14.2.1Concepts and Notations

The mass flow ratio /x, sometimes also called mixture ratio, is defined as


H = —’, (14.1)


Where Ths is the mass flow of solid particles (kg/s) and Mg is the mass flow of gas. In pneumatic conveying systems the gas is almost always air.

Let us then consider an arbitrarily selected point in a pipe in which gas and solid particles are flowing. The flow of the mixture of gas and solid parti cles need not be homogeneous, i. e., the concentration of particles may vary across the cross-section of the pipe. This means that the mixture ratio /x should generally be regarded as a function of position in the pipe, and there­fore the definition, Eq. (14.1), should be replaced by

Iu = (14.2)


Where M’l and M’g are the mass flow densities (kg/m2 s) of solids and gas, re spectively, at a certain point in the pipe.

The flows of gas and solid particles are assumed to be parallel. This means that the velocity of gas V and the velocity of solid particles C are both in the di­rection of the pipeline. On the other hand, because we do not assume a homo­geneous flow at this stage of the theory, the absolute values of V and C may vary across the cross-section of the pipe.

A differential volume element DV in the flow field contains a mass of gas Dmy and a mass of solids Dms. The corresponding volumes taken by gas and

Solids are denoted by DVg and DVs. The sum of these partial volumes is the to­

Tal volume of the mixture,

DV= dVg + dVs. (14.3;

The real density of the gas at the point of the volume element DV is

№ “ Tv, ,14’4′

And the corresponding apparent or partial density of the gas is

Dm„ , ,

Pg~~dv- (,4-5)

The real density PG is the mass of the gas divided by the volume occupied by the gas. The apparent density or partial density Pg is the mass of the gas di­vided by the total volume of the solid-gas mixture.

We define the corresponding densities for solid particles analogous to Eqs. (14.4) and (14.5),


,,s-— ,14.(1..


Dm. , , .

Ps = If ‘ !

Ps is the real density of the solid particles and Ps is the partial density of parti­cles in the mixture.

The volume fraction of the gas in the mixture, or void fraction or voidage, as it is also called, is defined as


From Eqs. (14.4) and (14.5) it follows immediately that

^ = (J>, (14.9;

Pg ‘

And from Eqs. (14.6), (14.7), and (14.3) it follows that

& = 1 -<f>. (14.10)


On the other hand, if Dl is a differential length in the direction of the flow, the differential volumes can be written as

DV = dA dl

DVg = DAg dl (14.11)

DVs = DAs dl

Where DAg is the cross-sectional differential area through which the gas flows and correspondingly DAs is the area for flowing solid particles. The sum of these two partial areas is

DA = DAg + dAS. (14.12)

On the basis of Eqs. (14.11), the volume fraction of gas can also be written in the form


4 = ^. (14.13)

DA ‘

This interpretation of <Ј we shall need in connection with linear momentum equations.

With the aid of the partial densities Pg and Ps, the mass flow densities can be expressed as follows:

Mg = pgv (14.14)

N% = psc, (14.15)

Where V is the velocity of the gas and C is the velocity of the conveyed material. The total mass flow density is the sum of these two fluxes,

Nipgv + Psc. (14.16)

Substituting Eqs. (14.14) and (14.15) into Eq. (14.2), we obtain

M = = (14.17)

V Pg v Pg

Where the last equality follows from Eqs. (14.9) and (14.10).

When the mixture ratio P, and the velocity ratio Vie are known, the void frac­tion <f> can be determined from Eq. (14.17). As an example of this we have cal­culated <p = <fi(p-, c/v) for the case pG = 1.2 kg/m3 and Ps = 1500 kg/ni" Since mostly C s V and DVg^ d V, it follows that 0 Ј C/v Ј land 0 Ј < 1 .

However, in some cases C can be greater than V, for instance in downward pneu­matic conveying or after a pipestepping. The results of the calculations are shown in Fig. 14.1.




C! v






FIGURE 14.1 The void fraction <j> — c/v)in the case where = 1.2 Kg/m5

And pj = 1500 kg/m3.



The curves in Fig. 14.1 are not drawn down to zero, as the void fraction in a mixture with solid particles can not be zero. Equation (14.17) makes sense only in connection with equations of motion and particle size.