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Models - Calculation of heat flow

Last modified by Omid Abouali on 2026/10/14 15:33

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Calculation of heat flow

Calculation of heat flow

The temperature fields are computed not just in the air within the tunnel system (tunnel sections, platforms and plenums) but also in the surrounding walls. Temperatures of trains are integrated taking the heat exchange with air into account. Effects such as heat from equipments, occupants and condensation/evaporation are included.

Tunnel section heat balance (RailSection, RoadSection)

The heat transport along the tunnel section is described by the discretised advection equation. The exchange with the tunnel wall is modeled with convection and radiation.

$\rho_il_iA_i\frac{dE_i}{dt}+\mathrm{\Delta}(m_{dot}E)_i=Q_{air_i}+Q_{cond_i}+Q_{veh_i}+Q_{equi_i}-Q_{evap_i}+Q_{fire_i}+Q_{trans_i}-Q_{pipe_i}$

where

E_i	air enthalpy ( = c_pAir·T_i ) in segment i (J/kg)	EntAir
ρ_i	density of air in segment i (kg/m³)	Rho
Δ(m_dot E)_i	(if m_{dot_i+}_1/2>0 then m_{dot_i}_+1/2E_i else m_{dot_i}_+1/2E_i₊₁ ) -(if m_{dot_i-}_1/2>0 then m_{dot_i}_-1/2E_i-₁ else m_{dot_i}_-1/2E_i )
m_{dot_i-}_1/2	air mass flow at node between segment i-1 and i (kg/s)

Q_{air_i}	=	$l_i\cdot(4\cdot A_i/d_i)\cdot U_i\cdot(T_{wall_i}-T_i)$	QAir
T_i	=	air temperature in segment i (˚C)	TAir
T_{wall_i}	=	wall temperature in segment i (˚C)	TWall
U_i	=	total heat transfer coefficient in segment i containing convection and radiation depending on temperature, air velocity, extinction coefficient and hydraulic diameter (see below) (W/ m²˚C)	UTot
Q_{cond_i}	=	heat from condensation of moisture in segment i (= 0 in IDA RTV) (W)
Q_{veh_i}	=	heat from cars/trains in segment i (see 2.2.7) (W)	Qtraffic
Q_{equi_i}	=	heat from equipment in segment i (W)
Q_{evap_i}	=	heat for evaporation of liquid water mist from equipment in segment i (= 0 in IDA RTV) (W)
Q_{trans_i}	=	= heat from transverse ventilation in segment i (see 2.1.2) (W)
m_{TrIn_i}	=	total supplied air mass flow into segment i (kg/s)	m_trin_tot
m_{TrOut_i}	=	total exhausted air mass flow from segment i (kg/s).	m_trout
T_{TrIn_i}	=	temperature of supplied airflow in segment i (˚C)	TWall
Q_{fire_i}	=	f_air · q_fire ·δ_i = heat from fire to air in segment i (W)
q_fire	=	fire power (see 2.2.6) (W)	FirPower
f_air	=	fraction of fire power directly to air (the rest goes to wall)	fire2air
δ_i	=	1 if fire in segment i, otherwise 0
l_i	=	length of tunnel segment i (m)	lTun
A_i	=	cross section area of tunnel segment i (m²).	aTun
Q_{pipe_i}		Q_airP · l heat to cooling pipes in segment i (see 2.2.9) (W)	QAirPl2r QAirPr2l

The heat transfer coefficient U_i depends on radiation and convection

$U=U_{rad}+U_{conv}$

with

$U_{rad}=em_{gas}\cdot\sigma\cdot(K_{air}^2+K_{wall}^2)\cdot(K_{air}+K_{wall})$

and

$U_{conv}=k_1\cdot m a x{(}h,U_{film})$

where

h	Nu_ ٠ *     k_air* / d = convective heat transfer coefficient (W/ m²˚C)
U_film	heat transfer coefficient at low air velocities computed as a function of the temperature difference between air and wall. From IDA Indoor Climate and Energy 3.0 (W/ m²˚C)
k₁	factor (e.g. for surface enlargement)	k1
Nu	$0.7\cdot(\lambda_T/8)\cdot Re/(1+1.592\cdot(15.217\cdot(\lambda_T/4)\cdot Re^{1/5}-1)/Re^{1/8})$ Nusselt number (based on work by Nunner 1956)	Nusselt
Re	\|v_air\|٠ρ_air٠d / μ_air = Reynolds number	Reynold
Pr	c_pAir٠μ_air / k_air = Prandtl number
_λ T	Darcy friction factor for tunnel wall
c_pAir	specific heat of air (J/kg˚C)	CP_AIR
k_air(T)	thermal conductivity in air. Approximated by a 3-degree polynomial in T_air fitted to table data in “Fundamentals of Heat and Mass Transfer” (W/ m˚C)
μ_air(T)	dynamic viscosity in air. Approximated by a 3-degree polynomial in T_air fitted to table data in “Fundamentals of Heat and Mass Transfer” (Pa s)
K_air	(T_air+273.15) = absolute temperature of air (K)
K_wall	absolute temperature of wall (K)
em_gas	1 – e^-ext٠d = gas emission	GasEmiss
ext	extinction coefficient (m^-1)	extinc
σ	5.6697٠10^-8 = constant of Stefan-Boltzmann (W/ m²K⁴)
d	hydraulic diameter (m)	dTun
v_air	air velocity (m/s).	VAir

Platform (Platform)

The air in the platform is assumed to be well mixed. The temperature is computed using

$c_{pAir}V\frac{d(\rho_{dry}T)}{dt}=\sum_{j} Q_j+Q_{air}-Q_{evap}+Q_{occ}+\sum_{j} Q_{equi_j}+Q_{train}\cdot(1-f_{TES})+Q_{fire}$

where

T	temperature on platform (˚C)	TAir
c_pAir	specific heat of air (J/kg˚C)	CP_AIR
ρ_dry	density of dry air = ρ ٠(1 + w)^-1 (kg/m³)	RhoDry
ρ	density of air (kg/ m³)	Rho
w	humidity ratio	xHum
V	volume of platform (m³)	VEncl
Q_j	heat flow from tunnel/leak j into platform (W)	Q [kW]
Q_air	$A_{wall}\cdot U\cdot(T_{wall}-T)+\sum Q_{win_j}$ = heat flow from wall and windows into platform (W)	QAir [kW]
U	U_rad + U_conv = total heat transfer coefficient. U_rad and U_conv are computed as in sections (W/ m²˚C)	Utot
A_wall	wall area (= 2 · length · (width+height)) (m²)	Awall
T_wall	wall temperature (˚C)	Twall
Q_{win_j}	$k_1U_{film_j}\cdot A_{win_j}\cdot(T_{win_j}-T)$ = heat from window j (W)	win[j].QAir
A_{win_j}	area of window j (m²)	win[j].area
k₁	factor	k1
U_{film_j}	heat transfer coefficient at low air velocities computed as a function of the temperature difference between air and window From IDA Indoor Climate and Energy 3.0 (W/ m²˚C)
T_{win_j}	temperature of window j (˚C)	win[j].TGlass
Q_evap	heat for evaporation of water emission from equipment (W)
Q_occ	s_occ ·N_occ ·1.8 ·(f_cl ·h_cl ·(T_cl – T) + 0.0014 ·58 ·MET·(34 - T)) = heat from occupants (W), from (ISO / DIS 7730)	QOcc2air [kW]
N_occ	maximum number of persons on platform	nOcc
s_occ	occupancy coefficient (schedule)	schedOcc
T_cl	$35.7-0.028\cdot MET\cdot58-0.155\cdot CLO\cdot f_{cl}\cdot(3.96\cdot10^{-8}\cdot((T_{cl}+273.15)^4-(T_{wall}+273.15)^4)+(h_{cl}\cdot(T_{cl}-T_{wall}))$ = temperature of clothes (˚C), from (ISO / DIS 7730)	TCl
f_cl	1 + 1.290·(0.155·CLO) if CLO < 0.500125 else 1 + 0.645·(0.155·CLO) = area increase factor, from (ISO / DIS 7730)	fCl
h_cl	max(2.38·max(0.0001, \| T_cl - T \|)^0.25 , 12.1·v^0.5 ) = heat transfer coefficient clothes to air (W/ m²˚C), from (ISO / DIS 7730)	hCl
MET	activity, 1 Met = 58 W/m² (Met), from (ISO / DIS 7730)	MET
CLO	heat resistance of clothes, 1 clo = 0.155 m²˚C/W, from (ISO / DIS 7730)	CLO
v	(Σmax(0, m_{dot_j} )+ max(0, m_{dot_l}))/(·A) = mean air velocity (m/s)	AirVel
m_{dot_j}	mass flow from tunnel j (kg/s)	tunnel.m_dot
m_{dot_l}	mass flow from leak (kg/s)	leak.m_dot
A	cross section area of platform (m²)	VEncl
Q_{equi_j}	heat from equipment j (W)	equip.Q2air [kW]
Q_train	heat from trains (see 2.2.7, Qtrain) (W)	Qtraff [kW]
Q_fire	f_air · q_fire = heat from fire to air in platform (W)
q_fire	fire power (see 2.2.6) (W)	FirPow [MW]
f_air	fraction of firepower directly to air (the rest is radiated to the wall) (default = 0.7).
f_TES	fraction heat removed from train by trackway exhaust system, dependent of train speed through user given parameters (default = 0).	fracTES

Plenum (Plenum)

The air in the plenum is assumed to be well mixed. The temperature is computed using

$c_{pAir}V\frac{d(\rho_{dry}T)}{dt}=\sum_{j} Q_j+Q_{air}+\sum_{j} Q_{equi_j}+Q_{fire}$

where

T	temperature in plenum (˚C)	TAir
c_pAir	specific heat of air (J/kg˚C)	CP_AIR
ρ_dry	density of dry air =  ٠(1 + w)^-1 (kg/m³)	RhoDry
ρ	density of air (kg/ m³)	Rho
w	humidity ratio (= 0 in IDA RTV)	xHum
V	volume of Plenum (m³)	VEncl
Q_j	heat flow from tunnel j into Plenum (W)	Q [kW]
Q_air	$A_{wall}\cdot U\cdot(T_{wall}-T)$ = heat flow from wall into Plenum (W)	QAir [kW]
U	U_rad + U_film = total heat transfer coefficient. U_rad and U_film are computed as in sections (W/ m²˚C)	Utot
A_wall	wall area (= 2 · length · (width+height)) (m²)	Awall
T_wall	wall temperature (˚C)	Twall
m_{dot_j}	mass flow from tunnel j (kg/s)	tunnel.m_dot
A	cross section area of Plenum (m²)	VEncl
Q_{equi_j}	heat from equipment j (W)	equip.Q2air [kW]
Q_fire	f_air · q_fire = heat from fire to air in Plenum (W)
q_fire	fire power (see 2.2.6) (W)	FirPow [MW]
f_air	fraction of firepower directly to air (the rest is radiated to the wall) (default = 0.7).

Wall

When computing the temperature in the ground around sections/platforms/plenums not only the temperature of the air within the sections/platforms/plenums can be taken into account but also the ambient temperature and in a special case the temperature in neighboring sections/platforms/plenums. To achieve this without too much additional computation we use superposition of temperature fields. The cases we consider are:

One tunnel deep below ground surface (only case available in IDA Road Tunnel)
Two equivalent parallel tunnels deep below ground surface
One tunnel close to ground surface
Two equivalent parallel tunnels close to ground surface
One tunnel close to boundary with known temperature
One vertical tunnel close to ground surface
Two equivalent parallel vertical tunnels close to ground surface.

These cases require at most computation of three one-dimensional temperature fields, if we assume

1. The cross sections of sections/platforms can be approximated as being circular
2. Heat conduction in ground parallel to section/platform can be neglected in the temperature field caused by conditions in the section/platform
3. The thermal properties are homogeneous outside a thin region close to section/platform (not necessary in cases 1 and 5) where the properties can vary with radial coordinate. The distances to the neighbor section/platform and ground surface are large, so that the gradients of the temperature fields caused by these are small, i.e. within the thin layer the total temperature is well approximated by radial depth (and time) dependence.
4. The ground surface above each segment/platform is approximated by a horizontal plane. In simulations containing inclined tunnels be careful to get all tunnel parts defined as close to ground surface (cases 3, 4, 6, 7) beneath the ground surface.

The undisturbed ground temperature is computed using the one-dimensional heat equation with Dirichlet boundary conditions at ground surface and at a depth, such that variations at the ground level are handled in a correct way.

$\rho_{gr}c_{pgr}\frac{\partial T_z}{\partial t}=\frac{\partial}{\partial z}(k_{gr}\frac{\partial T_z}{\partial z})$

with boundary conditions

$T_z=T_{amb}(t)$ at ground surface z = 0

$T_z=\overline{T_{amb}}$ at bottom of computational domain

where

T_z	undisturbed temperature in ground (˚C)
ρ _gr	density in ground (kg/m³)
c_pgr	specific heat in ground (J/kg˚C)
k_gr	thermal conductivity in ground (W/m˚C).
$\overline{T_{amb}}$	mean of T_amb (˚C)	TWall0

Geothermal effects are modelled by a constant vertical temperature gradient in cases 3, 4, 6 and 7.

where

T_g	temperature in ground due to geothermal effects (equal to 0 at ground surface) (˚C)
gradT	geothermal temperature gradient (˚C/m)	geoTGrad
z	vertical coordinate starting from ground surface (m).

The temperature field caused by conditions in the section/platform is computed using the one-dimensional conduction-advection equation in cylindrical coordinates. This is done around each segment in sections and around each platform. The thermal properties are allowed to vary radially by using layers with different properties. The size of the computational domain in radial direction is estimated by looking at the 1-dimensional heat equation with constant properties. To achieve ‘equal’ properties in all layers the thickness of each layer is adjusted by dimensional analysis. Given a typical time scale one can from the analytical solution of the time dependent heat equation decide where to set the outer boundary condition (Dirichlet or noflow B.C.) with an error smaller than a given tolerance as compared to an infinite domain.

$2\pi r\rho_{wall}(r)c_{pWall}(r)\frac{\partial T_r}{\partial t}=2\pi\frac{\partial}{\partial r}(k_{wall}(r)r\frac{\partial T_r}{\partial r})+m_{seep}c_{pWat}\frac{\partial T_r}{\partial r}$

with boundary conditions

$-2\pi k_{wall}(r_0)r_0\frac{\partial T_r}{\partial r}=\alpha(T_{air}-T_{wall})+Q_{wall}/l$ at wall surface r₀

$\frac{\partial T_r}{\partial r}=0$ or T_r=0 at outer boundary of computational domain

where

T_r	temperature change in wall caused by conditions in segment/platform/plenum (˚C)
T_wall	temperature in wall surface (˚C)
T_wall(r)	density in wall (kg/m³)
c_pWall(r)	specific heat in wall (J/kg˚C)
c_pWat	specific heat of water (J/kg˚C)	CP_WAT
m_seep	water seepage/meter out of wall (= 0 in IDA RTV) (kg/sm)	mSeep
k_wall(r)	thermal conductivity in wall (W/m˚C)
α	U· A_wall / l = heat transfer coefficient between wall and segment/platform/plenum (W/m˚C)	Alfa
U	total heat transfer coefficient containing convection and radiation depending on temperature, extinction coefficient and hydraulic diameter (see 2.2.1-3) (W/ m²˚C)	UTot
A_wall	wall area of segment/platform/plenum (m²)	aTun/AWall
l	length of segment/platform/plenum (m)	lTun/length
r₀	half the hydraulic diameter (m)	rPlat
T_air	air temperature in segment/platform/plenum (˚C)	TAir

Q_wall = Q_fire + Q_equip + Q_evap + Q_occ - Q_pipe + Q_window (W)

Q_fire	(1-f_air )· q_fire = heat from fire to wall in segment/platform/plenum (W)
q_fire	fire power (see 2.2.6) (W)
Q_equip	s_equip · q_equipNom = radiation heat from equipment to wall (W)	Q2wall [kW]
s_equip	schedule
q_equipNom	nominal radiation load (W)	Q2wallNom [kW]
Q_evap	HF_vap · m_cond · l = heat from air to wall through condensation/evaporation (= 0 in IDA RTV) (W)
m_cond	humidity flow between air and wall (= 0 in IDA RTV) (see 2.4) (kg/sm)	Humf
HF_vap	2.501·10⁶ = water vaporization heat at 0 ˚C (J/kg)	HF_VAP
Q_occ	s_occ ·N_occ·1.8·(3.96E-8 ·f_cl ·((T_cl + 273.15)⁴ - (T + 273.15)⁴))= radiation from occupants to wall in platform (Q_occ = 0 in sections and plenums) (W)	QOcc2wall [kW]
N_occ	maximum number of persons on platform	nOcc
s_occ	occupancy coefficient (schedule)	schedOcc
T_cl	temperature of clothes (see 2.2.2) (˚C)	TCl
f_cl	area increase factor	fCl
f_air	fraction of firepower directly to air (the rest goes to the wall).
Q_pipe	Q_wallP · l in sections, otherwise 0 heat to cooling pipes (see 2.2.9) (W)	QWallPl2r QWallPr2l
Q_window	$\sum{Q_{dir_j}+Q_{diff_j}+Q_{back_j}}+Q_{win_j}$ = heat radiation through and from windows to wall (platform only) (W)
Q_{dir_j}	direct inward radiation through window j (platform only) (W)	winRad[j]. Q_dir
Q_{diff_j}	diffuse inward radiation through window j (platform only) (W)	winRad[j]. Q_diff
Q_{back_j}	- outward radiation through window j (platform only). Computed from Q_dir and Q_diff (W)	winRad[j]. Q_back
Q_{win_j}	$5\cdot A_{win_j}\cdot(T_{win_j}-T_{wall})$ = long wave radiation from window j (platform only) (W)	win[j].Qlw
A_{win_j}	area of window j (m²)	win[j].area
T_{win_j}	temperature of window j (˚C).	win[j].TGlass

If there are homogeneous material properties in the ground then the surface temperature T_wall of the segment/platform is e.g. in case 1

$T_{wall}=T_r(r_0)+T_\infty$ TGround

and in case 4

$T_{wall}=T_r(r_0)+T_z(z_b)+T_g(z_b)+T_r(r_1)-T_r(2\cdot z_b)-T_r(\sqrt{r_1^2+4\cdot z_b^2})$ TGround

where

r₀	radius of cross section in segments/platforms (m)
r₁	distance between centers of segments/platforms (m)	SupR(1)
z_b	distance from ground surface to center of segment/platform (m).

The partial differential equations are discretised using finite difference methods in space. The discretisation is governed by a grid factor (quotient between consecutive cell lengths) to yield finer grid at segment/platform and ground surfaces. The systems of ordinary differential equations to solve are of the form

$\mathbit{D}\cdot•T=K⋅T+q$

and with the Backward Euler method in time we get

$\mathbit{D}\cdot(\mathbit{T}^n-\mathbit{T}^{n-1})/\mathrm{\Delta}t_n=\mathbit{K}\cdot\mathbit{T}^n+\mathbit{q}^n$

$(\mathbit{D}/\mathrm{\Delta}t_n-\mathbit{K})\cdot\mathbit{T}^n=\mathbit{D}/\mathrm{\Delta}t_n\cdot\mathbit{T}^{n-1}+\mathbit{q}^n$

where

Tⁿ	vector of temperatures (T_r or T_z) in nodes at time t_n (˚C)	TGroundz, TGroundr
D	diagonal matrix of heat capacities
K	tridiagonal matrix, discretised conduction/advection operators
Δt_n	t_n - t_n-1 = time step (sec)
qⁿ	vector of sources at time t_n.

In cases where there is a thin layer at the segment/platform wall with other material properties than in the surrounding ground the above system has to be modified. Partitioning the system into 3, where partition 1 contains the thin layer nodes, 2 the first node outside the thin layer and 3 the rest of the nodes in the homogeneous region we get

$\begin{bmatrix} \mathbf{D}_{1} & & \\ & \mathbf{D}_{2} & \\ & & \mathbf{D}_{3} \\ \end{bmatrix} \cdot \overset{\bullet}{\begin{bmatrix} \mathbf{T}_{r1} \\ \mathbf{T}_{r2} \\ \mathbf{T}_{r3} \\ \end{bmatrix}} = \begin{bmatrix} \mathbf{K}_{11} & \mathbf{K}_{12} & \\ \mathbf{K}_{21} & \mathbf{K}_{22} & \mathbf{K}_{23} \\ & \mathbf{K}_{32} & \mathbf{K}_{33} \\ \end{bmatrix} \cdot \begin{bmatrix} \mathbf{T}_{r1} \\ \mathbf{T}_{r2} \\ \mathbf{T}_{r3} \\ \end{bmatrix} + \begin{bmatrix} \mathbf{q}_{1} \\ \mathbf{q}_{2} \\ \mathbf{q}_{3} \\ \end{bmatrix}\]$ .

According to our assumptions the temperature fields caused by the undisturbed ground, geothermal gradient, neighbour segment/platform and their images add up to a contribution, which can be considered to have negligible gradients in the thin layer and its vicinity. In case 1 this contribution is

$T_{sum}=T_\infty$

and in case 4

$T_{sum}=T_z(z_b)+T_g(z_b)+T_r(r_1)-T_r(2\cdot z_b)-T_r(\sqrt{r_1^2+4\cdot z_b^2})$ .

Thus the total field is here given by

$T_{tot}=T_r+T_{sum}$

and we have to solve for the total field in the thin layer and for the radial field outside in the homogeneous region, i.e. the vector [ T_tot1, T_r2, T_r3]^T . This is achieved by integrating the modified system

$\begin{bmatrix} \mathbf{D}_{1} & & \\ & \mathbf{D}_{2} & \\ & & \mathbf{D}_{3} \\ \end{bmatrix} \cdot \overset{\bullet}{\begin{bmatrix} \mathbf{T}_{tot1} \\ \mathbf{T}_{r2} \\ \mathbf{T}_{r3} \\ \end{bmatrix}} = \begin{bmatrix} \mathbf{K}_{11} & \mathbf{K}_{12} & \\ \mathbf{K}_{21} & \mathbf{K}_{22} & \mathbf{K}_{23} \\ & \mathbf{K}_{32} & \mathbf{K}_{33} \\ \end{bmatrix} \cdot \begin{bmatrix} \mathbf{T}_{tot1} \\ \mathbf{T}_{r2} \\ \mathbf{T}_{r3} \\ \end{bmatrix} + \begin{bmatrix} \mathbf{q}_{1} \\ \mathbf{q}_{2} \\ \mathbf{q}_{3} \\ \end{bmatrix} + \begin{bmatrix} & \mathbf{K}_{12} & \\ - \mathbf{K}_{21} & & \\ & & \\ \end{bmatrix} \cdot \begin{bmatrix} T_{sum} \cdot \mathbf{1}_{1} \\ T_{sum} \cdot \mathbf{1}_{2} \\ 0 \\ \end{bmatrix}$

This turns out to be a minor modification of the original system since the ‘conductivity’ matrix K is triangular. One term is added in the equation at outermost node in the thin layer region and one is subtracted in the next equation.

Heater (Heater)

The heater tries to increase temperature of air flowing in specified direction (forward or reverse) to desired T_set with available power Q_max.

$10\cdot\frac{d\mathrm{\Delta T}}{dt}+\mathrm{\Delta T}=\frac{Q_{max}}{max{(}{Qreq}_{max}\cdot\mathrm{\Delta}T_{req}}$ if Q_req > 0

= 0 otherwise

$\mathrm{\Delta}T_{act}=min{(}\mathrm{\Delta T},\frac{Q_{max}}{max{(}{Qreq}_{max}\cdot\mathrm{\Delta}T_{req})}$ if Q_req > 0

= 0 otherwise

T_outlet = T_inlet + ΔT_act

where

ΔT	integrated temperature rise (˚C)	dT
ΔT_req	T_set - T_inlet (˚C)
ΔT_act	actual temperature rise (˚C)	dTActual
T_set	desired temperature at outlet (˚C)	TSet
T_inlet	inlet temperature (˚C)	inlet.T
T_outlet	outlet temperature (˚C)	outlet.T
Q_max	available power (W)	QMax
Q_req	m_dot·c_pAir·ΔT_req = required power (W)	QRequired
Q_act	m_dot·c_pAir·ΔT_act = actual power (W).	QActual

Fire model

Fires can be specified in sections, platforms and plenums. In a section the fire is limited to one segment. In a segment the developed firepower q_fire is limited by temperature and the available air. The air limitation is based on the fact that air contains 23.2 % by weight of oxygen and the energy developed per consumed kg oxygen is 13.1 MJ thus giving an approximate maximum firepower of 0.232·13.1· m_dot = 3.04·m_dot MW if 2-dimensional effects are omitted.

$q_{fire}=\ q_{sup}f_{limT}f_{limAir}\ f_{red}\ {cq}_1\$

where

q_sup	q_sup(t) = supplied maximum fire power (W)	FirPow [MW]
f_limT	if T_i <1500 1 else if T_i >2000 0.1 else 0.1 + 0.9(1+ cos(π (T_i – 1500)/500))/2 = factor in interval [0.1 : 1] limiting power for high temperatures
f_limAir	if \|3.04m_dot\| > q_sup 1 else 1 - f_red (1+ cos(π 3.04 m_dot/ q_sup ))/2 = factor in interval [1- f_red : 1] limiting power due to oxygen supply
f_red	reduction factor (default = 0.9)	redF
cq₁	factor used for numerical reasons, integrated by a 1^st order ODE with time constant τ_fire from zero to one if q_sup > 0 in a segment and back to zero when q_sup = 0.	CQ1

In a platform/plenum the firepower is

$q_{fire}=q_{sup}$

The firepower q_fire is distributed to the air and the wall according to

f_air · q_fire to air and

(1-f_air ) · q_fire to wall.

Train heat model

The trains traveling in the tunnel system are exchanging heat with the air. Each train is modeled with thermal masses for the body and the brakes with direct heat transfer to air, exchanging heat through braking and regeneration losses, motor losses, friction losses and on-board heat sources.

$C_{body}\frac{dT_{body}}{dt}+Q_{body}=0$

$Q_{body}=H_{body}\cdot A_{body}\cdot(T_{body}-T_{air})$

where

T_body	temperature of vehicle body (˚C)	TTrain
T_air	air temperature (˚C)	Tair
Q_body	heat transfer from vehicle to air (W)	Qbody [kW]
C_body	heat capacity of vehicle (J/˚C)	TrainHeatCap
H_body	max(5.0, (if T_body > T_air then 3.62 else 3.75) ٠\|v_rel\|^0.8) = heat transfer coefficient, based on the Dittus-Boelter equation (“Fundamentals of Heat and Mass Transfer”) (W/ m²˚C)	Htrain
A_body	area of vehicle body (m²)	TrainArea
v_rel	air velocity relative vehicle body (m/s).

$C_{brake}\frac{dT_{brake}}{dt}+Q_{brake}=q_{brake}\cdot(1-f_{regen})$

$Q_{brake}=H_{brake}\cdot A_{brake}\cdot(T_{brake}-T_{air})$

where

T_brake	temperature of vehicle brakes (˚C)	Tbrake
Q_brake	heat transfer from vehicle brakes to air (W)	Qbrake [kW]
C_brake	heat capacity of vehicle brakes (J/˚C)	BrakeHeatCap
H_brake	U_rad + U_conv + h_conv٠v_veh = total heat transfer coefficient (W/ m²˚C)	Htotbrake
U_conv	convective heat transfer coefficient (W/ m²˚C)	Hbrake
h_conv	convective heat transfer coefficient per m/s train speed (W/ m²˚C)	Hbrake1
U_rad	$em\cdot\sigma\cdot(K_{air}^2+K_{brake}^2)\cdot(K_{air}+K_{brake})$ = radiative heat transfer coefficient, K_brake and K_air Kelvin temperatures (W/ m²˚C)	Ubrake
em	emission coefficient	Emiss
σ	5.6697٠10^-8 = constant of Stefan-Boltzmann (W/ m²K⁴)
A_brake	area of vehicle brakes (m²)	BrakeArea
q_brake	F_brake · v_veh = braking power of vehicle (W)	BrakeP [kW]
F_brake	F_tract - F_loc = braking force (N)	BrakeF [kN]
f_regen	quotient regenerative braking power to actual braking power.	RegenRat
F_tract	max(0, F_loc) = tractive force (N)	TractF [kN]
F_loc	F_drag+ F_roll + M_veh· g · sin(f_slope) + km_veh· M_veh · a_veh = locomotive force (N)	LocF [kN]
F_drag	aerodynamic drag force (see 2.1.1) (N)	DragF [kN]
F_roll	(A_veh+ B_veh· v_veh) · M_veh = rolling resistance force (N)	RollF [kN]
v_veh	velocity of vehicle (m/s)	x_dot
a_veh	acceleration of vehicle (m/s²)	x_dot_dot
g	9.80665 = acceleration of gravity (m/s²)
km_veh	kinetic energy coefficient	Km
f_slope	mean slope of tunnel	slope
M_veh	train total mass (incl. passengers) (kg)	totalMass
A_veh	coefficient (m/ s²)	A
B_veh	coefficient (s^-1).	B

The total heat from a train to the air in tunnel sections and platforms is

$Q_{train}=Q_{body}+Q_{brake}+Q_{loss}+Q_{frict}+Q_{clim}+Q_{condenser}+Q_{dieselHeat}+Q_{equip}$

where

Q_train	heat from train to air (W)	Qtrain2air [kW]
Q_loss	- τ_loss·dQ_loss/dt + Q_regen + Q_motor = time delayed regeneration and motor losses (W)	QLoss [kW]
τ_loss	time constant regeneration and motor loss equation (s)	tauLoss [h]
Q_regen	q_brake· f_regen· (1- ζ_regen) = regeneration losses (W)
ζ_regen	regeneration efficiency, braking power to power supply	EtaRegen
Q_motor	q_tract· (1/ζ_tract - 1) = motor losses of vehicle (W)
q_tract	F_tract · v_veh = tractive power of vehicle (W)	TractP [kW]
ζ_tract	traction system efficiency, power supply to propulsion power	EtaTract
Q_frict	(F_drag+ F_roll) · v_veh = friction losses of vehicle (W)
Q_clim	heat flux from interior train (see 2.2.8) (W)	QClim [kW]
Q_condenser	condenser heat from interior train (see 2.2.8) (W)	QCondenser [kW]
Q_dieselHeat	max(Q_dieselIdle, Q_diesel · (1 - ζ_diesel)) = heat emitted by diesel operation (W)	QDieselHeat [kW]
Q_dieselIdle	diesel idling emitted heat (W)	QDieselIdle [kW]
Q_diesel	Q_dieselSup / ζ_diesel = total diesel power (lower heating value) (W)	QDiesel [kW]
ζ_diesel	diesel to electric efficiency	EtaDiesel
Q_dieselSup	q_tract/ ζ_tract + Q_source + Q_supCool + Q_heat = heat emitted by diesel operation (W)	QDieselSup [kW]
Q_source	interior sources on vehicle (see 2.2.8) (W)	Qsource [kW]
Q_supCool	AC electric power (see 2.2.8) (W)	QSupCool [kW]
Q_heat	heating power (see 2.2.8) (W)	QHeat [kW]
Q_equip	heat from equipment (W).	equip.Q [kW]

Train compartment heating and air conditioning

The climate within trains is modeled taking occupants, internal heating/cooling sources and ventilation to the surrounding into account.

The interaction between occupants and the surrounding regarding heat and moisture is based on the three equations from (ISO / DIS 7730):

$f_{cl}=\begin{matrix}1.00+1.290\cdot0.155\cdot C L O&\\1.05+0.645\cdot0.155\cdot C L O&\\\end{matrix}\begin{matrix}CLO<0.500125\\CLO\geq0.500125\\\end{matrix}$

$h_{cl}=2.38\cdot m a x{(}0.001,\left|T_{cl}-T_{comp}\right|)^{0.25}$

$T_{cl}=35.7-0.028\cdot MET\cdot58-0.155\cdot CLO\cdot f_{cl}.\ (3.96.{10}^{-8}.((T_{cl}+273.15)^4-(T_{comp}+273.15)^4)+(h_{cl}.(T_{cl}-T_{comp}))$

and the following for heat exchange

$Q_{occCon}=s_{occ}\cdot N_{occ}\cdot1.8\cdot(f_{cl}\cdot h_{cl}\cdot(T_{cl}-T_{comp})+0.0014\cdot MET\cdot58\cdot(34-T_{comp}))$

$Q_{occRad}=s_{occ}\cdot N_{occ}\cdot1.8\cdot3.96\cdot10^{-8}\cdot f_{cl}\cdot((T_{cl}+273.15)^4-(T_{comp}+273.15)^4)$

$Q_{occVap}=s_{occ}\cdot N_{occ}\cdot1.8\cdot(3.05\cdot(5733-6.99\cdot MET\cdot58-p_{vap})+1.7.{10}^{-5}MET\cdot\ 58.(5867-p_{vap}))$

$Q_{occMet}=s_{occ}\cdot N_{occ}\cdot1.8\cdot MET\cdot58$

$Q_{occ}=Q_{occCon}+Q_{occRad}$

$Q_{sweat}=max{(}0,Q_{occMet}-(Q_{occ}+Q_{occVap}))$

and moist emission

$q_{occVapE}=(Q_{occVap}+Q_{sweat})/2430$

where

f_cl	area increase factor	fCl
h_cl	heat transfer coefficient clothes to air (W/ m²˚C)	hCl
T_cl	temperature of clothes (˚C)	TCl
T_comp	train compartment temperature (˚C)	TComp
N_occ	maximum number of persons on train	nOcc
s_occ	occupancy coefficient (schedule)	schedOcc
MET	activity, 1 Met = 58 W/m² (Met)	MET
CLO	heat resistance of clothes, 1 clo = 0.155 m²˚C/W	CLO
p_vap	p_amb· w/( f_mol + w) = partial vapour pressure (Pa)	PVap
w	humidity ratio of air in tunnel segment	xHum
f_mol	0.62198 = ratio of molecular masses of water and air
Q_occCon	convective heat transfer from occupants (W)	QOccCon [kW]
Q_occRad	radiation from occupants (W)	QOccRad [kW]
Q_occVap	evaporation power (W)	QOccVap [kW]
Q_occMet	power according to MET (W)	QOccMet [kW]
Q_occ	heat flux from occupants (W)	QOcc [kW]
Q_sweat	remaining power emitted as sweat (W)	QSweat [kW]
q_occVapE	vapour emission ventilated to tunnel (kg/s).	OccVapEmission

The cooling and heating are computed according to

T_comp		=		if Q_occ + Q_source – U_tot·(T_high – T_air) > 0 then	T_high
				elseif Q_occ + Q_source – U_tot·(T_low– T_air) < 0 then	T_low
				else	(Q_occ + Q_source) / U_tot + T_air

Q_cool	=		if T_comp = T_high then Q_occ + Q_source – U_tot·(T_high – T_air) else			0

Q_heat	=	if T_comp = T_low then U_tot·(T_low– T_air) – Q_occ – Q_source else 0

Q_clim	=	U_tot·(T_comp– T_air)

Q_supCool	=	Q_cool /COP

Q_condenser	=	(1 + COP)·Q_cool

where

Q_source	interior sources on vehicle (W)	Qsource [kW]
Q_cool	cooling power (W)	QCool [kW]
Q_heat	heating power (added to train power consumption) (W)	QHeat [kW]
Q_clim	heat flux from interior train to tunnel/platform air (W)	QClim [kW]
Q_supCool	AC electric power (added to train power consumption) (W)	QSupCool [kW]
Q_condenser	condenser heat to tunnel/platform air (W)	QCondenser [kW]
T_high	compartment cooling setpoint (˚C)	THi
T_low	compartment heating setpoint (˚C)	TLo
T_air	air temperature (˚C)	Tair
U_tot	U_wall· A_body + s_occ· N_occ· v_occ · 1.2 · c_pAir = total U-value including passenger ventilation (W/K)	Utot [kW/K]
U_wall	heat transfer coefficient indoor to ambient air (W/ m²˚C)	UWall
A_body	area of vehicle body (m²)	TrainArea
s_occ	occupancy coefficient (schedule)	schedOcc
N_occ	maximum number of persons on train	nOcc
v_occ	ventilation airflow per passenger (m³/s)	vOcc [l/s]
c_pAir	specific heat of air (J/kg˚C)	CP_AIR
COP	AC unit coefficient of performance .	COP

Cooling pipes

In sections cooling pipes can be inserted and connected to form cooling systems in parts of the tunnel system. Heat is exchanged with air and tunnel wall. In the present version pressure drops in the pipe systems are omitted and the (water) mass-flow is explicitly given together with temperature at the inlet. The temperature along a pipe system is computed by applying a discretisation to the equation

$\pi\frac{d^2}{4}\rho_{wl}c_{pW}\frac{\partial T_w}{\partial t}+m_{dot_w}c_{pW}\frac{\partial T_w}{\partial x}=Q_{airP}+Q_{wallP}$

with boundary conditions

$T(x_0,t)=T_{inlet}(t)$ at inlet x₀

$m_w(t)=m_{inlet}(t)$ at inlet x₀

where

T_w	water temperature (˚C)	TpipeL2R, TpipeR2L
T_air	air temperature (˚C)	TAir
T_wall	wall temperature (˚C)	TWall
ρ_w	density of water (kg/m³)	Rho_WAT
c_pW	specific heat of water (J/kg˚C)	CP_WAT
m_{dot_w}	water mass flow (kg/s)
Q_wallP	α_wall (T_wall - T_w) heat per length from tunnel wall to pipe (W/m)
Q_airP	α_air (T_air - T_w) heat per length from air to pipe (W/m)
α_wall	heat transfer coefficient per length between wall and pipe (W/m˚C)	AlfaW
α_air	α_air1 + α_air0 · \|v_air\| heat transfer coefficient per length between air and pipe (W/m˚C)
α_air1	constant part of heat transfer coefficient per length between air and pipe (W/m˚C)	AlfaA
α_air0	coefficient in heat transfer coefficient multiplying air velocity (Ws/m²˚C)	Alfa0
d	inner diameter of pipe (m)	DPipe
v_air	air velocity (m/s).	VAir