So assuming isothermal conditions, in the large vessel, you'll have compressed butane, whether "iso" or "n" is irrelevant at this point. So liquid butane is to be conveyed under pressure up the beginning of the piping where presumably you have a valve that may act as a throttling device, therefore, assume constant enthalpy before and after the valve. Past the valve the butane could start changing into a saturated vapor still under isothermal condition of 273dC. At end of piping another valve is encountered,before the small tank, and there initially butane will flash into a superheated vapor under constant enthalpy. When this small tank is being filled, its internal pressure will increase under assumed isothermal conditions; then conditions in this small tank will reach the saturated vapor line and then saturated vapor conditions will prevail inside of the small tank. The mass transfer rate of butane at saturated vapor condition will have by that time decrease substantially. When the butane quality reaches about 5% which is very close to the saturated liquid line then you will have your final condition. I think that is one scenario that you should investigate using time step analysis of Bernoulli's equations for non-compressible fluid within the large tank assumed to be infinitely large in comparison to the small tank; Bernoulli's equation for compressible fluids under isothermal condition between the discharge end of the tube and small tank; within the tube you would probably have a capillary tube such as found in small air conditioner where you could apply the Bernoulli's equations for compressible fluid under isothermal condition. Constant enthalpy reasoning can be apply at the inlet and outlet of both valves if these valves act as throttling devices.
Get a Mollier diagram for butane; The one that I found on line was for isobutane; and research the Bernoulli's equation for the conditions stated in any fluid mechanics book or Schaum's outline. You may have to assume an initial mass rate for Butane and probably have trial and error calculations in conjunction with time step analysis for which you can designate small time elements. I would also read about capillary tubes used in AC, to get an idea on pressure drops, materials used and flow rates.
