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Pneumatic pipe accounting for pressure loss and added heat due to flow resistance
The Pneumatic Resistive Tube block models the loss in pressure and heating due to viscous friction along a short stretch of pipe with circular cross section. Use this block with the Constant Volume Pneumatic Chamber block to build a model of a pneumatic transmission line.
The tube is simulated according to the following equations:
$${p}_{i}-{p}_{o}=\{\begin{array}{ll}\frac{R{T}_{i}}{{p}_{i}}\xb7\frac{32\mu L}{A{D}^{2}}\xb7G\hfill & \text{for}Re\text{}R{e}_{lam}\text{(laminarflow)}\hfill \\ f\xb7\frac{R{T}_{i}}{{p}_{i}}\xb7\frac{L}{D}\xb7\frac{{G}^{2}}{2{A}^{2}}\hfill & \text{for}Re\text{}R{e}_{turb}\text{(turbulentflow)}\hfill \end{array}$$
where
p_{i}, p_{o} | Absolute pressures at the tube inlet and outlet, respectively. The inlet and outlet change depending on flow direction. For positive flow (G > 0), p_{i} = p_{A}, otherwise p_{i} = p_{B}. |
T_{i}, T_{o} | Absolute gas temperatures at the tube inlet and outlet, respectively |
G | Mass flow rate |
μ | Gas viscosity |
f | Friction factor for turbulent flow |
D | Tube internal diameter |
A | Tube cross-sectional area |
L | Tube length |
Re | Reynolds number |
The friction factor for turbulent flow is approximated by the Haaland function
$$f={\left(-1.8{\mathrm{log}}_{10}\left(\frac{6.9}{\mathrm{Re}}+{\left(\frac{e}{3.7D}\right)}^{1.11}\right)\right)}^{-2}$$
where e is the surface roughness for the pipe material.
The Reynolds number is defined as:
$$\mathrm{Re}=\rho vD/\mu $$
where ρ is the gas density and v is the gas velocity. Gas velocity is related to mass flow rate by
$$G=\rho vA$$
For flows between Re_{lam} and Re_{turb}, a linear blend is implemented between the flow predicted by the two equations.
In a real pipe, loss in kinetic energy due to friction is turned into added heat energy. However, the amount of heat is very small, and is neglected in the Pneumatic Resistive Tube block. Therefore, q_{i} = q_{o}, where q_{i} and q_{o} are the input and output heat flows, respectively.
The gas is ideal.
The pipe has a circular cross section.
The process is adiabatic, that is, there is no heat transfer with the environment.
Gravitational effects can be neglected.
The flow resistance adds no net heat to the flow.
Internal diameter of the tube. The default value is 0.01 m.
Tube geometrical length. The default value is 10 m.
This parameter represents total equivalent length of all local resistances associated with the tube. You can account for the pressure loss caused by local resistances, such as bends, fittings, armature, inlet/outlet losses, and so on, by adding to the pipe geometrical length an aggregate equivalent length of all the local resistances. The default value is 0.
Roughness height on the tube internal surface. The parameter is typically provided in data sheets or manufacturer catalogs. The default value is 1.5e-5 m, which corresponds to drawn tubing.
Specifies the Reynolds number at which the laminar flow regime is assumed to start converting into turbulent flow. Mathematically, this value is the maximum Reynolds number at fully developed laminar flow. The default value is 2000.
Specifies the Reynolds number at which the turbulent flow regime is assumed to be fully developed. Mathematically, this value is the minimum Reynolds number at turbulent flow. The default value is 4000.
Use the Variables tab to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.