Relating collision theory to rate of reaction I want to relate the macroscopic advancement of a reaction, the rate of the reaction, with the microscopic collisions of the molecules. This is the type I have developed so far:

$$u = A^2 \times \frac{n_1}{m^2} \times\frac{n_2}{m^2}\times \frac{(m_1u_1^2+m_2u_2^2)}{E_\mathrm{act}}\times 1/t $$

where $u$ is the rate of reaction in $\pu{mol2 sec-1}$.

$A =$ the common area of the substances

$n_1/m^2 =$ the moles per square metre of the first substance 

$n_2/m^2 =$ the moles per square meter of the second substance

$u_1$ and $u_2 =$ average velocity of molecules of each substance

$E_\mathrm{act} =$ activation energy of the reaction

Now the molecules of the first substance must bump to the molecules of the second substance with the appropriate orientation. How can I add this to my type?
It is interesting to see you creating your own approach. The comments above provide good responses. I would recommend looking at Buttonwoods suggestion regarding collision theory and cross-sections. However, it appears none have answered your question directly, so I will provide my response to it here.

Question: "Now the molecules of the first substance must bump to the molecules of the second substance with the appropriate orientation. How can I add this to my type?"

My Response: You recognize that orientation is important when it comes to chemical reactions. There are many ways to handle this, I will provide three short suggestions and you can consider them or use them to come up with something else.

Arrhenius (constant) approach: The pre-exponential factor, $\alpha$, in the following Arrhenius equation absorbs two important concepts of chemical reactivity - 1) the frequency of collisions, 2) the orientation of the molecules.

$$u = \alpha e^{-E_a/RT}$$

Note, I changed from the conventional notation in order to match and not confuse with your own. In fact, $\alpha$ is defined as the frequency of properly oriented collisions. So a simple constant multiplier that accounts for orientation can help modify your equation.

Boolean logic: You can possibly simplify things further by using a simple selection function that tells you whether the molecules are in proper orientation. Here is the simplest you could use:

$$f(x) = 
\begin{cases}
1 \text{ if proper orientation}\\
0 \text{ if not proper orientation}
\end{cases}$$

Functional dependence: Last mentioned here would be the most creative approach to come up with a function that describes the orientation of your molecules. You will have to come up with the functional dependencies you need (i.e. a reaction coordinate) to help solve your problem. Here is an example of a reaction between atomic hydrogen with molecular hydrogen to make the same species using sinusoidal function where the independent variable (reaction coordinate) is the angle of the H-H bond relative to the hydrogen atom.

$$\text{Reaction:    } H-H + H^{\boldsymbol{\cdot}} \to H^{\boldsymbol{\cdot}} + H-H$$
$$\text{Function:    } f(\theta) = \frac{1}{2}\left[\cos\left(\frac{\theta}{2}\right)+1\right]$$

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The interpretation of such a function as this would be 100% likely to react if the $H$ atom is in the plane of the bond with $H_2$ while 0% likely to react if $H_2$ is vertical. This isn't an exact reality, but just an example.