To solve this problem, we need to use principles of fluid mechanics, specifically the Bernoulli equation and the Darcy-Weisbach equation for head loss due to friction in a pipe.

Given Data

• Oil: SAE 50W at 20°C
• Density ($\rho$) = 902 kg/m³
• Dynamic viscosity ($\mu$) = 0.86 kg/m·s
• Length of the tube ($L$) = 160 cm = 1.6 m
• Diameter of the tube ($D$) = 5 cm = 0.05 m
• Elevation difference: $z_1 = 2$ m, $z_2 = 0.8$ m
• Required: Flow rate ($Q$) in m³/hr

Approach

1. Bernoulli Equation: We will assume steady, incompressible flow between points (1) and (2). Since the oil flows from a tank, we can assume atmospheric pressure at both ends. We ignore kinetic energy due to negligible velocity change between the tanks. The main energy difference is due to the elevation difference and head loss.

2. Head Loss Calculation: Use the Darcy-Weisbach equation to calculate the head loss due to friction. The head loss $h_f$ in the pipe is given by: $h_f = f \frac{L}{D} \frac{V^2}{2g}$ where:

   • $f$ = Darcy friction factor (we can calculate it using the Reynolds number).
   • $L$ = Length of the pipe.
   • $D$ = Diameter of the pipe.
   • $V$ = Velocity of the fluid in the pipe.
   • $g$ = Acceleration due to gravity (9.81 m/s²).

3. Flow Rate Calculation: Use the continuity equation to relate the flow rate $Q$ and velocity $V$: $Q = A \cdot V$ where $A$ is the cross-sectional area of the pipe, $A = \frac{\pi D^2}{4}$.

4. Reynolds Number Calculation: To determine whether the flow is laminar or turbulent, calculate the Reynolds number: $\text{Re} = \frac{\rho V D}{\mu}$ If $\text{Re} < 2000$, the flow is laminar, and we use $f = \frac{64}{\text{Re}}$. If $\text{Re} > 4000$, the flow is turbulent, and we use an empirical correlation for $f$.

Let's perform the calculations in sequence.The estimated flow rate of SAE 50W oil in this setup is approximately 4.26 m³/hr.

Would you like a breakdown of the calculations, or do you have any further questions?

Here are some additional related questions to deepen understanding:

1. How would changing the pipe diameter affect the flow rate, assuming other factors remain constant?
2. What is the impact of oil viscosity on flow rate, and how would higher or lower viscosity alter this result?
3. How does the friction factor vary between laminar and turbulent flow in this context?
4. How could adding a pump between the two tanks influence the flow rate and pressure requirements?
5. How would you apply this calculation method to different fluids, such as water or gasoline?

Tip: Understanding how to adjust flow rate calculations for different pipe lengths and diameters can help you tackle diverse fluid dynamics problems more efficiently.