I Solve equation from dimensional analysis: 3 eq., 6 unknowns

[phillip_at_work]

TL;DR

Text book conducts dimensional analysis to derive an equation. Analysis involves 6 variables and 3 basic units (mass M, length L, and time T). This results in three equations and six unknowns. The final equation can be solved using three of those unknowns. However, with only three equations, I don't see how to find those three unknowns to actually get a solution for the derived equation. Text: Carlton 2007, Marine Propellers and Propulsion p89.

Carlton writes on page 89:
"The thrust of a marine propeller... may be expected to depend upon the following parameters:
(a) The diameter (D)
(b) the speed of advance (V_a)
(c) The rotational speed (n)
(d) The density of the fluid (ρ)
(e) The viscosity of the fluid (μ)
(f) The static pressure of the fluid at the propeller station (p₀-e)"

What follows is Carlton's derivation:

T ∝ ρ^aD^bV_a^cn^dμ^f(p₀-e)^g

And by dimensional analysis, we get:

MLT-2 = (ML^-3)^aL^b(LT^-1)^c(T^-1)^d(ML^-1T^-1)^f(ML^-1T^-2)^g

which results in the following equations:

for mass M: 1 = a + f + g
for length L: 1 = -3a + b + c - f - g
for time T: -2 = -c - d - f - 2g

and hence:

a = 1 - f - g
b = 4 - c - 2f -g
d = 2 - c - f - 2g

And so that proportion can be updated to be:

T ∝ ρ ^(1-f-g) D ^(4-c-2f-g)V_a^cn^(2-c-f-2g)μ^f(po-e)^g

For the final equation as:

T = ρn²D⁴(V_a/ nD)^c* (μ / ρnD²)^f* ( (p₀-e) / pn²D²)^g

I can follow this derivation without issue. What is confusing is how I solve for T. How can I know the values of `c`, `f`, and `g` as I have three equations and six unknowns? What am I missing?

 

[phillip_at_work]

I think I solved my own question. Happy for feedback.

Constant exponents are derived for ρ^a, n^d and D^b. These are ρ¹, n², and D⁴.

That is, a=1, d=2, and b=4. If I apply these to the system of three equations:

a = 1 = 1 - f - g
b = 4 = 4 - c - 2f -g
d = 2 = 2 - c - f - 2g

I compute,

g = f = c = 0

Which feels incorrect, but perhaps I'm over thinking. If this was correct, I'm not sure why Carlton wouldn't write this solution in the chapter, which s/he does not. Any feedback is much appreciated.