Engine oil (η = 0.20 Pa⋅s) passes through a fine 2.60-mm-diameter tube that is 8.60 cm long. What pressure difference is needed to maintain a flow rate of 7.8 mL/min ?
So what i did was
(7.8Ee-3 /60) = π P (1.3e-3) ^4 /(8* 0.086*0.2)
I get 1.99E5 but its not right, im not sure what im doing wrong
To solve this problem, we can apply Poiseuille's Law, which relates the pressure difference, flow rate, viscosity, and dimensions of a tube. The formula is as follows:
Flow rate (Q) = (π * ΔP * r^4) / (8 * η * L)
Where:
- Q is the flow rate (volume per unit of time)
- ΔP is the pressure difference across the tube
- r is the radius of the tube
- η is the viscosity of the fluid
- L is the length of the tube
Let's plug in the given values:
Flow rate (Q) = 7.8 mL/min = 7.8 × 10^(-6) m^3/s (since 1 mL = 10^(-6) m^3)
Viscosity (η) = 0.20 Pa⋅s
Tube diameter (d) = 2.60 mm = 2.60 × 10^(-3) m
Tube length (L) = 8.60 cm = 8.60 × 10^(-2) m
First, we need to find the radius of the tube (r):
r = d/2 = 2.60 × 10^(-3) m / 2 = 1.30 × 10^(-3) m
Now, rearrange the formula to solve for ΔP:
ΔP = (Q * 8 * η * L) / (π * r^4)
ΔP = (7.8 × 10^(-6) m^3/s * 8 * 0.20 Pa⋅s * 8.60 × 10^(-2) m) / (π * (1.30 × 10^(-3) m)^4)
Simplify the equation:
ΔP = 3.1715 / (π * 2.208 × 10^(-15))
ΔP ≈ 1.44 × 10^15 Pa
So, the pressure difference needed to maintain a flow rate of 7.8 mL/min is approximately 1.44 × 10^15 Pa.
To find the pressure difference needed to maintain a flow rate, you can use the Poiseuille's Law, which relates flow rate to pressure difference and other parameters.
The formula you used is correct, but there seems to be a mistake in the calculations. Let's break it down step-by-step:
Given:
- Flow rate (Q) = 7.8 mL/min = 7.8 × 10^(-6) m^3/s
- Viscosity of engine oil (η) = 0.20 Pa⋅s
- Tube diameter (d) = 2.60 mm = 2.60 × 10^(-3) m
- Tube length (L) = 8.60 cm = 8.60 × 10^(-2) m
Step 1: Convert flow rate from mL/min to m^3/s
Q = 7.8 × 10^(-6) m^3/s (already given)
Step 2: Calculate the radius of the tube
radius (r) = diameter / 2 = (2.60 × 10^(-3) m) / 2
Step 3: Calculate the pressure difference (ΔP)
ΔP = (8ηLQ) / (πr^4)
Substituting the given values:
ΔP = (8 * 0.20 Pa⋅s * 8.60 × 10^(-2) m * 7.8 × 10^(-6) m^3/s) / (π * (2.60 × 10^(-3) m / 2)^4)
Calculating the equation above will give you the correct pressure difference.