Control Volumes

In thermo, analysis often starts with the defining of boundaries around the system being analyzed. This boundary contains all that happens within the system and provides a reference point to compare to. Within this boundary is called a control volume.

Fig. 1 - Image of a control volume.

Now with systems, there are two types: closed systems and open systems.

• Closed System - no mass crosses the boundary, whereas energy, heat, etc. can cross.
• Open System - mass can cross the boundary and energy, heat, etc. can cross.

An example of a closed system would be a sealed bottle because it would not allow mass to enter or exit the boundary, but it would allow for energy transfer. For an open system, one example could be a plant where the plant absorbs water and nutrients from the soil and releases oxygen. Here, mass can flow in and out of the control volume.

System Properties

Inside of a system, it will have many properties such as pressure ($ P $), density ($ \rho $), temperature ($ T $), and volume ($ V $). These properties can be independent or dependent in systems such that they change given other factors over time; we will look at this in great detail for future lessons and tutorials!

With properties, there are two different types: intensive and extensive. Intensive properties are independent of mass such as pressure, temperature, and density. Extensive properties are otherwise dependent on mass such as mass, volume, and energy. Now notice with density ($ \rho $), it is defined as mass divided by volume, which are both extensive properties, but density is an intensive property. What is interesting about this is by dividing two extensive properties, their dependency on mass is cancelled such that an intensive property remains.

State Postulate

Pertaining to states of a system, there is a postulate that claims the state of a system is completely specified by 2 independent, intensive properties. This means with 2 known intensive properties such as pressure, temperature, and density, the state of a system can be found. In fact, if we had 2 intensive properties, we could use the ideal gas law to find the third. The ideal gas law is stated as

Where $ R $ is the gas constant and is unique for each gas. We can also solve for the internal energy of a system given its temperature and gas type by using

Here, $ e $ is the internal energy of a system, $ c_v $ is the specific heat in constant volume, and $ T $ is the temperature. $ c_v $ might look familiar from some topics in Introduction to Aerospace, but what it represents is the amount of heat required to increase the temperature by 1°C at a constant volume.

One thing we should certainly mention for states is quasi-equilibrium and non-quasi-equilibrium processes. Quasi-equilibrium processes occur very slowly, and the reason why is because they require enough time for the system to remain in internal physical thermodynamic equilibrium. Because of this, this type of process is considered ideal and reversible; this means the state in a system can be reversed and not affect anything else in the universe. As for non-quasi-equilibrium processes, these are faster processes and are hence, non-reversible. In this process type, the inertia of the gases becomes increasingly important.

Temperature

Temperature is going to be a very important part for the rest of this guide because it is a driving force for many changes in system states and stabilities. There are however intuitions that we can learn to help us throughout our studies. One of these intuitions is the Zeroth Law of Thermodynamics. This law states if there are three systems A, B, and C, each with some temperature and if the temperature of systems A and C are the same and the temperatures of B and C are the same, then the temperatures of systems A and B must be the same.

Fig. 2 - Example of the Zeroth Law of Thermodynamics

Another way to write this law is if $ (T_B = T_C ) \land (T_A = T_C ) \Rightarrow ( T_A = T_B ) $, where $ \land $ means and and $ \Rightarrow $ means to imply.

Since we are on the topic of temperature, we should also talk about different temperature scales. For English units, temperature is read in $ °F (Fahrenheit) and °R (Rankine) $, and for SI units, temperature is read in $ °C (Celcius) and K (Kelvin) $. Degrees Rankine and Kelvin are absolute temperature scales which means they have no negative values and 0 °R as well as 0 K are known as absolute zero. Degrees Fahrenheit and degrees Celcius are relative temperatures.

The conversions for these scales can be found below.

Pressure

The final property that we will talk about is pressure. Pressure is written as force over area: $ P = \frac{F}{A} $. Pressure is another important property for this guide as well. One classic example of pressure is static equilibrium of a body submerged in a liquid.

Fig. 3 - Static equilibrium example.

In Figure 3, there is a balance between pressures 2 and 1 on the top and bottom of the body, respectively. Now this difference can be written as

Where a pressure difference would result in a change in the z position. This is merely one pressure example of many, but it's fundamentals like these that will help us with our analysis!