Analysis of the Thermal Stability of a Vibrationally Non-Equilibrium Molecular Gas with Fixed Vibrational Energy Flowing in a Circular Tube


The thermal stability of a vibrationally non-equilibrium molecular gas with a fixed vibrational energy flowing in a circular tube with a constant surface temperature was theoretically analyzed. The analysis determined the boundaries for the existence of the vibrationally non-equilibrium gas state. It was shown that the upper bound for the existence of a vibrationally non-equilibrium state of a molecular gas increases linearly with increasing Reynolds number over the investigated range, 0 ≤ Re ≤ 550. It also increases when the length of the tube relative to its radius is decreased, and when the non-dimensional quantity, \({\delta ={\frac{\rho\varepsilon_{\rm eq}({\rm T}_{\rm S}){\rm r}_{0}^{2}}{\lambda\tau({\rm T}_{\rm S}){\rm T}_{\rm S}}}}\), is decreased. In this non-dimensional quantity, ρ is the density of the gas, TS is the tube surface temperature, τ is the gas vibrational relaxation time, ε eq is the gas vibrational energy at TS, and λ is the gas coefficient for heat conduction. To obtain a large storage of excess vibrational energy, low values of δ must be used.

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ε :

Excess gas vibrational energy

ε eq :

Equilibrium gas vibrational energy

ε c :

Critical excess gas vibrational energy


Gas translational temperature

TS :

Tube surface temperature

Tmax :

Maximum gas translational temperature along tube axis

Tv :

Gas vibrational temperature

τ :

Gas vibrational-relaxation time


Gas heat-conduction coefficient for equilibrium degrees of freedom

CV :

Gas specific-heat capacity for equilibrium degrees of freedom

ρ :

Gas density


Gas velocity


Gas pressure

μ :

Coefficient of gas viscosity

r0 :

Tube radius


Tube length

\({\gamma=\frac{{\rm L}}{{\rm r}_{\rm 0}}}\) :

Ratio of tube length to tube radius


Equilibrium gas vibrational energy at tube surface temperature Ts

\({\beta=\frac{\varepsilon_{\rm c}}{\varepsilon_{\rm eq} \left({{\rm T}_{\rm s}}\right)}}\) :

Non-dimensional critical vibrational energy


Gas vibrational relaxation time at tube surface temperature Ts

\({\delta=\frac{\rho\varepsilon_{\rm eq}\left({\rm T}_{\rm S}\right){\rm r}_{\rm 0}^{\rm 2}}{\lambda\tau\left({\rm T}_{\rm S}\right){\rm T}_{\rm S}}}\) :

Non-dimensional quantity containing gas parameters

\({\phi_{\rm f}}\) :

Heat flux out of the gas by flow

\({\phi_{\rm s}}\) :

Heat flux out of the gas by conduction

\({\phi=\phi_{\rm f}/\phi_{\rm s}}\) :

Non-dimensional heat flux


Reynolds number


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Correspondence to Salahdeen M. Younis.

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Younis, S.M. Analysis of the Thermal Stability of a Vibrationally Non-Equilibrium Molecular Gas with Fixed Vibrational Energy Flowing in a Circular Tube. Arab J Sci Eng 36, 137–143 (2011).

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  • Molecular energy transfer
  • Heat transfer