# Hydraulic calculation of pipelines

The hydrodynamic calculation of the flow of an incompressible fluid reduces to solving the Bernoulli equation for two consecutive sections:

*ρgh _{1} + P_{1} + α_{1}×w_{1}^{2}ρ / 2 = ρgh_{2} + P_{2} + α_{2}×w_{2}^{2}ρ / 2 + ΔP_{пот.}*, where:

*h _{1}, h_{2}* – the height of the starting and ending points of the pipeline;

*w*– flow rates at the start and end points of the pipeline;

_{1}, w_{2}*P*– hydrostatic pressures;

_{1}, P_{2}*α*– Coriolis coefficients taking into account the uneven distribution of velocities over the cross section;

_{1}, α_{2}*ΔP*– loss of pressure to overcome resistance.

_{пот.}The hydraulic online calculation, presented in this section allows you to calculate the flow characteristics of an incompressible fluid, as well as the flow of a compressible fluid or high pressure gas. Both calculations are performed for a straight pipe.

When solving such problems by the finite element method in the ANSYS software, it is extremely important that the mesh size in the wall layer of the pipeline does not exceed certain values in the radial direction. The algorithms in this section calculate the minimum size of the first cell, recommended by the developers with the value of the wall function Y ^{ + } = 30. In general, the value of the wall function should be within 30 < Y^{+} < 300.

## Hydrodynamic calculation of an incompressible fluid pipeline

When conducting a hydrodynamic calculation, the Reynolds number is determined:

*Re = W×D×ρ / μ;*, where

*μ* – dynamic fluid viscosity;

*W* – flow rate;

*D* – pipe diameter.

The thickness of the laminar sublayer is determined along the inner surface of the pipe:

*δ = 68,4×Re ^{-0.875}×D / 2 *

The friction coefficient Δ is determined depending on the roughness of the inner surface of the pipe:

*λ = 0,316×Re ^{-0.25}* at δ > Δ

*λ = 0,11(Δ / D + 68 / Re)*at δ < Δ

^{0.25}The D’Arcy formula determines pressure loss in straight sections:

*ΔP = λ×(L / D)×(W ^{2}ρ / 2)*

Pressure loss on local resistances:

*ΔP = ΣK _{i}×(W^{2}ρ / 2)*

Summarizing the results obtained, we obtain a total pressure loss in a certain section of the pipeline.

### Initial data:

Q – fluid flow rate in the pipeline, liters per second;

ρ – fluid density, kilogram / meter ^{3};

μ – dynamic fluid viscosity, pascal×second;

ΔH – elevation difference of the starting and ending points of the pipeline section, meters;

D – the inner diameter of the pipeline, millimeters;

L – pipeline length, meters;

ΣK_{i} – total local resistance coefficient;

Δ – absolute roughnessof the inner wall of the pipe, millimeters.

PIPELINE HYDRAULIC CALCULATION

Flow rate Q, l/sec

Fluid density ρ, kg/m^{3}

Dynamic fluid viscosity μ, Pa*sec

Pipeline elevation difference ΔH, m

Pipeline inner diameter D, mm

Pipeline length L, mm

Total coeff. of local resist ΣK_{i}

Absolute roughness Δ, mm

Static inlet pressure P_{s}, Pa

Dynamic pressure P_{d}, Pa

Total inlet pressure P, Pa

Friction pressure loss ΔP, Pa

Flow rate W, m/sec

Reynolds number Re

Coefficient of friction λ

Laminar sublayer thickness δ_{л}, mm

The size of the first cell of the mesh, mm

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## Calculation of a high pressure gas pipeline

When high pressure gases are transported in pipelines, due to pressure losses to overcome resistance, the specific volume of gas increases and its density decreases. Moreover, the change in pressure at the elementary length dL is:

*dP = – λ×(1/D)×(W ^{2} / 2)×ρdL*, wherein:

*W = W _{0}(TP_{0} / T_{0}P);*

*ρ = ρ _{0}(T_{0}P/ TP_{0});*

*ρ _{0}, W_{0}* – gas density and gas flow rate under normal physical conditions(NPC);

*T*

_{0}= 273°C;*P*

_{0}= 101300 Pa.Substituting the resulting expressions:

*pdP = – λ×(W _{0}^{2}ρ_{0} / 2D)×(T / T_{0})×P_{0}dL;*

After integration:

*(P _{fin}^{2} – P_{start}^{2}) / 2 = -λ(L / D)(W_{0}^{2}ρ_{0} / 2)(P_{0}T / T_{0});*

It’s easy to get pressure loss from here:

*ΔP = P _{start}(1 – (1 – λ(L / D)×(W_{0}^{2}ρ_{0})×(P_{0}T / P_{start}^{2}T_{0})) ^{1/2});*

*P _{start}* – absolute pressure at the starting point of the pipeline section.

The friction coefficient is the same as in the calculation of the flow of an incompressible fluid.

### Initial data:

Q – gas flow in the pipeline under normal physical conditions, in cubic meters per hour;

ρ – gas density under normal physical conditions, in kilogram / meter ^{3};

T – gas temperature, in °C;

μ – dynamic viscosity of gas at operating temperature, in pascals×sec;

D – the inner diameter of the pipeline, in millimeters;

L – pipeline length, in meters;

ΣK_{i} – total local resistance coefficient;

Δ – absolute roughness of the inner wall of the pipe, in millimeters.

P_{н} – overpressure at the inlet of the pipeline, in pascals;

GAS-DYNAMIC CALCULATION OF THE PIPELINE

Gas flow rate at NPC, Q, m^{3}/hour

Gas density at NPC, ρ_{0}, kg/m^{3}

Gas temperature Т, ^{0}C

Dynamic gas viscosity μ, Pa*sec

Pipeline inside diameter D, mm

Pipeline length L, m

Coefficient of local resistance ΣK_{i}

Pipe wall roughness Δ, mm

Inlet overpressure Р_{start}, Pa

Minimum overpressure at the inlet to the pipeline P_{min}, Pa

Pressure loss due to friction in the pipeline ΔP, Pa

Gas inlet flow velocity W_{start}, m/sec

Gas outlet flow velocity W_{fin}, m/sec

Reynolds number Re

Coefficient of friction λ

The size of the first cell of the mesh Y (Y^{+}=30), mm

©Copyright Caetec 2020

*Ref 12 *

## Other calculators

– pipeline strength calculation

– thin-walled axisymmetric shell

– thick-walled pipe under internal and external pressure

– free convection for a horizontal surface

– free convection for a vertical surface

– heat transfer coefficient of a flat wall

– heat transfer coefficient of the inner wall of the pipe

– heat transfer coefficient of the outer wall of the pipe

– heat transfer coefficient of the outer wall of the group of pipes

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