Uncompressible Fluid Flow in Pipe | Online Calculator

Fluid Dynamic Calculators

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.

Hydrodynamic of Pipelines
Hydrodynamic of Pipes

INITIAL DATA

Q - Volumetric flow rate;

ρ - Fluid density (see table below);

μ - Dynamic fluid viscosity (see table below);

ΔH - Altitude difference between the outlet and the inlet of the pipe;

D - Pipe inner diameter;

L - Pipe length;

Σξi - Total coefficient of hydraulic 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).

RESULTS DATA

Pstat - Static fluid pressure at the pipeline inlet.

Pd - Dynamic fluid pressure.

P - Total fluid pressure at the pipeline inlet.

ΔP - Frictional pressure loss in pipeline.

W - Average velocity of the flow.

Re - Reynolds number.

λ - Viscous friction coefficient.

δlam - Laminar sublayer thickness.

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 (Q)

Liquid density (ρ)

Dynamic viscosity (μ)

Height difference (ΔH)

Inner diameter (D)

Pipe length (L)

Coefficient of resistance (Σξi)

Absolute roughness (Δ)

Static pressure (Pstat)

Dynamic pressure (Pd)

Total pressure (P)

Pressure loss (ΔP)

Velocity (W)

Reynolds number (Re)

Friction coefficient (λ)

Laminar sublayer (δlam)

First cell size for CFD (Y)

BASIC FORMULAS

Reynolds number:

Re = W*D*ρ / μ

Laminar sublayer thickness:

δlam = 68.4*Re-0.875*D / 2

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

Pressure loss in straight pipeline sections according Darcy formula:

ΔPstraight = λ*(L / D)*(W2*ρ / 2)

Pressure loss at local resistances:

ΔPlocal = Σξi*(W2ρ / 2)

Total frictional pressure loss:

ΔP = ΔPstraight + ΔPlocal.

INITIAL DATA

Q - Volumetric flow rate;

ρ - Fluid density (see table below);

μ - Dynamic fluid viscosity (see table below);

ΔH - Altitude difference between the outlet and the inlet of pipe;

D - Pipe inner diameter;

L - Pipe length;

Σξi - Total coefficient of hydraulic 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).

RESULTS DATA

Pstat - Static fluid pressure at the pipeline inlet.

Pd - Dynamic fluid pressure in the pipeline.

P - Total fluid pressure at the pipeline inlet.

ΔP - Frictional pressure loss in pipeline.

W - Flow rate in pipeline.

Re - Reynolds number.

λ - Viscous friction coefficient.

δlam - Laminar sublayer thickness.

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

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

FLUID PROPERTIES

Fluid

Dynamic viscosity

Pa*s

(lb*s/ft2)

Density

Kg/m3

(lb/ft3)

Water

0.001

(0.0000208)

1005

(62.8)

Freon-11

0.00048

(0.00001)

1494

(93.4)

Freon-12

0.00028

(0.00000586)

1330

(83.1)

Freon-22

0.00025

(0.00000522)

1202

(75.1)

Freon-113

0.00074

(0.0000155)

1573

(98.3)

Kerosene

0.00256

(0.0000535)

804

(50.25)

Benzene

0.00069

(0.0000144)

879

(54.9)

Glycerol

1.393

(0.029)

1261

(78.8)

Ethanol

0.00122

(0.0000255)

789

(49.3)

Mercury

0.00159

(0.0000332)

13545

(846)

Linseed oil

0.044

(0.000919)

924

(57.75)

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