Example 17.5-1: Gas Absorption from Rising Bubbles

Bird, Stewart, and Lightfoot. Transport Phenomena. John Wiley and Sons. 1960.

 

A Project by Amanda Watford and Judy Hsii

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Problem:  Gas bubbles of component A are rising in liquid B at their terminal velocity, Vt, through a "clean" quiescent liquid.  In this problem, we wish to estimate the rate at which gas A bubbles are absorbed by liquid B.

 

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For moderate-sized bubbles in a liquid without surface-active agent, the bubbles experience a toroidal circulation.  Fresh liquid is encountered at the top of the bubble and the liquid moves downward as the bubble rises. The liquid leaves as it reaches the bottom of the bubble. Near the interface, the liquid is most likely laminar flow and keeps its identity. (The liquid behaves similar to liquid at the surface of a falling film.)  Contact time is deemed to be short as the bubble is traveling at terminal velocity, so penetration of dissolved gas A is small.  Therefore, the assumptions utilized in the previous section, Chapter 17.5 Notes, are valid.

Once again, we shall use Maple V to help derive the solution to the problem.  First, all assumptions will be cleared by using restart.

> restart;

We start with the result obtained in the previous section, Equation 17.5-17 of the BSL text, to estimate the rate of gas absorption and the change in the bubble size.

> WA := 2*W*cA0*(vmax*L*DAB/Pi)^(1/2);

[Maple Math]

WA is the total moles of A transferred per unit time from the gas to the liquid film. W is the width of the film, cA0 is the solubility of gas A in liquid B, vmax is the terminal velocity, and L is the length of the film.

Mass transfer is directly proportional to the square root of the diffusivity and is inversely proportional to the square root of the exposure time. t = L/vmax.  We use this relation to define vmax.  

> vmax:=L/t;

[Maple Math]

After defining vmax, WA is now changed to include this definition.  

> WA;

[Maple Math]

We now designate WA as a positive value and ask Maple to simplify the expression.   

> WA:=simplify(WA,assume=positive);

[Maple Math]

We are interested in the average rate of mass transfer of material A.  Thus, dividing WA, the total moles of A transferred per unit time from gas A to the liquid, by the width and length, W and L, respectively, we are left with NAavg, the average rate of mass transfer of A. The equation shown below is comparable to the equation shown as Equation 17.5-20 of the BSL text.  

> NA[avg]:=(WA/W/L);

[Maple Math]

To verify that the equations are the same, we will subtract the equation the text gives with the one derived.  By inspection, the expression is equal to zero.  

> difference:=simplify(NA[avg]-(4*DAB/Pi/t)^(1/2)*cA0,assume=positive);

[Maple Math]

Maple sometimes has a hard time simplify things that inherently can be simplified. Thus, we will expand the expression above, and then simplify it to show that the difference is in fact zero.   

> expand(%);

[Maple Math]

> simplify(%);

[Maple Math]

The time required for liquid B to slide from the top of the bubble to the bottom of the bubble is the exposure time.  For the gas bubble, D is the diameter.  Thus, for moderate bubbles of 0.3-0.5 cm in diameter, the exposure time can be approximated as tD/vt, where vt is the terminal velocity.

> t:=D/vt;

[Maple Math]

After defining the exposure time, NAavg simplifies to the expression given below.   

> NA[avg]:=eval(simplify(NA[avg]));

[Maple Math]

We now desire to compare it to the equation derived in the text for the absorption rate through the bubble-liquid interface, Equation 17.5-21.  Once again, we will subtract the equation given in the text from the one derived above.  If the equations are the same, then the difference should be zero. 

> differ:=simplify(NA[avg]-(4*DAB*vt/Pi/D)^(1/2)*cA0,assume=positive);

[Maple Math]

Again, Maple has a difficult time simplifying an expression that is inherently simplifiable. We will attempt to expand the expression then simplifying it see if the difference is zero.   

> expand(%);

[Maple Math]

> simplify(%);

[Maple Math]

The difference between the equations derived using Maple and the one the book gives as the solution, Equation 17.5-21, is zero. Thus, the two equations are the same.  The rate through the bubble liquid interface is verified.