Church of St Laurence, Upminster

Transcription

We just found out that in order to compute the speed of sound, we need to know how the density and pressure are related. Relations between such thermodynamic properties of a fluid are known as equations of state. In this unit, we'll go through the equations of state that will be used throughout the course. The first equation of state we're going to look at is one I hope most of you are familiar with: the ideal gas law. This tells us that the pressure (p) is equal to the product of the density (ρ), the temperature (T) and the gas constant (R). The gas constant for a specific gas is given by the universal gas constant divided by the molecular weight of a gas. Now, you could compute the sound speed just from the ideal gas law, provided that you assume that temperature was constant, and this is precisely what Newton did when he derived what's known as the isothermal sound speed. Recall that the sound speed squared is equal to the derivative of the pressure with respect to density, and here we are incorrectly assuming that this derivative is evaluated at constant temperature. Replacing p with ρ R T from the ideal gas law, then taking RT out the front of the derivative since both are constant, we find that the sound speed squared is just equal to RT. However, even the earliest experimental measurements of the sound speed were accurate enough to determine that this expression was incorrect. These experiments were performed by William Derham who had a telescope in the tower of the Church of St Laurence in Upminster, England. He had an assistant fire a shotgun from a visible point some miles away, and timed the delay between seeing the smoke and hearing the gunshot. After learning the correct expression for the sound speed in a few units time, you'll be given the task of evaluating whether the difference in travel time between a real sound wave and an isothermal one was measurable with a half second pendulum which is what they had to measure time when these experiments were done. The ideal gas law is not the only equation of state we need. Conservation of energy is usually expressed in terms of the internal energy (u) or the enthalpy (h), which is equal to u+RT, so we need to know how these are related to the other thermodynamic variables. To do this we use the specific heats of the gas. c_v is equal to δu δT at constant volume, and c_p is equal to δh δT at constant pressure. For gases at moderate pressures, these specific heats are only functions of temperature and the gas is called thermally perfect. A differential change in internal energy du is then simply given by c_v times the change in temperature dT, and dh is equal to c_p dT. The ratio of the two specific heats, which is given the symbol gamma (γ), is the second most important parameter governing compressible flow behaviour, after the Mach number (M). In the rest of the course we'll see that quantities such as the pressure ratio across shock waves or the factor by which temperature increases when a flow is brought to rest can be expressed in terms of M and gamma (γ) alone. The specific heats are also related to the gas constant (R), since by definition the derivative of enthalpy is equal to the derivative of the internal energy plus RT, which is equal to du + RdT. Replacing dh with c_p dT and du with c_v dT, then cancelling out the dT's, we see that c_p = c_v + R thus the gas constant is equal to the difference between the specific heats. This relationship is going to very useful in simplifying compressible flow relations derived later in the course. You will get a little practice in doing this in the problems following this video. For example, it's possible to completely eliminate the specific heats from any expression. Starting with R = c_p - c_v, then using the fact that c_v is equal to c_p / γ, we can factorise this and rearrange to find that c_p = γR / γ-1, and naturally c_v is equal to that same expression divided by γ. Finally, I want to mention a further simplification we can make to our equations of state. For air at low temperatures, below 1000 K, the specific heats are approximately constant, and the gas can be modelled as calorically perfect. This means that the change in internal energy between two states is c_v times the change in temperature and similarly ∆h is equal to c_p ∆T. Above 1000 K, the specific heats increase because the individual molecules that make up the gas begin to vibrate, as illustrated here for a molecule much larger than what you would typically find in air. The molecules in a gas undergo random thermal motions, and temperature is a measure of the energy of these translations. At low temperatures, the molecules also store energy by rotating, but collisions with other molecules are generally not strong enough to excite the molecules to vibrate. Once the temperature is high enough though, vibrational excitation starts to occur. This means that in order to raise the temperature of the gas, more energy is required because it will be soaked up by these vibrations in addition to random translational motions and rotation. If more energy is required for a given change in temperature, this means that specific heats have been increased. In the next unit, we will put these equations of state to use.