Torispherical Head Tank Volume

Calculating Tank Volume
Saving time, increasing accuracy
By Dan Jones, Ph.D., P.E.

C

alculating fluid volume in a horizontal or vertical cylindrical or elliptical tank can be complicated, depending on fluid height and the shape of the heads (ends) of a horizontal tank or the bottom of a vertical tank. Exact equations now are available for several commonly encountered tank shapes. These equations can be used to make rapid and accurate fluid-volume calculations. All equations are rigorous, but computational difficulties will arise in certain limiting configurations.

All volume equations give fluid volumes in cubic units from tank dimensions in consistent linear units. All variables defining tank shapes required for tank volume …show more content…

. h ≠ R, D; a ≠ 0, R, − R;

a R2 − x 2 2 r 2 − x 2 tan −1 dx − A f z a r 2 − R2 w a2 + R2 2|a|

(

)

.......... .

h ≠ R, D; a ≠ 0, R, − R;

a < 0.01D

r=

a≠0 ;

a = ± r − r 2 − R2

+ ( −) for convex (concave ) heads

w ≡R−h

y ≡ 2 R h − h2

z ≡ r 2 − R2

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Torispherical heads. In the Vf equation, use +(-) for convex(concave) heads.

Vf = A f L ±

2 [ 2 v 1,max − v 1 (h = D − h) + v 2,max + v 3,max ] .......... .......... .... h 2 ≤ h ≤ D

2 ( v 1,max + v 2 + v 3 ) .......... .......... .......... .......... .......... ..........

2 v1

.......... .......... .......... .......... .......... .......... .......... .......... .... 0 ≤ h ≤ h1 h1 < h < h 2

2kDh−h2

v1 ≡
0 kD cos α

n 2 sin −1 n 2 cos −1

n2 − w 2 − w n 2 − w 2 dx n g w − w n 2 − w 2 + g n 2 − g 2 dx − cos −1 n n
2

v2 ≡
0

g g2 + r w z r3 g2 − r w 2+ cos −1 + cos −1 − r g(w + r ) r 3 g (w − r ) v3 ≡ g cos −1

g2 − w 2 w3 w tan −1 − w r2 − 3 z g .......... .......... .......... .......... ..... 0.5 < f ≤ 10

+

w z g2 − w 2 6 g2 − x 2 z

+

wz 2 2 g (h − h1 ) − (h − h1 ) 2

(r

2

− x 2 tan −1

)

dx −

w

z 2 w 2 g cos −1 − w 2g(h − h1 ) − (h − h1 ) 2 g

0.5 < f < 10,000

v 2,max ≡ v 2 (h = h 2 ) v 3,max ≡ v 3 (h = h 2 ) =

v 1,max ≡ v 1 (h = h1 )

π a1
6

( 3g

2

2 + a1

)

a 1 ≡ r ( 1 − cos α ) r ≡ fD h 2 ≡ D − h1 w ≡R−h z ≡ r 2 − g 2 = f D cos α = r cos α

α ≡ sin −1

1−