# Modeling of Falling Velocity in a Tube

In the previous section we determined the equivalent particle diameter Dsm of a set of particles of different sizes, with the aid of which we can treat the mix­ture as composed of one size of particles, namely Dsm. The mean free-falling velocity of the mixture is the same.

So far we have considered the free-falling velocity in air which is at rest. But the falling phenomenon in a tube differs from this because the falling of solid particles itself causes an airflow upwards (see Fig. 14.7). The modeling idea of Fig. 14.7 is from Weber.2 The air volume replaced by the falling parti cle flows upward and this airflow rate is

VmAg = wsAs, (14.68)

Where Ws is the falling velocity of the particles in the tube, the quantity that we wish to determine, Ag is the cross-sectional area for airflow, and is the

 FIGURE 14.7 Falling of a solid particle in a tube.

Cross-sectional area of the solid particle. Equation (14.68) assumes that the flow is incompressible.

In the coordinate system moving upward with velocity Vm, the particle is falling as if the air around it were at rest. In other words, the free-falling veloc­ity is the sum of the two velocities

W\$Q = ws + vm, (14.69)

Solving for Vm from Eq. (14.68) we obtain

 4-1

 (14.70)

F

1J> =

Vm = T-M’s = —AWS =

AG Ag

Where we have used Eq. (14.13). Substituting Eq. (14.70) into Eq. (14.69), we get a very simple result for the falling velocity:

Ws = (j>w.So — (14.71)

Equation (14.30), whose analytical form was Eqs. (14.64) and (14.65), can be used for determining the free-falling velocity Wso in the case that the particles behave like separate particles but due to the great number of random colli­sions have one free-falling velocity Wso. Then Eq. (14.71) gives the correction to calculate the falling velocity in tube flow.

But, if the assumption of particles behaving separately is not adequate,

I. e., the particles form large clumps which fall at different velocity, then nei­ther Eq. (14.30) nor Eq. (14.71) is applicable. Then Eq. (14.71) has to be re­placed by an empirical correlation.

In the following we will give some empirical formulas for the correlation between Ws and m.’so, where Wso is to be understood as a proper mean free — falling velocity of the mixture, but is not necessarily given by the analysis based on Eq. (14.30).

Maude and Whitmore10 have presented a correlation

 FIGURE 14.8 Exponent Y in Eq. ( 14.72).1

 Re,,

Where Y is a function of Reynolds number Re^ defined by Eq. (14.20), i. e.,

7=7(Rerf). (14.73)

The function y(Rej) is graphically shown in Fig. 14.8.

However, the correlation between Ws and twso is essentially dependent on the flow pattern, and therefore the correlations, for example Eq. (14.72), are limited to distinctly specified cases. Figure 14.9 illustrates different types of vertical flow, each of which requires its own model for the correlation be­tween Ujs and Wso.

To conclude this section we express Brauer’s11 correlation

 A.
 W* = «’so /
 1.2
 L +
 1 + Ц^ Ф"
 1 , Tt/12 2 1 -ф
 (14.74)

If the volume fraction of air 4> is close to 1, the correlations of Figs. (14.71) and (14.74) are close to each other. The more <b deviates from 1, the greater is the difference in the ratio Ws/wso. The comparison of these two formulas is shown in Fig. 14.10.

Because of the great uncertainty between Ws and Wso, as Fig. 14.10 shows, we try to determine Ws directly on the basis of pressure loss measure­ments; this approach is presented in Section 14.3.3.
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# Modeling of Falling Velocity in a Tube

In the previous section we determined the equivalent particle diameter Dsm of a set of particles of different sizes, with the aid of which we can treat the mix­ture as composed of one size of particles, namely Dsm. The mean free-falling velocity of the mixture is the same.

So far we have considered the free-falling velocity in air which is at rest. But the falling phenomenon in a tube differs from this because the falling of solid particles itself causes an airflow upwards (see Fig. 14.7). The modeling idea of Fig. 14.7 is from Weber.2 The air volume replaced by the falling parti cle flows upward and this airflow rate is

VmAg = wsAs, (14.68)

Where Ws is the falling velocity of the particles in the tube, the quantity that we wish to determine, Ag is the cross-sectional area for airflow, and is the

 FIGURE 14.7 Falling of a solid particle in a tube.

Cross-sectional area of the solid particle. Equation (14.68) assumes that the flow is incompressible.

In the coordinate system moving upward with velocity Vm, the particle is falling as if the air around it were at rest. In other words, the free-falling veloc­ity is the sum of the two velocities

W\$Q = ws + vm, (14.69)

Solving for Vm from Eq. (14.68) we obtain

 4-1

 (14.70)

F

1J> =

Vm = T-M’s = —AWS =

AG Ag

Where we have used Eq. (14.13). Substituting Eq. (14.70) into Eq. (14.69), we get a very simple result for the falling velocity:

Ws = (j>w.So — (14.71)

Equation (14.30), whose analytical form was Eqs. (14.64) and (14.65), can be used for determining the free-falling velocity Wso in the case that the particles behave like separate particles but due to the great number of random colli­sions have one free-falling velocity Wso. Then Eq. (14.71) gives the correction to calculate the falling velocity in tube flow.

But, if the assumption of particles behaving separately is not adequate,

I. e., the particles form large clumps which fall at different velocity, then nei­ther Eq. (14.30) nor Eq. (14.71) is applicable. Then Eq. (14.71) has to be re­placed by an empirical correlation.

In the following we will give some empirical formulas for the correlation between Ws and m.’so, where Wso is to be understood as a proper mean free — falling velocity of the mixture, but is not necessarily given by the analysis based on Eq. (14.30).

Maude and Whitmore10 have presented a correlation

 FIGURE 14.8 Exponent Y in Eq. ( 14.72).1

 Re,,

Where Y is a function of Reynolds number Re^ defined by Eq. (14.20), i. e.,

7=7(Rerf). (14.73)

The function y(Rej) is graphically shown in Fig. 14.8.

However, the correlation between Ws and twso is essentially dependent on the flow pattern, and therefore the correlations, for example Eq. (14.72), are limited to distinctly specified cases. Figure 14.9 illustrates different types of vertical flow, each of which requires its own model for the correlation be­tween Ujs and Wso.

To conclude this section we express Brauer’s11 correlation

 A.
 W* = «’so /
 1.2
 L +
 1 + Ц^ Ф"
 1 , Tt/12 2 1 -ф
 (14.74)

If the volume fraction of air 4> is close to 1, the correlations of Figs. (14.71) and (14.74) are close to each other. The more <b deviates from 1, the greater is the difference in the ratio Ws/wso. The comparison of these two formulas is shown in Fig. 14.10.

Because of the great uncertainty between Ws and Wso, as Fig. 14.10 shows, we try to determine Ws directly on the basis of pressure loss measure­ments; this approach is presented in Section 14.3.3.
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The passage discusses the modeling of the falling velocity of particles in a tube, including the derivation of equations and empirical correlations for calculating the falling velocity. It does not contain any specific mathematical questions or answers to such questions.

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