Pipe (TL)
Rigid pipeline for liquid supply in heat-conducting systems.
blockType: AcausalFoundation.ThermalLiquid.Elements.Pipe
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Description
Block Pipe (TL) It is a segment of a pipeline with a fixed volume of liquid. The liquid experiences pressure loss due to viscous friction and heat exchange due to convection between the liquid and the pipe wall. Viscous friction follows from the Darcy-Weisbach equation, and the heat transfer coefficient follows from the ratio of the Nusselt numbers.
Hydraulic effects in the pipe
Block Pipe (TL) allows you to enable the effects of dynamic compressibility and inertia of the fluid. Including each of these effects can improve the accuracy of the model by increasing the complexity of the equations and potentially increasing the cost of modeling resources.:
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When dynamic compressibility is disabled, it is assumed that the liquid spends little time in the volume of the pipe. Therefore, there is no mass accumulation in the pipe, and the mass inflow is equal to the mass 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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With dynamic compressibility, an imbalance of mass inflow and mass outflow can lead to an accumulation or decrease of fluid in the pipe. As a result, the pressure in the pipe volume can rise and fall dynamically, which ensures a certain pliability of the system and modulates rapid pressure changes. This option is used by default.
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If dynamic compressibility is enabled, you can also enable fluid inertia. This effect leads to additional resistance to flow, in addition to the resistance caused by friction. This additional resistance is proportional to the rate of change of the mass flow rate. Accounting for fluid inertia slows down rapid flow changes, but it can also lead to overestimation and fluctuations in flow. This option is suitable for a very long pipe. Turn on the inertia of the liquid and connect several pipe segments in series to simulate the propagation of pressure waves along the pipe, such as in a water hammer.
Conservation of mass
The mass conservation equation for a pipe is as follows:
where
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— mass flow through port A;
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— mass flow through port B;
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— the volume of liquid in the pipe;
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— the density of the liquid in the pipe, depending on the temperature;
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— isothermal modulus of volumetric elasticity in the pipe;
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— coefficient of isobaric thermal expansion in the pipe;
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— the density of the liquid in the pipe, depending on the temperature;
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— the temperature of the coolant in the pipe.
Conservation of momentum
The table shows the momentum conservation equations for each half-pipe.
For half of the pipe adjacent to port A |
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For half of the pipe adjacent to port B |
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In the equations:
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— the cross-sectional area of the pipe;
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— the pressure of the liquid in the pipe;
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— liquid pressure at the port inlet A;
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— liquid pressure at the port inlet B;
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— pressure loss due to viscous friction between the center of the pipe volume and the port A;
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— pressure loss due to viscous friction between the center of the pipe volume and the port B.
Pressure losses due to viscous friction
The table shows the equations of pressure loss under viscous friction for each half of the pipe.
For the half pipe adjacent to port A |
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For half of the pipe adjacent to port B |
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In the equations:
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— pipe shape coefficient;
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— kinematic viscosity of the heat-conducting liquid in the pipe;
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— total equivalent length of local pipe losses;
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— hydraulic pipe diameter;
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— the Darcy coefficient of friction in half of the pipe adjacent to the port A;
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— the Darcy coefficient of friction in the half of the pipe adjacent to the port B;
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and — the Reynolds numbers for ports A and B, respectively;
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— the Reynolds number, above which the flow becomes turbulent;
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— the Reynolds number, below which the flow becomes laminar.
The Darcy friction coefficients follow from the Haaland approximation for the turbulent regime:
where
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— the Darcy coefficient of friction;
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— roughness of the pipe surface.
Energy conservation
The energy conservation equation for a pipe has the form:
where
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and — energy flow into the pipe through ports A and B, respectively;
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— the flow of heat entering the pipe through the pipe wall.
Heat flow through the wall
The heat flow between the heat-conducting liquid and the pipe wall is:
where
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— heat flow through the pipe wall;
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— the part of the heat flow attributed to convection at a non-zero flow rate;
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— thermal conductivity of the heat-conducting liquid in the pipe;
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— the surface area of the pipe wall, the product of the perimeter and length of the pipe;
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— the temperature on the pipe wall.
If we assume an exponential temperature distribution along the pipe, then convective heat transfer will be
where
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— average mass flow through port A to port B;
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— specific heat at an average temperature;
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— inlet temperature depending on the flow direction.
Heat transfer coefficient, , depends on the Nusselt number:
where — thermal conductivity at medium temperature.
The Nusselt number depends on the flow regime.
The Nusselt number in laminar flow mode is constant and is equal to the value of the parameter Nusselt number for laminar flow heat transfer.
The Nusselt number in the turbulent flow regime is calculated by the Gnelinsky ratio.:
where – the Darcy coefficient of friction with an average Reynolds number, , and – the Prandtl number calculated at an average temperature.
The average Reynolds number is calculated as:
where – dynamic viscosity, estimated at an average temperature.
When the average Reynolds number is between the values of the Laminar flow upper Reynolds number limit and Turbulent flow lower Reynolds number limit parameters, the Nusselt number follows a smooth transition between the laminar and turbulent values of the Nusselt number.
Ports
Non-directional
A — pipe inlet or outlet type:q[<br>] heat-conducting liquid
The port of the heat-conducting liquid corresponds to the inlet or outlet of the pipe. This port does not have its own direction.
B — pipe inlet or outlet type:q[<br>] heat-conducting liquid
The port of the heat-conducting liquid corresponds to the inlet or outlet of the pipe. This port does not have its own direction.
H is the temperature of the pipe wall
warm
A port related to the temperature of the pipe wall. This temperature may differ from the temperature of the heat-conducting liquid inside the pipe.
Parameters
Geometry
Pipe length — pipe length
5 m (by default)
The length of the pipe along the flow direction.
Cross-sectional area — cross-sectional area of the
0.01 m^2 (default)
The cross-sectional area of the pipe is normal to the flow direction.
Hydraulic diameter — hydraulic diameter of the
0.1128 m (default)
The diameter of an equivalent cylindrical tube with the same cross-sectional area.
Friction and Heat Transfer
Aggregate equivalent length of local resistances — the total length of all local resistances present in the pass pipe:q[<br>]1 m (default)
The total length of all local resistances present in the pipe.
Local resistances include bends, fittings, fittings, and pipe entrances and exits. 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 liquid inside the pipe depends only on the geometric length of the pipe, determined by the Pipe length parameter.
Internal surface absolute roughness — the absolute roughness of the inner surface of the pipe
15e-6 m (default)
The average depth of all surface defects on the inner surface of the pipe that affect pressure loss in a turbulent flow regime.
Laminar flow upper Reynolds number limit — the Reynolds number, when exceeded, the flow begins to transition from laminar to turbulent
2000 (default)
The Reynolds number, when exceeded, the flow begins to transition from laminar to turbulent.
This number is equal to the maximum Reynolds number corresponding to a fully developed laminar flow.
Turbulent flow lower Reynolds number limit — the Reynolds number below which the flow begins to transition from turbulent to laminar
4000 (default)
The Reynolds number, below which the flow begins to transition from turbulent to laminar.
This number is equal to the minimum Reynolds number corresponding to a fully developed turbulent flow.
Shape factor for laminar flow viscous friction — shape coefficient for viscous friction in laminar flow
64 (default)
A dimensionless coefficient reflecting the effect of the geometry of the pipe cross-section on viscous friction losses in the laminar flow regime.
Typical values are 64 for a circular section, 57 for a square section, 62 for a rectangular section with an aspect ratio of 2, and 96 for a thin annular section.
Nusselt number for laminar flow heat transfer — Nusselt number for laminar flow
3.66 (default)
The ratio of convective to conductive heat transfer in the laminar flow regime. Its value depends on the geometry of the pipe’s cross-section and the thermal boundary conditions on the pipe wall, such as constant temperature or constant heat flow.
The typical value is 3.66 for a circular section with a constant wall temperature.
Effects and Initial Conditions
Fluid dynamic compressibility — dynamic compressibility of liquid
enabled (by default) | turned off
Check the box for this parameter to take into account the dynamic compressibility of the fluid in the simulation.
Dynamic compressibility makes the density of a liquid dependent on pressure and temperature, which affects the transient response of the system on small time scales.
Fluid inertia — inertia of the fluid
disabled (by default) | enabled
Check the box for this parameter to take into account the inertia of the fluid flow in the simulation.
The inertia of the flow gives the liquid resistance to changes in mass flow.
Dependencies
To enable fluid inertia, check the box for the Fluid dynamic compressibility parameter.
Initial liquid pressure — liquid pressure at zero time
0.101325 MPa (default)
The pressure of the liquid in the pipe at the beginning of the simulation.
Dependencies
To turn on the liquid pressure at time zero, check the box for the Fluid dynamic compressibility parameter.
Initial liquid temperature — the temperature of the liquid at time zero
293.15 K (default)
The temperature of the liquid in the pipe at the beginning of the simulation.
Initial mass flow rate from port A to port B — mass flow rate at zero time
0 kg/s (default)
The mass flow of air from port A to port B at time zero.
Dependencies
To enable mass flow at time zero, check the box for the Fluid inertia parameter.