Table 2 Empirical formulas for DPH in PCs.

Miyauchi and Oya¹⁵

VPSPC^*

\(\:{\upphi\:}=4.93\times\:{10}^{2}{\psi\:}^{0.84}{V}_{d}^{2/3}\:for\:{\uppsi\:}<0.0031\:{m}^{11/12}{s}^{-1}\)

\(\:{\upphi\:}=3.42\times\:{10}^{6}{\psi\:}^{0.24}{V}_{d}^{2/3}\:for\:{\uppsi\:}>0.0031\:{m}^{11/12}{s}^{-1}\)

\(\:\psi\:=\frac{Af}{{\left(\beta\:h\right)}^{1/3}}{\left(\frac{{\mu\:}_{d}^{2}}{\sigma\:{\Delta\:}\rho\:}\right)}^{1/4}\)

Melnyk²⁴

HPSPC^*

\(\:{\upphi\:}=0.87{\left(1+\frac{{V}_{c}}{{V}_{d}}\right)}^{0.27}{\left(\frac{{A}^{2}{\rho\:}_{c}g}{\sigma\:}\right)}^{-0.26}{\left(\frac{{V}_{d}{\rho\:}_{c}}{\sigma\:g}\right)}^{0.12}{\left(\frac{{f}^{4}\sigma\:}{{\rho\:}_{c}g}\right)}^{-0.09}\)

Venkatanarasaiah and Verma,²⁵

VPSPC

\(\:{\upphi\:}={K}_{1}exp\left({K}_{2}\left|Af-{\left(Af\right)}_{m}\right|\right){V}_{d}^{1.02}{V}_{c}^{0.02}{\varDelta\:\rho\:}^{-0.23}{\mu\:}_{d}^{0.52}{d}^{-0.3}{\alpha\:}^{-0.4}{h}^{-0.4}\)

\(\:{\left(Af\right)}_{m}=9.69\times\:{10}^{-3}{\left(\frac{\sigma\:{\varDelta\:\rho\:}^{1/4}\alpha\:}{{\mu\:}_{d}^{3/4}}\right)}^{0.33}\)

K₁=116.5 and K₂ = 39.35 for no mass transfer

K₁=84.6 and K₂ = 42.56 for c→d mass transfer

K₁=92.0 and K₂ = 42.56 for d→c mass transfer

Panahinia et al.,¹⁴

HPSPC

\(\:\phi\:=0.101{\left(1+\frac{{Q}_{c}}{{Q}_{d}}\right)}^{0.124}{\left(\frac{{Af}^{4}{\rho\:}_{c}}{\sigma\:g}\right)}^{-0.286}{\left(\frac{{\rho\:}_{c}}{\varDelta\:\rho\:}\right)}^{-0.783}{\left(\frac{{\mu\:}_{d}^{4}g}{{\rho\:}_{c}{\sigma\:}^{3}}\right)}^{-0.071}{\left(\frac{{Af}^{3}{Q}_{d}{\rho\:}_{d}^{2}}{{\sigma\:}^{2}}\right)}^{0.282}\)

Mohammadi et al.,⁹

LPSPC^*

\(\:For\:horizontal\:section:\:\)

\(\:\phi\:=1.629{\left(\frac{Af}{{V}_{d}}\right)}^{-0.308}{\left(\frac{\varDelta\:\rho\:}{{\rho\:}_{c}}\right)}^{1.181}{\left(\frac{{\mu\:}_{c}}{{\mu\:}_{d}}\right)}^{0.863}{\left(\frac{{\mu\:}_{d}{V}_{d}}{\sigma\:}\right)}^{0.184}{\begin{array}{c}\:\\\:\left(1+\frac{{V}_{d}}{{V}_{c}}\right)\end{array}}^{-0.208}\)

\(\:For\:vertical\:section:\:\)

\(\:\:\phi\:=1.12\times\:{10}^{9}exp\left(11.53\left|Af-{\left(Af\right)}_{m}\right|\right){\left(\frac{Af}{{V}_{d}}\right)}^{0.486}{\left(\frac{\varDelta\:\rho\:}{{\rho\:}_{c}}\right)}^{6.475}{\left(\frac{{\mu\:}_{c}}{{\mu\:}_{d}}\right)}^{3.921}{\left(\frac{{\mu\:}_{d}{V}_{d}}{\sigma\:}\right)}^{1.241}{\begin{array}{c}\:\\\:\left(1+\frac{{V}_{d}}{{V}_{c}}\right)\end{array}}^{-0.34}\)