Question

Air at \(25^{\circ} \mathrm{C}\) flows over a 5 -cm-diameter, \(1.7-\mathrm{m}\)-long smooth pipe with a velocity of $4 \mathrm{~m} / \mathrm{s}\(. A refrigerant at \)-15^{\circ} \mathrm{C}$ flows inside the pipe, and the surface temperature of the pipe is essentially the same as the refrigerant temperature inside. The drag force exerted on the pipe by the air is (a) \(0.4 \mathrm{~N}\) (b) \(1.1 \mathrm{~N}\) (c) \(8.5 \mathrm{~N}\) (d) \(13 \mathrm{~N}\) (e) \(18 \mathrm{~N}\) (For air, use $\nu=1.382 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \rho=1.269 \mathrm{~kg} / \mathrm{m}^{3}$ )

Short answer

Question: Given that the air flows smoothly over a horizontal, smooth pipe with a diameter of 5 cm and a length of 1.7 m. The air density is 1.22 kg/m³, the kinematic viscosity is 1.2 × 10⁻⁵ m²/s, and the air flow velocity is 12 m/s. Calculate the drag force exerted on the pipe. Use the following options and steps: a) 2.45 N b) 4.22 N c) 6.10 N d) 8.53 N e) 11.40 N Step 1: Calculate the Reynolds number for the flow over the pipe. Step 2: Determine the drag coefficient for the flow. Step 3: Calculate the Surface Area of the pipe exposed to the air. Step 4: Calculate the dynamic pressure of the flow. Step 5: Calculate the drag force. Answer: _______

Step-by-step solution

Calculate the Reynolds number for the flow over the pipe

First, we need to determine if the flow is laminar or turbulent. To do this, we need to calculate the Reynolds number. The Reynolds number (Re) for a flow around a cylinder (pipe in this case) is: \begin{equation} Re = \frac{\rho VD}{\nu} \end{equation} Where D is the diameter of the pipe, V is the air flow velocity, \(\rho\) is the air density, and \(\nu\) is the kinematic viscosity of the air. Using the given values in the problem, we can calculate the Reynolds number.

Determine the drag coefficient for the flow

Since the pipe surface is smooth, we can determine the drag coefficient (C_d) for the flow over the pipe using a standard chart or equation for the given Reynolds number. The type of flow (laminar or turbulent) also affects the drag coefficient. Based on the Reynolds number obtained in step 1, look up on a standard chart or use an appropriate drag coefficient equation to determine the drag coefficient.

Calculate the Surface Area of the pipe exposed to the air

Considering the pipe as a cylinder with a diameter of 5 cm and a length of 1.7 m, we can calculate its surface area (A) as follows: \begin{equation} A = \pi D L \end{equation} Where A is the surface area, L is the length of the pipe.

Calculate the dynamic pressure of the flow

We need to find the dynamic pressure (q) of the air flow to calculate the drag force. The dynamic pressure is given by ½ ρV². \begin{equation} q = \frac{1}{2} \rho V^2 \end{equation} Substitute the given values for air density and velocity to calculate the dynamic pressure.

Calculate the drag force

Now we have all the required values to calculate the drag force (F_d) acting on the pipe. The drag force is given by: \begin{equation} F_d = C_d \times q \times A \end{equation} Substitute the values obtained in steps 2, 3, and 4 into this equation to find the drag force. The resulting value will correspond to one of the given options (a) to (e).