Relating collision theory to rate of reaction
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.