Question

Determine the stagnation temperature and stagnation pressure of air that is flowing at \(36 \mathrm{kPa}, 238 \mathrm{K},\) and \(325 \mathrm{m} / \mathrm{s}\).

Short answer

Question: Calculate the stagnation temperature and stagnation pressure of air flowing at a static pressure of 36 kPa, static temperature of 238 K, and velocity of 325 m/s. Answer: The stagnation temperature is approximately 422.10 K, and the stagnation pressure is approximately 64.23 kPa.

Step-by-step solution

Calculate Stagnation Temperature using \(T_0 = T + \frac{V^2}{2c_p}\)

We are given \(T=238K\), \(V=325\mathrm{m/s}\), and \(c_p\approx1006\mathrm{J/(kg.K)}\). Plugging in the values, we have: \(T_0 = 238 + \frac{(325)^2}{2(1006)}\) Calculating this: \(T_0 \approx 422.10K\) The stagnation temperature of air is approximately \(422.10K\).

Calculate Stagnation Pressure using \(P_0 = P\left(\frac{T_0}{T}\right)^{\frac{\gamma}{\gamma-1}}\)

We are given \(P=36\mathrm{kPa}\), \(\gamma\approx1.4\), \(T=238K\), and from step 1, \(T_0\approx422.10K\). Plugging in the values, we have: \(P_0 = 36\left(\frac{422.10}{238}\right)^{\frac{1.4}{1.4-1}}\) Calculating this: \(P_0 \approx 64.23\mathrm{kPa}\) The stagnation pressure of air is approximately \(64.23\mathrm{kPa}\). So, the stagnation temperature and stagnation pressure of air flowing at \(36\mathrm{kPa}, 238\mathrm{K},\) and \(325\mathrm{m/s}\) are approximately \(422.10\mathrm{K}\) and \(64.23\mathrm{kPa}\), respectively.