Critical Temperature and Pressure in Gas Flow Calculate the critical pressure and temperature of carbon dioxide for the flow conditions described in Example 17–3 (Fig. 17–19).

Question

Accepted Answer

For the flow discussed in Example 17–3, 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 (Table A–2a).
Analysis The ratios of critical to stagnation temperature and pressure are determined to be

[latex]\frac{T^{*}}{T_{0}}=\frac{2}{k+1}=\frac{2}{1.289+1}=0.8737[/latex]

[latex]\frac{P^{*}}{P_{0}}=\left(\frac{2}{k+1} ight)^{k /(k-1)}=\left(\frac{2}{1.289+1} ight)^{1.289 /(1.289-1)}=0.5477[/latex]

Noting that the stagnation temperature and pressure are, from Example 17–3, [latex] T_{0}=473 K ext { and } P_{0}=1400 kPa [/latex], we see that the critical temperature and pressure in this case are

[latex]\begin{aligned}&T^{*}=0.8737 T_{0}=(0.8737)(473 K )=413 K \&P^{*}=0.5477 P_{0}=(0.5477)(1400 kPa )=767 kPa\end{aligned}[/latex]

Discussion Note that these values agree with those listed in Table 17–1, 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 17–1

Variation of fluid properties in flow direction in duct described in Example 17–3 for [latex] \dot{m}=3 kg / s = ext { constant } [/latex]
P, kPa	T, K	V, m/s	ρ, kg/m³	c, m/s	A, cm²	Ma
1400	473	0	15.7	339.4	∞	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
* 767 kPa is the critical pressure where the local Mach number is unity.

Question 17.4: Critical Temperature and Pressure in Gas Flow Calculate the ...

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