Tank draining
Tank draining
(OP)
Hi all,
I have a question which has plagued me with quiet awhile.
If a tank (circular or other) has to be drained, what is the formula to calculate the total time taken to empty such a tank?
Also, if the outlet of the tank was connected to the inlet of another tank at the same level (same volume), can the time taken to bring the two tanks to the same level be calculated? (I am assuming as the other tank fills the difference in head will decrease, slowing down the flowrate)
I understand that there are a large amount of factors such as type of outlet, working fluid etc etc involved but a user friendly formula would be much appreciated, even a discussion on such a topic would be informative.
Thanks for your time
I have a question which has plagued me with quiet awhile.
If a tank (circular or other) has to be drained, what is the formula to calculate the total time taken to empty such a tank?
Also, if the outlet of the tank was connected to the inlet of another tank at the same level (same volume), can the time taken to bring the two tanks to the same level be calculated? (I am assuming as the other tank fills the difference in head will decrease, slowing down the flowrate)
I understand that there are a large amount of factors such as type of outlet, working fluid etc etc involved but a user friendly formula would be much appreciated, even a discussion on such a topic would be informative.
Thanks for your time





RE: Tank draining
I have not failed. I've just found 10,000 ways that won't work."-Edison "If Edison had a needle to find in a haystack, he would proceed at once to examine straw after straw until he found the object of his search. I was a sorry witness of such doings, knowing that a little theory and calculation would have saved 90% of his work.- Tesla
RE: Tank draining
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RE: Tank draining
In real life, most people wouldn't drain a full water tank; they'd draw the level down as far as possible before isolating it.
RE: Tank draining
A good guess for an average flow rate would be 1/2 of the discharge rate at the start.
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RE: Tank draining
I have looked at the example and formula supplied by efunda (thanks Bimr), and worked the solution for a typical size tank, 10m diameter, 10m depth. I calculated for the top level and worked a table for the falling values. I think i got a bit carried away with the 1 sec intervals but it is interesting to see all the same.
While this seems about as good an answer as one could get, i can't help but think that a integration/differentiation (as mentioned) method might be alot less work and even more accurate(if my method is correct!).
My skills in the calculus department are quiet weak, so if somebody has a strong handle on calculus it would be good to see if it can be done.
Also, following the values worked out from the tank drain, i will look at the idea of calculating the simultaneous filling of an equal tank as previously mentioned. I imagine that a few headaches will accompany this (calculus again might be the prefered option!). I have attached the workings........ appreciate any comments. Thanks
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RE: Tank draining
All you can integrate is the flowrate, once it is determined for each step and all the higher math you can possibly think of won't improve the accuracy of the Cd or Hl values you need to use to get that flowrate.
I have not failed. I've just found 10,000 ways that won't work."-Edison "If Edison had a needle to find in a haystack, he would proceed at once to examine straw after straw until he found the object of his search. I was a sorry witness of such doings, knowing that a little theory and calculation would have saved 90% of his work.- Tesla
RE: Tank draining
The time to drain is:
T = V/(4 A (sq rt h) )
V = Volume
A= Drain Area
h= height of liquid above drain
You also should have seen that the drain time varied with 1 over the square of the drain diameter.
RE: Tank draining
RE: Tank draining
The discharge coefficient is C, whose value is typically between 0.90 and 0.98.
So, there should be little difference between the numbers. You must be making an error.
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RE: Tank draining
Cd of as low as 0.61 might be used for square flush edges through a thin walled tank, 0.52 if the exit pipe sticks into the tank past the tank wall.
I think you have square edges.
What's the value now?
8000
I have not failed. I've just found 10,000 ways that won't work."-Edison "If Edison had a needle to find in a haystack, he would proceed at once to examine straw after straw until he found the object of his search. I was a sorry witness of such doings, knowing that a little theory and calculation would have saved 90% of his work.- Tesla
RE: Tank draining
Check 20 Aug post of bimr. The average flowrate is half the initial instantaneous flowrate. Actually you maximum flowrate initially and then zero in the last second. You can get the average flowrate by (initial flowrate+final flowrate)/2 or initial flowrate/2.
Now, Q = 0.98*0.007693*(2*9.81*10)0.5, which is 0.1056 cu.mtr/sec. Half of this is 0.0528cu.mtr/sec.
So, time taken will be 785/0.0528 = 14868 sec (approximately)
In your spreadsheet, you have emptied out only 779.26 cu.mtr after 13322 seconds. You have to still drain 5.74 cu.mtr. Even if we consider that flowrate at 13322nd second remains constant further, you still require about 5.74/0.00922 = 623 seconds.
RE: Tank draining
Time calculation based on average flowrate between initial and final conditions (i.e half of initial flowrate) can be established by integation as well.
If A is CS area of tank and a is CS area of nozzle, then the flowrate through nozzle will be Cdxax(2gZ)0.5.
If we consder a drop of dZ in liquid column height correspondingly, in a time dt then the volume flowrate out of the tank is AxdZ/dt.
Therefore, AxdZ/dt = Cdxax(2gZ)0.5
By rearranging the terms,
(A/(Cdxax(2g)0.5))dZ/Z = dt
Integrating both sides, we get
(A/(Cdxax(2g)0.5))x2xZ1/2 = t
Applying the boundary conditions of 0 and 10 meters for Z,
t = 78.5*2*(10-0)1/2/(Cdx0.00785x4.43)
or t = 14276.65/Cd
If our Cd is 0.98, then t = 14568 sec.
If Cd is 0.9, then t = 15863 sec.
The descripancy in my earlier post was due to the fact that I considered Cd twice. Otherwise, it is 0.0528*0.00785/0.007693 = 0.05388 cu.mtr/s
So, time will be 785/0.05388 = 14569 sec.
RE: Tank draining
It is not correct to apply the calculated velocity to the entire cross-sectional area of the nozzle. It must be applied to the area of the vena contracta.
Rounded Sharp-edged Short tube Borda
Cc 1.0 0.62 1.0 0.52
Cv 0.98 0.98 0.8 0.98
Cd = Cc x Cv = 0.62 * 0.98 = 0.61
Q = A x Cd x (2gH)^0.5
exit velocity = Cv * (2gH)^0.5
Cv = 0.98
exit velocity = 0.98 *(2 * 9.81 * D)^0.5 = 13.727 m/s
Cc = 0.62
Area of Vena Contracta = 3.14*((0.1m)^2)/4 * Cc = 0.00486 m2
Flowrate = 0.00486 m2 * 13.727 m/s = 0.06681 m3/s
Flowrate = Cv * Cc * (2gH)^0.5
Flowrate = Cd * (2gH)^0.5
where Cd = Cv * Cc
Flowrate = 0.66809 m3/s
"I'm all in favor of keeping dangerous weapons out of the hands of fools. Let's start with typewriters."
- Frank Lloyd Wright (1868-1959)
RE: Tank draining
"I'm all in favor of keeping dangerous weapons out of the hands of fools. Let's start with typewriters."
- Frank Lloyd Wright (1868-1959)
RE: Tank draining
RE: Tank draining
Ya 1 sec is a rediculous waste of computer resources.
Acceptable
Error calculated
in Q time time to reach
% step 0.1 meter depth
seconds Hours
1 60 5.85
1 120 5.83
1 240 5.8
1 300 5.5
0.05 10 5.87
"I'm all in favor of keeping dangerous weapons out of the hands of fools. Let's start with typewriters."
- Frank Lloyd Wright (1868-1959)
RE: Tank draining
The orifice coefficient for a short tube, square edge with fluid separation from the walls is 0.61.
One would think that 0.82 would be more correct.
Not really sure that the original post was concerned about the minutiae of orifice coefficients. I thought he was interested in the theoretical concept of calculating the time to drain a tank.
RE: Tank draining
"I'm all in favor of keeping dangerous weapons out of the hands of fools. Let's start with typewriters."
- Frank Lloyd Wright (1868-1959)