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**Design of packed tower**

The cross sectional view of the packed tower is shown in Figure 4.5.**Design of packed tower may be **

(I) on the basis of individual mass transfer coefficients or

(II) on the basis of overall mass transfer coefficient.**Figure 4.5: Cross sectional view of packed tower.**

The column is packed with packing materials (any type) to provide more contact between gas and liquid.

Let, G^{/ }and L^{/} are gas and liquid flow rate per unit area basis, mol/h.m^{2}. ā is specific interfacial contact area between gas and liquid, m^{2} /m^{3} . The mole fraction of solute in gas is y. Hence, solute flow rate in gas= G^{/}y mol/h.m^{2}

The decrease in solute flow rate over the thickness dh=d(G^{/}y) (4.1)

For a unit cross-sectional area (1m^{2} ), volume of differential section=1×dh m^{3} and interfacial area of contact in differential section= ā×1×dh m^{2}

If N_{A} is solute flux and k_{y} is individual gas-phase mass transfer coefficient, solute transfer through differential section= ā×dh×N_{A}.

Therefore,

-[G^{/}dy+ y dG^{/}]= ā×dh×N_{A} (4.2)

-G^{/}dy - y dG^{/} = ā×dh×N_{A} (4.3)

The change in total gas flow rate (dG^{/ }) is equal to rate of solute transfer (ā×dh×N_{A}) as carrier gas is not soluble, i.e.,

- dG^{/ }= ā×dh×N_{A} (4.4)

Putting the value of –dG^{/} in Equation 4.3, we have,

-G^{/}dy + ā×dh×N_{A} y= ā×dh×N_{A} (4.5)

-G^{/}dy = ā×dh×N_{A}(1-y)

(4.6)

Boundary conditions:

h=0; y=y_{1}

h=h_{T}; y=y_{2 }

Integration of Equation 4.6 gives the height of packed column as follows:

(4.7)

Interfacial solute concentration, yi is not known; hence the integration of the right hand side of Equation 4.7 is complicated.**STEP-BY-STEP PROCEDURE **

(1) For a particular gas-liquid system, draw equilibrium curve on X-Y plane.

(2) Draw operating line in X-Y plane (PQ) using material balance Equation.

Lower terminal Q (X_{2}, Y_{2}) and upper terminal P (X_{1}, Y_{1}) are placed in x-y plane. Overall mass balance Equation for the absorption tower is as follows:

(4.8)

If liquid mass flow rate, Ls is not known, minimum liquid mass flow rate (L_{s})_{min} is to be determined. L_{s} is generally 1.2 to 2 times the (L_{s})_{min }**Figure 4.6: Graphical determination of (Ls)min for absorption.**

In Figure 4.6, lower terminal of absorption tower is represented by Q (X_{2}, Y_{2}); i.e., bottom of the tower. Operating line is PQ. If liquid rate is decreased, slope of operating line (L_{s}/G_{s}) also decreases and operating line shifts from PQ to P^{/}Q, when touches equilibrium line. This operating line is tangent to equilibrium line.

The driving force for absorption is zero at P^{/} and is called “PINCH POINT”.

(3) A point A (x, y) is taken on the operating line. From the known value of k_{x} and k_{y} or k_{x}ā and k_{y}ā, a line is drawn with slope of k_{x}/k_{y} to equilibrium line, B(x_{i},y_{i}). Line AB is called “TIE LINE” and xi and yi are known for a set of values of x and y.

(4) Step (3) is repeated for other points in the operating line to get several (x_{i},y_{i}) sets for y_{1}≥y≥y_{2}.

(5) Calculate flow rate of gas G (kg/h) at each point as G=G_{s}(1+y).

(6) Calculate height of the packing h_{T} of Equation 4.7 graphically or numerically.

The height of the „stripping column‟ is also obtained in a similar way. For stripping, y_{2}>y_{1} and driving force is (y_{i}-y). The corresponding design Equation will be

(4.9)

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