Math Is Fun Forum

tigerfan

Member

Registered: 2007-03-09

Posts: 8

help me please

a room measures 3 m by 4.5 m by 6 m.  The heating and air conditioning ducts to and from the room are round with a diameter of .30 m, and the air in the room has to be exchanged every 12 minutes.  What is the necessary flow speed in the duct?  Assuming air density is constant.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: help me please

I am going to change the numbers so I don't do your homework for you.
Hopefully it will still help you, though.  Sorry to say that if this is not homework.

Now the new room I'm doing is in feet, so it is a rectangular cube with sides of 20 feet by 30 feet and a really high ceiling of 16 feet 8 inches.  Now multiply the 3 dimensions of the room together.
  20 times 30 times 16.6666666 = 10000 cubic feet in the room.
Now the ducts in my case are triangular ducts.
Each duct has three sides which are ONE foot each, so the triangle is equilateral.
The height of the triangle from the center of one side (the base), is the square root of
3 times one-half a foot, since there is a 30-60-90 triangle, two of them beside each other.
The triangle is a 60-60-60 triangle, but if you cut it in half the tall way, you get
two right triangles 60-90-30 triangles.  The sides of this triangle are:
one-half, one-half times 1.7320508 or so, and one.
The 1.7320508 number is the square root of 3.
Try multiplying 1.732 times 1.733 on your calculator.(Just about 3, huh?)

So what is the area (called a cross-section of something real and 3-D) or the
triangular pipe?  Well it is one-half the base(one) times the height(0.866 or so).
0.866 is about half of 1.732.
Area = 1/2 times (one, the base) times (height, sqrt of 3 divided by 2)
Area = 1/2 times 1 times sqrt(3) times 1/2
Area = sqrt(3) / 4
So the Area of the triangular pipe is about 0.433 square feet.
In your problem, it will be square meters.
Do you know the area of a circle?  It is Area = pi times diameter times diameter times 1/4
Also known as Area = pi times radius times radius

So now you have to empty the whole room and refill it at the same time.
This amount you empty and refill is just the volume of the room,
not twice the volume of the room.
Why?  Because one pipe empties the volume of the room.
One pipe fills the volume of the room.

Now you want to find out how long the triangular pipe would be to hold
all the air in the room, mostly nitrogen, surprisingly.
The pipe may be shorter, but this is a fun way to do it.
Because if you have a really long pipe that holds the whole room worth of air,
then this length of the pipe divided by the time required to refill the room
is probably your answer.

---

igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: help me please

To continue...
If want to put 10 thousand cubic feet of air into a pipe with a cross-cut area of
about 0.433 square feet, then you need a length of pipe, whereby when you
multiply the length of the pipe by 0.433, you get square feet to change into cubic feet.
So,
0.433 square feet times HOWlongFEETofPIPE = 10 thousand cubic feet
Divide both sides of equation by 0.433 to solve.
HowLongPipe = 10000/0.433, which equals about 23 thousand foot long pipe.
Check this in your head a bit.
0.433 was less than one, so answer is bigger than 10 thousand.
0.433 is like about half of one, so answer is about twice 10 thousand.

So now you have an approximate length of pipe of 23 thousand feet.

The question asks for the flow rate of the air, and this would be simply the
long length of pipe divided by the time it takes to fill it, I would think.
Because if the whole pipe was filled in one second, which is very fast,
then the flow speed would be 23000 feet per second.
If the whole room needs to be filled in 2 seconds, then
the flow rate or flow speed would be 23000 feet per 2 seconds.
"per" means "divide"
Like miles per hour, means miles divided by hours.
And "miles an hour" is the same thing, but just not as proper English.

So how many seconds are there in 12 minutes?
60 times 12 = 720 seconds.
So 23000/720 is 23000 long tube in 720 seconds.
Divide it on calculator or by hand.
Answer is about 32 feet per second.

The real answer with your numbers and a circle hopefully is more reasonable, like
around 2 feet per second.  That's just a rough guess, but who knows.
Hope I haven't confused you even more...  Bye.

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igloo myrtilles fourmis

tigerfan

Member

Registered: 2007-03-09

Posts: 8

Re: help me please

Thanks for the help.  You have been a great help.  You cleared it all up.