Answer:
Explanation:
Given that,
Omega is a function of the following
ω = f(b, h, v, ρ, k)
Where, all unit have a dimension of
ω = T^-1
b = L
h = L
V = LT^-1
ρ = FL^-4T²
k = FL
Then,
From the pie theorem
The required pi term is 6—3 = 3 terms,
So, we use V, p and b as a repeating term.
For first pi
π1 = ω•b^a•v^b•ρ^c.
Since
ω= T^-1, b = L, v = LT^-1 and
ρ= FL^-4T²
Since π is dimensionless then,
π = F^0•L^0•T^0
(T^-1)•(L^a)•(LT^-1)^b•(FL^-4T²)^c = F^0•L^0•T^0
Rearranging
F^0•L^0•T^0 = F^c•T^(2c-b-1)• L^(a+b-4c)
Comparing coefficient
c = 0
2c - b - 1 = 0
b = 2c - 1 = 0 - 1 = -1
a + b - 4c = 0
a = 4c - b = 0 - -1 = 0+1
a = 1
Then, a = 1, b = -1 and c = 0
So, π1 = ω•b^a•v^b•ρ^c.
π1 = ω•b^1•v^-1•ρ^0
π1 = ω•b / v
Check dimensions
ωb/v = (T^-1)L / LT^-1 = L^0•T^0 = 1
Then, π1 is dimensionless
For second pi
π2 = h•b^a•v^b•ρ^c.
Since
h = L, b = L, v = LT^-1 and ρ= FL^-4T²
Since π is dimensionless then,
π = F^0•L^0•T^0
(L)•(L^a)•(LT^-1)^b•(FL^-4T²)^c = F^0•L^0•T^0
Rearranging
F^0•L^0•T^0 = F^c•T^(2c-b)• L^(1+a+b-4c)
Comparing coefficient
c = 0
2c - b = 0
b = 2c = 0
1 + a + b - 4c = 0
a = 4c - b - 1 = 0 -0 - 1 = -1
a = -1
Then, a = -1, b = 0 and c = 0
So, π2 = h•b^a•v^b•ρ^c.
π2 = h•b^-1•v^0•ρ^0
π2 = h / b
Check dimensions
h / b = L / L = 1
Then, π2 is dimensionless
For third pi
π3 = k•b^a•v^b•ρ^c.
Since
k= FL, b = L, v = LT^-1 and ρ=FL^-4T²
Since π is dimensionless then,
π = F^0•L^0•T^0
(FL)•(L^a)•(LT^-1)^b•(FL^-4T²)^c = F^0•L^0•T^0
Rearranging
F^0•L^0•T^0 = F^(c+1)•T^(2c-b)• L^(1+a+b-4c)
Comparing coefficient
c + 1= 0
Then, c = -1
2c - b = 0
b = 2c = -2
1 + a + b - 4c = 0
a = 4c - b - 1 = -4 +2 - 1 = -3
a = -3
Then, a = -3, b = -2 and c = -1
So, π3 = k•b^a•v^b•ρ^c.
π3 = k•b^-3•v^-2•ρ^-1
Therefore,
π3 = k / b³•v²•ρ
Let check for dimension
π3 = FL / (L³• L²T^-2 • FL^-4T²)
π3 = FL / (L^(3+2-4) • T^(-2+2) •F)
π3 = FL / (L• T^(0) •F)
π3 = FL / LF = 1
π3 is also dimensions less
So.
I. There are three none dimensional pi
II. The none dimensional group are
π1 = ω•b / v
π2 = h / b
π3 = k / b³•v²•ρ
III. Reynolds Number. The Reynolds number is the ratio of inertial forces to viscous forces and it is dimensionless
So, the π3 can be considered as a Reynolds number
