Welcome back to Sports and Building Aerodynamics. In the first week on basic aspects of fluid flow. In this second module, we continue our focus on fluid properties. And, we start again, with the module question. If an elite cyclist wants to establish a new world hour record on an indoor velodrome. Which location should she or he choose? Is it Eindhoven in The Netherlands at 35 meters about mean sea level? Is it Högvalen, Sweden at 835 meters above mean sea level? Is it Mexico City at 2,250 meters above mean sea level. Or finally La Paz, Bolivia at 3,640 meters above mean sea level? Please hang on to your answer and we'll come back to this question later in this module. At the end of this module, you will understand the fluid property density. You'll understand how density changes as a function of temperature and pressure. You'll also understand how air density changes as a function of altitude. And how cycling aerodynamics is influenced by temperature and altitude. [BLANK_AUDIO]. We focused on physical properties also in the previous module. There we addressed velocity, pressure, and temperature. This module will be exclusively focused on density. First, some etymology. Density derives from the Latin densitas it means thickness, or densus, thick or dense. And is actually simply defined as mass divided by volume and the unit is kilogram per cubic meter. The density of air actually influences the air resistance of a cyclist. Imagine this cyclist here and we have only draw two forces. One is the force exerted by the wheel on the road which actually pushes the cyclist and the bicycle forward. And then the resistance which is to the largest part made up of the aerodynamic drag. And this aerodynamic drag can be calculated with this equation. Where D is the drag force, A is the frontal area of the cyclist and bicycle, CD is the drag coefficient, rho is the air density, U is the relative air speed. And then we divide this product by two. Some typical values. Often the frontal area and the drag coefficient are combined in the so-called drag area. And for a time-trial position of an elite cyclist this is about 0.211 square meters. Let's then assume a wind speed or a relative cycling speed, relative air speed, sorry, a cycling speed of 50 meters per second. And then for different densities we can calculate the drag force and you see indeed that the density has a very large impact on the drag force. The density also changes with temperature and with pressure. And actually when you increase the pressure you decrease the volume, so you increase the density. And these changes are large for gases typically, but small for liquids. And that is also indicated by the animations on this slide. However, also the air resistance of a cyclist will change with temperature and pressure, and we're going to look at that later. Let's first focus on how density of water changes as a function of temperature but with a constant pressure of one atmosphere. Then this is a demonstration of the change of the density. When you look at for example, the difference that we get whether we have 0 degrees C or 30 degrees C. You see that actually the difference in density is quite small. 0.4% change for a temperature difference of 30 degrees C. But let´s look at air. How the density of air changes as a function of temperature and then for the same values of temperature, so for 0 and 30 degrees you see that the difference is much more pronounced. Here we have a 10% change for this temperature difference of 30 degrees C. So this has implications for cycling. Let's focus again on the cyclist where we, earlier explained these two forces, where the drag force is indicated by this equation. We have some typical values here. Let's assume also for density now that you take 1.225 kilograms per cubic meter. Which is the value for the density of air at 15 degrees C and one atmosphere. Then you can calculate the drag force, which is this value. But let's now look at how this value changes with temperature and with pressure. This is actually, for a cyclist that's cycling at one atmosphere, this is about sea level. With a wind speed or a cycling speed actually, of 15 meters per second, so 54 kilometers an hour. You see that the drag force is very much influenced by temperature. However, the temperature range indicated here is not very realistic for cycling races. If you would translate this unrealistic range of temperatures, and of the drag force to the time gain that you could get over 50 kilometers, which is a typical time-trial distance, you can see that the difference in seconds is very large. However, as I mentioned before, the temperature range here is unrealistic. But if you look at a more realistic range, for example, between 10 and 20 degrees C. You see that the time difference, the time-gain you could get is 60 seconds and this is substantial. However of course, when cyclists are cycling together in a race, they are exposed to the same temperature conditions. However if you attempt a world hour record, this is different. You have to compete with records established before you in maybe different temperature conditions. Density of air also changes with pressure and let's look at these influences. While it changes with pressure, and therefore, because pressure changes with altitude, it also changes with altitude. And these are some important equations. This is the equation describing the variation of temperature with altitude, where L is the temperature lapse rate. Then we can express the change in pressure so the decrease in pressure with height by this equation. Where you see the gravitational acceleration appearing, the molar mass of dry air, and the ideal gas constant, R. And then finally the density at a given altitude h can be calculated with this equation. Where you again see the molar mass, the ideal gas constant, temperature, and pressure p. So let's use these equations and then calculate how density changes as a function of altitude and this is the graph illustrating that. So you see a very drastic decrease of density with altitude. If you translate that to drag forces, you'll see that also the drag force decreases very drastically with altitude. If you translate that again to a theoretical cycling speed that you could achieve in your world hour attempt. Then you see that also density really drastically increases the theoretical cycling speed. This effect is very, very pronounced. However, it's not realistic. Because actually when you're cycling at high altitude or doing sport, or performance in general at high altitude, actually your oxygen uptake is also decreased. And then you get these curves for the cycling speed, which seem rather flat but this is due to the fact that the right axis has a quite wide range. If you zoom in, you see that here also there's a clear benefit, there's a clear maximum of cycling at higher altitude. There's a curve that shows the situation after acclimatization when your oxygen uptake will be better and before acclimatization. I have to mention that these are the curves based on the biomedical characteristics of Chris Boardman at the time of his world record attempt. So let's go back to the module question then. If an elite cyclist wants to establish a new world hour record on an indoor velodrome, which location should she or he choose? And there are four locations being offered, so let's have a look. well, this is Eindhoven at only very limited height above mean sea level. And if you would do this world hour record attempt, so you cycle for one hour then you have this distance of 56.4 kilometers. If you do the same attempt in Högvalen, Sweden, your distance will be substantially larger. Even in Mexico City it increases even further but then in La Paz, Bolivia, it would be less. And the reason there is that the decreased oxygen uptake will become relatively more important. So clearly there is an optimum. And actually, the optimum is not even Mexico City. But again for the biomedical characteristics of Chris Boardman, there you find it's a bit more than 2,400 meters above mean sea level. In this module, we've learned about the fluid property density. How density changes as a function of temperature and pressure. How air density changes as a function of altitude. And how cycling aerodynamics is influenced by temperature and altitude. In the next module, we are going to focus on the fluid property viscosity. We are going to see how viscosity changes as a function of temperature and how viscosity influences cycling aerodynamics. Thank you very much for watching and I hope to see you again in the next module.
