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axisymmetric cooling tower


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Hello !

Please I need your help to solve this homework.

 

The figure shows a proposed design for the upper section of an axisymmetric cooling tower, which is to be made of concrete of density 2300 kg/m^3.

Find the minimum value of the shell thickness t if the tower must be capable of sustaining an internal pressure 2kPa at A and 5kPa at B without inducing tensile membrane stresses at these points. With your choice of t, do you expect there to be tensile stresses elsewhere in the tower and if so where?

 

Thank you

 

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I won't solve your structural problem for you, but I will share a story. The hyperbolic shape of natural draft cooling towers is entirely structural. The shape is not to induce or optimize the flow of air through the tower. The shape is to achieve the least materials and expense. The thickness of the concrete in the shell is the minimum required to cover and protect the rebar. This is not true at the bottom or veil where the shell sits on columns, which is much thicker. The shape is obtained by stretching a soap bubble between two circles of wire. The soap bubble naturally achieves the surface of least resistance. This is also the natural way to design HVAC transitions from round to rectangular or whatever. You can show this is the right shape using the calculus of variations. I have climbed to the top of several such towers and walked around the lip, dragging instruments to measure the temperature, pressure, and velocity. I have worked in the field with these impressive devices for forty years and am well known in the industry, having published many papers on thermal performance. I've been to the top of both (wet cement and dry metal) cooling towers in Schmehausen, Germany. Here's a picture of me standing next to one of the heat exchangers (like car radiators) in the dry metal tower and from the top of one looking down on the other.

bearded_tourist.jpg

a_giant_falls.jpg

big_claus.jpg

little_klaus.jpg

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On 2/11/2021 at 4:01 PM, dudleybenton said:

The soap bubble naturally achieves the surface of least resistance.

Dear Dudley,

I'm quite familiar with the soap bubble problem, but there the quantity minimized is the surface energy which is proportional to surface area. I fail to see the connection to the cooling tower problem. Would you elaborate, please?

DrD

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On 2/11/2021 at 11:01 PM, dudleybenton said:

I won't solve your structural problem for you, but I will share a story. The hyperbolic shape of natural draft cooling towers is entirely structural. The shape is not to induce or optimize the flow of air through the tower. The shape is to achieve the least materials and expense. The thickness of the concrete in the shell is the minimum required to cover and protect the rebar. This is not true at the bottom or veil where the shell sits on columns, which is much thicker. The shape is obtained by stretching a soap bubble between two circles of wire. The soap bubble naturally achieves the surface of least resistance. This is also the natural way to design HVAC transitions from round to rectangular or whatever. You can show this is the right shape using the calculus of variations. I have climbed to the top of several such towers and walked around the lip, dragging instruments to measure the temperature, pressure, and velocity. I have worked in the field with these impressive devices for forty years and am well known in the industry, having published many papers on thermal performance. I've been to the top of both (wet cement and dry metal) cooling towers in Schmehausen, Germany. Here's a picture of me standing next to one of the heat exchangers (like car radiators) in the dry metal tower and from the top of one looking down on the other.

bearded_tourist.jpg

a_giant_falls.jpg

big_claus.jpg

little_klaus.jpg

 Thank you, that's very interesting,  I have a question please :

we can find th tensile stresses with a minimum thickness?

 

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The soap bubble isn't exact, but it is approximately the optimal shape plus it's easy to visualize and it also works for shapes that aren't round. Ideally, the surface energy is minimized, which is proportional to the area and also the stresses. Consider this… >---< [abrupt necking at both ends with very narrow tube in between] would have less surface area than )===( [slight neck down on both ends with a tube in the middle having a minimum diameter close to that of the smaller end]. While the first example would have less surface area than the second, the stresses would be much larger because of the more abrupt change and smaller diameter central "tube". The soap bubble creates a natural surface with smooth transition and minimal stresses that result in less concrete and steel than a cylindrical stack. The rising air velocity slightly increases toward the throat but the head loss is minimal (at least for this part). Then the velocity slightly decreases between the throat and exit (as the shell flares out), recovering some of the lost head.

flow_inside_cooling_tower.gif

The stresses are quite interesting and the subject of much controversy. Not only must the veil carry and distribute the weight, then sit on top of the X slanted columns, it must be able to handle wind forces. The veil at the bottom where it sits on the columns is usually about 0.5 meter wide. The NRC demanded TVA produce such an analysis for the natural draft cooling towers at Watts Bar Nuclear Plant for safety reasons. The tower was designed and erected by Research-Cottrell. I was present for the arguments about design wind loads. I had the paper reports at one time, but I don't have them now. Most of my paper documents were lost before I could scan them to PDF. I know the original calculations were done "by hand" and that didn't satisfy the NRC. More calculations were done using a FEM structural code. It would be great to have that report and TVA, being a government agency, would have to give it to you, but I doubt they could find it now. Towers like this are still being built. Perhaps you can find an analysis for a newer tower, also mandated by safety concerns and, therefore, available to the public. It's worth searching the NRC web site. Perhaps they have this on file for some other nuclear plant with similar towers. They have some massive reports. It could be in an appendix or attachment, as this is deemed "safety-related".

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