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alex_amvdor

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- Homework Statement
- This is actually not an assigned homework question, rather it is for my research. I need to determine the thermal resistance of a set of stainless steel edge-welded vacuum bellows, but it seems like a problem that is very similar to something that would be assigned to an undergrad.

The edge-welded bellows can be treated as a set of 40 or so stacked hollow truncated circular cones. Each cone has a small end with radius ##R_1## and a large end with radius ##R_2##. They have a wall thickness of ##t##, and ##t<<R_1, R_2## so that the cross sectional area can be treated as $$A = 2 \pi Rt$$.

The cones' main axes are concentric with the z-axis.

The total length of the bellows as a whole is L, so the height of each cone is L/40, and the resistance of the complete bellows will be 40 times that of one cone.

- Relevant Equations
- $$R = \frac{l}{\lambda A}$$

Where l is the length traversed by the heat, i.e. ##\sqrt{h^2 + (R_2-R_1)^2}## from the bottom of a cone to the top.

##\lambda = 14.4 \frac{W}{m \cdot K}## for stainless steel

##R_1 = 9.525 mm##

##R_2 = 14.288 mm##

##t = 1.245 mm##

##L = 31.75 mm##

We can write our radius as a function of the height, z, of our cone: $$R(z) = \frac{R_2 - R_1}{h} z + R_1$$

Where h is the height of our cone, ##h = \frac{L}{40}##.

Our cross sectional area, $$A = 2 \pi R t$$ can then be written as $$A = 2 \pi t [\frac{R_2 - R_1}{h} z + R_1]$$

This I am all relatively sure about.Here is where I am not so sure:

For a small change in z, ##\Delta z##, we will have a corresponding change in the cross sectional area: $$\Delta A = 2 \pi t \frac{R_2-R_1}{h} \Delta z$$ which, using our equation for thermal resistance, would give us: $$\Delta R = \frac{\sqrt{(\Delta z)^2 + (\frac{R_2-R_1}{h} \Delta z)^2}}{2 \lambda \pi t \frac{R_2-R_1}{h} \Delta z}$$

We can then pull out the ##\Delta z##'s from the radical and cancel them, then clean the equation up a bit, giving $$\Delta R = \frac{\sqrt{1+(\frac{h}{R_2-R_1})^2}}{2 \lambda \pi t}$$

This result confuses me, as there is no ##\Delta z## dependence, and I am not entirely sure how to integrate it. My hope is to find an expression which I can integrate from ##z = 0## to ##z = h##. If anyone can point out a problem with my math, or give me some clues as to where to go from here, I would greatly appreciate it.

I realize I may be missing the use of the definition of a derivative, i.e. ##\frac{dR}{dz} = \lim_{\Delta z \to 0} \frac{R(z + \Delta z) - R(z)}{\Delta z}##, but I'm not really sure how to work this into the problem.

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