Question 10.21: Experiments have been carried out on the mass transfer of ac...
Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with aconstant fractional rate of surface renewal, s, but an upper limit on surface age equal to the life of the jet, \tau , show that theage frequency distribution function, ϕ(t)
, for this case is given by:
ϕ(t) = s exp(-st/[1 – exp(-st)]) for 0 <t<\tau
ϕ(t) =0 for t >\tau .
Hence, show that the enhancement, E, for the increase in value of the liquid-phase mass transfer coefficient is:
E = [(πs\tau )^{1/2} erf(s\tau )^{1/2}]/{2[1 – exp(-s\tau ]}
where E is defined as the ratio of the mass transfer coefficient predicted by conditions described above to the mass transfer coefficient obtained from the penetration theory for a jet with an undisturbed surface. Assume that the interfacial concentration of acetone is practically constant.
For the penetration theory:
\frac{∂C_{A}}{∂t} =D\frac{∂^{2}C_{A}}{∂y^{2}} (equation 10.66)
As shown in Problem 10.19, this equation can be transformed and solved to give:
\overline{C} _{A}=Ae^{\sqrt{(p/D)y} }+Be^{-\sqrt{(p/D)y} }
The boundary conditions are:
When y = 0, C_{A} = C_{Ai}, B = C_{Ai}/p
and when y = ∞, C_{A} = 0 and A = 0
∴ \overline{C} _{A}=\frac{C_{Ai}}{p} e^{-\sqrt{(p/D)}y }
\frac{d\overline{C}_{A} }{dy} =-C_{Ai}\sqrt{\frac{1}{D} } \sqrt{\frac{1}{p} } e^{-\sqrt{(p/D)}y }
From Volume 1, Appendix, Table 12, No 84, the inverse:
\frac{dC_{A}}{dy} =-C_{Ai}\sqrt{\frac{1}{D} } \sqrt{\frac{1}{\pi t} } e^{-y^{2}/4Dt}
At the surface:(N_{A})_{t}=-D\left(\frac{dC_{A}}{dy} \right) _{y=0}=C_{Ai}\sqrt{\frac{D}{\pi t} } at time t
The average rate over a time \tau is:
\frac{1}{\tau } C_{Ai}\sqrt{\frac{D}{\pi } } \int_{0}^{\tau }\frac{dt}{\sqrt{t} } =2C_{Ai}\sqrt{\frac{D}{\pi \tau } }
In general, C_{A0} ≠ 0 and N_{A} = 2(C_{Ai}- C_{A0})\sqrt{D/\pi \tau } for mass transfer without surface renewal.
Random surface renewal is discussed in Section 10.5.2 where it is shown that the age distribution function is:
= constant e^{-st} = ke^{-st}
where s is the rate of production of fresh surface per unit total area of surface.
If the maximum age of the surface is \tau , then
K\int_{0}^{\tau }e^{-st}dt=1
-\frac{k}{s} \left[e^{-st}\right] _{0}^{\tau }=1
1-e^{-s\tau}=s/k and K=\frac{s}{1-e^{s\tau }}
∴ the age distribution function is: \left(\frac{s}{1-e^{-st}} \right) e^{-st}
The mass transfer in time \tau is:
\int_{0}^{\tau }\sqrt{\frac{D}{\pi t} } (C_{Ai}-C_{A0})\frac{s}{1-e^{-st}}e^{-st} dt=\sqrt{\frac{D}{\pi t} } (C_{Ai}-C_{A0 })\frac{s}{1-e^{-st}}\int_{0}^{\tau }\frac{e^{-st}}{\sqrt{t} } dt
The integral is conveniently solved by substituting st = β² and \sqrt{t} =β/\sqrt{s} or s dt = 2β dβ and dt = 2β dβ/s
then: \int_{0}^{\tau } \frac{\sqrt{s} }{\beta } e^{-\beta }\frac{2\beta d\beta }{s} =\frac{2}{\sqrt{s} } \int_{0}^{\tau }e^{-\beta ^{2}}d\beta =\sqrt{\frac{\pi }{s} } erf\sqrt{s\tau } =(C_{Ai}-C_{A0})\sqrt{Ds} \frac{erf\sqrt{s\tau } }{1-e^{-st}}
The enhancement factor E is given by:
E=\frac{(C_{Ai}-C_{A0})\sqrt{Ds} \frac{erf\sqrt{s\tau } }{1-e^{-s\tau }} }{2\sqrt{\frac{D}{\pi \tau } }(C_{Ai}-C_{A0}) } =\frac{\sqrt{\pi s\tau }erf\sqrt{st} }{2(1-e^{-s\tau })}