Direct numerical simulation of film boiling

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In additional to the above analytical results about film boiling, numerical solutions on this topic were also reported in the recent literature (Esmaeeli and Tryggvason, 2004). Son and Dhir (1997) simulated film boiling for both two-dimensional and axisymmetric flows using a moving body-fitted coordinate system, which is limited to modest deformation of the phase boundary. This limitation was later overcome by Juric and Tryggvason (1998) using a front tracking method and Son and Dhir’s (1997) using a level set method. The subcooled film boiling was solved by Banerjee and Dhir (2001) using a level set method. The Volume of Fluid (VOF) method was employed to simulate film boiling by Welch and Wilson (2000). In addition to the above work that use one method, the front tracking method was also combined with the finite difference method (Esmaeeli and Tryggvason, 2001; 2003) and level contour technique (Shin and Juric, 2002). The direct numerical simulation of film boiling conducted by Esmaeeli and Tryggvason (2004) will be presented here.

The continuity, momentum, and energy equations for each phase in the boiling flow are

$\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot (\rho {\mathbf{V}}) = 0\qquad \qquad(1)$ $\frac{{\partial \rho {\mathbf{V}}}}{{\partial t}} + \nabla \cdot (\rho {\mathbf{VV}}) = - \nabla p + \rho {\mathbf{g}} + \nabla \cdot \mu (\nabla {\mathbf{V}} + \nabla {{\mathbf{V}}^T})\qquad \qquad(2)$ $\frac{{\partial \rho {c_p}T}}{{\partial t}} + \nabla \cdot (\rho {c_p}{\mathbf{V}}T) = \nabla \cdot (k\nabla T)\qquad \qquad(3)$

where the viscosity dissipation has been neglected in eq. (3). At the liquid-vapor interface, the following jump conditions must be satisfied:

${\rho _\ell }({{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}) \cdot {\mathbf{n}} = {\rho _v}({{\mathbf{V}}_v} - {{\mathbf{V}}_I}) \cdot {\mathbf{n}} = \dot m''\qquad \qquad(4)$ $\dot m''({{\mathbf{V}}_v} - {{\mathbf{V}}_\ell }) = ({{\mathbf{\tau }}_v} - {{\mathbf{\tau }}_\ell }) \cdot {\mathbf{n}} - ({p_v} - {p_\ell }){\mathbf{I}} \cdot {\mathbf{n}} + \sigma K{\mathbf{n}}\qquad \qquad(5)$ $\dot m''{h_{\ell v}} = {q''_I} = {\left. {{k_v}\frac{{\partial T}}{{\partial n}}} \right|_v} - {\left. {{k_\ell }\frac{{\partial T}}{{\partial n}}} \right|_\ell }\qquad \qquad(6)$

where ${{\mathbf{V}}_\ell }{\rm{ and }}{{\mathbf{V}}_v}$ are the fluid velocities at the liquid and vapor sides of the interface, and ${{\mathbf{V}}_I}$ is the interfacial velocity. The above momentum and energy equations and their jump conditions can be rewritten as the following “single-field” form:

$\frac{{\partial \rho {\mathbf{V}}}}{{\partial t}} + \nabla \cdot (\rho {\mathbf{VV}}) = - \nabla p + \rho {\mathbf{g}} + \nabla \cdot \mu (\nabla {\mathbf{V}} + \nabla {{\mathbf{V}}^T}) + \sigma \int_I {\delta ({\mathbf{r}} - {{\mathbf{r}}_I})K{{\mathbf{n}}_I}d{A_I}} \qquad \qquad(7)$ $\frac{{\partial \rho {c_p}T}}{{\partial t}} + \nabla \cdot (\rho {c_p}{\mathbf{V}}T) = \nabla \cdot (k\nabla T) - \left[ {1 - ({c_v} - {c_\ell })\frac{{{T_{sat}}}}{{{h_{\ell v}}}}} \right]\int_I {\delta ({\mathbf{r}} - {{\mathbf{r}}_I}){{{\mathbf{q''}}}_I}d{A_I}} \qquad \qquad(8)$

where $δ$ is a two- or three- dimensional delta-function, r is the position vector and r_l is the position of the interface. The subscript I in eqs. (7) and (8) represents interface. Although both liquid and vapor can be treated as incompressible flows, the incompressible assumption near the interface is invalid. In order to rewrite the continuity equation and the jump condition into a single-field equation, the velocity field can be rewritten as

${\mathbf{V}} = {{\mathbf{V}}_v}H + {{\mathbf{V}}_\ell }(1 - H)\qquad \qquad(9)$

where H is a Heaviside function that has a value of one in vapor but zero in the liquid phase. The gradient of the Heaviside function can be expressed in terms of the interfacial properties:

$\nabla H = \int_I {\delta ({\mathbf{r}} - {{\mathbf{r}}_I}){{\mathbf{n}}_I}d{A_I}} \qquad \qquad(10)$

which is zero everywhere except at the interface. The divergence of the velocity defined in eq. (9) is

$\nabla \cdot {\mathbf{V}} = H\nabla \cdot {{\mathbf{V}}_v} + (1 - H)\nabla \cdot {{\mathbf{V}}_\ell } + ({{\mathbf{V}}_v} - {{\mathbf{V}}_\ell }) \cdot \nabla H\qquad \qquad(11)$

Substituting eq. (10) into eq. (11) and considering liquid and vapor are incompressible ( $\nabla {{\mathbf{V}}_v} = \nabla {{\mathbf{V}}_\ell } = 0$ ), one obtains

$\nabla \cdot {\mathbf{V}} = \int_I {\delta ({\mathbf{r}} - {{\mathbf{r}}_I})({{\mathbf{V}}_v} - {{\mathbf{V}}_\ell }) \cdot {{\mathbf{n}}_I}d{A_I}} \qquad \qquad(12)$

Eliminating V_l in eq. (4) yields

$({{\mathbf{V}}_v} - {{\mathbf{V}}_\ell }) \cdot {\mathbf{n}} = \dot m''\left( {\frac{1}{{{\rho _v}}} - \frac{1}{{{\rho _\ell }}}} \right) = \frac{{q''}}{{{h_{\ell v}}}}\left( {\frac{1}{{{\rho _v}}} - \frac{1}{{{\rho _\ell }}}} \right)\qquad \qquad(13)$

Substituting eq. (13) into eq. (12), the single-field continuity equation becomes

$\nabla \cdot {\mathbf{V}} = \frac{{q''}}{{{h_{\ell v}}}}\left( {\frac{1}{{{\rho _v}}} - \frac{1}{{{\rho _\ell }}}} \right)\int_I {\delta ({\mathbf{r}} - {{\mathbf{r}}_I}){{q''}_I}d{A_I}} \qquad \qquad(14)$

Esmaeeli and Tryggvason (2004) solved the single-field governing equations (7), (8), and (14) using a second- order space-time accurate front-tracking/finite difference method on a staggered grid. They obtained a numerical solution of film boiling on a horizontal surface where a thin vapor film separates the liquid and the heated surface.

References

Banerjee, D., and Dhir, V.K., 2001, “Study of Subcooled Film Boiling on a Horizontal Disc: Part I - Analysis,” ASME Journal of Heat Transfer, Vol. 123, pp. 271-284.

Esmaeeli, A., and Tryggvason, G., 2001, “Direct Numerical Simulations of Boiling Flows,” Proceedings of the Fourth International Conference on Multiphase Flow, ICMF-2001, New Orleans, LA.

Esmaeeli, A., and Tryggvason, G., 2003, “Computations of Explosive Boiling in Microgravity,” Journal of Scientific Computing, Vol. 19, pp. 163-182.

Esmaeeli, A., and Tryggvason, G., 2004, “Computations of Film Boiling. Part I: Numerical Method,” International Journal of Heat and Mass Transfer, Vol. 47, pp. 5451-5461.

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.

Juric, D., and Tryggvason, G., 1998, “Computations of Boiling Flow,” International Journal of Multiphase Flow, Vol. 24, pp. 387-410.

Shin, S., and Juric, D., 2002, “Modeling Three-Dimensional Multiphase Flow Using a Level Contour Reconstruction Method for Front Tracking without Connectivity,” Journal of Computational Physics, Vol. 180, pp. 427-470.

Son, G., and Dhir, V.K., 1997, “Numerical Simulation of Saturated Film Boiling on a Horizontal Surface,” ASME Journal of Heat Transfer, Vol. 119, pp. 525-533.

Welch, S.W.J., and Wilson, J.J., 2000, “A Volume of Fluid Based Method for Fluid Flows with Phase Change,” Journal of Computational Physics, Vol. 160, pp. 662-682.

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Direct numerical simulation of film boiling

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