Sizing Orifices of an Injector

One of the biggest tasks to tackle with designing your own Liquid Rocket Engine is with designing your injector plate. This plate is composed of holes that will allow a certain amount of the two propellants to enter the combustion chamber. But coming up with the area of these orifice holes takes some background knowledge and some computing power. The only things you will need to know prior to this calculation is what your two propellants are, what your upstream pressure is of each of the propellants, and what the overall thrust of you are trying to have your engine generate. The most helpful tool you will learn from this lesson is how to run NASA CEA. 

NASA CEA is an amazing tool that handles a lot of the intensive thermodynamic and chemical mathematics of us so that we can just focus on designing and manufacturing our engine. Also, to try to get a very good approximation of what the results we will be getting from NASA CEA will be very difficult and to verify your own calculations will be an even more difficult process. NASA CEA however is highly utilized in industry and has been verified to be a good approximation.  
Making our Excel Spreadsheet
​Start with creating a Google Spreadsheet with the following layout:
This setup contains all of the starting layout you will need, you can just use the following link to copy the layout:
Running NASA CEA
Go ahead and input your Total Mass Flow Rate which you should know based off your feed system or simply estimate this right now. Then you may not know what your O/F ratio is until we run NASA CEA.

Go to the Following Website to run NASA CEA: https://cearun.grc.nasa.gov/
Make sure you have rocket selected and Hit the "Submit" button.
Ok, this section is for imputing your combustion chamber pressure. You may be wondering where I got 150 psia? So for the design I will be running, I am designing based on my "run tank" which is the tank which holds our propellant to be at 300 psia. Because we are at 300 psia, in an ideal case assuming that the pressures losses in our feed system are negligible (they compromise ~1% pressure loss) we need to make sure that through our injector plate that the flow does not try to go backwards, thus we need to choke the flow at the injector. This means that we need to have a ratio of Upstream Pressure/Downstream Pressure to be at least 2 which indicates choked flow. Thus, our downstream pressure (ie. the pressure in our combustion chamber) needs to be half of the pressure located on the input side of the injector plate. 
From what I've told you so far, input your Downstream Pressure, and change the units to "psia". Hit "Accept Input & Continue to Next Form." You may be wondering what those options are to the left for. You will see later on that NASA CEA is not just meant to take in a single set of data and generate output data for. Rather, we can use NASA CEA to create multiple outputs, where we vary the input parameter for each output and NASA CEA will just output all the different permutations in a text format.
Here we will input our fuel parameters. In my example we will use gaseous Hydrogen as our fuel, keep in mind the (L) indicates the fuel being in Liquid state. Choose from the following fuels and hit "Accept Fuel Selection & Continue"
Here we will state our oxidizer parameters. For my example run we will use Nitrous Oxide(N2O) as our oxidizer, but for your case indicate whatever oxidizer you are choosing to use. Hit "Accept Oxidizer Selection & Continue to next form."
Now we need to specify the output data specifications, which alone are already fine. No need to change anything, just hit "Submit input & Perform CEA Analysis."
The next thing we need to understand before analyzing the results is in looking at the material capabilities we will be using. For example, since we will be using stainless steel we need to consider the melting point and need to make sure our NASA CEA results for engine temperature stay well below that melting point. Looking at the chart to the left, you can see the melting point of different metals. Knowing what the Melting point of your material, we need to use a safety factor of 1.5 or greater. For our case, we are using stainless steel which as a melting point of 2500 F. So the maximum temperature we can have our engine reach is (2500 F) / 1.5 = 1666 F, and we can round that down to 1600 F. So as long as we stay under that value we will be fine. Further analysis can be done in analyzing the change in yield strength of the material at that temperature, but with this safety factor the yield strength should not have shifted considerably. 
Now that we understand our temperature range, we need to cover the first aspect NASA CEA will provide for us. This will be the O/F ratios. Essentially as we mentioned before, we don't really know what our ideal O/F ratios are, which is why NASA CEA is so powerful in being able to generate all permutations of our specified parameters. O/F has a direct correlation to the temperature of combustion similar to what you can see from the graph shown below. You can see that the highest point of temperature is known as the "Stocheometric Ratio" which also indicates that there exists just enough oxidizer to mix with all of the fuel need. The more oxidizer you add in after that point the temperature will start to decrease again which means there is wasted oxidizer. The more wasted propellant the less efficient your engine is since that propellant has the potential to add more energy to the system through combustion but is just exhausted out the nozzle. Higher the temperature the more efficient your engine is.
You can tell just from this exaggerated graph how to O/F ratio is related to the temperature the flame will be at and be transferred to the engine material. In reality we really are more constrained to the melting point of our material than the ability to have combustion at the Stoichiometric ratio which is generally the bigger bottleneck for most metals.
Ok, now we can go ahead and look at NASA CEA's results. As you scroll down you will be able to see each individual permutation of O/F ratios. 
You can see as indicated above where our two values of interest are: our O/F ratio, and the achieved chamber temperature, we are more so looking at the chamber temperature since it will tend to be higher than the temperature of the nozzle throat. So the result above tells us that at an O/F ratio of 0.1 the resultant temperature in the chamber will be approximately 350 F. Let's keep scrolling down and see how close we can get to our maximum temperature limitation.
From what you can see above we have reached the upper limit of our O/F range and our chamber temperature still could be closer to our maximum temperature. So we have to go back to the "Oxid/Fuel" tab at the top and make some changes as seen below:
Still too low.... 
Great we are close enough for now. But if you wanted to get even closer, we could refine the interval of our O/F parameters and rerun it. But we will go ahead and use this data: O/F ratio of 4.9
Back to our spreadsheet....
Ok now that we know our O/F ratio we can continue our spreadsheet. We will first enter in our total mass flow rate into the engine which will come from the feed system it self and what it is capable of providing. Input your O/F ratio. In the