A 7.9cm diameter horizontal pipe gradually narrows to 5.9cm . When water flows through this pipe at a certain rate, the gauge pressure in these two sections is 31.0kPa and 21.0kPa , respectively.
What is the volume rate of flow? in m3/s
please help!
To find the volume rate of flow through the pipe, we can use the principle of continuity, which states that the volume rate of flow of an incompressible fluid is constant at all points along a horizontal pipe.
The principle of continuity is based on the conservation of mass, which states that the mass of the fluid entering a certain cross-sectional area per unit time must equal the mass of the fluid leaving that area per unit time.
We can express this principle mathematically using the equation:
A1 * V1 = A2 * V2
Where:
A1 and A2 are the cross-sectional areas of the pipe at two different points,
V1 and V2 are the velocities of the fluid at those points.
In this case, we want to find the volume rate of flow, so we need to rearrange the equation:
A1 * V1 = A2 * V2
V1 = (A2 * V2) / A1
Now, we can use the fact that the area of a circle is given by the formula:
A = π * r^2
Where:
A is the area,
π is a mathematical constant approximately equal to 3.14159,
r is the radius.
Let's calculate the areas of the two cross-sections:
For the first section, with a diameter of 7.9 cm:
r1 = 7.9 cm / 2 = 3.95 cm = 0.0395 m
A1 = π * (0.0395 m)^2
For the second section, with a diameter of 5.9 cm:
r2 = 5.9 cm / 2 = 2.95 cm = 0.0295 m
A2 = π * (0.0295 m)^2
Now, we need to find the velocities V1 and V2. We can use Bernoulli's principle, which states that the sum of the pressures and kinetic energy per unit volume of a fluid is constant along a streamline.
For a horizontal pipe, the pressure term can be simplified to:
P + (1/2) * ρ * v^2 = constant
Where:
P is the pressure,
ρ is the density of water,
v is the velocity of water.
We can rearrange this equation to solve for v:
v = sqrt(2 * (constant - P) / ρ)
The constant term cancels out since we're comparing two different sections of the pipe. So, we can rewrite the equation as:
v1 = sqrt(2 * (P1 - P2) / ρ)
v2 = sqrt(2 * (P2 - P2) / ρ)
Now, let's calculate the velocities:
P1 = 31.0 kPa = 31,000 Pa
P2 = 21.0 kPa = 21,000 Pa
ρ = density of water = 1000 kg/m^3
v1 = sqrt(2 * (31,000 Pa - 21,000 Pa) / 1000 kg/m^3)
v2 = sqrt(2 * (21,000 Pa - 21,000 Pa) / 1000 kg/m^3)
Finally, we can substitute the values into the equation for V1 to find the volume rate of flow:
V1 = (A2 * v2) / A1
Now, we have all the necessary components to calculate the volume rate of flow. Plug in the values and simplify the equation to find the final answer.