Approximate analytical solutions of the kinetic and balance equations for intense boiling

Abstract

The process of intense boiling is theoretically studied on the basis of kinetic and balance equations for the bubble-size distribution function and system temperature. The kinetic equation for the bubble-size distribution function represents the first-order partial differential equation with a source term. The heat balance is spatially homogeneous and takes into account the heat exchange of the system with the external environment. This non-linear system is supplemented by the initial and boundary conditions. Namely, the initial bubble-size distribution and temperature are regarded as known, and the flux of bubbles overcoming the critical size is defined by the rate of nuclei formation. A non-linear integro-differential system of model equations is solved analytically by the integral Laplace transform and saddle-point methods. It is shown that the solution has a different form at $x\ge \tau$ and $x<\tau$ (here, x and $\tau$ are the dimensionless spatial and temporal variables). Also, we show that the initial bell-shaped distribution function decreases, and the liquid temperature increases with increasing time.

Appendix A: Approximate calculation of the integral contribution \(A(\tau )\)

Let us evaluate here the integral term $A(\tau )$ entering in expression (23) using the saddle-point technique [53, 54] for a Laplace-type integral. At first, introducing

$$\begin{aligned} \varphi (\tau )=\frac{\sqrt{\pi }}{2}\left( 1-\mathrm{erfc}\sqrt{a\tau } \right) -\sqrt{a\tau } \exp \left( -a\tau \right) , \end{aligned}$$

(A.1)

we rewrite $A(\tau )$ from (24) in the form of

$$\begin{aligned} A(\tau ) = \frac{R\left[ u(0)\right] \varphi (\tau ) }{a^{3/2}} -\frac{1}{a^{3/2}}\int \limits _0^\tau \varphi (\tau _1)\frac{\partial R\left[ u (\tau -\tau _1) \right] }{\partial \tau _1}\mathrm{d}\tau _1. \end{aligned}$$

(A.2)

Now, introducing the new variable $\xi =\tau -\tau _1$ and evaluating the derivative $\partial R/\partial \tau _1$, we rewrite the integral (A.2) as

$$\begin{aligned} A(\tau ) = \frac{R\left[ u(0)\right] \varphi (\tau ) }{a^{3/2}} +\frac{1}{a^{3/2}}\int \limits _0^\tau {{\tilde{f}}}(\xi ,\tau ) \exp \left[ \kappa S[u(\xi )] \right] \mathrm{d}\xi , \end{aligned}$$

(A.3)

where

$$\begin{aligned} {{\tilde{f}}}(\xi ,\tau )&= \kappa \varphi (\tau -\xi ) \frac{3u(\xi )+2}{u^3(\xi )\left[ u(\xi )+1 \right] ^2}\frac{\mathrm{d}u}{\mathrm{d}\xi },\\\&S[u(\xi )]=-\frac{1}{u^2(\xi )\left[ u(\xi )+1 \right] }. \end{aligned}$$

(A.4)

We assume that the function $S(\xi )$ has a maximum that is sharper, the larger the parameter $\kappa$ [53, 54]. The maximum of this function attains at the maximum point of dimensionless temperature $u_{\mathrm{max}}=u(\tau )$. Taking this into account, we evaluate the integral (A.3) using the saddle-point method [53, 54] as

$$\begin{aligned} A(\tau ) = \frac{R\left[ u(0)\right] \varphi (\tau ) }{a^{3/2}} +\frac{1}{a^{3/2}\kappa }\exp \left[ \kappa S[u(\tau )] \right] \sum \limits _{k=0}^\infty \frac{a_k}{\kappa ^k}, \end{aligned}$$

(A.5)

where

$$\begin{aligned} a_k= & {} \frac{(-1)^{k+1}}{k!}\varGamma (k+1)\left[ h(\xi )\frac{\mathrm{d}}{\mathrm{d}\xi } \right] ^k\left[ {{\tilde{f}}}(\xi ,\tau ) h(\xi ) \right] _{\xi =\tau },\\ h(\xi )= & {} \frac{1}{S^\prime [u(\xi )]}. \end{aligned}$$

Here, $S[u(\xi )]$ is given by expression (A.4), and $\varGamma (k+1)$ is the Euler Gamma function.

Taking into account that all coefficients $a_k$ are proportional to $\varphi (\tau )$ from (A.1), and $\varphi (0)=0$, we conclude that the second term in (A.5) vanishes. Therefore, $A(\tau )$ can be estimated as

$$\begin{aligned} A(\tau ) = \frac{R\left[ u(0)\right] \varphi (\tau ) }{a^{3/2}} . \end{aligned}$$

(A.6)

An important point is that the function $A(\tau )$ from (A.6) is independent on $u(\tau )$. This enables us to use the temperature dependence (26) in the case $x<\tau$.