how to calculate the maximum obtainable directivity for a microphone array? Introduction

The relationship between the maximum obtainable directivity and aperture of an antenna is described by:

$$ D = \frac{4\pi A_e}{\lambda^2} $$

Where $D$ is the directivity, $A_e$ the effective aperture and $\lambda$ the used wavelength. The derivation can be found in the following document: pdf.

However, if the antenna does not scan as a sphere but in $x, y$ but not $z$. Following the same path as the above-mentioned document the equation becomes:

$$ D = \frac{2\pi A_e}{\lambda^2}  $$

The aperture of a lossless isotropic antenna is then also $A_e =  \frac{\lambda²}{2 \pi}$ instead of $\frac{\lambda²}{4 \pi}$ 

Problem

I cannot manage to use that equation for a microphone array using delay-and-sum beamforming.

For example, if I have an array of 25 microphones spaced as a grid and that my setup has the following properties:


Smallest distance $d$ between them is $0.01$ meters. 
Amount of orientations $o$ that it looks at is $64$.
Frequency $f$ to be detected is 8 kHz.


The wavelength $\lambda = \frac{v}{f} = \frac{343}{8000} = 0.0429 $ 

The aperture $A$ is $A = \frac{\lambda^2}{precision} = \frac{0.0429^2}{\frac{2\pi}{64}} = 0.0187 $

Thus:

$$ D = \frac{2\pi A_e}{\lambda^2} =  \frac{2\pi A}{\lambda^2} = \frac{2 \pi \cdot 0.0184}{0.0429^2} = 62 $$

I rounded the numbers for the simplicity of formulating my problem, if you do not round then you get $64$ instead of $62$ which is the number of orientations that it looks at furthermore the results cannot be true because:


It does not involve the number of microphones used.
It does not involve the smallest spacing between microphones.
I probably made a mistake somewhere.


How can I calculate the maximum obtainable directivity for an array of sensors?
When using a microphone array for baseband acoustic signals, Array Gain is a preferred performance metric over Directivity Index. 

In geophysics, the term “stacking” is often preferred over the equivalent “delay-and-sum beam former. It is certainly more descriptive because you essentially time align signals like a stack and add them coherently.  

If the noise is independent at each microphone, for the aligned stack, the upper bound on coherent gain is $10 Log N$. 

Directivity is a function of the beam pattern which is a function of a single frequency.  For an RF system with an obvious carrier, the pattern at that nominal frequency is an often used way to characterize gain. 

The stacked array gain is baseband. It is the upper bound on gain.