- #1
colinjohnstoe
- 13
- 4
I am doing some multi-fluid hydrodynamic modelling and I have a quick question. I think I know the answer, but I am not convinced. One of the things that I need to know is the specific heat ratio, ##\gamma##, for the gas and my question is, how does one calculate this from the values of each species in the mixture.
At a given point, I know for each species the specific heat ratio, ##\gamma_i##, and number density, ##n_i##. The thermal energy density is
$$\epsilon = \frac{p}{\gamma-1},$$
where ##p## is the thermal pressure. The thermal energy density is equal to the sums of the values for each individual species
$$\epsilon = \sum_i \epsilon_i = \sum_i \frac{p_i}{\gamma_i - 1}.$$
Inserting ##p = n k_B T## (where ##n = \sum_i n_i##) and ##p_i = n_i k_B T## (therefore assuming all species have the same temperature) gives
$$\frac{n k_B T}{\gamma - 1} = \sum_i \frac{n_i k_B T}{\gamma_i - 1},$$
which can be rearranged to give
$$\gamma = \frac{n}{\sum_i \left( \frac{n_i}{\gamma_i - 1} \right) } + 1.$$
This ##\gamma## is the value that I want. Am I correct here or have I made a mistake somewhere?
At a given point, I know for each species the specific heat ratio, ##\gamma_i##, and number density, ##n_i##. The thermal energy density is
$$\epsilon = \frac{p}{\gamma-1},$$
where ##p## is the thermal pressure. The thermal energy density is equal to the sums of the values for each individual species
$$\epsilon = \sum_i \epsilon_i = \sum_i \frac{p_i}{\gamma_i - 1}.$$
Inserting ##p = n k_B T## (where ##n = \sum_i n_i##) and ##p_i = n_i k_B T## (therefore assuming all species have the same temperature) gives
$$\frac{n k_B T}{\gamma - 1} = \sum_i \frac{n_i k_B T}{\gamma_i - 1},$$
which can be rearranged to give
$$\gamma = \frac{n}{\sum_i \left( \frac{n_i}{\gamma_i - 1} \right) } + 1.$$
This ##\gamma## is the value that I want. Am I correct here or have I made a mistake somewhere?