Isothermal compressible flow in horizontal pipe

[db725]

Homework Statement

A compressor is required to drive acetylene gas at 1.85 kg/s through a horizontal pipe, 68.7 m long. The maximum pressure that may be developed by the compressor is to be found, and gas pressure at the delivery end of the pipeline must be 470 kPa. The system is to operate isothermally at 20 degrees.

If a pipeline has a diameter of :0.0924 m, to what pressure must the gas be at the compressor if delivery pressure is to remain the same?

Homework Equations

Data:
Molecular weight = 26 gram/mol
Gas Constant = 8.314 Jmol^(-1)K^(-1)
Fanning friction factor =0.002


I have attached the equation for horizontal, isothermal ideal gas flow in a pipe of constant cross section.

The Attempt at a Solution

The answer is : 527.5 kPa

I understand the logic behind the question and the process that needs to be followed but I seem to have difficulty with the part where we need to use the 'iteration' method.

I have attached my working out. My question is with this iteration method when do we know when to stop? Sometimes in questions you find the pressure in your first attempt and in others in order to get the answer you need to do 3 steps. In this question it took me a 2 step process. Can someone please help me? I am preparing for exams and I need to use this method in questions like these but I don't know when to stop or when my answer is right. Is there a way of checking or knowing I am on the right track or how many steps I need to find the right answer?

Thanks for the all the help in advance.

 

[RTW69]

On your first guess you get rid of the ln term so can find P1 directly. Plug P1 into the complete equation and see how close you are to 0. Guess a new P1, compare to 0, Guess a new P1 compare to 0. You get the idea.

 

[db725]

Not sure if I understand it correctly but doesn't each new estimate for P1 satisfy the zero condition to begin with? So the variation for each P1 isn't massive because they all have to satisfy the equation. So how would I go about knowing if i have the right pressure?

 

[SteamKing]

If you plug your value of P1 obtained from the calculations on the second attachment back into the formula shown in the first attachment, evaluating the formula shows a net result of about -68200, which alas is not zero. What RTW is trying to tell you is that you must guess a new value of the unknown pressure P1 (different from 527 kPa) and re-evaluate the formula to obtain the sum of the terms. If this sum is different from zero, a new trial P1 value must be guessed and the formula evaluation repeated until you are satisfied that each new trial value of P1 does not change in value to be significant.