## A GENERAL THEORY OF THE HYDRAULIC

TRANSPORT OF SOLIDS IN
FULL SUSPENSION

By A C Bonapace

**III. CONDITIONS DEFINING PARTICLE SUSPENSION**

The condition of a particle transported in full suspension is expressed
by the following relationship:

(3.1)

i.e.

(3.2)

with Eqs. (3.1) and (3.2) related to μ >< 0.. The case μ < 0 will be associated to a particle buoyant in the fluid.

Eq. (3.1) written in explicit form results as below:

(3.3)

Considering a particle of diameter the above expression (3.3) can be written in the form:

(3.4)

Eq. (3.4) for = constant is just a linear relationship of against itself.

For a friction velocity:

(3.5)

one defines the field of variation of between the limits:

defined, e.g. for a smooth pipe, by the relation:

(3.6)^{#}

i.e. by

(3.7)

Hence Eq. (3.4) will be considered inside the field of variation of the ratio as given below:

(3.8)

being

(3.9)

Hence with Eq. (3.4) as an experimental datum, and for a particle inside the interval (3.8), this equation insures full suspension of the
particle by the stream.

In Fig. 1 Eq. (3.4) has been represented by the segment AB in a linear relationship (continuous line). The locus AB intersects the axis of the ordinates at the value Y = 0,0607 for 1 as expected.Hence the slope of the AB locus is 0,0607. Next to AB
one has drawn (in dotted line) the locus corresponding to a slope 0,0551
in agreement with the Sh experimental value 0,0551 as per expression
(2.10). This value defines the width of the band of Shields experiments
on the lower side.

Let one introduce now the condition:

μ → μ_{10} → 0 (3.10)

by which the particle becomes "indifferent" to the force due to gravity. Let one write in correspondence of Eq (3.10):

(3.11)

In the representation Ω_{10} versus the corresponding plane is herein denoted as "generating plane". Let one write the above equations
referred to the generating plane with the same reference number but with a
star sign as shown below:

(3.1.^{*})

(3.2.^{*})

(3.3.^{*})

(3.4.^{*})

(3.5.^{*})

(3.6.^{*})

(3.7.^{*})

(3.8.^{*})

With gravitational forces playing a marginal role inside the generating plane, only forces of inertia acting upon the particle ought to be
considered.

Let one introduce in the generating plane the following reference condition:

(3.12)

Thus with Eq. (3.12) into (3.3.*) one gets:

(3.13)

-----------------------------------------------------------------------------------------------------------------

# At the value Re_{*μo} = ω_{*μo}_{*μo} = 2,50 of the variable, the Sh function for a particle bed in a flume has increased to its original value 0,0607. Then, let one consider the
flow in a pipe, where for _{μo} = 4_{*μo} one will get Eq. (3.6). Hence the Sh function
for pipe flow will acouire the value 0,0607 at R'e_{*μo} = 10.

Previous | Next