A pipe carrying 20°C water has a diameter of 3.9 cm. Estimate the maximum flow speed if the flow must be streamline. (The density of water is 1.00 ✕ 10^3 kg/m3.) ans in cm/s
To estimate the maximum flow speed in a streamline flow, we can use the principle of continuity, which states that the mass flow rate remains constant along a streamline. The formula for mass flow rate is:
Mass flow rate = density × cross-sectional area × velocity
In this case, we have a pipe carrying water with a diameter of 3.9 cm, which means the radius (r) is half of the diameter, or 1.95 cm (0.0195 m).
First, we need to convert the units for the radius and density into SI units:
Radius (r) = 0.0195 m
Density (ρ) = 1000 kg/m^3
Next, we can calculate the cross-sectional area of the pipe using the formula for the area of a circle:
Cross-sectional area (A) = π × r^2
A = π × (0.0195 m)^2
A ≈ 0.001196 m^2
Now we can rearrange the mass flow rate formula to solve for velocity:
Velocity = Mass flow rate / (density × cross-sectional area)
We want to estimate the maximum flow speed, so we assume that the flow is streamlined, which means the flow is at its maximum for the given conditions.
To find the maximum flow speed, we consider the velocity when the mass flow rate is at its maximum. This occurs when the fluid velocity at any point within the pipe is at its maximum, which corresponds to the streamline flow condition.
For streamline flow, the velocity at any point is given by:
Velocity = (2 × Pressure difference) / (density × constant)
Since we don't have any information about the pressure difference or the constant, we assume that the velocity is at its maximum value, meaning that the pressure difference is also at its maximum value.
So, the maximum flow speed can be estimated by replacing the pressure difference with the maximum value:
Velocity = (2 × Maximum pressure difference) / (density × constant)
Since the question does not provide any specific information about the pressure difference or the constant, we cannot calculate the exact maximum flow speed. However, we know that streamline flow is the most efficient flow with the highest speed possible, so the maximum flow speed for a streamline flow is generally close to the speed of sound (approximately 343 m/s in dry air at 20°C).
To convert this speed to cm/s:
Maximum flow speed ≈ 343 m/s × (100 cm/1 m) = 34300 cm/s
Therefore, the estimated maximum flow speed in cm/s is approximately 34300 cm/s.
I am not going to try to do this for you. Your text has likely different caalculations for crritical Reynolds number and water viscosity than mine.
Look in your text for dritical transition Reynolds number from laminar to turbulent flow in pipe.
Calculate that number using your text values for Re, viscosity versus temp (in degrees K likely add 273 for 293 K) and diameter of 0.039 meter
solve for v
https://en.wikipedia.org/wiki/Reynolds_number