I'm trying to find an equation that will give me a required pipe diameter, the variables I have are the following:
I'm not sure if its possible with these variables but any help with finding an equation is appreciated.
In this field, you can usually find expressions for calculating pressure change $\Delta P$ for a given geometry and mass flowrate $\dot{m}$: $$\Delta P = R(\dot{m})\cdot \dot{m} = \frac{\dot{m}}{C(\dot{m})}$$
The relation is expressed either using conductivity $C$ or resistance $R$. Both of these can be basically constants for laminar flow, but in practice, the flow will likely be turbulent, so both $C$ and $R$ will be functions of mass flowrate $\dot{m}$ (or volumetric flowrate or mean fluid velocity).
Apart from this, $C$ and $R$ are also functions of geometry and in general, you will not be able to find direct expression for pipe diameter. What you can do, is basically trial and error, i.e. you can guess different diameters and evaluate resulting pressure change $\Delta P$, which you did not listed as an input, but usually there is a given value which you should not exceed.
Calculating diameter I would recommend bisection method for this problem. This method requires lower and upper boundary on a variable you are trying to find, which could be diameter $d$. However, only diameter you might be able to calculate directly would be for laminar flow $d_{lam}$ and the diameter you are looking for will be higher than this, so you would have only one useful boundary for bisection: $$d\in [d_{lam}; \infty)$$
Although $\infty$ is not a valid boundary for bisection method, you are not obliged to look directly for $d$, instead, you can look for $1/d$, for which you have useful boundaries: $$1/d\in (0; 1/d_{lam}]$$
