∆P

∆P

Accounting for pressure losses on local resistances

V – flow velocity

Specifics of the Stokes program

Pressure losses in the pipeline section consist of losses due to friction along length of pipe, differences in hydrostatic pressure due to the difference in the altitudes of the beginning and end of the pipe, and losses on local resistances.

All equations are given for the basic SI units of measurement: length – m, pressure – Pa, volumetric flow – m^{3}/s, kinematic viscosity – m^{2}/s.

Losses along length of low pressure gas pipeline or pipelines with liquids are calculated according to the Darcy-Weisbach equation:

∆P_{L} = 8 λ Q_{0}^{2} ρ_{0} L / D^{5} π^{2} (1.1),

For high pressure gas pipelines:

∆P_{L} = P_{1} - √P_{1}^{2} - 16 P_{atm} λ L Q_{0}^{2} ρ_{0} / π^{2} D^{5} (1.2),

where

∆P_{L} – pressure loss along length, Pa

P_{atm} – normal atmospheric pressure = 101325 Pa

P_{1} – absolute pressure in start of pipe, Pa;

λ – friction factor;

Q_{0} – volumetric flow (for gases – at normal pressure), m^{3}/с;

ρ_{0} – densiti of the gas/liquid at normal atmospheric pressure, kg/m3;

L – length of the calculated pipe, m;

D – hydraulic diameter (for circular pipes = inner D), m.

First we need to calculate the Reynolds number:

Re = V D / ν = 4 Q_{0} / π D ν_{0} (2),

where

V – flow velocity,

ν – kinematic viscosity (for gases ν_{0} – viscosity at normal atmospheric pressure), m^{2}/s.

The formula for calculating λ is chosen depending on the obtained value of Re:

laminar flow | unstable flow | turbulent flow | |||

criterion | Re < 2000 | 2000 <= Re < 4000 | Re >= 4000 | ||

(Re * n / D) < 23 | (Re * n / D) >= 23 | ||||

Re <= 100000 | Re > 100000 | ||||

λ = | 64 / Re | 0.0025 Re^{⅓} |
0.3164 / Re^{¼} |
1 / (1.82 log(Re) - 1.64)^{2} |
0.11 ((68 / Re) + (n / D))^{¼} |

(3.1) | (3.2) | (3.3) | (3.4) | (3.5) |

where

n – the roughness of the surface of the pipe, m.

If the beginning and end of the pipe are at different heights (parameter **Z – altitude**), it is necessary to take into account the hydrostatic pressure according to the formula:

∆P_{h} = (ρ - ρ_{в}) g ∆h = (ρ - ρ_{в}) g (h_{2} - h_{1}) (4),

where

ρ – average density of the pumped liquid/gas, kg/m^{3};

ρ_{в} – air density at normal atmospheric pressure = 1,293 kg/m^{3};

∆h = h_{2} - h_{1} – difference in altitude between the end and the beginning of the pipe, m;

g – the local acceleration due to gravity, 9,81 m/s^{2}.

The density of the pumped liquid does not depend on pressure and is equal to the density at normal pressure ρ_{0}.

The density of the pumped gas depends on the compression ratio of the gas in the pipe:

ρ = ρ_{0} P_{avg} / P_{atm} (5),

where

ρ_{0} – density of the gas at normal atmospheric pressure, kg/m^{3};

P_{avg} – average *absolute* pressure in the pipe, Pa.

Average absolute pressure of the gas in pipe calculated to the formula from STO GP GR 12.2.2-1-2013 (p. В.5):

P_{avg} = ⅔ (P_{1} + P_{2}^{2} / (P_{1} + P_{2})) (6),

where

P_{2} – absolute pressure in the end of pipe.

P_{2} is calculated by the formula: P_{2} = P_{1} - ∆P_{L} - ∆P_{h}.

But since we calculate P_{2} for calculation ∆P_{h}, we discard ∆P_{h} and count P_{2} according to the formula:

P_{2} = P_{1} - ∆P_{L} (7),

This simplification is acceptable within the limits of the applicability of the Stokes: SP 42-101-2003 requires account ∆P_{h} for low pressure only. In gas pipelines for the medium and high pressure ∆P_{h} << ∆P_{L}, therefore, it could even be completely neglected.

Pressure losses at each local resistance are not calculated separately in Stokes.

In SP 42-101-2003, p. 3.30 pressure losses at local resistances (elbows, tees, valves, etc) can be taken into account by increasing the actual length of the pipe by 5-10%. Pipes in Stokes has parameter **Coeff. local losses** — L multiplied by this coefficient before substituting into the equation (1):

L_{calc.} = L × k (8),

Average flow velocity calculated by formula:

V = 4 * Q / D^{2} π (9),

where

Q – volumetric flow, m^{3/с.}

For liquids Q = Q_{0}, for gases Q is less than P/P_{atm} times. It is usually required to understand the maximum speed, so we use P_{2} (absolute pressure at the end of the pipe) as a P – because the velocity is maximum there:

Q = Q_{0} P_{atm} / P_{2} (10)

- If ∆P > P
_{1}, than P_{2}= 0 – It is believed that there is an absolute vacuum at the end of the pipe (manometric pressure = -101325 Pa); - The change in atmospheric pressure with altitude is not taken into account;
- Gases in calculations are considered ideal (compressibility factor = 1), liquids – absolutely incompressible.