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Fluid Dynamic Calculators

# 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.

### INITIAL DATA

Q0 - Volumetric flow rate under standard conditions (T = 273.15K, P = 100 kPa);

ρ0 - Gas density under standard conditions (see table below);

T - Gas tempetature;

μ - Dynamic gas viscosity at temperature T (see table below);

D - Pipe inner diameter;

L - Pipe length;

Σξi - Total coefficient of resistance of the fittings and of local changes at the pipe cross-section (see table below);

Δ - Absolute roughness of the inner surface of the pipe (see table below).

Pover - Overpressure at pipe inlet. Before input, you should calculate Minimal Overpressure Pmin, that provides the requared gas flow.

### RESULTS DATA

Pmin - Minimal Overpressure, that provides the requared gas flow.

ΔP - Frictional pressure loss in pipeline.

Winlet - Average velocity of the flow at pipeline inlet.

Woutlet - Average velocity of the flow at pipeline outlet.

Re - Reynolds number.

λ - Viscous friction coefficient.

Y - Size of the first cell near the pipe wall for calculating the flow characteristics by CFD analysis. Wall Function Y+ = 30.

Volumetric flow rate (Q0)

Gas density (ρ0)

Gas temperature (Т)

Dynamic viscosity (μ)

Inner diameter (D)

Pipe length (L)

Coefficient of resistance (Σξi)

Absolute roughness (Δ)

Overpressure (Рover)

Minimal overpressure (Pmin)

Pressure loss (ΔP)

Flow rate at inlet (Winlet)

Flow rate at outlet (Woutlet)

Reynolds number (Re)

Friction coefficient (λ)

First cell size for CFD, (Y)

### BASIC FORMULAS

pdP = - λ × (W02ρ0 / 2D) × (T / T0) × P0dL

After integration from Pinlet to Poutlet and from 0 to L:

(Poutlet2 - Pinlet2) / 2 = -λ(L / D)(W02ρ0 / 2)(P0T / T0)

Pressure loss:

ΔP = Pinlet*(1 - [1 - λ(L/D)*(W02ρ0)*(P0T / Pinlet2T0)] 1/2);

Pinlet - absolute pressure in pipe inlet.

Viscous friction coefficient, depending on absolute roughness Δ of the inner surface of the pipe:

λ = 0.316*Re -0.25 at δlam > Δ;

λ = 0.11(Δ / D + 68 / Re) 0.25 at δlam < Δ

### INITIAL DATA

Q0 - Volumetric flow rate under standard conditions (T = 273.15K, P = 100 kPa);

ρ0 - Gas density under standard conditions (see table below);

T - Gas tempetature;

μ - Dynamic gas viscosity at temperature T (see table below);

D - Pipe inner diameter;

L - Pipe length;

Σξi - Total coefficient of resistance of the fittings and of local changes at the pipe cross-section (see table below);

Δ - Absolute roughness of the inner surface of the pipe (see table below).

Pover - Overpressure at pipe inlet. Before input, you should calculate Minimal Overpressure Pmin, that provides the requared gas flow.

### RESULTS DATA

Pmin - Minimal Overpressure, that provides the requared gas flow.

ΔP - Frictional pressure loss in pipeline.

Winlet - Average velocity of the flow at pipeline inlet.

Woutlet - Average velocity of the flow at pipeline outlet.

Re - Reynolds number.

λ - Viscous friction coefficient.

Y - Size of the first cell near the pipe wall for calculating the flow characteristics by CFD analysis. Wall Function Y+ = 30.

### FLUID PROPERTIES

#### Fluid properties at 20ºC (68 ºF)

 Fluid Density Kg/m³ (lb/ft³) Dynamic viscosity Pa*s (lb*s/ft²) Volumetric expansion 1/ºC (1/ºF) Thermal conductivity W/m*ºC (W/in*ºF) Specific heat J/kg*ºC (J/lb*ºF) Water 998 (62.3) 0.001 (0.0000209) 0.00021 (0.00016) 0.599 (0.084) 4182 (1053) Air 1.205 (0.075) 0.000018 (0.000000377) 0.00365 (0.00202) 0.0259 (0.00036) 1005 (253) Water vapor 0.01 (0.00063) 0.0000096 (0.0000002) 0.00365 (0.00202) 0.0178 (0.00025) 1859 (468) Engine Oil SAE 15W-40 879 (54.9) 0.287 (0.006) 0.00655 (0.0036) 0.134 (0.00190) 2039 (513)

#### Fluid properties at 40ºC (104 ºF)

 Fluid Density Kg/m³ (lb/ft³) Dynamic viscosity Pa*s (lb*s/ft²) Volumetric expansion 1/ºC (1/ºF) Thermal conductivity W/m*ºC (W/in*ºF) Specific heat J/kg*ºC (J/lb*ºF) Water 992 (62) 0.00065 (0.0000135) 0.00021 (0.00016) 0.635 (0.0090) 4170 (1050) Air 1.128 (0.07) 0.000019 (0.000000397) 0.00365 (0.00202) 0.0276 (0.00039) 1005 (253) Water vapor 0.05 (0.0031) 0.0000104 (0.000000217) 0.00365 (0.00202) 0.0195 (0.000275) 1860 (468) Engine Oil SAE 15W-40 866 (54.1) 0.091 (0.0019) 0.00655 (0.0036) 0.131 (0.00184) 2106 (530)

#### Fluid properties at 60ºC (140 ºF)

 Fluid Density Kg/m³ (lb/ft³) Dynamic viscosity Pa*s (lb*s/ft²) Volumetric expansion 1/ºC (1/ºF) Thermal conductivity W/m*ºC (W/in*ºF) Specific heat J/kg*ºC (J/lb*ºF) Water 963 (60.2) 0.00046 (0.0000096) 0.00021 (0.00016) 0.659 (0.0093) 4184 (1054) Air 1.060 (0.066) 0.000020 (0.00000042) 0.00365 (0.00202) 0.0290 (0.00041) 1005 (253) Water vapor 0.14 (0.0088) 0.0000112 (0.00000023) 0.00365 (0.00202) 0.0212 (0.00029) 1870 (471) Engine Oil SAE 15W-40 853 (53.3) 0.038 (0.00079) 0.00655 (0.0036) 0.129 (0.00180) 2165 (545)

#### Fluid properties at 80ºC (176 ºF)

 Fluid Density Kg/m³ (lb/ft³) Dynamic viscosity Pa*s (lb*s/ft²) Volumetric expansion 1/ºC (1/ºF) Thermal conductivity W/m*ºC (W/in*ºF) Specific heat J/kg*ºC (J/lb*ºF) Water 972 (60.7) 0.00035 (0.0000073) 0.00021 (0.00016) 0.675 (0.0095) 4196 (1057) Air 1.00 (0.063) 0.000021 (0.00000044) 0.00365 (0.00202) 0.0305 (0.00043) 1009 (254) Water vapor 0.29 (0.018) 0.0000119 (0.00000025) 0.00365 (0.00202) 0.0229 (0.00032) 1880 (474) Engine Oil SAE 15W-40 841 (52.6) 0.019 (0.00039) 0.00655 (0.0036) 0.127 (0.00178) 2227 (561)

### HYDRAULIC RESISTANCES

 # Local resistances Figure Coefficient ξ 1 Entering a hole with sharp edges ξ = 0.5 2 Channel exit ξ = 1.0 3 Smooth elbow at 90° r / b ξ 0.5 1.2 0.75 0.38 1 0.19 2 0.12 5 0.08 4 Smooth elbow from 30° to 180° Value ξ (#3) multiplied by К ratio α° К 30 0.5 60 0.8 90 1 120 1.2 150 1.3 180 1.4 5 Sharp elbow without rounding α° ξ 30 0.6 60 1 90 1.2 120 1.4 180 1.7 6 Sudden narrowing F2 / F1 ξ 0.1 0.5 0.5 0.3 0.9 0.1 7 Sudden expansion F2 / F1 ξ 0.1 0.8 0.5 0.3 0.9 0.01 8 Partially open damper Opening, % ξ 10 230 30 17 50 4 70 1 90 0.2 100 0.1 9 Throttle α° ξ 10 0.52 30 3.9 50 32.6 70 151 10 Diaphragm F2 / F1 ξ 0.1 246 0.2 51 0.3 18 0.4 8 0.6 2 0.7 1 0.8 0.3 11 Channel bundle entrance Holes: square ξ = 2..2.5 circular ξ = 3..3.5 rectangular ξ=1.5..2 12 Valve h / d ξ 0.15 9 0.2 4.5 0.3 2.1 0.4 1.6 0.45 1.5 13 Transfer valve ξ = 2 14 Channel niche ξ = 0.1..1 and increases with increasing h / d 15 Pipe cross (merge flow) W / Wk ξ 0.1 1.5 0.3 1.4 0.5 1.2 0.7 0.9 0.9 0.5 1.0 0.2 16 Tee with counter flow At W1 = W2 = W3 ξ = 3 17 Dispensing tee Wb* ξb Wa db/da 0.35 0.58 1.0 0.6 3.2 4.0 6.2 0.8 1.9 2.5 4.5 1.0 1.6 2.1 3.6 1.2 1.4 1.6 3.4 1.4 1.2 1.4 2.8 ξa at db/da = 1 -0.2 -0.1 0 0.12 0.34 18 Collecting tee Wb* ξb Wa db / da 0.35 0.58 1.0 0.6 -3.8 -1.6 0.1 0.8 -1.0 0 0.6 1.0 -0.6 0 1.2 1.4 0.4 0.4 1.3

### HYDRAULIC RESISTANCES

 # Local resistances Figure Coefficient ξ 1 Entering a hole with sharp edges ξ = 0.5 2 Channel exit ξ = 1.0 3 Smooth elbow at 90° r / b ξ 0.5 1.2 0.75 0.38 1 0.19 2 0.12 5 0.08 4 Smooth elbow from 30° to 180° Value ξ (#3) multiplied by К ratio α° К 30 0.5 60 0.8 90 1 120 1.2 150 1.3 180 1.4 5 Sharp elbow without rounding α° ξ 30 0.6 60 1 90 1.2 120 1.4 180 1.7 6 Sudden narrowing F2 / F1 ξ 0.1 0.5 0.5 0.3 0.9 0.1 7 Sudden expansion F2 / F1 ξ 0.1 0.8 0.5 0.3 0.9 0.01 8 Partially open damper Opening, % ξ 10 230 30 17 50 4 70 1 90 0.2 100 0.1 9 Throttle α° ξ 10 0.52 30 3.9 50 32.6 70 151 10 Diaphragm F2 / F1 ξ 0.1 246 0.2 51 0.3 18 0.4 8 0.6 2 0.7 1 0.8 0.3 11 Channel bundle entrance Holes: square ξ = 2..2.5 circular ξ = 3..3.5 rectangular ξ=1.5..2 12 Valve h / d ξ 0.15 9 0.2 4.5 0.3 2.1 0.4 1.6 0.45 1.5 13 Transfer valve ξ = 2 14 Channel niche ξ = 0.1..1 15 Pipe cross (merge flow) W / Wk ξ 0.1 1.5 0.3 1.4 0.5 1.2 0.7 0.9 0.9 0.5 1.0 0.2 16 Tee with counter flow At W1 = W2 = W3 ξ = 3 17 Dispensing tee Wb* ξb Wa db/da 0.35 0.58 1.0 0.6 3.2 4.0 6.2 0.8 1.9 2.5 4.5 1.0 1.6 2.1 3.6 1.2 1.4 1.6 3.4 1.4 1.2 1.4 2.8 ξa at db/da = 1 -0.2 -0.1 0 0.12 0.34 18 Collecting tee Wb* ξb Wa db / da 0.35 0.58 1.0 0.6 -3.8 -1.6 0.1 0.8 -1.0 0 0.6 1.0 -0.6 0 1.2 1.4 0.4 0.4 1.3

### ROUGHNESS OF PIPES

 Pipes material Pipe condition Δ, mm Non-ferrous metals New 0 ÷ 0.002 Seamless steel pipes New 0.01 ÷ 0.02 After exploitation 0.15 ÷ 0.3 Welded steel pipes New 0.03 ÷ 0.1 With little corrosion 0.1 ÷ 0.2 Moderately corroded 0.3 ÷ 0.7 With significant corrosion 0.8 ÷ 1.5 With deposits on the walls 2 ÷ 4 Riveted steel pipes New 0.5 ÷ 3 Galvanized steel pipes New 0.1 ÷ 0.2 After exploitation 0.4 ÷ 0.7 Cast iron pipes New uncoated 0.2 ÷ 0.5 After exploitation 0.5 ÷ 1.5 Very old 1.5÷3 Plywood pipes New 0.05 ÷ 0.2 Concrete pipes New 0.03 After exploitation 0.5