Uncompressible Fluid Flow in Pipeline
Hydrodynamic calculation of the pipeline makes it possible to determine the dynamic nature of fluid motion, pipe section diameter, pump output, and loss of system pressure. The calculation of incompressible flow is reduced to solving the Bernoulli equation for two successive cross-sections:
ρgh1 + P1 + α1×w12ρ / 2 = ρgh2 + P2 + α2×w22ρ / 2 + ΔPloss.
h1, h2 - heights of pipe inlet and outlet points;
w1, w2 - flow velocities at pipe inlet and outlet points;
P1, P2 - hydrostatic pressures;
α1, α2 - Coriolis coefficients, taking into account the uneven distribution of velocities over the section;
ΔPloss. - frictional pressure loss.
In this calculation, an unbranched pipeline with an inner diameter D, length L, and height difference between outlet and inlet of the pipe ΔH is considered. An incompressible fluid with flow rate Q, density ρ and dynamic viscosity μ passes through the pipeline. Under condition of the calculation, the pipeline may include fittings with the total local loss coefficient Σξi (in the absence of fittings Σξi = 0). Depending on the pipe material, the wall roughness value Δ is set.
Following the calculation, the average flow velocity W and Reynolds number Re are determined. The following are to be calculated next: frictional pressure loss ΔP, static pressure Pstat at the pipeline inlet depending on the height difference and the magnitude of the pressure loss, as well as dynamic fluid pressure Pd determined by the flow rate. When solving this problem, the viscous friction coefficient λ, thickness of the laminar sublayer δlam and total fluid pressure P can also be determined.
When performing finite element calculations, it is extremely important that the mesh size in the near-wall layer of the pipeline does not exceed certain values in the radial direction. The algorithms in this section calculate the minimum the size of the first cell Y recommended by software developers at the value of the wall function Y+ = 30. In the general case, the value of the wall function should lie within 30 < Y+ < 300.