Problem Statement
In 1:30 model of a spillway, the velocity and discharge are 1.5m/s and 2 m³/s. Find the corresponding velocity and discharge in the prototype.
Given Data
Solution Approach
To find the prototype velocity and discharge, we’ll use Froude’s model law. This law is applicable for spillways where gravitational forces are dominant. The Froude number must be the same in both the model and prototype.
Calculations
Velocity in Prototype
Step 1: According to Froude model law, the Froude number must be the same in both model and prototype:
Where:
Vm = velocity in model
Vp = velocity in prototype
gm = gp = 9.81 m/s² (gravitational acceleration)
Lm = characteristic length in model
Lp = characteristic length in prototype
Step 2: Applying the scale ratio: Lp = 30 × Lm
Step 3: Solving for Vp:
Velocity in Prototype (Vp) = 8.216 m/s
Discharge in Prototype
Step 4: The ratio of discharges can be expressed as:
Where:
Qp = discharge in prototype
Qm = discharge in model
Ap = cross-sectional area in prototype
Am = cross-sectional area in model
Step 5: Substituting the known values:
Step 6: Calculating Qp:
Discharge in Prototype (Qp) = 9859 m³/s
Detailed Explanation
Froude Model Law
Froude’s model law is used in hydraulic modeling when gravitational forces are dominant, such as in open channel flows, spillways, and hydraulic structures. The Froude number represents the ratio of inertial forces to gravitational forces.
Scale Ratios in Hydraulic Models
For a 1:30 scale model, the relationship between model and prototype parameters follows specific scale ratios:
- Length ratio: Lr = Lp/Lm = 30
- Velocity ratio: Vr = Vp/Vm = √Lr = √30 = 5.477
- Area ratio: Ar = Ap/Am = Lr² = 30² = 900
- Discharge ratio: Qr = Qp/Qm = Lr²·√Lr = 30²·√30 = 900·5.477 = 4929.3
Verification
We can verify our calculations using the scale ratio formulas directly:
- Velocity in prototype: Vp = Vm·√Lr = 1.5·√30 = 8.216 m/s
- Discharge in prototype: Qp = Qm·Lr²·√Lr = 2·30²·√30 = 2·4929.3 = 9858.6 m³/s ≈ 9859 m³/s
Practical Significance
The results indicate that the prototype spillway will experience a significantly higher velocity and discharge compared to the model. This is crucial information for:
- Structural design to withstand hydrodynamic forces
- Erosion protection measures downstream
- Energy dissipation structures sizing
- Flood routing and management
Applications in Hydraulic Engineering
Physical modeling with Froude similarity is widely used in:
- Dam spillway design
- River training works
- Bridge hydraulics
- Coastal structures
- Hydraulic structures such as weirs, stilling basins, and energy dissipators
The calculated prototype discharge of 9859 m³/s represents a significant flood event that would need careful consideration in the design of the spillway and downstream protection measures.
