Question 12.4: Critical Temperature and Pressure in Gas Flow Calculate the ...
Critical Temperature and Pressure in Gas Flow
Calculate the critical pressure and temperature of carbon dioxide for the flow conditions described in Example 12–3 (Fig. 12–19).
For the flow discussed in Example 12–3, the critical pressure and temperature are to be calculated.
Assumptions 1 The flow is steady, adiabatic, and one-dimensional. 2 Carbon dioxide is an ideal gas with constant specific heats.
Properties The specific heat ratio of carbon dioxide at room temperature is k = 1.289.
Analysis The ratios of critical to stagnation temperature and pressure are determined to be
\begin{aligned}&\frac{T^{*}}{T_{0}}=\frac{2}{k+1}=\frac{2}{1.289+1}=0.87337 \\&\frac{P^{*}}{P_{0}}=\left(\frac{2}{k+1}\right)^{k /(k-1)}=\left(\frac{2}{1.289+1}\right)^{1.289 /(1.289-1)}=0.5477\end{aligned}
Noting that the stagnation temperature and pressure are, from Example 12–3, T_{0}=473 K \text { and } P_{0}=1400 kPa , we see that the critical temperature and pressure in this case are
\begin{gathered}T^{*}=0.87337 T_{0}=(0.87337)(473 K )=413 K \\P^{*}=0.5477 P_{0}=(0.5477)(1400 kPa )=767 k P a\end{gathered}
Discussion Note that these values agree with those listed in Table 12–1, as expected. Also, property values other than these at the throat would indicate that the flow is not critical, and the Mach number is not unity.
| TABLE 12–1 |
| Variation of fluid properties in flow direction in the duct described in Example 12–3 for \dot{m}=3 kg / s =\text { constant } |
| \begin{array}{ccccccl}P, kPa & T, K & V, m / s & \rho, kg / m ^{3} & c, m / s & A, cm ^{2} & Ma \\\hline 1400 & 473 & 0 & 15.7 & 339.4 & \infty & 0 \\1200 & 457 & 164.5 & 13.9 & 333.6 & 13.1 & 0.493 \\1000 & 439 & 240.7 & 12.1 & 326.9 & 10.3 & 0.736 \\800 & 417 & 306.6 & 10.1 & 318.8 & 9.64 & 0.962 \\767^{*} & 413 & 317.2 & 9.82 & 317.2 & 9.63 & 1.000 \\600 & 391 & 371.4 & 8.12 & 308.7 & 10.0 & 1.203 \\400 & 357 & 441.9 & 5.93 & 295.0 & 11.5 & 1.498 \\200 & 306 & 530.9 & 3.46 & 272.9 & 16.3 & 1.946\end{array} |
| * 767 kPa is the critical pressure where the local Mach number is unity. |