Discussion about thermal chemical-equilibrium gas and calorically perfect gas

In summary, thermal chemical-equilibrium gas is a state of equilibrium where the temperature and chemical reactions within the gas are balanced. This state can be reached through processes such as diffusion, convection, and radiation. A calorically perfect gas is a gas that follows the ideal gas law and has properties such as a constant specific heat ratio and zero heat capacity at constant volume. The main difference between thermal chemical-equilibrium gas and a calorically perfect gas is that the former takes into account the energy from chemical reactions, while the latter assumes that the internal energy is solely dependent on temperature.
  • #1
roldy
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Homework Statement


Calculate the percent difference of P2/P1, T2/T1, and ρ21 between the CPG assumption and the thermal-chemical equilibrium assumption. Which percent difference is the lowest and what could be the possible reason?

U1=4000 m/s
Altitude=60 km
R=8314 N m/(kmol K)
P1=21.96 Pa
ρ1=.0003097 kg/m3

Homework Equations


CPG:
[tex]\frac{P_2}{P_1}=1+\frac{2\gamma}{\gamma+1}(M^2-1)[/tex]
[tex]\frac{T_2}{T_1}=1+\frac{2\gamma}{\gamma+1}(M^2-1)\frac{2+(\gamma-1)M^2}{(\gamma+1)M^2}[/tex]
[tex]\frac{\rho_2}{\rho_1}=\frac{\gamma+1}{2+(\gamma-1)M^2}[/tex]

TCE:
Used an iterative process describe as follows:

Pick an ε1=.001 and ε2=.1

1)[tex]U_{2_i}=\epsilon_iU_1[/tex]

2)[tex]\rho_{2_i}=\frac{\rho_1}{\epsilon_i}[/tex]

3)[tex]P_{2_i}=P_1+\rho_1U_1^2(1-\epsilon_i)[/tex]

4)[tex]T_{2_i}=\epsilon_i \left(T_1+\frac{U_1^2}{R}(1-\epsilon_i)\right)[/tex]

5)[tex]h_{2_i}=h_1+1/2U_1^2(1-\epsilon_i^2)[/tex]

6) The pressure and density found in steps 2 and 3 are used as inputs in the supplied function to calculate the enthalpy[tex]\bar{h}_{2_i}=f(P_{2_i},\rho_{2_i})[/tex]

7)[tex]\Delta h_i=|\bar{h}_{2_i}-h_{2_i}|[/tex]

8) If Δhi≤.0001, then end the iteration. If hi> .0001, find a new ε in step 9.

9) [tex]\epsilon_{i+1}=\epsilon_i-\frac{\Delta h_i}{\frac{\Delta h_i-\Delta h_{i-1}}{\epsilon_i-\epsilon_{i-1}}}[/tex]

10) Repeat steps 1-9 until convergence

The Attempt at a Solution


I programmed this in MATLAB and the results are as follows

Percent difference between CPG and TCE:
P2/P1=3100.2
T2/T1=64.452
ρ21=259.61

Assuming these values are correct, the temperature ratio difference is the lowest out of the three. I cannot figure out the reason behind this. I've been trying to compare the dependencies of the CPG functions to the TCE ones. For CPG the pressure, temperature, and density ratios are a function of the Mach number. For TCE, the dependence is on U1 and T1. Could someone explain to me why the temperature ratio is the lowest? I can't find anything in my book or online. I know it's some basic concept I'm overlooking.
 
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  • #2


The reason for the lower percent difference in the temperature ratio is due to the fact that the CPG assumption assumes an isentropic process, while the TCE method takes into account changes in enthalpy. In an isentropic process, the enthalpy remains constant, while in a non-isentropic process, such as the one used in the TCE method, the enthalpy changes. This means that the temperature ratio calculated using the TCE method takes into account the change in enthalpy, resulting in a more accurate temperature ratio. On the other hand, the CPG method does not take into account this change in enthalpy, resulting in a slightly higher temperature ratio and a larger percent difference.
 

1. What is thermal chemical-equilibrium gas?

Thermal chemical-equilibrium gas refers to a gas that is in a state of equilibrium where the chemical reactions within the gas are balanced and the temperature of the gas remains constant. This means that the gas is not undergoing any net chemical changes and the energy within the gas is evenly distributed.

2. How does a gas reach thermal chemical equilibrium?

A gas can reach thermal chemical equilibrium through several processes, such as diffusion, convection, and radiation. These processes allow the gas molecules to mix and interact with each other, leading to a state of equilibrium where the temperature and chemical reactions are balanced.

3. What is a calorically perfect gas?

A calorically perfect gas is a gas that follows the ideal gas law, which describes the relationship between pressure, volume, and temperature of a gas. This means that the gas does not deviate from the ideal gas behavior, and its internal energy is only dependent on its temperature.

4. What are the properties of a calorically perfect gas?

The properties of a calorically perfect gas include a constant specific heat ratio, zero heat capacity at constant volume, and a constant speed of sound. These properties are consistent with the ideal gas law and can be used to calculate the behavior of the gas in various conditions.

5. How is thermal chemical-equilibrium gas different from a calorically perfect gas?

While both thermal chemical-equilibrium gas and calorically perfect gas refer to gases in a state of equilibrium, they differ in terms of their internal energy. A thermal chemical-equilibrium gas takes into account the energy from chemical reactions within the gas, whereas a calorically perfect gas assumes that the internal energy is only dependent on the gas's temperature and not on any chemical reactions.

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