# Film Boiling Analysis

## Film boiling over a vertical plate

When the excess temperature increases beyond the minimum heat flux point, boiling enters the film boiling regime, characterized by a continuous vapor film covering the entire surface. The major thermal resistance is in this vapor layer. Unlike the vapor film in the transition boiling regime, that in the film boiling regime is stable. Since the liquid is separated from the heating surface by a stable vapor film, heat transfer in this regime can be obtained by analyzing evaporation at the liquid-vapor interface. Figure 1 shows film boiling from a heated vertical flat plate in a motionless pool of subcooled liquid. Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. Drag force from the vapor causes the adjacent liquid to flow upward as well. Therefore, a velocity boundary layer forms in the liquid phase adjacent to the vapor film. Successful solution of the film boiling problem requires solutions of vapor and liquid flow as well as heat transfer in the vapor phase and at the interface. Since the liquid phase is subcooled, there is also heat transfer in the liquid. The vapor layer thickness is usually small compared to the length scale of the vertical flat plate, so boundary layer approximations are applicable to the vapor film. It is further assumed that the vapor flow is laminar and two-dimensional. The continuity, momentum, and energy equations in the vapor layer are $\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1)$ ${u_v}\frac{{\partial {u_v}}}{{\partial x}} + {v_v}\frac{{\partial {u_v}}}{{\partial y}} = {\nu _v}\frac{{{\partial ^2}{u_v}}}{{\partial {y^2}}} + \frac{{g({\rho _\ell } - {\rho _v})}}{{{\rho _v}}}\qquad \qquad(2)$ ${u_v}\frac{{\partial {T_v}}}{{\partial x}} + {v_v}\frac{{\partial {T_v}}}{{\partial y}} = {\alpha _v}\frac{{{\partial ^2}{T_v}}}{{\partial {y^2}}}\qquad \qquad(3)$

Vapor flow drags the liquid adjacent to the vapor film, but the rest of the liquid phase is stationary. Therefore, boundary layer theory can be applied to describe the motion of the liquid, i.e., $\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4)$ ${u_\ell }\frac{{\partial {u_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {u_\ell }}}{{\partial y}} = {\rho _\ell }g{\beta _\ell }({T_\ell } - {T_\infty }) + {\nu _\ell }\frac{{{\partial ^2}{u_\ell }}}{{\partial {y^2}}}\qquad \qquad(5)$

There will also be a thermal boundary layer in the liquid phase, and its temperature satisfies ${u_\ell }\frac{{\partial {T_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {T_\ell }}}{{\partial y}} = {\alpha _\ell }\frac{{{\partial ^2}{T_\ell }}}{{\partial {y^2}}}\qquad \qquad(6)$

The boundary conditions at the heated wall (y = 0) are $y = 0:{\rm{ }}{u_v} = {v_v} = 0,{\rm{ }}T = {T_w}\qquad \qquad(7)$

The boundary conditions at the liquid-vapor interface (y = δ) are necessary to couple the fluid flows in the liquid and vapor phases. The velocities for both phases are the same due to the nonslip condition, i.e. $y = \delta \begin{array}{*{20}{c}} : & {{u_v} = {u_\ell }} \\ \end{array}\qquad \qquad(8)$

The shear stresses calculated from the liquid and vapor phases are the same as required by force balance, i.e., ${\left( {\mu \frac{{\partial u}}{{\partial y}}} \right)_v} = {\left( {\mu \frac{{\partial u}}{{\partial y}}} \right)_\ell }\begin{array}{*{20}{c}} , & {y = \delta } \\ \end{array}\qquad \qquad(9)$

The mass balance at the liquid-vapor interface can be obtained by applying conservation of mass for a control volume shown in Fig. 2, i.e., ${\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_v} = {\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_\ell }\qquad \qquad(10)$

The temperature at the liquid-vapor interface is equal to the saturation temperature: ${T_v} = {T_\ell } = {T_{sat}}\begin{array}{*{20}{c}} , & {y = \delta } \\ \end{array}\qquad \qquad(11)$

The energy balance at the liquid-vapor interface is $- {k_v}\frac{{\partial {T_v}}}{{\partial y}} + {q''_{wR}} = {\rho _v}\left( {{u_v}\frac{{d\delta }}{{dx}} - {v_v}} \right) - {k_\ell }\frac{{\partial {T_\ell }}}{{\partial y}}\qquad \qquad(12)$

where q''wR is heat flux due to radiation from the heat wall to the interface. The boundary condition in the liquid far from the heated surface is ${u_\ell } \to 0\begin{array}{*{20}{c}} , & {y \to \infty } \\ \end{array}\qquad \qquad(13)$

For a case in which the liquid is saturated ( ${T_\ell } = {T_{sat}}$), the first term on the right-hand side of eq. (5), representing buoyancy force in the liquid phase, will disappear. The energy equation (6) for the liquid phase will not be needed. The similarity solution for the film boiling in a saturated liquid was obtained by Koh (1962) and will be presented here. Defining the stream functions in the vapor and liquid phases as $\begin{array}{l} {u_v} = \frac{{\partial {\psi _v}}}{{\partial y}}\begin{array}{*{20}{c}} {} & {{v_v} = - \frac{{\partial {\psi _v}}}{{\partial x}}} \\ \end{array} \\ {u_\ell } = \frac{{\partial {\psi _\ell }}}{{\partial y}}\begin{array}{*{20}{c}} {} & {{v_\ell } = - \frac{{\partial {\psi _\ell }}}{{\partial x}}} \\ \end{array} \\ \end{array}\qquad \qquad(14)$

the continuity equations (1) and (4) are automatically satisfied by the stream functions defined by eq. (14). To obtain the similarity solution, the following similarity variables are defined: $\eta = {\left( {\frac{g}{{4\nu _v^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}\frac{y}{{{x^{1/4}}}}\qquad \qquad(15)$ ${\psi _v} = 4{\nu _v}{\left( {\frac{g}{{4\nu _v^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}{x^{3/4}}f(\eta )\qquad \qquad(16)$ $\theta = \frac{{T - {T_{sat}}}}{{{T_w} - {T_{sat}}}}\qquad \qquad(17)$

where η is the independent similarity variable and f(η) is the nondimensional stream function in the vapor phase. Substituting eqs. (15) and (16) into eq. (14), the velocity components in x- and y- directions are respectively ${u_v} = \frac{{\partial {\psi _v}}}{{\partial y}} = \frac{{\partial {\psi _v}}}{{\partial \eta }}\frac{{\partial \eta }}{{\partial y}} = 4{\nu _v}{\left( {\frac{g}{{4\nu _v^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/2}}{x^{1/2}}f'(\eta )\qquad \qquad(18)$ ${v_v} = - \frac{{\partial {\psi _v}}}{{\partial x}} = - \frac{{\partial {\psi _v}}}{{\partial \eta }}\frac{{\partial \eta }}{{\partial x}} = {\nu _v}{\left( {\frac{g}{{4\nu _v^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}{x^{ - 1/4}}\left[ {\eta f'(\eta ) - 3f(\eta )} \right]\qquad \qquad(19)$

Substituting eqs. (18) – (19) into eq. (2), the momentum equation in the vapor phase becomes $f''' + 3ff'' - 2{(f')^2} + 1 = 0\qquad \qquad(20)$

Substituting eqs. (15) and (17) into eq. (3), the nondimensional energy equation in the vapor layer is obtained: $\theta '' + 3{\Pr _v}f\theta ' = 0\qquad \qquad(21)$

The momentum equation in the liquid phase can also be transformed into an ordinary differential equation: $F’’’ + 3FF – 2(F’{)^2} = 0 \qquad \qquad (22)$

where the similarity variables in the liquid phase are $\xi = {\left( {\frac{g}{{4\nu _\ell ^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}\frac{y}{{{x^{1/4}}}}\qquad \qquad(23)$ ${\psi _\ell } = 4{\nu _\ell }{\left( {\frac{g}{{4\nu _\ell ^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}{x^{3/4}}F(\xi )\qquad \qquad(24)$

Therefore, the film boiling problem can be described with a set of ordinary differential equations (20) – (22). To solve these ordinary differential equations, the boundary conditions specified by eqs. (1) – (13) must be rewritten in terms of similarity variables. Substituting the nondimensional variables defined by eqs. (15) – (17) and eqs. (23) – (24) into eq. (7) – (13), the boundary conditions in terms of nondimensional variables become $f(0) = f'(0) = 0\qquad \qquad(25)$ $\theta (0) = 1\qquad \qquad(26)$ $F({\xi _\delta }) = {\left( {\frac{{{\rho _v}{\mu _v}}}{{{\rho _\ell }{\mu _\ell }}}} \right)^{\frac{1}{2}}}f({\eta _\delta })\qquad \qquad(27)$ $F'({\xi _\delta }) = f'({\eta _\delta })\qquad \qquad(28)$ $F''({{\zeta}_{\delta}})= {{(\frac{{{\rho}_v}{{\mu}_v}}{{{\rho}_{\ell}}{{\mu}_{\ell}}})}^{1/2}}f''({n_{\delta}}) \qquad \qquad (29)$ $\theta ({\eta _\delta }) = 0\qquad \qquad(30)$ $F'(\infty ) = 0\qquad \qquad(31)$

The film boiling problem is now described by a set of three ordinary differential equations with their corresponding boundary conditions. The film boiling problem is not closed because there is no equation that governs the thickness of the vapor film. It is therefore necessary to develop another equation to find the vapor film thickness. This additional equation can be found by performing the energy balance at the interface. In the control volume shown in Fig. 3, the mass flow rate per unit width of the surface at x is ${\Gamma _v}(x) = \int_0^\delta {{\rho _v}{u_v}dy} \qquad \qquad(32)$

The mass flow rate at x+dx is obtained by ${\Gamma _v}(x + dx) = {\Gamma _v}(x) + \frac{{d{\Gamma _v}}}{{dx}}dx\qquad \qquad(33)$

The increase of vapor mass flow rate is due to evaporation at the interface and is related to heat transfer as $dq' = {h_{\ell v}}d{\Gamma _v} = {h_{\ell v}}\frac{d}{{dx}}\left( {\int_0^\delta {{\rho _v}{u_v}dy} } \right)dx\qquad \qquad(34)$

On the other hand, heat transfer at the interface can be obtained by applying Fourier’s law at the interface, i.e., $dq' = {k_v}{\left. {\frac{{\partial T}}{{\partial y}}} \right|_{y = \delta }}dx\qquad \qquad(35)$

If all the heat transferred to the system goes into latent heat of vaporization, one can combine eqs. (34) and (35) to obtain the energy balance at the liquid-vapor interface. ${\left. {\frac{{\partial T}}{{\partial y}}} \right|_{y = \delta }} = \frac{{{h_{\ell v}}}}{{{k_v}}}\frac{d}{{dx}}\int_0^\delta {{\rho _v}{u_v}dy} \qquad \qquad(36)$

Equation (36) can be rewritten in terms of the similarity variables defined in eqs. (15) – (17), i.e., $\frac{{3f({\eta _\delta })}}{{\theta '({\eta _\delta })}} = \frac{{{c_{p,v}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}{{\Pr }_v}}}\qquad \qquad(37)$

At this point, the film boiling problem is mathematically closed and can be solved numerically. This is a boundary value problem that can be solved by using the shooting method. The local heat transfer coefficient is ${h_x} = - \frac{1}{{{T_w} - {T_{sat}}}}{k_v}{\left. {\frac{{\partial T}}{{\partial y}}} \right|_{y = 0}} = - {k_v}{\left. {\frac{{\partial \theta }}{{\partial y}}} \right|_{y = 0}}\qquad \qquad(38)$

Substituting eq. (15) into eq. (38), the local heat transfer coefficient becomes ${h_x} = - {k_v}{\left( {\frac{g}{{4\nu _v^2}}\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{1/4}}\frac{{\theta '(0)}}{{{x^{1/4}}}}\qquad \qquad(39)$

It follows that the local Nusselt number is of the form ${\rm{N}}{{\rm{u}}_x} = \frac{{{h_x}x}}{{{k_v}}} = [ - \theta '(0)]{\left[ {\frac{{({\rho _\ell } - {\rho _v})g{x^3}}}{{4{\rho _v}\nu _v^2}}} \right]^{1/4}}\qquad \qquad(40)$

where θ'(0) is the first-order derivative of nondimensional temperature at η = 0. It is a function of Prv and $({\rho _v}{\mu _v})/({\rho _\ell }{\mu _\ell })$, and it can be obtained by numerical solution of the problem. The effect of subcooling on boiling from a vertical plate was studied by Sparrow and Cess (1962).

## Forced convection laminar film boiling

Forced convection laminar film boiling from a horizontal constant temperature plate for the cases of saturated and subcooled liquids is solved by liquid and vapor boundary layer equations by Cess and Sparrow (1961a, b). The result for saturated film boiling is presented below: $\frac{1}{2}{K^{1/2}} = {\rm{Nu}}{{\mathop{\rm Re}} ^{ - 1/2}}\left( {\frac{{{\mu _v}}}{{{\mu _\ell }}}} \right){\left[ {1 + {\pi ^{1/2}}{\rm{Nu}}{{{\mathop{\rm Re}} }^{ - 1/2}}\left( {\frac{{{\mu _v}}}{{{\mu _\ell }}}} \right)} \right]^{1/2}}\qquad \qquad(41)$

where $K = \left( {\frac{{{\rho _v}{\mu _v}}}{{{\rho _\ell }{\mu _\ell }}}} \right)\frac{{{{\Pr }_v}{h_{\ell v}}}}{{{c_{p,v}}({T_w} - {T_{sat}})}}\qquad \qquad(42)$ ${\rm{Nu}} = \frac{{hx}}{k}\qquad \qquad(43)$ ${\mathop{\rm Re}} = \frac{{{u_\infty }x}}{{{\nu _\ell }}}\qquad \qquad(44)$

where x is distance from leading edge, and ${u_\infty }$ is liquid forced velocity far from the plate. Equation (41) is valid for ${10^{ - 3}} \le K \le 1$. Gravity is negligible in the above analysis since forced convection was the driving force and therefore the result for ${u_\infty } = 0$ is not meaningful. Liquid subcooling will increase both heat transfer and plate shear.

The thermophysical properties of both liquid and vapor in the above analysis were assumed to be independent from the temperature. The influence of constant-properties assumption was examined by Nishikawa et al. (1976) by numerically solving the two-phase boundary layer equations. They confirmed that, except at high system pressure, the results based on the constant-properties assumption reasonably agreed with the results based on variable properties. Nakayama (1986) also provided additional analysis related to forced convective film subcooled boiling for cases of constant surface temperature.

The analysis developed above do not account for heat transfer due to radiation from the heating surface across the vapor layer to the liquid. When the surface temperature is very high, the radiation across the vapor film contributes more and more to the overall heat transfer rate from the heating surface to the liquid. The rate of vapor production increases and the vapor layer thickens when radiation is significant. A detailed analysis of the radiation heat transfer must be performed in order to accurately predict the film boiling heat transfer.

Since the vapor is virtually transparent to infrared radiation emitted by the heating surface and liquid, the vapor can be treated as a nonparticipating medium; this assumption significantly simplifies the modeling of radiation heat transfer. The vapor films in film boiling are usually thin, allowing the radiation heat transport to be modeled as the radiation exchange between two parallel plates. Heat transfer due to radiation acts independently from the heat transfer from conduction and convection. The total heat transfer can be found by adding the contribution from radiation directly to the contributions from conduction and convection. Since the heating surface and the interface are treated as two parallel plates with different temperatures, the contribution from radiation can be obtained by ${q''_{rad}} = \frac{{{\sigma _{SB}}(T_w^4 - T_{sat}^4)}}{{\frac{1}{{{\varepsilon _s}}} + \frac{1}{{{\varepsilon _I}}} - 1}}\qquad \qquad(45)$

where σSB is the Stefan-Boltzmann constant, and ${\varepsilon _s}$ and ${\varepsilon _I}$ are the emissivity of the heating surface and the interface, respectively. The effective heat transfer coefficient due to radiation is then obtained by ${h_{rad}} = \frac{{{{q''}_{rad}}}}{{{T_w} - {T_{sat}}}} = \frac{{{\sigma _{SB}}(T_w^2 + T_{sat}^2)({T_w} + {T_{sat}})}}{{\frac{1}{{{\varepsilon _s}}} + \frac{1}{{{\varepsilon _I}}} - 1}}\qquad \qquad(46)$

The overall heat transfer coefficient for film boiling can be obtained by (Bromley, 1950) $h = {h_{con}}{\left( {\frac{{{h_{con}}}}{h}} \right)^{1/3}} + {h_{rad}}\qquad \qquad(47)$

where hcon is the convective heat transfer coefficient determined by eq. (39). Equation (47) is not convenient for determining the overall heat transfer coefficient h, since h appears on both sides of the equation. Bromley (1950) further recommended that eq. (47) could be simplified to $h = {h_{con}} + 0.75{h_{rad}}\qquad \qquad(48)$

The difference between the overall heat transfer coefficient obtained by eqs. (47) and that obtained by eq. (48) is within 5%.

## Film boiling near a cylinder or sphere

Film boiling near an immersed body, such as a cylinder or sphere, can be solved by the integral approximation method. The average Nusselt number for film boiling on a horizontal cylinder (Bromley, 1950) or sphere (Lienhard and Dhir, 1973) is given by $\overline {{\rm{Nu}}} = \frac{{\bar hD}}{{{k_v}}} = C{\left[ {\frac{{{\rho _v}g({\rho _\ell } - {\rho _v}){{h'}_{\ell v}}{D^3}}}{{{\mu _v}{k_v}}}} \right]^{\frac{1}{4}}}\qquad \qquad(49)$

where ${h'_{\ell v}} = {h_{\ell v}}\left[ {1 + \frac{{0.4{c_{pv}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}}} \right]\qquad \qquad(50)$

The correlation constants C for horizontal cylinders and spheres are 0.62 and 0.67, respectively. The vapor properties are evaluated at the film temperature, Tf = (Tw + Tsat) / 2. Equation (49) is valid for $0.8 < {\lambda _c}/D \le 8$ where ${\lambda _c} = 2\pi {\{ \sigma /[({\rho _\ell } - {\rho _v})g]\} ^{1/2}}$ is critical wavelength, or as long as the diameter of the horizontal cylinder or sphere is much greater than the vapor film thickness.

## Film boiling over a horizontal, large flat plate

Another useful geometric configuration is film boiling over a horizontal, large flat plate, with the dense liquid overlaying a less-dense vapor. Vapor leaves the film in the form of detaching bubbles, which determines the heat transfer rate. Berenson (1961) performed an analysis of film boiling on a horizontal heat surface in which he assumed that the bubbles are released at the node of the Taylor wave. The correlation resulting from the Taylor instability analysis is given by $h = 0.425{\left\{ {\left[ {\frac{{k_v^3g{\rho _v}({\rho _\ell } - {\rho _v}){{h'}_{\ell v}}}}{{{\mu _v}({T_w} - {T_{sat}})}}} \right]{{\left[ {\frac{{g({\rho _\ell } - {\rho _v})}}{\sigma }} \right]}^{\frac{1}{2}}}} \right\}^{\frac{1}{4}}}\qquad \qquad(51)$

which is in good agreement with experimental results obtained by using n-pentane and carbon tetrachloride. Klimenko (1981) correlated the film boiling from a horizontal plate ${\rm{Nu}} = \left\{ \begin{array}{l} 0.19{\left[ {{\rm{Ga}}\left( {{\rho _\ell }/{\rho _v} - 1} \right)} \right]^{1/3}}\Pr _v^{1/3}{f_1}{\rm{ Ga}}\left( {{\rho _\ell }/{\rho _v} - 1} \right) < {10^8}{\rm{ (Laminar)}} \\ 0.0086{\left[ {{\rm{Ga}}\left( {{\rho _\ell }/{\rho _v} - 1} \right)} \right]^{1/2}}\Pr _v^{1/3}{f_2}{\rm{ Ga}}\left( {{\rho _\ell }/{\rho _v} - 1} \right) > {10^8}{\rm{(Turbulent)}} \\ \end{array} \right.\qquad \qquad(52)$

where ${f_1} = \left\{ \begin{array}{l} 1{\rm{ J}}{{\rm{a}}_v} \ge 0.71 \\ 0.89{\rm{Ja}}_v^{ - 1/3}{\rm{ J}}{{\rm{a}}_v} < 0.71 \\ \end{array} \right.\qquad \qquad(53)$ ${f_2} = \left\{ \begin{array}{l} 1{\rm{ J}}{{\rm{a}}_v} \ge 0.5 \\ 0.71{\rm{Ja}}_v^{ - 1/2}{\rm{ J}}{{\rm{a}}_v} < 0.5 \\ \end{array} \right.\qquad \qquad(54)$

where ${\rm{Nu}} = \frac{{h{\lambda _c}}}{k}\qquad \qquad(55)$ ${\rm{J}}{{\rm{a}}_v} = \frac{{{c_{p,v}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}}\qquad \qquad(56)$ ${\rm{Ga}} = \frac{{g\lambda _c^3}}{{2\nu _v^2}}\qquad \qquad(57)$

are Nusselt, Jakob, and Galileo numbers, respectively. All vapor properties are evaluated at (Tw + Tsat) / 2. Equation (52) is valid for $7 \times {10^4} <$ ${\rm{Ga}}({\rho _\ell }/{\rho _v} - 1) < 3 \times {10^8},$ $0.69 < {\Pr _v} < 3.45,$ 0.14 < Jav < 32.26, 0.0045 < p / pcrit < 0.98 and up to 21.7g. The effect of plate size can be accounted for by $\frac{h}{{{h_\infty }}} = \left\{ {\begin{array}{*{20}{c}} {1,{\rm{ }}L/{\lambda _c} > 5} \\ {2.9{{({\lambda _c}/L)}^{0.67}},{\rm{ }}L/{\lambda _c} < 5} \\ \end{array}} \right.\qquad \qquad(58)$

where L is the minimum dimension of the horizontal plate.

## References

Berenson, P.J., 1961, “Film Boiling Heat Transfer from a Horizontal Surface,” ASME Journal of Heat Transfer, Vol. 83, pp. 351-356.

Bromley, L.A., 1950, “Heat Transfer in Stable Film Boiling,” Chemical Engineering Progress, Vol. 46, pp. 221-227.

Cess, R.D., and Sparrow, E.M., 1961b, “Subcooled Forced-Convection Film Boiling on a Flat Plate,” ASME Journal of Heat Transfer, Vol. 83, pp. 377-379.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Klimenko, V.V., 1981, “Film Boiling on a Horizontal Plate–New Correlation,” International Journal of Heat and Mass Transfer, Vol. 24, pp. 69-79.

Koh, J.C.Y., 1962, “Analysis of Film Boiling on Vertical Surfaces,” ASME Journal of Heat Transfer, Vol. 84, p. 55-62.

Lienhard, J.H., and Dhir, V.K., 1973, Extended Hydrodynamic Theory of the Peak and Minimum Heat Fluxes, NASA CR-2270.

Nakayama, A., 1986, “Subcooled Forced Convection Film Boiling on Plane and Axisymmetric Bodies in the Presence of Pressure Gradient,” AIAA Journal, Vol. 24, pp. 230-236.

Nishikawa, K., Ito, T., and Matsumoto, K., 1976, “Investigation of Variable Thermophysical Property Problem Concerning Pool Film Boiling from Vertical Plate with Prescribed Uniform Temperature,” International Journal of Heat and Mass Transfer, Vol. 19, pp. 1173-1181.

Sparrow, E.M., and Cess, R.D., 1962, “The Effect of Subcooled Liquid on Laminar Film Boiling,” ASME Journal of Heat Transfer, Vol. 84, pp. 149-156.