The 3.0 cm -diameter water line in the figure splits into two 1.0 cm -diameter pipes. All pipes are circular and at the same elevation. At point A, the water speed is 2.0 m/s and the gauge pressure is 50 kPa. What is the gauge pressure at point B?
To find the gauge pressure at point B, we need to apply the principle of conservation of mass and the Bernoulli's equation.
Step 1: Apply the principle of conservation of mass.
The principle of conservation of mass states that the mass of fluid entering a junction is equal to the mass of fluid leaving the junction. In this case, since the water line splits into two pipes at point A, the mass flow rate at point A must be equal to the sum of the mass flow rates at point B.
Since we know the water speed at point A (2.0 m/s) and the diameter of the water line at point A (3.0 cm), we can use the formula for the mass flow rate to calculate it:
Mass flow rate at A = (ρ * A * V)
Where:
ρ is the density of water (approximately 1000 kg/m^3)
A is the cross-sectional area of the water line at A (π * (diameter/2)^2)
V is the water speed at A (2.0 m/s)
Substituting the given values, we can calculate the mass flow rate at A.
Step 2: Apply Bernoulli's equation.
Bernoulli's equation relates the pressure, velocity, and height of a fluid in a streamline. Along a streamline, the total mechanical energy of the fluid (pressure energy, kinetic energy, and potential energy) remains constant.
Bernoulli's equation is given as:
P + (1/2)ρV^2 + ρgh = constant
Where:
P is the pressure of the fluid
ρ is the density of the fluid
V is the speed of the fluid
g is the acceleration due to gravity
h is the elevation of the fluid
We can apply Bernoulli's equation between points A and B, assuming the elevation and gravitational potential energy remains constant.
P_A + (1/2)ρV_A^2 = P_B + (1/2)ρV_B^2
Substituting the known values, we can solve for P_B, the gauge pressure at point B.
Step 3: Calculate the gauge pressure at point B.
After substituting the known values into Bernoulli's equation, the equation will simplify to:
P_B = P_A + (1/2)ρ(V_A^2 - V_B^2)
Substituting the values given in the problem:
P_A = 50 kPa (convert to Pascals by multiplying by 1000)
V_A = 2.0 m/s
V_B = ? (unknown)
Using the mass flow rate equation and the principle of conservation of mass, we can relate V_B to V_A:
(ρ * A_A * V_A) = (ρ * A_B * V_B)
Simplifying, we find:
A_A * V_A = A_B * V_B
Substituting A_A = π * (diameter_A/2)^2 and A_B = π * (diameter_B/2)^2, we can cancel out ρ and solve for V_B:
(diameter_A/2)^2 * V_A = (diameter_B/2)^2 * V_B
Substituting the given diameters (diameter_A = 3.0 cm and diameter_B = 1.0 cm) and V_A = 2.0 m/s, we can calculate V_B.
Finally, substitute V_B into the equation for P_B to find the gauge pressure at point B.