Problem Statement
Calculate the weight of a ball of diameter 50mm which is just supported in a vertical air stream which is flowing at a velocity of 10 m/s. Take density of air = 1.25kg/m³ and kinematic viscosity = 15 stokes.
Given Data
Solution Approach
To determine the weight of the ball, we need to recognize that when the ball is just supported in the vertical air stream, the drag force acting upward on the ball exactly balances the weight of the ball acting downward. We’ll calculate the Reynolds number to determine the appropriate drag coefficient, then find the drag force which equals the weight of the ball.
Calculations
Reynolds Number Calculation
Step 1: Calculate the Reynolds number (Re) using the formula:
Since Re = 333.33 falls between 5 and 1000, we can use the drag coefficient (CD) value of 0.4 for a sphere in this flow regime.
Step 2: Calculate the drag force using the drag equation:
Step 3: Since the ball is just supported in the air stream, the drag force exactly equals the weight of the ball:
Weight of the ball = 0.049 N
Detailed Explanation
Physical Principles
This problem involves the balance of forces on a spherical object in a fluid stream. When the ball is just supported in the vertical air stream, it reaches a state of equilibrium where the upward drag force exactly equals the downward gravitational force (weight).
Reynolds Number and Flow Regime
The Reynolds number (Re = 333.33) indicates that the flow around the ball is in the transitional regime between laminar and turbulent flow. In this range (5 < Re < 1000), the drag coefficient for a sphere is approximately constant at CD = 0.4.
Drag Force Analysis
The drag force on a spherical object in a fluid stream is given by the equation:
Where:
- CD is the drag coefficient (0.4 in this case)
- ρ is the fluid density (1.25 kg/m³)
- A is the projected area of the sphere (π/4 × D² = 0.001963 m²)
- V is the fluid velocity (10 m/s)
Force Balance
In this equilibrium condition:
Practical Applications
This principle of force balance between drag and weight is utilized in various engineering applications:
- Fluid bed reactors where solid particles are suspended in a fluid stream
- Aerodynamic testing of objects in wind tunnels
- Design of parachutes and air resistance devices
- Particle classification systems in mineral processing
- Pneumatic conveying systems in industrial applications
Mass of the Ball
If we wanted to determine the mass of the ball, we could use the relation:
Understanding these principles of fluid dynamics and force equilibrium is crucial in aerospace engineering, mechanical design, and environmental studies where objects interact with fluid flows.
