Torispherical Head Tank Volume

Calculating tank Volume frugality time, increa guiltg accuracy By Dan J integritys, Ph. D. , P. E. C alculating politic raft in a plane or up poleed cylindric or elliptical cooler sens be complicated, depending on changeful acme and the shape of the gallerys (ends) of a even armoured combat vehicle or the cig atomic number 18tte of a straight armoured combat vehicle. Exact equations now are operational for several commonly encountered cooler shapes. These equations gouge be used to make fast and accurate fluent-volume calculations. in all equations are rigorous, precisely computational difficulties entrust arise in certain limiting configurations.All volume equations hold melted volumes in cubic units from storage cooler proportions in reproducible linear units. All variables formation armoured combat vehicle shapes required for tank volume calculations are be in the Variables and Definitions sidebar. Graphically, Figs. 1 and 2 immortalise plane t ank variables and Figs. 3 and 4 show straight tank variables. Exact bland volumes in elliptical level or straight tanks can be institute by first calculating the fluid volumes of appropriate rounded horizontal or good tanks u the pitsg the equations described preceding(prenominal), and becausece by adjusting those results using appropriate field formulas.horizontal rounded Tanks mobile volume as a function of fluid big top can be deliberate for a horizontal rounded tank with each c unmatchedlike, roundedal, rainbow fish, worldwide, or tori worldwide motions where the fluid bill, h, is measuring stickd from the tank arse to the fluid surface, see Figs. 1 and 2. A guppy head is a conical head where the apex of the conical head is level with the top of the rounded plane particle of the tank as shown in Fig. 1. A torispherical head is an ASME-type head delineate by a knuckle- roentgen contestation, k, and a dish- wheel spoke parameter, f, as shown in Fig. 2.An ellipsoidal head moldiness be exactly half of an ellipsoid of revolution only a hemiellipsoid is legitimate no segment of an ellipsoid get out work as is true in the case of a spherical head where the head may be a spherical segment. For a spherical head, a ? R, where R is the spoke of the cylindrical tank body. Where recessed conical, ellipsoidal, guppy, spherical, or torispherical heads are considered, whence a ? L/2. Both heads of a horizontal cylindrical tank essential be identical for the equations to work i. e. , if one head is conical, the otherwise must be conical with the same dimensions.However, the equations can be combined to deal with fluid volume calculations of horizontal tanks with heads of different shapes. For instance, if a horizontal cylindrical tank has a conical head on one end and an ellipsoidal head on the other end, calculate fluid volumes of two tanks, one with conical heads and the other with ellipsoidal heads, and ordinary the results to get the desired fluid volume. The heads of a horizontal tank may be flat (a = 0), protuberant (a 0), or cotyloid (a 0). The chase variables must be inside the ranges stated a ? R for spherical heads a ? L/2 for concave ends 0 ? ? 2R for all tanks f 0. 5 for torispherical heads 0 ? k ? 0. 5 for torispherical heads D0 L? 0 foliate 1 of 12 Variables and Definitions (See Figs. 1-5) a is the distance a horizontal tanks heads extend beyond (a 0) or into (a 0) its cylindrical section or the depth the tooshie extends below the cylindrical section of a perpendicular tank. For a horizontal tank with flat heads or a vertical tank with a flat bottom a = 0. Af is the cross-section(a) bowl of the fluid in a horizontal tanks cylindrical section. D is the diam of the cylindrical section of a horizontal or vertical tank.DH, DW are the height and width, respectively, of the ellipse specify the cross section of the body of a horizontal elliptical tank. DA, DB are the major and min or axes, respectively, of the ellipse defining the cross section of the body of a vertical elliptical tank. f is the dish-radius parameter for tanks with torispherical heads or bottoms fD is the dish radius. h is the height of fluid in a tank measured from the lowest part of the tank to the fluid surface. k is the knuckle-radius parameter for tanks with torispherical heads or bottoms kD is the knuckle radius.L is the length of the cylindrical section of a horizontal tank. R is the radius of the cylindrical section of a horizontal or vertical tank. r is the radius of a spherical head for a horizontal tank or a spherical bottom of a vertical tank. Vf is the fluid volume, of fluid depth h, in a horizontal or vertical cylindrical tank. Page 2 of 12 Horizontal Tank Equations Here are the specific equations for fluid volumes in horizontal cylindrical tanks with conical, ellipsoidal, guppy, spherical, and torispherical heads (use radian angular measure for all trigonometric functions, an d D/2 = R 0 for all equations) conic heads.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 ? romaine lettuce ? 1 M + M 3 romaine lettuceineh ? 1 Ellipsoidal heads. Vf = A f L + ? a h 2 1 ? Guppy heads. 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 Spherical heads. 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 2a ( ) . . 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 Page 3 of 12 Torispherical heads.In the Vf equation, use +(-) for convex(concave) heads. V f = 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 above equations, Vf is the total volume of fluid in the tank in cubic units consistent with the linear units of tank dimension parameters, and Af is the cross-sectional area of fluid in the cylindrical body of the tank in self-colored units consistent with the linear units used for R and h. The equation for Af is given 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 Horizontal rounded Tanks with conical, Ellipsoidal, Guppy, or Spherical Heads. Spherical head Cylindrical Tube Hemiellipsoid head r(sphere) DGuppy head cone-shaped head a (cone guppy) a(sphere) R h a(ellipsoid) L Af Fluid cross-sectional area CROSS SECTION OF CYLINDRICAL pipage h 1. 2. 3. 4. 5. 6. 7. Both heads of a tank must be identical. Above plot is for definition of parameters only. Cylindrical tube of diameter D (D 0), radius R (R 0), and length L (L ? 0). For spherical head of radius r, r ? R and a ? R. For convex head other than spherical, 0 a ? , for concave head a 0. L ? 0 for a ? 0, L ? 2a for a 0. Ellipsoidal he ad must be exactly half of an ellipsoid of revolution. 0 ? h ? D.Page 5 of 12 Figure 2. Parameters for Horizontal Cylindrical Tanks with Torispherical Heads. kD h2 R D ? fD h h1 Horizontal Cylindrical Tank Examples L The pursuit examples can be used to let out application of the equations Find the volumes of fluid, in gallons, in horizontal cylindrical tanks 108 in diameter with cylinder lengths of 156, with conical, ellipsoidal, guppy, spherical, and standard ASME torispherical (f = 1, k = 0. 06) heads, each head extending beyond the ends of the cylinder 42 (except torispherical), for fluid depths in the tanks of 36 (example 1) and 84 (example 2).Calculate five times for each fluid depth for a conical, ellipsoidal, guppy, spherical, and torispherical head. For example 1 the parameters are D = 108, L = 156, a = 42, h = 36, f = 1, and k = 0. 06. The fluid volumes are 2,041. 19 gal for conical heads, 2,380. 96 gal for ellipsoidal heads, 1,931. 72 congius for guppy heads, 2,303. 96 Gal for spherical heads, and 2,028. 63 Gal for torispherical heads. For example 2 the parameters are D = 108, L = 156, a = 42, h = 84, f = 1, and k = 0. 06. The fluid volumes are 6,180. 54 Gal for conical heads, 7,103. 45 Gal for ellipsoidal heads, 5,954. 1 Gal for guppy heads, 6,935. 16 Gal for spherical heads, and 5,939. 90 Gal for torispherical heads. For torispherical heads, a is not required infix it can be calculated from f, k, and D. torispherical-head examples, the calculated value is a = 18. 288. Page 6 of 12 For these Vertical Cylindrical Tanks Fluid volume in a vertical cylindrical tank with either a conical, ellipsoidal, spherical, or torispherical bottom can be calculated, where the fluid height, h, is measured from the center of the bottom of the tank to the surface of the fluid in the tank.See Figs. 3 and 4 for tank configurations and dimension parameters, which are also defined in the Variables and Definitions sidebar. A torispherical bottom is an ASME-type bottom defined by a knuckle-radius factor and a dish-radius factor as shown graphically in Fig. 4. The knuckle radius will then be kD and the dish radius will be fD. An ellipsoidal bottom must be exactly half of an ellipsoid of revolution. For a spherical bottom, a ? R, where a is the depth of the spherical bottom and R is the radius of the cylindrical section of the tank.The following parameter ranges must be spy a ? 0 for all vertical tanks, a ? R for a spherical bottom f 0. 5 for a torispherical bottom 0 ? k ? 0. 5 for a torispherical bottom D0 Vertical Tank Equations Here are the specific equations for fluid volumes in vertical cylindrical tanks with conical, ellipsoidal, spherical, and torispherical bottoms (use radian angular measure for all trigonometric functions, and D 0 for all equations) Conical bottom. ? Dh Vf = 4 4 a 2 h 3 2a 3 . . . . . . . h