Calculating Tank Substance Saving era, increasing foresight By Dan Jones, Ph. D. , P. E. C alculating running substance in a vapid or perpendicular cylindrical or suggestive tank can be confused, depending on running summit and the figure of the commanders (ends) of a vapid tank or the deep of a perpendicular tank. Exact equations now are available for diverse regularly encountered tank figures. These equations can be used to execute swift and respectful running-substance calculations. All equations are arduous, but computational difficulties get prepare in established limiting configurations.
All substance equations produce running substances in stable units from tank magnitude in compatible rectirectidirect units. All variables defining tank figures required for tank substance calculations are defined in the “Variables and Definitions” sidebar. Graphically, Figs. 1 and 2 profession vapid tank variables and Figs. 3 and 4 profession perpendicular tank variables. Exact running substances in suggestive vapid or perpendicular tanks can be plant by foremost wary the running substances of alienate cylindrical vapid or perpendicular tanks using the equations feeling over, and then by adjusting those results using alienate punishment formulas.
Horizontal Cylindrical Tanks Running substance as a duty of running summit can be conducive for a vapid cylindrical tank after a while either conical, ellipsoidal, guppy, round, or toriround commanders where the running summit, h, is gauged from the tank deep to the running deportment, see Figs. 1 and 2. A guppy commander is a conical commander where the sharp-end of the conical commander is equalize after a while the top of the cylindrical ateion of the tank as professionn in Fig. 1. A toriround commander is an ASME-type commander defined by a knuckle-radius parameter, k, and a platter-radius parameter, f, as professionn in Fig. 2.
An ellipsoidal commander must be accurately half of an ellipsoid of shape; singly a hemiellipsoid is substantial – no “segment” of an ellipsoid get toil as is gentleman in the contingency of a round commander where the commander may be a round limb. For a round commander, |a| ? R, where R is the radius of the cylindrical tank substance. Where scooped conical, ellipsoidal, guppy, round, or toriround commanders are considered, then |a| ? L/2. Both commanders of a vapid cylindrical tank must be selfsame for the equations to toil; i. e. , if one commander is conical, the other must be conical after a while the selfsame magnitude.
However, the equations can be completely to chaffer after a while running substance calculations of vapid tanks after a while commanders of irrelative figures. For illustration, if a vapid cylindrical tank has a conical commander on one end and an ellipsoidal commander on the other end, estimate running substances of two tanks, one after a while conical commanders and the other after a while ellipsoidal commanders, and mediocre the results to get the desired running substance. The commanders of a vapid tank may be tasteless (a = 0), relievant (a > 0), or scooped (a < 0). The forthcoming variables must be after a whilein the ranges stated: • • • • • • • |a| ? R for round commanders |a| ? L/2 for scooped ends 0 ? ? 2R for all tanks f > 0. 5 for toriround commanders 0 ? k ? 0. 5 for toriround commanders D>0 L? 0 Page 1 of 12 Variables and Definitions (See Figs. 1-5) a is the protraction a vapid tank's commanders expand further (a ; 0) or into (a ; 0) its cylindrical ateion or the profoundness the deep expands adown the cylindrical ateion of a perpendicular tank. For a vapid tank after a while tasteless commanders or a perpendicular tank after a while a tasteless deep a = 0. Af is the perverse-sectional area of the running in a vapid tank's cylindrical ateion. D is the perverseing of the cylindrical ateion of a vapid or perpendicular tank.
DH, DW are the summit and width, respectively, of the omission defining the perverse ateion of the substance of a vapid suggestive tank. DA, DB are the superior and junior axes, respectively, of the omission defining the perverse ateion of the substance of a perpendicular suggestive tank. f is the platter-radius parameter for tanks after a while toriround commanders or deeps; fD is the platter radius. h is the summit of running in a tank gauged from the lowest sever of the tank to the running deportment. k is the knuckle-radius parameter for tanks after a while toriround commanders or deeps; kD is the knuckle radius.
L is the protraction of the cylindrical ateion of a vapid tank. R is the radius of the cylindrical ateion of a vapid or perpendicular tank. r is the radius of a round commander for a vapid tank or a round deep of a perpendicular tank. Vf is the running substance, of running profoundness h, in a vapid or perpendicular cylindrical tank. Page 2 of 12 Vapid Tank Equations Here are the local equations for running substances in vapid cylindrical tanks after a while conical, ellipsoidal, guppy, round, and toriround commanders (use radian incurved gauge for all trigonometric dutys, and D/2 = R > 0 for all equations): Conical commanders.
Vf = A f L + K .......... ... 0 ? h < R 2 aR2 ? ? / 2 ......... h = R 3 ? ? K ....... R < h ? 2 R 1 ? 2 M 1 ? M2 M M= R? h R K ? cos ? 1 M + M 3 cosh ? 1 Ellipsoidal commanders. Vf = A f L + ? a h 2 1 ? Guppy commanders. h 3R Vf = A f L + 2aR2 2a h cos ? 1 1 ? + 2 Rh ? h 2 (2 h ? 3 R )(h + R ) 3 R 9R Round commanders. 3R 2 + a 2 6 ? a 3R 2 + a 2 3 h ? a h2 1 ? 3R Vf = A f L + a a ?a ( ( ) ) .......... .......... .......... .......... .......... .......... .......... .......... ..... h = R, .......... .......... .......... .......... .......... .......... .......... .......... .... h = D, a ? R a ? R .......... .......... .......... .......... .......... .......... .......... .. h = 0 or a = 0, R, ? R 2 2r3 R2 ? r w R2 + r w z R cos ? 1 2+ + cos ? 1 ? 3 R (w ? r ) R(w + r ) r r ? 2 w r2 ? R cos ? 1 w R a ? 0. 01D y 4w y z w3 tan ? 1 + 3 z 3 .......... . 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 relievant (scooped ) commanders w ? R? h y ? 2 R h ? h2 z ? r 2 ? R2 Page 3 of 12 Toriround commanders.
In the Vf equation, use +(-) for relievant(concave) commanders. 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 ? 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 ? ) 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 ) ? a1 6 ( 3g 2 2 + a1 ) a 1 ? r ( 1 ? cos ? ) r ? fD h 2 ? D ? h1 w ? R? h z ? r 2 ? g 2 = f D cos ? = r cos ? ? ? sin ? 1 1? 2k = cos ? 1 2 (f ? k ) 4 f 2 ? 8 f k + 4k ? 1 2 (f ? k ) h1 ? k D (1 ? sin ? ) n ? R ? k D + k 2D 2 ? 2 g ? f D sin ? = r sin ? In the over equations, Vf is the whole substance of running in the tank in stable units compatible after a while the rectirectidirect units of tank size parameters, and Af is the perverse-sectional area of running in the cylindrical substance of the tank in clear units compatible after a while the rectirectidirect units used for R and h. The equation for Af is producen by: A f = R 2 cos ? 1 R? h ? (R ? h) 2 R h ? h 2 R Page 4 of 12 Figure 1. Parameters for Vapid Cylindrical Tanks after a while Conical, Ellipsoidal, Guppy, or Round Heads. Round commander Cylindrical Tube Hemiellipsoid commander r(sphere) D
Guppy commander Conical commander a (cone; guppy) a(sphere) R h a(ellipsoid) L Af Running perverse-sectional area CROSS SECTION OF CYLINDRICAL TUBE h 1. 2. 3. 4. 5. 6. 7. Both commanders of a tank must be selfsame. Over diagram is for determination of parameters singly. Cylindrical tube of perverseing D (D > 0), radius R (R > 0), and protraction L (L ? 0). For round commander of radius r, r ? R and |a| ? R. For relievant commander other than round, 0 < a < ? , for scooped commander a < 0. L ? 0 for a ? 0, L ? 2|a| for a < 0. Ellipsoidal commander must be accurately half of an ellipsoid of shape. 0 ? h ? D.
Page 5 of 12 Figure 2. Parameters for Vapid Cylindrical Tanks after a while Toriround Heads. kD h2 R D ? fD h h1 Vapid Cylindrical Tank Examples L The forthcoming models can be used to stop impression of the equations: Find the substances of running, in gallons, in vapid cylindrical tanks 108" in perverseing after a while cylinder protractions of 156", after a while conical, ellipsoidal, guppy, round, and “standard” ASME toriround (f = 1, k = 0. 06) commanders, each commander expanding further the ends of the cylinder 42" (ate torispherical), for running profoundnesss in the tanks of 36" (model 1) and 84" (model 2).
Calculate five eras for each running profoundness – for a conical, ellipsoidal, guppy, round, and toriround commander. For model 1 the parameters are D = 108", L = 156", a = 42", h = 36", f = 1, and k = 0. 06. The running substances are 2,041. 19 Gal for conical commanders, 2,380. 96 Gal for ellipsoidal commanders, 1,931. 72 Gal for guppy commanders, 2,303. 96 Gal for round commanders, and 2,028. 63 Gal for toriround commanders. For model 2 the parameters are D = 108", L = 156", a = 42", h = 84", f = 1, and k = 0. 06. The running substances are 6,180. 54 Gal for conical commanders, 7,103. 45 Gal for ellipsoidal commanders, 5,954. 1 Gal for guppy commanders, 6,935. 16 Gal for round commanders, and 5,939. 90 Gal for toriround commanders. For toriround commanders, ‘a’ is not required input; it can be conducive from f, k, and D. torispherical-commander models, the conducive appraise is ‘a’ = 18. 288". Page 6 of 12 For these Perpendicular Cylindrical Tanks Running substance in a perpendicular cylindrical tank after a while either a conical, ellipsoidal, round, or toriround deep can be conducive, where the running summit, h, is gauged from the core of the deep of the tank to the deportment of the running in the tank.
See Figs. 3 and 4 for tank configurations and size parameters, which are too defined in the “Variables and Definitions” sidebar. A toriround deep is an ASME-type deep defined by a knuckle-radius element and a platter-radius element as professionn graphically in Fig. 4. The knuckle radius get then be kD and the platter radius get be fD. An ellipsoidal deep must be accurately half of an ellipsoid of shape. For a round deep, |a| ? R, where a is the profoundness of the round deep and R is the radius of the cylindrical ateion of the tank.
The forthcoming parameter ranges must be observed: • • • • a ? 0 for all perpendicular tanks, a ? R for a round deep f > 0. 5 for a toriround deep 0 ? k ? 0. 5 for a toriround deep D>0 Perpendicular Tank Equations Here are the local equations for running substances in perpendicular cylindrical tanks after a while conical, ellipsoidal, round, and toriround deeps (use radian incurved gauge for all trigonometric dutys, and D > 0 for all equations): Conical deep. ? Dh Vf = 4 4 a 2 h 3 2a 3 .......... .......... .......... . .......... .......... .......... h