Partial Liquid Volumes in Horizontal Vessels

Notes & ExperienceThe following are some guidelines and experienced hints for the design and utilization of process vessels.This information is never taught nor discussed in University courses or academic circles. It has beenhistorically expected that graduate engineers will learn this information using their own efforts.1)Always try to design around existing or available standard materials such as:a.Standard pipe caps. These are usually available off-the-shelf in carbon steel, as well as stainless,in sizes up to 42" and in various pipe schedule thicknesses.b.Standard seamless pipe. This is basic material that can be readily found available today. Alwaysmake this your first priority in selecting the vessel shell because of the convenience of eliminatingany plate rolling, longitudinal weld seam, and reducing the possibility of stress relief. Thisoption should be rejected only if required alloy, wall thickness, or diameter is not available.2)Own a copy of Eugene Megyesy's "Pressure Vessel Handbook" as published by Pressure VesselHandbook Publishing Inc.; P.O. Box 35365; Tulsa, OK 74153. This is probably the most useful andpractical engineering book ever published in the USA. It clearly belongs on every process plantengineer's desk. Study it thoroughly and your project problems will start to fade away.3)Ellipsoidal 2:1 heads have, by definition, 50% of the volumetric capacity of a hemispherical head with thesame internal diameter.Ellipsoidal heads are designed and fabricated on the basis of using the inside diameter as their nominaldiameter.These type of heads are used in preference to ASME Flanged and Dished heads for pressures in therange of 100 psig and for most vessels designed for pressures over 200 psig. Their inside depth of dish(IDD) is defined as half of the minor axis and is equal to 1/4 of the inside diameter of the head.4)ASME F&D (also called Torispherical) heads are designed and fabricated in the USA on the basis of usingthe outside diameter as their nominal diameter.Flanged and dished heads are inherently shallower (smaller IDD) than comparable ellipsoidal heads.These heads (like the ellipsoidal) are formed from a flat plate into a dished shape consisting of two radii:the "crown" radius or radius of the dish and the inside-corner radius, sometimes referred to as the"knuckle" radius. Because of the relative shallow dish curvature, ASME F&D heads are subject tohigher localized stresses at the knuckle radius as compared to the ellipsoidal type. The pressure ratingof these heads is increased by forming the head so that the knuckle radius is made at least equal to 3times the plate thickness. For code construction, the radius should in no case be less than 6% of theinside diameter.ASME F&D heads are used for pressure vessels in the general range of from 15 to about 200 psig .Although these heads may be used for higher pressures, for pressures in excess of 200 psig it may bemore economical to use an ellipsoidal type.5)The straight flange that forms part of each vessel head is part of the cylindrical vessel portion and shouldbe accounted for as such in calculating the vessel volume. These flanges vary in length depending on thehead thickness. A typical head flange length is about 1.5" to 2".6)Try to stay away from the immediate area of the knuckle radius with respect to locating nozzles or doingother welding, cutting or grinding. The need to locate a nozzle, insulation ring, clips or other itemnear the knuckle radius should be consulted with a mechanical or fabrication engineer.7)Be aware of the fact that the outside diameter of the cylindrical section may be bigger than that of thehead if a flush fit is required between the two inside diameters. This occurs because the required headthickness for a given design pressure is usually less than for the corresponding cylindrical section.This is especially true in the case of Hemispherical heads.8)Hemispherical heads are the strongest of the formed heads for a given thickness. A sphere is thestrongest known vessel shape. However, the main trade-off here is that all spheres have to be fabricatedas welded spherical segments. This requires more manual intensive work and results in a higher cost.9)Always be cognizant of the need for vessel entry into a vessel as well as vessel internal parts such astrays, baffles, agitators, dip pipes, downcomers, separator vanes, demister pads, etc. Sometimes theseitems directly affect not only the height of a vessel, but also the diameter. A chemical engineer shouldtake these factors into consideration even though they normally are not considered while doing processcalculations and simulations. Often, if not in the majority of cases, these factors and items are thecontrolling parameters that practically establish the diameter and height of the fabricated vessel regardlessof what the simulation program output states.10)As you consider the physical dimensions of a process vessel, always keep in mind that you must have,as a minimum, certain required nozzles built into the vessel - besides those required for basic processoperations. Many times some of these nozzles are not identified early in a project and their introductionlater requires costly change orders or, worse, vessel field modifications after the vessel is installed. Someof these nozzles are: manways, inspection ports, drains, cleaning (spraying) ports, auxiliary levelinstrument nozzle, liquid temperature probe, sample(s) probe, top head vents, critical high and low levelprobes, etc. Process Chemical Engineers are the best qualified to identify this need and specify thelocation and size. Never expect to lift a vessel by its nozzles. Lifting lugs are required for this, and aqualified structural or mechanical engineer should be commissioned to design this critical need.11)Do not forget to allow for insulation support rings. You must allow sufficient nozzle length so that anyrequired vessel insulation can be applied in the field without obstructing nozzle flanges and bolts.It is always advisable for the process Chemical Engineer to participate in the specification of the ultimateinsulation requirements and type since he/she are the most informed people of the temperature rangesand insulation types compatible with the vessel material, temperature, and service. Again, if this is notconsidered initially and is found to be required later, project timing and costs will suffer due to field vesselmodifications that could involve an ASME "R" stamp procedure.12)This Workbook was originally compiled to organize and utilize the techniques, formulas, basic data,and other information that I saved and used over the course of approximately 40 years of experiencein Chemical Engineering. Users will probably find it useful for carrying out day-to-day process plantprojects such as:1.Calculating the maximum volume capacity of a vessel;2.Calculating the partial volumes of a vessel at different levels ("Strapping" a vessel);3.Calculating the required vessel size for a given partial volume;4.Calculating the surface area of a vessel for primer, painting and insulation purposes;5.Calculating the location of critical liquid levels on a vessel for alarms and shutdown;6.Calculating the weight of a process vessel for cost estimates or foundation work;7.Calculating the "Line Pack", or volume content, of a piping system with fittings.There are probably more uses or applications for this Workbook, but the above should suffice toindicate the utilitarian value of this information to a Process or Project Engineer - especially in anoperating process plant in the field. Most of the basic information contained here was kept by me foryears in notes, 3-ring binders, between pages of text books, in formal calculations, etc. Thanks to Godfor giving me the good common sense to save and document this information and for giving us the digitalcomputer and a spreadsheet to organize and distribute it for use and exploitation by others. I hope thishelps others - especially young, striving, and determined engineers who earnestly want to do a successfuland safe project.Arthur Montemayor

&LArt Montemayor&CVessel Design Tips&RAugust 21, 2000Rev: 2(05-05-03)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Partial-Filled HorizontalVesselVOLUMES IN PARTIALLY FILLED HORIZONTAL VESSELSName:General Purpose TankItem No:T-C-15Vessel VolumeFlat HeadsUnit2:1 Ellip. HeadsCase:Partial Vol108,573in3137,52662.83ft379.59Tank Inside Dia. in =48.00470.0gal595.4Cylindrical Length, in =60.0Liquid Height, in =48.00Hemi HeadsUnitF & D HeadsL/D =1.3H/D =1.0000166,479in3120,48996.34ft369.73720.7gal521.6Cylindrical radius =r=24.00in.Chord Length =CL=0.0in.Segment Area =Aseg=1,810in2U. S. GallonsCylindrical Volume =Vcyl=108,573in3470.0F & Dished Volume =VFD=11,915in351.6Ellipsoidal Volume =Vell=28,953in3125.3Spherical Volume =Vsph=57,906in3250.7ac0.0000G0.0h0.0no foul0.0000foul0.0000

&LArt Montemayor&CPartially-Filled Horizontal Vessels&RMay 15, 1998Rev:1(01/22/00)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &AThe calculated Partial liquid capacity uses the following relationships found in the literature: Circular Sector = Circular Segment + Triangular Segment Partial Cylindrical volume = (Circular Segment)(Cylindrical length) Partial Cylindrical volume = r2[ACOS(r-H/r)] - (r-H)(2rH-H2)0.5 Volume of ASME F&D Head = 0.215483 H2 (1.5d -H) Volume of 2:1 Ellipsoidal Head = 0.5236 H2 (1.5d- H) Volume of Hemispherical Head = 2 (Volume of Ellipsoidal)

The Ellipsoidal and Spherical volumes calculated for the corresponding heads agree accurately (well within 99.5%) with fabricator's data of full volume capacity (Trinity Industries, Inc. Head Division; Catalog No. 7962M).Steps:(1) Enter the required information in the YELLOW cells;(2) The calculated results appear in RED numbers.Reference: Chemical Engineers' Handbook; Perry & Chilton; 5th Edition; P.6-87To obtain the total volumetric capacity of a process vessel, the volumetric capacity of the vessel heads must be calculated separately and added to the vessel's cylindrical volume.

The five types of formed vessel heads most frequently used are: 1. Hemispherical 2. 2:1 Ellipsoidal 3. ASME F&D (Torispherical) 4. Standard Dished (a misnomer, since there are no existing standards for dished heads) 5. Conical

The Standard Dished head is not suited for pressure vessels and, consequently, does not comply with the ASME Pressure Vessel Code. It is restricted to pressures less than 15 psig. The ASME F&D head is usually restricted to pressure vessels designed for less than 200 psig. Above this design pressure the 2:1 Ellipsoidal head is usually employed, with the Hemispherical head reserved for those applications that require the maximum in pressure resistance and mechanical integrity.

To obtain the partially-filled liquid contents' volume of a horizontal tank requires the determination of the partial volume of the two vessel heads as well as the cylindrical partial volume. The contents of a partially-filled vessel are arrived at by adding the partial contents of the Cylindrical portion and both heads:

Partial Volume = (Total Cylinder volume)(Zc) + (Total Heads' volume)(Ze)

where, Zc = Cylindrical partial volume coefficient Ze = Heads' partial volume coefficient

The cylindrical partial volume can be expressed by the following explicit analytical expressions:

1) V1 = {r2cos-1[(r-h/r]-(r-h)(2rh-h2)0.5}L .........(Kowal,G.; Chem. Eng; pp. 130-132; 6/11/73)2) V2 = 0.00433 L{pd2/8-[(0.5d-h)(dh-h2)0.5 + 0.25d2arcSine(0.25d-0.5h)]} ..........(Caplan, F.; Hydrocarbon Processing; July 1968)3) V3 = L r2[(a/57.30) - sinacosa] ..........(Chem. Engrs. Handbook; Perry/Chilton; 5th ed.; p.6-86)where, V1 = in3 V2 = gal V3 = in3 r = vessel's inside radius, in. h = depth of liquid content in the horizontal head, in. L = total straight, cylindrical, horizontal length, in. a = 1/2 of the total angle subtended by the chord forming the liquid level, degrees

The partial volumes of horizontal-oriented heads (except for Hemi-heads) are not defined in a mathematically exact formula but can be expressed by the following analytical expressions found in an article by F. Caplan in Hydrocarbon Processing; July 1968:

VDH = 0.0009328 h2 (1.5d - h) .......................Volume of a dished-only head, in US gallons VEll = 0.00226 h2 (1.5d - h) .......................Volume of 2:1 Ellipsoidal head, in US gallons VHH = 2 VEll .......................Volume of Hemispherical head, in US gallonswhere, h = depth of liquid content in the horizontal head, in. d = inside diameter of the horizontal head, in.The calculation of the partially-filled cylindrical portion of a horizontal vessel is straight-forward and can be done using the analytical expressions noted above. The equation given by Caplan (V2) should be very accurate since it is directly derived from an exact mathematical model presented in C.R.C. Standard Mathematical Tables; 12th Ed.(1959); p. 399.

The partial volume of heads is open to inaccuracies and while the analytical equations are suitable for estimating, the method usually used is the Ze method for determining the liquid fraction of the entire head. For this purpose, the Doolittle [Ind. Eng. Chem. 21, p. 322-323 (1928)] equation is used:

Vpartial = 0.00093 h2 (3r - h)

where, Vpartial = partial volume, gallons h = depth of liquid in both heads, in. r = inside radius of the horizontal heads, in.

(Note that this is the same equation offered by Caplan, above, for a dished-only head. His equation for an ellipsoidal head, although of the same form, is 142% in excess of the basic Doolittle relationship.)

Doolittle made some simplifying assumptions which affect the accuracy of the volume given by his equation, but the equation is satisfactory for determining the volume as a fraction of the entire head. This fraction, calculated by Doolittle's formula, is given in the Table listed above and regressed in the accompanying Chart. The Table or the resulting 3rd order polynomial equation,

Ze = -2 (h/d)3 + 3 (h/d)2 - 0.0016 (h/d) + 0.0001

can be used to arrive at a partial volume of standard dished, torispherical (ASME F&D), ellipsoidal, and hemispherical heads with an error of less than 2% of the entire head's volume.

Conical heads' volumes are defined by the exact mathematical expression for a truncated cone:

Vc = p h (D2 + dD + d2) / 12

where, Vc = total conical volume, cu. ft. h = height of the cone, ft d = diameter of the small end, ft D = diameter of the large end, ft When a tank volume cannot be calculated, or when greater precision is required, calibration may be necessary. This is done by draining (or filling) the tank and measuring the volume of liquid. The measurement may be made by weighing, by a calibrated fluid meter (i.e., Micro Motion Coriolis flowmeter), or by repeatedly filling small measuring tanks which have been calibrated by weight. From the known fluid density at the measured temperature, the equivalent volume can be quickly converted from the measured fluid mass.The Doolittle relationship can be applied to Horizontal and Vertical-oriented Ellipsoidal (and F&D) vessel heads. However, it is important to note that the H/D ratio that sets the fractional Coefficient, Ze, is measured differently in both cases. Refer to the above illustrations of Ellipsoids oriented horizontally and vertically.

For Horizontal Vessel Heads:

In this case, note that the H/D ratio represents the Liquid depth divided by the Major Axis (internal diameter) of the Ellipsoidal heads.

For Vertical Vessel Heads:

The H/D ratio corresponding to this orientation is the Liquid depth divided by the Minor Axis, not the Major Axis (internal diameter) of the Ellipsoidal heads. This means that the Inside Depth of Dish (IDD) must be known. The IDD is the depth of the head at its center and includes the inside corner radius but not the straight flange or nominal thickness of the head. Characteristic IDD's for various types of heads are:

Standard dished head: OD / 7 (Note: This is only approximate, since no standards exist for dished heads) ASME F&D head: OD / 6 Ellipsoidal, 2:1 head: ID / 4 Hemispherical head: ID / 2

An analytical equation for the partial volume of vertical oriented, "standard" dished heads at various depths is:

V = 0.01363 H2 L - 0.004545 H3 ......................(Chemical Processing Nomographs;Dale S. Davis; Chemical Publishing Co.;1969; p. 276)

where, V = liquid volume in the dish, gallons (excluding flanged section) H = liquid depth in the dish, inches L = radius of the dish, inches (usually equal to the tank ID, minus 6 inches)Horizontal vessel diameter (D)(major axis)minor axis(D)Horizontal Vessel Heads' orientationVertical Vessel Heads' orientationHH

Horizontal Tank StrappingCALIBRATION DATA FOR HORIZONTAL TANK WITH FORMED HEADSTank Inside Diameter (ID)inches90.000=7.5000ftTank length, tan/tanfeet7.167=86inchesTank HeadTypePressure1) Std. dish (non-pressure)< 15 psigNote: Place an "x" in only one of the2) Torispherical (ASME F&D)< 200 psig5 head options available. If more than3) Ellipsoidal (2:1)> 200 psigxone option contains an "x", the4) Ellipsoidal (non-std)Variesprogram will use the first one it finds.5) HemisphericalTo SuitHead type selected:2:1 EllipsoidalHead Volume =55.22cu.ft.Inside depth of head (IDD):inches20NOT REQUIRED FOR THIS HEAD TYPEHead thickness:inches0.375NOT REQUIRED FOR THIS HEAD TYPENumber of calibration increments:90.000(max 200)Calibration curve for90.0in. dia tank,7.167ft tan/tan,2:1 EllipsoidalheadsLiquid DepthLiquid Volume ContentH/DAngle radInchesCentimetersFt3US GalsLiters12.540.675.0190.0111110.21125.081.9314.5550.0222220.29937.623.6027.01020.0333330.367410.165.6042.01590.0444440.425512.707.9059.32240.0555560.476615.2410.4778.52970.0666670.522717.7813.2899.63770.0777780.565820.3216.32122.44630.0888890.605922.8619.57146.85550.1000000.6441025.4023.02172.66530.1111110.6801127.9426.65199.97560.1222220.7141230.4830.46228.58640.1333330.7481333.0234.44258.39760.1444440.7801435.5638.58289.31,0940.1555560.8111538.1042.87321.51,2150.1666670.8411640.6447.30354.71,3410.1777780.8701743.1851.87389.01,4700.1888890.8991845.7256.57424.21,6040.2000000.9271948.2661.39460.41,7400.2111110.9552050.8066.33497.51,8800.2222220.9822153.3471.38535.32,0240.2333331.0082255.8876.54574.02,1700.2444441.0342358.4281.80613.52,3190.2555561.0602460.9687.15653.62,4710.2666671.0852563.5092.60694.52,6250.2777781.1102666.0498.13736.02,7820.2888891.1352768.58103.74778.12,9410.3000001.1592871.12109.43820.73,1020.3111111.1832973.66115.19863.93,2660.3222221.2073076.20121.02907.63,4310.3333331.2313178.74126.91951.83,5980.3444441.2543281.28132.86996.43,7660.3555561.2783383.82138.861,041.43,9370.3666671.3013486.36144.911,086.84,1080.3777781.3243588.90151.011,132.54,2810.3888891.3473691.44157.141,178.64,4550.4000001.3693793.98163.321,224.94,6300.4111111.3923896.52169.521,271.44,8060.4222221.4153999.06175.761,318.24,9830.4333331.43740101.60182.011,365.15,1600.4444441.45941104.14188.291,412.25,3380.4555561.48242106.68194.591,459.45,5170.4666671.50443109.22200.901,506.75,6950.4777781.52644111.76207.211,554.15,8740.4888891.54945114.30213.531,601.56,0540.5000001.57146116.84219.851,648.96,2330.5111111.59347119.38226.161,696.26,4120.5222221.61548121.92232.471,743.56,5910.5333331.63849124.46238.771,790.86,7690.5444441.66050127.00245.051,837.86,9470.5555561.68251129.54251.301,884.87,1240.5666671.70552132.08257.541,931.57,3010.5777781.72753134.62263.741,978.17,4770.5888891.75054137.16269.922,024.47,6520.6000001.77255139.70276.062,070.47,8260.6111111.79556142.24282.152,116.17,9990.6222221.81857144.78288.202,161.58,1710.6333331.84158147.32294.202,206.58,3410.6444441.86459149.86300.152,251.18,5090.6555561.88760152.40306.042,295.38,6760.6666671.91161154.94311.872,339.08,8410.6777781.93462157.48317.632,382.29,0050.6888891.95863160.02323.322,424.99,1660.7000001.98264162.56328.932,467.09,3250.7111112.00765165.10334.462,508.59,4820.7222222.03166167.64339.912,549.39,6360.7333332.05667170.18345.262,589.59,7880.7444442.08268172.72350.522,628.99,9370.7555562.10769175.26355.682,667.610,0840.7666672.13370177.80360.732,705.510,2270.7777782.16071180.34365.672,742.510,3670.7888892.18772182.88370.492,778.710,5040.8000002.21473185.42375.192,814.010,6370.8111112.24274187.96379.762,848.210,7660.8222222.27175190.50384.192,881.510,8920.8333332.30176193.04388.482,913.611,0130.8444442.33177195.58392.622,944.611,1310.8555562.36278198.12396.602,974.511,2440.8666672.39479200.66400.413,003.111,3520.8777782.42780203.20404.043,030.311,4550.8888892.46281205.74407.493,056.211,5520.9000002.49882208.28410.743,080.511,6440.9111112.53683210.82413.783,103.311,7310.9222222.57684213.36416.593,124.411,8100.9333332.61985215.90419.163,143.711,8830.9444442.66686218.44421.463,160.911,9480.9555562.71787220.98423.463,176.012,0050.9666672.77488223.52425.133,188.512,0520.9777782.84289226.06426.393,197.912,0880.9888892.93090228.60427.063,203.012,1071.0000001.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.000.0000000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.00000.000.000.0000.000

&LArt Montemayor&CHorizontal Storage TankVolume Calibration&RNovember 11, 1999Rev: 1(03/12/00)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &ASteps:(1) Enter the required information in the YELLOW cells;(2) The calculated results appear in RED numbers.

Partial HorizontalHorizontal Cylindrical Tank with Ellipsoidal or Hemispherical HeadsTotal tank volume =(Total volume in two heads) + (Total volume in cylindrical section)=( 1/6 p K1 D3 ) + ( 1/4 p D2 L )K1 =2b/DZe =H1/DZc =H1/DPartial tank volume =( 1/6 p K1 D3 ) ([f(Ze)]) + ( 1/4 p D2 L ) ([fZc)])f(Zc) = Horizontal cylinder coefficient (from Doolittle tables)or,f(Ze) = Ellipsoidal coefficient (from Doolittle tables)or,Where,For Ellipsoidal 2:1 heads,b =(1/4) DK1 =1/2

&LArt Montemayor&CHorizontal Cylindrical TankPartial Volume Determination&RMay 5, 2001Rev: 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &AH1D/2D/2aLDbbOct 31, 1999www.about.com

The volume V of a liquid in a horizontal cylindrical tank is:

V = LD2 (2Z-sin(2Z)) /8

Where,Z = arccos(1-2h/D) h = height of liquid in the horizontal cylindrical tank D = diameter of the tank L = length of the tank

Note that the result of the arccos-function has to be taken in radians.

Bernhard Spang

Partial VerticalVertical Cylindrical Tank with Ellipsoidal or Hemispherical HeadsTotal tank volume =(Total volume in two heads) + (Total volume in cylindrical section)=( 1/6 p K1 D3 ) + ( 1/4 p D2 L )Partial tank volume =( 1/6 p K1 D3 ) ([f(Ze)]) + ( 1/4 p D2 H3)K1 =2b/DZe =(H1 + H2)/K1Df(Ze) =Ellipsoidal coefficient (from Doolittle tables)or,

&LArt Montemayor&CVertical Cylindrical TankPartial Volume Determination&RMay 05, 2001Rev: 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &ALbH1H1bH3DH3H2

Partial Cylind. Vol.Zcf(Zc)0.0000000.0000000.0500000.0186920.1000000.0520440.1500000.0940610.2000000.1423780.2500000.1955010.3000000.2523150.3500000.3119180.4000000.3735300.4500000.4364450.5000000.5000000.5500000.5635550.6000000.6264700.6500000.6880820.7000000.7476850.7500000.8044990.8000000.8576220.8500000.9059390.9000000.9479560.9500000.9813081.0000001.000000

&LArt Montemayor&CRegression of Doolittle Partial Volume Coefficient&RMay 15, 1998Rev: 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Partial Cylind. Vol.

f(Zc)H/D = Zcf(Zc)Coefficients for Partial Volumes of Horizontal Cylindersy = -1.144x3 + 1.716x2 + 0.4365x - 0.0043R2 = 1

Hds Partial Vol.H/DVol. Fraction0.020.00120.040.00470.060.01040.080.01820.100.02800.120.03970.140.05330.160.06860.180.08550.200.10400.220.12390.240.14510.260.16760.280.19130.300.21600.320.24200.340.26800.360.29500.380.32300.400.35200.420.38100.440.41000.460.44000.480.47000.500.50000.520.53000.540.56000.560.59000.580.61900.600.64800.620.67700.640.70500.660.73200.680.75800.700.78400.720.80870.740.83240.760.85490.780.87610.800.89600.820.91450.840.93140.860.94670.880.96030.900.97200.920.98180.940.98960.960.99530.980.99881.001.0000

Data Source:

NGPSA Engineering Data Book9th Edition; 1972; p. 13-7&LArt Montemayor&C&"Arial,Bold"&12Doolittle Equation for Parially-Filled Vessel Heads&RMay 27, 1998Rev: 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Hds Partial Vol.

Liquid Depth/Head ID, (H/D)Volumetric FractionVolume Fraction of Horizontal Vessel Headsy = -2.0026x3 + 3.004x2 - 0.0016x + 0.0001R2 = 1

Partial Ellip. Vol.Zef(Ze)0.0000.0000000.0500.0072500.1000.0280000.1500.0607500.2000.1040000.2500.1562500.3000.2160000.3500.2817500.4000.3520000.4500.4252500.5000.5000000.5500.5747500.6000.6480000.6500.7182500.7000.7840000.7500.8437500.8000.8960000.8500.9392500.9000.9720000.9500.9927501.0001.000000

Reference: Chemical Engineers' Handbook; Perry & Chilton; 5th Edition; P.6-87

To obtain the total volumetric capacity of a process vessel, the volumetric capacity of the vessel heads must be calculated separately and added to the vessel's cylindrical volume.

The five types of formed vessel heads most frequently used are: 1. Hemispherical 2. 2:1 Ellipsoidal 3. ASME F&D (Torispherical) 4. Standard Dished (a misnomer, since there are no existing standards for dished heads) 5. Conical

The Standard Dished head is not suited for pressure vessels and, consequently, does not comply with the A.S.M.E. Pressure Vessel Code. It is restricted to pressures less than 15 psig. The ASME F&D head is usually restricted to pressure vessels designed for less than 200 psig. Above this design pressure the 2:1 Ellipsoidal head is usually employed, with the Hemipherical head reserved for those applications that require the maximum in pressure resistance and mechanical integrity.

To obtain the partially-filled liquid contents' volume of a horizontal tank requires the determination of the partial volume of the two vessel heads as well as the cylindrical partial volume. The contents of a partially-filled vessel are arrived at by adding the partial contents of the Cylindrical portion and both heads:

Partial Volume = (Total Cylinder volume)(Zc) + (Total Heads' volume)(Ze)

where, Zc = Cylindrical partial volume coefficient Ze = Heads' partial volume coefficient

The cylindrical partial volume can be expressed by the following explicit analytical expressions:

1) V1 = {r2cos-1[(r-h/r]-(r-h)(2rh-h2)0.5}L .........(Kowal,G.; Chem. Eng; pp. 130-132; 6/11/73)2) V2 = 0.00433 L{pd2/8-[(0.5d-h)(dh-h2)0.5 + 0.25d2arcSine(0.25d-0.5h)]} ..........(Caplan, F.; Hydrocarbon Processing; July 1968)3) V3 = L r2[(a/57.30) - sinacosa] ..........(Chem. Engrs. Handbook; Perry/Chilton; 5th ed.; p.6-86)where, V1 = in3 V2 = gal V3 = in3 r = vessel's inside radius, in. h = depth of liquid content in the horizontal head, in. L = total straight, cylindrical, horizontal length, in. a = 1/2 of the total angle subtended by the chord forming the liquid level, degrees

The partial volumes of horizontal-oriented heads (except for Hemi-heads) are not defined in a mathematically exact formula but can be expressed by the following analytical expressions:(From Caplan, F.; Hydrocarbon Processing; July 1968)

VDH = 0.0009328 h2 (1.5d - h) .......................Volume of a dished-only head, in US gallons VEll = 0.00226 h2 (1.5d - h) .......................Volume of 2:1 Ellipsoidal head, in US gallons VHH = 2 VEll .......................Volume of Hemispherical head, in US gallons

where, h = depth of liquid content in the horizontal head, in. d = inside diameter of the horizontal head, in.The calculation of the partially-filled cylindrical portion of a horizontal vessel is straight-forward and can be done using the analytical expressions noted above. The equation given by Caplan (V2) should be very accurate since it is directly derived from an exact mathematical model presented in C.R.C. Standard Mathematical Tables; 12th Ed.(1959); p. 399.

The partial volume of heads is open to inaccuracies and while the analytical equations are suitable for estimating, the method usually used is the Ze method for determining the liquid fraction of the entire head. For this purpose, the Doolittle [Ind. Eng. Chem. 21, p. 322-323 (1928)] equation is used:

Vpartial = 0.00093 h2 (3r - h)

where, Vpartial = partial volume, gallons h = depth of liquid in both heads, in. r = inside radius of the horizontal heads, in.

(Note that this is the same equation offered by Caplan, above, for a dished-only head. His equation for an ellipsoidal head, although of the same form, is 142% in excess of the basic Doolittle relationship.)

Doolittle made some simplifying assumptions which affect the accuracy of the volume given by his equation, but the equation is satisfactory for determining the volume as a fraction of the entire head. This fraction, calculated by Doolittle's formula, is given in the Table listed above and regressed in the accompanying Chart. The Table or the resulting 3rd order polynomial equation,

Ze = -2 (h/d)3 + 3 (h/d)2 - 0.0016 (h/d) + 0.0001

can be used to arrive at a partial volume of standard dished, torispherical (ASME F&D), ellipsoidal, and hemispherical heads with an error of less than 2% of the entire head's volume.

Conical heads' volumes are defined by the exact mathematical expression for a truncated cone:

Vc = p h (D2 + dD + d2) / 12

where, Vc = total conical volume, cu. ft. h = height of the cone, ft d = diameter of the small end, ft D = diameter of the large end, ft When a tank volume cannot be calculated, or when greater precision is required, calibration may be necessary. This is done by draining (or filling) the tank and measuring the volume of liquid. The measurement may be made by weighing, by a calibrated fluid meter (i.e., Micro Motion Coriolis flowmeter), or by repeatedly filling small measuring tanks which have been calibrated by weight. From the known fluid density at the measured temperature, the equivalent volume can be quickly converted from the measured fluid mass.The Doolittle relationship can be applied to Horizontal and Vertical-oriented Ellipsoidal (and F&D) vessel heads. However, it is important to note that the H/D ratio that sets the fractional Coefficient, Ze, is measured differently in both cases. Refer to the above illustrations of Ellipsoids oriented horizontally and vertically.

For Horizontal Vessel Heads:

In this case, note that the H/D ratio represents the Liquid depth divided by the Major Axis (internal diameter) of the Ellipsoidal heads.

For Vertical Vessel Heads:

The H/D ratio corresponding to this orientation is the Liquid depth divided by the Minor Axis, not the Major Axis (internal diameter) of the Ellipsoidal heads. This means that the Inside Depth of Dish (IDD) must be known. The IDD is the depth of the head at its center and includes the inside corner radius but not the straight flange or nominal thickness of the head. Characteristic IDD's for various types of heads are:

Standard dished head: OD / 7 (Note: This is only approximate, since no standards exist for dished heads) ASME F&D head: OD / 6 Ellipsoidal, 2:1 head: ID / 4 Hemispherical head: ID / 2

An analytical equation for the partial volume of vertical oriented, "standard" dished heads at various depths is:

V = 0.01363 H2 L - 0.004545 H3 ......................(Chemical Processing Nomographs;Dale S. Davis; Chemical Publishing Co.;1969; p. 276)

where, V = liquid volume in the dish, gallons (excluding flanged section) H = liquid depth in the dish, inches L = radius of the dish, inches (usually equal to the tank ID, minus 6 inches)Horizontal vessel diameter (D)(major axis)minor axis(D)Horizontal Vessel Heads' orientationVertical Vessel Heads' orientationHH&LArt Montemayor &CRegression of Doolittle Partial Volume Coefficient&RMay 15, 1998Rev: 1(02/25/01)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Partial Ellip. Vol.

H/D = Zef(Ze)Coefficients for Partial Volumes in Ellipsoids & Spheresy = -2x3 + 3x2 + 1.156504905E-15x - 1.11143497E-16R2 = 1

Hds Vol & Surf Area Internal DiameterVolume of one head, Ft3Internal Surface Area of one head, Ft2InchesFtHemisphereEllipsoidalASME F&DStandard F&DHemisphereEllipsoidalASME F&DStandard F&D121.00000.260.130.080.051.571.000.930.79141.16670.420.210.130.092.141.361.261.07161.33330.620.310.200.132.791.781.651.40181.50000.880.440.280.183.532.252.091.77201.66671.210.610.390.254.362.782.582.18221.83331.610.810.510.335.283.363.122.64242.00002.091.050.670.436.284.003.713.14262.16672.661.330.850.557.374.704.363.69282.33333.331.661.060.688.555.455.064.28302.50004.092.051.300.849.826.265.804.91322.66674.962.481.581.0211.177.126.605.59342.83335.952.981.901.2212.618.047.456.31363.00007.073.532.251.4514.149.018.367.07383.16678.314.162.651.7015.7510.049.317.88403.33339.704.853.091.9917.4511.1210.328.73423.500011.225.613.582.3019.2412.2611.389.62484.000016.768.385.343.4325.1316.0214.8612.57544.500023.8611.937.614.8831.8120.2718.8015.90605.000032.7216.3610.446.7039.2725.0323.2219.64665.500043.5621.7813.898.9247.5230.2828.0923.76726.000056.5528.2718.0411.5856.5536.0433.4328.27786.500071.9035.9522.9314.7266.3742.3039.2333.18847.000089.8044.9028.6418.3876.9749.0545.5038.48907.5000110.4555.2235.2322.6188.3656.3152.2344.18968.0000134.0467.0242.7527.44100.5364.0759.4350.271028.5000160.7880.3951.2832.92113.4972.3367.0956.751089.0000190.8595.4360.8739.07127.2381.0975.2263.621149.5000224.46112.2371.5945.96141.7690.3583.8170.8812010.0000261.80130.9083.5053.60157.08100.1192.8678.5412610.5000303.07151.5396.6662.05173.18110.37102.3886.5913211.0000348.45174.23111.1471.34190.07121.13112.3695.0313811.5000398.16199.08126.9981.52207.74132.40122.81103.8714412.0000452.39226.19144.2992.62226.19144.16133.72113.1015012.5000511.33255.66163.09104.69245.44156.42145.09122.7215613.0000575.17287.59183.45117.76265.46169.19156.93132.7316213.5000644.12322.06205.44131.88286.28182.45169.24143.1416814.0000718.38359.19229.12147.08307.88196.22182.01153.9417414.5000798.13399.06254.56163.41330.26210.48195.24165.1318015.0000883.57441.79281.81180.90353.43225.25208.94176.7218615.5000974.91487.45310.94199.60377.38240.52223.10188.6919216.00001,072.33536.17342.02219.55402.12256.28237.72201.0619816.50001,176.04588.02375.09240.78427.65272.55252.81213.8320417.00001,286.22643.11410.24263.34453.96289.32268.37226.9821017.50001,403.08701.54447.51287.26481.06306.59284.38240.5321618.00001,526.81763.41486.97312.60508.94324.36300.87254.4722218.50001,657.62828.81528.69339.38537.61342.63317.81268.8022819.00001,795.68897.84572.73367.64567.06361.40335.22283.5323419.50001,941.21970.60619.14397.44597.30380.67353.10298.6524020.00002,094.401,047.20668.00428.80628.32400.44371.44314.16Note: The Volume and Surface Area attributable to a head's straight flange is not included in this data.The Internal Diameter is used in calculating the Surface Area; therefore, the resultant Area is slightly less than theactual external surface area.References and Sources:(1)Pressure Vessel Handbook; Eugene F. Megyesy; 8th Edition; Pressure Vessel HandbookPublishing, Inc.(2)Process Vessel Design; L.E. Brownell & E.H. Young; John Wiley & Sons; N.Y.; 1959(3)A. Montemayor personal files

Data Source:

NGPSA Engineering Data Book9th Edition; 1972; p. 13-9

NOTE: These capacity coefficients apply for the volume of 2 ellipsoidal or hemispherical heads..not the volume for 1 head!!&LArt Montemayor&CPressure Vessel Heads&RAugust 04, 1998Rev:1(08/21/00)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Mfr's Hds' VolDiameter ftHead Volume in Cubic FeetHead Volume in U.S. GallonsEllipsoidalASME F&DHemisphericalDishedEllipsoidalASME F&DHemisphericalDished1.000.1310.0820.2620.0530.9800.6131.9600.3961.500.4420.2770.8840.1823.3062.0726.6131.3612.001.0470.6562.0950.4307.8324.90715.6723.2172.502.0451.3704.0910.84215.29810.24830.6036.2993.003.5352.2167.0691.45426.44416.57752.88010.8773.505.6133.69211.2252.31041.98827.61883.96917.2804.008.3785.25516.7563.44862.67239.310125.34425.7934.5011.9287.76723.8574.90989.22858.101178.46336.7225.0016.36410.26432.7256.733122.41176.780244.80050.3665.5021.77913.80343.5578.963162.918103.254325.82967.0486.0028.27618.07256.54811.636211.519135.188423.00887.0436.5035.95122.35171.89614.794268.932167.197537.819110.6677.0044.90229.08189.79718.477335.890217.541671.728138.2187.5055.22634.903110.44722.727413.119261.093826.201170.0108.0067.02543.239134.04127.582501.382323.4501,002.696206.3288.5080.39450.764160.77833.083601.389379.7411,202.703247.4789.0095.43259.098190.85239.271713.881442.0841,427.672293.7689.50112.23770.821224.46046.188839.591529.7781,679.078345.51010.00130.90881.172261.79953.871979.260607.2091,958.393402.983

&LArt Montemayor&C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &AData source:

Trinity Industries, Inc. Head Division Catalog No. 7962M; Page 15

Ellipsoidal Curve FitI. D., inchesVol. Gallons120.98183.31247.833015.303626.444241.994862.675489.2360122.4166162.9272211.5278268.9384335.8990413.1296501.38102601.39108713.88114839.59120979.261261133.611321303.391381489.331441692.161501912.611562151.431622409.341682687.081742985.391803304.991863646.631924011.041984398.95Ellipsoidal Head Inside Diameter =120inches2044811.092105248.21Volume of Single Ellipsodial Head =979.26Gallons =130.907Ft32165711.032226200.292286716.732347261.072407834.06

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Ellipsoidal Curve Fit

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &AInside Diameter, inchesVolume, gallons2:1 Ellipsoidal Head Volumey = 0.000566699x3.000000133R2 = 1

Ellipsoidal HeadsInches60.00Approximate area for nozzle attachmentStart of Knuckle Radiusmm1524Inside DepthKnuckle Radius(= I.D./4)Inches12.9518.75Inchesmm329476mm24.55InchesDish RadiusNote:67.84InchesVerify all dimension624mm1723mmwith vendor drawings75Inches1905mmNOTE:Ellipsoidal 2:1 heads are fabricated and measured using the Internal Diameter (ID) of the head.Note that this measurement convention is opposite to that of the ASME F&D head.Any cylindrical shell fabricated to fit these heads must conform to or match the ID dimension.

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)&LArt Montemayor&C2:1 Ellipsoidal Heads&RMay 21, 2003Rev: 1&CPage &P of &N&RFileName: &FWorksheet: &ATangent LineStraightFlange(Varies)2" Nom.51mm2:1 Elliptical HeadKey In the Head I.D.Enter the Ellipsoidal Head's ID in Inches

ASME F&D Curve FitI. D., inchesVolume, gal.120.61182.07244.913010.253616.584227.624839.315458.106076.7866103.2572135.1978167.2084217.5490261.0996323.45102379.74108442.08114529.78120607.21126714.90132809.04138934.151441,015.271501,227.021561,361.281621,504.821681,712.891741,879.891802,057.211862,312.531922,515.83ASME F&D Head Inside Diameter =84inches1982,730.512043,078.42Volume of Single ASME F&D Head =205.29Gallons =27.443Ft32103,324.022163,582.122223,853.002284,187.612344,700.902405,025.88

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic File: &FWorkSheet: &A

ASME F&D Curve Fit

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &NVolume, gal.Inside Diameter, inchesVolume, gallonsASME F&D HEAD VOLUMEy = 0.000292744x3.0378R2 = 0.9996

ASME F&D HeadsAll Dimensionsare in Inches (mm)Verify all dimensionwith vendor drawingsNOTE:ASME F&D heads are fabricated and measured using the Outside Diameter (OD) of the head.Note that this measurement convention is opposite to that of the Ellipsoidal head.Any cylindrical shell fabricated to fit these heads must conform to or match the OD dimension.Not all wall thicknesses are shown. Interpolate for approximate inside depth O.D. dish IDDInches (Flanged & Dished Head ASME Table)Millimeters (Flanged & Dished Head ASME Table)O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38241.634.5010610411140.50241.634.4426"13610411130.63241.884.5066016610481140.75242.254.6919610571190.38261.754.8110660441220.50261.754.7528"13660441210.63261.884.7571116660481210.75262.254.9419660571250.38301.884.8810762481240.50301.884.8130"13762481220.63301.884.8176216762481220.75302.255.0019762571270.38302.005.5610762511410.50302.005.5032"13762511400.63302.005.3881316762511370.75302.255.5019762571400.38342.135.5610864541410.50342.135.503413864541400.63302.136.0086416762541520.75302.256.0619762571540.38362.255.9410914571510.50362.255.8836"13914571490.63362.255.8191416914571480.75362.255.751991457146O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38362.386.5010914601650.50362.386.4438"13914601640.63362.386.3896516914601620.75362.386.3819914601620.38402.506.63101016641680.50402.506.5640"131016641670.63362.506.94101616914641760.75362.507.0019914641780.38402.637.19101016671830.50402.637.1342"131016671810.63402.637.061067161016671790.75402.637.00191016671780.38423.008.00101067762030.50423.008.7542"131067762220.63423.008.691219161067762210.75423.008.63191067762190.38543.258.94101372832270.50483.259.7554"131219832480.63483.259.751372161219832480.75483.259.63191219832450.38603.6310.00101524922540.50603.639.8860"131524922510.63543.6310.691524161372922720.75543.6310.6319137292270O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38664.0011.001016761022790.50604.0010.9466"1315241022780.63604.0011.7516761615241022980.75604.0011.631915241022950.38724.3812.001018291113050.63724.3811.8872"1618291113020.75724.3811.8818291918291113020.88664.3812.632216761113210.38784.7513.001019811213300.50724.7513.8178"1318291213510.75724.7513.6919811918291213481.00724.7513.502518291213430.38845.1314.001021341303560.63845.1313.8884"1621341303530.88845.1313.7521342221341303491.00845.1313.692521341303480.38905.5015.131022861403840.50845.5015.8190"1321341404020.75845.5015.6922861921341403991.00845.5015.562521341403950.38965.8816.131024381494100.50905.8816.8896"1322861494290.88905.8816.6324382222861494221.25905.8816.44322286149418O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.50966.1317.881324381564540.75966.1317.69102"1924381564491.00966.1317.5625912524381564461.13906.1318.502922861564700.501026.5018.881325911654800.751026.5018.75108"1925911654761.001026.5018.5627432525911654711.13966.5019.442924381654940.501086.8819.881327431755050.751086.8819.75114"1927431755021.001086.8819.6328962527431754991.251086.8819.503227431754950.501147.2520.881328961845300.881147.2520.69120"2228961845261.251087.2521.4430483227431845451.631087.2521.254127431845400.501207.6321.881330481945560.881207.6321.69126"2230481945511.251207.6321.5032003230481945461.381147.6322.313528961945670.751268.0022.811932002035790.881208.0023.69132"2230482036021.251208.0023.4433533230482035951.631208.0023.25413048203591O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.631328.3823.941633532136081.001328.3823.75138"2533532136031.381328.3823.5635053533532135981.751328.3823.384433532135940.631328.7525.881633532226571.001328.7525.63144"2533532226511.381328.7525.4436583533532226461.751328.7525.194433532226400.751449.3827.751936582387051.131449.3827.50156"2936582386991.501449.3827.3139623836582386941.881449.3827.064836582386870.7514410.1331.811936582578081.1314410.1331.50168"2936582578001.5014410.1331.3142673836582577951.8814410.1331.134836582577910.8817010.8831.442243182767991.2517010.8831.251803243182767941.6317010.8831.0045724143182767872.0017010.8830.815143182767830.8817011.6335.442243182959001.2517011.6335.19192"3243182958941.6317011.6334.9448774143182958872.0017011.6334.75514318295883O.D"T""R1""R2"IDDASMEO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.8817012.2539.5622431831110051.2517012.2539.38204"32431831110001.6317012.2539.1951824143183119952.0017012.2538.945143183119891.0017012.6341.8125431832110621.3817012.6341.63210"35431832110571.7517012.6341.31533444431832110492.2517012.6341.0057431832110411.0017013.0044.2525431833011241.3817013.0044.00216"35431833011181.7517013.0043.69548644431833011102.0017013.0043.5051431833011051.0018013.7546.5625457234911831.3818013.7546.31228"35457234911761.7518013.7546.06579144457234911702.0018013.7545.695145723491161

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)&LArt Montemayor&CASME Flanged and Dished Heads&RMay21, 2003Rev: 0&CPage &P of &N&RFileName: &FWorksheet: &ATangent LineStraightFlange(Varies)2" Nom.51mmWall Thickness "T"Knuckle Radius"R2"Dish Radius"R1"Area for nozzle attachmentO.D. - (R2+T)x2Outside Diameter (O.D.)Flanged and Dished Head (ASME)Inside Depthof Dish"IDD"26"28"30"32"34"36"38"40"42"48"54"60"66"72"78"84"90"96"102"108"114"120"126"132"138"144"156"168"180"192"204"210"216"228"

Hemispherical Curve FitInternal Diameter, ftHemispherical Volume, cu. Ft.1.000.2621.500.8842.002.0952.504.0913.007.0693.5011.2254.0016.7564.5023.8575.0032.7255.5043.5576.0056.5486.5071.8967.0089.7977.50110.4478.00134.0418.50160.7789.00190.8529.50224.4610.00261.799Hemispherical Head Inside Diameter =120.000inchesVolume of Single Hemispherical Head =261.900Ft3=1,959.1Gallons

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic FileName: &F WorkSheet: &A

Hemispherical Curve Fit

&LArt Montemayor&C&A&RSeptember 12, 1997Rev 0&CPage &P of &NHemispherical Volume, cu. Ft.Inside Diameter, FtVolume, CuFtHEMISPHERICAL HEAD VOLUMEy = 0.2619x2.9997R2 = 1

Dished Curve FitInternal Diameter, ftDished Volume, Ft31.000.0531.500.1822.000.4302.500.8423.001.4543.502.3104.003.4484.504.9095.006.7335.508.9636.0011.6366.5014.7947.0018.4777.5022.7278.0027.5828.5033.0839.0039.2719.5046.18810.0053.871Dished Head Inside Diameter =120.000inchesVolume of Single Dished Head =53.60Ft3=401.0Gallons

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &A

Dished Curve Fit

&LArt Montemayor &C&A&RSeptember 12, 1997Rev 0&CPage &P of &NDished Volume, Ft3Inside Diameter, FtVolume, CuFtDISHED HEAD VOLUMEy = 0.0536x3.0033R2 = 1

Dished HeadsAll Dimensionsare in Inches (mm)Verify all dimensionwith vendor drawingsNOTE:F & D heads are fabricated and measured using the Outside Diameter (OD) of the head.Any cylindrical shell fabricated to fit these heads must conform to or match the OD dimension.Not all wall thicknesses are shown. Interpolate for approximate inside depth O.D. dish IDDInches (Flanged & Dished Head Table)Millimeters (Flanged & Dished Head Table)O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38241.134.2510610291080.50241.504.3826"13610381110.63241.884.5066016610481140.75242.254.6919610571190.38264.504.50106601141140.50264.634.6328"136601181180.63264.754.75711166601211210.75264.944.94196601251250.38304.504.50107621141140.50304.634.6330"137621181180.63304.814.81762167621221220.75305.005.00197621271270.38305.005.00107621271270.50305.195.1932"137621321320.63305.315.31813167621351350.75305.505.50197621401400.38345.005.00108641271270.50345.195.1934138641321320.63335.445.44864168381381380.75306.066.06197621541540.38365.255.25109141331330.50365.445.4436"139141381380.63365.635.63914169141431430.75365.755.7519914146146O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38361.135.8110914291480.50361.506.0038"13914381520.63361.886.1396516914481560.75362.256.3119914571600.38401.135.81101016291480.50401.505.9440"131016381510.63361.886.69101616914481700.75362.256.8819914571750.38421.136.06101067291540.50421.506.2542"131067381590.63421.886.381067161067481620.75402.256.81191016571730.38481.136.88101219291750.50481.507.0042"131219381780.63481.887.191219161219481830.75482.257.38191219571870.38541.137.69101372291950.50541.507.8154"131372381980.63541.888.001372161372482030.75542.258.19191372572080.38601.138.50101524292160.50601.508.6360"131524382190.63601.888.811524161524482240.75602.258.9419152457227O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.38661.139.31101676292360.50661.509.4466"131676382400.63661.889.631676161676482450.75662.259.75191676572480.38721.1310.06101829292560.63721.8810.3872"161829482640.88722.6310.691829221829672721.13723.3811.00291829862790.38781.1310.88101981292760.63781.8811.1978"161981482840.88782.6311.501981221981672921.13783.3811.81291981863000.38841.1311.69102134292970.63841.8811.8884"162134483020.88842.6312.312134222134673131.13843.3812.63292134863210.38901.1312.50102286293180.63841.8813.7590"162134483490.88842.6314.002286222134673561.13843.3814.31292134863630.50961.5013.44132438383410.75962.2513.7596"192438573491.00963.0014.062438252438763571.25963.7514.3832243895365O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.501021.5014.25132591383620.75962.2515.50102"192438573941.00963.0015.752591252438764001.25963.7516.06322438954080.501081.5015.06132743383830.751082.2515.38108"192743573911.001023.0016.562743252591764211.251023.7516.81322591954270.501141.5015.88132896384030.751142.2516.19114"192896574111.001083.0017.382896252743764411.251083.7517.63322743954480.501201.5016.69133048384240.881202.6317.13120"223048674351.251203.7517.593048323048954471.631204.8818.064130481244590.501261.5017.50133200384450.881202.6318.81126"223048674781.251203.7519.253200323048954891.631204.8819.694130481245000.631321.8818.44163353484680.881322.6318.75132"223353674761.131323.3819.003353293353864831.501324.5019.50383353114495O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.631321.8820.13163353485110.881322.6320.44138"223353675191.131323.3820.693505293353865261.501324.5021.133833531145370.631441.8820.00163658485080.881442.6320.31144"223658675161.131443.3820.633658293658865241.501444.5021.133836581145370.751442.2523.69193658576021.131443.3824.13156"293658866131.501444.5024.5039623836581146221.881445.6324.944836581436330.751702.2523.13194318575881.131703.3823.56168"294318865981.501704.5024.0642673843181146111.881705.6324.194843181436140.881702.6326.69224318676781.251703.7527.13180324318956891.631704.8827.5645724143181247002.001706.0028.005143181527110.881702.6330.50224318677751.251703.7530.88192"324318957841.631704.8831.2548774143181247942.001706.0031.63514318152803O.D"T""R1""R2"IDDO.D"T" (mm)"R1"(mm)"R2"(mm)IDD(mm)0.881702.6334.63224318678801.251703.7535.00204"324318958891.631704.8835.3151824143181248972.001706.0035.635143181529050.881702.6339.06224318679921.251703.7539.50216"3243189510031.631704.8839.75548641431812410102.001706.0040.0051431815210160.881802.6341.252245726710481.251803.7541.50228"3245729510541.631804.8841.75579141457212410602.001806.0042.0051457215210670.881802.6346.252245726711751.251803.7546.44240"3245729511801.631804.8846.63609641457212411842.001806.0046.815145721521189

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)&LArt Montemayor&CFlanged and Dished Heads&RMay 21, 2003Rev: 0&CPage &P of &N&RFileName: &FWorksheet: &ATangent LineStraightFlange(Varies)2" Nom.51mmWall Thickness "T"Knuckle Radius"R2"Dish Radius"R1"Area for nozzle attachmentO.D. - (R2+T)x2Outside Diameter (O.D.)Flanged and Dished HeadInside Depthof Dish"IDD"26"28"30"32"34"36"38"40"42"48"54"60"66"72"78"84"90"96"102"108"114"120"126"132"138"144"156"168"180"192"204"216"228"240"

Cylindrical Tank VolumeDiam., in.Gal./in.Diam., in.Gal./in.Diam., in.Gal./in.Diam., in.Gal./in.15.00.76528.02.666529.1949631.33415.50.81728.52.762539.5519832.65316.00.87029.02.859549.91410034.00016.50.92629.52.9595510.28510235.37317.00.983303.0605610.66210436.77417.51.041313.2675711.04710638.20218.01.102323.4825811.43810839.65718.51.164333.7035911.83511041.14019.01.227343.9306012.24011242.64919.51.293354.1656213.07011444.18620.01.360364.4066413.92611645.75020.51.429374.6556614.81011847.34121.01.499384.9106815.72212048.96021.51.572395.1717016.66012250.60522.01.646405.4407217.62612452.27822.51.721415.7157418.61812653.97823.01.799425.9987619.63812855.70523.51.878436.2877820.68513057.46024.01.958446.5828021.76013259.24124.52.041456.8858222.86113461.05025.02.125467.1948423.99013662.88625.52.211477.5118625.14613864.74926.02.298487.8348826.32914066.64026.52.388498.1639027.54014268.55727.02.479508.5009228.77714470.50227.52.571518.8439430.04214672.47414874.473

&LArt Montemayor&CCylindrical Vessel Volume Relationship&RMay 27, 1998Rev:1(06/06/01)&CPage &P of &N&RElectronic FileName: &FWorkSheet: &ACylindrical Volumes of Vessels --- expressed as Gallons of Liquid Content per inch length of Cylinder.

Source: Chemical Engineers' Handbook; Perry & Chilton; 5th Edition; p. 6-86Oct 31, 1999www.about.com

The volume, V, of a liquid in a horizontal cylindrical tank with flat heads is:

V = LD2 (2Z-sin(2Z)) /8

Where,Z = arccos(1-2h/D) h = height of liquid in the horizontal cylindrical tank D = diameter of the tank L = length of the tank

Note that the result of the arccos-function has to be taken in radians.

Bernhard Spang

Fittings' VolumesVOLUMETRIC CAPACITY FOR BUTT-WELDED FITTINGSAll volumes expressed in cubic inchesReference: Piping Engineering; Tube Turns Division of Chemetron Corp.; Nov. 1971; p.47Nominal Pipe Size, in.90o Elbows180o Returns45o ElbowsTeesCapsCrossesStub EndsNominal Pipe Size, in.ReducersTeesLong RadiusShort RadiusLong RadiusShort RadiusLong RadiusFull-size outletsFull-size outletsLap JointConcentric & Eccentricwith Reducing OutletStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongStandardX-StrongLarge endSmall endStandardX-StrongStandardX-Strong11/20.70.61.41.10.40.20.80.60.30.20.90.73/81.51.32.72.33/40.90.81.91.50.50.41.61.30.60.41.61.31/21.81.42.82.412.01.71.44.13.42.71.00.83.52.91.10.93.52.93/42.11.83.02.61-1/44.43.72.98.77.45.92.21.87.56.51.81.59.58.26.05.11-1/41/22.62.25.85.11-1/27.26.24.84.214.412.59.68.33.63.112.410.82.52.015.513.78.17.13/43.12.66.05.4215.813.910.59.231.627.821.118.37.97.022.219.73.93.227.724.720.117.813.73.16.35.7352.246.734.831.1104.493.369.662.226.123.365.258.711.19.480.572.944.439.61-1/21/24.03.49.48.34119.8108.379.972.0239.7216.6159.9143.959.954.1135.8123.624.220.8166.5152.476.469.03/44.63.99.68.66408.1368.3272.1245.4816.1736.6544.3490.8204.0184.2413.2367.177.365.7501.3441.0231.1208.515.34.59.99.18942.3860.3628.3572.71,884.71,720.61,256.71,145.5471.1430.1881.1811.2148.5122.31,061.9983.0400.2365.31-1/46.55.610.89.9101,856.91,758.21,238.01,172.23,713.83,516.52,476.02,344.3928.5879.11,675.41,594.9295.6264.42,010.41,920.6788.5746.623/47.66.616.715.2123,195.93,064.22,130.72,042.86,391.86,128.54,261.54,085.61,598.01,532.12,816.92,712.3517.0475.03,371.93,255.91,131.01,084.318.57.416.915.6144,545.94,376.13,030.72,917.49,091.88,752.26,061.45,834.82,273.02,188.13,809.33,681.0684.6640.04,171.54,043.71,654.61,592.81-1/410.08.817.616.5166,882.26,658.44,301.44,161.513,764.513,316.98,602.88,323.03,441.13,329.25,804.35,634.3967.6911.06,311.76,144.02,191.82,120.61-1/211.410.018.417.4189,906.59,621.76,054.05,879.919,813.119,243.412,108.011,759.84,953.34,810.88,396.58,179.21,432.61,363.09,081.38,868.12,804.52,723.83150.945.52013,707.513,353.89,366.89,125.127,415.026,707.618,733.618,250.26,853.86,676.911,701.311,429.42,026.41,938.012,634.112,368.23,492.53,402.41-1/420.217.951.846.42218,365.217,935.836,730.923,914.59,182.78,967.914,348.014,049.22,784.12,682.91-1/221.919.452.947.32423,995.623,482.414,664.014,350.347,991.146,964.729,327.928,700.711,997.811,471.220,647.220,249.73,451.03,313.022,189.421,802.95,094.74,985.7225.522.755.149.52630,644.830,041.061,289.640,054.615,322.415,020.523,912.323,493.24,014.33,884.12-1/229.526.358.652.63047,449.746,642.131,648.831,110.594,899.462,189.463,297.762,221.035,985.035,442.55,163.15,006.441-1/237.333.4108.097.73469,490.168,449.0110,260.534,745.034,224.552,836.752,135.9241.837.5110.499.93682,695.481,526.655,130.354,351.0176,155.6110,260.5108,702.141,347.740,763.362,945.162,157.57,010.86,811.52-1/246.641.8113.9103.242132,116.7130,520.288,077.887,013.466,054.865,260.194,207.093,209.010,936.310,666.7354.448.9119.7108.648198,322.0196,203.040,971.040,538.0145,443.0144,092.013,480.013,157.03-1/262.155.9125.9114.562-1/211.9100.6334.1301.83123.7111.3340.0307.43-1/2134.8121.5346.3313.74147.1132.8354.1321.290o Elbows:5175.3158.7375.7361.8V = P2D2A/883716.9655.53-1/2221.7201.8722.6661.0V = Volume4235.6215.0730.6668.7D = Inside diameter5269.6245.8753.0690.5A = Center to face distance6309.2280.9791.1719.4104385.9362.01,373.01,300.05428.3401.41,396.01,323.06476.8444.81,432.01,350.08586.0546.71,506.01,426.0180o Returns:125639.4606.42,318.02,224.0V = P2D2O/86697.7658.82,348.02,250.08827.0779.82,430.02,329.0V = Volume10993.8947.72,567.02,468.0D = Inside diameter1461,4961,4191,9921,915A = Center to center distance81,7381,6462,5872,488102,0411,9522,9222,816122,3822,2882,9763,0221662,6212,53282,3212,2123,3963,28345o Elbows:102,6562,5523,8033,701V = P2D2A/8123,0292,9204,8914,741143,2893,1755,0544,902V = Volume1884,3184,191D = Inside diameter103,4133,2914,8494,711A = Center to face distance123,8213,6954,9935,013144,1043,9736,1475,976164,5984,4587,1806,9852085,3605,214106,0105,855Full Size Outlet Tees:126,2486,0596,2046,221V = (PD2/2) (C + M/2) - D3/3146,9226,7177,6067,414167,9747,7478,8778,657V = Volume189,4049,15010,1639,916D = Inside diameter221012,02711,745C= Center to end of run1212,22611,944M = center to end of branch147,3337,12912,39412,110168,0557,84112,62012,337188,8488,62212,99512,710209,7119,4741,345,813,17224108,7018,519Pipe Caps:1214,97214,603V = (PD2/4) (E - t - D/12)1410,97910,745168,6378,41912,79512,526V = Volume189,4519,22114,62814,325D = Inside diameter2010,33410,09416,47716,141E = length2216,60616,287t = wall thickness261220,06219,6681420,23319,8371620,46520,0701820,84620,449Crosses:2021,31620,919V = (PD2/2) (C + M) - (2/3)D32221,88721,4872422,56522,164V = Volume301430,28329,768D = Inside diameter1630,52030,006C= Center to end of run1830,90830,392M = center to end of branch2031,38630,8692231,96431,4452432,65232,1312633,45832,9362834,10233,583Concentric & eccentric reducers:341644,38543,7241844,77944,1162045,26544,6002245,85145,1842446,54845,8792647,36446,6932848,01847,3513049,05848,3893250,24249,571361652,70151,9581853,09852,3532053,58752,8412254,17753,4292454,87854,1282655,70054,9472856,35955,6103030,53930,07157,40456,6533232,31431,83158,59457,8423434,17633,69859,94059,185422076,74675,8252277,17276,2532432,03131,56177,63976,7232633,61733,13278,89977,9722879,57478,6503037,05336,54080,63679,7103238,90238,37581,46280,5403440,84040,22982,34181,4253642,86642,31183,27682,3654822125,186123,8742485,14383,984125,667124,3582689,35488,163126,693125,37728127,344126,0313098,30597,047128,044126,73632103,044101,753129,561128,24534107,959106,635130,462129,15136113,050111,693131,419130,11338118,317116,927133,521132,20740123,760122,337134,710133,40242129,378127,923135,960134,66044135,173133,685138,742137,43246141,144139,622140,256138,954

&LArt Montemayor&RJune 02, 1999Rev: 0&CPage &P of &N&RElectronic FileName: &FWorkSheet: &AA

Torispherical Heads

F & D Partial VolumeVERTICAL TANK BOTTOM TORISPHERICAL HEAD VOLUME CALCULATIONTank Internal Diameter(3)D2,134mm =84.02inchesCrown RadiusRi2,134mm =84.02inches% Knuckle Radius6.55%Knuckle Radiusri139.8mm =5.50inchesb =D/2 - ri927.2mma =b Ri / (Ri - ri)992.2mmc =((Ri - ri)2 - b2)1,765.6mm =Sin-1 (a / Ri)0.484radians27.7x =Ri Cos - c123.7mm =4.87inchesz =Ri - c - x244.7mm =9.63inchesh =x + z368.4mm =14.51inchesApprox. Head Volume =p / 6 * z (3a2 + z2)+ p / 3 * x ((D/2)2 + (D/2)a + a2)approximate calculation for knuckle section=386.1+412.3=798.4litres =210.91US galsVolume of partially filled Torispherical head:Level in End dish:Liquid Height "h" (mm)Sector AreaKnuckle AreaTotal Head Volume"z""r"Volume (1)"x""r"Volume (2)litres%0%00000992000%10%373739590992091%20%74745563609920365%30%11111167880099208010%40%1471477791420992014218%50%1841848672210992022128%60%2212219463160992031640%70%258245992386131,0004142754%80%295245992386501,02216054668%90%332245992386871,04528366984%100%368.452459923861241,067412798.4100%Notes:(1)Sector volume =PI / 6 * "z" (3 * "r"2 + "z"2)(2)Knuckle volume =PI / 3 * "x" ("r"2 + "r" * a + a2)(3)Torispherical (also called ASME F&D) heads are designed and fabricated in the USA on the basis of using theoutside diameter as their nominal diameter.

&LArt Montemayor&CVolume of a Partially Filled Torispherical Bottom Head&RJuly 20, 2003Rev: 1&CPage &P of &N&RFileName: &FWorkSheet: &AabxRirichz

F & D Partial Volume

Level of FillVolume of FillVertical Torispherical Tank Head Volume

Vertical F&D Head VolumeI.D. =84.0inchesk =Knuckle-Radius (kD)0.06inchesf =dish-radius parameter (fD)1kD =5.04inchesfD =84inchesa =0.487radiansCos a =0.883683Sin a =0.468085a1 =9.7706inchesAcos a =1.062004Asin a =0.508792a2 =4.4538inchesD1 =78.6382978723s =5.565583703t =8.90758.9075u(h) =5.5831947325Limits of the Equation0.00h9.779.77h14.2214.22hToph =15.35379inh =15.35379inh =24inV =58,420in3 =252.90GallonsV =54,255in3 =234.87GallonsV =102,183in3 =442.35Gallons

&LArt Montemayor&CVertical Tank Bottom Torispherical Head Volume&RSeptember 30, 2004Rev: 0&CPage &P of &N&RFileName: &FWorkSheet: &A

Reference ArticleThe following article appeared in "Chemical Processing" magazine on Novermber 17, 2002; pp. 46-50:Computing Fluid Tank VolumesUpdated equations allow engineers to calculate the fluid volumes of many tanks quickly and accuratelyBy Dan Jones, Ph.D., P.E.Calculating fluid volume in a horizontal or vertical cylindrical tank or elliptical tank can be complicated, dependingon fluid height and the shape of the heads (ends) of a horizontal tank or the bottom of a vertical tank. Exactequations now are available for several commonly encountered tank shapes. These equations allow rapid andaccurate fluid-volume calculations.All volume equations give fluid volumes in cubic units from tank dimensions in consistent linear units. Allvariables defining tank shapes required for tank volume calculations are defined in the Variables and Definitionssidebar. Fig. 1 and Fig. 2 graphically illustrate horizontal tank variables, and Fig. 3 and Fig. 4 graphicallyillustrate vertical tank variables.Exact fluid volumes in elliptical horizontal or vertical tanks can be determined by calculating the fluid volumes ofappropriate cylindrical horizontal or vertical tanks using the equations described above, and then by adjustingthose results using appropriate correction formulas.Horizontal cylindrical tanksFluid volume as a function of fluid height can be calculated for a horizontal cylindrical tank with either conical,ellipsoidal, guppy, spherical or torispherical heads where the fluid height, h, is measured from the tank bottom tothe fluid surface. A guppy head is a conical head with its apex level with the top of the cylindrical section of thetank, as shown in Fig. 1. A torispherical head is an American Society of Mechanical Engineers (ASME-type)head defined by a knuckle-radius parameter, k, and a dish-radius parameter, f, as shown in Fig. 2.An ellipsoidal head must be exactly half of an ellipsoid of revolution; only a hemi ellipsoid is valid - no segmentof an ellipsoid will work, as is true in the case of a spherical head that can be a spherical segment. For aspherical head, |a| < R, where R is the radius of the cylindrical tank body. For concave conical, ellipsoidal,guppy, spherical or torispherical heads, |a| < L/2.Figure 1. Parameters for Horizontal Cylindrical Tanks with Conical, Ellipsoidal, Guppy or Spherical Heads1.Both heads of a tank must be identical. Above diagram is for definition of parameters only.2.Cylindrical tube of diameter D (D > 0), radius R (R > 0) and length L (L > 0)3.For spherical head of radius r, r > R and |a| < R4.For convex head other than spherical, 0 < a < a , for concave head a < 05.L > 0 for a > 0, L > 2|a| for a < 06.Ellipsoidal head must be exactly half of an ellipsoid of revolution7.0 < h < D.Both heads of a horizontal cylindrical tank must be identical for the equations to work; i.e., if one head is conical,the other must be conical with the same dimensions. However, the equations can be combined to calculate thefluid volume of a horizontal tank 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 otherend, calculate fluid volumes of two tanks, one with conical heads and the other with ellipsoidal heads, andaverage the results to get the desired fluid volume. The heads of a horizontal tank can be flat (a = 0), convex(a > 0) or concave (a < 0).The following variables must be within the ranges stated:|a| < R for spherical heads.|a| < L/2 for concave ends.0 < h < 2R for all tanks.f > 0.5 for torispherical heads.0 < k < 0.5 for torispherical heads.D > 0.L > 0.Variables used in Volumetric Equations and their DefinitionsaThis is the distance a horizontal tank's heads extend beyond (a>0) or into (a