It is found experimentally that the terminal settling velocity uo of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle - weight of displaced fluid), W; fluid density, �, and fluid viscosity, µ. Obtain a relationship for uo using dimensional analysis
To obtain a relationship for terminal settling velocity uo using dimensional analysis, we need to identify the relevant dimensions and create a dimensionless equation.
Let's start by identifying the variables and their dimensions:
1. Particle diameter, d: Dimension = Length [L]
2. Buoyant weight of the particle, W: Dimension = Force [F]
3. Fluid density, ρ: Dimension = Mass/Volume = [M]/[L]^3
4. Fluid viscosity, μ: Dimension = Force × Time/Length^2 = [F] × [T]/[L]^2
Now, let's express the dimensions in terms of base dimensions:
1. Length [L]
2. Mass [M]
3. Time [T]
To find a relationship among these variables, we can construct dimensionless groups using the Buckingham π theorem. Since we have 4 variables and 3 base dimensions, we can create 4 - 3 = 1 dimensionless group.
Let's define our dimensionless group π:
π = (uo^a) × (d^b) × (W^c) × (ρ^d) × (μ^e)
Substituting the dimensions into the equation for π, we get:
[L^a] × [L^b] × [F^c] × ([M]/[L]^3)^d × ([F] × [T]/[L]^2)^e
Simplifying the equation, we can equate the powers of each base dimension to zero:
1. For Length:
a + b - 3d - 2e = 0
2. For Mass:
c + d - e = 0
3. For Time:
e = 0
We now have a set of three equations with three unknowns (a, b, and c). By solving them, we can find the relationship for uo.
Solving the equations:
e = 0 (from Time)
From the second equation, d = e = 0
Substituting into the first equation:
a + b = 0
b = -a
Substituting these results into the third equation:
c + 0 - 0 = 0
c = 0
Therefore, the relationship for uo using dimensional analysis can be expressed as:
uo ∝ d^(-a) × W^(-a) × ρ^0 × μ^0
Taking all the terms to the power of -a and simplifying, we get:
uo ∝ d^(-a) × W^(-a)
Thus, the relationship for terminal settling velocity uo using dimensional analysis is:
uo = K × d^(-a) × W^(-a)
Note: The constant K will depend on additional factors such as fluid properties, but the relationship for uo will be inversely proportional to the particle diameter (d) and the buoyant weight of the particle (W).
To obtain a relationship for the terminal settling velocity (uo) using dimensional analysis, we need to identify the relevant dimensions and construct a dimensionless group that encompasses all the variables mentioned (particle diameter, buoyant weight, fluid density, and fluid viscosity).
Let's denote the dimensions as follows:
- Length: [L]
- Mass: [M]
- Time: [T]
The relevant dimensions for the variables in question are:
- Particle diameter (d): [L]
- Buoyant weight (W): [M][L][T]⁻²
- Fluid density (ρ): [M][L]⁻³
- Fluid viscosity (µ): [M][L][T]⁻¹
To obtain a dimensionless group, we can combine these dimensions using multiplication and division.
Let's construct the following dimensionless group:
G = (uo / d^n) ρ^m µ^p W^q
where n, m, and p, and q are the unknown exponents we need to determine.
We want the dimension of G to be dimensionless, so if we equate the dimensions of G with the dimensionless dimension [1], we get:
[L][T]⁻¹^n [L]⁻^(3m) [M]⁻^(m+p) [L]^(m+n-q) [T]^(m+p-2q) = [1]
Equating the powers of each dimension on both sides of the equation, we get a system of equations:
For length [L]: n + m + n - q = 0 (1)
For time [T]: -n + (m + p) - 2q = 0 (2)
For mass [M]: m + p = 0 (3)
Solving this system of equations, we can determine the values of n, m, p, and q.
From equation (1), we find: 2n + m - q = 0
From equation (2), we find: -2n + m + p = 0
Solving these two equations simultaneously, we get:
n = -p/4
m = q/4
Now, substituting these values into equation (3), we find:
p = -2q
Finally, substituting the obtained values of n, m, and p into the original dimensionless group G, we get:
G = (uo / d^(-p/4)) ρ^(q/4) µ^(-q/2) W^q
This relationship for the terminal settling velocity can be obtained through dimensional analysis by constructing the dimensionless group as described and solving for the appropriate exponents. However, the specific values of these exponents will depend on further experimental analysis or theoretical considerations.