# Piston effect in underground networks

The symbols used in this page are explained in a nomenclature at the end of the page.

Knowledge of air flows and overpressure in underground stations is necessary to assess user comfort and safety. As part of the work carried out on air quality in underground stations, we are trying to estimate the order of magnitude of the quantity of air displaced at the station platforms following the arrival and departure of a train

# 1. Concept of pressure losses

When a fluid is moving in a conduit, part of its energy is dissipated by friction corresponding to a pressure drop. Two types of pressure drop can be considered:

• Linear pressure drops. The variation on pressure can be modeled with Darcy's law :

$$\Delta P =\frac{\rho\lambda Lv^2}{2D_h}$$

Where $P$ is the pressure, $v$ the fluid velocity, $L$ is the le duct length, $D_h$ is the hydraulic diameter and $\lambda$ a linear pressure drop coefficient.

• The singular pressure losses that are modelled by means of a coefficient $\xi$  that can be estimated, for example, by using abacus (Idelchik tables).

The variation in pressure caused by the passage of a singularity can be written as:

$$\Delta P_\text{tot} =\frac{\rho v^2}{2 }\left(\frac{ \lambda L }{D_h}+\xi\right)$$

# 2. Equation of the phenomenon

The calculation of the piston effect in an underground station requires modelling a network of tunnels in which a train will move along a linear path and mark a number of stops. It is therefore a question of modelling transient flows in 1D, driven by the pressure drop induced by the movement of the train.

The equations used to establish this simplified model are based on the principles of conservation of mass and quantity of motion (recalled here) from which it is possible to compute the speed of the air put in motion by the train.

For this purpose, it is necessary to apply the generalized Bernouilli theorem to a fluid control volume surrounding the train.

This yields the equation governing the evolution of the air flow in a tunnel:

$$\rho C \frac{\mathrm{d}Q_s}{\mathrm{d}t} =\underbrace{\Delta P_\text{sing}+\Delta P_\text{lin}+\Delta P_\text{fan}+\Delta P_\text{piston}}_{w(Q_s)}+\Delta P_\text{ext}$$

Where $\Delta P_\text{sing}$ corresponds to the pressure difference generated at the level of a singularity, $\Delta P_\text{lin}$ corresponds to the pressure difference induced by friction in a constant cross-section, $\Delta P_\text{fan}$ corresponds to the pressure difference due to mechanical ventilation, $\Delta P_\text{piston}$ corresponds to the pressure difference caused by the piston effect due to the movement of the train and $\Delta P_\text{ext}$ is the pressure difference between the two ends of the studied duct segment.

$C$ is the inertial term for the air column:

• In the absence of a train: $C=\frac{L_t}{A_t}$
• In the presence of trains: $C=\frac{L_t-l_v}{A_t}+\frac{l_v}{A_t-a_v}$

By noting $L_t$ the length of the tunnel segment, $l_v$ that of the vehicle in the segment, $A_t$ the section of the tunnel segment and $a_v$ that of the vehicle in the segment.

The state of the art provides expressions of these different $\Delta P$  provided in the following table:

Without train With train
$\Delta P_\text{lin}$ $-\frac{\rho f_t L_t P_t |Q_S|Q_S}{8 A_t^3}$ $-\frac{\rho f_t L_t P_t |Q_S|Q_S}{8 A_t^3}$
$\Delta P_\text{sing}$ $-k\rho\frac{|Q_S|Q_S}{2 A_t^2}$ $-k\rho\frac{|Q_S|Q_S}{2 A_t^2}$
$\Delta P_\text{fan}$ $\rho\frac{ |Q_\text{fan}|\left(v_\text{fan}-\frac{Q_S}{A_t}\right)}{A_t}$ $\rho\frac{ |Q_\text{fan}|\left(v_\text{fan}-\frac{Q_S}{A_t}\right)}{A_t}$
$\Delta P_\text{piston}$ 0 $\frac{\rho (\textcolor{blue}{K_{vB}}+\textcolor{red}{K_{vF}})(A_t U_v-Q_S)^2}{2 A_t^2}+ \frac{\rho f_t l_v P_t |(a_v U_v-Q_S)|(a_v U_v-Q_S)}{8 (A_t-a_v)^3}+\frac{\rho \lambda_v l_v P_v |(a_t U_v-Q_S)|(a_t U_v-Q_S) }{8 (A_t-a_v)^3}+\frac{\rho a_v l_v}{A_t-a_v}\frac{dU_v}{dt}$
$\textcolor{blue}{+\rho \frac{Q_S^2}{2 A_t^2}-\rho\frac{(a_v U_v-Q_S)^2}{2 (A_t-a_v)^2}}\textcolor{red}{-\rho\frac{Q_S^2}{2 A_t^2}+\rho\frac{(a_v U_v-Q_S)^2}{2 (A_t-a_v)^2}}+\frac{\rho f_t l_v P_t |Q_S|Q_S}{8 A_t^3}$

Red terms must be taken into account when the front of the train is present in the segment studied. The blue terms must be introduced when the rear of the train present in the said segment.

Note: In steady state and in the absence of a train, the equation governing the piston effect can be reduced to the following equation:

$$\Delta P = Z Q_s^2$$

This equation is commonly used for the study of hydraulic networks using the "Z method" but given the importance of the transient effects produced by the passage of the train, this method is not applicable in this context.

# Network resolution

Since the equations of the phenomenon are now established for all the segments of the network, it is appropriate to reorder them in order to reach a system of equations of the form:

$$\rho\left[\underline{\underline{C}}\right]\cdot\left[\underline{\dot{Q_s}}\right]=\left[\underline{\omega}(Q_s)\right]$$

To do this, we use the knot and mesh methods (similar to those used in electrical engineering) illustrated below: The transient system can then be resolved by using a explicit resolution scheme.

# 4. Presentation of the SPES1D software

The SPES1D software (for Subway Piston Effect Simulation in 1D) was developed by our experts to solve the transitory problem of trains operating in tunnels and underground stations.

This software introduces three categories of segments to cover a wide range of cases:

• the duct-type element. This is the most basic element. It can be used both for tunnels, stations (if the train does not stop) and various ducts (junctions between two passenger buildings, underground passages,...).
• the station type element. This element is used to mark a stop. The stop is such that the centre of the element is coincident with the centre of the train when the train's speed is zero.
• the fan-type element with ventilation. This element allows for the effects of mechanical ventilation to be taken into account.

It is then possible to determine the evolution of all network variables during the train movement. Flow rate on the train route at the station entrance and direction of the associated flows

# 5. Validation

The software developed, SPES1D, was validated after comparison with the Sajben study (3). In his article, Sajben studies the piston effect of a train positioned in a ring and whose propulsive power is constant. In particular, he observes that in the configuration studied and under established conditions, when the train reaches 21 m/s, the air in the tunnel is displaced by 8.2 m/s. The same configuration was modelled with our SPES1D software. As a result, under established operating conditions and for a train moving at 21 m/s, the speed of the air displaced is also 8.2 m/s.

# 6. Nomenclature

Symbol Meaning
$A$ Section [m2]
$A_t$ Tunnel section [m2]
$a_v$ Train section [m2]
$C$ Inertial term of the air column
$D_H$ Hydraulic diameter[m]
$K_{vb}$ Coefficient of sudden pressure drop due to air expansion following the passage of the train
$K_{vf}$ Coefficient of sudden pressure drop due to air contraction as the train passes through
$P$ Pressure [Pa]
$\Delta P_{ext}$ Pressure difference between the ends of the segment [Pa]
$\Delta P_{sing}$ Pressure variation within a singularity [Pa]
$\Delta P_{lin}$ Pressure difference due to friction [Pa]
$\Delta P_{fan}$ Pressure difference induced by mechanical ventilation [Pa]
$\Delta P_{piston}$ Pressure difference induced by piston effect [Pa]
$f_t$ Regular pressure drop coefficient for the tunnel
$L$ Length [m]
$L_t$ Tunnel length [m]
$l_v$ Train length [m]
$\lambda$ Regular pressure drop coefficient
$\lambda_v$ Regular pressure drop coefficient at the train level
$Q$ Flow rate [kg/m3]
$Q_s$ Air flow rate due to piston effect [kg/m3]
$U_v$ Train speed [m/s]
$v$ Air velocity [m/s]
$\rho$ Air density [kg/m3]
$Z$ Hydraulic resistance
$\xi, k$ Singular pressure drop coefficients

# 7. References

1. Idelchik, I. E. (1986). Handbook of hydraulic resistance. Washington, DC, Hemisphere Publishing Corp., 1986, 662 p. Translation.
2. Part 1, User’s Manual. Subway Environmental Design Handbook Volume II: Subway Environment Simulation Computer Program, Version 4. US Department of Transportation, Research and Special Programs Administration, 1997.
3. Sajben, M. (1971). Fluid Mechanics of Train-Tunnel Systems in Unsteady Motion. AIAA Journal, 9(8), 1538-1545.
4. Prince, J. (2015). Coupled 1D-3D simulation of flow in subway transit networks.