Calculating combustion temperature of mixture

[The Sparrow]

Hey guys, I'm trying to write a small article that very simply explains some of the math involved with a rocket engine. I'm not looking for a very accurate answer so I tried the following method

1) Stoichiometry on the fuel which was sugar and saltpeter and their respective enthalpies of formation which I have as follows:

Sugar (sucrose) - 2221.5 kJ/mol (5 mol)
+
Saltpeter - 494 kJ/mol (48 mol)
=
Water - 285 kJ/mol (36 mol)
Nitrogen - 0 kJ/mol (24 mol)
CO2 - 393 kJ/mol (55 mol)
Potassium Carbonate 1150.8 kJ/mol (24 mol)

Giving me 24720 joules for 5 moles of sugar combined with 48 moles saltpeter.

5 moles of sugar and 48 moles of saltpeter weigh about 6.722 kilograms. So I divided 24720 kJ/Mol by that to get:

3677 Kilo-joules released for every kilogram of sugar/saltpeter mixture.

2) Now we know the amount of energy that will be released, we can work out how hot it will get. For example, if you switch on the kettle, it takes a certain amount of energy from the wall socket transferred to the water in the kettle to get it up to a certain temperature. The ratio of energy to rise in temperature is called a the heat capacity. And for water it's like 2.2 or something. That means for one kilogram of water to be raised by one degree Celsius, you need 2.2 Kilo-joules of energy.Every substance has its own heat capacity, and I'll find the average heat capacity of the burnt hot exit gasses below so we can predict the temperature. Heat capacity of

Carbon Dioxide: 1.247 kilojoules to raise 1kg by a degree

Nitrogen: 1.04 kilojoules to raise 1kg by a degree

Water: 2.28 kilojoules to raise 1kg by a degree

Potassium Carbonate (salt): 1.3768 kilojoules to raise 1kg by a degreeI'm going to get the overall heat capacity of the exit gass mixture by multiplying the heat capacities of those 4 things by the fraction that they contribute to the total mass output. If we had 1 kg of mixture that was burned, then we will have 0.493 kg salt, 0.36kg Carbon Dioxide, 0.0964kg Water, 0.05kg Nitrogen. (This is the heat capacity at an expected 1800 degree temperature)

Here is where I have the problem with the result I get:
So, we have 3677 Killojoules, and we start off with 293 degrees Kelvin (room temperature). We raise it one degree for every 1.4 killojoules so we end up with a temperature of 2899 + 293 K which is almost twice the expected temperature.

I haven't done these calculations much and I think there may be several things wrong with what I'm doing. Firstly, for simplicity I assumed constant heat capacity but didn't expect the error to be 80%. I may also be doing this completely wrong and have no idea what garbage I'm punching into Mathcad.

If you guys have any suggestions on where my fault lies it would be a big help.

Thanks in advance

 

[Achy47]

Hello,

I appreciate your effort to explain the math involved with a rocket engine. However, I believe there are a few errors in your calculations that may be leading to the incorrect result.

Firstly, in your stoichiometry calculation, you have not taken into account the mass of the reactants. In order to calculate the energy released per kilogram of sugar/saltpeter mixture, you need to divide the energy released per mole by the molar mass of the mixture (which is approximately 184 g/mol). This will give you a more accurate value for the energy released per kilogram of mixture.

Secondly, in your calculation for the heat capacity of the exhaust gases, you have not taken into account the heat capacity of the unreacted fuel (sugar and saltpeter). This will affect the overall heat capacity of the exhaust gases and may change the expected temperature.

Lastly, as you mentioned, the heat capacity of gases can vary with temperature. Therefore, it is important to use the specific heat capacity at the expected temperature (1800 degrees) rather than at room temperature.

I suggest double-checking your calculations and taking into account the above points to see if it changes your result. Also, it may be helpful to consult a textbook or online resources for more accurate values of heat capacities and molar masses.

I hope this helps and good luck with your article!