = 0 or k – 0 and the last term indicates the presence of the wall. Substituting Eq. (18) into Eq. (30) gives
DWðr; h; T w Þ ¼ DG þ
prlv r2 3
2 þ 3 cos h cos3 h ;
ð31Þ
where the first term DG on the right-hand side of Eq. (31) depends on k ¼ 0 or k – 0. For DG = 0 at equilibrium, Eq. (31) reduces to the expression given by Eq. (3) for DWc which is the change in availability for heterogeneous nucleation at equilibrium. Substituting Eq. (25) into Eq. (31) gives the change in the availability for k – 0, 2 3 DWðr; h; T w Þ ¼ qv pð1þ12cos hÞ r ð5 3 cos hÞhfg ðT w T s Þ=T s R lnðpv =pl Þ½ð5 3 cos hÞT w þ ð3 cos hÞT s g þprlv r 2 ð2 þ 3 cos h cos3 hÞ=3 ð32Þ
Fig. 3. Variation of rc with Tw at various h for heterogeneous nucleation of water in pool boiling.
Table 1 Summary of critical radius and the change in availability obtained for heterogeneous and homogeneous nucleation. Heterogeneous nucleation
Homogeneous nucleation
k – 0, h – 0
k – 0, h = 0
k = 0, h – 0 Ref. [1]
k = 0, h = 0 Ref. [1]
rc
2Rrlv T s ½ð5 3 cos hÞT w þ ð3 cos hÞT s Eq. (26): pl hfg ðT w T s Þð5 3 cos hÞ
2Rrlv T s ðT w þ T s Þ Eq. (26) with h = 0: pl hfg ðT w T s Þ
2rlv T s Eq. (1): qv hfg ðT w T s Þ
Eq. (1):
DW c
Eq. (3):
Eq. (3) with h = 0:4prlv r 2c =3
Eq. (3):
prlv r 2c ð2 þ 3 cos h cos3 hÞ=3
prlv r2c ð2 þ 3 cos h cos3 hÞ=3
2rlv T s qv hfg ðT ‘ T s Þ Eq. (2): 4prlv r 2c =3
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measurements of critical radii of a few micrometers at low fluid inlet velocity given in [10]. Also, the critical radius for onset of homogeneous nucleation given by Eq. (1) is presented as a short dotted line in Fig. 3, showing that the critical radius for onset of homogeneous nucleation is smaller than heterogeneous nucleation when the liquid temperature of homogeneous nucleation is equal to the wall temperature of heterogeneous nucleation. 4.2. Critical wall temperature gradient for onset of heterogeneous nucleation Eq. (27) for critical wall temperature gradient kc(rc) is presented in Fig. 4, showing that the increase in Tw and h lead to the increase of kc(rc). Since k ¼ ð@T l =@ x Þw = (Tw Ts)/det , the value of k is related to the superheated liquid layer thickness det , which is equal to the bubble height for critical wall temperature gradient kc. At a specific Tw, nucleation takes place only if k 6 kc ðrc Þ, i.e., T l ðxb Þ P T s . When k > kc(rc) at a specific Tw, which means det is too thin, boiling nucleation will be suppressed and the embryo would not grow spontaneously. 4.3. Nucleation heat flux We now introduce the dimensionless heat flux and the dimensionless superheat as,
~ONB ¼ q
qONB rlv ; khfg qv T s
DT~ ONB ¼
ðT w T s ÞONB ; Ts
ð34Þ
ð35Þ
Thus, Eqs. (28) and (8) can be expressed in dimensionless form as follows:
~ONB ¼ q
pl ð5 3 cos hÞDT~ 2ONB h i 2qv RT s ð1 þ cos hÞ ð5 3 cos hÞDT~ ONB þ ð8 4 cos hÞ
ð36Þ
~ONB ¼ q
DT~ 2ONB : 12:8
ð37Þ
~ONB versus DT~ ONB Eq. (36) with contact angles of 0°, 60° and 120° for q in pool boiling of water are presented in Fig. 5. When the contact angle is increased, it can be concluded from Fig. 5 that: (i) more heat flux is needed for ONB at the same wall temperature, and (ii) the wall temperature at ONB decreases for the same heat flux. Hsu’s
Fig. 5. Comparison of nucleation heat flux in pool boiling of water based on the present model and Hsu’s model as well as experimental data.
model given by Eq. (37) is also plotted for comparison. It can be seen from Fig. 5 that the nucleation heat flux given by Hsu’s model for an equivalent contact angle of 53.1° is slightly lower than the present model given by Eq. (36) for a contact angle of 60° at the same wall superheat. This can be explained based on the criteria for onset of nucleation given by the two models as follows: Hsu’s criteria for onset of nucleation is Tl(xb) P Ts(pv), while the criteria for onset of nucleation in the present model is Tl(xb) P Ts(pl). Since Ts(pv) > Ts(pl), the temperature at the tip of the bubble required for boiling inception is higher for Hsu’s model than the present model. This means that for nucleate bubble with the same rb, the superheated liquid layer thickness det is larger for Hsu’s model than the present model as can be seen from Fig. 2. Consequently, according to k ¼ ðT w T s ðpl ÞÞ=det , it can be seen that for the same heat flux at ONB (i.e., the same wall temperature gradient k), the required wall superheat is larger for Hsu’s model. In other words, for the same wall superheat, the wall temperature gradient is smaller in Hsu’s model (i.e., smaller heat flux). The above explanations show that Hsu’s boiling inception criteria [4] is more restrictive for onset of heterogeneous nucleation than the present model, leading to a higher superheat required for ONB at the same heat flux. Experimental results obtained by various investigators [11–13] for inception of pool boiling of water are also presented in Fig. 5 for comparison purposes. These experimental results include (i) four data points obtained by Petrovic et al. [11] under four different subcooled conditions, (ii) two data points obtained by Li and Peterson [12] for degassed and gas-dissolved water, respectively, and (iii) nine data points obtained by Li and Peterson [13] for using nine different heated surfaces. Except for three data points obtained by Li and Peterson [13], most of the other experimental data are shown to be in good agreement with the present model and Hsu’s model, although detailed comparisons are not possible because the contact angles of heater surface in the experiments were not reported explicitly in the papers.
4.4. Change in availability during heterogeneous nucleation
Fig. 4. Relation between kc(rc) and Tw at various h for heterogeneous nucleation of water in pool boiling.
Fig. 6 shows the change of availability for heterogeneous nucleation with k – 0 versus embryo radius r for h = 0°, 60°, 120° for water at pl = 101,325 Pa, Ts = 373 K and Tw = 127 °C, which was computed according to Eq. (33). As shown in this figure, the value of DW has a local maximum at r = rc, which is the critical radius for heterogeneous nucleation shown in Fig. 3, corresponding to a state of unstable equilibrium. When an embryo has a radius r less than rc, any increase of r corresponds to an increase in DW, indicating
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Fig. 6. Variations of the change in availability with bubble radius at various h for heterogeneous nucleation of water in pool boiling.
that the embryos would collapse spontaneously. On the other hand, when an embryo has a radius greater than rc, any increase in r corresponds to a decrease of DW. This indicates that the embryo would grow spontaneously, resulting in heterogeneous nucleation. Fig. 6 also shows that DW required for bubble nucleation decreases with the increases of contact angle, indicating that large contact angles will favor nucleation. This is consistent with the results presented in Fig. 3, where it is shown that required wall superheat decreases with the increase of contact angle (i.e., a more hydrophobic surface) at given rc. The change in availability for homogeneous nucleation given by Eq. (29b) is also presented in Fig. 6 for comparison. It can also been seen from Fig. 6 that the critical radius for homogeneous nucleation is smaller than heterogeneous nucleation, which is consistent with the results presented in Fig. 3. It is shown from Fig. 6 that the change in availability of homogeneous nucleation (with Tl = Tw) can be smaller or larger than heterogeneous nucleation with temperature gradient (k – 0) depending on its contact angle. The change in availability for heterogeneous nucleation at equilibrium given by Eq. (3) for any k can be expressed in dimensionless form as
Dwc ¼
DWc ¼ 2 þ 3 cos h cos3 h ; prlv rc =3
ð38Þ
where the right-hand side depends only on the contact angle. Eq. (38) is presented in Fig. 7, which shows that the change in dimensionless availability at equilibrium decreases from a value of 4 (a perfectly hydrophilic surface with h = 0°) to zero (a perfectly hydrophobic surface with h = 180°) implying that nucleation becomes easier with increasing contact angle h (i.e., a more hydrophobic surface). Note that the case of h = 0° also corresponds to homogeneous nucleation. Fig. 8 shows the effects of kc on DW versus r given by Eq. (33) for water at pl = 101,325 Pa and Ts = 373 K and at h = 60°. It can be seen that DW increases with decrease of kc which corresponds to lower wall superheat as shown in Fig. 4. This indicates that a decrease in kc will suppress bubble nucleation. For the special case of kc = 0 (meaning that Tw = Ts(pl) from Eq. (24), i.e., the wall is at saturated temperature), DW approaches infinity and nucleation will not take place. 4.5. Comparisons of models for onset of heterogeneous nucleation We now compare the present model with two other existing models by Hsu [3] as well as by Wu et al. [8]. The temperature dis-
Fig. 7. The effect of contact angle h on the change in dimensionless availability for onset of heterogeneous nucleation in pool boiling.
Fig. 8. Relation between the availability and bubble radius for various kc for heterogeneous nucleation of water in pool boiling.
tributions of the three models are sketched in Fig. 2. Since the Young-Laplace equation was used in the three different models, the mechanical equilibrium criteria is satisfied in these three models. Table 2 is a summary of temperatures of the liquid and vapor and the change in Gibbs free energy for the three models for heterogeneous nucleation with liquid temperature gradient k – 0. Although the present model and those by Wu et al. [8] are based on similar assumptions of linear liquid temperature distribution and starting from the second law of thermodynamics, the two models differ from each other due to two important aspects: (i) for the vapor temperature: the present model assumes that Tv(x) = Tl(x) while in the model by Wu et al. [8], the vapor temperature was obtained based on Eq. (10) as Tv = Tw kxb/2 with xb being the bubble projection height from the wall; (ii) bubble growth and collapse is considered from the point of view of the change in Gibbs free energy DG and availability DW in present model while it was considered based on DH in the paper by Wu et al. [8], where H is different from the availability function. In Hsu’s model, the vapor temperature is given by Eq. (6c), which means that the chemical potential for vapor is equal to that for the liquid at tip of the bubble. As the liquid temperature is greater than the vapor temperature along the bubble interface except the top point, it can be concluded that DG = Gv Gl < 0 through Eq. (17). In the model considered by Wu et al. [8], the
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X. Quan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4762–4769 Table 2 Comparison of three models for heterogeneous nucleation. Model
Hsu’s model [4]
Wu et al.’s model [8]
Present model
Liquid-phase temperature distribution
Eq. (4b) and (4c): Tl(x) = Tw kx (for 0 < x < dt) where k ¼ ðT w T 1 Þ=dt 2rlv T s Eq. (6c): T v ¼ T s þ ¼ const r b qv hfg dG < 0
Eq. (4b) and (4c): Tl(x) = Tw kx (for 0 < x < dt) where k ¼ ðT w T 1 Þ=dt
Eqs. (20) and (21): Tl(x) = Tw kx (for 0 < x < det ) where k ¼ ðT w T s Þ=det Eq. (21): Tv(x) = Tl(x) = Tw kx dG = 0
Vapor-phase temperature distribution Change in Gibbs free energy (dG) during nucleation
Eq. (10): T v ¼ T w kxb =2 ¼ const dG > 0
vapor temperature was obtained using Eq. (10) based on a thermal equilibrium for flat interface without taking into consideration the effects of capillary pressure difference due to curve interface. Therefore, the vapor temperature given in model by Wu et al. [8] is overestimated, and the embryo bubble is likely to collapse because of the net condensation along the interface, i.e., DG = Gv Gl > 0.
tact angle becomes larger, it requires smaller nucleation temperature for same heat flux and thus is more favorable for nucleation; (iv) the wall nucleation temperature increases with the increase of the temperature gradient at the wall, and (v) the wall temperature gradient must be smaller than the critical wall temperature gradient given by Eq. (27) in order for heterogeneous nucleation to occur.
5. Conclusions
Acknowledgements
In this paper, a thermodynamic analysis based on Gibbs free energy and availability has been carried out to study onset of heterogeneous boiling nucleation in the superheated liquid layer, with wall temperature gradient k taken into consideration. Closed-form solutions for the critical radius of bubble and heat flux at onset of nucleate boiling (ONB) are obtained in terms of wall superheat and contact angle. It is shown that (i) For k = 0, the critical radius obtained by setting the generalized expression given by Eq. (23) equal to zero, recovers to the classical homogeneous nucleation theory given by Eq. (1), and for h = 0 the change in availability at equilibrium given by Eq. (3) for heterogeneous nucleation with a wall temperature gradient (k – 0) reduces to that of the classical homogeneous nucleation theory given by Eq. (2). (ii) The predicted heterogeneous nucleation heat flux on a wall with a contact angle of 60° obtained from the present model is shown to be slightly higher than those obtained based on Hsu’s model [4] for an equivalent contact angle of 53.1°; and (iii) existing experimental data for nucleation heat flux of water versus superheated wall temperatures in pool boiling at ONB are shown in reasonably good agreement with those predicted by the present theory. All of these verify the validity of the present model. The results of this analysis also show that: (i) the critical radius for heterogeneous nucleation with temperature gradient in the superheated liquid layer near the wall (k – 0) is larger than those for homogeneous nucleation with the liquid temperature equal to the wall temperature; (ii) for heterogeneous nucleation with a wall temperature gradient (k – 0), the critical radius at ONB decreases with the increase of contact angle (i.e., a more hydrophobic surface), leading to easy occurrence of bubble nucleation; while for heterogeneous nucleation without a temperature gradient (k = 0), the critical radius is independent of contact angle; (iii) as the con-
This work was supported by National Natural Science Foundation of China through Grant No. 51036005, and by Shanghai Municipal Science and Technology Commission through Grant No. 08JC1411100. References [1] V.P. Carey, Liquid-Vapor Phase-Change Phenomena, Hemisphere, Washington, 1992. [2] D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth, Heinemann, Oxford, 2000. [3] H. Steiner, A. Kobor, L. Gebhard, A wall heat transfer model for subcooled boiling flow, Int. J. Heat Mass Transfer 48 (2005) 4161–4173. [4] Y.Y. Hsu, On the size range of active nucleation cavities on a heating surface, ASME J. Heat Transfer 84 (1962) 207–216. [5] J. Li, P. Cheng, Bubble cavitation in a microchannel, Int. J. Heat Mass Transfer 47 (2004) 2689–2698. [6] A.E. Bergles, W.M. Rohsenow, The determination of forced-convection surfaceboiling heat transfer, J. Heat Transfer 86 (1964) 365–372. [7] N. Basu, G.R. Warrier, V.K. Dhir, Onset of nucleate boiling and active nucleation site density during subcooled flow boiling, J. Heat Transfer 124 (2002) 717– 728. [8] D. Wu, Y.Y. Duan, Z. Yang, Thermodynamic model for heterogeneous bubble nucleation in a temperature gradient, Appl. Phys. Lett. 97 (2010) 081911. [9] E.J. Davis, G.H. Anderson, The incipience of nucleate boiling in forced convection flow, AIChE J. 12 (1966) 774–780. [10] S.G. Kandlikar, V. Mizo, M. Cartwright, E. Ikenze, Bubble nucleation and growth characteristics in subcooled flow boiling of water national heat transfer conference HTD-342, ASME (1997), 11–18. [11] S. Petrovic, T. Robinson, R.L. Judd, Marangoni heat transfer in subcooled nucleate pool boiling, Int. J. Heat Mass Transfer 47 (2004) 5115–5128. [12] J. Li, G.P. Peterson, Microscale heterogeneous boiling on smooth surfaces-from bubble nucleation to bubble dynamics, Int. J. Heat Mass Transfer 48 (2005) 4316–4332. [13] C. Li, G.P. Peterson, Parametric study of pool boiling on horizontal highly conductive microporous coated surfaces, J. Heat Transfer 129 (2007) 1465– 1475.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A thermodynamic analysis for heterogeneous boiling nucleation on a superheated wall Xiaojun Quan, Gang Chen, Ping Cheng ⇑ MOE Key Laboratory for Power Machinery and Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 January 2011 Received in revised form 20 May 2011 Accepted 20 May 2011 Available online 28 June 2011
A thermodynamic model based on Gibbs free energy and availability is developed for onset of heterogeneous nucleation on heated surfaces with different wettabilities in pool boiling. Different from classical nucleation theory, this model takes into consideration the temperature gradient in the superheated liquid layer adjacent to the wall as well as the contact angle between the liquid and the wall. Using Gibbs free energy equilibrium condition, a closed form solution is obtained on the critical radius for onset of heterogeneous boiling nucleation on walls with different wettabilities. Effects of contact angles and wall temperatures on the critical radius, the wall temperature gradient of the superheated liquid layer and the heat flux at onset of heterogeneous nucleate boiling are illustrated. These effects on the change of availability during the heterogeneous nucleation process, representing the energy barrier for the occurrence of the first-order phase transition, are also discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Heterogeneous nucleation Thermodynamic equilibrium Critical radius Wall temperature Contact angle
1. Introduction In the classical theory of homogeneous nucleation [1] in a superheated liquid at a uniform temperature Tl and pressure pl, the critical radius for onset of homogeneous nucleation is
r Homo ¼ 2r‘v T s =qv hfg DT c
ð1Þ
with Ts being the saturation temperatures corresponding to pl and DT = Tl Ts. The change in availability (defined as W = U + p0V T0S with p0 and T0 being the pressure and temperature at a reference condition) at equilibrium during homogeneous nucleation is
DWHomo ¼ WHomo ðT l ; pl Þ W0 ðT l ; pl Þ ¼ c c
4prlv r 2c : 3
ð2Þ
For heterogeneous nucleation on a perfectly smooth wall at a wall temperature Tw equal to the temperature of the superheated liquid at a uniform temperature of Tl, i.e., no temperature gradient, it is known [1,2] that the critical radius is also given by Eq. (1). The change in availability during heterogeneous nucleation at equilibrium is given by
DWc ¼
prlv r2c 3
ð2 þ 3 cos h cos3 hÞ:
ð3Þ
The above equation with h = 0 reduces to homogeneous nucleation given by Eq. (2). Comparing Eqs. (2) and (3), it is apparent that (i) the homogeneous nucleation would require more work to achieve ⇑ Corresponding author. Tel./fax: +86 21 34206337. E-mail address: [email protected] (P. Cheng). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.05.026
than heterogeneous nucleation in a superheated liquid at uniform temperature; (ii) the classical homogeneous nucleation theory given by Eqs. (1) and (2) is identical to heterogeneous nucleation theory when the wall temperature is equal to the uniform liquid temperature (Tw = Tl) and the contact angle is zero (h = 0), i.e., a perfectly hydrophilic wall. 2. Existing models for heterogeneous nucleation with wall temperature gradient When a wall exists in the superheated liquid, however, a wall temperature gradient k always exists in the superheated liquid during heterogeneous nucleation as shown in Fig. 1, where the solid curve is the temperature distribution in the thermal boundary layer dt which increases from a bulk temperature of T1 to Tw at the wall. The superheated liquid layer [3] has a thickness det (where det < dt for subcooled boiling with T 1 < T s as seen from Fig. 1) within which the fluid temperature is higher than the saturated temperature Ts corresponding to the pressure of the liquid. In the following, we will first review existing models for heterogeneous nucleation with liquid temperature gradient near the wall taken into consideration. 2.1. Hsu’s model Hsu [4] was apparently the first one to study the onset of heterogeneous nucleation in a cavity, having mouth radius of rca, on a superheated wall at Tw (where Tw > Ts(pl)) in pool boiling with liquid temperature gradient near the wall taken into consideration.
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Nomenclature cp g G h hfg k m p q r R s S T U V x
specific heat (J/(kg K)) specific Gibbs free energy (J/kg) Gibbs free energy (J) specific enthalpy (J/kg) latent heat (J/kg) temperature gradient (K/m) mass (kg) pressure (Pa) heat flux (W/m2) bubble radius (lm) gas constant (J/(kg K)) specific entropy (J/(kg K)) entropy (J/K) temperature (K) internal energy (J) volume (m3) projection height from the heated wall (m)
h
deg
H
H = U + pV TwS (J)
density (kg/m3) surface tension of liquid and vapor (N/m) availability (J)
q rlv W
Subscripts 0 reference condition, initial state b bubble c critical state ca cavity l liquid s saturated state v vapor w wall 1 bulk liquid Superscript Homo homogeneous nucleation
Greek symbols dt thermal boundary layer thickness (m) superheated liquid layer thickness (m) det k thermal conductivity (W/(m K)) l specific chemical potential (J/kg)
Acronyms ONB onset of nucleate boiling
and since Tl = Tv, we have
T s ðpv Þ ¼ T v ¼ T l ¼ T s ðpl Þ þ
Fig. 1. Formation of an embryo bubble in a superheated liquid layer with thickness det on a heated wall.
He assumed that a linear temperature distribution exists in the thermal boundary layer (with a thickness of dt) of the liquid adjacent to the wall as,
x ; T l ðxÞ T 1 ¼ ðT w T 1 Þ 1 dt
ð0 6 x 6 dt Þ:
ð4aÞ
or
T l ðxÞ ¼ T w kx with k ¼
ðT w T 1 Þ ; dt
ð4b; cÞ
where k is the temperature gradient of the liquid near the wall, x is the distance from the heated wall, T1 is the liquid temperature at the edge of the thermal boundary layer. Hsu [4] proposed that the onset of nucleation occurs when the temperature at tip of the bubble, Tl(xb), is at least equal to the saturation temperature corresponding to the pressure inside the bubble Ts(pv), i.e.,
T l ðxb Þ P T s ðpv Þ;
ð5Þ
where xb is the projection height of the vapor bubble formed on the heated wall (see Fig. 1) and Ts(pv) is the saturation temperature corresponding to the vapor pressure inside bubble. The value of Ts(pv) can be determined approximately as follows. The Clausius–Clapeyron equation gives:
T l ¼ T s ðpl Þ þ
2rlv T s ðpl Þ : rb qv hfg
ð6aÞ
Making the approximation
T s ðpv Þ ffi T v
ð6bÞ
2rlv T s ðpl Þ ; r b qv hfg
ð6cÞ
which is a constant independent of x. Fig. 2(a) is a sketch showing the temperature profiles in the bubble given by Eq. (6c) (represented by a vertical straight line) and in the thermal boundary layer given by Eq. (4a) or (4b) (represented by an inclined straight line) given by Hsu’s model [4]. Note that for a bubble embryo to grow, the condition of Tl P Tv (both of which are greater than saturation temperature of the liquid Ts(pl)) must be satisfied on the vapor/liquid interface. It can be seen from Fig. 2(a) that the condition of Tl > Tv on the vapor/liquid interface is always satisfied in Hsu’s model [4]. For a truncated bubble sitting on a cavity mouth with a radius of rca, Hsu [4] assumed that the height and radius of the bubble are related to the radius of the month rca by xb = 2rca and rb = 1.25rca. The values of rca for a given wall superheat DTw = Tw Ts can be obtained by imposing Eq. (5), i.e., letting Tl(xb) from Eq. (4a) equal to Ts(pv) from Eq. (6c), resulting in a second-order algebraic equation in terms of rca which has two real roots if
dt P
12:8rlv T s ðT w T 1 Þ
qv hfg ðDT w Þ2
ð7Þ
:
Assuming that heat is transferred by conduction in the superheated layer and substituting the above expression into the heat conduction equation gives the required heat flux for onset of nucleation 2
qONB ¼
kðT w T 1 Þ khfg qv ðT w T s ÞONB ¼ ; dt 12:8rlv T s
ð8Þ
which is the expression originally given in Hsu’s paper [4]. Note that for a truncated bubble sitting on a cavity mouth with radius of rca and assuming that the bubble is of spherical shape, the bubble height xb and bubble radius rb are related to rca and contact angle h by xb = rca(1 + cosh)/sinh and rb = rca/sinh. A comparison of these expressions with Hsu’s assumptions of xb = 2rca and rb = 1.25rca shows that sin h = 0.8 and cosh = 0.6, i.e., h = 53.1°. Thus, Hsu’s assumptions of xb = 2rca and rb = 1.25rca are equivalent to the case
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tribution of the thermal boundary layer given by Eq. (4b) is an inclined straight line which intercepts each other at x = xb/2, where Tl(xb/2) = Tv. It can be seen from this figure that Tv > Tl(x) in the upper half of the vapor bubble (xb =2 < x 6 xb ), and Tv < Tl(x) for the lower half of the vapor bubble (0 6 x < xb =2). It follows that the condensation takes place along the upper half of vapor/liquid interface while evaporation takes place along the lower half of the interface. Based on second law of thermodynamics and defining a function H = U + plV TwS, Wu et al. [8] on discussed the growth and collapse of a bubble based DH. In this paper, we study heterogeneous boiling nucleation taking into consideration the temperature gradient in the superheated liquid adjacent to the heated wall and contact angle of the wall based on consideration of the changes in Gibbs free energy (G) and availability function (W) after an embryo is formed. At thermodynamic equilibrium condition, a closed-form expression is obtained for embryo’s critical radius at onset of heterogeneous nucleation in a superheated liquid layer. The change in the availability function DW is used to investigate the growth and collapse of an embryo bubble.
3. Heterogeneous nucleation with wall temperature gradient In this section, we will develop a thermodynamic model for heterogeneous nucleation on a wall with wall temperature gradient and wettability taken into consideration from the view point of changes in Gibbs free energy and availability. 3.1. Change in Gibbs free energy
Fig. 2. Comparisons of previous models by Hsu [4] and Wu et al. [8] with the present model.
of a contact angle of 53.1°. Li and Cheng [5] extended Hsu’s theory to include the effects of contact angle and dissolved gas on heterogeneous nucleation in pool boiling, and they showed that the nucleation temperature decreased with the increase in the contact angle or amount of gas dissolved in water. Bergles and Rohsenow [6] as well as Basu et al. [7] have applied Hsu’s model for onset of nucleation in flow boiling. 2.2. Wu et al.’s model Most recently, Wu et al. [8] developed a thermodynamic model for heterogeneous bubble nucleation on a wall with temperature gradient taken into consideration. They assumed that the liquid temperature is linear within the thermal boundary layer which is also given by Eq. (4b). Based on the assumptions that (i) temperature inside the bubble Tv is uniform and (ii) the net heat transfer through the bubble interface is equal to zero, i.e.,
Z klv Alv
ðT l T v Þ dAlv ¼ 0: dlv
ð9Þ
Wu et al. [8] obtained the following expression for temperature of the vapor bubble
T v ¼ T w krb ð1 þ coshÞ=2 ¼ T w kxb =2;
During a constant temperature and pressure process, the second law of thermodynamics given by dS P 0 (where S is the entropy) leads to dG 6 0 where G is the Gibbs free energy defined as:
G ¼ U þ pV TS;
where U is the internal energy. For bubble nucleation which is a constant pressure and constant temperature process,
dG ¼ Gv Gl 6 0;
ð12Þ
where Gv(pv, Tv) = Uv + pvVv TvSv and Gl(pl, Tl) = Ul + plVl TlSl, respectively. It is noted that the chemical potential l is defined as
l ¼ h Ts:
ð13Þ
Therefore, the chemical potential on a mass basis is equal to the specific Gibbs function (l = G/m = g) for a pure substance. Since the temperature of superheated liquid Tl is equal to that of superheated vapor Tv at thermal equilibrium, the change in Gibbs free energy between vapor and liquid is:
Dg ¼ g v g l ¼ lv ll ¼ ðhv hl Þ T l ðsv sl Þ;
ð14Þ
where
hv hl ¼ hfg þ ðcpv cpl ÞðT l T s Þ;
ð15Þ
hfg Tl p sv sl ¼ R ln v : þ cpv cpl ln Ts Ts pl
ð16Þ
Substituting Eqs. (15) and (16) into Eq. (14) gives
Dg ¼ g v g l ¼ Dl ¼
ð10Þ
where Tw is the wall temperature and xb = rb(1 + cosh) is the bubble projection height from the wall with h being the contact angle. Fig. 2(b) is a sketch showing that the temperature profile in the bubble given by Eq. (10) is a vertical line and the temperature dis-
ð11Þ
hfg p DT þ RT l ln v ; Ts pl
ð17Þ
where DT = Tl Ts with Ts being the saturation temperature corresponding to the liquid pressure pl, and hfg is the latent heat; pv and pl are related by the Young-Laplace equation given by
pv ¼ pl þ 2rlv =r
ð18Þ
X. Quan et al. / International Journal of Heat and Mass Transfer 54 (2011) 4762–4769
if the embryo bubble is in mechanical equilibrium with the surrounding superheated liquid. Note that if gv < gl, i.e., Dg < 0, then nucleation will take place. Conversely, if gv > gl, i.e., Dg > 0, then the evaporation is impossible and condensation will take place instead. Setting Dg = 0 in Eq. (17) gives the critical radius for homogeneous nucleation given by Eq. (1). To investigate the critical condition for heterogeneous bubble nucleation with k – 0 based on Gibbs free energy consideration, we now make the following assumptions: (i) A necessary condition for bubble nucleation is that the temperature at the tip of the bubble (at x = xb) is at least equal to the saturation temperature corresponding to the liquid pressure
T l ðxb Þ P T s ðpl Þ
ð19Þ
where xb is the projection height of bubble from the heated wall (see Fig. 1). (ii) The temperature of the superheated liquid Tl within the superheated layer is assumed to be linear [3] given by Eq. (4b) with k being the temperature gradient at the wall given by,
k ¼ ðT w T s Þ=det
ð20Þ
(iii) The temperature of vapor bubble Tv(x) in the differential control distance dx is assumed to be equal to the liquid phase outside of the bubble Tl(x) at the same distance x from the heated wall, i.e.,
T v ðxÞ ¼ T l ðxÞ ¼ T w kx:
ð21Þ
which is the ‘‘local thermal equilibrium’’ assumption along the vapor/liquid interface. Fig. 2(c) is a sketch showing the temperature profiles of the vapor and the superheated liquid in the present model where Tv(x) = Tl(x) = Tw kx is an inclined straight line with temperature of vapor bubble greater than the saturation temperature Ts(pl) everywhere along the vapor/liquid interface except at the tip of the bubble where Tv(xb) = Tl(xb) = Ts(pl). We can speculate the evaporation and condensation processes along the interface based on Eq. (17), where the second term is always positive as pv > pl. At the tip of the bubble x = xb where Tl = Ts (i.e., DT = 0), the first term in Eq. (17) vanishes. It follows that lv > ll near the tip which means that the interface of the embryo bubble near the bubble tip will be condensed due to the higher vapor chemical potential compared with that of the liquid. Away from the tip where DT > 0, the first term in Eq. (17) becomes more negative at smaller x. At some point at x where the first term is larger than the second term in Eq. (17) so that lv < ll and therefore evaporation will take place along the embryo bubble interface near the wall. At steady state, the total mass of condensation is equal to the total mass of evaporation along the interface so that the radius of the embryo bubble remains constant. Since the liquid temperature Tl(x) given by Eq. (21) is a function of x within the superheated liquid layer, the change in the Gibbs free energy before and after a bubble is formed on the heated surface can be obtained by integrating Eq. (17) with respect to x from x = 0 to x = xb to give:
Z
DG ¼ Gv ðT l ; pv Þ Gl ðT l ; pl Þ ¼ ½g v ðT l ; pv Þ g l ðT l ; pl Þdmv Z ¼ qv DgdV v h Z xb i h p ¼ qv fg ðT l T s Þ RT l ln v p r2 ðr cos h xÞ2 dx: Ts pl 0 ð22Þ
4765
Substituting Eqs. (18) and (21) to Eq. (22) and integrating with respect to x from x = 0 to x = xb give,
q pð1 þ cos hÞ2 r3 DG ¼ v 12 ½ð8 4 cos hÞðT w T s Þ krð1 þ cos hÞð3 cos hÞhfg =T s R lnðpv =pl Þ½ð8 4 cos hÞT w krð1 þ cos hÞð3 cos hÞÞ ð23Þ which has taken into consideration temperature variations in the embryo bubble and in the superheated liquid layer with a thickness of det . Note that if Gv < Gl or DG < 0, then the embryo bubble will grow. Conversely, if Gv > Gl or DG > 0, then the bubble will collapse. It is relevant to note that the assumption for onset of nucleation in the present model given by Eq. (19) differs from Hsu’s criteria given by Eq. (5). Since pv > pl, i.e. Ts(pv) > Ts(pl), therefore it can be seen from Eq. (17) that gl is always greater than or equal to gv at the liquid/vapor interface for Hsu’s criteria given by Eq. (5), which means that the nucleation condition along the vapor/liquid interface is always satisfied by Hsu’s criteria, i.e., DG < 0. While for present model, onset of nucleation take places when DG = 0. The above implies that the criteria for onset of nucleation given by Hsu [4] is more strict than that given in the present model. It has also been pointed out by Davis and Anderson [9] that Hsu’s criteria provides an upper limit to the wall superheat required to initiate nucleate boiling. In the present model, the critical radius can be obtained by setting Eq. (23) to zero (DG = 0) for the following two cases: (i) For heterogeneous nucleation at uniform liquid temperature (Tw = Tl = Tv), i.e., for the case of k = 0, the critical radius can be determined by setting Eq. (23) to zero, which reduces to Eq. (1) for homogenous nucleation. This means that the critical radius for heterogeneous nucleation without a temperature gradient (i.e., k = 0) is the same as that of homogeneous nucleation [1]. (ii) For the case of heterogeneous nucleation with a temperature gradient (k – 0) in the superheated liquid, the critical temperature gradient at the wall with temperature of Tw for the embryo bubble with radius r is given by Eq. (20) by substituting det ¼ xb to give
kc ðrÞ ¼ ðT w T s Þ=xb ¼ ðT w T s Þ=½rð1 þ coshÞ:
ð24Þ
Substituting Eq. (24) for the term k in Eq. (23) gives the change in Gibbs free energy during heterogeneous nucleation with k – 0: cos hÞ2 r 3 DG ¼ qv pð1þ 12 ð5 3 cos hÞhfg ðT w T s Þ=T s R lnðpv =pl Þ½ð5 3 cos hÞT w þ ð3 cos hÞT s g
ð25Þ
At equilibrium, the embryo’s Gibbs free energy must be equal to the liquid’s Gibbs free energy, i.e., Gv(Tv, pv) = Gl(Tl, pl). Thus, setting Eq. (25) to zero gives the critical radius for onset of nucleation for heterogeneous nucleation with k – 0:
rc ¼
2Rrlv T s ½ð5 3 cos hÞT w þ ð3 cos hÞT s pl hfg ðT w T s Þð5 3 cos hÞ
ð26Þ
The critical temperature gradient for critical radius rc for onset of heterogeneous nucleation with k – 0 can be obtained by substituting Eq. (26) into Eq. (24) to give,
kc ðr c Þ ¼
hfg pl ð5 3 cos hÞðT w T s Þ2 2Rrlv T s ð1 þ cos hÞ½ð5 3 cos hÞT w þ ð3 cos hÞT s
ð27Þ
Consequently, the heat flux at ONB for heterogeneous nucleation with k – 0 can be obtained as,
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qONB
¼ kkc ðr c Þ khfg pl ð5 3 cos hÞðT w T s Þ2ONB
¼ 2Rrlv T s ð1 þ cos hÞ ð5 3 cos hÞðT w ÞONB þ ð3 cos hÞT s ð28Þ
where k is the thermal conductivity of the superheated liquid.
Solving Tw from Eq. (24) in terms of kc and substituting it into Eq. (32) gives the change in availability in terms of (r, h, kc) for heterogeneous nucleation with k – 0 as, hÞ2 r 3 DWðr; h; K c Þ ¼ qv pð1þcos ð5 3 cos hÞkc rhfg ð1 þ cos hÞ=T s 12 R lnðpv =pl Þ½kc rð5 3 cos hÞð1 þ cos hÞ þ ð8 4 cos hÞT s g þprlv r2 ð2 þ 3 cos h cos3 hÞ=3
ð33Þ
3.2. Change in availability It should be noted that heating the liquid phase to a supersaturated state is only a necessary condition for the boiling nucleation to occur. However, there is no guarantee that phase transition from liquid to vapor will actually take place due to the fact that the metastable state is separated from the truly stable one by an energy barrier. The change of availability DW represents such an energy barrier for the occurrence of the first-order phase transition, and an increase of which will cause onset of nucleation more difficult. The change in availability for homogeneous nucleation is [1]:
4. Results and discussion 4.1. Critical radius for onset of heterogeneous nucleation
DWHomo ¼ W W0 ¼ Gv ðT v ; pv Þ Gl ðT l ; pl Þ þ ðpl pv ÞV v þ 4prlv r 2 :
ð29aÞ
Substituting Eqs. (18) and (17) (where the latter was obtained under the thermal equilibrium condition of Tl = Tv) into Eq. (29a) gives:
DWHomo ¼
Table 1 summarizes the results obtained for the critical radius and change in availability at equilibrium for the cases of heterogeneous nucleation with and without liquid temperature gradient and contact angle as well as homogeneous nucleation with liquid temperature equal to the wall temperature. Note that the change in availability for heterogeneous nucleation (at the same h) at equilibrium is the same for both k = 0 and k – 0 as mentioned earlier.
hfg 4pqv r 3 p 4prlv r 2 DT þ RT l ln v : 3 pl Ts 3
ð29bÞ
At equilibrium, the first term on the right-hand side of the above equation vanishes which reduces to the expression given by Eq. (2). On the other hand, the change in availability for heterogeneous nucleation for any k is [1]:
DW ¼ W W0 ¼ Gv ðT v ; pv Þ Gl ðT l ; pl Þ þ ðpl pv ÞV v 1 1 þ 4pr 2 rlv ð1 þ cos hÞ þ cos hð1 cos2 hÞ ; 2 4
Fig. 3 shows the variation of the critical radius for onset of heterogeneous nucleation of water on a wall at different wall temperatures with different contact angles, which was computed from Eq. (26) for water at atmospheric pressure (pl = 101,325 Pa, Ts = 373 K). It can be seen that the critical embryo bubble radius rc of water decreases with increase of Tw and the contact angle h (i.e., a more hydrophobic surface). Consequently, the value of rc for heterogeneous nucleation with temperature gradient (k – 0), depends on the contact angle, which is different from the results of heterogeneous nucleation without temperature gradient (i.e. k = 0), where rc is independent of contact angle for the same working fluid [1]. At a given contact angle, rc decreases sharply at first with increasing Tw and decreases gradually with further increase in Tw. The predicted values of rc shown in Fig. 3 are in fair agreement with
ð30Þ
where the first two terms on the right-hand side of Eq. (30) depending on k = 0 or k – 0 and the last term indicates the presence of the wall. Substituting Eq. (18) into Eq. (30) gives
DWðr; h; T w Þ ¼ DG þ
prlv r2 3
2 þ 3 cos h cos3 h ;
ð31Þ
where the first term DG on the right-hand side of Eq. (31) depends on k ¼ 0 or k – 0. For DG = 0 at equilibrium, Eq. (31) reduces to the expression given by Eq. (3) for DWc which is the change in availability for heterogeneous nucleation at equilibrium. Substituting Eq. (25) into Eq. (31) gives the change in the availability for k – 0, 2 3 DWðr; h; T w Þ ¼ qv pð1þ12cos hÞ r ð5 3 cos hÞhfg ðT w T s Þ=T s R lnðpv =pl Þ½ð5 3 cos hÞT w þ ð3 cos hÞT s g þprlv r 2 ð2 þ 3 cos h cos3 hÞ=3 ð32Þ
Fig. 3. Variation of rc with Tw at various h for heterogeneous nucleation of water in pool boiling.
Table 1 Summary of critical radius and the change in availability obtained for heterogeneous and homogeneous nucleation. Heterogeneous nucleation
Homogeneous nucleation
k – 0, h – 0
k – 0, h = 0
k = 0, h – 0 Ref. [1]
k = 0, h = 0 Ref. [1]
rc
2Rrlv T s ½ð5 3 cos hÞT w þ ð3 cos hÞT s Eq. (26): pl hfg ðT w T s Þð5 3 cos hÞ
2Rrlv T s ðT w þ T s Þ Eq. (26) with h = 0: pl hfg ðT w T s Þ
2rlv T s Eq. (1): qv hfg ðT w T s Þ
Eq. (1):
DW c
Eq. (3):
Eq. (3) with h = 0:4prlv r 2c =3
Eq. (3):
prlv r 2c ð2 þ 3 cos h cos3 hÞ=3
prlv r2c ð2 þ 3 cos h cos3 hÞ=3
2rlv T s qv hfg ðT ‘ T s Þ Eq. (2): 4prlv r 2c =3
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measurements of critical radii of a few micrometers at low fluid inlet velocity given in [10]. Also, the critical radius for onset of homogeneous nucleation given by Eq. (1) is presented as a short dotted line in Fig. 3, showing that the critical radius for onset of homogeneous nucleation is smaller than heterogeneous nucleation when the liquid temperature of homogeneous nucleation is equal to the wall temperature of heterogeneous nucleation. 4.2. Critical wall temperature gradient for onset of heterogeneous nucleation Eq. (27) for critical wall temperature gradient kc(rc) is presented in Fig. 4, showing that the increase in Tw and h lead to the increase of kc(rc). Since k ¼ ð@T l =@ x Þw = (Tw Ts)/det , the value of k is related to the superheated liquid layer thickness det , which is equal to the bubble height for critical wall temperature gradient kc. At a specific Tw, nucleation takes place only if k 6 kc ðrc Þ, i.e., T l ðxb Þ P T s . When k > kc(rc) at a specific Tw, which means det is too thin, boiling nucleation will be suppressed and the embryo would not grow spontaneously. 4.3. Nucleation heat flux We now introduce the dimensionless heat flux and the dimensionless superheat as,
~ONB ¼ q
qONB rlv ; khfg qv T s
DT~ ONB ¼
ðT w T s ÞONB ; Ts
ð34Þ
ð35Þ
Thus, Eqs. (28) and (8) can be expressed in dimensionless form as follows:
~ONB ¼ q
pl ð5 3 cos hÞDT~ 2ONB h i 2qv RT s ð1 þ cos hÞ ð5 3 cos hÞDT~ ONB þ ð8 4 cos hÞ
ð36Þ
~ONB ¼ q
DT~ 2ONB : 12:8
ð37Þ
~ONB versus DT~ ONB Eq. (36) with contact angles of 0°, 60° and 120° for q in pool boiling of water are presented in Fig. 5. When the contact angle is increased, it can be concluded from Fig. 5 that: (i) more heat flux is needed for ONB at the same wall temperature, and (ii) the wall temperature at ONB decreases for the same heat flux. Hsu’s
Fig. 5. Comparison of nucleation heat flux in pool boiling of water based on the present model and Hsu’s model as well as experimental data.
model given by Eq. (37) is also plotted for comparison. It can be seen from Fig. 5 that the nucleation heat flux given by Hsu’s model for an equivalent contact angle of 53.1° is slightly lower than the present model given by Eq. (36) for a contact angle of 60° at the same wall superheat. This can be explained based on the criteria for onset of nucleation given by the two models as follows: Hsu’s criteria for onset of nucleation is Tl(xb) P Ts(pv), while the criteria for onset of nucleation in the present model is Tl(xb) P Ts(pl). Since Ts(pv) > Ts(pl), the temperature at the tip of the bubble required for boiling inception is higher for Hsu’s model than the present model. This means that for nucleate bubble with the same rb, the superheated liquid layer thickness det is larger for Hsu’s model than the present model as can be seen from Fig. 2. Consequently, according to k ¼ ðT w T s ðpl ÞÞ=det , it can be seen that for the same heat flux at ONB (i.e., the same wall temperature gradient k), the required wall superheat is larger for Hsu’s model. In other words, for the same wall superheat, the wall temperature gradient is smaller in Hsu’s model (i.e., smaller heat flux). The above explanations show that Hsu’s boiling inception criteria [4] is more restrictive for onset of heterogeneous nucleation than the present model, leading to a higher superheat required for ONB at the same heat flux. Experimental results obtained by various investigators [11–13] for inception of pool boiling of water are also presented in Fig. 5 for comparison purposes. These experimental results include (i) four data points obtained by Petrovic et al. [11] under four different subcooled conditions, (ii) two data points obtained by Li and Peterson [12] for degassed and gas-dissolved water, respectively, and (iii) nine data points obtained by Li and Peterson [13] for using nine different heated surfaces. Except for three data points obtained by Li and Peterson [13], most of the other experimental data are shown to be in good agreement with the present model and Hsu’s model, although detailed comparisons are not possible because the contact angles of heater surface in the experiments were not reported explicitly in the papers.
4.4. Change in availability during heterogeneous nucleation
Fig. 4. Relation between kc(rc) and Tw at various h for heterogeneous nucleation of water in pool boiling.
Fig. 6 shows the change of availability for heterogeneous nucleation with k – 0 versus embryo radius r for h = 0°, 60°, 120° for water at pl = 101,325 Pa, Ts = 373 K and Tw = 127 °C, which was computed according to Eq. (33). As shown in this figure, the value of DW has a local maximum at r = rc, which is the critical radius for heterogeneous nucleation shown in Fig. 3, corresponding to a state of unstable equilibrium. When an embryo has a radius r less than rc, any increase of r corresponds to an increase in DW, indicating
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Fig. 6. Variations of the change in availability with bubble radius at various h for heterogeneous nucleation of water in pool boiling.
that the embryos would collapse spontaneously. On the other hand, when an embryo has a radius greater than rc, any increase in r corresponds to a decrease of DW. This indicates that the embryo would grow spontaneously, resulting in heterogeneous nucleation. Fig. 6 also shows that DW required for bubble nucleation decreases with the increases of contact angle, indicating that large contact angles will favor nucleation. This is consistent with the results presented in Fig. 3, where it is shown that required wall superheat decreases with the increase of contact angle (i.e., a more hydrophobic surface) at given rc. The change in availability for homogeneous nucleation given by Eq. (29b) is also presented in Fig. 6 for comparison. It can also been seen from Fig. 6 that the critical radius for homogeneous nucleation is smaller than heterogeneous nucleation, which is consistent with the results presented in Fig. 3. It is shown from Fig. 6 that the change in availability of homogeneous nucleation (with Tl = Tw) can be smaller or larger than heterogeneous nucleation with temperature gradient (k – 0) depending on its contact angle. The change in availability for heterogeneous nucleation at equilibrium given by Eq. (3) for any k can be expressed in dimensionless form as
Dwc ¼
DWc ¼ 2 þ 3 cos h cos3 h ; prlv rc =3
ð38Þ
where the right-hand side depends only on the contact angle. Eq. (38) is presented in Fig. 7, which shows that the change in dimensionless availability at equilibrium decreases from a value of 4 (a perfectly hydrophilic surface with h = 0°) to zero (a perfectly hydrophobic surface with h = 180°) implying that nucleation becomes easier with increasing contact angle h (i.e., a more hydrophobic surface). Note that the case of h = 0° also corresponds to homogeneous nucleation. Fig. 8 shows the effects of kc on DW versus r given by Eq. (33) for water at pl = 101,325 Pa and Ts = 373 K and at h = 60°. It can be seen that DW increases with decrease of kc which corresponds to lower wall superheat as shown in Fig. 4. This indicates that a decrease in kc will suppress bubble nucleation. For the special case of kc = 0 (meaning that Tw = Ts(pl) from Eq. (24), i.e., the wall is at saturated temperature), DW approaches infinity and nucleation will not take place. 4.5. Comparisons of models for onset of heterogeneous nucleation We now compare the present model with two other existing models by Hsu [3] as well as by Wu et al. [8]. The temperature dis-
Fig. 7. The effect of contact angle h on the change in dimensionless availability for onset of heterogeneous nucleation in pool boiling.
Fig. 8. Relation between the availability and bubble radius for various kc for heterogeneous nucleation of water in pool boiling.
tributions of the three models are sketched in Fig. 2. Since the Young-Laplace equation was used in the three different models, the mechanical equilibrium criteria is satisfied in these three models. Table 2 is a summary of temperatures of the liquid and vapor and the change in Gibbs free energy for the three models for heterogeneous nucleation with liquid temperature gradient k – 0. Although the present model and those by Wu et al. [8] are based on similar assumptions of linear liquid temperature distribution and starting from the second law of thermodynamics, the two models differ from each other due to two important aspects: (i) for the vapor temperature: the present model assumes that Tv(x) = Tl(x) while in the model by Wu et al. [8], the vapor temperature was obtained based on Eq. (10) as Tv = Tw kxb/2 with xb being the bubble projection height from the wall; (ii) bubble growth and collapse is considered from the point of view of the change in Gibbs free energy DG and availability DW in present model while it was considered based on DH in the paper by Wu et al. [8], where H is different from the availability function. In Hsu’s model, the vapor temperature is given by Eq. (6c), which means that the chemical potential for vapor is equal to that for the liquid at tip of the bubble. As the liquid temperature is greater than the vapor temperature along the bubble interface except the top point, it can be concluded that DG = Gv Gl < 0 through Eq. (17). In the model considered by Wu et al. [8], the
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Hsu’s model [4]
Wu et al.’s model [8]
Present model
Liquid-phase temperature distribution
Eq. (4b) and (4c): Tl(x) = Tw kx (for 0 < x < dt) where k ¼ ðT w T 1 Þ=dt 2rlv T s Eq. (6c): T v ¼ T s þ ¼ const r b qv hfg dG < 0
Eq. (4b) and (4c): Tl(x) = Tw kx (for 0 < x < dt) where k ¼ ðT w T 1 Þ=dt
Eqs. (20) and (21): Tl(x) = Tw kx (for 0 < x < det ) where k ¼ ðT w T s Þ=det Eq. (21): Tv(x) = Tl(x) = Tw kx dG = 0
Vapor-phase temperature distribution Change in Gibbs free energy (dG) during nucleation
Eq. (10): T v ¼ T w kxb =2 ¼ const dG > 0
vapor temperature was obtained using Eq. (10) based on a thermal equilibrium for flat interface without taking into consideration the effects of capillary pressure difference due to curve interface. Therefore, the vapor temperature given in model by Wu et al. [8] is overestimated, and the embryo bubble is likely to collapse because of the net condensation along the interface, i.e., DG = Gv Gl > 0.
tact angle becomes larger, it requires smaller nucleation temperature for same heat flux and thus is more favorable for nucleation; (iv) the wall nucleation temperature increases with the increase of the temperature gradient at the wall, and (v) the wall temperature gradient must be smaller than the critical wall temperature gradient given by Eq. (27) in order for heterogeneous nucleation to occur.
5. Conclusions
Acknowledgements
In this paper, a thermodynamic analysis based on Gibbs free energy and availability has been carried out to study onset of heterogeneous boiling nucleation in the superheated liquid layer, with wall temperature gradient k taken into consideration. Closed-form solutions for the critical radius of bubble and heat flux at onset of nucleate boiling (ONB) are obtained in terms of wall superheat and contact angle. It is shown that (i) For k = 0, the critical radius obtained by setting the generalized expression given by Eq. (23) equal to zero, recovers to the classical homogeneous nucleation theory given by Eq. (1), and for h = 0 the change in availability at equilibrium given by Eq. (3) for heterogeneous nucleation with a wall temperature gradient (k – 0) reduces to that of the classical homogeneous nucleation theory given by Eq. (2). (ii) The predicted heterogeneous nucleation heat flux on a wall with a contact angle of 60° obtained from the present model is shown to be slightly higher than those obtained based on Hsu’s model [4] for an equivalent contact angle of 53.1°; and (iii) existing experimental data for nucleation heat flux of water versus superheated wall temperatures in pool boiling at ONB are shown in reasonably good agreement with those predicted by the present theory. All of these verify the validity of the present model. The results of this analysis also show that: (i) the critical radius for heterogeneous nucleation with temperature gradient in the superheated liquid layer near the wall (k – 0) is larger than those for homogeneous nucleation with the liquid temperature equal to the wall temperature; (ii) for heterogeneous nucleation with a wall temperature gradient (k – 0), the critical radius at ONB decreases with the increase of contact angle (i.e., a more hydrophobic surface), leading to easy occurrence of bubble nucleation; while for heterogeneous nucleation without a temperature gradient (k = 0), the critical radius is independent of contact angle; (iii) as the con-
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