Hello. Your question is very open ended and therefore it's hard to answer to it even for somebody knowledgeable in the area. It's comparable to asking “How many batteries you need to hurt somebody?” without specifying battery types, condition of the subject, definition of being
hurt, etc...
1. How much pressure should the compressed air have for it to push someone?
By
pushing do you mean making a force upon the person or moving it overcoming the static friction between him and the surface he lies on?. If it's just making a force, then there's no lower limit. Even the pressure from gently blowing is enough. If it's displacing the person, then it obviously depends on the specific situation. It isn't the same to displace a man with a mass of 120 kg lying supine in an asphalt road than it is to displace a 50 kg woman standing on a ice-skating rink. In both cases you'd have to determine it experimentally, or at least determine the relevant parameters experimentally. Numerically, it's not a readily tractable problem, see below.
2. How big does he diameter of the outlet has to be to be able to push someone without inflicting damage? (of course not taking into consideration what could happen to the poor guy after getting blown)
This questions adds more ambiguity and open-endness to the previous one. Let's recall that pressure is experienced by objects as a force distributed continuously though the whole or a part of its surface. To answer your question we would first need to know the pressure of the
hose, the force needed to “push someone” with the open ended condition of “without inflicting damage” and how that person and the air nozzle interacts: I.e: distance, angle, part of body, area of nozzle opening, etc... If you want to do it numerically in an accurate answer, you will find yourself having to solve the
Navier-Stokes equations or making simplifications hard to justify. The problem would be a bit easier with water, because given the fluid density, flow rate and speed and assuming that it loses all of its momentum in the person being hit by the water jet it's trivial to compute the force (force=density×speed×flow rate) but such a force isn't evenly distributed or even time-invariant, and the shape of the body part being hit is also affected by the force, which alters the force distribution forming a feedback loop. It's worser with your problem concerning an air jet. An air jet isn't so well behaved, it diverges rapidly and creates turbulence. The interaction with the air already present in the environment is very complex. The chaotic and bad-behaved problems (including yours and the slightly easier water jet problem) makes it hard to give a meaningful answer even for well defined conditions, and even those are not provided in your question.
Let me note that I'm not at all an expert in fluid dynamics. Having said this, in order to compute the nozzle speed of air as a function of the pressure in the hose, you'd have to apply
https://en.wikipedia.org/wiki/Bernoulli's_principle]Bernoulli's[/PLAIN] equation combined with the fact that the air is at atmospheric absolute pressure (or zero gauge pressure) once it has exit the nozzle. Note that the air pressure in the hose isn't the same as the pressure in the tank (due to the fact the air is being accelerated when entering the hose from the tank, and that again explained by Bernoulli's equation) and the pressure read by workshop type pressure gauges is only the excess from atmospheric pressure. It's very conveniently called gauge pressure. Note that Bernoulli's equation is sometimes described as being an instance of conservation of energy, but this a possibly misleading explanation. When a sample of fluid accelerates a work is being done on it, so that its energy increases; since the energy increases; hence the energy is not conserved for a sample of fluid as it passes from the tank to the hose or the tank to the nozzle. Computing the nozzle speed is as far as you can go to solving this problem without performing experiments, and even then you need data not provided by your question (Area of the nozzle opening and hose pressure). If you're interested in computing this, look up the relevant information on the Internet. The linking article in Wikipedia is a good starting point though only a small part is relevant for the problem in question. Or alternatively, pick an university-level introductory physics book which talks about fluid dynamics, for instance:
Fundamentals of Physics by Halliday and Resnick, 10th edition, p. 401 onwards.
To summarize: Your questions are too open ended to answer, and even if they were well defined, it would be very hard to give an answer by numerical methods (What some people would call “theoretical approach”).
I hope that this helps.
Regards.