(41) Bulk Cylindrical Tank

10152018, 04:31 PM
Post: #41




RE: (41) Bulk Cylindrical Tank
(10142018 10:15 PM)Dieter Wrote: The paper uses a "different" formula for the slant bottom section and assumes a spherical dome top with a given radius (which is different from the calculated radius). This is something in the paper that I don't understand: Why would they determine the radius R graphically or by measurement when it can be calculated? (10152018 12:33 PM)Dieter Wrote: 2. If the dome top section is actually spherical there is exactly one solution: the radius R must be 90". From your excellent sketch I concluded: \( \begin{align*} a^2+r^2=R^2=(a+v_0)^2&=a^2+2av_0+v_0^2 \\ r^2&=2av_0+v_0^2 \end{align*} \) And thus: \(2a=\frac{r^2v_0^2}{v_0}\) Let \(x\) be the radius of the surface in the dome on level \(v\). Then: \( \begin{align*} x^2 &= R^2(a+v)^2 \\ &= R^2a^22avv^2 \\ &= r^2(2a+v)v \end{align*} \) And thus: \(\left ( \frac{x}{r} \right )^2=1\left ( \frac{2a}{r} + \frac{v}{r} \right )\frac{v}{r}\) But since: \(\frac{2a}{r}=\frac{r^2v_0^2}{rv_0}=\frac{r}{v_0}\frac{v_0}{r}\) We end up with: \(\left ( \frac{x}{r} \right )^2=1\left ( \frac{r}{v_0}\frac{v_0}{r} + \frac{v}{r} \right )\frac{v}{r}\) Here's the modified program for the HP42S: Code: 00 { 72Byte Prgm } Of course the constant \(\frac{r}{v_0}\frac{v_0}{r}\) could be stored in another variable instead of calculating it again and again: Code: 30 RCL "r" Examples: 195 XEQ "TANK" 13,318.1905 198 XEQ "TANK" 13,509.0116 228 XEQ "TANK" 14,488.2093 Here I used \(10^{12}\) for ACC. Do these results agree with your calculations? Kind regards Thomas 

10152018, 05:52 PM
Post: #42




RE: (41) Bulk Cylindrical Tank
(10152018 12:33 PM)Dieter Wrote: Take a look at this sketch which should be quite exactly to scale: Very nice sketch. Thanks ! To get the 1943 gallon dome, I assumed the author was not stupid. There must be a reason he measured R of 78" (not calculated 90") If what he was doing is do get a fit with same cross section area, it all make sense. With R=90", blue dotted line always underestimated true dome volume. With R=78", bluedotted line will go pass the Dome Top. This overestimated area should cancelled out the lower underestimated part. Also, 6" liquid volume is better approximated as cylinder of 6" high. (Red line almost vertical) For Elliptical Dome Top, I assumed bottom part of dome is a cyclinder, top part as elliptical dome. Matching both tank height and tank volume. The revised dimensions is Height = (slant height = 12) + (cylinder height = 190.67) + (elliptical dome height = 25.33) With this revised numbers, the tank should match better to actual shape ... Quote:Now imagine a radius of 75,22". That's not much more than the tank radius r, so the dome top section would be much higher than v_{0} = 36" ... That R were for a different dome shape, Spherical Segment Dome Top. The dome is actually flat at v0 = 36", with radius = sqrt(2320) ~ 48.17" You can confirm with this (set a1=72, a2=48.17, h=36): https://rechneronline.de/pi/sphericalsegment.php 

10152018, 07:06 PM
Post: #43




RE: (41) Bulk Cylindrical Tank
(10152018 04:31 PM)Thomas Klemm Wrote: This is something in the paper that I don't understand: Why would they determine the radius R graphically or by measurement when it can be calculated? You take the words right out of my mouth. ;) (10152018 04:31 PM)Thomas Klemm Wrote: We end up with: Fine. (10152018 04:31 PM)Thomas Klemm Wrote: 195 XEQ "TANK" Yes, these are exactly my results for a spherical dome top. Both with the classic formula, determining R first, as well as with yours where R is not explicitely calculated. Dieter 

10152018, 07:36 PM
Post: #44




RE: (41) Bulk Cylindrical Tank
(10152018 05:52 PM)Albert Chan Wrote: To get the 1943 gallon dome, I assumed the author was not stupid. So the 78" have been measured, and not calculated. (10152018 05:52 PM)Albert Chan Wrote: If what he was doing is do get a fit with same cross section area, it all make sense. How do you know if the blue dotted line (R=90) underestimates the true dome volume? How do you know the true dome value in the first place? But maybe this is another argument for an elliptical dome top. ;) (10152018 05:52 PM)Albert Chan Wrote: With R=78", bluedotted line will go pass the Dome Top. Here is another picture: Red line: elliptical dome top Blue line: spherical dome top, R=90" Green line: spherical dome top, R=78" I don't think that things cancel out here. Even if the liquid level in the dome is limited to v_{0}. "Mr. Chan, I am not convinced". ;) (10152018 05:52 PM)Albert Chan Wrote: With this revised numbers, the tank should match better to actual shape ... Since noone knows the actual shape we cannot conclude this. All we know is that we know nothing. #) Dieter 

10152018, 08:37 PM
Post: #45




RE: (41) Bulk Cylindrical Tank
(10152018 05:52 PM)Albert Chan Wrote: There must be a reason he measured R of 78" (not calculated 90") I just came across another problem. The formula for the spherical dome top include R and Vo as parameters. We have already seen that R=78 does not match Vo=36. But if you do the calculation with these two contradicting parameters, as suggested in the paper and shown in the results in table 253, you get volumes that are less (!) than those with R=90 and V0=36. This is because you do not (!) calculate the volume within the green line in the sketch above. To calculate the latter Vo would have to be adjusted to 48". Here are some results for different ombinations of R and Vo. Code: level R=90 Vo=36 R=78 Vo=36 R=78 Vo=48 Ellipsoid I wonder what the figures in the R=78 and Vo=36 column mean. They are the lowest of all possible parameter combinations. Dieter 

10152018, 09:56 PM
(This post was last modified: 10152018 10:09 PM by Albert Chan.)
Post: #46




RE: (41) Bulk Cylindrical Tank
(10152018 07:36 PM)Dieter Wrote: Since noone knows the actual shape we cannot conclude this. Well put ... To make everything consistent, the dome have to be very flat on top, but possible. (even flatter than ellipsoidal dome, the red line, in post 44) The problem with the setup is that graphically fitted R is then thrown away. The maximum v is 36", so the carefully fitted volume is wasted ... Worst, this wasted piece affected calculated dome volume (badly underestimated). I like the ellipsoidal dome setup, since the shape is adjustable. (If v0 is allowed to change, like what I did in post 36) 

10162018, 08:57 PM
Post: #47




RE: (41) Bulk Cylindrical Tank
(10152018 09:56 PM)Albert Chan Wrote: The problem with the setup is that graphically fitted R is then thrown away. The advantage is that no cumbersome and inaccurate graphically fitted R is required. (10152018 09:56 PM)Albert Chan Wrote: The maximum v is 36", so the carefully fitted volume is wasted ... Apropos "carefully fitted": what about this idea?
In our example the tank may have an allowed liquid level of 215" with a volume of 14500 gallons. For an elliptical dome top this means Vo = 34,8942". This way we get: Code: level volume How do you like that? Dieter 

10172018, 12:24 AM
(This post was last modified: 10172018 03:33 AM by Albert Chan.)
Post: #48




RE: (41) Bulk Cylindrical Tank
(10162018 08:57 PM)Dieter Wrote: Apropos "carefully fitted": what about this idea? Good idea. Now, we only need 4 inputs, instead of 5:
v0 = 3 * (H  h0/2  V/(Pi r²)) h1 = H  h0  v0 Edit: I would avoid using (in, gallon) units, and use (in, in³) instead. This way, the code also work in other units, say (ft, ft³), (cm, cm³), (m, m³), ... 

10172018, 11:33 PM
(This post was last modified: 10182018 01:37 PM by SlideRule.)
Post: #49




RE: (41) Bulk Cylindrical Tank
My last post on this thread, enjoy!
extract from volume 5, issue 4 (April 2016) International Journal of Research in Engineering and Technology SemiElliptical Head on a Vertical Tank & Torispherical Head in Vertical Tank. BEST! SlideRule edit: [ missing ('s & )'s restored ] 

10182018, 02:10 AM
Post: #50




RE: (41) Bulk Cylindrical Tank
Here are some parentheses: (, (, (, ), ), )
Just sprinkle them here and there in the document where they are missing. Cheers Thomas 

10182018, 01:11 PM
Post: #51




RE: (41) Bulk Cylindrical Tank
(10182018 02:10 AM)Thomas Klemm Wrote: Here are some parentheses: (, (, (, ), ), ) LOL!! I thought it was me at first (or maybe a bad scan?)! Is that some kind of 'special' notation, or just an incredibly bad job of proofreading? Bob Prosperi 

10182018, 01:19 PM
(This post was last modified: 10182018 01:20 PM by SlideRule.)
Post: #52




RE: (41) Bulk Cylindrical Tank
(10182018 01:11 PM)rprosperi Wrote: … thought it was me …at first … maybe a bad scan?) … 'special' notation … bad job of proofreading?One [ ( ] was missing in the original, the others were dropped in the extract process. A first, for me, using Acrobat XI Pro. I'ld like to blame it on my very recent cataract surgery but most likely just negligent proof reading. BEST! SlideRule 

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