Gas Flow through a Converging–Diverging Duct Carbon dioxide flows steadily through a varying cross-sectional area duct such as a nozzle shown in Fig. 12–12 at a mass flow rate of 3 kg/s. The carbon dioxide enters the duct at a pressure of 1400 kPa and 200°C with a low velocity, and it expands in

Question

Gas Flow through a Converging–Diverging Duct

Carbon dioxide flows steadily through a varying cross-sectional area duct such as a nozzle shown in Fig. 12–12 at a mass flow rate of 3 kg/s. The carbon dioxide enters the duct at a pressure of 1400 kPa and 200°C with a low velocity, and it expands in the nozzle to a pressure of 200 kPa. The duct is designed so that the flow can be approximated as isentropic. Determine the density, velocity, flow area, and Mach number at each location along the duct that corresponds to a pressure drop of 200 kPa.

Accepted Answer

Carbon dioxide enters a varying cross-sectional area duct at specified conditions. The flow properties are to be determined along the duct.
Assumptions 1 Carbon dioxide is an ideal gas with constant specific heats at room temperature. 2 Flow through the duct is steady, one-dimensional, and isentropic.
Properties For simplicity we use [latex] c_{p}=0.846 kJ / kg \cdot K ext { and } k=1.289 [/latex] throughout the calculations, which are the constant-pressure specific heat and specific heat ratio values of carbon dioxide at room temperatures. The gas constant of carbon dioxide is R = 0.1889 kJ/kg · K.
Analysis We note that the inlet temperature is nearly equal to the stagnation temperature since the inlet velocity is small. The flow is isentropic, and thus the stagnation temperature and pressure throughout the duct remain constant. Therefore,

[latex] T_{0} \cong T_{1}=200^{\circ} C =473 K [/latex]

and

[latex] P_{0} \cong P_{1}=1400 kPa [/latex]

To illustrate the solution procedure, we calculate the desired properties at the location where the pressure is 1200 kPa, the first location that corresponds to a pressure drop of 200 kPa.
From Eq. 12–5,

[latex] T=T_{0}\left(\frac{P}{P_{0}} ight)^{(k-1) / k}=(473 K )\left(\frac{1200 kPa }{1400 kPa } ight)^{(1.289-1) / 1.289}=457 K [/latex]

[latex] \frac{P_{0}}{P}=\left(\frac{T_{0}}{T} ight)^{k /(k-1)} [/latex] (12–5)

From Eq. 12–4,

[latex]\begin{aligned}V &=\sqrt{2 c_{p}\left(T_{0}-T ight)} \&=\sqrt{2(0.846 kJ / kg \cdot K )(473 K -457 K )\left(\frac{1000 m ^{2} / s ^{3}}{1 kJ / kg } ight)} \&=164.5 m / s\end{aligned}[/latex]

[latex] T_{0}=T+\frac{V^{2}}{2 c_{p}} [/latex] (12–4)

From the ideal-gas relation,

[latex] ho=\frac{P}{R T}=\frac{1200 kPa }{\left(0.1889 kPa \cdot m ^{3} / kg \cdot K ight)(457 K )}=13.9 kg / m ^{3} [/latex]

From the mass flow rate relation,

[latex] A=\frac{\dot{m}}{ ho V}=\frac{3 kg / s }{\left(13.9 kg / m ^{3} ight)(164.5 m / s )}=13.1 imes 10^{-4} m ^{2}=13.1 cm ^{2} [/latex]

From Eqs. 12–11 and 12–12,

[latex]\begin{aligned}c &=\sqrt{k R T}=\sqrt{(1.289)(0.1889 kJ / kg \cdot K )(457 K )\left(\frac{1000 m ^{2} / s ^{2}}{1 kJ / kg } ight)}=333.6 m / s \Ma &=\frac{V}{c}=\frac{164.5 m / s }{333.6 m / s }=0.493\end{aligned}[/latex]

[latex] c=\sqrt{k R T} [/latex] (12–11)

[latex] Ma =\frac{V}{c} [/latex] (12–12)

The results for the other pressure steps are summarized in Table 12–1 and are plotted in Fig. 12–13.
Discussion Note that as the pressure decreases, the temperature and speed of sound decrease while the fluid velocity and Mach number increase in the flow direction. The density decreases slowly at first and rapidly later as the fluid velocity increases.

TABLE 12–1

Variation of fluid properties in flow direction in the duct described in Example 12–3 for [latex] \dot{m}=3 kg / s = ext { constant } [/latex]

[latex]\begin{array}{ccccccl}P, kPa & T, K & V, m / s & ho, 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}[/latex]

* 767 kPa is the critical pressure where the local Mach number is unity.

Question 12.3: Gas Flow through a Converging–Diverging Duct Carbon dioxide ...

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