## A GENERAL THEORY OF THE HYDRAULIC

TRANSPORT OF SOLIDS IN
FULL SUSPENSION

By A C Bonapace

**V. COMBINED EQUILIBRIUM
EQUATIONS**

(b) Outside the generating plane one will consider, with μ ≠ 0 and x → 0, i.e. for a single particle of excess density μ, the system
of equations (3.3) and (4.5).

Further with relations (4.3), (4.4) and (3.12) one obtains:

(5.1)

From Eq. (5.1) one obtains:

f_{μo} = 0,1214_{μ} (1 + μ)^{1−3α} (5.2)

Division of by yields on account of Eqs. (4.3), (4.4) and
(3.12):

(5.3)

i.e.

(5.4)

(c) Outside the generating plane let one consider with μ → μ_{10} → 0,
a certain volumetric concentration x > 0.

Referring at first to a particle having the form of a cylindrical body of diameter , let one consider capable of approaching the diameter of
the pipe .

Let one express the volumetric concentration for such a particle by the ratio square as below:

(5.5)

Further one introduces the following assumption:

that the fluid stream and the particle proceed at the same nominal velocity of the mixture, i.e. no slippage between the solid and the liquid phases can
occur. This assumption can be verified on a system of fully suspended
particles by an exchange of momentum between particles and fluid and
vice versa, resulting in a perfect mixing of the two phases. From this one
infers that such a system allows the definition of the solid concentration as
per Eq. (5.5) both from the Lagrangian and from the Eulerian point of
view, as each other identical.

Hence from Eq. (5.5) one gets for the definition of the particle size in the generating plane:

(5.6)

(cf. Eq. 3.8)

From Eq (4.8) one gets:

(5.7)

Eq. (5.7) defines ε by its term at the right as the "relative" volumetric concentration.

By means of Eqs. (4.6), (4.7) and (4.8) let one write an expression analogous to Eq. (5.3). Thus taking into account condition (3.12) one
gets:

(5.8)

with

(5.9)

and

(5.10)

Hence addition of x particles to the system has produced a decrease
in the actual linear section available for transport as per Eq. (5.10).

Let one consider the case x → 1, i.e. that of a cylindrical particle approaching the dimension of the pipe.

Hence one can define the volumetric concentration x instead of expression (5.5), by a linear concentration as given below:

(5.11)

Let one write Eq. (5.9) with ε and x in place of ε and x as below:

(5.12)

being below

(5.13)

In Eq. (5.13) one has purposely introduced a parameter c of expression

(5.14)

Eq. (5.14) is to be warranted by the following analytical development.

With c in the role of a suitable undefined multiplier of x let one write instead of (5.7):

(5.15)

and from (5.12):

(5.16)

On account of the arbitrariness of c let one write

(5.17)

i.e.

1 – x = 1 – (cx)^{2}

i.e.

(5.14)(Repeated)

As expected.

With (5.13) into (5.12) one gets for the velocity Ω_{1ε} :

(5.19)

In order to simplify notation let one put from now onward

= Ω_{lx} and write:

For the stream velocity

(5.20)

For the hydraulic gradient I_{lx}

(5.21)

The graphical representation of versus x is shown in Fig. 2 by the locus I.

This locus has its origin at the point {0, 1} on the axis of the ordinates and its end on the axis of the abscissae at point {1, 0}.

At these two points the geometrical tangent to the locus coincides with the corresponding axis.

In Fig. 2 the locus II represents the velocity ratio

(5.20), the locus III represents the hydraulic gradient ratio s per Eq. (5.21).

The analysis so far developed has been referred to a cylindrical particle of diameter δ. Let one extend below the obtained results to a
spherical particle of the same diameter δ.

One gets from the volumetric ratio between the two particles

. Hence a spherical particle of the size of the pipe will occupy a volume 2/3 = 0,6666 of the cylindrical particle of the same diameter.

Referring this ratio to the concentration x^{1/2} one obtains for x → 1 and 0,816 =

(5.22)

In Fig 3 one has represented the ratio of a cylindrical particle and the ratio

of a spherical particle, this for any value of 0 < x < 1. The diagram for the spherical particle starts for = 1 at the point of abscissa x = 0,816.

From Fig 3 one can infer the following:

Given a certain volumetric concentration x on the axis of the
abscissae, referred to a cylindrical particle, the square root volumetric
concentration x^{1/2} of a spherical particle to reckon with, is 1,11 times
greater than the square root concentration x^{1/2} of a cylindrical particle.
Hence when dealing with spherical particles one has to consider volumetric
concentrations as below:

(5.23)

In Fig. 3 one has shown in Insert (a) four particles in a compact assembly inside a cylindrical body of diameter δ and length 4δ. In Insert
(b) the particles have lost their alignment and proceed inside the pipe
under the influence of the stream:

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