Gas Flow in Pipeline
As in calculating the incompressible flow, the gas-dynamic calculation is reduced to solving the Bernoulli equation for two successive cross-sections:
P1 + w12ρ / 2 = P2 + w22ρ / 2 + ΔPloss.
w1, w2 - flow velocities at pipe inlet and outlet points;
P1, P2 - hydrostatic pressures;
ΔPloss. - frictional pressure loss.
When compressible fluids and gases flowing, the pressure along the length of the pipeline decreases due to energy losses for motion. At that, the gas density decreases while its volume increases. For this reason, formulas determining the pressure loss in pipelines with an incompressible fluid are not suitable for gas calculations. The change in pressure in the gas pipeline at an elementary length dL is equal to:
dP = - λ*(1 / D)*(W2 / 2)*ρdL
W = W0(TP0 / T0P); - flow rate under standard conditions;
ρ = ρ0(T0P/ TP0) - gas density under standard conditions.
T0 = 273°C;
P0 = 101300 Pa.
In this calculation, an unbranched pipeline with an inner diameter D and length L is considered. A gas flow with flow rate under standard conditions Q0, density under standard conditions ρ0 and dynamic viscosity μ, determined at service temperature T is passing through a 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.
Based on the results of calculation, the minimum overpressure Pmin necessary for ensuring the required gas flow rate is determined at the pipeline inlet. At that, friction losses go down as the overpressure increases. After determining the minimum overpressure, the overpressure at the inlet of the pipeline Pover > Pmin should be specified for further calculations of the flow characteristics. Next, the average flow velocities at the inlet Winlet and outlet Woutlet of the pipeline and the Reynolds number Re are to be calculated. Then frictional pressure loss ΔP and viscous friction coefficient λ are calculated.
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.