The derivation of thermal relaxation time between two-phase bubbly flow

Abstract

Thermal relaxation time constant is derived analytically for the relaxed model with unequal phase-temperatures of a vapour bubble at saturation temperature and a non-steady temperature field around the growing vapour bubble. The energy and state equation are solved between two finite boundary conditions. Thermal relaxation time perform a good agreement with Mohammadein (in Doctoral thesis, PAN, Gdansk, 1994) and Moby Dick experiment in terms of non-equilibrium homogeneous model (Bilicki et al. in Proc R Soc Lond A428:379–397, 1990) for lower values of initial void fraction. Thermal relaxation is affected by Jacob number, superheating, initial bubble radius and thermal diffusivity.

Appendix

Evaluation of $\bar{T}_{{\text{l}}} (\theta _{{\text{T}}} )$

The average temperature $\bar{T}_{{\text{l}}} (t_{{\text{i}}} )$ is estimated from temperature distribution around the growing of vapour bubble between two boundaries R(t _i) and R _m respectively as follows

$$\bar{T}_{{\text{l}}} (t_{{\text{i}}} ) = \frac{1} {{V_{{\text{l}}} }}{\int\limits_{R(t_{{\text{i}}} )}^{R_{{\text{m}}} } {{\left( {4\pi r^{2} } \right)}T_{{\text{l}}} (r,t_{{\text{i}}} ){\text{d}}r} },\quad t_{{\text{i}}} \;{\text{is any}}\;{\text{time}}{\text{.}}$$

(24)

where

$$\bar{T}_{{\text{l}}} (t_{{\text{i}}} ).$$

(25)

Integrating by parts, Eq. 24 becomes

$$\begin{aligned} \bar{T}_{{\text{l}}} (t_{{\text{i}}} ) = & \frac{1} {{R^{3}_{{\text{m}}} - R^{3} (t_{{\text{i}}} )}} \\ & \quad \times {\left[ {R^{3}_{{\text{m}}} T_{0} - R^{3} (t_{{\text{i}}} )T_{{\text{l}}} (R(t_{{\text{i}}} ),t_{{\text{i}}} ) - {\int\limits_{R(t_{{\text{i}}} )}^{R_{{\text{m}}} } {r^{3} \frac{{{\text{d}}T_{{\text{l}}} (r,t_{{\text{i}}} )}}{{{\text{d}}r}}{\text{d}}r} }} \right]}, \\ \end{aligned} $$

(26)

where

$$T_{{\text{l}}} (R_{{\text{m}}} ,t_{{\text{i}}} ) = T_{0} .$$

(27)

At, t _i =θ_T, $\frac{{{\text{d}}T_{{\text{l}}} (r,\theta _{{\text{T}}} )}} {{{\text{d}}r}} = \frac{{\partial T_{{\text{l}}} (r,\theta _{{\text{T}}} )}} {{\partial r}},$ then

$$ \begin{aligned} \bar{T}_{{\text{l}}} (\theta _{{\text{T}}} ) &= \frac{1} {{R^{3}_{{\text{m}}} - R^{3} (\theta _{{\text{T}}} )}} \\ \,&\quad \times {\left[ {R^{3}_{{\text{m}}} T_{0} - R^{3} (\theta _{{\text{T}}} )T_{{\text{l}}} (R(\theta _{{\text{T}}} ),\theta _{{\text{T}}} ) - {\int\limits_{R(\theta _{{\text{T}}} )}^{R_{{\text{m}}} } {r^{3} \frac{{\partial T_{{\text{l}}} (r,\theta _{{\text{T}}} )}} {{\partial r}}{\text{d}}r} }} \right]}. \\ \end{aligned} $$

(28)