Discharge and Jet Height from a Nozzle
Problem Statement
A jet of water issues vertically upward from a nozzle mounted at a height of 0.2 m. The nozzle has an inlet diameter of 100 mm and an outlet diameter of 40 mm. If the pressure at the inlet is 20 kPa above atmospheric, determine the discharge (Q) and the height to which the jet will rise. Assume no friction.
Given Data
| Inlet Diameter (d₁) | 100 mm = 0.1 m |
| Area at Inlet (A₁) | A₁ = (π/4) × (0.1)² ≈ 0.007854 m² |
| Outlet Diameter (d₂) | 40 mm = 0.04 m |
| Area at Outlet (A₂) | A₂ = (π/4) × (0.04)² ≈ 0.001257 m² |
| Inlet Pressure (P₁) | 20 kPa above atmosphere = 20000 N/m² |
| Outlet Pressure (P₂) & At Jet (P₃) | Atmospheric (gauge = 0) |
| Nozzle Elevation | 0.2 m |
| Acceleration due to Gravity (g) | 9.81 m/s² |
| Specific Weight of Water (γ) | 9810 N/m³ |
1. Relating Inlet and Outlet Velocities
From the continuity equation:
A₁V₁ = A₂V₂
Therefore, V₁ = (A₂/A₁) V₂ ≈ 0.16 V₂.
2. Applying Bernoulli’s Equation Between Inlet and Outlet
Taking the datum at the inlet (Z₁ = 0), Bernoulli’s equation gives:
P₁/γ + V₁²/(2g) + Z₁ = P₂/γ + V₂²/(2g) + Z₂
Substituting the values (with Z₂ = 0.2 m, P₂ = 0):
20000/9810 + V₁²/(2×9.81) + 0 = 0 + V₂²/(2×9.81) + 0.2
Rearranging:
V₁² – V₂² = -36.076
(Using 2g = 19.62 and 20000/9810 ≈ 2.04, the calculation leads to this relation.)
3. Determining Velocities and Discharge (Q)
Substituting V₁ = 0.16V₂ into the equation:
(0.16V₂)² – V₂² = -36.076
0.0256V₂² – V₂² = -36.076 ⇒ -0.9744V₂² = -36.076
Solving gives: V₂ ≈ 6.08 m/s and hence V₁ ≈ 0.97 m/s.
The discharge is then:
Q = A₁ × V₁ ≈ 0.007854 m² × 0.97 m/s ≈ 0.007618 m³/s.
4. Determining the Height of the Jet
Apply Bernoulli’s equation between the outlet (point 2) and the highest point of the jet (point 3) with datum at the outlet:
P₂/γ + V₂²/(2g) + Z₂ = P₃/γ + V₃²/(2g) + Z₃
Since P₂ = P₃ = 0 and V₃ = 0:
V₂²/(2g) + 0 = h
Substituting V₂ ≈ 6.08 m/s:
h = (6.08)²/(2×9.81) ≈ 1.88 m.
Jet Height ≈ 1.88 m
Physical Interpretation
In this problem, the energy provided by the pressure at the inlet is converted into kinetic energy at the nozzle outlet. The decrease in cross-sectional area causes an increase in velocity according to the continuity principle. Bernoulli’s equation is used to relate the pressure energy at the inlet to the kinetic and potential energy at the outlet and at the maximum jet height.
The calculated discharge shows how much water flows through the nozzle, while the height represents the maximum vertical rise of the water jet (above the nozzle exit) due to its initial kinetic energy.
Detailed Explanation for Students
Step 1: Use the continuity equation to relate velocities in sections of different cross-sectional areas.
Step 2: Apply Bernoulli’s equation between the inlet and outlet. With the inlet pressure known and the outlet at atmospheric pressure, set up the energy balance (accounting for the nozzle height) to relate the inlet and outlet velocities.
Step 3: Substitute the velocity relation from the continuity equation into the Bernoulli equation to solve for the velocities. Use these to compute the discharge.
Step 4: Apply Bernoulli’s equation between the outlet and the maximum jet height. With the velocity at the highest point being zero, the kinetic energy at the outlet is completely converted into potential energy, allowing you to solve for the jet height.
