Hi solorider, Using the term "liquid" in front of "nitrogen" and saying it is at 1000 psi is confusing at the very least. Nitrogen, like any fluid, has some pressure and temperature that it boils at. For nitrogen at atmospheric pressure (14.7 psia) that temperature is around -320 F. When it's inside a tank such as you're suggesting, the gas that boils off increases the pressure of the tank (assuming it can't get out. So as we add heat, the pressure rises. The temperature also rises. If the gas and vapor are in thermal equilibrium, meaning they are at the same temperature because they're in close, physical contact, the liquid is boiling and the gas is at the same temperature as it bubbles out of the liquid.
As the pressure of this boiling mixture increases, the temperature also increases. So at atmospheric pressure (14.7 psia) the temperature is -320 F. But as the pressure rises to the critical pressure of 492 psia, the temperature rises to -232 F. So the temperature has come up a total of 88 degrees F. The same thing happens in the radiator of your car. If you have the cap on the radiator, the pressure increases and the temperature at which the coolant boils at also increases. All liquids do this AFAIK. They all will need to get hotter in order to boil at a higher temperature.
In addition to an increase in temperature, the density of the boiling liquid decreases and the density of the gas which is boiling off increases as the pressure and temperature increases. For nitrogen at atmospheric pressure, the density of the liquid is 50 lbm/ft3 and the density of the gas is 3.5 lbm/ft3. When the pressure reaches 492 psia, the density of the liquid is 24 lbm/ft3 and the density of the gas is 16 lbm/ft3. Note how close together the density of the gas and liquid is as we approach the critical pressure. Also remember the dramatic rise in temperature.
In fact, all the properties change including surface tension. As pressure and temperature increase, the surface tension decreases until it finally fades away at the critical pressure.
When nitrogen gets up to the critical pressure of just over 492 psia, the density of the liquid and gas are the same and the surface tension between the two disappears. There is no longer a boundary between the liquid and gas and there is no difference in density. At this critical pressure and just above it, you could heat up the nitrogen at constant pressure and the nitrogen (assuming it could expand) would warm up and become less dense, but there wouldn't be any liquid. Similarly, you could remove heat from the nitrogen and make it colder and more dense, but there still wouldn't be any liquid/gas interface because there is no surface tension above the critical pressure.
For a restriction such as a round hole in the tank 10" in diameter, the flow rate out of the hole is a function of the density of the fluid. Assuming a discharge coefficient for the hole of 0.6 (which is fairly typical of a choked orifice, though it could be a bit higher) the flow rate is a function of density which is also a function of temperature. For the sake of argument, we know the LIN has warmed up from -320 F to -230 F at around 500 psi, so let's just use -200 F as an example at 1000 psig. At -200 F, the flow is around 3100 lbm/s. It could be much warmer or even much colder. It depends on how the nitrogen got up to 1000 psi. You talk about wanting to put LIN into keep the pressure up to 1000 psi, but you don't say how. The only realistic way is to pump it in (with a very large, multi stage centrifugal pump). If it was pumped up to that point for example, you have a roughly isentropic compression of the nitrogen. If we assume 100% isentropic efficiency for our pump and assuming our pump went from atmospheric pressure to 1000 psig, the temperature of this 1000 psi nitrogen would be -317 F which is MUCH denser and colder than the nitrogen that simply warmed up. In this case, the flow could be almost 10,000 lbm/s.
So if you want to know how much nitrogen comes out, it's somewhere between about 1000 lbm/s and 10,000 lbm/s and the actual flow rate will depend on what density the supercritical gas is that's blowing out. There is no way to determine what that density is without knowing 'how it got there' so to speak. If it warms up and blows off, it could be fairly warm. If it is being replenished by a pump, it would be very cold. The actual flow rate depends on density which depends on how it got to that point.