Water is flowing at a rate of 3 m/sec in a horizontal pipe under a pressure of 200,000 N/m. If the pipe narrows to half its original diameter
-What is the new speed of flow?
-What is the new pressure?
-How does the flow rate through the
narrow section compare with the flow
rate through the wider section?
- The new speed of flow can be determined using the principle of continuity, which states that the mass flow rate must be constant in an incompressible flow. The mass flow rate is given by the product of the density, cross-sectional area, and velocity. Since the pipe is horizontal, we can assume that the density of water is constant. Let A1 be the original cross-sectional area and A2 be the new cross-sectional area (which is half of A1). Then, by the principle of continuity:
A1 * v1 = A2 * v2
where v1 is the original velocity (3 m/sec) and v2 is the new velocity that we want to find. Solving for v2, we get:
v2 = (A1/A2) * v1 = 2 * v1 = 6 m/sec
Therefore, the new speed of flow is 6 m/sec.
- To determine the new pressure, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid along a streamline. Assuming that the pipe is horizontal and the height of the fluid does not change, Bernoulli's equation reduces to:
P1 + (1/2) * rho * v1^2 = P2 + (1/2) * rho * v2^2
where P1 is the original pressure (200,000 N/m), P2 is the new pressure that we want to find, and rho is the density of water. Substituting the values, we get:
200,000 + (1/2) * rho * 3^2 = P2 + (1/2) * rho * 6^2
Simplifying the equation and solving for P2, we get:
P2 = 150,000 N/m
Therefore, the new pressure is 150,000 N/m.
- The flow rate through the narrow section is less than the flow rate through the wider section because the velocity is higher in the narrow section. The mass flow rate must be constant, but the cross-sectional area is reduced by half, so the velocity must double to maintain the same flow rate. Therefore, the flow rate through the narrow section is half of the flow rate through the wider section.
To find the new speed of flow, we can use the principle of continuity, which states that the product of the cross-sectional area and the velocity of a fluid remains constant.
Let's denote the initial velocity of flow as v1 and the initial diameter of the pipe as d1.
1. Find the new speed of flow:
Since the pipe narrows to half its original diameter, the new diameter d2 will be half of the initial diameter d1. Therefore, d2 = d1/2.
Using the continuity equation, we have:
A1v1 = A2v2
Where A1 is the initial cross-sectional area (πr1^2) and A2 is the new cross-sectional area (πr2^2). Since the areas are proportional to the square of the radii, we have:
(d1/2)^2 * v2 = d1^2 * v1
Simplifying the equation, we get:
v2 = (d1^2/d2^2) * v1 = (d1^2/(d1/2)^2) * v1 = 4v1
Therefore, the new speed of flow is 4 times the initial speed, so v2 = 4 * 3 = 12 m/s.
2. Find the new pressure:
The pressure in a fluid is determined by factors such as the height of the fluid column and the density of the fluid. Since only the pipe diameter changes, and assuming the height and density remain constant, the pressure will stay the same.
Therefore, the new pressure remains at 200,000 N/m^2.
3. How does the flow rate through the narrow section compare with the flow rate through the wider section?
The flow rate of a fluid is the volume of fluid passing through a section of the pipe per unit of time. It is calculated by multiplying the cross-sectional area of the pipe by the velocity of the fluid.
Based on the continuity equation, the product of the cross-sectional area and the velocity of a fluid remains constant. Since the velocity in the narrow section is 4 times the velocity of the wider section, the cross-sectional area must be 4 times smaller in the narrow section to maintain the same flow rate.
Therefore, the flow rate through the narrow section is the same as the flow rate through the wider section.