To calculate the volume flow rate of the motor oil when it's at -30°C, we can use the Poiseuille's Law for laminar flow through a pipe, which is given by:

$Q = \frac{\pi r^4 \Delta P}{8 \eta L}$

Where:

• $Q$ is the volume flow rate (m³/s)
• $r$ is the radius of the tube (m)
• $\Delta P$ is the pressure difference (Pa)
• $\eta$ is the viscosity of the oil (N⋅s/m²)
• $L$ is the length of the tube (m)

Step-by-Step Calculation:

1. Convert the given values into consistent units:

   • Radius $r = 5.2 \, \text{mm} = 5.2 \times 10^{-3} \, \text{m}$
   • Length $L = 2.2 \, \text{m}$
   • Pressure difference $\Delta P = 115 \, \text{kPa} = 115 \times 10^3 \, \text{Pa}$
   • Viscosity $\eta = 6.4 \, \text{N⋅s/m}^2$ (at -30°C)

2. Substitute the values into Poiseuille's law:

$Q = \frac{\pi (5.2 \times 10^{-3})^4 (115 \times 10^3)}{8 (6.4) (2.2)}$

3. Solve the expression:

First, calculate the power of $r^4$:

$r^4 = (5.2 \times 10^{-3})^4 = 7.38 \times 10^{-11} \, \text{m}^4$

Now substitute this into the equation:

$Q = \frac{\pi \times (7.38 \times 10^{-11}) \times (115 \times 10^3)}{8 \times 6.4 \times 2.2}$

$Q = \frac{3.9 \times 10^{-6}}{112.64}$

$Q = 3.46 \times 10^{-8} \, \text{m}^3/\text{s}$

4. Convert to a more practical unit (liters per second):

Since $1 \, \text{m}^3 = 1000 \, \text{L}$:

$Q = 3.46 \times 10^{-8} \, \text{m}^3/\text{s} \times 1000 \, \text{L/m}^3 = 3.46 \times 10^{-5} \, \text{L/s}$

Thus, the volume flow rate when the oil is at -30°C is:

$Q \approx 3.5 \times 10^{-5} \, \text{L/s}$

Final Answer:

$Q \approx 0.000035 \, \text{L/s}$

This is the volume flow rate produced by the pump when the oil is at -30°C.

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Would you like to explore more details, or is there anything specific you'd like clarified? Here's a list of questions related to this:

1. How would the flow rate change if the oil's viscosity were lower at a higher temperature?
2. How does the pressure difference across the tube impact the volume flow rate?
3. What effect would increasing the tube length have on the flow rate?
4. What role does the radius of the tube play in fluid dynamics in this case?
5. How would the flow rate differ in turbulent flow compared to laminar flow?

Tip: Always check that all units are consistent before plugging values into equations.