New equations for the calculation of Bingham plastic flow through circular pipes have been derived by rearrangement of McMllen's equations as follows.
Total volume of flow rate: (1)
Average Nelocity in the pipe: (2)
Velocity at the central plug (maximum velocity):
(3)
Velocity at any point from pipe wall to central plug boundary:
(4)
In constructing the curve indicated by Equation(4), use is made of parallel displacement of the "imaginary velocity distribution curve" indicated by the following equation.
(5)
Apparent viscosity is represented by:
(6)
where
(7)
(8)
(9)
(10)
Another method has been proposed for computing pressure loss during laminar flow of Bingham plastic materials.
The equations used in this method are as follows:
(11)
(12)
(13)
In determining pressure loss in a pipe of radius R, when Bingham plastic material flows at a given average velocity ua, it is recommended to apply either Equation (11) to a/φ or Equation (12) to φ/a, so as to obtain φ by the use of Fig.2 or a-φ table, which will be published in full shortly
8).
When φ is known, the pressure loss can be calculated by means of Equation (13).
One more method is given for the evaluation of the two plastic constants.
In two pairs of p and V the following equations hold:
(14)
(15)
where
(16)
(17)
By means of a simple trial-and-error procedure, a1, a2, φ1 and φ2 can be easily obtained.
When a
1 and φ
1 are known, yield value, τ
y, and plastic viscosity, η, can be calculated by means of the following equations.
(18)
(19)
It is supposed that the use of a-φ table
8) together with the equations derived by the author is more convenient than that of the method previously proposed by E.L. McMillen
6) and developed by B.0.A. Hedstrom
4).
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