# Nucleation and Inception

(Difference between revisions)
 Revision as of 20:42, 31 May 2010 (view source)← Older edit Revision as of 21:19, 31 May 2010 (view source)Newer edit → Line 4: Line 4:
${p_g} + {p_v} - {p_\ell } = \frac{{2\sigma }}{{{R_b}}} \qquad \qquad(1)$
${p_g} + {p_v} - {p_\ell } = \frac{{2\sigma }}{{{R_b}}} \qquad \qquad(1)$
- (10.1) - where ${R_b}$ is the radius of the vapor bubble. Equation (10.1) indicates that the pressure of the vapor and gas must be greater than the pressure of the liquid to maintain a balance of forces in the vapor-interface-liquid system. Although these events occur on a molecular scale, we may attempt to visualize the physical implications of this imbalance of pressures across the boundary separating vapor and liquid phases. The higher vapor pressure in the bubble causes an increased number of molecules to strike the interface where they are absorbed by the liquid phase. To maintain the mass balance that equilibrium requires, there must be a corresponding increase in the number of molecules emitted through the interface from the liquid. Under the given constraints, this can be accomplished only by increasing the temperature of the liquid side of the interface. Consequently, the liquid near the bubble’s surface must be superheated to a temperature above the saturation temperature that corresponds to the prevailing bulk liquid pressure. To put these qualitative observations on a quantitative basis, consider that the vapor pressure for a curved bubble interface can be expressed as (Section 2.6.3) + where ${R_b}$ is the radius of the vapor bubble. Equation (1) indicates that the pressure of the vapor and gas must be greater than the pressure of the liquid to maintain a balance of forces in the vapor-interface-liquid system. Although these events occur on a molecular scale, we may attempt to visualize the physical implications of this imbalance of pressures across the boundary separating vapor and liquid phases. The higher vapor pressure in the bubble causes an increased number of molecules to strike the interface where they are absorbed by the liquid phase. To maintain the mass balance that equilibrium requires, there must be a corresponding increase in the number of molecules emitted through the interface from the liquid. Under the given constraints, this can be accomplished only by increasing the temperature of the liquid side of the interface. Consequently, the liquid near the bubble’s surface must be superheated to a temperature above the saturation temperature that corresponds to the prevailing bulk liquid pressure. To put these qualitative observations on a quantitative basis, consider that the vapor pressure for a curved bubble interface can be expressed as
${p_v} = {p_{v,sat}}(T)\exp \left[ {\frac{{ - 2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(2)$
${p_v} = {p_{v,sat}}(T)\exp \left[ {\frac{{ - 2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(2)$
- (10.2) - where ${p_{v,sat}}(T)$ is the normal saturation pressure corresponding to temperature, $T$. Since $2\sigma {\rho _v}/[{p_{v,sat}}(T){R_b}{\rho _\ell }] \ll 1$, eq. (10.2) can be approximated as + where ${p_{v,sat}}(T)$ is the normal saturation pressure corresponding to temperature, $T$. Since $2\sigma {\rho _v}/[{p_{v,sat}}(T){R_b}{\rho _\ell }] \ll 1$, eq. (2) can be approximated as
${p_v}\textstyle.\over= {p_{v,sat}}(T)\left[ {1 - \frac{{2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(3)$
${p_v}\textstyle.\over= {p_{v,sat}}(T)\left[ {1 - \frac{{2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(3)$
- (10.3) Combining eqs. (10.1) and (10.3) yields Combining eqs. (10.1) and (10.3) yields
${p_g} + {p_{v,sat}}(T) - {p_\ell }\textstyle.\over= \frac{{2\sigma }}{{{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) \qquad \qquad(4)$
${p_g} + {p_{v,sat}}(T) - {p_\ell }\textstyle.\over= \frac{{2\sigma }}{{{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) \qquad \qquad(4)$
- (10.4) Recalling the fundamental relationship between pressure and temperature differences in the two-phase region, the Clapeyron equation, Recalling the fundamental relationship between pressure and temperature differences in the two-phase region, the Clapeyron equation,
$\frac{{dp}}{{dT}} = \frac{{{h_{\ell v}}}}{{T\left( {1/{\rho _v} - 1/{\rho _\ell }} \right)}} \qquad \qquad(5)$
$\frac{{dp}}{{dT}} = \frac{{{h_{\ell v}}}}{{T\left( {1/{\rho _v} - 1/{\rho _\ell }} \right)}} \qquad \qquad(5)$
- (10.5) and assuming that the fluid obeys the ideal gas law, i.e., and assuming that the fluid obeys the ideal gas law, i.e.,
$p = \rho {R_g}T \qquad \qquad(6)$
$p = \rho {R_g}T \qquad \qquad(6)$
- (10.6) and that ${\rho _\ell } \gg {\rho _v}$, we arrive at and that ${\rho _\ell } \gg {\rho _v}$, we arrive at
$\frac{{dp}}{p} = \frac{{{h_{\ell v}}}}{{{R_g}{T^2}}}dT \qquad \qquad(7)$
$\frac{{dp}}{p} = \frac{{{h_{\ell v}}}}{{{R_g}{T^2}}}dT \qquad \qquad(7)$
- (10.7) - For a pressure change from ${p_\ell }$ to ${p_{v,sat}}(T),$ the corresponding saturation temperature changes from ${T_{sat}}$ to $T$. So by integrating eq. (10.7) between these limits, we obtain + For a pressure change from ${p_\ell }$ to ${p_{v,sat}}(T),$ the corresponding saturation temperature changes from ${T_{sat}}$ to $T$. So by integrating eq. (7) between these limits, we obtain
$\ln \frac{{{p_{v,sat}}(T)}}{{{p_\ell }}} = \frac{{{h_{\ell v}}}}{{{R_g}{T_{sat}}}}\frac{{T - {T_{sat}}}}{T} \qquad \qquad(8)$
$\ln \frac{{{p_{v,sat}}(T)}}{{{p_\ell }}} = \frac{{{h_{\ell v}}}}{{{R_g}{T_{sat}}}}\frac{{T - {T_{sat}}}}{T} \qquad \qquad(8)$
- (10.8) - Substituting eq. (10.4) into eq. (10.8) gives + Substituting eq. (4) into eq. (8) gives
$\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{h_{\ell v}}}}\ln \left[ {1 + \frac{{2\sigma }}{{{p_\ell }{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) - \frac{{{p_g}}}{{{p_\ell }}}} \right] \qquad \qquad(9)$
$\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{h_{\ell v}}}}\ln \left[ {1 + \frac{{2\sigma }}{{{p_\ell }{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) - \frac{{{p_g}}}{{{p_\ell }}}} \right] \qquad \qquad(9)$
- (10.9) - Considering that $2\sigma /({p_\ell }{R_b}) - {p_g}/{p_\ell } \ll 1$ and again that ${\rho _\ell } \gg {\rho _v}$, eq. (10.9) can be further simplified as + Considering that $2\sigma /({p_\ell }{R_b}) - {p_g}/{p_\ell } \ll 1$ and again that ${\rho _\ell } \gg {\rho _v}$, eq. (9) can be further simplified as
$\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{p_\ell }{h_{\ell v}}}}\left( {\frac{{2\sigma }}{{{R_b}}} - {p_g}} \right) \qquad \qquad(10)$
$\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{p_\ell }{h_{\ell v}}}}\left( {\frac{{2\sigma }}{{{R_b}}} - {p_g}} \right) \qquad \qquad(10)$
- (10.10) - For the case with no noncondensable gas (${p_g} = 0$), eq. (10.10) can be rearranged to yield the critical vapor bubble radius: + For the case with no noncondensable gas (${p_g} = 0$), eq. (10) can be rearranged to yield the critical vapor bubble radius:
${R_b} = \frac{{2\sigma {T_{sat}}}}{{{h_{\ell v}}{\rho _v}\Delta T}} \qquad \qquad(11)$
${R_b} = \frac{{2\sigma {T_{sat}}}}{{{h_{\ell v}}{\rho _v}\Delta T}} \qquad \qquad(11)$
- (10.11) - This critical radius is necessary for a bubble to exist at $\Delta T$ above the saturation temperature that corresponds to the prevailing vapor pressure. Any bubble with a radius less than the size given by eq. (10.11) will collapse, but a bubble with a radius equal to ${R_b}$ will grow in a spontaneous fashion. It should be noted from eq. (10.10) that the size of the equilibrium vapor nucleus becomes smaller as the superheat ($\Delta T$) increases. This identifies the basic mechanism for the increase of $q''$ with rising excess temperature between points A and C of Fig. 10.3, namely, the increased bubble density that results from the activation of more and more nucleation sites at ever-smaller surface imperfections. + This critical radius is necessary for a bubble to exist at $\Delta T$ above the saturation temperature that corresponds to the prevailing vapor pressure. Any bubble with a radius less than the size given by eq. (11) will collapse, but a bubble with a radius equal to ${R_b}$ will grow in a spontaneous fashion. It should be noted from eq. (10) that the size of the equilibrium vapor nucleus becomes smaller as the superheat ($\Delta T$) increases. This identifies the basic mechanism for the increase of $q''$ with rising excess temperature between points A and C of Fig. 10.3, namely, the increased bubble density that results from the activation of more and more nucleation sites at ever-smaller surface imperfections. - Equation (10.10) shows that the presence of noncondensable gas reduces the superheat required to generate bubbles. The subject of superheat was presented in Section 2.6.5. The bubble growth is spontaneous if the excess temperature is greater than the value given in eq. (10.10). Any real surface always has some small cavities with very small inclined angles where liquid can only partially fill the cavities, so opportunities for the formation of gas or vapor pockets are common. Equation (10.10) gives the superheat required to initiate nucleate boiling from a pre-existing nucleus with a radius of ${R_b}$. [[#References|Griffith and Wallis (1960)]] suggested that it is also applicable for nucleate boiling from a surface with a micro cavity whose mouth radius is ${R_b}$. Alternative studies by [[#References|Mizukami (1975), Nishio (1985)]], and [[#References|Wang and Dhir (1993a)]] suggested that the superheat required to initiate nucleate boiling is determined by the instability of vapor nuclei in the cavity. If the curvature of the liquid-vapor interface, $K$, increases with increasing vapor volume ($dK/d{V^*} > 0$, where ${V^*}$ is dimensionless vapor volume), the vapor nucleus is stable. On the other hand, the vapor bubble embryo or nucleus is unstable if $K$ decreases with increasing vapor volume. For + Equation (10) shows that the presence of noncondensable gas reduces the superheat required to generate bubbles. The subject of superheat was presented in Section 2.6.5. The bubble growth is spontaneous if the excess temperature is greater than the value given in eq. (10). Any real surface always has some small cavities with very small inclined angles where liquid can only partially fill the cavities, so opportunities for the formation of gas or vapor pockets are common. Equation (10) gives the superheat required to initiate nucleate boiling from a pre-existing nucleus with a radius of ${R_b}$. [[#References|Griffith and Wallis (1960)]] suggested that it is also applicable for nucleate boiling from a surface with a micro cavity whose mouth radius is ${R_b}$. Alternative studies by [[#References|Mizukami (1975), Nishio (1985)]], and [[#References|Wang and Dhir (1993a)]] suggested that the superheat required to initiate nucleate boiling is determined by the instability of vapor nuclei in the cavity. If the curvature of the liquid-vapor interface, $K$, increases with increasing vapor volume ($dK/d{V^*} > 0$, where ${V^*}$ is dimensionless vapor volume), the vapor nucleus is stable. On the other hand, the vapor bubble embryo or nucleus is unstable if $K$ decreases with increasing vapor volume. For - [[Image:poiling_p_(2).png|thumb|400 px|alt=Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity|Figure 10.4 Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity [[#References|(Wang and Dhir, 1993a)]].]] + [[Image:poiling_p_(2).png|thumb|400 px|alt=Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity|Figure 1 Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity [[#References|(Wang and Dhir, 1993a)]].]] the nucleate boiling to initiate from the wall, the wall superheat must be sufficient to ensure the vapor nuclei are unstable. the nucleate boiling to initiate from the wall, the wall superheat must be sufficient to ensure the vapor nuclei are unstable. - Figure 10.4 shows the dependence of the nondimensional modified curvature of the interface on the dimensionless volume of the vapor nucleus in a spherical cavity with a mouth angle of 30°. The noncondensable gas is not present and the results at different contact angles are shown. The dimensionless volume of the vapor nucleus is defined as ${V^*} = {V_b}/{V_c}$ where ${V_b}$ is the volume of the nucleus and ${V_c}$ is the volume of the nucleus at point A, where the curvature is zero. It can be seen from Fig. 10.4 that the nucleus is stable between points A and D, since $dK/d{V^*} > 0$. When the contact angle is less than 90°, the maximum curvature, ${K_{max}}$, is 1. For the cases in which $\theta > {90^ \circ }$, the maximum curvature becomes $\sin \theta$. Nucleation occurs beyond the point where the curvature reaches its maximum; the corresponding superheat is [[#References|(Wang and Dhir, 1993a)]] + Figure 1 shows the dependence of the nondimensional modified curvature of the interface on the dimensionless volume of the vapor nucleus in a spherical cavity with a mouth angle of 30°. The noncondensable gas is not present and the results at different contact angles are shown. The dimensionless volume of the vapor nucleus is defined as ${V^*} = {V_b}/{V_c}$ where ${V_b}$ is the volume of the nucleus and ${V_c}$ is the volume of the nucleus at point A, where the curvature is zero. It can be seen from Fig. 1 that the nucleus is stable between points A and D, since $dK/d{V^*} > 0$. When the contact angle is less than 90°, the maximum curvature, ${K_{max}}$, is 1. For the cases in which $\theta > {90^ \circ }$, the maximum curvature becomes $\sin \theta$. Nucleation occurs beyond the point where the curvature reaches its maximum; the corresponding superheat is [[#References|(Wang and Dhir, 1993a)]]
$\Delta T = T - {T_{sat}} = \frac{{4\sigma {T_{sat}}}}{{{p_v}{h_{\ell v}}{D_c}}}{K_{\max }} \qquad \qquad(12)$
$\Delta T = T - {T_{sat}} = \frac{{4\sigma {T_{sat}}}}{{{p_v}{h_{\ell v}}{D_c}}}{K_{\max }} \qquad \qquad(12)$
- (10.12) where ${D_c}$ is the diameter of the cavity mouth. where ${D_c}$ is the diameter of the cavity mouth. - Example 10.1 A steam bubble with a radius of 5 $\mu$m is surrounded by liquid water at 120 ˚C. Will this bubble grow or collapse? + Example 1 A steam bubble with a radius of 5 $\mu$m is surrounded by liquid water at 120 ˚C. Will this bubble grow or collapse? Solution: The liquid pressure is ${p_\ell } = 1{\rm{ atm}} = 1.013 \times {10^5}{\rm{ Pa}}{\rm{.}}$ The saturation temperature at this pressure is ${T_{sat}} = 100{\rm{ }}{}^{\rm{o}}{\rm{C}} = 373.15{\rm{ K}}{\rm{.}}$ Solution: The liquid pressure is ${p_\ell } = 1{\rm{ atm}} = 1.013 \times {10^5}{\rm{ Pa}}{\rm{.}}$ The saturation temperature at this pressure is ${T_{sat}} = 100{\rm{ }}{}^{\rm{o}}{\rm{C}} = 373.15{\rm{ K}}{\rm{.}}$ The properties of water at this temperature are $\sigma = 58.9 \times {10^{ - 3}}{\rm{ N/m,}}$ ${h_{\ell v}} =$ $2251.2{\rm{ kJ/kg}},$ The properties of water at this temperature are $\sigma = 58.9 \times {10^{ - 3}}{\rm{ N/m,}}$ ${h_{\ell v}} =$ $2251.2{\rm{ kJ/kg}},$ And ${\rho _v} = 0.5974{\rm{ kg/}}{{\rm{m}}^{\rm{3}}}.$ And ${\rho _v} = 0.5974{\rm{ kg/}}{{\rm{m}}^{\rm{3}}}.$ - Therefore, the critical bubble radius can be determined from eq. (10.11), i.e., + Therefore, the critical bubble radius can be determined from eq. (11), i.e.,
$\begin{array}{l} [itex]\begin{array}{l} Line 84: Line 72: Since the radius of the bubble, 5 [itex]\mu$m, is greater than the critical bubble radius, the bubble will grow. Since the radius of the bubble, 5 $\mu$m, is greater than the critical bubble radius, the bubble will grow. - Figure 10.5 shows an idealized model of a surface crack containing a vapor pocket as it expands under the influence of evaporation. The vapor-liquid interface is idealized as spherical in shape. As the vapor pocket expands upward in the surface crack, its interface takes a concave shape and the radius of curvature, ${R_b}$, increases as the interface moves toward the lip of the cavity. Once the incipient bubble reaches the lip of the interface, the radius of curvature begins to decrease. It reaches its minimum value (approximately) as the liquid- vapor interface becomes a hemisphere with a radius equal to that of the cavity mouth, ${R_c}$. Beyond this point, any further evaporation pushes the liquid-vapor interface out of the surface cavity, and the interface radius of curvature begins to increase + Figure 2 shows an idealized model of a surface crack containing a vapor pocket as it expands under the influence of evaporation. The vapor-liquid interface is idealized as spherical in shape. As the vapor pocket expands upward in the surface crack, its interface takes a concave shape and the radius of curvature, ${R_b}$, increases as the interface moves toward the lip of the cavity. Once the incipient bubble reaches the lip of the interface, the radius of curvature begins to decrease. It reaches its minimum value (approximately) as the liquid- vapor interface becomes a hemisphere with a radius equal to that of the cavity mouth, ${R_c}$. Beyond this point, any further evaporation pushes the liquid-vapor interface out of the surface cavity, and the interface radius of curvature begins to increase - [[Image:boiling_j_(5).jpg|thumb|400 px|alt=Variation of bubble radius as the bubble grows within and emerges from an idealized cavity|Figure 10.5 Variation of bubble radius as the bubble grows within and emerges from an idealized cavity.]] + [[Image:boiling_j_(5).jpg|thumb|400 px|alt=Variation of bubble radius as the bubble grows within and emerges from an idealized cavity|Figure 2 Variation of bubble radius as the bubble grows within and emerges from an idealized cavity.]] - [[Image:boiling_j_(6).jpg|thumb|400 px|alt=Liquid microlayer under a vapor bubble at a nucleation site|Figure 10.6 Liquid microlayer under a vapor bubble at a nucleation site.]] + [[Image:boiling_j_(6).jpg|thumb|400 px|alt=Liquid microlayer under a vapor bubble at a nucleation site|Figure 3 Liquid microlayer under a vapor bubble at a nucleation site.]] - again. Consideration of this sequence of radius variation, along with eq. (10.10), indicates that the superheat required to form a vapor bubble from this cavity can be determined by substituting ${R_b} = {R_c}$ into eq. (10.10). In normal circumstances, of course, this variation deviates from the idealized model in Fig. 10.5 and is a function of cavity angle $\phi$ and contact angle $\theta$. + again. Consideration of this sequence of radius variation, along with eq. (10), indicates that the superheat required to form a vapor bubble from this cavity can be determined by substituting ${R_b} = {R_c}$ into eq. (10). In normal circumstances, of course, this variation deviates from the idealized model in Fig. 2 and is a function of cavity angle $\phi$ and contact angle $\theta$. - In reality, the shape of a vapor bubble is not hemispherical, and there is a microlayer (a very thin liquid layer, see Fig. 10.6) that is responsible for more than half of heat transfer in some cases. [[#References|Plesset and Sadhal (1979)]] provided the average liquid film thickness $\bar \delta$ in terms of bubble lift time ${t_0}$ and liquid kinematic viscosity by the following equation + In reality, the shape of a vapor bubble is not hemispherical, and there is a microlayer (a very thin liquid layer, see Fig. 3) that is responsible for more than half of heat transfer in some cases. [[#References|Plesset and Sadhal (1979)]] provided the average liquid film thickness $\bar \delta$ in terms of bubble lift time ${t_0}$ and liquid kinematic viscosity by the following equation
$\bar \delta = \frac{8}{7}{(3{\nu _\ell }{t_0})^{1/2}} \qquad \qquad(13)$
$\bar \delta = \frac{8}{7}{(3{\nu _\ell }{t_0})^{1/2}} \qquad \qquad(13)$
- (10.13) The contribution of the microlayer to the overall heat transfer obviously depends on the fraction of area occupied by the microlayer. Despite its significant importance, there is no detailed treatment of the microlayer related to nucleate boiling. The contribution of the microlayer to the overall heat transfer obviously depends on the fraction of area occupied by the microlayer. Despite its significant importance, there is no detailed treatment of the microlayer related to nucleate boiling.

## Revision as of 21:19, 31 May 2010

Nucleation, or bubble initiation, in typical industrial applications is characterized by the cyclic formation of vapor bubbles at preferred sites on the solid heating surface of the system. Any surface, regardless of how highly polished, contains surface irregularities – micro-cracks, cavities, or boundaries between solid crystals – that can trap small gas pockets. These serve as nucleation sites for vapor bubbles. The trapped gas and/or vapor, known as the bubble embryo, grows by acquiring mass via evaporation from the nearby liquid. It grows until it reaches a critical size, at which point the forces tending to cause separation of bubbles from the heating surface overcome the adhesive forces. When the force imbalance favors the separating forces, the bubble is released from the surface. This process, by which solid surface imperfections with entrapped gases promote the formation of bubble embryos, is known as heterogeneous nucleation. Pure liquids that have been thoroughly degassed may still experience bubble formation at molecular vapor clusters within the bulk liquid, (i.e., away from any solid surface) by a process known as homogeneous nucleation. However, the degree of superheat required for homogeneous nucleation is substantially larger than that for heterogeneous nucleation. The common observation of bubble formation at the low superheat indicated by point A of Fig. 10.3, coupled with the repetitive formation of bubbles at selected points on heater surfaces, confirms that heterogeneous nucleation is by far the more common process. Since the trapped gases and/or vapors are central to the nucleation event, a consideration of the thermodynamics of vapor bubbles immersed in a liquid is appropriate. We will find that it justifies the expectation that the presence of gas or vapor pockets enhances bubble formation. We begin by considering a vapor bubble, including the possibility of noncondensable gas, immersed in its own liquid. Recall from Chapter 2 (see Section 2.6.3) that Laplace-Young equation must hold in order for the vapor/gas and liquid to be in equilibrium: ${p_g} + {p_v} - {p_\ell } = \frac{{2\sigma }}{{{R_b}}} \qquad \qquad(1)$

where Rb is the radius of the vapor bubble. Equation (1) indicates that the pressure of the vapor and gas must be greater than the pressure of the liquid to maintain a balance of forces in the vapor-interface-liquid system. Although these events occur on a molecular scale, we may attempt to visualize the physical implications of this imbalance of pressures across the boundary separating vapor and liquid phases. The higher vapor pressure in the bubble causes an increased number of molecules to strike the interface where they are absorbed by the liquid phase. To maintain the mass balance that equilibrium requires, there must be a corresponding increase in the number of molecules emitted through the interface from the liquid. Under the given constraints, this can be accomplished only by increasing the temperature of the liquid side of the interface. Consequently, the liquid near the bubble’s surface must be superheated to a temperature above the saturation temperature that corresponds to the prevailing bulk liquid pressure. To put these qualitative observations on a quantitative basis, consider that the vapor pressure for a curved bubble interface can be expressed as ${p_v} = {p_{v,sat}}(T)\exp \left[ {\frac{{ - 2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(2)$

where pv,sat(T) is the normal saturation pressure corresponding to temperature, T. Since $2\sigma {\rho _v}/[{p_{v,sat}}(T){R_b}{\rho _\ell }] \ll 1$, eq. (2) can be approximated as ${p_v}\textstyle.\over= {p_{v,sat}}(T)\left[ {1 - \frac{{2\sigma {\rho _v}}}{{{p_{v,sat}}(T){R_b}{\rho _\ell }}}} \right] \qquad \qquad(3)$

Combining eqs. (10.1) and (10.3) yields ${p_g} + {p_{v,sat}}(T) - {p_\ell }\textstyle.\over= \frac{{2\sigma }}{{{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) \qquad \qquad(4)$

Recalling the fundamental relationship between pressure and temperature differences in the two-phase region, the Clapeyron equation, $\frac{{dp}}{{dT}} = \frac{{{h_{\ell v}}}}{{T\left( {1/{\rho _v} - 1/{\rho _\ell }} \right)}} \qquad \qquad(5)$

and assuming that the fluid obeys the ideal gas law, i.e., $p = \rho {R_g}T \qquad \qquad(6)$

and that ${\rho _\ell } \gg {\rho _v}$, we arrive at $\frac{{dp}}{p} = \frac{{{h_{\ell v}}}}{{{R_g}{T^2}}}dT \qquad \qquad(7)$

For a pressure change from ${p_\ell }$ to pv,sat(T), the corresponding saturation temperature changes from Tsat to T. So by integrating eq. (7) between these limits, we obtain $\ln \frac{{{p_{v,sat}}(T)}}{{{p_\ell }}} = \frac{{{h_{\ell v}}}}{{{R_g}{T_{sat}}}}\frac{{T - {T_{sat}}}}{T} \qquad \qquad(8)$

Substituting eq. (4) into eq. (8) gives $\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{h_{\ell v}}}}\ln \left[ {1 + \frac{{2\sigma }}{{{p_\ell }{R_b}}}\left( {1 + \frac{{{\rho _v}}}{{{\rho _\ell }}}} \right) - \frac{{{p_g}}}{{{p_\ell }}}} \right] \qquad \qquad(9)$

Considering that $2\sigma /({p_\ell }{R_b}) - {p_g}/{p_\ell } \ll 1$ and again that ${\rho _\ell } \gg {\rho _v}$, eq. (9) can be further simplified as $\Delta T = T - {T_{sat}} = \frac{{{R_g}{T_{sat}}T}}{{{p_\ell }{h_{\ell v}}}}\left( {\frac{{2\sigma }}{{{R_b}}} - {p_g}} \right) \qquad \qquad(10)$

For the case with no noncondensable gas (pg = 0), eq. (10) can be rearranged to yield the critical vapor bubble radius: ${R_b} = \frac{{2\sigma {T_{sat}}}}{{{h_{\ell v}}{\rho _v}\Delta T}} \qquad \qquad(11)$

This critical radius is necessary for a bubble to exist at ΔT above the saturation temperature that corresponds to the prevailing vapor pressure. Any bubble with a radius less than the size given by eq. (11) will collapse, but a bubble with a radius equal to Rb will grow in a spontaneous fashion. It should be noted from eq. (10) that the size of the equilibrium vapor nucleus becomes smaller as the superheat (ΔT) increases. This identifies the basic mechanism for the increase of q'' with rising excess temperature between points A and C of Fig. 10.3, namely, the increased bubble density that results from the activation of more and more nucleation sites at ever-smaller surface imperfections. Equation (10) shows that the presence of noncondensable gas reduces the superheat required to generate bubbles. The subject of superheat was presented in Section 2.6.5. The bubble growth is spontaneous if the excess temperature is greater than the value given in eq. (10). Any real surface always has some small cavities with very small inclined angles where liquid can only partially fill the cavities, so opportunities for the formation of gas or vapor pockets are common. Equation (10) gives the superheat required to initiate nucleate boiling from a pre-existing nucleus with a radius of Rb. Griffith and Wallis (1960) suggested that it is also applicable for nucleate boiling from a surface with a micro cavity whose mouth radius is Rb. Alternative studies by Mizukami (1975), Nishio (1985), and Wang and Dhir (1993a) suggested that the superheat required to initiate nucleate boiling is determined by the instability of vapor nuclei in the cavity. If the curvature of the liquid-vapor interface, K, increases with increasing vapor volume (dK / dV * > 0, where V * is dimensionless vapor volume), the vapor nucleus is stable. On the other hand, the vapor bubble embryo or nucleus is unstable if K decreases with increasing vapor volume. For

the nucleate boiling to initiate from the wall, the wall superheat must be sufficient to ensure the vapor nuclei are unstable. Figure 1 shows the dependence of the nondimensional modified curvature of the interface on the dimensionless volume of the vapor nucleus in a spherical cavity with a mouth angle of 30°. The noncondensable gas is not present and the results at different contact angles are shown. The dimensionless volume of the vapor nucleus is defined as V * = Vb / Vc where Vb is the volume of the nucleus and Vc is the volume of the nucleus at point A, where the curvature is zero. It can be seen from Fig. 1 that the nucleus is stable between points A and D, since dK / dV * > 0. When the contact angle is less than 90°, the maximum curvature, Kmax, is 1. For the cases in which $\theta > {90^ \circ }$, the maximum curvature becomes sinθ. Nucleation occurs beyond the point where the curvature reaches its maximum; the corresponding superheat is (Wang and Dhir, 1993a) $\Delta T = T - {T_{sat}} = \frac{{4\sigma {T_{sat}}}}{{{p_v}{h_{\ell v}}{D_c}}}{K_{\max }} \qquad \qquad(12)$

where Dc is the diameter of the cavity mouth.

Example 1 A steam bubble with a radius of 5 μm is surrounded by liquid water at 120 ˚C. Will this bubble grow or collapse?

Solution: The liquid pressure is ${p_\ell } = 1{\rm{ atm}} = 1.013 \times {10^5}{\rm{ Pa}}{\rm{.}}$ The saturation temperature at this pressure is Tsat = 100oC = 373.15K. The properties of water at this temperature are $\sigma = 58.9 \times {10^{ - 3}}{\rm{ N/m,}}$ ${h_{\ell v}} =$ 2251.2kJ / kg, And ρv = 0.5974kg / m3. Therefore, the critical bubble radius can be determined from eq. (11), i.e., $\begin{array}{l} {R_b} = \frac{{2\sigma {T_{sat}}}}{{{h_{\ell v}}{\rho _v}\Delta T}} \\ = \frac{{2 \times 58.9 \times {{10}^{ - 3}} \times 373.15}}{{2251.2 \times {{10}^3} \times 0.5974 \times (120 - 100)}} = 1.63 \times {10^{ - 6}}{\rm{ m}} = 1.63{\rm{ \mu m}} \\ \end{array}$

Since the radius of the bubble, 5 μm, is greater than the critical bubble radius, the bubble will grow.

Figure 2 shows an idealized model of a surface crack containing a vapor pocket as it expands under the influence of evaporation. The vapor-liquid interface is idealized as spherical in shape. As the vapor pocket expands upward in the surface crack, its interface takes a concave shape and the radius of curvature, Rb, increases as the interface moves toward the lip of the cavity. Once the incipient bubble reaches the lip of the interface, the radius of curvature begins to decrease. It reaches its minimum value (approximately) as the liquid- vapor interface becomes a hemisphere with a radius equal to that of the cavity mouth, Rc. Beyond this point, any further evaporation pushes the liquid-vapor interface out of the surface cavity, and the interface radius of curvature begins to increase

again. Consideration of this sequence of radius variation, along with eq. (10), indicates that the superheat required to form a vapor bubble from this cavity can be determined by substituting Rb = Rc into eq. (10). In normal circumstances, of course, this variation deviates from the idealized model in Fig. 2 and is a function of cavity angle φ and contact angle θ. In reality, the shape of a vapor bubble is not hemispherical, and there is a microlayer (a very thin liquid layer, see Fig. 3) that is responsible for more than half of heat transfer in some cases. Plesset and Sadhal (1979) provided the average liquid film thickness $\bar \delta$ in terms of bubble lift time t0 and liquid kinematic viscosity by the following equation $\bar \delta = \frac{8}{7}{(3{\nu _\ell }{t_0})^{1/2}} \qquad \qquad(13)$

The contribution of the microlayer to the overall heat transfer obviously depends on the fraction of area occupied by the microlayer. Despite its significant importance, there is no detailed treatment of the microlayer related to nucleate boiling.

## References

Griffith, P., and Wallis, J.D., 1960, “The Role of Surface Conditions in Nucleate Boiling,” Chemical Engineering Progress Symposium, Ser. 56, No. 30, pp. 49-63.

Mizukami, K., 1975, “Entrapment of Vapor in Re-Entrant Cavities,” Letters in Heat and Mass Transfer, Vol. 2, pp. 279-284.

Nishio, A., 1985, “Stability of Pre-Existing Vapor Nucleus in Uniform Temperature Field,” Transactions of JSME, Series B, Vol. 54-303, pp. 1802-1807.

Plesset, M.S., and Sadhal, S.S., 1979, “An Analytical Estimate of the Microlayer Thickness in Nucleate Boiling,” ASME Journal of Heat Transfer, Vol. 101, pp. 180-182.

Wang, C.H., and Dhir, V.K., 1993b, “Effect of Surface Wettability on Active Nucleation Site Density During Pool Boiling of Water on a Vertical Surface,” ASME Journal of Heat Transfer, Vol. 115, pp. 659-669.