## A GENERAL THEORY OF THE HYDRAULIC

TRANSPORT OF SOLIDS IN
FULL SUSPENSION

By A C Bonapace

**IV. MINIMUM ENERGY FUNCTIONS DEFINED IN THE**

EXPERIMENTAL FIELD

From the analysis (by elementary methods) of a minimum energy dissipative function, certain simple relations have been obtained by the author as discussed below:

Omitting the rather long analytical derivation they are introduced as"acceptable" experimental expressions.

(a) In the generating plane this relation is simply given by

Ω_{10} x Δ_{10}° = RE_{10} = constant (4.1)

being RE_{10} in Eq. (4.1) the Reynolds number of the stream in the generating plane.

The relation (4.1) is to be interpreted as follows:

For a particle in equilibrium under a force of inertia and a hydraulic force in vicinity of the boundary, the equilibrium is maintained along the
hyperbola RE_{10} = constant.

(b) outside the generating plane let one consider the conditions μ ≠ 0 and x → 0, i.e. one deals with a single small particle of excess density μ , which is present in the system.

Hence one writes (from the theory) the condition of inertial equilibrium for the particle as follows:

Ω_{μo} x = RE_{10} (1 + μ) = RE_{μo} (4.2)

with RE_{10} defined by Eq. (4.2).

Introducing the exponents α and 1-α one puts

Ω_{μo} = Ω_{10} (1 + μ)^{α} (4.3)

= Δ° (1 + μ)^{1−α} (4.4)

Hence from Eqs. (4.3), (4.4) into (4.2) one gets with (4.1):

(1 + μ)^{α} (1 + μ)^{1−α} = 1 + μ (4.5)

Eq. (4.5) is the first equilibrium equation derived from the minimum energy functional principle applied outside the generating plane.

(c) Equally outside the generating plane for a concentration x > 0 but for a particle of excess density μ_{10} → 0, i.e. for a particle of
size let one denote by ε an increment in stream velocity. Then one puts:

Ω_{1x} = Ω_{10} (1 + ε) (4.6)

Δ_{1x}° = Δ_{0}° (1 – x) (4.7)

Eqs. (4.6) and (4.7) express the following condition:

To an increase in stream velocity to 1 + ε there correspond a decrease in stream size

to 1 – x.

Hence by multiplication of (4.6) and (4.7) one obtains on account of
(4.1)

(1 + ε) (1 – x) = 1 (4.8)

Eq. (4.8) is the second equilibrium equation derived from the minimum energy functional principle, outside the generating plane.

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