# Estimation of Transportation Velocity of Solid Particles in Pipe Row at Various Inclination Angles

The important factor in the pressure loss calculations will be c, the velocity of the conveyed material, and therefore it is important to know this term. It is clear that with the same gas velocity V, the velocity of the conveyed mate­rial will be different at various pipe inclination angles S. In this section we (B) V » V < 10 ‘

 (c) Wsq > V > ws Ws > wso //<30 (a) wso < v < ws Ws > !<„, FIGURE 14.9 Different flow patterns in vertical transport.2  Derive an equation for the velocity difference V — c at various angles 8 (see Fig. 14.11), when its value is known in horizontal and vertical pipes. We be­gin by considering the force balance of a solid particle in a pipeline as shown in Fig. 14.11.

The velocity difference V — c causes a drag force Fd, which can be de­scribed according to Eq. (14.19):

(14.75)

The other forces affecting the particle are the gravitational force and friction; hence the force balance is

 (14.76) MTt = — c)2 — MgsmS — ЯftngcosS,

Where

 4 M = р^тг
 S V У And is the friction coefficient.

In a steady state the acceleration term vanishes, and then it follows from

(14.76) that

,2

FifCosф = A(vC) — sin 8 Where the constant A is

Hf=A(v-ch)1, (14.79)

Where we have denoted

Ch = c(8 = 0), (14.80)

I. e., Ch is the velocity of material in a horizontal line.

Then, substituting S = 7t/2 into Eq. (14.77) we get

A{v — c,,)1 = 1, (14.81)

Where

Cv = C.(8 = 77/2) (14.82)

Is the velocity of material in a vertical line. Now, from Eqs. (14.79) and (14.81) we can solve for the constants A and ^-and then, substituting them into Eq. (14.77), we obtain

(v-cY = {v — ch) cos <5 + (v — cvYSin 8 ,

Or

VC = [(v — ch)2cos8 + (v — c,,)2sin 8]1 ~. (14.83)

Equation (14.83) gives us an estimate for the velocity C at various pipeline angles 8 when the corresponding values for vertical and horizontal lines are known. It is an estimate, of course, because its derivation was based on the force balance of a single particle and not on the balance of the mixture of par­ticles as a whole.

The velocity difference in the vertical line is usually greater than the fall­ing velocity of the particles in the tube,

V — c„^ws, (14.84)

And can be estimated by the method discussed in Section 14.3.1, where we also present how to determine the corresponding velocity difference in the horizontal line, V — ch.

Assuming that VCv = wso, we get from Eq. (14.81)

 W A = -4-, (14.8.5)

So

Substituting this into Eq. (14.79) we get

(14.86)

Which can be used for estimating Ch, provided that the friction coefficient Hf is known.

To give an example of Eq. (14.83) we have calculated the velocity differ­ence V — c for wood chips, when V — ch 8.0 m/s and V — cv = 20.0 m/s (see Section 14.3.3). The results are shown in Fig. 14.12. Angle ( 0 ) FIGURE 14.12 The velocity difference for wood chips at various pipeline angles based on Eq, (14.83).