Terminal velocity http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
11.4 Terminal velocity
Another common application of the drag force equation is finding the steady-state speed of a body that is falling
through a fluid. When a body is dropped, it accelerates under the action of gravity. As the speed of the falling
body increases, the drag increases until the upward force (drag) equals the net downward force (weight minus
buoyant force). Once the forces are balanced, the body moves at a constant speed called the terminal velocity,
which is identified as the maximum velocity attained by a falling body.
To find terminal velocity, balance the forces acting on the object, and then solve the resulting equation. In
general this process is iterative as illustrated by Example 11.4.
EXAMPLE 11.4 TERMI AL VELOCITY OF A SPHERE I
WATER
A 20 mm plastic Sphere (S = 1.3) is dropped in water. Determine its terminal velocity. Assume
T = 20°C.
Problem Definition
Situation: A smooth sphere (D = 0.02 m, S = 1.3) is falling in water.
Find: Terminal velocity (m/s) of the sphere.
Properties: Water (20°C), Table A.5, ½ = 1 × 10-6 m2/s, Á = 998 kg/m3, and Å‚ = 9790 N/m3.
Plan
This problem requires an iterative solution because the terminal velocity equation is implicit.
1. Apply force equilibrium.
2. Develop an equation for terminal velocity.
3. To solve the terminal velocity equation, set up a procedure for iteration.
4. To implement the iterative solution, build a table in a spreadsheet program.
Solution
1. Force equilibrium
· Sketch a free-body diagram.
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· Apply force equilibrium (vertical direction):
2. Terminal velocity equation
· Analyze terms in the equilibrium equation:
· Solve for V0
3. Iteration 1
· Initial guess: V0 = 1.0 m/s
· Calculate Re:
· Calculate CD using Eq. (11.9):
· Find new value of V0 (use equation from step 2):
4. Iterative solution
· As shown, use a spread sheet program to build a table. The first row shows the results of
iteration 1.
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· The terminal velocity from iteration 1 V0 = 0.413 m/s is used as the initial velocity for
iteration 2.
· The iteration process is repeated until the terminal velocity reaches a constant value of
V0 = 0.44 m/s. Notice that convergence is reached in two iterations.
Iteration # Initial V0 Re CD ew V0
(m/s) (m/s)
1 1.000 20000 0.456 0.413
2 0.413 8264 0.406 0.438
3 0.438 8752 0.409 0.436
4 0.436 8721 0.409 0.436
5 0.436 8723 0.409 0.436
6 0.436 8722 0.409 0.436
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
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