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.

 

[DeeNos]

I would like to point out that the given problem is a classic example of isothermal compressible flow in a horizontal pipe. In this type of flow, the gas temperature remains constant throughout the pipe, which means that the ideal gas equation can be used to calculate the pressure at any point. Additionally, the given data also provides the necessary information, such as molecular weight, gas constant, and friction factor, to solve the problem using the given equation.

Now, coming to the iteration method, it is a numerical technique used to solve problems where the exact solution cannot be obtained analytically. In this case, the pressure at the compressor needs to be iterated until it matches the given delivery pressure. The number of iterations required can vary depending on the problem and the chosen initial guess. In some cases, the answer can be obtained in one or two iterations, while in others, it may take more steps. The key is to keep iterating until the answer converges and stops changing significantly.

In order to check if you are on the right track, you can compare your answer with the given answer and see if they match. If they do not match, then you need to keep iterating until you get the correct answer. Additionally, you can also try changing your initial guess and see if it affects the number of iterations required.

In conclusion, the iteration method is a useful tool for solving problems in which the exact solution cannot be obtained analytically. It may require some trial and error, but with practice, you will be able to determine when to stop and how many steps are needed to obtain the correct answer. It is important to carefully follow the given equation and use the provided data to ensure accurate results.