Pipe (P)

Block appearance in REPEAT VISION

 The unit describes flow of air through the pipe section and takes into account pressure losses during friction against the rough pipe wall

 Optionally, the unit can take into account compression of air, elastic deformations of pipe walls, inertia.

 The pipe can be divided into several sections.

Pressure losses due to friction:

 If the Reynolds number of the flow is less than the upper limit of the Reynolds number for laminar flow, then the pressure losses are calculated by the formula (1) :

(1)\[ \Delta P_{fric} = \frac{\nu \cdot \lambda}{2\cdot D^2\cdot S}\cdot L \ \cdot \dot{m} \]

where:

 \( u\) - kinematic viscosity, [m\(^2\)/s];

 \(\lambda\) - Darcy friction coefficient, which is equal to 64;

 \(L\) - pipe length, [m];

 \(D\) - pipe diameter, [m];

 \(S\) - pipe cross-section area, [m\(^2\)];

 \(m\) - mass flow rate of liquid, [kg/s].

 If Reynolds number of flow is greater than the lower limit of the Reynolds number for turbulent flow, then pressure losses are calculated by the formula (2):

(2)\[ \Delta P_{fric} = \frac{f}{2\cdot \rho \cdot S^2} \cdot \frac{L}{D} \cdot \mid \dot{m} \mid \cdot \dot{m} \]

where:

 \(f = \{-1.8 \cdot \ log_{10}[\frac{6.9}{Re} + (\frac{\varepsilon}{3.7\cdot D_h})^{1.11}] \}^2\) ;

 \( ho\) – liquid density, [kg/m\(^3\)];

 \(S\) – pipe cross-section area, [m\(^2\)];

 \(L\) – pipe length, [m];

 \(D\) – pipe diameter, [m];

 \(\dot{m}\) – mass flow rate of liquid, [kg/s];

 \(Re\) – Reynolds number;

 \(\varepsilon\) – pipe roughness;

 \(D_h\) – hydraulic diameter of the pipe, [m].

 If the Reynolds number corresponds to a transient process, then the resistance coefficient is linear interpolation between the boundary points of the turbulent and laminar flow condition.

Mass conservation principle:

 If liquid is incompressible, then flowrate at the pipeline inlet and outlet is the same. If compressed liquid mode is selected, then each pipe segment is divided into three parts - two pipes connected by a capacitive unit. The mass conservation principle in a capacitive unit is determined by the ratio (3) :

(3)\[ \dot{m_A} + \dot{m_B} = \dot{P_n} \cdot \frac{\Delta \rho_n}{\Delta P_n}\ \cdot V \cdot \dot{m_A} \]

 \(\dot{m_A}\) – mass flow rate of liquid in the “left” pipe, [kg/s];

 \(\dot{m_B}\) – mass flow rate of liquid in the “right” pipe, [kg/s];

 \(P_n\) – pressure in the capacitive unit, [Pa];

 \( ho_n\) – density of liquid in the capacitive unit, [kg/m\(^3\)];

 \(V\) – pipe segment volume, [m\(^3\)].

 If deformations of pipeline walls are taken into account, then the term associated with the volumetric expansion is added to the mass conservation principle (4) :

(4)\[ \dot{m_A} + \dot{m_B} = \dot{P_n}\cdot \frac{\Delta \rho_n}{\Delta P_n}\ \cdot V + \rho_n \cdot \dot{V} \]

Momentum conservation law

 The total pressure differential between pipe ends is determined by friction losses and elevation differences (5) :

(5)\[ \Delta P = \Delta P_{fric} + \rho \cdot \Delta z \cdot g \]

 \(P_{fric}\) – friction pressure losses, [Pa];

 \( ho\) – liquid density, [kg/m\(^3\)];

 \(\Delta z\) – difference of elevations, [m];

 \(g\) – gravity acceleration, [m/s\(^2\)].

 If liquid inertia is taken into account, then the total pressure differential is determined by the formula (6):

(6)\[ \Delta P = \Delta P_{fric} + \rho \cdot \Delta z \cdot g + \ddot{m} \cdot \frac{L}{2\cdot S} \]

 \(L\) – pipe length, [m];

 \(S\) – pipe cross-section area, [m\(^2\)].

Pipe specifications

 If deformable walls of the pipeline are selected, then the volumetric expansion of the pipe is calculated by the formula (7) :

(7)\[ \dot{V} = \frac{\pi}{2}\cdot D \cdot L\cdot \frac{D(p)-D}{\tau} \]

where:

 \(\dot{V}\) – volumetric expansion, [m\(^3\)];

 \(L\) – pipe length, [m];

 \(D(P)\) – pipeline diameter depending on pressure, [m];

 \(D\) – pipeline diameter at the previous integration step, [m];

 \( au\) – volumetric expansion time constant.

 Change in pipe diameter, as well as volumetric expansion, is limited by the time constant of volumetric expansion.

 Pipe diameter depends on the static pressure in the pipe segment (8):

(8)\[D(P) = D_n \cdot(1 + \epsilon_{hoop})\]

where:

 \(\epsilon_{hoop} = \frac{P\cdot D_n}{2\cdot E \cdot h} \cdot [1 - \frac{\nu}{2}]\)

 \(P\) – static pressure inside the pipe, [Pa];

 \(D_n\) – nominal pipe diameter, [m];

 \(E\) – Young’s modulus of the pipe wall;

 \(h\) – pipe wall thickness, [m];

 \( u\) – Poisson’s ratio.

1.  Properties

Property	Default value	Description	Units of measurement
Gas compressibility	Off	-	-
Gas inertia, available when Gas compressibility - On	Off	-	-
Pipe length	5	-	m
Pipe diameter	0.1	-	m
Elevation gain from port A to port B	0	-	m
Gravitational acceleration	9.81	-	м/с\(^2\)
Internal surface absolute roughness	15e-6	-	m
Laminar flow upper Reynolds number limit	2000	-	-
Turbulent flow lower Reynolds number limit	4000	-	-
Initial gas gauge pressure	0	-	Pa
Initial gas temperature in the pipe	20	-	°C
Initial mass flow rate, available when Gas inertia - On	0	-	kg/s

2.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
Input	Input	-	-	Pneumatics
Heat Port	Output	-	-	Heat exchange
Output	Output	-	-	Pneumatics

In order to see the connection capabilities of the block, you need to go to page Description of block ports

3.  Examples of use

 Examples of using this block can be found in the following sections:

• Tutorials;

• How-to guides;

• Description of Demo.

Tutorials

Library	Tutorial name
Pneumatics	Gas outflow to a cylinder through a tube with a narrowing

How-to guides

Section	Guide name
Machine learning approximation	Approximation model of gas flow through a nozzle
Solving optimization tasks	Maximizing gas production rate in the gas well cluster collection system