Calculate the volume of a square to circular cross section transition

[Dave]

Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?

Any help with this would be greatly appreciated.

Dave

[William Elliot]

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.

What's a uniform transition?

[[Mr.] Lynn Kurtz]

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, there are lots of ways of transitioning from a square to a
circle. You aren't going to like the equations, but here is a picture
where d = 8, s = 1 showing some cross sections of the transition
contours using the l^p unit balls. It's too late right now to send
more, I have to hit the sack.

Loading Image...

--Lynn

[[Mr.] Lynn Kurtz]

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph of
the surface. You have an area function A(x) with A(0) = s^2 and A(L) =
pi D^2 / 4. So you can easily construct the linear function A(x) and
the volume would be:

V = int[0..L] A(x) dx

--Lynn

[Dave]

I don't think you can "easily construct the linear function A(x)" even if
you know the end point conditions.
Is the answer as simple as the average cross section times the length, i.e.
((s^2 +piD^2/4)/2)xL?
Dave

Post by [Mr.] Lynn Kurtz

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph of
the surface. You have an area function A(x) with A(0) = s^2 and A(L) =
pi D^2 / 4. So you can easily construct the linear function A(x) and
V = int[0..L] A(x) dx
--Lynn

[[Mr.] Lynn Kurtz]

On Tue, 9 Oct 2007 18:31:05 -0400, "Dave" <***@complaninc.com>
wrote:

Please don't top post [corrected]. My reply is at the bottom.

Post by Dave

Post by [Mr.] Lynn Kurtz

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph of
the surface. You have an area function A(x) with A(0) = s^2 and A(L) =
pi D^2 / 4. So you can easily construct the linear function A(x) and
V = int[0..L] A(x) dx
--Lynn

I don't think you can "easily construct the linear function A(x)" even if
you know the end point conditions.
Is the answer as simple as the average cross section times the length, i.e.
((s^2 +piD^2/4)/2)xL?

It is indeed easy to construct a linear function through two points.
and if this is a calculus text exercise, I would expect the teacher
would want to see that formula and see the integral worked out.

You didn't answer whether this is a calculus book exercise and whether
"uniform transition" means linear. *If* the answer to that is yes,
then yes, that formula is correct, because the integral giving the
volume can be interpreted as the area under the straight line y = A(x)
betweem x = 0 and x = L, which is a trapezoid. Or is this a real world
problem with a real pipe of unknown shape? Where did the problem come
from? It makes a difference.

--Lynn

[Dave]

Post by [Mr.] Lynn Kurtz
Please don't top post [corrected]. My reply is at the bottom.

Post by Dave

Post by [Mr.] Lynn Kurtz

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph of
the surface. You have an area function A(x) with A(0) = s^2 and A(L) =
pi D^2 / 4. So you can easily construct the linear function A(x) and
V = int[0..L] A(x) dx
--Lynn

I don't think you can "easily construct the linear function A(x)" even if
you know the end point conditions.
Is the answer as simple as the average cross section times the length, i.e.
((s^2 +piD^2/4)/2)xL?

It is indeed easy to construct a linear function through two points.
and if this is a calculus text exercise, I would expect the teacher
would want to see that formula and see the integral worked out.
You didn't answer whether this is a calculus book exercise and whether
"uniform transition" means linear. *If* the answer to that is yes,
then yes, that formula is correct, because the integral giving the
volume can be interpreted as the area under the straight line y = A(x)
betweem x = 0 and x = L, which is a trapezoid. Or is this a real world
problem with a real pipe of unknown shape? Where did the problem come
from? It makes a difference.
--Lynn

Lynn,
Thanks for your reply and request for clarification so here goes...

It's not a textbook exercise although it would probably be a pretty good
one. It's been 46 years since I studied calculus so I don't have some
instructor waiting for my answer no matter how hard I wish that was the
case! And I no longer have the math skills (or maybe persistance!) to solve
the problem on my own.

A friend making a ductwork transition piece from square to round posed the
question during an unfortunate relapse into scholarly inquiry.
My engineering training curiosity was also tweaked by the question and I'd
like to know the exact answer but I'll sleep well tonight no matter what.
So it's not exactly a "real world problem" but it arose out of some real
world activity.

To me the reference to a uniform transition just means that it progresses
smoothly from square to round. Whatever that mathematically implies is fine
with me. But back to my previous post - just exactly what is the A(x)
function that needs to be integrated between A(0) = s^2 and A(L) = piD^2/4?
I don't care what assumption you make about what linear means. Just pick
something that satisfies the end point conditions and looks like a piece of
square to round ductwork that any sheet metal worker can easily fabricate.

Dave

[[Mr.] Lynn Kurtz]

Post by Dave

Post by [Mr.] Lynn Kurtz
Please don't top post [corrected]. My reply is at the bottom.

Post by Dave

Post by [Mr.] Lynn Kurtz

Post by Dave
Given a square cross section of side 's' and a downstream circular cross
section of diameter 'd', how do you calculate the volume of a uniform
transition from the square cross section to the circular cross section over
a transition length of 'L'?
Any help with this would be greatly appreciated.
Dave

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph of
the surface. You have an area function A(x) with A(0) = s^2 and A(L) =
pi D^2 / 4. So you can easily construct the linear function A(x) and
V = int[0..L] A(x) dx
--Lynn

I don't think you can "easily construct the linear function A(x)" even if
you know the end point conditions.
Is the answer as simple as the average cross section times the length, i.e.
((s^2 +piD^2/4)/2)xL?

It is indeed easy to construct a linear function through two points.
and if this is a calculus text exercise, I would expect the teacher
would want to see that formula and see the integral worked out.
You didn't answer whether this is a calculus book exercise and whether
"uniform transition" means linear. *If* the answer to that is yes,
then yes, that formula is correct, because the integral giving the
volume can be interpreted as the area under the straight line y = A(x)
betweem x = 0 and x = L, which is a trapezoid. Or is this a real world
problem with a real pipe of unknown shape? Where did the problem come
from? It makes a difference.
--Lynn

Lynn,
Thanks for your reply and request for clarification so here goes...
It's not a textbook exercise although it would probably be a pretty good
one. It's been 46 years since I studied calculus so I don't have some
instructor waiting for my answer no matter how hard I wish that was the
case! And I no longer have the math skills (or maybe persistance!) to solve
the problem on my own.
A friend making a ductwork transition piece from square to round posed the
question during an unfortunate relapse into scholarly inquiry.
My engineering training curiosity was also tweaked by the question and I'd
like to know the exact answer but I'll sleep well tonight no matter what.
So it's not exactly a "real world problem" but it arose out of some real
world activity.
To me the reference to a uniform transition just means that it progresses
smoothly from square to round. Whatever that mathematically implies is fine
with me. But back to my previous post - just exactly what is the A(x)
function that needs to be integrated between A(0) = s^2 and A(L) = piD^2/4?
I don't care what assumption you make about what linear means. Just pick
something that satisfies the end point conditions and looks like a piece of
square to round ductwork that any sheet metal worker can easily fabricate.
Dave

If you really want the volume, you could build one, block off one end,
and fill it with water. Then measure the quantity (or weight) of the
water. Someone stretching sheet metal from one shape to the other
isn't going to have a nice formula to work with, but the assumption
that A(x) is linear should give a decent approximation. Consider:

A(x) = s^2 +(1/L) ( piD^2 / 4 - s^2 ) x

When x = 0 get A(0) = s^2 and when x = L you get:

A(L) = s^2 + piD^2/4 - s^2 = piD^2/4

If you actually work the integral of A(x) from 0 to L you will get the
trapezoidal area formula you have above. Try it. :-)

--Lynn

[William Elliot]

Post by Dave

Post by [Mr.] Lynn Kurtz
Please don't top post [corrected]. My reply is at the bottom.

Also trim out stuff that no longer needed.

http://oakroadsystems.com/genl/unice.htm

Post by Dave

Post by [Mr.] Lynn Kurtz

Post by [Mr.] Lynn Kurtz

Post by Dave
Given a square cross section of side 's' and a downstream circular
cross section of diameter 'd', how do you calculate the volume of a
uniform transition from the square cross section to the circular
cross section over a transition length of 'L'?

Of course, if this is just a calculus book exercise, then the phrase
"uniform transition" might mean the area cross section changes
*linearly*. In that case there is no need for the equation or graph
of the surface. You have an area function A(x) with A(0) = s^2 and
A(L) = pi D^2 / 4. So you can easily construct the linear function
V = int[0..L] A(x) dx

To me the reference to a uniform transition just means that it progresses
smoothly from square to round. Whatever that mathematically implies is fine
with me. But back to my previous post - just exactly what is the A(x)
function that needs to be integrated between A(0) = s^2 and A(L) = piD^2/4?
I don't care what assumption you make about what linear means. Just pick
something that satisfies the end point conditions and looks like a piece of
square to round ductwork that any sheet metal worker can easily fabricate.

A(x) = s^2 + pi.d^2 x/4L