Calling all thermo whiz's...

[slickjohannes]

Here's the problem statement:

A 0.5m^3 rigid tank has equal volumes of saturated refrigerant vapor and refrigerant liquid at 312 deg K. Additional refrigerant is then slowly introduced into the tank until the total mass of refrigerant (liquid and vapor) is 400 kg. Some vapor is bled off to maintain the original temperature and pressure. After adding the refrigerant, the refridgerant is saturated at 312 deg K and 0.9334 MPa. Under these saturated conditions, v_f=0.000795 m^3/kg and v_g=0.01872 m^3/kg.

And the question:

What was the mass of refrigerant added to the tank?

If someone can explain the solution, it would be much appreciated...

 

[IRstuff]

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[slickjohannes]

Actually, this is not homework (but I see why you would think so). This is a problem from FE review material (a VERY popular piece of review material). The solution for the problem is given as 75kg. I don't know enough to say I agree with how that number is arrived at.

My solution, which assumes a negligible amount is "bled off", is 72 kg. If my memory serves me right, and it has been 7+ years since I thought about thermodynamics... when something is done slowly, like mass added to a rigid tank, you assume that there is no temperature change of the system (the temperature change I speak of would be due to the compression of the vapor in the tank). If there is no temperature change of the system, then there is no pressure rise (can't have one without the other, right?). If there is no pressure rise, then adding mass without changing volume just pushes you closer to the saturated liquid point (v_f) WHEN YOU ARE UNDER THE VAPOR DOME.

The solution indicates that one must first calculate the volume that is bled off. This is done by taking the final volume of the vapor (which we known from solving for the quality using the final mass and volume, or .5m^3/400kg) and then subtracting from the initial volume of the vapor:

0.25m^3 - 0.19m^3 = 0.06m^3

Then the solution goes on to use the 0.06m^3 to find the resultant mass of vapor that was bled off.

What I'm asking is, what is the change in vapor volume due to (i.e. the 0.25m^3 to 0.19m^3)? Is it truly "bled off"? Or, is this the result of the quality of the mixture being pushed more toward the saturated liquid point (v_f)???

 

[Latexman]

Is it a multiple choice question? Are all other choices < 72 kg?

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[slickjohannes]

Choices are 14kg, 22kg, 72kg, and 75kg. I guess you could say that since you bled off some gas, the amount of mass you added must be more than 72kg. Still doesn't explain why it would be exactly 75kg though...

What am I missing here???

 

[Latexman]

I calculated 72.2 kg plus the blowoff. 75 kg is the only solution.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[slickjohannes]

Latexman- are you saying that 75kg is the answer, merely for the fact that 75kg is larger than 72kg? Or did you actually calculate a blow off of ~3 kg of vapor? The step in the solution covering the calculation of the additional mass is very specific in that the blow off is [initial vapor volume - final vapor volume]/v_g. With actual values...

[0.25 m^3 - 0.19 m^3]/0.01872 m^3/kg= 3.2kg

 

[Latexman]

"Latexman- are you saying that 75kg is the answer, merely for the fact that 75kg is larger than 72kg?" Yes, it is indeterminant as stated. It is written to confuse you too. The conditions and properties of the refrigerant that is added is not known. The conditions and properties at the state where the tank has exactly 400 kg mass is not known. We only know the conditions and properties at the original temperature and pressure and after obtaining 400 kgs AND some vapor is bled off to maintain the original temperature and pressure. The sentence "After adding the refrigerant, the refridgerant is saturated at 312 deg K and 0.9334 MPa. Under these saturated conditions, vf=0.000795 m^3/kg and vg=0.01872 m^3/kg." is not literally correct based on "Some vapor is bled off to maintain the original temperature and pressure."

It's a convoluted problem as written.

So, let's ASSuME the refrigerant added is 312 deg K and 0.9334 MPa too.

400 kg - 0.25/0.000795 - 0.25/0.01872 = 72.18 kg added IF the refrigerant added is 312 deg K and 0.9334 MPa too.

Since we know some refrigerant was bled off, it must be greater than 72.18 kg.

75 kg is the only viable answer.



Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[slickjohannes]

The statement "slowly introduced" screams quasiequilibrium process. I think this is the result of revising a an old problem to be a new problem with new answers (as is commonly done when authors create new editions of their texts).

I'm going to report it as errata: it is misleading at best, bad analysis at worse.

 

[Latexman]

I think that's reading too much into it. It's a thermo problem that also tests logic skills. An engineer needs both, so it's fair game.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[slickjohannes]

I will, respectfully, disagree. The solution uses the difference between vapor volumes to find the EXACT weight that is supposedly bled off.

Logic dictates that there must be more than 72 kg added to the tank if some of it was bled off, but science does not dictate that it was exactly 3 kg that was bled off, thus the book's solution to the problem is at fault.

 

[Latexman]

If it wasn't multiple choice with those exact selections, I'd agree with you.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[1gibson]

"..the book's solution to the problem is at fault."

Prove it. Prove that the book's solution is incorrect (or define "at fault" in that context) with only the given information.

Can't do it? Well now who is incorrect, you or the book?


You should learn the difference between should, would and could.

If you only pick the answers that "should" be correct, nobody will care what your piece of paper says when something doesn't work out during testing due to erroneous assumptions.

If you only pick the answers that "would" be correct "if only..." then you are just wasting peoples' time.

Sometimes in the real world, you have to pick the answer that "could" be correct, because it works, and you can't prove that it is incorrect.


Should, would, and/or could you agree?

 

[BernardRhodes]

A cubic metre of liquid inflow adds 1/0.000795 kilograms = 1257.86

As that cubic metre leaves as vapour to make room for the liquid out goes 1/0.01872 kilograms = 53.42

The net effect is to add 1204.44 to the trapped mass for every 1257.86 of inflow.

So to add the 72.18 noted upthread requires an inflow of 72.18 * 1257.86/1204.44 = 75.38

 

[ivymike]

I get the same answer as BR

total volume 0.5 m3
liquid density 1257.861635 kg/m3
gas density 53.41880342 kg/m3
intial vol each phase 0.25 m3
mass gas initially 13.35470085 kg
mass liquid initially 314.4654088 kg
total mass initially 327.8201097 kg

final mass 400 kg
v_tot = v_gas + v_liq
m_tot = (dens gas)*v_gas + (dens liq) * v_liq
v_gas = 0.5 - v_liq

final conditions
0.309860689 v_liq
0.190139311 v_gas
0.059860689 displaced vgas
3.197686386 displaced mgas

added mass is 400 - 327.82 + 3.20 = 75.38kg

 

[ivymike]

I gather from re-reading some comments above that people are seeing the statement "Additional refrigerant is then slowly introduced into the tank" and thinking "well it depends on how slowly ... if you add it fast, you'll affect the answer!"

I don't think that's the point of the problem?

What I'm asking is, what is the change in vapor volume due to (i.e. the 0.25m^3 to 0.19m^3)? Is it truly "bled off"? Or, is this the result of the quality of the mixture being pushed more toward the saturated liquid point (vf)???

Why would the quality (or vapor pressure) change if pressure and temperature are held constant?

 

[chicopee]

I am thinking that if the problem had been submitted several decades ago when slide rules were used instead of more accurate calculators than 75Kg would be the correct answer.