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Thermal Simulation of a Metro System

J. Amaya, G. Poblete, R. Roman, J. A. Sanchez

Centro de Modelamiento Matematico, Universidad de Chile, Av. Blanco Encalada2120 Piso 7, Santiago de Chile

Abstract

This document contains the basic assumptions and equations to describe the thermalbalance model in a metro system. This model is based on a system of equations,arising from classic mechanics and designed to be a part of a more detailed energyefficiency model. The goal of this model is to provide a rigorous justification forthe construction of a software tool for simulation and optimization of a generalmetro line system. The software can be used to study the behavior of energy andheat exchanges in the tunnel, in order to propose efficient solutions for passengerscomfort.

Key words:Metro, Underground, Railway, Temperature, Tunnel, Thermal comfort, Heat load,Modeling

1 Introduction

The operation of underground railway systems can generate enough heat toraise tunnel and station temperatures as much as 811 oC above ambient tem-perature[1]. In London, where ambient temperature can reach 30 oC and overin summer, temperatures of more than 37 oC have been recorded on sometrains within the London underground tube network [2]. It is obvious thatpassenger comfort will be difficult to achieve in such conditions. The thermalcomfort conditions in underground railway environments is a complex problemwith many approaches. However, little has been written on the complications

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and difficulties associated with cooling an underground railway system. Thedeep and, sometimes, narrow tunnels make cooling of the underground rail-way environment differ from those normally encountered in conventional airconditioning and also the process is energy and capital intensive[3].

A train that runs on tracks out in the open does not significantly changesthe thermal balance of the environment. The losses that it dissipates do notaffect the outside conditions in a measurable way. But external conditionscan and do affect the interior environment in a train. In effect, heat gainedby radiation through glazing or by indirectly heating up the train skin cansignificantly impact the interior conditions. In a tunnel system the situation isquite different. The energy exchanges between the train and surroundings cansignificantly change ambient conditions. The model described in this documentpermits the analysis of this phenomenon in the following ways:

Energy balances: to categorize energy exchanges and properly identify theimpact on energy consumption as well as thermal balance. Prediction of ambient conditions: identifies the conditions that will lead to

uncomfortable situations in a metro line. Evaluation of the impact of changes in the system: how a change in operation

practices, technology or other variables, affects energy issues in the system.

2 Energy Balance Model

The whole system can be conceived as a series of nodes interconnected bytracks. Each node is a station and the tracks link these stations. From athermal and mass transfer point of view, all nodes are linked. Normally airenters stations, flows along corridors, platforms and tunnels, to be evacuatedat ventilation stations inside the tunnel. Air gains heat and moisture fromwaste heat in the system and moisture released by passengers and other watersources.

In what concerns the calculation core, the model evaluates separately eachtrack segment. This procedure has been conceived in order to accurate measurethe energy release by the trains. In fact, the goal is to know how this energyis related whit operation procedures, ambient conditions, trains technology,track profiles, etc.

The Energy Balance Model was constructed to evaluate the heat transfertaking place between the surrounding soil, inside air and inner loads. Assumingaverage peak loads and normal train running conditions, the tunnel and itscontent was split in several control volumes where the following loads wereconsidered in the analysis; see Fig 1:

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Train: is a thermal energy source. This energy comes from: rolling and airfriction, braking losses, auxiliary power dissipated into the tunnel and pas-senger thermal load.

Air in tunnel: it heats up in response to the thermal energy dissipated bysuccessive trains. It is also a sink as it is evacuated from the tunnel.

Earth around the tunnel and stations: it is a thermal sink, since it canabsorb energy from the air.

Fig. 1. Thermal balance model sketch

The terms in Figure 1 are:

Symbol

x = length of segments in m

xi = train actual position, in m

xi1 = train previous position, in m

Sj = actual stationSj+1 = next station

The system, is divided in discrete segments x of length 1 [m], where the speedand the air temperature are assumed constant, generating control volumeswhere the energy balance could be calculated, as shown below, in equation(1).

QT = QA + QG (1)

with:

QT = heat flow generated inside the line, in W

QA = heat flow absorbed by the air, in W

QG = heat flow absorbed by the ground, in W

The Energy Balance Model uses the following assumptions:

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The model evaluates two systems: The Single Train System and TheLine System. The first system, measures the performance of a single trainrunning in a specific way. As each traveling way has its own speed profile,they are treated separately in each iteration. Also we recognize two travelingways: LR 1 and RL 2 . The second system evaluated, is in charge of the line

energy balance and thus, temperature predictions. A train running in a specific way generates a Heat Profile over the track.

This Heat profile is added with the heat rejected in each station in orderto generate a unique heat profile a cross the whole line, for a specific timeinterval.

Each track section (the tunnel that interconnect two stations) is isolatedfrom the rest of the tunnels, in the sense that stations work as boundaryconditions at the beginning of each iteration for temperature predictions,and this generic situation is extended to the general system.

As stations also have specific properties (length, surface area, auxiliary

power, people coming in and out, etc.) and, more over, they are affectedby the track sections that are in both sides of it, a line will be considered asa non-homogeneous tunnel. This means that stations will linked the tracksections in order to create a unique tunnel, but with different geometricalproperties. The non homogeneous tunnel has been conceived as a semi in-finite solid with variable surface temperature, as boundary condition. Theground that surrounds the tunnel acts as a sink and can store energy fromambient conditions along the year, keeping ground temperature almost con-stant depending on the season.

3 Heat Flow Generated Inside the Line QT

The total thermal load along the track will depend on how many trains passduring a given time interval, the type of the train, its speed, the number oftransported passengers, stations technology, etc. In order to stabilize the flows

inside the tunnel, we consider a one hour regimen. Therefore the line heat flowwill be the energy dissipated in it by all the thermal loads, for an hour regimen(ie: QT = QT/3600).

1 LR: from left to right, with each new station at the right of the previous one2 RL: from right to left, is the opposite way to go over the line

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3.1 Heat Profile Release by the Trains

To calculate the heat release by the trains we used mean operation valuesduring this hour, in order to have, for each traveling direction, a single heat

profile over a line section (QZug for his abbreviation in German). The totalamount of heat release by the trains will be this value multiplied by the trainsfrequency, as shown below.

QT = fLRQZug(LR) + fRLQZug (RL) + QS (2)

with:

fLR = train frequency for the Left-Right direction, in trains/hour

QZug(LR) = heat dissipated bye the trains running in the Left-Right direc-tion in one hour, in J

fRL = train frequency for the Right-Left direction, in trains/hour

QZug(RL) = heat dissipated bye the trains running in the Right-Left direc-tion in one hour, in J

QS = heat dissipated bye the stations in one hour, in J

Details of how the train heat profile is computed are in section 6 explained.

3.2 Heat Release in Stations

In each station, besides the heat release bye the trains at the departure andat the arriving of them, other heat sources are present. Sources like the heatcarried by the air from outside, the amount of people circulating in the stationand the auxiliaries required for them, are now involved (see equation 3).

QS = SAUX + AirLoad + QSpeople (3)

with:

QS = heat release in the station, in J

SAUX = station auxiliary power, in W

AirLoad = station air thermal load, in W

QSpeople = heat release bye the people in the station, in J

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The stations have variable auxiliary loads. Auxiliaries are modeled with aconstant value plus another term, which increase with the amount of peoplein the station. In order to calculate the station auxiliary power, we suppose asensibility to the stations amount of people, with an escalation rate of 50%.

AuxSi = MinAuxSi +

[MaxAuxSi MinAuxSi ] dLD PSi0, 5 MaxPSi + PSi

(4)

The outside air conditions are fundamental in the stations inner energy bal-ance. Its from this point where there influence the whole line.

AirLoad = mairCp,A [TSi Tout] (5)

The heat dissipated by the people in the stations is hard to evaluate. The

prior is caused by the people flow undefination and its chaotic behavior. Somecomplex approximations were tried for modeling this important heat load,without improving accuracy. Therefore a simple approach has been taken,were the user introduce the average amount of people in the station, for agiven time interval.

QSpeople = Hw PSi (6)

With:

AuxSi = station (Si) auxiliary load, in W

dLD = station (Si) auxiliary system sensibility to the amount of people in thestation, set as 1,5 or with a 50% of escalation rate

Hw = human load walking, set as 140 W

MaxAuxSi = station (Si) maximum auxiliary load aloud, in W

MaxPSi = maximum number of people on the station (Si)

MinAuxSi = station (Si) minimum auxiliary load aloud, in W

mair = station (Si) air mass flow , in kg/s

PSi = average number of people on station (Si)

TSi = station (Si) air temperature, inoC

Tout = outside air temperature, inoC

Besides we use the following assumptions:

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(1) The Heat profile generated by the trains and rejected by the stations isdissipated to the air.

(2) Air transfer heat by convection to the non-homogeneous tunnel walls andfloor.

(3) Heat is transferred to the ground by conduction.

(4) Radiative exchanges (short wave), between train and tunnel, are assumedto be negligible.

4 Air Heat Flow (QA)

The final factor, which cannot be accounted for by a steady-state heat bal-ance approach, is the thermal interaction between air within the tunnel andthe surrounding ground. The tunnel lining and surrounding ground have a

large thermal mass and are strongly coupled to the air within the system. Inits most fundamental form, the problem under consideration is described byNewtons law of cooling: the heat flux between the air within a tunnel andthe surrounding soil is proportional to the difference between the temperatureof the air and of the tunnel wall surface. The magnitude of the heat trans-fer or flux for a given temperature difference is dictated by the heat transfercoefficient h, a parameter that is a function of the nature of the air flow andproperties of the wall surface.

We will use the linear solution to this complex problem [5], where the air heat

flux is related just to the air temperature over the tunnel longitudinal axis.

Air heat flux is simply calculated with equation (7), using the stations initialtemperatures as boundary condition at the beginning of each iteration 2.

Fig. 2. Air flow inside the tunnel

It is supposed, that for security reasons, air enters trough the stations corri-dors, goes over the platforms and exit in the tunnels vents shafts. (See figure2)

Modeling the air flow inside the tunnel is quite a problem, it has to take

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in to account the air extractor system, the piston effect of the trains, thechimney effect due to different air densities between the tunnel and externalambient air, as well as different elevations of stations. In order to predict airmovements generated by trains moving through the system, our model use asbasis, an aerodynamic network model of the tunnels, stations, passageways

and ventilation shafts.

QA = wAACp,A(Ti Ti1) (7)

where:

wA = air flow inside the tunnel taking in to account air renovation, in m3/s

A = air density = 1.177 kg/m3

Ti = air temperature at xi, inC

Ti1 = air temperature at xi1, inC

Air properties change according to the pressure and temperature inside thetunnel but, at the temperature range that we are working, this propertiesdo not varied significantly. We display below the typical values used (for at-mospheric pressure and 300 K):

air kinematic viscosity () = 1.857 kg/(ms)

air thermal conductivity (A,T) = 0.02623 W/(mK)

The flows are fully turbulent under the generic conditions considered, and givegeneric turbulent flow profiles. We then use the so-called Reynolds analogyto relate the air flow velocity problem.

5 Ground Heat Flow (QG)

Earth temperature at the wall of the tunnel depends firstly on the heat trans-fer from earth surface to deeper layers. Secondly the air in the tunnel itselfinfluences the earth temperature at the tunnel wall [5].

The calculations are based on approximations for the earth temperature whichvaries with the season of the year and depth under surface. Heat transfer coef-ficients for the heat flow between air, tunnel wall and earth are estimated frommaterial coefficients, flow properties and geometric parameters. The followingrestrictions are made for the current version of the program:

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homogeneous earth is situated above and around the tunnel ground properties are constant

To calculate the heat exchange in the tunnel the total length of the tunnelis divided into segments which are treated step by step. Each segment is

supposed to carry air of constant temperature so that heat exchange in thesegment leads to a jump in temperature at the border between two segments.The heat exchange for each segment is:

QG = xUL (Ti TE,W) (8)

where:

UL = heat transfer factor per length of wall between bulk air and wall, in W/(mK)

TE,W = earth temperature at the wall of the tunnel, in C

It is necessary to introduce a correction factor [7] to represent the influenceof the tunnel on the earth temperature. Then, comparing the heat flow fromthe earth surface to the tunnel, with the heat flow through the tunnel wall [6],the corrected earth temperature at the wall of the tunnel is:

TE,W =UTE,0 + Ti

U + 1(9)

where U is the conductance ratio of heat transfer from earth surface to tunneland from airflow to tunnel wall.

This parameter U is defined to take into account thermal conductivity ofthe earth, heat transfer coefficient between the airflow and the earth at thetunnel wall as well as the geometric configuration:

U =2

UL

1

lnS0R0

+

S0R0

2 1

(10)

with:

= ground thermal conductivity = 1.5 W/(mK)

S0 = depth of tunnel center under surface, in m

R0 = hydraulic radius of tunnel, in m

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The heat transfer coefficient per length of wall of tunnel UL depends only onhi, the heat transfer coefficient at the inner surface of the tunnel, in the form:

UL = 2R0hi (11)

The heat transfer coefficient at the inner surface of the tunnel hi (which ismeasured in W/(m2K)) depends on flow properties, dimensions of the tunneland material properties of the air in the tunnel [5], in the form:

hi =A,TN u

2R0(12)

The Nusselt number N u of air in a tunnel depends on Reynolds number Re and

thus on flow rate. For turbulent airflow Gnielinski [8] proposes the followingapproximation:

N u = 0.0214(R0.8e 100)P0.4r (13)

with:

Pr = Prandtl number of air

Re = Reynold number

The Reynolds number is basically the ratio of the inertial force of the mediumover it viscous force.

Re =2VaR0

(14)

where Va is air speed. The Prandtl number of air is taken as a constant (typ-ically Pr = 0.72).

Pr =

Cp

A,T (15)

The earth temperature at the wall of the tunnel not influenced by the tunnel(denoted TE,0) is calculated from the ambient air temperature with its meanvalue Tm and its maximum value Tmax, assuming a sinusoidal temperaturevariation throughout the year.

TE,0 = Tm (Tmax Tm)ecos(As ) (16)

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A parameter describes the thermal depth of the tunnel. Heat flows fromair to earth surface without resistance.

= S0

ct0

(17)

where:

As = season constant (As=0 for summer and As=0.5 for winter)

c = volumetric heat capacity of ground, in J/m3K

t0 = duration of year, in s (1 year = 31.5 106 s)

6 Train Model

A train along a track section has a distinct speed versus distance profile,depending on the trajectory of the train (L-R or R-L), that is defined by thesystem operator. This input and the track profile are the base, from where thekinematics and the energy consumption of the train are calculated.

6.1 Previous Calculations for Cars

Ones a train is build in to the software, it calculates the train total PassengerCapacity, Inertial Mass and its Length. With these global variables set, wecan calculate the car total weight, power consumption and heat rejected.

The model is constructed in such a way that each car has constant auxil-iary power consumption, related with pneumatic and lightning systems, plus

a variable auxiliary load generated by the HVACs.

The HVAC power consumption, as well as the heat rejected by the car to thetunnel, is a function of the amount of people on the car, the car inner comforttemperature, ambient conditions and the car technology.

The first thing, in order to calculate the cooling power needed to maintain theselected comfort temperature, is to evaluate all the heat sources in the car.

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These are: The Passenger Load, The Air Load 3 and The Conduction Load 4 .

PLoad = Pass Hs (18)

ALoad = RAirACp,A [Tout TComfort] (19)

CLoad = k CarSurface [Tout TComfort] (20)

with:

PLoad = passenger thermal load, in W

Hs = human load standing, set as 110 W

ALoad = car air thermal load, in W

RAir = car air renovation, in m3/s

A = air density, in kg/m3

Cp,A = air specific heat = 1007 J/(kgK)

TSout = outside temperature, inoC

TComfort = train inner comfort temperature, inoC

CLoad = car conduction load, in W

k = car global heat transfer coefficient, in W/(m K)

CarSurface = car surface area, in m2

Equation 21 shows the thermal power needed to be absorbed by the car HVACsystem to maintain comfort conditions.

CarCool = PLoad + ALoad + CLoad + Caraux (21)

with:

Caraux = car constant auxiliary power, in W

The heat dissipated by the HVAC system is a function of its COP and the heatrejected by the car to the line is a resultant of all the heat sources considering

3 the heat exchange generated with the outside air by air renovation4 the heat exchange generated with the outside trough the cars surface

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also HVACs, as shown in equation 22.

QCar =

1 +1

COP

CarCool (22)

with:

QCar = heat rejected by the car, in W

COP = HVACs coefficient of performance

Finally, the total auxiliary power could be calculated as shown in equation 23.

CarAUX =CarCool

COP+ Caraux (23)

with:

CarAUX = car total auxiliary power, in W

And the heat flow release by each car, and thus the train, will be:

QPass&Aux =

(QCar ALoad + Caraux) (24)

For the case, that the cooling power is different then cero. When there are noHVACs systems on the cars, we used the following simple approach.

QPass&Aux =

(PLoad + Caraux) (25)

With:

dotQPass&Aux = train heat flow rejected to the tunnel, in W

All these parameters are used by the calculation core, witch is in charge of themechanical power, needed to move the train, predictions.

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6.2 Kinematics

From the speed profile all the basics kinematics are calculated as show inequations 26 and 27.

The traveling time is calculated as:

t(xi) = t(xi1) +2

V(xi) + V(xi1)(xi xi1) (26)

With:

t(xi) = train traveling time to the actual position, in s

t(xi1) = train traveling time to the previous position, in s

V(xi) = train actual speed, in m/s

V(xi1) = train previous speed, in m/s

If one knows the speed V(xi) and the speed V(xi1), then the accelerationcan be simply calculated from the speed differences.

a(xi) =V(x

i) V(x

i1)

t (27)

With:

a(xi) = train instant acceleration, in m/s2

t = time interval between train positions (t(xi) t(xi1)), in s

The software also uses a maximum acceleration and jerk limit to overcomenoise in the acceleration profile.

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6.3 Power calculations

In order to achieve a specific speed, the train has to overcome all the forcesthat are against its motion. Therefore the inlet power can be conceived as the

result of multiplying these forces with the train speed.

Power(xi) = F(xi)V(xi)

F(xi) = FI(xi) + FR(xi) + FD(xi) + FC(xi) + FS(xi)

where:

Power(xi) = power generated by the motors at xi, in W

F(xi) = forces over the train at xi, in N

The force needed to accelerate the train can be broken down into the followingcomponents:

6.3.1 Inertial Forces (FI)

To move forward, the train must provide enough energy to overcome the trainsinertia, which is directly related to its weight.

FI = (MT + MI)a(xi)where:

FI = inertial force, in N

MT = train total weight, in kg

MI = train inertial mass, in kg

6.3.2 Rolling Friction Force (FR)

It is conceived, as force necessarily to move the wheels forward and is directlyproportional to the weight of the load supported by the wheels. The magnitudeof the friction force is [9]:

Davis formula term for Rolling Friction:

FR = M

T(A + B V(xi)) (28)

where:

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FR = rolling friction force, in N

MT = Train total weight, in ton

A = Davis formula A empirical parameter for the train

B = Davis formula B empirical parameter for the train

6.3.3 Aerodynamic Drag (FD)

The force exerted on a train moving inside a tunnel, depends in a complexway upon the velocity of the train relative to the air, the viscosity and densityof air, the shape of the train, the roughness of its surface and the tunnel

cross-sectional area[10].

FD = C D v2 (29)

with:

FD = aerodynamic drag force, in N

C = Davis formula A empirical parameter for the tunnel

D = Davis formula D empirical parameter for the train

For small values of the Reynolds number (called laminar flow since the flowis nonturbulant) the drag coefficient is inversely proportional to the velocity.This means that the drag force is only proportional to the train velocity.

When the flow is turbulent the Reynolds number is large, and the drag co-efficient D is approximately constant. This is the quadratic model of fluidresistance, in that the drag force is dependent on the square of the velocity.

6.3.4 Force due to Track Curve (FC)

This force is a result of the change in the accelerating vector.

FC = MT gE

r(30)

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with:

FC = force due track curve in N

E = Davis formula E empirical parameter for the train

g = gravity acceleration, in m/s2

r = curve radius, in m

6.3.5 Force due to Track Slope (FS)

Not always this force plays against motion. It depends on the direction inwhich the train is moving, down or up stream.

FS = MT gdh

dl

(31)

with:

FS = force due track slope, in N

dh = track height difference with equal slope and curve radius, in m

dl = track length with equal slope and curve radius, in m

6.4 Energy Usage

The software use a linear approach to calculate the energy used in motion.This means that no matter what speed the train has, we consider the sametraction chain efficiency. Therefore the power needed by the electrical motorsis only a function of the mechanical power and thus, function of the forcesagainst its motion.

EDrive =Power

t (32)

The total energy usage will came from the energy needed for motion and theused in auxiliaries, but with the caution of taking in to account the energyrecovered in the breaking phase ER.

EZug = EDrive + ZugAUXt ER (33)

ER = (Ek(xi) + Ep(xi))(RECOV) (34)

These energy terms are calculated as:

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Ek(xi) =1

2MTV

2(xi)

Ep(xi) = gMTdh

dlx

with:

EDrive = energy used for motion, in J

= traction chain efficiency

ER = recovered energy, in J

RECOV = recovery rate, 0 RECOV 1

Ek(xi) = kinetic energy at xi, in J

Ep(xi) = potential energy difference at xi, in J

6.5 Train Heat(QZug)

The heat generated by a train can be split in four terms, according on accel-erating rate, the use of brakes, friction loses and constant heat values relatedwith the passenger load and the use of auxiliaries.

Equation 35 shows the different train heat sources.

QZug = QFriction + QBrakes + QPass&Aux + QMotors (35)

with:

QZug = train heat losses, in J

QFriction = train friction losses, in J

QBrakes = train breaking losses, in J

QPass&Aux = passenger and auxiliary on the train losses , in J

QMotors = train electrical motor losses, in J

6.5.1 Heat dissipated by the electrical motors (QMotors)

Electric and mechanical inefficiencies are taking in to account as the trainincreases its acceleration. QMotors are energy loses related with the differences

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between the motors electrical power consumption and power necessary formotion.

The train was modeled as a dynamic system that moves along rails. The trainhas a certain mass (tare weight, load and rolling inertial mass). Forces that

act on the train are: inertia, rolling friction, air drag, forces due to curvatureand forces due to slope. From the equations its possible to calculate the trainrequired force to accelerate, the input power and thus the motors energy losses.

QMotors=

Power

Power

t

6.5.2 Heat dissipated in the breaking phase (QBrakes)

As the train brakes one part of the energy needed to decrease its speed is reinjected to the lines and the other is dissipated as heat.

In this situation we have two cases. The first on consider only the work ofregenerative brakes and the braking efficiency related to it.

Brakes= Ek(xi) + Ep(xi) ER

Brakes= |Ek(xi) + Ep(xi)| (1 RECOV)

The second case, suppose that for a particular speed regenerative brakes arealmost useless and mechanical breaks are used. Thus this specific speed isconsidered as input.

Brakes = |[Ek(xi) Ek(xi1)] + Ep(xi)|

where:

Ek(xi1) = kinetic energy at xi1, in J

6.5.3 Heat generated by friction (QFriction)

This term considers friction from wheels to rails or train to air. The amountof energy dissipated on friction is calculated as the power wasted due to Aero-dynamic Drag Forces and Curve Forces, plus the Rolling Friction Force work.

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QFriction = (FD + FC+ FR) V(xi)t

6.5.4 Heat Rejected by passengers and auxiliaries (QPass&Aux)

The fourth term considers the heat generated by electrical auxiliaries 5 andpassengers.

QPass&Aux = QPass&Aux t

A Software Inputs

Its important to understand that the Software use a lot of physical, technicaland operational parameters, which in the case that they ware irresponsiblyfilled, the output data will be corrupted.

With this warning expressed, the calculation method and all the parameters 6

involved in it, are further presented.

When a line is created four kinds of parameters are required:

A.1 General Parameters

These take care of ground and air thermal properties, climatological statisticsand human standard parameters, such as weight and thermal load.These values are not to be fixed between iterations, and should be coherentwith the location of the line.

A.2 Line Parameters

These are the stations and tunnels geometrical parameters, plus the track andspeed profile for each section.The first step to build a line is to create the stations and the tunnels thatlinked them. Following the nodes and arcs analogy, the first station created

5 lights, ventilators, cooling systems and other electrical systems6 All the input data is in International Standard Units (IS).

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represents the first node and the next station will be placed at the right of it.These means that a the line is created from Left to Right.For each station the following data is request:

Length

Mean Section Area Mean Perimeter Minimum Auxiliary Power Maximum Auxiliary Power

Ones two station are created the tunnel that interconnect them appear. Nowthe track profile needs to be created. The track profile is define between sta-tions middle point and is discretizated in segments with equal slope and curveradius. Two speed profiles are asked, one for the trains running from thefirst station to the second one, or from Left-Right, and another one for the

trains running from Right-Left. Besides this information, also the next datais needed:

Davis Formula C Parameter Total Air flow inside Tunnel Station A Shaft Influence Tunnel Cross Section Area Tunnel Mean Perimeter

A.3 Train Parameters

The software creates a train data base from where we can choose. When atrain is build in to the software, some properties are given from the cars thatcompose it, and others are inherent of the train.

Each car has its own:

Constant Auxiliary power

Air Renovation HVAC coefficient of performance COP HVAC maximum cooling Power Heat transfer coefficient (k) Car surface Inertial mass Maximum number of passengers aloud on the car Car tar weight Length

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And the train has the following specific data:

Minimum speed at which regenerative brakes work Motor & Traction chain efficiency Breaking energy recovery rate (RECOV)

Davis Formula A Parameter Davis Formula B Parameter Davis Formula D Parameter Davis Formula E Parameter

A.4 Case Parameters

After the line and the trains are created we can run a simulation with spe-cific operational parameters such as the amount of people in the stations,

the amount of passengers traveling en each direction, type of trains, trainfrequency and dwell time, inner comfort temperature, year season and ini-tial station temperature. The before mention parameters, generate, what wecall, a case study and can be selected for a whole line or just for a line segment.

List of notations

Symbol Description

A =Davis formula A empirical parameter for the train

a(xi) =train instant acceleration, in m/s2

ALoad =car air thermal load, in W

As =season constant (As=0 for summer and As=0.5 for winter)

AirLoad =station air thermal load, in W

AuxSi =station (Si) auxiliary load, in W

B =Davis formula B empirical parameter for the train

C =Davis formula A empirical parameter for the tunnel

CLoad =car conduction load, in W

Cp,A =air specific heat = 1007 J/(kgK)

Caraux =car constant auxiliary power, in W

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Symbol Description

CarAUX =car total auxiliary power, in W

CarSurface =car surface area, in m2

COP =HVAC coefficient of performance

D =Davis formula D empirical parameter for the train

dh =track height difference with equal slope and curve radius, in m

dl =track length with equal slope and curve radius, in m

dLD =station (Si) auxiliary system sensibility to the amount of people in thestation, set as 1,5 or with a 50

E =Davis formula E empirical parameter for the trainEp(xi) =potential energy difference at xi, in J

=traction chain efficiency

EDrive =energy used for motion, in J

ER =recovered energy, in J

Ek(xi1) =kinetic energy at xi1, in J

Ek(xi) =kinetic energy at xi, in J

F(xi) =forces over the train at xi, in N

fLR =train frequency for the Left-Right direction, in trains/hour

fRL =train frequency for the Right-Left direction, in trains/hour

FC =force due track curve, in N

FD =aerodynamic drag force, in N

FI =inertial force, in N

FR =rolling friction force, in N

FS =force due track slope, in N

g =gravity acceleration, in m/s2

Hs =human load standing, set as 110 W

Hw =human load walking, set as 140 W

k =car global heat transfer coefficient, in W/(mK)

=ground thermal conductivity = 1.5 W/(mK)

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Symbol Description

mair =station (Si) air mass flow , in kg/s

MaxAuxSi =station (Si) maximum auxiliary load aloud, in W

MinAuxSi =station (Si) minimum auxiliary load aloud, in W

MaxPSi =maximum number of people on the station (Si)

MI =train inertial mass, in kg

MT =Train total weight, in kg

MT =Train total weight, in ton

PLoad =passenger thermal load, in W

Power(xi) =power generated by the motors at xi, in W

PSi =average number of people on station (Si)

Pr =Prandtl number of air

QA =heat flow absorbed by the air, in W

QCar =heat rejected by the car, in W

QG =heat flow absorbed by the ground, in W

QT =heat flow generated inside the line, in W

QBrakes =train breaking losses, in J

QFriction =train friction losses, in J

QMotors =train electrical motor losses, in J

QPass&Aux =passenger and auxiliary on the train losses , in J

QS =heat dissipated bye the stations in one hour, in J

QSpeople =heat release bye the people in the station, in J

QZug (LR) =heat dissipated bye the trains running in the Left-Right direction in

one hour, in J

QZug (RL) =heat dissipated bye the trains running in the Right-Left direction inone hour, in J

QS =heat release in the station, in J

r =curve radius, in m

A =air density = 1.177 kg/m3

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Symbol Description

A =air density, in kg/m3

c =volumetric heat capacity of ground, in J/m3K

R0 =hydraulic radius of tunnel, in m

RAir =car air renovation, in m3/s

Re =Reynold number

RECOV =recovery rate, 0 e 1

S0 =depth of tunnel center under surface, in m

SAUX =station auxiliary power, in W

Sj =actual station

Sj+1 =next station

t(xi1) =train traveling time to the previous position, in s

t(xi) =train traveling time to the actual position, in s

t =time interval between train positions (t(xi) t(xi1)), in s

t0 =duration of year, in s (1 year = 31.5 106 s)

TComfort =train inner comfort temperature, inoC

TE,W =earth temperature at the wall of the tunnel, inC

Ti1 =air temperature at xi1, inC

Tout =outside air temperature, inoC

TSi+1 =next station (Si+1) initial temperature, inoC

TSi =station (Si) air temperature, inoC

Ti =air temperature at xi, inC

UL =heat transfer factor per length of wall between bulk air and wall, in

W/(mK)

V(xi1) =train previous speed, in m/s

V(xi =train actual speed, in k/h

wA =air flow inside the tunnel taking in to account air renovation, in m3/s

x =length of segments, in m

xi1 =train previous position, in m

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Symbol Description

xi =train actual position, in m

References

[1] F. Ampofo, G. Maidment, J. Missenden; Underground railway environment inthe UK Part 2: Investigation of heat load; Applied Thermal Engineering, vol.24 (2004), pp. 633-645.

[2] P. Walker; Lotsa fun in the Hot Tubes tonight; The Rail Engineer, vol. 6 (2006),

pp. 38-39.

[3] F. Ampofo, G. Maidment, J. Missenden; Underground railway environment inthe UK Part 1: Review of thermal comfort; Applied Thermal Engineering, vol.24 (2004), pp. 633-645.

[4] W. M. Rohsenow and J.P. Hartnett, Handbook of Heat Transfer, McGraw-HillBoock Compaby, New York, 1973.

[5] St. Benkert, F.D. Heidt, D. Scholer, Calculation tool for earth heat exchangersGAEA, Department of Physics, University of Siegen, D-57068 Siegen,Germany.

[6] K.J. Albers, Untersuchungen zur Auslegung von Erdwarmeaustauschern furdie Konditionierung der Zuluft fur Wohngebaude, Ph. D. Thesis, UniversitatDortmund, Dortmund, 1991.

[7] H.L. von Cube, Die Projektierung von erdverlegten Rohrschlangen furHeizwarmepumpen (Erdreich- Warmequelle), Klima + Kalte-Ingenieur, 1977.

[8] V. Gnielinski, Neue Gleichungen fur den Warmeund den Stoffubergang inturbulent durchstromten Rohren und Kanalen, Forschung im Ingenieur-Wesen,41, 1975.

[9] Dan B. Marghitu, Mechanical Engineers Handbook, ACADEMIC PRESS, 2001.

[10] Edward H. Smith, Mechanical Engineer s Reference Book, Buttenvorth-Heinemann,Twelfth edition, 2000.

[11] John H. Lienhard IV and John H. Lienhard, Mechanical A Heat TransferTextBook, Phlogiston Press,Third edition, 2000.

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