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I'm building a rocket engine and need to know the temperature of the gases inside the combustion chamber (ie the stagnation temperature).

The combustion chamber pressure is 5 atmospheres. The engine works with ethanol and pure oxygen (inlet pressure: 10atm; levels out to 5atm once in the chamber) mixed at a perfect stoichiometric ratio.

For now, knowing the adiabatic flame temperature of ethanol under these conditions will suffice ie I don't need to take into account dissociation effects or the heat sinking properties of the metal walls of the combustion chamber.

My research has shown that one has to do the following in order to calculate the adiabatic flame temperature:

Set up an equation with the sum of enthalpies of all reactants on one side, and the sum of enthalpies of products on the other.

Enthalpy is a function of temperature. Therefore, in order to find the enthalpy of the products, one will have to experiment with different values for the temperature until the equation is equalized.

The value for the temperature that satisfies the equation is the adiabatic flame temperature.

I used this site primarily as a reference: http://braeunig.us/space/thermo.htm

The issue is, however, that enthalpy is actually a function of not only temperature, but pressure too. The site mentioned above however assumes that the pressure of the products is 1atm ie 0.1 MPa (aka reference pressure). In other words, its calculations don't apply to my case, as my combustion occurs in a much higher pressure environment.

How is the enthalpy of a substance at a given pressure & temperature calculated? Alternatively, where can I find tables that document this data if calculations aren't possible? I'm still too young to have an advanced background in thermodynamics, so please take it easy on me. I appreciate any help. Thank you!

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  • $\begingroup$ In my judgment, it is really borderline whether you need to correct for the effect of pressure on enthalpy for the gases at 5 atm and high temperature. I would start out by taking each component and determining its compressibility factor z at its partial pressure in the mixture. $\endgroup$ – Chet Miller Apr 7 at 11:15
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For a mixture of gases, the effect of pressure on enthalpy is described by the equation $$dH=\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$ This equation has to be integrated between low pressure and the pressure of the mixture. To apply this equation fundamentally, one needs to know the equation of state of the gas mixture: V = V(T,P). Moran et al (Fundamentals of Engineering Thermodynamics) have presented methods for getting the equation of state behavior of a real gas mixture based on the behavior of the pure components at the same temperature and total pressure.

For a pure gas, the integral in the above equation is evaluated for you, and expressed in a generalized corresponding states plot of the Residual enthalpy as a function of the reduced temperature and pressure of the species: Smith and Van Ness (Introduction to Chemical Engineering Thermodynamics). This plot can also be used for mixtures if an appropriate (approximate) mixing rule is employed.

One such mixing rule is Kaye's rule which starts by calculating a molar average critical temperature and critical pressure for the mixture. Based on this, the mixture can be treated as a pure species having the same critical temperature and pressure, and the generalized plot can be used for this.

Another mixing rule approach is to use so-called Additive Volume Rule, which is equivalent to the Sterling Rule. Here, the generalized plot is used for each individual species in the mixture, with the residual enthalpy evaluated as if it were a pure species at the same temperature and total pressure as the mixture; in this case, the value for the individual species residual enthalpy is also taken to be equal to the partial molar enthalpy of the species in the mixture. So, the residual enthalpy of the mixture is equal to the molar average of the residual enthalpies of the individual components.

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