# An Empirical Approach for Calculating The Pressure Drop in Pneumatic Transport

The pressure loss for pure gas flow in a pipe of length Dl is

 DP =

(14.87)

Where D is the diameter of the pipe. The pipe friction coefficient Af; can be calculated from

AG = 0.3164 Re"1/4 (14.88)

For Reynolds numbers 2320 < Re < 105, or from

L/JTG = 2.0 log10(Re7v;)-0.8, (14.89)

 VD

 Re =

 (14.90)

 Where V is the kinematic viscosity of the gas. For calculating the pressure loss in pneumatic transport, the following simply modified version of Eq. (14.87) is often presented in the literature:

Which is valid for fully turbulent flow. The Reynolds number is defined by

Equation (14.91) contains only the mass flow ratio Jx as a characteristic number of the mechanics of similitude of the mixture. All the other impor­tant factors, such as particle size, solid density, etc., are contained in the ad­ditional pressure-loss coefficient of the solid particles, s, which is determined separately for each material.

But how can we estimate the pressure-loss coefficient A5? Stegmaierl~ has summarized horizontal transport for several fine-granular solids by a correla­tion which contains some characteristics of the material. The same idea has been used by Weber,13 who has found a correlation of the pressure-loss coeffi­cient for vertical pneumatic conveyance based on data measured by Flatow.14 In order to express these models, we first introduce two dimensionless numbers

Fr = (14.921

And

Fr, = ^5Q, ! 14.931

Asg

Where g is the acceleration due to gravity (9.82 m/s2) and Ws0 is the free-fall­ing velocity of a solid particle of diameter Ds. The numbers Fr and Fr, are called Froude numbers related to the pipe and solids, respectively.

The mathematical model developed by Stegmaier for horizontal transport

Is

 1,- 0.25
 2.1 /x ’Fr Fr
 O. i

 (14.94)

Equation (14.94) is an average value for the most solids. This has a rather high standard deviation, which can be seen from Fig. 14.13.

If the relatively high standard deviation is not acceptable, each type of solid can be correlated separately. This is a standard approach in the literature but we shall not repeat it here.

It is known from experience with vertical pneumatic transport that the in­fluence of weight prevails at low velocities, but as the velocity increases fric­tion gains importance. Therefore, in the calculation of the pressure loss one must find not only the weight of the solids, which could be set up theoreti­cally, but also an empirical relationship for vertical transport from the mea­sured data. A correlation of the pressure-loss coefficient for vertical pneumatic conveyance according to data measured by Flatow14 has been developed by Weber,13 and the result is

1 _ V/c , Iv/c _ (14.95)

1200 Fr

The correlation of Eq. (14.95) with Flatow’s measurements is rather good, as shown in Fig. 14.14. Its standard deviation is about 15%.

 Symbol Solid Ds (mm) Ps Kg/m3 D (mm) Ref. ■ Totalit 40 2200 40 Bohnet * Sand 69 2650 8 Bohnet D Quarzpowder 15 2640 70 Miiller О Catalyst 70 1500 8 Bohnet О Sand 6У 2650 40 Bohnet ■ Catalyst 70 1500 40 Bohnet V Flyash 24 2360 40 Bohnet Д Totalit 40 2200 8 Bohnet О Alumina 45 2480 40 Lippert * Ferrous sulfate waste 112 4100 40 Lippert • Ferrous sulfate waste 46 4100 40 Lippert
 1F.+0 I

 IE+00

 IE-02

 IE-03

 D,(mm) D (mm) + Polystrole 1/2.7 50/100/200 0.5/27 Glass Spheres 1.21 50/100/200 0.5/19 Ф Steel spheres 1.13 50/100/200 0.5/12

5 10 50

■ FIGURE 14.14 Correlation of the pressure loss coefficient for vertical pneumatic conveyance based on Flatow’s data according to Weber.13