Pipe (IL)
Rigid pipework for fluid flow in isothermal liquid systems.
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Description
The Pipe (IL) block models the dynamics of isothermal liquid flow in a pipe. The block accounts for viscous friction losses and can also account for dynamic compressibility and inertia of the fluid.
The pipe contains a constant volume of fluid. The pressure loss is due to viscous friction and is described by the Darcy-Weisbach equation.
The set of block parameters varies depending on the Fluid dynamic compressibility and Fluid Inertia parameters.
Tube effects
This block allows you to include the effects of dynamic compressibility and fluid inertia. Including each of these effects can improve the accuracy of the model at the cost of more complex equations and potentially increased modelling time:
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When the option to account for the dynamic compressibility of the fluid is turned off, it is assumed that the fluid passes through the pipe in a short period of time, so there is no mass accumulation in the pipe and the inflow of mass is equal to its outflow. This is the simplest option. It is suitable when the mass of liquid in the pipe is a negligible fraction of the total mass of liquid in the system.
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When the option to account for the dynamic compressibility of the fluid is enabled, an imbalance of mass inflow and outflow can result in an increase or decrease in the amount of fluid in the pipe. As a result, the pressure in the pipe can rise and fall, which will provide a certain amount of pliability to the system and result in rapid pressure changes. This option is enabled by default.
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If the option to account for the dynamic compressibility of the fluid is enabled, the option to account for fluid inertia can also be enabled. This effect results in additional hydraulic resistance on top of the resistance due to friction. This additional resistance is proportional to the rate of change in mass flow rate. Accounting for fluid inertia slows down rapid changes in flow rate, but can also cause surges and fluctuations in flow rate. This option is suitable for very long pipe. Enable the fluid inertia option and connect several pipe segments in series to simulate the propagation of pressure waves along the pipe, such as in a water hammer phenomenon.
Mass conservation
The mass conservation equation for the pipe is:
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where:
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- mass flow rates through ports A and B.
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- volume of liquid in the pipe.
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- pressure inside the pipe.
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- the density of the liquid inside the pipe.
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- bulk modulus of elasticity of the liquid inside the pipe.
The fluid can be a mixture of pure liquid and a small amount of air, as specified by the Liquid Properties (IL) block connected to the circuit. The equations used to calculate and , and the densities at the ends of the pipe and in the Darcy-Weisbach equations for each half of the pipe depend on the isothermal liquid model chosen.
Pulse balance
The momentum conservation equations for each half of the pipe:
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For the half of the pipe adjacent to the port A
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For half of the pipe adjacent to the port B
where:
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and are fluid pressures at the ends of pipe A and B respectively.
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and - pressure losses on viscous friction between the pipe centre and ports A and B.
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- length of the pipe.
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- is the cross-sectional area of the pipe.
Pressure losses due to viscous friction
Viscous friction pressure loss equations for each half of the pipe:
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For the half of the pipe adjacent to the port A
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For half of the pipe adjacent to the port B
where:
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- is the pipe shape coefficient used to calculate the Darcy friction coefficient in the laminar flow regime.
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- dynamic viscosity of the liquid in the pipe.
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- total equivalent length of local resistances in the pipe.
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- hydraulic diameter of the pipe.
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and - Darcy friction coefficients in the pipe halves adjacent to ports A and B.
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and - Reynolds numbers at ports A and B.
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- Reynolds number, at exceeding of which the flow passes into turbulent flow regime.
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- Reynolds number, below which the flow switches to laminar flow regime.
When the Reynolds number is between and , the flow is in a transition state between laminar and turbulent flow regimes. Pressure losses due to viscous friction in the transition region are smooth between those in the laminar flow regime and those in the turbulent flow regime.
The block calculates the Reynolds numbers at ports A and B based on the mass flow rate through the corresponding port:
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The Darcy friction coefficients follow the Haaland approximation for the turbulent flow regime:
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where:
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- Darcy’s coefficient of friction.
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- roughness of the pipe surface.
Ports
Non-directional
A - inlet or outlet port
isothermal liquid
The isothermal liquid port corresponds to the inlet or outlet of the pipe. This unit has no internal directionality.
B - inlet or outlet port
isothermal liquid
The isothermal liquid port corresponds to the inlet or outlet of the pipe. This unit has no internal directionality.
Parameters
Pipe length - pipe length
`5.0 m (by default)
Pipe length along the flow direction.
Cross-sectional area - cross-sectional area of the pipe
0.01 m² (by default)
The cross-sectional area of the pipe perpendicular to the direction of flow.
Hydraulic diameter - diameter of an equivalent cylindrical pipe with the same cross-sectional area
`0.1128 m (by default).
The diameter of an equivalent cylindrical pipe with the same cross-sectional area.
Friction
Aggregate equivalent length of local resistances - total length of all local resistances present in the pipe
`1.0 m (By default)
The total length of all local resistances present in the pipe. Local resistances include bends, fittings, fittings, inlets and outlets of the pipe. The effect of local resistances is to increase the effective length of the pipe section. This length is added to the geometric length of the pipe only for friction calculations. The volume of fluid inside the pipe depends only on the geometric length of the pipe, defined by the Pipe length parameter.
Internal surface absolute roughness - average depth of all surface defects on the internal surface of the pipe
`15e-6 m (By default).
The average depth of all surface defects on the internal surface of the pipe that affect pressure losses in turbulent flow regime.
Laminar flow upper Reynolds number limit - Reynolds number above which the flow starts to change from laminar to turbulent flow regime
2000 (By default).
Reynolds number above which the flow starts to change from laminar to turbulent flow regime. This number is equal to the maximum Reynolds number corresponding to stationary laminar flow.
Turbulent flow lower Reynolds number limit - Reynolds number below which the flow starts to change from turbulent to laminar flow regime
`4000 (by default).
Reynolds number below which the flow starts to change from turbulent to laminar flow regime. This number is equal to the minimum Reynolds number corresponding to stationary turbulent flow.
Laminar friction constant for Darcy friction factor - coefficient of hydraulic friction in laminar flow regime for Darcy friction factor
`64.0 (by default).
Dimensionless coefficient determining the effect of pipe cross-section geometry on viscous friction losses in laminar flow regime. Typical values: 64.0 for circular cross section, 57.0 for square cross section, 62.0 for rectangular cross section with aspect ratio 2 and 96.0 for thin annular cross section.
Effects and Initial Conditions
Fluid dynamic compressibility - consideration of fluid dynamic compressibility
On (By default) | Off
Determines whether dynamic compressibility of the fluid is taken into account. Dynamic compressibility makes the fluid density pressure dependent, which affects the transient response of the system over small time intervals.
Fluid inertia - account for fluid inertia
Off (By default) | On
Determines whether fluid inertia is taken into account. Flow inertia resists changes in mass flow rate.
Dependencies
To use this parameter, select the Fluid dynamic compressibility checkbox.
Initial liquid pressure is the liquid pressure at zero point in time
`0.101325 MPa (by default).
Liquid pressure in the pipe at the initial moment of time.
Dependencies
To use this parameter, select the Fluid dynamic compressibility checkbox.
Initial mass flow rate from port A to port B - mass flow rate at time zero
0.0 (By default).
Mass flow rate from port A to port B at the initial moment of time.
Dependencies
To use this parameter, select the checkbox for Fluid inertia.