ID 122

Lifetime of a Tensioned Liquid Following a Depressurizing Impulse

 

Abstract:

We consider a unit volume of pure liquid at temperature $T_s$, density $\rho_0$ and pressure $P_0$ which could be negative (tensile). Spontaneous nucleation takes place and the tensile pressure is relieved due to vapor generation at a constant temperature.

The objective of this analysis is to calculate the pressure recovery transient. The time to reach saturation is defined as the life-time of a liquid under tensile.

For a constant volume apparatus, as of the Berthelot tube \cite{ref:Chapman1975}, the average density, $\rho$, is constant, yet the liquid density, $\rho_L$, as well as its isothermal compressibility, $\beta_T$, are pressure dependent; thereby are time varying in the current analysis.

Following Skripov \cite{ref:Skripov1974} and Elias and Shusser~\cite{ref:Elias2005}, we derive the void fraction rate: \begin{equation} \frac{d\alpha}{dt} = 4\pi \int_{0}^{t}J(t')\left[r(t,t')\right]^2 \frac{dr(t,t')}{dt}dt' \end{equation}

Where $\alpha$ is the void fraction, $r(t,t')$ is the radius of the vapor bubble at any time $t$, from its time of "birth" ($t'$) to the time of integration. $J$ is the nucleation flux, represented by an exponential term \cite{ref:Carey1992}. \begin{flalign} J(t) = I_0 \exp\left(-\frac{B}{\Delta P(t)^2}\right) \end{flalign} \begin{flalign} I_0 =1.44\cdot 10^{40} \sqrt{\frac{\rho_L^2 \sigma}{M^3}} ; \hspace{10mm} B=\frac{1.213\cdot 10^{24} \sigma^3}{T}\nonumber \end{flalign}

Where $\sigma$ is the surface tension and $M$ is the molecular weight.

Finally, the fourth order differential equation is expanded and reorganized as a set of four first order equations. These are being solved numerically to yield the pressure recovery history and the transient void fraction during the relaxation process.

Results indicate that the recovery time or lifetime depends strongly on the degree of superheat. A negative pressure of the order of a thousand bar is relaxed in microseconds, while negative pressure of the order of a hundred bars is relaxed in milliseconds.