# Explain the equation describing total energy of the electrons in HET

I have this equation describing total energy of the electrons in HET (Hall-effect thruster):

$$\frac{\partial (\frac{1}{2}mn(v_{ex}+v_{e\vartheta})^2 + \frac{3}{2}nT_e)}{\partial t}+\frac{\partial (\frac{1}{2}mnv_{ex}(v_{ex}+v_{e\vartheta})^2 + \frac{5}{2}nv_{ex}T_e)}{\partial x}=S$$

I do not understand terms $$\frac{3}{2}nT_e$$ and $$\frac{5}{2}nv_{ex}T_e$$ and also I do not understand the whole $$\frac{\partial (...)}{\partial x}$$ part.

Could you please explain or at least provide me with some link to explanation?

English is not my native language. I should have say Control volume form of the first law : https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node19.html

$$\frac{dE(V)}{dt}+{{\left( {{D}_{v}}\left( h+\frac{1}{2}{{v}^{2}} \right) \right)}_{s}}-{{\left( {{D}_{v}}\left( h+\frac{1}{2}{{v}^{2}} \right) \right)}_{e}}=\text{other terms}$$

With $${{D}_{v}}=Sv$$ the volume flow rate.

Here, you just have to apply to a small volume element between $$x$$ and $$x+dx$$ :

$$E(V)=\left( u+\frac{1}{2}{{v}^{2}} \right)Sdx$$

and $${{\left( {{D}_{v}}\left( h+\frac{1}{2}{{v}^{2}} \right) \right)}_{x+dx}}-{{\left( {{D}_{v}}\left( h+\frac{1}{2}{{v}^{2}} \right) \right)}_{x}}=\frac{\partial }{\partial x}\left( {{D}_{v}}\left( h+\frac{1}{2}{{v}^{2}} \right) \right)dx=\frac{\partial }{\partial x}\left( v\left( h+\frac{1}{2}{{v}^{2}} \right) \right)Sdx$$

Conclusion $$\frac{\partial }{\partial t}\left( u+\frac{1}{2}{{v}^{2}} \right)+\frac{\partial }{\partial x}\left( v\left( h+\frac{1}{2}{{v}^{2}} \right) \right)=...$$

I am not at all a plasma specialist.

Simply, I recognize the local first principle applied to an open system in one dimension: $$\frac{\partial ({{e}_{c}}+u)}{\partial t}+\frac{\partial ({{v}_{x}}({{e}_{c}}+h))}{\partial x}=\text{source term}$$

The term $$\frac{1}{2}mn{{({{v}_{ex}}+{{v}_{e\vartheta }})}^{2}}+\frac{3}{2}n{{T}_{e}}$$ is the volumic density of energy.

$$\frac{1}{2}mn{{({{v}_{ex}}+{{v}_{e\vartheta }})}^{2}}$$ is the cinetic volumic density of energy.

$$\frac{3}{2}nT$$ is the thermal volumic density, product of the density particle by the thermal energy by particle. Probably, temperature is measured with the Boltzman constant included $${{k}_{B}}T$$ .

$$\frac{5}{2}nT$$ is the volumic density of enthalpy. You have to use the enthalpy and not the internal energy to take into account the pressure work on the system.

For a perfect gas, the molar enthalpy is $$h=u+Pv=u+n{{k}_{B}}T=\frac{5}{2}n{{k}_{B}}T$$

Hope it can help and sorry for my poor english.

• Thank you, included Boltzmann constant and enthalpy explain a lot. Could you please give me more info about that “local first principle applied to an open system”? I can’t find it anywhere. – Andrej Jan 28 at 8:51
• I added an explanation about the control volume form of the first law for a differential volume. – Vincent Fraticelli Jan 28 at 9:36