Question 10.4: In an experimental wetted wall column, pure carbon dioxide i......

For the flow of a vertical film of fluid, the mean velocity of flow is governed by equation 3.87 in which sinΦ is put equal to unity for a vertical surface:

u=\frac{\rho g\sin\phi s^{2}}{3\mu}        (equation 3.87)

where s is the thickness of the film.
The flowrate per unit perimeter (ρgs³/3μ) = 3 x 10^{-4} m²/s

and: s=\left({\frac{3\times10^{-4}\times10^{-3}\times3}{1000\times9.81}}\right)^{1/3}

= 4.51 × 10^{-4} m

The velocity u_{x} at a distance y’ from the vertical column wall is given by equation 3.85 (using y’ in place of y) as:

u_{x}={\frac{\rho g\sin\theta}{\mu}}(s y-{\textstyle{\frac{1}{2}}}y^{2})              (3.85)

The free surface velocity u_{s} is given by substituting s for y’ or:

Thus: {\frac{u_{x}}{u_{s}}}=2\left({\frac{y^{\prime}}{s}}\right)-\left({\frac{y^{\prime}}{s}}\right)^{2}=1-\left(1-{\frac{y^{\prime}}{s}}\right)^{2}

When u_{x}/u_{s} = 0.95, that is velocity is 95 per cent of surface velocity, then:

and the distance below the free surface is y = s – y’ = 1.010 x 10^{-4} m
The relationship between concentration C_{A}, time and depth is:

{\frac{C_{A}-C_{A o}}{C_{A i}-C_{A o}}}=\operatorname{erfc}\,\,\left({\frac{y}{2{\sqrt{D t}}}}\right)                  (equation 10.108)

The time at which concentration reaches 0.01 of saturation value at a depth of 1.010 x 10^{-4} m is given by:

Thus: \operatorname{erf}\ \left({\frac{1.305}{\sqrt{t}}}\right)=0.99

Using Tables of error functions (Appendix A3, Table 12):

and: t = 0.51s
The surface velocity is then u_{s}={\frac{\rho g s^{2}}{2\mu}}

= \frac{1000\times9.81\times(4.51\times10^{-4})^{2}}{2\times10^{-3}}

= 1 m/s

and the maximum length of column is: = (1 x 0.51) = \underline{\underline{0.51\ m}}