Question

Bio-oil flows through a duct with a radius of \(2.5 \mathrm{~cm}\) at ambient temperature with a velocity \(\left(\mathrm{v}_{1}\right)\) of \(10 \mathrm{~m} \cdot \mathrm{s}^{-1}\); the duct is followed by a permeable wall part with suction. At the end of this section (with the same radius), the velocity \(\left(\mathrm{v}_{2}\right)\) has dropped to \(8 \mathrm{~m} \cdot \mathrm{s}^{-1}\). If \(p_{1}=140 \mathrm{kPa}\), estimate \(p_{2}\) for the case that wall friction is negligible. What happens to \(p_{2}\) in the case of significant friction?

Short answer

Answer: If there were significant friction, the new pressure (p2) in the duct would be lower than the case with negligible friction. This is because high friction causes a pressure drop in the fluid flow. The actual value of p2 will depend on the specific value of the work done due to friction.

Step-by-step solution

Write Bernoulli's Equation for flow through the duct

Bernoulli's Equation is given by: $$ p_{1} + \frac{1}{2} \rho v_{1}^{2} = p_{2} + \frac{1}{2} \rho v_{2}^{2} + w $$ Where \(p_{1}\) and \(p_{2}\) are the pressures at the upstream and downstream of the pipe (in Pa), and \(v_{1}\) and \(v_{2}\) are the velocities of the fluid flow (in m/s) at these pressure points, and \(\rho\) is the fluid density (in kg/m³), and \(w\) represents the work done (in J) due to friction, which is negligible in the first case.

Calculate initial kinetic energy

Calculate the initial kinetic energy term \(\frac{1}{2} \rho v_{1}^{2}\), where \(v_{1} = 10 \,m/s\): $$ \frac{1}{2} \rho v_{1}^{2} = \frac{1}{2}\rho (10\, m/s)^{2} $$

Calculate final kinetic energy

Calculate the final kinetic energy term \(\frac{1}{2} \rho v_{2}^{2}\), where \(v_{2} = 8 \,m/s\): $$ \frac{1}{2} \rho v_{2}^{2} = \frac{1}{2}\rho (8\, m/s)^{2} $$

Rearrange Bernoulli's Equation to find p2 with negligible friction

Since we are assuming friction to be negligible for this case, the work done \(w=0\). We can plug the values we obtained in steps 2 and 3 into Bernoulli's Equation, and rearrange to find p2: $$ p_{2} = p_{1} + \frac{1}{2} \rho v_{1}^{2} - \frac{1}{2} \rho v_{2}^{2} $$

Estimate p2 with given values

Now we can plug in the given values for \(p_1 = 140\, kPa\) (convert to Pa), \(v_{1} = 10\, m/s\), and \(v_{2} = 8\, m/s\). Note that we do not know the bio-oil density, but that's okay because the term will cancel out in the equation: $$ p_{2} = 140,000\, Pa + \frac{1}{2} \rho(10\, m/s)^{2} - \frac{1}{2} \rho(8\, m/s)^{2} $$ $$ p_{2} = 140,000\, Pa + \frac{1}{2} \rho(100 - 64)\, (m^2/s^2) $$ $$ p_{2} = 140,000\, Pa + 18 \rho\, (m^2/s^2) $$ We can see, in this case, that \(p_{2}\) is dependent on the fluid density, and we cannot estimate an exact value.

Discuss the case of significant friction

If there is significant friction, the value of \(w\) won't be zero in the Bernoulli's Equation, which means it will contribute to the energy balance. High friction causes a pressure drop in the fluid flow, which means \(p_{2}\) will be lower than the case with negligible friction. The actual value of \(p_{2}\) will depend on the specific value of the work done due to friction (\(w\)).

Key concepts

Fluid Dynamics

Fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the science of liquids and gases in motion. This area of physics is crucial when analyzing how fluids behave under various forces and in different conditions. In the context of the exercise, fluid dynamics helps us understand how bio-oil behaves as it flows through a duct.

When a fluid travels through a duct, its behavior can be described by the principles of conservation of mass and conservation of energy. Bernoulli's equation is one such principle derived from these conservation laws, which relates the velocity of the fluid, its pressure, and its height at different points along the flow. Essentially, as the fluid's velocity changes due to duct geometry or other factors, the pressure within the fluid also changes; this phenomenon is typically observable in both incompressible and steady-flow conditions.

Kinetic Energy in Fluid Flow

Kinetic energy in fluid flow refers to the energy that a fluid possesses due to its motion. It's part of the total energy of a flowing fluid and can be quantified using the equation \(\frac{1}{2} \rho v^2\), where \(\rho\) represents the fluid density and \(v\) its velocity. The higher the velocity, the greater the kinetic energy associated with the fluid particles.

In our exercise example, as the bio-oil flows through the duct, we calculate the kinetic energy at two different points using the respective velocities. This is key to applying Bernoulli's equation, which includes kinetic energy as one of its terms. By initially calculating the kinetic energy at the entry and exit points of the bio-oil flow, we prepare to understand how its pressure changes along the way.

Pressure Drop Due to Friction

When dealing with fluid flow in real-world applications, friction is often an important factor. It refers to the resistance that the fluid encounters as it moves along the walls of a pipe or duct. Due to this friction, energy from the fluid flow is dissipated as heat, resulting in what is known as a pressure drop.

In Bernoulli's equation, this pressure drop due to friction is accounted for by the work term \(w\). If the impact of friction is negligible, then \(w\) is zero, which simplifies the application of Bernoulli's equation. However, when friction cannot be ignored, we must consider this work term because it represents energy loss that affects the fluid's pressure. As seen in the step-by-step solution, significant wall friction means the downstream pressure \(p_2\) will be lower than if friction were negligible because the frictional work done on the fluid reduces its total energy.