Victor Fuentes http://www.victorfuentes.com Victor Fuentes personal site Tue, 14 Feb 2012 15:29:04 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 A traffic problem and Reynolds transport theorem http://www.victorfuentes.com/archives/772 http://www.victorfuentes.com/archives/772#comments Sun, 12 Feb 2012 23:11:04 +0000 admin http://www.victorfuentes.com/?p=772 Mathematical models about problems like public transportation regulation or traffic jams allow to analyses and simulate solutions before implement it. The Fluid Mechanical theory can simulate situations as traffic jams.

Since the traffic within the cities across the world is a huge problem, there is an untiring search of ways to improve the traffic regulation. The state-of-the-art of solutions involves complex mathematical models from other fields such as fluid mechanics.

Imagine we have a traffic density problem to analyze and city authorities would like to implement any kind of control system such a signaling regulation. In a normal situation we have cars that are moving within the area, going in and out, stopping momentarily, parking and so; we want to study the behavior of traffic and within an urban area. Actually this is a problem in every downtown of any city. To do that it’s necessary to model the traffic within the area in order to know how to regulate the signaling sequence or time between red and green lights.

There are many ways to model mathematically a system or situation like the exposed, but for this article we will take the Reynolds transport theorem. We will introduce the issue and explain how the Reynolds theorem could help to model the system, also we explain briefly the theorem theory.

To make the mathematical model we must considerate a number of elements which we want to explain mathematically. Since we are talking about traffic and city areas, we must to considerate the elements that compose this, so we must to take cars as traffic elements and streets as city area elements. In more detail about traffic we considerate the number of cars that passes through a determinate point during a time lapse, this is the flow rate. The area elements, this is the streets, we could considerate the number of streets, its lanes numbers and lanes widths. As a mathematical model we can go as further as we want, by example determining the asphalt quality in order to perform a comfort index to simulate the car speeds but this is out of our article scope.

Let’s see the theory. The Reynolds transport theorem relates the time rate change of elements of a system with the addition of the time rate change of elements within a control volume (in this case a surface) plus the total amount of elements passing through the inlet and outlet areas.

(1)   \begin{equation*} \frac{DB_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{cv} \rho bd \textnormal{V} + \int_{cs} \rho b \textbf{V} \cdot \hat{n} ̂\delta A \end{equation*}

Translating this equation to the problem itself, we must to consider a control volume, as said, the city area object of the analysis. Let N the number of cars, we consider Nsys the total number of cars in the system and Na the number of cars in the area. At time t=t0, where t0 is the initial time, the number of cars in the system and the area are the same, thus Nsys = Na. Accordingly the Reynolds transport theorem the equation expressing the relation between the number of cars in the system and the number of cars in the area would be

(2)   \begin{equation*} \frac{DN_{sys}}{Dt}=\frac{\partial N_a}{\partial t}+\dot{n}_{out}-\dot{n}_{in} \end{equation*}

where

    \[\dot{n}_{out}\]

and

    \[\dot{n}_{in}\]

represent the net rate (cars per unit time) at which cars leave and enter the area. Let be the perimeter of the area the control surface, there are a defined number streets crossing this perimeter, then the total cars number crossing into and out the area is given by

    \[ \dot{n}_{out}=\sum\limits_{i} n_{i_{out} \]

    \[ \dot{n}_{in}=\sum\limits_{i} n_{i_{in} \]

where ni represents the rate of cars along the i-th street crossing the perimeter of the area.

In a similar fashion we must to express the number of cars that are within the city area or control area. Let n the number of cars per area unit, we express the value of number of cars per unit area in terms of location and time n=n(x,y,t). Thus, we obtain the total of cars in the control area at any time by this equation.

(3)   \begin{equation*} N_a=\int_{a} ndA \end{equation*}

Thus the time rate change of the number of cars in the neighborhood is

(4)   \begin{equation*} \frac{\partial N_a}{\partial t}=\frac{\partial}{\partial t} \int_{a} ndA \end{equation*}

and combining we obtain

(5)   \begin{equation*} \frac{DN_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{a} ndA + \sum\limits_{i} \dot{n}_{i_{out}} - \sum\limits_{i} \dot_{i_{in}} \end{equation*}

Applied to our problem, we see that the rate at which the number of cars changes with time is equal to the rate at which the number of cars in our neighborhood changes with time plus the difference of the rate at which cars enters and exits the neighborhood. Hence we obtain a relationship based in the Reynolds transport theorem.

Let’s compare the two equations, by one hand the Reynold transport theorem and by other hand the obtained.

(6)   \begin{equation*} \frac{DB_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{cv} \rho bd \textnormal{V} + \int_{cs} \rho b \textbf{V} \cdot \hat{n} ̂\delta A \end{equation*}

(7)   \begin{equation*} \frac{DN_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{a} ndA + \sum\limits_{i} \dot{n}_{i_{out}} - \sum\limits_{i} \dot_{i_{in}} \end{equation*}

We could see a remarkably similarity but not equality, so we need to interpret the equation members to understand with exactitude how we apply the Reynold equation to our problem.

Since we are basing our approach in a relation between an area or volume in space and a mass composed by elements we can express a relationship between the extensive property and the mass and the intensive property, thus

    \[ B=mb \]

where b represent the amount of any parameter per unit of mass. If we express the fluid particles as infinitesimal size δV and a mass as ρδV , this summation takes the form of an integration over all the particles in the system and can be written as

(8)   \begin{equation*} B_s=\lim_{\delta \textnormal{V}} \sum\limits_{i} b_i(\rho_{i} \delta \textnormal{V}) = \int_{S} \rho bd \textnormal{V} \end{equation*}

where the s means the system specification. Since we want to evaluate the problem over the time, we must to express the rate at which the momentum of a system changes with time, the rate at which the mass of a system changes with time, and so on. Thus, we derive respect the time as shown:

(9)   \begin{equation*} \frac{dB_{sys}}{dt}=\frac{d(\int_{sys} \rho bd \textnormal{V})}{dt} \end{equation*}

And now we adapt this law to a control volume Bcv approach:

(10)   \begin{equation*} \frac{dB_{cv}}{dt}=\frac{d(\int_{cv} \rho bd \textnormal{V})}{dt} \end{equation*}

as we see, the difference is the limits of the integration, in first the limits are the system itself and in second the control volume. The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property for a system and that for a control volume.

At time t we consider that the extensive property and the intensive property equals, so we have

    \[ B_{sys}(t)=B_{cv}(t) \]

Since at time t+δt there is a variation on the fixed control volume, we must to subtract a quantity that exits from the original control volume and add the quantity that enters in. If we call Bo the exiting quantity and Bi the entering quantity, the equation results as

(11)   \begin{equation*} B_{sys}(t+\delta t)=B_{cv} (t+ \delta t)-B_o (t+\delta t)+B_i (t+\delta t) \end{equation*}

Thus, the change in the amount of B in the system in the time interval δt divided by this time interval is given by

(12)   \begin{equation*} \frac{DB_{sys}}{Dt}=\frac{\partial B_{cv}}{\partial t}+\dot{B}_o - \dot{B}_i \end{equation*}

where

(13)   \begin{equation*} \dot{B}_o=\lim_{\delta t \to 0} \frac{B_o(t+\delta t)}{\delta t}=\rho_2 b_2 A_2 v_2 \end{equation*}

and

(14)   \begin{equation*} \dot{B}_i=\lim_{\delta t \to 0} \frac{B_i(t+\delta t)}{\delta t}=\rho_1 b_1 A_1 v_1 \end{equation*}

and where ρ is the density of the mass, A the area through the mass enters and exits the control volume, v the velocity and b the intensive property. Thus, the rate on inflow of the property B into the control volume is Bi, and consequently the rate on outflow of property B is Bo. We could, also express the equation as

(15)   \begin{equation*} \frac{DB_{sys}}{Dt}=\frac{\partial B_{cv}}{\partial t}+\rho_2 b_2 A_2 v_2 - \rho_1 b_1 A_1 v_1 \end{equation*}

If we go further, we could detail the angle of particles motion and the normal surface vector, then the amount of the property B carried across the area element δA in the time interval δt is given by

(16)   \begin{equation*} \delta B=b \rho \delta \textnormal{V} = b \rho (v cos \theta \delta t) \delta A \end{equation*}

The rate at which B is carried out of the control volume across the small area element δA, denoted δBo, is

(17)   \begin{equation*} \delta \dot{B}_o=\lim_{\delta t \to 0} \frac{\rho b \delta \textnormal{V}}{\detal t}=\lim_{\delta t \to 0} \frac{(\rho b v cos \theta \delta t) \delta A}{\delta t} = \rho b v cos \theta \delta A \end{equation*}

By integrating over the entire outflow portion of the control surface, cso, we obtain

(18)   \begin{equation*} \dot{B}_o=\int_{cs_o}d\dot{B}_o=\int_{cs_o}\rho bvcos\theta \delta A \end{equation*}

The quantity vcosθ is the component of the velocity normal to the area element δA and can be written as vcosθ=V·ñ. Hence, an alternate form of the outflow rate is

(19)   \begin{equation*} \dot{B}_o=\int_{cs_o} \rho b \textbf{V} \cdot \hat{n} \delta A \end{equation*}

Similarly we arrange the inflow on the control surface csi, we find that the inflow rate of B into the control volume is

(20)   \begin{equation*} \dot{B}_i=- \int_{{cs}_i} \rho bvcos \theta \delta A = - \int_{{cs}_i} \rho b \textbf{V} \cdot \hat{n} \delta A \end{equation*}

Finally, we obtain the net flow rate of parameter B across the whole control surface:

(21)   \begin{eqnarray*} \dot{B}_o-\dot{B}_i=\int_{{cs}_o} \rho b \textbf{V} \cdot \hat{n} \delta A - (- \int_{{cs}_i} \rho b \textbf{V} \cdot \hat{n} \delta A)\\ = - \int_{cs} \rho b \textbf{V} \cdot \hat{n} \delta A \end{eqnarray*}

where the integration is over the entire control surface. By combining we obtain

(22)   \begin{equation*} \frac{dB_{sys}}{Dt}=\frac{\partial B_{cv}}{\partial t} +\int_{cs}\rho bd \textbf{V} \cdot \hat{n} \delta A \end{equation*}

Also, we can rewrite Bcv so we get this equation:

(23)   \begin{equation*} \frac{dB_{sys}}{Dt}=\frac{\partial}{\partial t}\int_{cv}\rho bd \textnormal{V} + \int_{cs} \rho b \textbf{V} \cdot \hat{n} \delta A \end{equation*}

And now we can link the two equations, by one hand the general form of Reynold transportation theorem equation and the one from our issue. Is obvious that Nsys that refers to the cars of the systems equals to Bsys, thus

(24)   \begin{equation*} \frac{DB_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{cv} \rho bd \textnormal{V} + \int_{cs} \rho b \textbf{V} \cdot \hat{n} ̂\delta A \end{equation*}

is equivalent to

(25)   \begin{equation*} \frac{DN_{sys}}{Dt}=\frac{\partial}{\partial t} \int_{a} ndA + \sum\limits_{i} \dot{n}_{i_{out}} - \sum\limits_{i} \dot_{i_{in}} \end{equation*}

Is obvious that the first term at left refers to the control volume, the general approximation, in our case we need an control area, so there is an adaption of the integration from the density of the particles by volume to number of cars by area. The second term at left refers at the integration on a control surface (the area which serves as entrance and exit of the elements) of the mass flow rate by unit area (expressed by density and velocity related the normal of the surface). In the specific case of traffic in the city, we assume that the cars enter and leave the control area by the streets and they do in a normal way, that means, driving straight the lanes, so it’s not necessary to express the angle of particles motion and is enough with the difference between the in and out traffic rate per unit time as expressed at the rightest term of the equation.

]]> http://www.victorfuentes.com/archives/772/feed 0 High-speed train aerodynamic analysis http://www.victorfuentes.com/archives/715 http://www.victorfuentes.com/archives/715#comments Mon, 12 Sep 2011 22:51:39 +0000 admin http://www.victorfuentes.com/?p=715 In the railway high-speed lines exploitation arises some problems such as aerodynamic drag forces.

The circulation of trains at commercial speeds of up 300 km / h involves a number of difficulties in the field of aerodynamics. On the one hand the energy cost of a train traveling at 300 km / h, on the other hand, the interaction with the environment and the crossing with other trains running in opposite directions.

In this article I expose a very specific issue, its analysis and a solving proposal. This article is an executive summary of a Master Thesis Master about the high-speed train aerodynamic. The Thesis was presented on the Master of Computer Integrated Manufacturing and Engineering at the Universitat Politècnica de Catalunya on July the 20th of 2011.

The scope of study is the high speed line linking Barcelona with Madrid. The train on which to perform the analysis is the serie 103, with a top speed of 350 km / h, however at the time of study, the maximum allowed speed is 300 km / h that is the speed to be used as reference.

CAD modeling

The CAD model was based on design draws. For this specific analysis, I took only the extreme car.

Based on Heine et al. I followed a simplification strategy. I determined only to model the car body and the bogies. The simplification rate can be checked on the images. All the CAD model was realized in Generative Shape Design, a Catia module.

 

CAD model views

CAE analysis.

The CAE analysis was done with Star CCM+ software. As any CAE analysis, it was necessary to do by three basic steps: pre-processing, calculation or processing and post-processing or results. Two issues arose within this analysis, first of them the processing time and the exactitude on results.

These are closely related. The most detailed is the model (measured in polygons) more times takes to process the calculation. Is very important to balance correctly between time and detail in business point of view.

Importing and setting up the mesh

The first step is to import the mesh, in this case in IGES format.

The strategy to follow is to split all geometry and gather in three different regions types. In this case, all geometry resulted split and gathered in this named regions:

  • Wall
  • Base
  • Inlet
  • Outlet
  • Train

Those regions are of one of these types:

  • Wall
  • Inlet
  • Outlet

Where wall regions are assigned with ‘slip’ or ‘non-slip’ property.  This property configures a passive geometry in which the fluid has or not a viscous behavior. Inlet means the origin of the fluid stream and outlet where towards the fluid stream moves.

Refining the mesh

This analysis requires a mesh refining. The refining must to be done around the area to study, in this case the hood of the train. This refining is done creating different bounding boxes with the property of give more or less detail to the mesh, either the train or the fluid volume. In this case I created three boxes, the first one around the hood, the second one along the car body and the last one on the under-frame.

Refinning the mesh

Setting up initial and boundary conditions

The simulation physics was set as constant density for the equation of state and turbulent for the viscous regime. Finally, was set the K-epsilon as turbulence model. As discussed above, calculation is performed in two phases. First we calculate the aerodynamic behavior towards the train in the direction of travel, collecting aerodynamic data footprint in the region known as outlet. Subsequently, this aerodynamic footprint  data is established as the as the source of the behavior of the fluid region designated as inlet, so that it simulates the end car of the train composition. All this in two scenarios: open field and tunnel. After the calculations, in which the steps needed until the convergence vary between 250 and 2000, an aerodynamic analysis was done.

Results and Conclusions.

The first analysis had the mission to study the flow around the hood of the train, as commented I was looking for some laminar to turbulent flow around the rear car of the train composition. As shown in the next figures, the flow is laminar around the upper part of the hood, but the turbulences produced by the shapes around the under-frame and bogies continue along the lowest part of the hood, the under deflector or spoiler. In the second and third figure show accurately the flow velocity streams.

1st CAE Analysis

In order to improve the behavior of the train arranging the deflector shape and achieving a better drag coefficient, the spoiler was modified conveniently. The aim was to get a new spoiler with no incidence on the frontward movement but good working in the backward movement. A subtle modification on the corner part, where the spoiler joints the car body in order to get a ‘conduction shape’ and a longer spoiler was modeled. The next figures show the original and modified spoiler.

Spoiler modifications

After the modifications the model was aerodynamic analyzed again, in order to check how the flow streams are around the spoiler and to know the difference with the drag force.

2nd CAE Analysis

Finally, as the figures show, the improvement is notorious. The absence of turbulence streams around the spoiler results in a better flow leak around the hood and in an extreme environment such a tunnel, the improvement is considerable. The improvement in numbers is as follows. The specific aerodynamic coefficient used in the general equation varies from 0.553 to 0.549. Although apparently the improvement may seem small, the continued exploitation of a high-speed throws a amount of saved energy to take into account. In fact, in the normal exploitation of 54 trains at day over a year the energy saved is about 0.8 MWh.

]]> http://www.victorfuentes.com/archives/715/feed 0 TVM – A driver view http://www.victorfuentes.com/archives/667 http://www.victorfuentes.com/archives/667#comments Tue, 05 Oct 2010 20:49:44 +0000 admin http://www.victorfuentes.com/?p=667 The TVM (Transmission Voie-Machine) is the protection system implemented in the French railway high-speed lines. This system presents in the driver cab the maximun speeds indications and speed restriction orders in every moment.

The TVM is the answer to the problem that occurs when driving at speeds above 220 km/h. This speed is the maximum considered that the indications and orders are displayed to the driver via semaphore signals, light signals or markers located on the track side.

With the opening of the first high speed line between Paris and Lyon, it was set up a protection system called TVM-300, named after it allows the maximun speed up to 300 km/h.

The design of the system was based on the experience of the 200 km/h exploitation with the BAL system (Block Lumineux Automatique). However, in the TVM and given the speed of movement, block sections or cantons in which the track is divided are greater (2100 m in flat and slightly sloping as they can reach 35 mil per meter). Since the BAL system indicates the maximum speed through the orders of signals and limitations, the TVM should do the same in the cabin, too, in the BAL system, signs and combinations thereof give the orders requiring the driver to slow down or stop at appropriate points, either a speed limitation or the presence of a train ahead.

For this, the TVM cab signal provides the driver with clear, concise and unambiguous, the maximum speed that should be driving at all times, the speed at which to circulate in the next block or section, and other supplementary orders. For so, the drivers have a markers references in the points aside the track in which they must comply with the speeds, the circulation under TVM implies the presence of this block markers indicating a transition from one block to the next.

Franchisable block marker

Non Franchisable block marker

ERTMS Franchisable block marker

Later, with the design and construction of high-speed line called ‘Atlantic’ linking Paris to Tours and Le Mans, the implemented system was the TVM-430, allowing maximum commercial speed of 300 km/h or upper.

The TVM consists of two equipments, one based on land and the other on board. Both equipements implement ADA compiled software, robust programming language used on critical systems.

The TVM circulation orders are transmitted by the rails on an ac signal in every block. Four different frequencies are used for transmission, which are 1700 Hz, 2000 Hz, 2300 Hz and 2600 Hz, used both alternately in each block in the same way as in adjacent tracks of the same block. Each train has four antennas mounted two and two on the extremes, working only the two mounted before to the first axe in the direction of travel. These antennas pick up the signal encoded into a digital word of 27 bits, each bit corresponds to one of 27 frequencies encoded in the transport signal flowing through the rails. This message or code consists of the following fields:

  • Codes of speed. There are four different information about the speed: the speed at the current point, the safe speed at that point, the speed at which the train must move at the entrance of the next block and the speed at which they must move at the exit of the next block.
  • Gradient. Gradient information is determined from the slope and length of the block. This information allows the onboard system calculate speed limits for each block.
  • Length of the canton. The lengths of each block may vary, though sometimes small, these variations are important for the calculation of safety and braking distances.
  • Network Code. The networking code determines the interpretation of speed codes that are displayed in the cockpit indicators based on the calculations of onboard equipment.
  • Checking error. The message integrity is checked and sometimes corrected.

In addition, in a timely manner, through beacons placed between the rails, the train receive various information such as:

  • Entrance and exit of hight speed lines.
  • Arming and disarming the TVM system.
  • Close air intakes for ventilation.
  • Raising and lowering pantographs.
  • The opening of circuit breakers.

One of the differences between the TVM-300 and TVM-430 is in the first the updates to the speed information is done at the entrance of each block, thus the maximum velocity profile is stepped up, while the TVM-430 system allows to update throughout the block itself that is circulated by the board system can calculate a deceleration curve depending on the speed of the canton in which you are moving and the speed at to be circular in the next. As the monitoring system speed much more rigorous.

Although the regulations (Règlement S 1 ª – Titre II SNCF) is equal for both the TVM-300 and TVM-430, the difference in displays in each series of train, and the circulation in different lines results in a different mode of presentation of the sequences of instructions. It is common for both systems the existence of two types of orders or information that is given to the driver via the display:

  • Speed indications.
  • Speed warnings.

The speed indications are presented under the following forms:

 

Normal speed. Allows the driver to run at the indicated speed indicated or  the line top if is shown ‘VL VL VL’
 

Speed limitation. Tells the driver to run at the indicated speed limit.
 

Caution. Order the driver stop at the first Nf block.

The warning indications are presented as following:

 

Warning and speed limitation. Tells the driver to slow down and run at the indicated speed limit.
 

Stop. Order the driver to stop at the first block sign.
• stop if the poster is Nf.
• resuming march with caution if the poster is F.

The presentation of these orders in flashing mode indicates the driver the following speed order may be more restrictive.

The difference between on-board equipment is the number of indicators of the displays. Thus, while the TVM-300 has 2×3 indicators for speed indications and 1×3 indicators for additional information, the TVM-430 has 2×9 for speed indications and 1×3 indicators for additional information.

As for the instructions on the display, the TVM-300 system speed information except for the warnings, are presented in fixed state, while the TVM-430 system speed information except the information ’000 ‘and’ RRR ‘, are presented in fixed state or flashing but when driving on lines fitted with TVM-300, occurs the same as what mentioned at the beginning.

 

 

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