RGCook

August 19, 2011 at 4:47pm

OK kiaghi7,

 

First, you are using the equation for the area of a circle but you really should be using circumference times length to calculate the effective surface area available to heat transfer. Another mistake in your example to me is that you used the diameter in your calculation, not radius. That is moot. Hear me out now;

The effective heat transfer area is not cross section but surface area of the tubes. Therefore, you will want to calculate the surface area of three small pipes that will fit inside one large pipe.

It can be shown that for three small pipes of radius = 1, the smallest large diameter circule that will contain them is 2.154. Units are not important.

Let's assume that the length of the pipes is unit for the sake of convenience. Therefore, the surface are of the large pipe, unit long is 2 x pi x (2.154/2) = 6.77 sq units

The surface area of the three smaller pipes that would fit inside would be 3 x 2 x pi x (1/2) = 9.42 sq units.

We can see that three smaller tubes affords an areal transfer area that is (9.42 - 6.77) /6.77 = 39% higher.

But that's not the whole story. The packing density of the three small pipes is higher than the larger pipe. If you draw a circle with three small ones inside it, you will see that you can actually overlap the large circles to make the small ones pack together. I won't do the math but you effectively pick up another 30% area by doing this. But I will leave it out under the pretense that it would be more effective to not tightly pack the smaller tubes but rather leave a gap bewteen them to optimize the conduction away from the heat source.

I know this stuff because in my industry, when you buy a heat exchanger to cool something, generally, what happens is the vendor will quote you a big exchanger that contains a million small tubes inside it. That way, you maximize your transfer area and keep the exchanger size down. But in cases wihere the fluid could plug or scale up, having all those small tubes is not a good thing when it comes time to clean them out.

I can send you a diagram that helps explain this better. In the end, you are wrong but your challenge is accepted and appreciated.