The percentage of carbon dioxide in the air

In summary: I know some who are like "air goes in, air goes out so there is no change, so 0 is applied") The equation I ended up with was:\frac{dx}{dt}=2.5(?unit/cu ft)(5000 cu ft/min) - 0.
  • #1
Beez
32
0
Hello. I need a help again for my differential equations class.
The problem is as follows:
The air in a room whose volume is 10,000 cu ft tests 0.15% carbon dioxide. Starting at t=0, outside air testing 0.05% carbon dioxide is admitted at the rate of 5000 cu ft/min.
(a) What is the percentage of carbon dioxide in the air in the room after 3 min?
(b) When does the air in the room test 0.1% carbon dioxide.

This problem is given after I studied the mathematical formulation of
[tex]\frac{dx}{dt}=IN - OUT[/tex]

I tried to write an equation using the above formula, but since there is no air coming out, I did not know how to write the euqation. The equation I wrote was

[tex]\frac{dx}{dt} = 2.5 ( ? unit/cu ft)(5000 cu ft/min) - 0[/tex]

I obtained 2.5 from multiplying 5000 by 0.0005. I don't even know if this is right. What I was trying to find was the amount of carbon dioxide in 1 cu ft.
Since no air is coming out, 0 is applied in the OUT section.
This does not seem to be right since the density of the air should be increased but there is no place that represent the change in my equation.


Will someone help me?
 
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  • #2
Beez said:
Hello. I need a help again for my differential equations class.
The problem is as follows:
The air in a room whose volume is 10,000 cu ft tests 0.15% carbon dioxide. Starting at t=0, outside air testing 0.05% carbon dioxide is admitted at the rate of 5000 cu ft/min.
(a) What is the percentage of carbon dioxide in the air in the room after 3 min?
(b) When does the air in the room test 0.1% carbon dioxide.

This problem is given after I studied the mathematical formulation of
[tex]\frac{dx}{dt}=IN - OUT[/tex]

I tried to write an equation using the above formula, but since there is no air coming out, I did not know how to write the euqation. The equation I wrote was

[tex]\frac{dx}{dt} = 2.5 ( ? unit/cu ft)(5000 cu ft/min) - 0[/tex]

I obtained 2.5 from multiplying 5000 by 0.0005. I don't even know if this is right. What I was trying to find was the amount of carbon dioxide in 1 cu ft.
Since no air is coming out, 0 is applied in the OUT section.
This does not seem to be right since the density of the air should be increased but there is no place that represent the change in my equation.


Will someone help me?

I think you have the IN rate right (no OUT rate). That's 2.5 cu ft of carbon dioxide (100% pure) per minute entering a 10,000 cu ft room that already has how many cu ft of carbon dioxide in it already? That's it. Just 2.5. Now just solve the ODE like that for cu ft of carbon dioxide in room as a function of time. Pretty sure anyway. :smile:
 
  • #3
Beez said:
Hello. I need a help again for my differential equations class.
The problem is as follows:
The air in a room whose volume is 10,000 cu ft tests 0.15% carbon dioxide. Starting at t=0, outside air testing 0.05% carbon dioxide is admitted at the rate of 5000 cu ft/min.
(a) What is the percentage of carbon dioxide in the air in the room after 3 min?
(b) When does the air in the room test 0.1% carbon dioxide.
altho not mentioned, any practical application (or air system) would be withdrawing air at the same volumetric rate that air is entering. of course, concentration of CO2 leaving is not equal to that entering.

let x(t) be the actual pure volume of CO2 in the room at time t.
then x(0) = 10,000*0.0015 = 15 cu ft pure CO2 (the t=0 initial condition)

at t=0, pure CO2 begins entering the room at the rate of 5000*0.0005 = 2.5 cu ft/min = IN RATE (pure CO2)

at t=0, inside room air (presumably instantaneously mixed with the entering air) begins leaving the room with concentration x(t)/10,000 of CO2 at the rate of 5000 cu ft/min (= air entry volumetric rate), so the rate at which pure CO2 leaves the room = -5000*x(t)/10,000 = -x(t)/2 cu ft/min = -OUT RATE (pure CO2)

dx(t)/dt = IN RATE (pure CO2) - OUT RATE (pure CO2)
dx(t)/dt = 2.5 - x(t)/2
or:
dx/dt + x/2 = 2.5

however, CO2 concentration c(t) = x(t)/10000 so that
dc/dt + c/2 = 2.5/10000 with initial condition c(0) = 15/10000

now you can determine concentration c(t) using Ordinary Differential Equation techniques given in your textbook.
 
Last edited:
  • #4
I didn't know that

Thanks for your help again Saltydog and thanks for the detailed explanation for the problem, Geosonel!

I did not know the air is going out at the same rate (I thought the air density would increase = this concept got me). (Not all students are as smart as you, so I really think the textbook should mentioned the fact either somewhere in the textbook or in the problem :frown: )

Thanks to you, now I am sure I can solve the equation. If I can't, I will come back here again (hopefully not).

Thanks!
 
  • #5
geosonel said:
altho not mentioned, any practical application (or air system) would be withdrawing air at the same volumetric rate that air is entering. of course, concentration of CO2 leaving is not equal to that entering.

Thanks Geosonel. Didn't know that. Sorry if I messed you up Beez. :blushing:
 
  • #6
Hello! I have a problem: I need to calibrate a CO2 senzor and I can't do it right today. I need to know the percentage of CO2 in a 12 sqm room under normal conditions. Can you help me or give me an idea? Thanks
 

What is the current percentage of carbon dioxide in the air?

The current percentage of carbon dioxide in the air is approximately 0.04%. This has been steadily increasing due to human activities such as burning fossil fuels and deforestation.

Why is the percentage of carbon dioxide in the air important?

The percentage of carbon dioxide in the air is important because it is a greenhouse gas that traps heat in the Earth's atmosphere. An increase in carbon dioxide can lead to global warming and climate change.

How is the percentage of carbon dioxide in the air measured?

The percentage of carbon dioxide in the air is measured using instruments such as infrared gas analyzers or gas chromatographs. These instruments can detect the concentration of carbon dioxide in the air by measuring the absorption of infrared light.

What are the natural sources of carbon dioxide in the air?

Natural sources of carbon dioxide in the air include respiration by animals and plants, volcanic eruptions, and decomposition of organic matter. These sources have been in balance for thousands of years, but human activities have caused an increase in carbon dioxide levels.

What are the consequences of high levels of carbon dioxide in the air?

High levels of carbon dioxide in the air can lead to a range of consequences, including global warming, climate change, ocean acidification, and negative impacts on human health. It can also affect plant growth and alter ecosystems.

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