Air enters a 4 -cm-square galvanized steel duct with \(p_{0}=\) \(150 \mathrm{kPa}, T_{0}=400 \mathrm{K},\) and \(V_{1}=120 \mathrm{m} / \mathrm{s},\) (Note: \(\mu=2.2 \times 10^{-5}\) \(\left.\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) (a) Compute the maximum possible duct length for these conditions. (b) If the actual duct length is 0.75 times the maximum value, calculate the mass flow rate, the exit pressure, and the stagnation pressure. (c) If the actual duct length is 1.3 times the maximum value for the stated conditions, compute the new mass flow rate and inlet velocity. Assume a low back pressure and use the same value of \(f\) as used in the maximum possible duct length case.

Step by step solution

01

Calculate the maximum possible duct length

To calculate the maximum possible duct length for these conditions, we need to use the formula based on the concept of Reynolds number. The maximum duct length (\(L_{max}\)) can be calculated using the relation \(Re = \rho V D / \mu\), where \(\rho\) is the density of air, \(V\) is the air velocity, \(D\) is the duct diameter and \(\mu\) is the dynamic viscosity. Since we have all the quantities here, we can substitute these values into the formula and calculate the maximum duct length.

02

Calculate the Mass Flow Rate, Exit Pressure, and Stagnation Pressure for 0.75 times the Maximum Length

In this case, the duct length is 0.75 times the maximum length. We can firstly derive the mass flow rate (\(\dot{m}\)) using the relation \(\dot{m} = \rho V A\), where \(A\) is the cross-sectional area of the duct. After we find \(\dot{m}\), we can calculate the exit pressure from the Bernoulli’s equation. Lastly, we can get the stagnation pressure using the formula \(p_{0} = p_{e} + 0.5\rho V^{2}\), where \(p_{e}\) is the exit pressure.

03

Compute the New Mass Flow Rate and Inlet Velocity for 1.3 times the Maximum Length

In this sub-part, the duct length is 1.3 times the maximum length. We should recalculate the mass flow rate and the inlet velocity. The same equations will be used for mass flux (\(\dot{m} = \rho V A\)) and the Bernoulli’s equation to find the new inlet velocity. The calculated mass flow rate and inlet velocity that were calculated earlier will have to be replaced with the new calculated values.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!