What is the formula for estimating the maximum stationary thrust of a rotor as function of its diameter?
The dimensional approach is simple... It can be safely assumed that the lift $L$ is a function of the input power $P$, the diameter $D$ of the rotor and the air density $\rho$.

Thus, $L = f(P,D,\rho)$

where $f$ is a function to be determined.

From dimensional analysis, the lift $L$ can be easily derived:

The variables are Lift $L$, dimensions $MLT^{–2}$; Power $P$, dimensions  $ML^2T^{–3}$; Rotor diameter $D$, dimensions $L$ and air density $\rho$, dimensions $ML^{–3}$

The variables form a non-dimensional product $k$

$k = L^a\cdot P^b\cdot D^c\cdot \rho^d$  where $a,b,c,d$ are numbers to be determined.

Let’s form now a parallel product $k^*$ with the dimensions:

$k^* = (MLT^{–2})^a  (ML^2T^{–3})^b  (L)^c (ML^{–3})^d$

Clearly, $k^* = M^0 L^0 T^0$... We now take the exponents for each dimension:

$a  + b + d = 0 \\
a + 2b + c – 3d = 0 \\
–2a – 3b = 0$

We make $a = 1$, since $L$ is the variable we’re going to solve for. 

$b = –2/3 \\ 
d = –1/3 \\
c = –2/3$

Then, 

$k = L^a\cdot P^b\cdot D^c\cdot \rho^d \rightarrow  k = L\cdot P^{–2/3}\cdot D^{–2/3}\cdot \rho^{–1/3}$ 

Solving for $L$

$L = k\cdot P^{2/3}\cdot D^{2/3}\cdot \rho^{1/3}$

where $k$ is a constant to be determined...