2004-corapcioglu-Rise velocity of an air bubble in porous media: Theoretical studies

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根据受力分析建立气泡运动方程

• buoyancy
  

  $$F_b = (\rho_f-\rho_g)g\frac43\pi R_b^3$$

• surface tension
  

  $$F_{st} = 2\pi R'\sigma\sin\theta$$
  where $R'$ is the equivalent radius of a pore throat through which a bubble can pass in a particular arrangment

• drag force

  • Ergun equation:Kozeny equation + Burke-Plummer equation
  • drag force = viscous energy losses + kinetic losses
    $$F_d=A[\frac{150\mu_bu_b}{d_p^2}\frac{(1-n)^2}{n^3}+\frac{1.75\rho_gu_b}{d_p}\frac{1-n}{n^3}]\frac43\pi R_b^3$$
    where $n$ is porosity

• additional inertial force

• steady state velocity
  

  $$u_b = \frac{\mu_b}{\rho_g}\frac{1-n}{d_p}\left[-42.86\pm\sqrt{1836.74-\frac{0.57}{A}\frac{\rho_gd_p^3n^3}{\mu_b^2(1-n)^3}(\frac32\frac{R'}{R_b^3}\sigma\sin\theta-(\rho_f-\rho_g)g})\right]$$
  where codfficient A is fitted by data.