Measurement of Viscosity Vertical Falling Ball Equation and Calculator

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Measurement of Viscosity Vertical Falling Ball Equation and Calculator

Fluids Dynamics and Engineering
Hydraulic and Pneumatic Knowledge

Viscosity of Fluid with Falling Sphere at Terminal Velocity Equation and Calculator

Measurement of Viscosity in a Vertical Falling Ball Viscometer Calculator

If a small sphere is allowed to fall from rest in a viscous fluid, it will accelerate until it reaches a constant velocity-the terminal velocity. When this steady-state condition has been reached the sum of all the forces acting on the sphere must be zero. The force of gravity on the solid acts in the direction of fall, and the buoyant and kinetic forces act in the opposite direction:

Preview: Viscosity of Fluid with Falling Sphere at Terminal Velocity Calculator

Eq. 1
(4/3) · π · µ · r³ · ρ_s · g = 4/3 · π · R³ · ρ · g + 6 · π · µ ·R · v_t

Here ρ_f, and ρ_s are the densities of the solid sphere and the fluid. Solving this equation for the terminal velocity gives

Eq. 2 (see Eq. 5)
µ = (2/9) · r² · ( ρ_s - ρ_f ) · g / v_t

Eq. 3
µ = ( g · d² ) / ( 18 · l) · ( ρ_s - ρ_f ) · t

Eq. 4
µ = ( γ_s - γ_f ) · d² / ( 18 · v_t )

Eq. 5 (see Eq. 2)
Reynolds number
Re = d · v_∞ · ρ / µ

This result may be used only if the Reynolds number is less than about 0.1.

This method provides an apparently simple method for determining viscosity. However, it is difficult to keep a homogeneous sphere from rotating during its descent, and if it does rotate, then Eq. 2 cannot be used. Sometimes weighted spheres are used in order to preclude rotation; then the left side of Eq. 1 has to be replaced by m, the mass of the sphere, times the gravitational acceleration.

Where:

Re = Reynolds Number (dimensionless)
v_t = Terminal Velocity of sphere (m/s)
ρ_f = Density of Fluid (g/m³)
ρ_s = Density of Sphere (g/m³)
µ = Viscocity of Fluid (cP)
g = gravity (m/s²)
d = Diameter (m)
R = radius (m)
l = length of fall (m)
t = time sphere passing the length of l
γ_s = specific weight sphere
γ_f = specific weight fluid
t = time sphere passing the length of l

Source:

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second ed.). John Wiley & Sons, Chapter: 2, Page: 61.
Ping Yuan and Ben-Yuan Lin, American Laboratory October, 2008