Formulations about Boiling in Porous Media

文献阅读整理

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confinement number

这个无量纲数 confinement number 看上去是描述受限行为的。

$$\begin{equation} Co=\frac{(\sigma/g(\rho_l - \rho_g))^{0.5}}{D_h} \end{equation}$$
当Co数大于0.5，应考虑通道的约束作用。
(来源 FLOW PATTERN, HEAT TRANSFER AND PRESSURE DROP BEHAVIORS OF MICRO-CHANNEL FLOW BOILING)

气泡动力学

合并气泡的等效直径

使用气泡照片的面积($A_b$)和周长($P_b$)，可以计算一个等效直径。公式为：

$$\begin{equation} d_e=1.55\cdot \frac{A_b^{0.625}}{P_b^{0.25}} \end{equation}$$
(来源 Enhancing thermal stability and uniformity in boiling heat transfer using micro-nano hybrid surfaces (MNHS))
(原始文献 Equivalent diameters of rectangular and oval ducts)

有效成核半径（effective nucleation site）

$$\begin{equation} r_{eff,min} = \frac{\delta_tD_2}{2D_1}\cdot\frac{\Delta T_w}{\Delta T_w+\Delta T_{sub}}\cdot[-\sqrt{1- \frac{8D_1 \gamma T_{sat}(P_l)(\Delta T_w+\Delta T_{sub})}{\rho_v l \delta_t^2(\Delta T_w)}}] \end{equation}$$
$$\begin{equation} r_{eff,max} = \frac{\delta_tD_2}{2D_1}\cdot\frac{\Delta T_w}{\Delta T_w+\Delta T_{sub}}\cdot[+\sqrt{1- \frac{8D_1 \gamma T_{sat}(P_l)(\Delta T_w+\Delta T_{sub})}{\rho_v l \delta_t^2(\Delta T_w)}}] \end{equation}$$

符号	含义
$\Delta T_w$	superheat
$\Delta T_{sub}$	degree of subcooling
$\delta_t$	thermal boundary layer thickness
$T_{sat}$	liquid saturation temperature
$\rho_v$	liquid density
$l$	latent heat

(来源 Enhancing thermal stability and uniformity in boiling heat transfer using micro-nano hybrid surfaces (MNHS))
(原始文献 Onset of nucleate boiling and active nucleation site density during subcooled flow boiling，怀疑这也是引用，但其中应有原始文献出处)

汽泡脱离频率

我也是佛了，这玩意也有文献，可以的。

$$\begin{equation} f_d=\frac1{t_w+t_g} \end{equation}$$
显然，等于等待时间和生长时间之和的倒数。
(来源 Realizing highly coordinated, rapid and sustainable nucleate boiling in microchannels on HFE-7100)
(原始文献 Characteristics of bubble departure frequency in a low-pressure subcooled boiling flow)

汽泡脱离直径与频率之积为常数

$$\begin{equation} fD_d=Const. \end{equation}$$
when heat flux changes
这其实相当诡异,因为HF增加时，单位时间产生的蒸汽按D的3次方增加，fD=C，则HF应和D的2次方成正比。即$D_d\propto \sqrt{HF}$。但实验来看不见的如此。详见来源文献Fig.11以及以Fig.11开头的一段论述。
(来源 Realizing highly coordinated, rapid and sustainable nucleate boiling in microchannels on HFE-7100)
(原始文献 Bubble dynamics during boiling in enhanced surface microchannels)

CHF相关理论和公式

CHF的触发，是同时受到流动不稳定性和表面性质（wettability，roughness，porosity）影响的复杂过程[1]。对CHF的预测有如下常见公式：

Kandlikar CHF correlation

饱和沸腾：

$$\begin{equation} q_{C,SAT}''=h_{fg} \sqrt{\rho_g}(\frac{1+cos\beta_r}{16})[\frac{\sigma(1+cos\beta_r)}{H_b}+(\rho_l - \rho_g)g\frac{H_b}{2}cos\phi]^{0.5} \end{equation}$$
过冷沸腾：
$$\begin{equation} q_{C，SUB}''=q_{C，SAT}''(1+\frac{\Delta T_{SUB}}{\Delta T_{SAT}}) \end{equation}$$
$\beta_r$是后退角，$\phi$是加热面取向角
这一公式只考虑了wettability的影响

(来源 Theoretical Model to Predict Pool Boiling CHF Incorporating Effects of Contact Angle and Orientation)

Ramilison CHF correlation

$$\begin{equation} q_{CHF}''=0.0366(\pi-\beta_r)^3R_q^{0.125}\frac\pi{24}\rho^{0.5}h_{fg}(\sigma g(\rho_f-\rho_g))^{0.25} \end{equation}$$
$R_q$: the root mean square (rms),$\beta_r$: 后退角
这一公式同时考虑了wettalibity和roughness

(来源 Separate effects of surface roughness, wettability, and porosity on the boiling critical heat flux)
(原始文献 Surface Factors Influencing Burnout on Flat Heaters)

Polezhaev & Kovalev CHF correlation

$$\begin{equation} q_{CHF}''=0.5\epsilon^{2.28}h_{fg}(\frac{\sigma\rho_f\rho_g}{(\rho_f+\rho_g)R_{bg}})^{0.5} \end{equation}$$
$R_{bg}$: the experimentally determined breakthrough ra-dius, which can be considered approximately half the parti-cle diameter for a porous layer composed of densely packedparticles.
$\epsilon$ 是啥也不晓得
这一公式同时考虑了porosity 和pore size

(来源 Separate effects of surface roughness, wettability, and porosity on the boiling critical heat flux)
(原始文献 Y. Polezhaev and S. Kovalev, Therm. Eng.37(12), 617–620 (1990)) 我估计这个公式我一般用不上，暂时先不查它的篇名了

考虑semi-wicking的CHFcorrelation

$$\begin{equation} q_{wick-CHF}''=Bl_{lg}\rho_l(1-\phi)W^2/\lambda^2+q_{kandlikar-CHF}'' \end{equation}$$
$B$:经验系数，$l$:latent heat,$\phi$:porosity,$W$:下一条中$t^{0.5}$的系数，$\lambda$:rayleigh-Taylor interficial wavelength，$q_{kandlikar-CHF}''$:上一条中第一个公式
(来源 Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer)

微柱阵列中的液体渗吸（或者说，semi-wicking）

$$\begin{equation} x=(\frac{2}{3\beta}\cdot\frac{cos\theta^*-cos\theta_c}{cos\theta_c}\cdot\frac{\gamma h}{\mu}\cdot t)^{0.5} \end{equation}$$
(来源 Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer)
(原始文献 Rough wetting（bico的文章，包含这个公式的推导）)

HTC构成与计算

沸腾HTC的构成（分配）

$$\begin{equation} q_{total}''=q_{advection}''+q_{nucleation}''+q_{evaporation}'' \end{equation}$$
emmm这个公式不是重点，重点是还有个2006年的综述专门研究这个分配的（原始文献），想起yangronggui的那篇文章
(来源 Realizing highly coordinated, rapid and sustainable nucleate boiling in microchannels on HFE-7100)
(原始文献 Heat transfer and wall heat flux partitioning during subcooled flow nucleate boiling—a review)

Thome的三区流动沸腾模型

$$\begin{equation} h_{film} = \frac1{t_{film}}\int_0^{t_{film}}\frac{k_l}{\delta(t)}dt=\frac{k_l}{\delta_0-delta_{min}}ln(\frac{\delta_0}{\delta_{min}}) \end{equation}$$
$k_l$:liquid thermal conductivity ; $\delta_0$:initial thickness ; $\delta_{min}$: minimum possible film thickness before the liquid film ruptures（破裂）
本公式的使用关键在于液膜的厚度。为此需要另外几个公式。详见本公式来源。
(来源 Local, transient heat transfer measurements for flow boiling in a microchannel with a pin fin)
(原始文献 Heat transfer model for evaporation in microchannels. Part I: presentation of the model)
(原始文献 Heat transfer model for evaporation in microchannels. Part II: comparison with the database)

表面粗糙度的表征

$R_a$: the arithmetic average of the absolute value of surface feature heights
$R_q$: the root mean square (rms) of the surface feature heights
$R_z$: the average distance of the five highest peaks to lowest five valleys

(来源 Separate effects of surface roughness, wettability, and porosity on the boiling critical heat flux)