## A GENERAL THEORY OF THE HYDRAULIC

TRANSPORT OF SOLIDS IN
FULL SUSPENSION

By A C Bonapace

**II. HYDRAULIC PREMISES**

One defines below certain well-known expressions related to the flow of streams, using at first dimensional symbols and thereafter nondimensional
ones.
For flow in pipes one denotes a pipe diameter by D, a stream
velocity by V, a friction factor by f, the kinematic viscosity by ν, the size

of an excrescence on a wall by k, the diameter of a particle by d and the
acceleration due to gravity by g.
Denoting by v∗ the friction velocity of the stream at the boundary,
this is expressed by the well-known relation:

(2.1)

Non-dimensional symbols will now be introduced for physical parameters having dimensions of a velocity and of a length respectively.
Considering e.g. a stream velocity V one transforms it into a non
dimensional velocity Ω by means of division by the group (gv)^{1/3} (dimensionally
equivalent to$$
i.e.

(2.2)

Analogously for the pipe of diameter D one transforms it to a nondimensional diameter Δ° by multiplication by the group (dimensionally equivalent to ) i.e.

(2.3)

In a short notation defining the above transformations by an arrow one represents the transformed non-dimensional quantity by means of a letter
of the Greek alphabet. Hence one will write:

For velocities:

(2.4)

v → ω

For linear dimensions

d → δ (2.5)

k →

Considering the hydraulic gradient of a stream, this can be expressed for a pipe by the well-known Darcy-Weisbach equation. Writing
the expression at the left in dimensional symbols and at the right in nondimensional
ones, one gets:

(2.6)

In the present paper one will deal with the following stream boundaries:

a) A smooth boundary, investigated by Prandtl [9].

b) A boundary of non-uniform roughness investigated by Colebrook
and White [1].

Boundaries of uniform roughness are often associated in open channel hydraulics with particle beds of constant size. Experiments by Shields
[11] did define for these deformable boundaries conditions of incipient
motion of a particle by a critical drag force.

Defining by the ratio between the solid and the fluid densities and expressing the above ratio in function of an excess density μ, let one write:

(2.7)

has been represented versus the Reynolds number Re* of the particle at the boundary, expressed by:

(2.8)^{#}

has been represented versus the Reynolds number Re* of the particle at the boundary, expressed by:

(2.9)

A well-known functional correspondence in the form of a narrow band containing all the experimental points could be so defined by Shields.
He found that for Re* = 1000 experimentation could not be carried on
properly as the particle bed became unstable, i.e. particles show a tendency
to leave the boundary for the stream. At Re* = 1000 Shields' functional
values resulted to be inside the interval:

0,0551 < Sh < 0,0607 (2.10)

The present work will be closely related to this condition as discussed later.
Beside the definition of the functional Sh as per Eq. (2.8) one will
make use of an other functional SH of expression:

(2.11)

where SH is referred to the pipe diameter by substitution in Eq. (2.8) of d → δ with D → Δ°.

The following subscripts will be used in the context in order to
denote a quantity associated with various conditions existing inside the
stream.

Denoting by x the volumetric concentration of the solid particles transported let one define an "indifferent" particle one of the same density
as the fluid. Hence for 1+μ → 1 i.e. for μ → μ_{10} → 0, being μ10 a very
small excess density one will use the subscript 1 for:

1 + μ → 1 (a)

Further for one solid particle only present in the system (of rather small diameter) one puts for the concentration

x → 0 (b)

Hence combining the two conditions in subscript one writes:

(2.12)

Further for 1 + μ → 1 and x > 0 one writes:

(2.13)

For μ ≠ 0 but x → 0, i.e. for one small particle of excess densityμ present in the system let one put:

(2.14)

Finally for the concentration x > 0 and excess density μ ≠ 0 one writes:

(2.15)

In the analytical discussion of the experimental results retrieved from the technical literature (all dealing with water streams) in absence of a water
temperature associated with a certain test, the author has assumed
19°C with a corresponding kinematic viscosity for water equal to 1x10^{-6} . Hence the numerical values of the transformation constants (2.2) and (2.3) are for water as below:

(2.16)

(2.17)

As well-known, values of friction factors are available in the technical literature in graphical forms.

In case (a) the determination has been carried out by Prandtl in case (b) calculations are due to Moody [7].

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*# Considering a stream over a particle bed in a flume of depth H, with particles of size d* as in Shields' experiments, one can pass to an equivalent stream flow in a pipe of
diameter D = 4H, with particles of size d = 4d* as given by Eq. (2.8).*

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