## A GENERAL THEORY OF THE HYDRAULIC

TRANSPORT OF SOLIDS IN
FULL SUSPENSION

By A C Bonapace

**VIII. CONCLUSIONS**

In Ch. I one has outlined the scope and the results of the investigation.

In Ch. II one has introduced non-dimensional quantities expressing a
velocity and a length. This has made the ensuing analytical development
independent of the kinematic viscosity and of the acceleration due to
gravity. Hydraulic laws considered have been referred to pipe flow in
presence of a smooth boundary or of a boundary of non-uniform roughness.

The hydraulic gradient of a stream as per Eq. (2.6), the Shields
function Sh, as per Eq. (2.8) (referred to a particle of size ) and the
Shields function SH as per (Eq. 2.11), (referred to a pipe diameter Δ°)
have been formally introduced and associated with the numerical values of
Eq. (2.10).

In Ch. III critical values defining particle suspension have been
expressed by means of relationships (3.1), (3.2) and more explicitly by
(3.3) and (3.4).

In all these relationships with μ ≠ 0 the role of the particle size resulted "irrelevant". For μ → μ_{10} → 0, i.e. with μ_{10} a small quantity,
one has defined an "indifferent" particle and a "generating" plane. In this
plane the force due to gravity plays a marginal role, so only the force of
inertia of a particle remains conspicuous.

In Ch. IV one has introduced the concept of a minimum energy dissipative function to be (ultimately) associated to the hydraulic gradient
.

(a) In the generating plane this has been expressed simply by Eq. (4.1).

(b) Outside the generating plane for μ ≠ 0 and x → 0 i.e. for a very small particle (cf. Eq. (3.8) for = ) the associated equation is
(4.5). Hence the condition x → 0 introduced above has limited considerably
the variational field (3.8) to

(c) Outside the generating plane for μ → μ_{10} → 0 and x ≠ 0, relations (4.6) and (4.7) yield the second equation (4.8) insuring particle
equilibrium against forces of inertia.

In Ch. V at (b) conditions of equilibrium under gravitational forces as per Eq. (3.4) have been combined with the conditions of equilibrium for
inertial forces (4.3), (4.4) and (4.5) for μ ≠ 0 but x → 0, i.e. for a particle
of a small diameter

In (c) one has departed from condition of equilibrium in the generating plane as per Eq. (4.1) and further discussed condition of equilibrium
(4.8).

The analysis has required an insight definition of the volumetric concentration x. This was referred at first to a cylindrical body, alternatively
to a spherical particle. The concentration x for a cylindrical body
has been expressed by the square ratio of the diameters as per Eq. (5.5).
A relative concentration has been defined by means of Eq. (5.7) and
finally a linear concentration x expressed by the linear ratio of the
diameters.

Analytical development led to an expression of the stream velocity Ω_{lx} as ratio as per Eq. (5.20). Hence an expression of the corresponding hydraulic gradient as ratio as per Eq (5.21)

Where dealing with a spherical particle the linear concentration of the particle has been obtained from that at the same diameter cylinder
inscribing the particle, i.e. by multiplication by the factor β = 1,11, as
given by Eq. (5.23).

In Ch. VI worked examples have been carried out by applying the literary expressions established in the analysis.

In Ch. VII experimental results have been discussed in a μ ≠ 0 and x > 0 field, with velocities and hydraulic gradients defined as below:

(8.1)

(8.2)

(8.3)

A remark: Fig 1 should be always referred to a very small solid concentration, e.g. to one particle only present in the system. This applies
also to the case , i.e. to spherical particle of the size of the pipe, for which the local concentration approaches the value 0,666.

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