Water of density 1000 kg/m3 flows through a tube as shown in figure1a below.At section 1 the pressure is 200 kN/m2, the velocity is 5 m/s and the pipe diameter is0.12m. The pipe diameter at section 2 is 0.065m.Calculate: i) the velocity and pressure at section 2
Question
Water of density 1000 kg/m3 flows through a tube as shown in figure1a below.At section 1 the pressure is 200 kN/m2, the velocity is 5 m/s and the pipe diameter is0.12m. The pipe diameter at section 2 is 0.065m.Calculate: i) the velocity and pressure at section 2
Solution
To solve this problem, we will use two principles of fluid dynamics: the continuity equation and Bernoulli's equation.
- The Continuity Equation: This principle states that the mass flow rate must be conserved at all points along a streamline. In other words, the product of the cross-sectional area (A) and the velocity (v) at any two points in a pipe must be equal. Mathematically, this is expressed as A1v1 = A2v2.
Given:
- Diameter at section 1 (D1) = 0.12 m, so the area at section 1 (A1) = π(D1/2)^2 = π(0.12/2)^2 = 0.0113 m^2.
- Velocity at section 1 (v1) = 5 m/s.
- Diameter at section 2 (D2) = 0.065 m, so the area at section 2 (A2) = π(D2/2)^2 = π(0.065/2)^2 = 0.0033 m^2.
We can now calculate the velocity at section 2 (v2) using the continuity equation:
v2 = A1v1/A2 = (0.0113 m^2 * 5 m/s) / 0.0033 m^2 = 17.1 m/s.
- Bernoulli's Equation: This principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Mathematically, this is expressed as P1 + 0.5ρv1^2 = P2 + 0.5ρv2^2.
Given:
- Pressure at section 1 (P1) = 200 kN/m^2 = 200,000 N/m^2.
- Density of water (ρ) = 1000 kg/m^3.
We can now calculate the pressure at section 2 (P2) using Bernoulli's equation:
P2 = P1 + 0.5ρv1^2 - 0.5ρv2^2 = 200,000 N/m^2 + 0.51000 kg/m^3(5 m/s)^2 - 0.51000 kg/m^3(17.1 m/s)^2 = 200,000 N/m^2 - 131,050 N/m^2 = 68,950 N/m^2 = 68.95 kN/m^2.
So, the velocity at section 2 is 17.1 m/s and the pressure at section 2 is 68.95 kN/m^2.
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