I was hoping that I could say this is physically feasible, but after running the numbers, it unfortunately seems not. Let me start by listing the various claims on the Kickstarter:
- Bore diameter: 8 inches
- Maximum pressure: 60,000 psi
- Tube length: 45 feet
- Altitude: 274.9 m above sea-level
- Muzzle velocity: 1500 m/s (mentioned in comments, I may relax this later)
- Maximum altitude of the payload is taken to be 100 km by the "space" qualifier
The bore diameter kills it. You can't do it with these parameters. I'll show this by establishing a maximum and minimum mass for the shot based on obvious physical constraints. I will show that these requirements conflict, and can not be satisfied by this device.
Maximum mass
This is simple, because we're given a maximum pressure from combustion. The core concept of the project is that it will be multi-chambered. All this does is allow them to sustain that driving pressure over a longer segment of the barrel than a single combustion gun would allow. So the actual pressures might be lower than this, but they can't be higher. This easily constrains the kinetic energy that the projectile can have. The calculation below is simple, but this only introduces conservatism. I'm not introducing back-pressure from the air or anything - so I'm being as generous as possible here.
$$ E = F \times d = \left( P A \right) \times d = (60,000 \text{ psi})\pi(4\text{ in})^2(45 \text{ ft}) = 184 MJ $$
This seems simple enough. If we take the muzzle velocity to be 1500 m/s, then we find that the maximum mass that can be shot is 163.5 kg. The payload must be lighter than this, or else the gun can not accelerate it to the required speed.
Minimum mass
In the OP I gave the required velocity needed to make it to the Karman line in the absence of air resistance. Note that the currently quoted muzzle velocity is 100 m/s higher than this. Earth's atmosphere is clustered mostly around the first 10 km of sea-level. So one approximation we can make it to imagine that atmosphere as a thin sheet that the bullet must make it through, in other words, we don't credit the loss of speed due to gravitational potential along this track.
If we take the mass-thickness of the atmosphere to be $\mu$, we can write the energy lost to drag as follows:
$$ \Delta E =\frac{1}{2} \mu v^2 A C_d = 150.2 MJ $$
A generous assumption I made was to take the $v$ in this equation to be the velocity needed to reach the Karman line, 1.4 km/s. That's the lower of the options, so this will underestimate the impact of drag. The area (again) comes from the bore diameter. The drag coefficient can be looked up. I'll take it to be a perfect sphere, with a coefficient of 0.47.
This is only the energy lost to drag. The projectile has to get past this and then still have enough energy to make it to the Karman line. This is straightforward to set up mathematically. With this, we establish a minimum mass.
$$ \frac{1}{2} m v_1^2 - \Delta E = \frac{1}{2} m v_2^2 \\
m = \frac{ \Delta E }{ \frac{1}{2} ( v_1^2 - v_2^2 ) } = 1,035.9 kg$$
Projectile must be more massive than this in order to preserve its momentum as it makes it past the atmosphere. The contradiction is obvious. It needs to be this massive, but the gun can't supply the energy in order to get it there because the barrel simply isn't large enough.
You can't fix this by tweaking the parameters either. In the above equation, I used 1500 and 1400 m/s for the v1 and v2. But make v1 larger. It doesn't reach parity with the other requirement until well over 2,000 m/s. We all agree that the kinetic energy of the explosives aren't enough the achieve this. It's outright physically impossible. See many other questions on this site about the subject for elaboration to why. This is why HARP moved on to light gas guns - to get closer to orbital velocities.
Compared to others
The tank gun example I cited also fails. Its parameters come out even worse than the Starfire. With the possible exception some heavy munitions of battleships, the answer to the question I asked is more or less "no".
The accepted answer has detail on the HARP firing that did what this project is trying to do. So it's possible, but there is a critical difference - they used a 16 inch bore. The theme of my calculations is that an 8 inch bore is too small. If you double the bore diameter, you quadruple the maximum energy you can put into it. I believe it was also longer, which puts it even further away from the energetic constraint here.
It looks like the Starfire project might be able to launch the payloads a few kilometers up, but it can't go further than this. It can't physically happen. I'm eager to hear any challenges to the math I've used and the assumptions therein. But as it looks now, this isn't getting anywhere close to space.
