4.1a. Void fraction near the walls of packed beds.
Consider a cylindrical vessel with a diameter of 305 mm packed with solid spheres with a diameter of 50 mm.
a) From equation (4-1), calculate the asymptotic porosity of the bed.
Solution
Answer
b) Estimate the void fraction at a distance of 100 mm from the wall.
Answer
4.2b. Void fraction near the walls of packed beds.
Because of the oscillatory nature of the void-fraction radial variation of packed beds, there are a number of locations close to the wall where the local void fraction is exactly equal to the asymptotic value for the bed. For the bed described in Example 4.1, calculate the distance from the wall to the first five such locations.
Solution
The local porosity near the wall equals the asymptotic porosity when J0(ard) = 0. From
Example 4.1,
Initial estimates of the roots can be obtained from Fig. 4.4
Initial estimate
Initial estimate
Initial estimate
Initial estimate
Initial estimate
4.3c. Void fraction near the walls of packed beds.
(a) Show that the radial location of the maxima and minima of the function described by Equation (4-1) are the roots of the equation
Solution
b) For the packed bed of Example 4.1, calculate the radial location of the first five maxima, and of the first five minima; calculate the amplitude of the void fraction oscilations at those points.

Solution
Initial estimates of the roots can be obtained from Fig. 4.4
Initial estimate
Initial estimate
Repeating this procedure, the following results are obtained:
Maxima: Minima
r, mm (r*) Amplitude, % r,mm (r*) Amplitude, %
20.8 (1.04) 37.2 11.3 (0.57) -56.7
39.6 (1.98) 21.1 30.2 (1.51) -27.3
58.3 (2.92) 13.5 49.0 (2.45) -16.7
77.1 (3.85) 9.2 67.7 (3.39) -11.1
95.8 (4.79) 6.4 86.4 (4.32) -7.6
c) Calculate the distance from the wall at which the absolute value of the porosity fluctuations has been dampened to less than 10% of the asymptotic bed porosity.

Solution
From the results of part (b), this must happen at r* between 3.39 and 3.85
(d) What fraction of the cross-sectional area of the packed bed is characterized by porosity fluctuations which are within 10% of the asymptotic bed porosity?

Solution
From part (c)
(f) For the packed bed of Example 4.1, estimate the average void fraction by numerical integration of equation (4-65) and estimate the ratio eav/eb.
Solution
4.4c. Void fraction near the walls of annular packed beds.
Annular packed beds (APBs) involving the flow of fluids are used in many technical and engineering applications, such as in chemical reactors, heat exchangers, and fusion reactor blankets. It is well known that the wall in a packed bed affects the radial void fraction distribution. Since APBs have two walls that can simultaneously affect the radial void fraction distribution, it is essential to include this variation in transport models. A correlation for this purpose was recently formulated (Mueller, G. E., AIChE J., 45, 2458-60, Nov. 1999). The correlation is restricted to randomly packed beds in annular cylindrical containers of outside diameter Do, inside diameter Di, equivalent diameter De = Do Ð Di, consisting of equal-sized spheres of diameter dp, with diameter aspect ratios of 4 ² De/dp ² 20. The correlation is
Consider an APB with outside diameter of 140 mm, inside diameter of 40 mm, packed with identical 10-mm diameter spheres.
(a) Estimate the void fraction at a distance from the outer wall of 25 mm.

Solution
(b) Plot the void fraction, as predicted by Eq. (4-66), for r* from 0 to R*.
(c) Show that the average porosity for an APB is given by
(d) Estimate the average porosity for the APB described above.

Solution
4.5a. Minimum liquid mass velocity for proper wetting of packing.
A 1.0-m diameter bed used for absorption of ammonia with pure water at 298 K is packed with 25-mm plastic Intalox saddles. Calculate the minimum water flow rate, in kg/s, neded to ensure proper wetting of the packing surface.

Solution
For plastic packing, vL,min = 1.2 mm/s.
From Steam Tables
4.6a. Minimum liquid mass velocity for proper wetting of packing.
Repeat Problem 4.5, but using ceramic instead of plastic Intalox saddles.

Solution
For ceramic packing, vL,min = 0.15 mm/s.
From Steam Tables
4.7b. Specific liquid holdup and void fraction in first-generation random packing.
Repeat Example 4.2, but using 25-mm ceramic Berl saddles as packing material.

Solution
From Table 4.1:
From Example 4.2:
4.8b. Specific liquid holdup and void fraction in structured packing.
Repeat Example 4.2, but using Montz metal B1-200 structured packing (very similar to the one shown in Figure 4.3). For this packing, a = 200 mÐ1, e = 0.979, Ch = 0.547 (Seader and Henley, 1998).

Solution
From Example 4.2:
4.9b. Specific liquid holdup and void fraction in first-generation random packing.
A tower packed with 25-mm ceramic Raschig rings is to be used for absorbing benzene vapor from a dilute mixture with an inert gas using a wash oil at 300 K. The viscosity of the oil is 2.0 cP and its density is 840 kg/m3. The liquid mass velocity is L' = 2.71 kg/m2-s. Estimate the liquid holdup, the void fraction, and the hydraulic specific area of the packing.

Solution
From Table 4.1:
4.10b. Pressure drop in beds packed with first-generation random packings.
Repeat Example 4.3, but using 15-mm ceramic Raschig rings as packing material. Assume that, for this packing, Cp = 1.783.


Solution
Packed Column Design Program
This program calculates the diameter of a packed
column to satisfy a given pressure drop criterium,
and estimates the volumetric mass-transfer coefficients.
Enter data related to the gas and liquid streams
Enter liquid flow rate, mL, in kg/s
Enter gas flow rate, mG, in kg/s
Enter liquid density, in kg/m3
Enter gas density, kg/m3
Enter liquid viscosity, Pa-s
Enter gas viscosity, Pa-s
Enter temperature, T, in K
Enter total pressure, P, in Pa
Enter data related to the packing
Enter packing factor, Fp, in ft2/ft3
Enter specific area, a, m2/m3
Introduce a units conversion factor in Fp
Enter porosity, fraction
Enter loading constant, Ch
Enter pressure drop constant, Cp
Enter allowed pressure drop, in Pa/m
Calculate flow parameter, X
Calculate Y at flooding conditions
Calculate gas velocity at flooding, vGf
As a first estimate of the column diameter, D, design for 70% of flooding
Calculate gas volume flow rate, QG, in m3/s
Calculate liquid volume flow rate, QL, in m3/s
Calculate effective particle size, dp, in m
Iterate to find the tower diameter for the given pressure drop
Column diameter, in meters
Fractional approach to flooding