Chapter 12.9, Problem 1E

To determine

The specific heat ratio using molar analysis.

Answer to Problem 1E

The specific heat ratio using molar analysis is $k_m=\frac{m{f}_{{\text{H}}_2}\cdot c_{p,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{p,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{p,C\text{O}}}{m{f}_{{\text{H}}_2}\cdot c_{v,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{v,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{v,C\text{O}}}$.

Explanation of Solution

Write the expression to calculate the specific heat ratio of mixture $k_m$ contains $H_2,O_{2}andCO$ using the following relation from molar analysis.

  $k_m=\frac{{\left(c_p\right)}_m}{{\left(c_v\right)}_m}$                                                                                                             (I)

Here, the specific heat at constant pressure of the mixture is ${\left(c_p\right)}_m$, the specific heat at constant volume of the mixture is ${\left(c_v\right)}_m$.

Write the expression for specific heat at constant pressure of the mixture ${\left(c_p\right)}_m$.

  ${\left(c_p\right)}_m=m{f}_{{\text{H}}_2}\cdot c_{p,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{p,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{p,C\text{O}}$                                                     (II)

Here, the mass fraction of hydrogen is $m{f}_{{\text{H}}_2}$, the mass fraction of oxygen is $m{f}_{{\text{O}}_2}$, the mass fraction of carbon monoxide is $m{f}_{\text{CO}}$,  the specific heat capacity at constant pressure of hydrogen at 300 K is $c_{p,{\text{H}}_2}$, the specific heat capacity at constant pressure of oxygen at 300 K is $c_{p,{\text{O}}_2}$, and the specific heat capacity at constant pressure of carbon monoxide at 300 K is $c_{p,C\text{O}}$.

Write the expression for specific heat at constant volume of the mixture ${\left(c_v\right)}_m$.

  ${\left(c_v\right)}_m=m{f}_{{\text{H}}_2}\cdot c_{v,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{v,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{v,C\text{O}}$                                                   (III)

Here, the specific heat capacity at constant volume of hydrogen at 300 K is $c_{v,{\text{H}}_2}$,the specific heat capacity at constant volume of oxygen at 300 K is $c_{v,{\text{O}}_2}$, specific heat capacity at constant volume of carbon monoxide at 300 K is $c_{v,C\text{O}}$.

Conclusion:

Substitute Equation (II) and (III) in (I).

  $k_m=\frac{m{f}_{{\text{H}}_2}\cdot c_{p,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{p,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{p,C\text{O}}}{m{f}_{{\text{H}}_2}\cdot c_{v,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{v,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{v,C\text{O}}}$

Thus, the specific heat ratio using molar analysis is $k_m=\frac{m{f}_{{\text{H}}_2}\cdot c_{p,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{p,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{p,C\text{O}}}{m{f}_{{\text{H}}_2}\cdot c_{v,{\text{H}}_2}+m{f}_{{\text{O}}_2}\cdot c_{v,{\text{O}}_2}+m{f}_{\text{CO}}\cdot c_{v,C\text{O}}}$.