vessel volumes

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Art Montemayor Vessel Design Tips August 21, 2000 Rev: 2(05-05-03) Page 1 of 83 Electronic FileName: document.xls WorkSheet: Notes & Experience The following are some guidelines and experienced hints for the design and utiliza This information is never taught nor discussed in University courses or academic c historically expected that graduate engineers will learn this information using th 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 in sizes up to 42" and in various pipe schedule thicknesses. b. Standard seamless pipe. This is basic material that can be readily found make this your first priority in selecting the vessel shell because of th any plate rolling, longitudinal weld seam, and reducing the possibility o option should be rejected only if required alloy, wall thickness, or diam 2) Handbook Publishing Inc.; P.O. Box 35365; Tulsa, OK 74153. This is probably t practical engineering book ever published in the USA. It clearly belongs on ev engineer's desk. Study it thoroughly and your project problems will start to f 3) Ellipsoidal 2:1 heads have, by definition, 50% of the volumetric capacity of a same internal diameter. diameter. These type of heads are used in preference to ASME Flanged and Dished heads for range of 100 psig and for most vessels designed for pressures over 200 psig. T 4) ASME F&D (also called Torispherical) heads are designed and fabricated in the U Flanged and dished heads are inherently shallower (smaller IDD) than comparable These heads (like the ellipsoidal) are formed from a flat plate into a dished s the "crown" radius or radius of the dish and the inside-corner radius, sometime "knuckle" radius. Because of the relative shallow dish curvature, ASME F&D hea higher localized stresses at the knuckle radius as compared to the ellipsoidal of these heads is increased by forming the head so that the knuckle radius is m times the plate thickness. For code construction, the radius should in no case inside diameter. ASME F&D heads are used for pressure vessels in the general range of from 15 to Although these heads may be used for higher pressures, for pressures in excess more economical to use an ellipsoidal type. 5) The straight flange that forms part of each vessel head is part of the cylindri be accounted for as such in calculating the vessel volume. These flanges vary head 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 other welding, cutting or grinding. The need to locate a nozzle, insulation ri near the knuckle radius should be consulted with a mechanical or fabrication en 7) Be aware of the fact that the outside diameter of the cylindrical section may b head if a flush fit is required between the two inside diameters. This occurs thickness for a given design pressure is usually less than for the correspondin This is especially true in the case of Hemispherical heads. Own a copy of Eugene Megyesy's "Pressure Vessel Handbook " as published by Press Ellipsoidal heads are designed and fabricated on the basis of using the inside (IDD) is defined as half of the minor axis and is equal to 1/4 of the inside di the outside diameter as their nominal diameter.

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Page 1: Vessel Volumes

Art Montemayor Vessel Design Tips August 21, 2000Rev: 2(05-05-03)

Page 1 of 61 Electronic FileName: document.xlsWorkSheet: Notes & Experience

The 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 been historically 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 eliminating any plate rolling, longitudinal weld seam, and reducing the possibility of stress relief. This option should be rejected only if required alloy, wall thickness, or diameter is not available.

2)Handbook Publishing Inc.; P.O. Box 35365; Tulsa, OK 74153. This is probably the most useful and practical engineering book ever published in the USA. It clearly belongs on every process plant engineer'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.

diameter.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

4) ASME F&D (also called Torispherical) heads are designed and fabricated in the USA on the basis of using

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 to higher 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 be more 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 the head 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 doing other welding, cutting or grinding. The need to locate a nozzle, insulation ring, clips or other item near 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 the head if a flush fit is required between the two inside diameters. This occurs because the required head thickness for a given design pressure is usually less than for the corresponding cylindrical section.This is especially true in the case of Hemispherical heads.

Own a copy of Eugene Megyesy's "Pressure Vessel Handbook" as published by Pressure Vessel

Ellipsoidal heads are designed and fabricated on the basis of using the inside diameter as their nominal

(IDD) is defined as half of the minor axis and is equal to 1/4 of the inside diameter of the head.

the outside diameter as their nominal diameter.

Page 2: Vessel Volumes

Art Montemayor Vessel Design Tips August 21, 2000Rev: 2(05-05-03)

Page 2 of 61 Electronic FileName: document.xlsWorkSheet: Notes & Experience

8) Hemispherical heads are the strongest of the formed heads for a given thickness. A sphere is the strongest 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 as trays, 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 should take these factors into consideration even though they normally are not considered while doing process calculations and simulations. Often, if not in the majority of cases, these factors and items are the controlling 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 process operations. 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 level instrument nozzle, liquid temperature probe, sample(s) probe, top head vents, critical high and low level probes, etc. Process Chemical Engineers are the best qualified to identify this need and specify the location 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 ultimate insulation 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 vessel modifications 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 experience in Chemical Engineering. Users will probably find it useful for carrying out day-to-day process plant projects 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 to indicate 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 for years in notes, 3-ring binders, between pages of text books, in formal calculations, etc. Thanks to God for giving me the good common sense to save and document this information and for giving us the digital computer and a spreadsheet to organize and distribute it for use and exploitation by others. I hope this helps others - especially young, striving, and determined engineers who earnestly want to do a successful and safe project.

Arthur Montemayor

Page 3: Vessel Volumes

Art Montemayor Partially-Filled Horizontal Vessels May 15, 1998Rev:1(01/22/00)

Page 3 of 61 Electronic FileName: document.xlsWorkSheet: Partial-Filled HorizontalVessel

VOLUMES IN PARTIALLY FILLED HORIZONTAL VESSELS

Name: General Purpose Tank

Item No: T-C-15 Vessel Volume

Flat Heads Unit

Case: Partial Vol

3,381,604 3,901,853

1,956.95 2,258.02

Tank Inside Dia. in = 126.00 14,638.9 gal 16,891.1

Cylindrical Length, in = 276

Liquid Height, in = 120.00 Hemi Heads Unit F & D Heads

L/D = 2.2

H/D = 0.9524 4,422,102 3,595,708

2,559.09 2,080.8519,143.3 gal 15,565.8

Cylindrical radius = r = 63.00 in.Chord Length = CL = 53.7 in.

Segment Area = Aseg = 12,252 U. S. Gallons

Cylindrical Volume = = 3,381,604 14,638.9

F & Dished Volume = = 214,104 926.9

Ellipsoidal Volume = = 520,249 2,252.2

Spherical Volume = = 1,040,498 4,504.3

2:1 Ellip. Heads

in3

ft3

in3

ft3

in2

Vcyl in3

VFD in3

Vell in3

Vsph in3

Steps:(1) Enter the required information in the YELLOW cells;(2) The calculated results appear in RED numbers.

Page 4: Vessel Volumes

Art Montemayor Horizontal Storage TankVolume Calibration

November 11, 1999Rev: 1(03/12/00)

Page 4 of 61 Electronic FileName: document.xlsWorkSheet: Horizontal Tank Strapping

CALIBRATION DATA FOR HORIZONTAL TANK WITH FORMED HEADS

Tank Inside Diameter (ID) inches 120.000 = 10.0000 ftTank length, tan/tan feet 23.000 = 276 inches

Tank HeadType Pressure 1) Std. dish (non-pressure) < 15 psig Note: Place an "x" in only one of the 2) Torispherical (ASME F&D) < 200 psig 5 head options available. If more than 3) Ellipsoidal (2:1) > 200 psig x one option contains an "x", the 4) Ellipsoidal (non-std) Varies program will use the first one it finds. 5) Hemispherical To Suit

Head type selected: 2:1 Ellipsoidal Head Volume = 130.90 cu.ft.Inside depth of head (IDD): inches 20 NOT REQUIRED FOR THIS HEAD TYPE

Head thickness: inches 0.375 NOT REQUIRED FOR THIS HEAD TYPENumber of calibration increments: 120.000 (max 200)

Calibration curve for 120.0 in. dia tank, 23.000 ft tan/tan, 2:1 Ellipsoidal heads

Steps:(1) Enter the required information in the YELLOW cells;(2) The calculated results appear in RED numbers.

Page 5: Vessel Volumes

Art Montemayor Horizontal Storage TankVolume Calibration

November 11, 1999Rev: 1(03/12/00)

Page 5 of 61 Electronic FileName: document.xlsWorkSheet: Horizontal Tank Strapping

Liquid Depth Liquid Volume Content

Inches Centimeters US Gals Liters1 2.54 2.38 17.9 682 5.08 6.78 50.9 1923 7.62 12.51 93.9 3554 10.16 19.33 145.0 5485 12.70 27.08 203.1 7686 15.24 35.67 267.5 1,0117 17.78 45.01 337.6 1,2768 20.32 55.05 412.9 1,5619 22.86 65.75 493.1 1,864

10 25.40 77.05 577.9 2,18411 27.94 88.93 666.9 2,52112 30.48 101.34 760.1 2,87313 33.02 114.27 857.1 3,24014 35.56 127.69 957.7 3,62015 38.10 141.58 1,061.8 4,01416 40.64 155.90 1,169.3 4,42017 43.18 170.66 1,279.9 4,83818 45.72 185.82 1,393.6 5,26819 48.26 201.36 1,510.2 5,70920 50.80 217.29 1,629.7 6,16021 53.34 233.57 1,751.8 6,62222 55.88 250.19 1,876.4 7,09323 58.42 267.15 2,003.6 7,57424 60.96 284.42 2,133.2 8,06325 63.50 302.00 2,265.0 8,56226 66.04 319.87 2,399.0 9,06827 68.58 338.02 2,535.2 9,58328 71.12 356.45 2,673.4 10,10529 73.66 375.13 2,813.5 10,63530 76.20 394.06 2,955.5 11,17231 78.74 413.23 3,099.3 11,71532 81.28 432.63 3,244.8 12,26533 83.82 452.25 3,391.9 12,82134 86.36 472.08 3,540.6 13,38435 88.90 492.11 3,690.9 13,95136 91.44 512.34 3,842.5 14,52537 93.98 532.74 3,995.6 15,10338 96.52 553.32 4,149.9 15,68739 99.06 574.07 4,305.5 16,27540 101.60 594.97 4,462.3 16,86741 104.14 616.02 4,620.2 17,46442 106.68 637.22 4,779.1 18,06543 109.22 658.54 4,939.1 18,67044 111.76 680.00 5,100.0 19,27845 114.30 701.57 5,261.7 19,88946 116.84 723.25 5,424.3 20,50447 119.38 745.03 5,587.7 21,12248 121.92 766.90 5,751.8 21,74249 124.46 788.87 5,916.5 22,36450 127.00 810.91 6,081.8 22,989

Ft3

Page 6: Vessel Volumes

Art Montemayor Horizontal Storage TankVolume Calibration

November 11, 1999Rev: 1(03/12/00)

Page 6 of 61 Electronic FileName: document.xlsWorkSheet: Horizontal Tank Strapping

51 129.54 833.03 6,247.7 23,61652 132.08 855.20 6,414.0 24,24553 134.62 877.44 6,580.8 24,87554 137.16 899.73 6,748.0 25,50755 139.70 922.06 6,915.5 26,14056 142.24 944.43 7,083.2 26,77557 144.78 966.82 7,251.2 27,40958 147.32 989.24 7,419.3 28,04559 149.86 1,011.67 7,587.5 28,68160 152.40 1,034.11 7,755.8 29,31761 154.94 1,056.55 7,924.1 29,95362 157.48 1,078.98 8,092.3 30,58963 160.02 1,101.39 8,260.4 31,22464 162.56 1,123.79 8,428.4 31,85965 165.10 1,146.15 8,596.2 32,49366 167.64 1,168.49 8,763.6 33,12767 170.18 1,190.77 8,930.8 33,75868 172.72 1,213.01 9,097.6 34,38969 175.26 1,235.19 9,263.9 35,01870 177.80 1,257.31 9,429.8 35,64571 180.34 1,279.35 9,595.1 36,27072 182.88 1,301.31 9,759.8 36,89273 185.42 1,323.19 9,923.9 37,51274 187.96 1,344.97 10,087.3 38,13075 190.50 1,366.65 10,249.9 38,74476 193.04 1,388.22 10,411.6 39,35677 195.58 1,409.67 10,572.5 39,96478 198.12 1,431.00 10,732.5 40,56979 200.66 1,452.19 10,891.4 41,17080 203.20 1,473.24 11,049.3 41,76681 205.74 1,494.15 11,206.1 42,35982 208.28 1,514.89 11,361.7 42,94783 210.82 1,535.47 11,516.1 43,53184 213.36 1,555.88 11,669.1 44,10985 215.90 1,576.10 11,820.8 44,68286 218.44 1,596.13 11,971.0 45,25087 220.98 1,615.96 12,119.7 45,81388 223.52 1,635.58 12,266.9 46,36989 226.06 1,654.98 12,412.4 46,91990 228.60 1,674.15 12,556.1 47,46291 231.14 1,693.08 12,698.1 47,99992 233.68 1,711.77 12,838.3 48,52993 236.22 1,730.19 12,976.4 49,05194 238.76 1,748.34 13,112.6 49,56695 241.30 1,766.22 13,246.6 50,07296 243.84 1,783.79 13,378.4 50,57197 246.38 1,801.07 13,508.0 51,06098 248.92 1,818.02 13,635.2 51,54199 251.46 1,834.65 13,759.9 52,012

100 254.00 1,850.93 13,882.0 52,474101 256.54 1,866.85 14,001.4 52,925102 259.08 1,882.40 14,118.0 53,366103 261.62 1,897.56 14,231.7 53,796

Page 7: Vessel Volumes

Art Montemayor Horizontal Storage TankVolume Calibration

November 11, 1999Rev: 1(03/12/00)

Page 7 of 61 Electronic FileName: document.xlsWorkSheet: Horizontal Tank Strapping

104 264.16 1,912.31 14,342.3 54,214105 266.70 1,926.64 14,449.8 54,620106 269.24 1,940.52 14,553.9 55,014107 271.78 1,953.94 14,654.6 55,394108 274.32 1,966.87 14,751.5 55,761109 276.86 1,979.29 14,844.7 56,113110 279.40 1,991.16 14,933.7 56,450111 281.94 2,002.47 15,018.5 56,770112 284.48 2,013.16 15,098.7 57,073113 287.02 2,023.20 15,174.0 57,358114 289.56 2,032.55 15,244.1 57,623115 292.10 2,041.14 15,308.5 57,866116 294.64 2,048.89 15,366.6 58,086117 297.18 2,055.70 15,417.8 58,279118 299.72 2,061.43 15,460.8 58,442119 302.26 2,065.83 15,493.8 58,566120 304.80 2,068.22 15,511.6 58,634

Page 8: Vessel Volumes

Art Montemayor Horizontal Cylindrical TankPartial Volume Determination

May 5, 2001Rev: 0

Page 8 of 61 Electronic FileName: document.xlsWorkSheet: Partial Horizontal

Horizontal Cylindrical Tank with Ellipsoidal or Hemispherical Heads

Total tank volume = (Total volume in two heads) + (Total volume in cylindrical section)

=

2b/D

Ze =

Zc =

Partial tank volume =

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) D

1/2

( 1/6 p K1 D3 ) + ( 1/4 p D2 L )

K1 =

H1/D

H1/D

( 1/6 p K1 D3 ) ([f(Ze)]) + ( 1/4 p D2 L ) ([fZc)])

K1 =

H1

D/2 D/2a

L

D

b b

f ( Zc )=(α −sin (α ) cos (α )π )

f ( Ze )=−( H1

D )2

(−3+2 H1

D )α=2 A tan( H1

√(2 H1D2 )− H1

2 ) α is in radians

Page 9: Vessel Volumes

Art Montemayor Horizontal Cylindrical TankPartial Volume Determination

May 5, 2001Rev: 0

Page 9 of 61 Electronic FileName: document.xlsWorkSheet: Partial Horizontal

Oct 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

Page 10: Vessel Volumes

Art Montemayor Vertical Cylindrical TankPartial Volume Determination

May 05, 2001Rev: 0

Page 10 of 61 Electronic FileName: document.xlsWorkSheet: Partial Vertical

Vertical Cylindrical Tank with Ellipsoidal or Hemispherical Heads

Total tank volume = (Total volume in two heads) + (Total volume in cylindrical section)

=

Partial tank volume =

2b/D

Ze =f(Ze) = Ellipsoidal coefficient (from Doolittle tables)

or,

( 1/6 p K1 D3 ) + ( 1/4 p D2 L )

( 1/6 p K1 D3 ) ([f(Ze)]) + ( 1/4 p D2 H3)

K1 =

(H1 + H2)/K1D

L

bH1H1

b

H3

D

H3

H2

f ( Ze )=− ( H1+H2

2b )2 (−3+(H1+H2

b ))

Page 11: Vessel Volumes

Art Montemayor Regression of Doolittle Partial Volume Coefficient May 15, 1998Rev: 0

Page 11 of 61 Electronic FileName: document.xlsWorkSheet: Partial Cylind. Vol.

0.000000 0.0000000.050000 0.0186920.100000 0.0520440.150000 0.0940610.200000 0.1423780.250000 0.1955010.300000 0.2523150.350000 0.3119180.400000 0.3735300.450000 0.4364450.500000 0.5000000.550000 0.5635550.600000 0.6264700.650000 0.6880820.700000 0.7476850.750000 0.8044990.800000 0.8576220.850000 0.9059390.900000 0.9479560.950000 0.9813081.000000 1.000000

Zc f(Zc)

0.000000 0.200000 0.400000 0.600000 0.800000 1.000000 1.2000000.000000

0.200000

0.400000

0.600000

0.800000

1.000000

1.200000

f(x) = − 1.14403 x³ + 1.71604 x² + 0.436493 x − 0.00425379R² = 0.999951432394297

Coefficients for Partial Volumes of Horizontal Cylinders

H/D = Zc

f(Z

c)

Data Source:

NGPSA Engineering Data Book9th Edition; 1972; p. 13-7

Page 12: Vessel Volumes

Art Montemayor Doolittle Equation for Parially-Filled Vessel Heads May 27, 1998Rev: 0

Page 12 of 61 Electronic FileName: document.xlsWorkSheet: Hds Partial Vol.

H/D Vol. Fraction0.02 0.00120.04 0.00470.06 0.01040.08 0.01820.10 0.02800.12 0.03970.14 0.05330.16 0.06860.18 0.08550.20 0.10400.22 0.12390.24 0.14510.26 0.16760.28 0.19130.30 0.21600.32 0.24200.34 0.26800.36 0.29500.38 0.32300.40 0.35200.42 0.38100.44 0.41000.46 0.44000.48 0.47000.50 0.50000.52 0.53000.54 0.56000.56 0.59000.58 0.61900.60 0.64800.62 0.67700.64 0.70500.66 0.73200.68 0.75800.70 0.78400.72 0.80870.74 0.83240.76 0.85490.78 0.87610.80 0.89600.82 0.91450.84 0.93140.86 0.94670.88 0.96030.90 0.97200.92 0.98180.94 0.98960.96 0.99530.98 0.99881.00 1.0000

0.00 0.20 0.40 0.60 0.80 1.000.00

0.20

0.40

0.60

0.80

1.00

1.20

f(x) = − 2.00261 x³ + 3.00397 x² − 0.00155356 x + 0.000108007R² = 0.999999755835569

Volume Fraction of Horizontal Vessel Heads

Liquid Depth/Head ID, (H/D)

Vo

lum

etri

c F

ract

ion

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.

Page 13: Vessel Volumes

Art Montemayor Doolittle Equation for Parially-Filled Vessel Heads May 27, 1998Rev: 0

Page 13 of 61 Electronic FileName: document.xlsWorkSheet: Hds Partial Vol.

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.

Page 14: Vessel Volumes

Art Montemayor Doolittle Equation for Parially-Filled Vessel Heads May 27, 1998Rev: 0

Page 14 of 61 Electronic FileName: document.xlsWorkSheet: Hds Partial Vol.

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.

Page 15: Vessel Volumes

Art Montemayor Doolittle Equation for Parially-Filled Vessel Heads May 27, 1998Rev: 0

Page 15 of 61 Electronic FileName: document.xlsWorkSheet: Hds Partial Vol.

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' orientation Vertical Vessel Heads' orientation

H

H

Page 16: Vessel Volumes

Art Montemayor Regression of Doolittle Partial Volume Coefficient May 15, 1998Rev: 1(02/25/01)

Page 16 of 61 Electronic FileName: document.xlsWorkSheet: Partial Ellip. Vol.

0.000 0.0000000.050 0.0072500.100 0.0280000.150 0.0607500.200 0.1040000.250 0.1562500.300 0.2160000.350 0.2817500.400 0.3520000.450 0.4252500.500 0.5000000.550 0.5747500.600 0.6480000.650 0.7182500.700 0.7840000.750 0.8437500.800 0.8960000.850 0.9392500.900 0.9720000.950 0.9927501.000 1.000000

Ze f(Ze)

0.000 0.200 0.400 0.600 0.800 1.000 1.2000.000000

0.200000

0.400000

0.600000

0.800000

1.000000

1.200000

f(x) = − 2 x³ + 3.00000000000001 x² − 2.56061827796986E-15 x + 3.11227034484436E-16R² = 1

Coefficients for Partial Volumes in Ellipsoids & Spheres

H/D = Ze

f(Z

e)

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!!

Page 17: Vessel Volumes

Art Montemayor Pressure Vessel Heads August 04, 1998Rev:1(08/21/00)

Page 17 of 61 Electronic FileName: document.xlsWorkSheet: Hds Vol & Surf Area

Internal DiameterInches Ft Hemisphere Ellipsoidal ASME F&D Standard F&D Hemisphere Ellipsoidal ASME F&D Standard F&D

12 1.0000 0.26 0.13 0.08 0.05 1.57 1.00 0.93 0.7914 1.1667 0.42 0.21 0.13 0.09 2.14 1.36 1.26 1.0716 1.3333 0.62 0.31 0.20 0.13 2.79 1.78 1.65 1.4018 1.5000 0.88 0.44 0.28 0.18 3.53 2.25 2.09 1.7720 1.6667 1.21 0.61 0.39 0.25 4.36 2.78 2.58 2.1822 1.8333 1.61 0.81 0.51 0.33 5.28 3.36 3.12 2.6424 2.0000 2.09 1.05 0.67 0.43 6.28 4.00 3.71 3.1426 2.1667 2.66 1.33 0.85 0.55 7.37 4.70 4.36 3.6928 2.3333 3.33 1.66 1.06 0.68 8.55 5.45 5.06 4.2830 2.5000 4.09 2.05 1.30 0.84 9.82 6.26 5.80 4.9132 2.6667 4.96 2.48 1.58 1.02 11.17 7.12 6.60 5.5934 2.8333 5.95 2.98 1.90 1.22 12.61 8.04 7.45 6.3136 3.0000 7.07 3.53 2.25 1.45 14.14 9.01 8.36 7.0738 3.1667 8.31 4.16 2.65 1.70 15.75 10.04 9.31 7.8840 3.3333 9.70 4.85 3.09 1.99 17.45 11.12 10.32 8.7342 3.5000 11.22 5.61 3.58 2.30 19.24 12.26 11.38 9.6248 4.0000 16.76 8.38 5.34 3.43 25.13 16.02 14.86 12.5754 4.5000 23.86 11.93 7.61 4.88 31.81 20.27 18.80 15.9060 5.0000 32.72 16.36 10.44 6.70 39.27 25.03 23.22 19.6366 5.5000 43.56 21.78 13.89 8.92 47.52 30.28 28.09 23.7672 6.0000 56.55 28.27 18.04 11.58 56.55 36.04 33.43 28.2778 6.5000 71.90 35.95 22.93 14.72 66.37 42.30 39.23 33.1884 7.0000 89.80 44.90 28.64 18.38 76.97 49.05 45.50 38.4890 7.5000 110.45 55.22 35.23 22.61 88.36 56.31 52.23 44.1896 8.0000 134.04 67.02 42.75 27.44 100.53 64.07 59.43 50.27

102 8.5000 160.78 80.39 51.28 32.92 113.49 72.33 67.09 56.75108 9.0000 190.85 95.43 60.87 39.07 127.23 81.09 75.22 63.62114 9.5000 224.46 112.23 71.59 45.96 141.76 90.35 83.81 70.88120 10.0000 261.80 130.90 83.50 53.60 157.08 100.11 92.86 78.54126 10.5000 303.07 151.53 96.66 62.05 173.18 110.37 102.38 86.59132 11.0000 348.45 174.23 111.14 71.34 190.07 121.13 112.36 95.03138 11.5000 398.16 199.08 126.99 81.52 207.74 132.40 122.81 103.87144 12.0000 452.39 226.19 144.29 92.62 226.19 144.16 133.72 113.10150 12.5000 511.33 255.66 163.09 104.69 245.44 156.42 145.09 122.72156 13.0000 575.17 287.59 183.45 117.76 265.46 169.19 156.93 132.73162 13.5000 644.12 322.06 205.44 131.88 286.28 182.45 169.24 143.14

Volume of one head, Ft3 Internal Surface Area of one head, Ft2

Page 18: Vessel Volumes

Art Montemayor Pressure Vessel Heads August 04, 1998Rev:1(08/21/00)

Page 18 of 61 Electronic FileName: document.xlsWorkSheet: Hds Vol & Surf Area

168 14.0000 718.38 359.19 229.12 147.08 307.88 196.22 182.01 153.94174 14.5000 798.13 399.06 254.56 163.41 330.26 210.48 195.24 165.13180 15.0000 883.57 441.79 281.81 180.90 353.43 225.25 208.94 176.72186 15.5000 974.91 487.45 310.94 199.60 377.38 240.52 223.10 188.69192 16.0000 1,072.33 536.17 342.02 219.55 402.12 256.28 237.72 201.06198 16.5000 1,176.04 588.02 375.09 240.78 427.65 272.55 252.81 213.83204 17.0000 1,286.22 643.11 410.24 263.34 453.96 289.32 268.37 226.98210 17.5000 1,403.08 701.54 447.51 287.26 481.06 306.59 284.38 240.53216 18.0000 1,526.81 763.41 486.97 312.60 508.94 324.36 300.87 254.47222 18.5000 1,657.62 828.81 528.69 339.38 537.61 342.63 317.81 268.80228 19.0000 1,795.68 897.84 572.73 367.64 567.06 361.40 335.22 283.53234 19.5000 1,941.21 970.60 619.14 397.44 597.30 380.67 353.10 298.65240 20.0000 2,094.40 1,047.20 668.00 428.80 628.32 400.44 371.44 314.16

Note: 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 the actual external surface area.

References and Sources:(1) Pressure Vessel Handbook; Eugene F. Megyesy; 8th Edition; Pressure Vessel Handbook

Publishing, Inc.(2) Process Vessel Design; L.E. Brownell & E.H. Young; John Wiley & Sons; N.Y.; 1959

(3) A. Montemayor personal files

Page 19: Vessel Volumes

Art Montemayor Mfr's Hds' Vol September 12, 1997Rev 0

Page 19 of 61 Electronic FileName: document.xlsWorkSheet: Mfr's Hds' Vol

Head Volume in Cubic Feet Head Volume in U.S. GallonsEllipsoidal ASME F&D Hemispherical Dished Ellipsoidal ASME F&D Hemispherical Dished

1.00 0.131 0.082 0.262 0.053 0.980 0.613 1.960 0.3961.50 0.442 0.277 0.884 0.182 3.306 2.072 6.613 1.3612.00 1.047 0.656 2.095 0.430 7.832 4.907 15.672 3.2172.50 2.045 1.370 4.091 0.842 15.298 10.248 30.603 6.2993.00 3.535 2.216 7.069 1.454 26.444 16.577 52.880 10.8773.50 5.613 3.692 11.225 2.310 41.988 27.618 83.969 17.2804.00 8.378 5.255 16.756 3.448 62.672 39.310 125.344 25.7934.50 11.928 7.767 23.857 4.909 89.228 58.101 178.463 36.7225.00 16.364 10.264 32.725 6.733 122.411 76.780 244.800 50.3665.50 21.779 13.803 43.557 8.963 162.918 103.254 325.829 67.0486.00 28.276 18.072 56.548 11.636 211.519 135.188 423.008 87.0436.50 35.951 22.351 71.896 14.794 268.932 167.197 537.819 110.6677.00 44.902 29.081 89.797 18.477 335.890 217.541 671.728 138.2187.50 55.226 34.903 110.447 22.727 413.119 261.093 826.201 170.0108.00 67.025 43.239 134.041 27.582 501.382 323.450 1,002.696 206.3288.50 80.394 50.764 160.778 33.083 601.389 379.741 1,202.703 247.4789.00 95.432 59.098 190.852 39.271 713.881 442.084 1,427.672 293.7689.50 112.237 70.821 224.460 46.188 839.591 529.778 1,679.078 345.510

10.00 130.908 81.172 261.799 53.871 979.260 607.209 1,958.393 402.983

Diameter ft

Data source:

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

Page 20: Vessel Volumes

Art Montemayor Ellipsoidal Curve Fit September 12, 1997Rev 0

Page 20 of 61 Electronic FileName: document.xlsWorkSheet: Ellipsoidal Curve Fit

I. D., inches Vol. Gallons12 0.9818 3.3124 7.8330 15.3036 26.4442 41.9948 62.6754 89.2360 122.4166 162.9272 211.5278 268.9384 335.8990 413.1296 501.38

102 601.39108 713.88114 839.59120 979.26126 1133.61132 1303.39138 1489.33144 1692.16150 1912.61156 2151.43162 2409.34168 2687.08174 2985.39180 3304.99186 3646.63192 4011.04198 4398.95 Ellipsoidal Head Inside Diameter = 126 inches204 4811.09

210 5248.21 Volume of Single Ellipsodial Head = 1133.61 Gallons = 151.541216 5711.03222 6200.29228 6716.73234 7261.07240 7834.06

Ft3

0 50 100 150 200 250 300

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

f(x) = 0.000567137020246092 x^2.99984269005698R² = 0.999999987165382

2:1 Ellipsoidal Head Volume

Inside Diameter, inches

Vo

lum

e,

ga

llo

ns

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)

Page 21: Vessel Volumes

Art Montemayor 2:1 Ellipsoidal Heads May 21, 2003Rev: 1

Page 21 of 61 FileName: document.xlsWorksheet: Ellipsoidal Heads

Inches 100.80 Approximate area for nozzle attachment

Start of Knuckle Radius mm 2560

Inside DepthKnuckle Radius (= I.D./4)

Inches 21.76 31.5 Inches

mm 553 800 mm

41.24 Inches Dish RadiusNote: 113.97 Inches

Verify all dimension 1047 mm 2895 mmwith vendor drawings

126 Inches

3200.4 mm

NOTE: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.

Tangent Line

StraightFlange(Varies)2" Nom.51mm

2:1 Elliptical Head

Key In the Head I.D.

G25
Enter the Ellipsoidal Head's ID in Inches
Page 22: Vessel Volumes

Art Montemayor ASME F&D Curve Fit September 12, 1997Rev 0

Page 22 of 61 Electronic File: document.xlsWorkSheet: ASME F&D Curve Fit

I. D., inches Volume, gal.12 0.6118 2.0724 4.9130 10.2536 16.5842 27.6248 39.3154 58.1060 76.7866 103.2572 135.1978 167.2084 217.5490 261.0996 323.45

102 379.74108 442.08114 529.78120 607.21126 714.90132 809.04138 934.15144 1,015.27150 1,227.02156 1,361.28162 1,504.82168 1,712.89174 1,879.89180 2,057.21186 2,312.53192 2,515.83 ASME F&D Head Inside Diameter = 84 inches198 2,730.51

204 3,078.42 Volume of Single ASME F&D Head = 205.29 Gallons = 27.443210 3,324.02216 3,582.12222 3,853.00228 4,187.61234 4,700.90240 5,025.88

Ft3

0 50 100 150 200 250 300

0

1,000

2,000

3,000

4,000

5,000

6,000

f(x) = 0.000365045341166082 x^2.99577371837685R² = 0.999929547908438

ASME F&D HEAD VOLUME

Inside Diameter, inches

Vo

lum

e,

ga

llo

ns

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)

Page 23: Vessel Volumes

Art Montemayor ASME Flanged and Dished Heads May21, 2003Rev: 0

Page 23 of 61 FileName: document.xlsWorksheet: ASME F&D Heads

All Dimensions

Verify all dimensionwith vendor drawings

NOTE: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 IDD

O.D "T" "R1" "R2" IDD ASME O.D0.38 24 1.63 4.50 10 610 41 1140.50 24 1.63 4.44 26" 13 610 41 1130.63 24 1.88 4.50 660 16 610 48 1140.75 24 2.25 4.69 19 610 57 1190.38 26 1.75 4.81 10 660 44 1220.50 26 1.75 4.75 28" 13 660 44 1210.63 26 1.88 4.75 711 16 660 48 1210.75 26 2.25 4.94 19 660 57 1250.38 30 1.88 4.88 10 762 48 1240.50 30 1.88 4.81 30" 13 762 48 1220.63 30 1.88 4.81 762 16 762 48 1220.75 30 2.25 5.00 19 762 57 1270.38 30 2.00 5.56 10 762 51 1410.50 30 2.00 5.50 32" 13 762 51 1400.63 30 2.00 5.38 813 16 762 51 1370.75 30 2.25 5.50 19 762 57 1400.38 34 2.13 5.56 10 864 54 1410.50 34 2.13 5.50 34 13 864 54 1400.63 30 2.13 6.00 864 16 762 54 1520.75 30 2.25 6.06 19 762 57 1540.38 36 2.25 5.94 10 914 57 1510.50 36 2.25 5.88 36" 13 914 57 1490.63 36 2.25 5.81 914 16 914 57 1480.75 36 2.25 5.75 19 914 57 146

O.D "T" "R1" "R2" IDD ASME O.D0.38 36 2.38 6.50 10 914 60 1650.50 36 2.38 6.44 38" 13 914 60 1640.63 36 2.38 6.38 965 16 914 60 1620.75 36 2.38 6.38 19 914 60 1620.38 40 2.50 6.63 10 1016 64 1680.50 40 2.50 6.56 40" 13 1016 64 1670.63 36 2.50 6.94 1016 16 914 64 1760.75 36 2.50 7.00 19 914 64 1780.38 40 2.63 7.19 10 1016 67 1830.50 40 2.63 7.13 42" 13 1016 67 1810.63 40 2.63 7.06 1067 16 1016 67 1790.75 40 2.63 7.00 19 1016 67 1780.38 42 3.00 8.00 10 1067 76 2030.50 42 3.00 8.75 42" 13 1067 76 222

are in Inches (mm)

Inches (Flanged & Dished Head ASME Table) Millimeters (Flanged & Dished Head ASME Table)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

Tangent Line

StraightFlange(Varies)2" Nom.51mm

Wall Thickness "T"Knuckle Radius

"R2"

Dish Radius"R1"

Area for nozzle attachment

O.D. - (R2+T)x2

Outside Diameter (O.D.)

Flanged and Dished Head (ASME)

Inside Depthof Dish"IDD"

26"

28"

30"

32"

34"

36"

38"

40"

42"

48"

Page 24: Vessel Volumes

Art Montemayor ASME Flanged and Dished Heads May21, 2003Rev: 0

Page 24 of 61 FileName: document.xlsWorksheet: ASME F&D Heads

0.63 42 3.00 8.69 1219 16 1067 76 2210.75 42 3.00 8.63 19 1067 76 2190.38 54 3.25 8.94 10 1372 83 2270.50 48 3.25 9.75 54" 13 1219 83 2480.63 48 3.25 9.75 1372 16 1219 83 2480.75 48 3.25 9.63 19 1219 83 2450.38 60 3.63 10.00 10 1524 92 2540.50 60 3.63 9.88 60" 13 1524 92 2510.63 54 3.63 10.69 1524 16 1372 92 2720.75 54 3.63 10.63 19 1372 92 270

O.D "T" "R1" "R2" IDD ASME O.D0.38 66 4.00 11.00 10 1676 102 2790.50 60 4.00 10.94 66" 13 1524 102 2780.63 60 4.00 11.75 1676 16 1524 102 2980.75 60 4.00 11.63 19 1524 102 2950.38 72 4.38 12.00 10 1829 111 3050.63 72 4.38 11.88 72" 16 1829 111 3020.75 72 4.38 11.88 1829 19 1829 111 3020.88 66 4.38 12.63 22 1676 111 3210.38 78 4.75 13.00 10 1981 121 3300.50 72 4.75 13.81 78" 13 1829 121 3510.75 72 4.75 13.69 1981 19 1829 121 3481.00 72 4.75 13.50 25 1829 121 3430.38 84 5.13 14.00 10 2134 130 3560.63 84 5.13 13.88 84" 16 2134 130 3530.88 84 5.13 13.75 2134 22 2134 130 3491.00 84 5.13 13.69 25 2134 130 3480.38 90 5.50 15.13 10 2286 140 3840.50 84 5.50 15.81 90" 13 2134 140 4020.75 84 5.50 15.69 2286 19 2134 140 3991.00 84 5.50 15.56 25 2134 140 3950.38 96 5.88 16.13 10 2438 149 4100.50 90 5.88 16.88 96" 13 2286 149 4290.88 90 5.88 16.63 2438 22 2286 149 4221.25 90 5.88 16.44 32 2286 149 418

O.D "T" "R1" "R2" IDD ASME O.D0.50 96 6.13 17.88 13 2438 156 4540.75 96 6.13 17.69 102" 19 2438 156 4491.00 96 6.13 17.56 2591 25 2438 156 4461.13 90 6.13 18.50 29 2286 156 4700.50 102 6.50 18.88 13 2591 165 4800.75 102 6.50 18.75 108" 19 2591 165 4761.00 102 6.50 18.56 2743 25 2591 165 4711.13 96 6.50 19.44 29 2438 165 4940.50 108 6.88 19.88 13 2743 175 5050.75 108 6.88 19.75 114" 19 2743 175 5021.00 108 6.88 19.63 2896 25 2743 175 4991.25 108 6.88 19.50 32 2743 175 4950.50 114 7.25 20.88 13 2896 184 5300.88 114 7.25 20.69 120" 22 2896 184 5261.25 108 7.25 21.44 3048 32 2743 184 5451.63 108 7.25 21.25 41 2743 184 5400.50 120 7.63 21.88 13 3048 194 5560.88 120 7.63 21.69 126" 22 3048 194 5511.25 120 7.63 21.50 3200 32 3048 194 5461.38 114 7.63 22.31 35 2896 194 5670.75 126 8.00 22.81 19 3200 203 5790.88 120 8.00 23.69 132" 22 3048 203 6021.25 120 8.00 23.44 3353 32 3048 203 5951.63 120 8.00 23.25 41 3048 203 591

O.D "T" "R1" "R2" IDD ASME O.D0.63 132 8.38 23.94 16 3353 213 6081.00 132 8.38 23.75 138" 25 3353 213 6031.38 132 8.38 23.56 3505 35 3353 213 5981.75 132 8.38 23.38 44 3353 213 5940.63 132 8.75 25.88 16 3353 222 6571.00 132 8.75 25.63 144" 25 3353 222 6511.38 132 8.75 25.44 3658 35 3353 222 6461.75 132 8.75 25.19 44 3353 222 640

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

48"

54"

60"

66"

72"

78"

84"

90"

96"

102"

108"

114"

120"

126"

132"

138"

144"

Page 25: Vessel Volumes

Art Montemayor ASME Flanged and Dished Heads May21, 2003Rev: 0

Page 25 of 61 FileName: document.xlsWorksheet: ASME F&D Heads

0.75 144 9.38 27.75 19 3658 238 7051.13 144 9.38 27.50 156" 29 3658 238 6991.50 144 9.38 27.31 3962 38 3658 238 6941.88 144 9.38 27.06 48 3658 238 6870.75 144 10.13 31.81 19 3658 257 8081.13 144 10.13 31.50 168" 29 3658 257 8001.50 144 10.13 31.31 4267 38 3658 257 7951.88 144 10.13 31.13 48 3658 257 7910.88 170 10.88 31.44 22 4318 276 7991.25 170 10.88 31.25 180 32 4318 276 7941.63 170 10.88 31.00 4572 41 4318 276 7872.00 170 10.88 30.81 51 4318 276 7830.88 170 11.63 35.44 22 4318 295 9001.25 170 11.63 35.19 192" 32 4318 295 8941.63 170 11.63 34.94 4877 41 4318 295 8872.00 170 11.63 34.75 51 4318 295 883

O.D "T" "R1" "R2" IDD ASME O.D0.88 170 12.25 39.56 22 4318 311 10051.25 170 12.25 39.38 204" 32 4318 311 10001.63 170 12.25 39.19 5182 41 4318 311 9952.00 170 12.25 38.94 51 4318 311 9891.00 170 12.63 41.81 25 4318 321 10621.38 170 12.63 41.63 210" 35 4318 321 10571.75 170 12.63 41.31 5334 44 4318 321 10492.25 170 12.63 41.00 57 4318 321 10411.00 170 13.00 44.25 25 4318 330 11241.38 170 13.00 44.00 216" 35 4318 330 11181.75 170 13.00 43.69 5486 44 4318 330 11102.00 170 13.00 43.50 51 4318 330 11051.00 180 13.75 46.56 25 4572 349 11831.38 180 13.75 46.31 228" 35 4572 349 11761.75 180 13.75 46.06 5791 44 4572 349 11702.00 180 13.75 45.69 51 4572 349 1161

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

156"

168"

180"

192"

204"

210"

216"

228"

Page 26: Vessel Volumes

Art Montemayor Hemispherical Curve Fit September 12, 1997Rev 0

Page 26 of 61 Electronic FileName: document.xls WorkSheet: Hemispherical Curve Fit

1.00 0.2621.50 0.8842.00 2.0952.50 4.0913.00 7.0693.50 11.2254.00 16.7564.50 23.8575.00 32.7255.50 43.5576.00 56.5486.50 71.8967.00 89.7977.50 110.4478.00 134.0418.50 160.7789.00 190.8529.50 224.46

10.00 261.799

Hemispherical Head Inside Diameter = 120.000 inches

Volume of Single Hemispherical Head = 261.900 = 1,959.1 Gallons

Internal Diameter, ft

Hemispherical Volume, cu. Ft.

Ft3

0.00 2.00 4.00 6.00 8.00 10.00 12.00

0

50

100

150

200

250

300

f(x) = 0.261930185982272 x^2.99973562561194R² = 0.999999997076694

HEMISPHERICAL HEAD VOLUME

Inside Diameter, Ft

Vo

lum

e,

Cu

Ft

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)

Page 27: Vessel Volumes

Art Montemayor Dished Curve Fit September 12, 1997Rev 0

Page 27 of 61 Electronic FileName: document.xlsWorkSheet: Dished Curve Fit

1.00 0.0531.50 0.1822.00 0.4302.50 0.8423.00 1.4543.50 2.3104.00 3.4484.50 4.9095.00 6.7335.50 8.9636.00 11.6366.50 14.7947.00 18.4777.50 22.7278.00 27.5828.50 33.0839.00 39.2719.50 46.188

10.00 53.871

Dished Head Inside Diameter = 120.000 inches

Volume of Single Dished Head = 53.60 = 401.0 Gallons

Internal Diameter, ft

Dished Volume, Ft3

Ft3

0.00 2.00 4.00 6.00 8.00 10.00 12.00

0.000

10.000

20.000

30.000

40.000

50.000

60.000

f(x) = 0.0535515347642583 x^3.00327204731088R² = 0.999997566306414

DISHED HEAD VOLUME

Inside Diameter, Ft

Vo

lum

e,

Cu

Ft

Reference: Trinity Industries, Inc.Head DivisionNavasota, TXProduct & ServicesCatalog # 7962M (1996)

Page 28: Vessel Volumes

Art Montemayor Flanged and Dished Heads May 21, 2003Rev: 0

Page 28 of 61 FileName: document.xlsWorksheet: Dished Heads

All Dimensions

Verify all dimensionwith vendor drawings

NOTE: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 IDD

O.D "T" "R1" "R2" IDD O.D0.38 24 1.13 4.25 10 610 29 1080.50 24 1.50 4.38 26" 13 610 38 1110.63 24 1.88 4.50 660 16 610 48 1140.75 24 2.25 4.69 19 610 57 1190.38 26 4.50 4.50 10 660 114 1140.50 26 4.63 4.63 28" 13 660 118 1180.63 26 4.75 4.75 711 16 660 121 1210.75 26 4.94 4.94 19 660 125 1250.38 30 4.50 4.50 10 762 114 1140.50 30 4.63 4.63 30" 13 762 118 1180.63 30 4.81 4.81 762 16 762 122 1220.75 30 5.00 5.00 19 762 127 1270.38 30 5.00 5.00 10 762 127 1270.50 30 5.19 5.19 32" 13 762 132 1320.63 30 5.31 5.31 813 16 762 135 1350.75 30 5.50 5.50 19 762 140 1400.38 34 5.00 5.00 10 864 127 1270.50 34 5.19 5.19 34 13 864 132 1320.63 33 5.44 5.44 864 16 838 138 1380.75 30 6.06 6.06 19 762 154 1540.38 36 5.25 5.25 10 914 133 1330.50 36 5.44 5.44 36" 13 914 138 1380.63 36 5.63 5.63 914 16 914 143 1430.75 36 5.75 5.75 19 914 146 146

O.D "T" "R1" "R2" IDD O.D0.38 36 1.13 5.81 10 914 29 1480.50 36 1.50 6.00 38" 13 914 38 1520.63 36 1.88 6.13 965 16 914 48 1560.75 36 2.25 6.31 19 914 57 1600.38 40 1.13 5.81 10 1016 29 1480.50 40 1.50 5.94 40" 13 1016 38 1510.63 36 1.88 6.69 1016 16 914 48 1700.75 36 2.25 6.88 19 914 57 1750.38 42 1.13 6.06 10 1067 29 1540.50 42 1.50 6.25 42" 13 1067 38 1590.63 42 1.88 6.38 1067 16 1067 48 1620.75 40 2.25 6.81 19 1016 57 1730.38 48 1.13 6.88 10 1219 29 1750.50 48 1.50 7.00 42" 13 1219 38 1780.63 48 1.88 7.19 1219 16 1219 48 183

are in Inches (mm)

Inches (Flanged & Dished Head Table) Millimeters (Flanged & Dished Head Table)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

Tangent Line

StraightFlange(Varies)2" Nom.51mm

Wall Thickness "T"Knuckle Radius

"R2"

Dish Radius"R1"

Area for nozzle attachment

O.D. - (R2+T)x2

Outside Diameter (O.D.)

Flanged and Dished Head

Inside Depthof Dish"IDD"

26"

28"

30"

32"

34"

36"

38"

40"

42"

48"

Page 29: Vessel Volumes

Art Montemayor Flanged and Dished Heads May 21, 2003Rev: 0

Page 29 of 61 FileName: document.xlsWorksheet: Dished Heads

0.75 48 2.25 7.38 19 1219 57 1870.38 54 1.13 7.69 10 1372 29 1950.50 54 1.50 7.81 54" 13 1372 38 1980.63 54 1.88 8.00 1372 16 1372 48 2030.75 54 2.25 8.19 19 1372 57 2080.38 60 1.13 8.50 10 1524 29 2160.50 60 1.50 8.63 60" 13 1524 38 2190.63 60 1.88 8.81 1524 16 1524 48 2240.75 60 2.25 8.94 19 1524 57 227

O.D "T" "R1" "R2" IDD O.D0.38 66 1.13 9.31 10 1676 29 2360.50 66 1.50 9.44 66" 13 1676 38 2400.63 66 1.88 9.63 1676 16 1676 48 2450.75 66 2.25 9.75 19 1676 57 2480.38 72 1.13 10.06 10 1829 29 2560.63 72 1.88 10.38 72" 16 1829 48 2640.88 72 2.63 10.69 1829 22 1829 67 2721.13 72 3.38 11.00 29 1829 86 2790.38 78 1.13 10.88 10 1981 29 2760.63 78 1.88 11.19 78" 16 1981 48 2840.88 78 2.63 11.50 1981 22 1981 67 2921.13 78 3.38 11.81 29 1981 86 3000.38 84 1.13 11.69 10 2134 29 2970.63 84 1.88 11.88 84" 16 2134 48 3020.88 84 2.63 12.31 2134 22 2134 67 3131.13 84 3.38 12.63 29 2134 86 3210.38 90 1.13 12.50 10 2286 29 3180.63 84 1.88 13.75 90" 16 2134 48 3490.88 84 2.63 14.00 2286 22 2134 67 3561.13 84 3.38 14.31 29 2134 86 3630.50 96 1.50 13.44 13 2438 38 3410.75 96 2.25 13.75 96" 19 2438 57 3491.00 96 3.00 14.06 2438 25 2438 76 3571.25 96 3.75 14.38 32 2438 95 365

O.D "T" "R1" "R2" IDD O.D0.50 102 1.50 14.25 13 2591 38 3620.75 96 2.25 15.50 102" 19 2438 57 3941.00 96 3.00 15.75 2591 25 2438 76 4001.25 96 3.75 16.06 32 2438 95 4080.50 108 1.50 15.06 13 2743 38 3830.75 108 2.25 15.38 108" 19 2743 57 3911.00 102 3.00 16.56 2743 25 2591 76 4211.25 102 3.75 16.81 32 2591 95 4270.50 114 1.50 15.88 13 2896 38 4030.75 114 2.25 16.19 114" 19 2896 57 4111.00 108 3.00 17.38 2896 25 2743 76 4411.25 108 3.75 17.63 32 2743 95 4480.50 120 1.50 16.69 13 3048 38 4240.88 120 2.63 17.13 120" 22 3048 67 4351.25 120 3.75 17.59 3048 32 3048 95 4471.63 120 4.88 18.06 41 3048 124 4590.50 126 1.50 17.50 13 3200 38 4450.88 120 2.63 18.81 126" 22 3048 67 4781.25 120 3.75 19.25 3200 32 3048 95 4891.63 120 4.88 19.69 41 3048 124 5000.63 132 1.88 18.44 16 3353 48 4680.88 132 2.63 18.75 132" 22 3353 67 4761.13 132 3.38 19.00 3353 29 3353 86 4831.50 132 4.50 19.50 38 3353 114 495

O.D "T" "R1" "R2" IDD O.D0.63 132 1.88 20.13 16 3353 48 5110.88 132 2.63 20.44 138" 22 3353 67 5191.13 132 3.38 20.69 3505 29 3353 86 5261.50 132 4.50 21.13 38 3353 114 5370.63 144 1.88 20.00 16 3658 48 5080.88 144 2.63 20.31 144" 22 3658 67 5161.13 144 3.38 20.63 3658 29 3658 86 5241.50 144 4.50 21.13 38 3658 114 5370.75 144 2.25 23.69 19 3658 57 602

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

54"

60"

66"

72"

78"

84"

90"

96"

102"

108"

114"

120"

126"

132"

138"

144"

Page 30: Vessel Volumes

Art Montemayor Flanged and Dished Heads May 21, 2003Rev: 0

Page 30 of 61 FileName: document.xlsWorksheet: Dished Heads

1.13 144 3.38 24.13 156" 29 3658 86 6131.50 144 4.50 24.50 3962 38 3658 114 6221.88 144 5.63 24.94 48 3658 143 6330.75 170 2.25 23.13 19 4318 57 5881.13 170 3.38 23.56 168" 29 4318 86 5981.50 170 4.50 24.06 4267 38 4318 114 6111.88 170 5.63 24.19 48 4318 143 6140.88 170 2.63 26.69 22 4318 67 6781.25 170 3.75 27.13 180 32 4318 95 6891.63 170 4.88 27.56 4572 41 4318 124 7002.00 170 6.00 28.00 51 4318 152 7110.88 170 2.63 30.50 22 4318 67 7751.25 170 3.75 30.88 192" 32 4318 95 7841.63 170 4.88 31.25 4877 41 4318 124 7942.00 170 6.00 31.63 51 4318 152 803

O.D "T" "R1" "R2" IDD O.D0.88 170 2.63 34.63 22 4318 67 8801.25 170 3.75 35.00 204" 32 4318 95 8891.63 170 4.88 35.31 5182 41 4318 124 8972.00 170 6.00 35.63 51 4318 152 9050.88 170 2.63 39.06 22 4318 67 9921.25 170 3.75 39.50 216" 32 4318 95 10031.63 170 4.88 39.75 5486 41 4318 124 10102.00 170 6.00 40.00 51 4318 152 10160.88 180 2.63 41.25 22 4572 67 10481.25 180 3.75 41.50 228" 32 4572 95 10541.63 180 4.88 41.75 5791 41 4572 124 10602.00 180 6.00 42.00 51 4572 152 10670.88 180 2.63 46.25 22 4572 67 11751.25 180 3.75 46.44 240" 32 4572 95 11801.63 180 4.88 46.63 6096 41 4572 124 11842.00 180 6.00 46.81 51 4572 152 1189

"T" (mm) "R1"(mm) "R2"(mm) IDD(mm)

156"

168"

180"

192"

204"

216"

228"

240"

Page 31: Vessel Volumes

Art Montemayor Cylindrical Vessel Volume Relationship May 27, 1998Rev:1(06/06/01)

Page 31 of 61 Electronic FileName: document.xlsWorkSheet: Cylindrical Tank Volume

Diam., in. Gal./in. Diam., in. Gal./in. Diam., in. Gal./in. Diam., in. Gal./in.15.0 0.765 28.0 2.666 52 9.194 96 31.33415.5 0.817 28.5 2.762 53 9.551 98 32.65316.0 0.870 29.0 2.859 54 9.914 100 34.00016.5 0.926 29.5 2.959 55 10.285 102 35.37317.0 0.983 30 3.060 56 10.662 104 36.774

17.5 1.041 31 3.267 57 11.047 106 38.20218.0 1.102 32 3.482 58 11.438 108 39.65718.5 1.164 33 3.703 59 11.835 110 41.14019.0 1.227 34 3.930 60 12.240 112 42.64919.5 1.293 35 4.165 62 13.070 114 44.186

20.0 1.360 36 4.406 64 13.926 116 45.75020.5 1.429 37 4.655 66 14.810 118 47.34121.0 1.499 38 4.910 68 15.722 120 48.96021.5 1.572 39 5.171 70 16.660 122 50.60522.0 1.646 40 5.440 72 17.626 124 52.278

22.5 1.721 41 5.715 74 18.618 126 53.97823.0 1.799 42 5.998 76 19.638 128 55.70523.5 1.878 43 6.287 78 20.685 130 57.46024.0 1.958 44 6.582 80 21.760 132 59.24124.5 2.041 45 6.885 82 22.861 134 61.050

25.0 2.125 46 7.194 84 23.990 136 62.88625.5 2.211 47 7.511 86 25.146 138 64.74926.0 2.298 48 7.834 88 26.329 140 66.640

26.5 2.388 49 8.163 90 27.540 142 68.55727.0 2.479 50 8.500 92 28.777 144 70.50227.5 2.571 51 8.843 94 30.042 146 72.474

148 74.473

Cylindrical Volumes of Vessels --- expressed as Gallons of Liquid Content per inch length of Cylinder.

Source: Chemical Engineers' Handbook; Perry & Chilton; 5th Edition; p. 6-86

Page 32: Vessel Volumes

Art Montemayor June 02, 1999Rev: 0

Page 32 of 61 Electronic FileName: document.xlsWorkSheet: Fittings' Volumes

VOLUMETRIC CAPACITY FOR BUTT-WELDED FITTINGSAll volumes expressed in cubic inches

Reference: Piping Engineering; Tube Turns Division of Chemetron Corp.; Nov. 1971; p.47

Tees CapsLong Radius Short Radius Long Radius Short Radius Long Radius Full-size outlets

Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong Standard X-Strong Standard

1/2 0.7 0.6 1.4 1.1 0.4 0.2 0.8 0.6 0.3 3/4 0.9 0.8 1.9 1.5 0.5 0.4 1.6 1.3 0.6

1 2.0 1.7 1.4 4.1 3.4 2.7 1.0 0.8 3.5 2.9 1.11-1/4 4.4 3.7 2.9 8.7 7.4 5.9 2.2 1.8 7.5 6.5 1.81-1/2 7.2 6.2 4.8 4.2 14.4 12.5 9.6 8.3 3.6 3.1 12.4 10.8 2.5

2 15.8 13.9 10.5 9.2 31.6 27.8 21.1 18.3 7.9 7.0 22.2 19.7 3.93 52.2 46.7 34.8 31.1 104.4 93.3 69.6 62.2 26.1 23.3 65.2 58.7 11.14 119.8 108.3 79.9 72.0 239.7 216.6 159.9 143.9 59.9 54.1 135.8 123.6 24.26 408.1 368.3 272.1 245.4 816.1 736.6 544.3 490.8 204.0 184.2 413.2 367.1 77.38 942.3 860.3 628.3 572.7 1,884.7 1,720.6 1,256.7 1,145.5 471.1 430.1 881.1 811.2 148.5

10 1,856.9 1,758.2 1,238.0 1,172.2 3,713.8 3,516.5 2,476.0 2,344.3 928.5 879.1 1,675.4 1,594.9 295.612 3,195.9 3,064.2 2,130.7 2,042.8 6,391.8 6,128.5 4,261.5 4,085.6 1,598.0 1,532.1 2,816.9 2,712.3 517.014 4,545.9 4,376.1 3,030.7 2,917.4 9,091.8 8,752.2 6,061.4 5,834.8 2,273.0 2,188.1 3,809.3 3,681.0 684.616 6,882.2 6,658.4 4,301.4 4,161.5 13,764.5 13,316.9 8,602.8 8,323.0 3,441.1 3,329.2 5,804.3 5,634.3 967.618 9,906.5 9,621.7 6,054.0 5,879.9 19,813.1 19,243.4 12,108.0 11,759.8 4,953.3 4,810.8 8,396.5 8,179.2 1,432.620 13,707.5 13,353.8 9,366.8 9,125.1 27,415.0 26,707.6 18,733.6 18,250.2 6,853.8 6,676.9 11,701.3 11,429.4 2,026.422 18,365.2 17,935.8 36,730.9 23,914.5 9,182.7 8,967.9 14,348.0 14,049.2 2,784.124 23,995.6 23,482.4 14,664.0 14,350.3 47,991.1 46,964.7 29,327.9 28,700.7 11,997.8 11,471.2 20,647.2 20,249.7 3,451.026 30,644.8 30,041.0 61,289.6 40,054.6 15,322.4 15,020.5 23,912.3 23,493.2 4,014.330 47,449.7 46,642.1 31,648.8 31,110.5 94,899.4 62,189.4 63,297.7 62,221.0 35,985.0 35,442.5 5,163.134 69,490.1 68,449.0 110,260.5 34,745.0 34,224.5 52,836.7 52,135.936 82,695.4 81,526.6 55,130.3 54,351.0 176,155.6 110,260.5 108,702.1 41,347.7 40,763.3 62,945.1 62,157.5 7,010.842 132,116.7 130,520.2 88,077.8 87,013.4 66,054.8 65,260.1 94,207.0 93,209.0 10,936.348 198,322.0 196,203.0 40,971.0 40,538.0 145,443.0 144,092.0 13,480.0

Nominal Pipe

Size, in.

90o Elbows 180o Returns 45o Elbows

Page 33: Vessel Volumes

Art Montemayor June 02, 1999Rev: 0

Page 33 of 61 Electronic FileName: document.xlsWorkSheet: Fittings' Volumes

V = VolumeD = Inside diameterA = Center to face distance

V = VolumeD = Inside diameterA = Center to center distance

V = VolumeD = Inside diameterA = Center to face distance

Full Size Outlet Tees:

V = Volume

90o Elbows:

V = P2D2A/8

180o Returns:

V = P2D2O/8

45o Elbows:

V = P2D2A/8

V = (PD2/2) (C + M/2) - D3/3

A

Page 34: Vessel Volumes

Art Montemayor June 02, 1999Rev: 0

Page 34 of 61 Electronic FileName: document.xlsWorkSheet: Fittings' Volumes

D = Inside diameterC= Center to end of runM = center to end of branch

Pipe Caps:

V = VolumeD = Inside diameterE = lengtht = wall thickness

Crosses:

V = VolumeD = Inside diameterC= Center to end of runM = center to end of branch

Concentric & eccentric reducers:

V = (PD2/4) (E - t - D/12)

V = (PD2/2) (C + M) - (2/3)D3

Page 35: Vessel Volumes

Art Montemayor June 02, 1999Rev: 0

Page 35 of 61 Electronic FileName: document.xlsWorkSheet: Fittings' Volumes

Caps Crosses Stub Ends Reducers TeesFull-size outlets Lap Joint Concentric & Eccentric with Reducing Outlet

X-Strong Standard X-Strong Standard X-Strong Large end Small end Standard X-Strong Standard X-Strong

10.2 0.9 0.7 3/8 1.5 1.3 2.7 2.30.4 1.6 1.3 1/2 1.8 1.4 2.8 2.40.9 3.5 2.9 3/4 2.1 1.8 3.0 2.61.5 9.5 8.2 6.0 5.1

1-1/41/2 2.6 2.2 5.8 5.1

2.0 15.5 13.7 8.1 7.1 3/4 3.1 2.6 6.0 5.43.2 27.7 24.7 20.1 17.8 1 3.7 3.1 6.3 5.79.4 80.5 72.9 44.4 39.6

1-1/2

1/2 4.0 3.4 9.4 8.320.8 166.5 152.4 76.4 69.0 3/4 4.6 3.9 9.6 8.665.7 501.3 441.0 231.1 208.5 1 5.3 4.5 9.9 9.1

122.3 1,061.9 983.0 400.2 365.3 1-1/4 6.5 5.6 10.8 9.9264.4 2,010.4 1,920.6 788.5 746.6

2

3/4 7.6 6.6 16.7 15.2475.0 3,371.9 3,255.9 1,131.0 1,084.3 1 8.5 7.4 16.9 15.6640.0 4,171.5 4,043.7 1,654.6 1,592.8 1-1/4 10.0 8.8 17.6 16.5911.0 6,311.7 6,144.0 2,191.8 2,120.6 1-1/2 11.4 10.0 18.4 17.4

1,363.0 9,081.3 8,868.1 2,804.5 2,723.8

3

1 50.9 45.51,938.0 12,634.1 12,368.2 3,492.5 3,402.4 1-1/4 20.2 17.9 51.8 46.42,682.9 1-1/2 21.9 19.4 52.9 47.33,313.0 22,189.4 21,802.9 5,094.7 4,985.7 2 25.5 22.7 55.1 49.53,884.1 2-1/2 29.5 26.3 58.6 52.65,006.4

4

1-1/2 37.3 33.4 108.0 97.72 41.8 37.5 110.4 99.9

6,811.5 2-1/2 46.6 41.8 113.9 103.210,666.7 3 54.4 48.9 119.7 108.613,157.0 3-1/2 62.1 55.9 125.9 114.5

6

2-1/2 11.9 100.6 334.1 301.83 123.7 111.3 340.0 307.4

3-1/2 134.8 121.5 346.3 313.7

Nominal Pipe Size, in.

Page 36: Vessel Volumes

Art Montemayor June 02, 1999Rev: 0

Page 36 of 61 Electronic FileName: document.xlsWorkSheet: Fittings' Volumes

64 147.1 132.8 354.1 321.2

5 175.3 158.7 375.7 361.8

8

3 716.9 655.53-1/2 221.7 201.8 722.6 661.0

4 235.6 215.0 730.6 668.75 269.6 245.8 753.0 690.56 309.2 280.9 791.1 719.4

10

4 385.9 362.0 1,373.0 1,300.05 428.3 401.4 1,396.0 1,323.06 476.8 444.8 1,432.0 1,350.08 586.0 546.7 1,506.0 1,426.0

12

5 639.4 606.4 2,318.0 2,224.0

6 697.7 658.8 2,348.0 2,250.08 827.0 779.8 2,430.0 2,329.0

10 993.8 947.7 2,567.0 2,468.0

14

6 1,496 1,419 1,992 1,9158 1,738 1,646 2,587 2,488

10 2,041 1,952 2,922 2,81612 2,382 2,288 2,976 3,022

16

6 2,621 2,5328 2,321 2,212 3,396 3,283

10 2,656 2,552 3,803 3,701

12 3,029 2,920 4,891 4,74114 3,289 3,175 5,054 4,902

18

8 4,318 4,19110 3,413 3,291 4,849 4,71112 3,821 3,695 4,993 5,01314 4,104 3,973 6,147 5,97616 4,598 4,458 7,180 6,985

20

8 5,360 5,21410 6,010 5,85512 6,248 6,059 6,204 6,221

14 6,922 6,717 7,606 7,41416 7,974 7,747 8,877 8,65718 9,404 9,150 10,163 9,916

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22

10 12,027 11,74512 12,226 11,94414 7,333 7,129 12,394 12,11016 8,055 7,841 12,620 12,33718 8,848 8,622 12,995 12,71020 9,711 9,474 1,345,813,172

24

10 8,701 8,51912 14,972 14,603

14 10,979 10,74516 8,637 8,419 12,795 12,52618 9,451 9,221 14,628 14,32520 10,334 10,094 16,477 16,14122 16,606 16,287

26

12 20,062 19,66814 20,233 19,83716 20,465 20,07018 20,846 20,44920 21,316 20,919

22 21,887 21,48724 22,565 22,164

30

14 30,283 29,76816 30,520 30,00618 30,908 30,39220 31,386 30,86922 31,964 31,44524 32,652 32,13126 33,458 32,93628 34,102 33,583

34

16 44,385 43,72418 44,779 44,11620 45,265 44,60022 45,851 45,18424 46,548 45,87926 47,364 46,69328 48,018 47,35130 49,058 48,389

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34

32 50,242 49,571

36

16 52,701 51,95818 53,098 52,35320 53,587 52,84122 54,177 53,42924 54,878 54,12826 55,700 54,94728 56,359 55,61030 30,539 30,071 57,404 56,65332 32,314 31,831 58,594 57,84234 34,176 33,698 59,940 59,185

42

20 76,746 75,82522 77,172 76,25324 32,031 31,561 77,639 76,72326 33,617 33,132 78,899 77,97228 79,574 78,65030 37,053 36,540 80,636 79,71032 38,902 38,375 81,462 80,54034 40,840 40,229 82,341 81,42536 42,866 42,311 83,276 82,365

48

22 125,186 123,87424 85,143 83,984 125,667 124,35826 89,354 88,163 126,693 125,37728 127,344 126,03130 98,305 97,047 128,044 126,73632 103,044 101,753 129,561 128,24534 107,959 106,635 130,462 129,15136 113,050 111,693 131,419 130,11338 118,317 116,927 133,521 132,20740 123,760 122,337 134,710 133,40242 129,378 127,923 135,960 134,66044 135,173 133,685 138,742 137,43246 141,144 139,622 140,256 138,954

Page 39: Vessel Volumes

Profiles of Torispherical Dished Heads

The volume calculator assumes the head profile to be a perfect ellipse, which is correct for a semi-ellipsoidal head but only approximate for a Torispherical profile. Torispherical heads can have different profiles depending on the relationship between: - Knuckle radius, Spherical Radius and Diameter.

Two typical Torispherical profiles are shown below in Red, and the true ellipse for the same diameter and head height is shown in Blue. Treating a Torisphere as an ellipse for volume calculation will generally give a slight under estimate of the volume. The error will depend on the relationship between: - Knuckle radius, Spherical Radius and Diameter used.

Page 40: Vessel Volumes

Art Montemayor Volume of a Partially Filled Torispherical Bottom Head July 20, 2003Rev: 1

Page 40 of 61 FileName: document.xlsWorkSheet: F & D Partial Volume

VERTICAL TANK BOTTOM TORISPHERICAL HEAD VOLUME CALCULATION

D 2,134 mm = 84.02 inches

Crown Radius 2,134 mm = 84.02 inches% Knuckle Radius 6.55%

Knuckle Radius 139.8 mm = 5.50 inches

b = 927.2 mm

a = 992.2 mm

c = 1,765.6 mm

ß = 0.484 radians27.7 °

x = 123.7 mm = 4.87 inches

z = 244.7 mm = 9.63 inchesh = x + z 368.4 mm = 14.51 inches

Approx. Head Volume =approximate calculation for knuckle section

= 386.1 + 412.3= 798.4 litres = 210.91 US gals

Volume of partially filled Torispherical head:

Level in End dish:Sector Area Knuckle Area Total Head Volume

"z" "r" "x" "r" litres %0% 0 0 0 0 0 992 0 0 0%

10% 37 37 395 9 0 992 0 9 1%20% 74 74 556 36 0 992 0 36 5%30% 111 111 678 80 0 992 0 80 10%40% 147 147 779 142 0 992 0 142 18%50% 184 184 867 221 0 992 0 221 28%60% 221 221 946 316 0 992 0 316 40%70% 258 245 992 386 13 1,000 41 427 54%80% 295 245 992 386 50 1,022 160 546 68%90% 332 245 992 386 87 1,045 283 669 84%

100% 368.45 245 992 386 124 1,067 412 798.4 100%

Notes:

(1) Sector volume =

(2) Knuckle volume =(3) Torispherical (also called ASME F&D) heads are designed and fabricated in the USA on the basis of using the

Tank Internal Diameter(3)

Ri

ri

D/2 - ri

b Ri / (Ri - ri)

((Ri - ri)2 - b2)½

Sin-1 (a / Ri)

Ri Cos ß - c

Ri - c - x

p / 6 * z (3a2 + z2) + p / 3 * x ((D/2)2 + (D/2)a + a2)

Liquid Height "h" (mm) Volume (1) Volume (2)

PI / 6 * "z" (3 * "r"2 + "z"2)

PI / 3 * "x" ("r"2 + "r" * a + a2)

outside diameter as their nominal diameter.

a

b

x

ß

Ri

ric

hz

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0%

20%

40%

60%

80%

100%f(x) = − 0.376151834147061 x³ + 1.44533007217208 x² − 0.0663660420954087 xR² = 0.99996485707352

Vertical Torispherical Tank Head Volume

Level of Fill

Vo

lum

e o

f F

ill

Page 41: Vessel Volumes

Art Montemayor Vertical Tank Bottom Torispherical Head Volume September 30, 2004Rev: 0

Page 41 of 61 FileName: document.xlsWorkSheet: Vertical F&D Head Volume

I.D. = 84.0 inchesk = Knuckle-Radius (kD) 0.06 inchesf = dish-radius parameter (fD) 1

kD = 5.04 inchesfD = 84 inches

a = 0.487 radians 0.883683 0.468085a1 = 9.7706 inches 1.062004 0.508792a2 = 4.4538 inchesD1 = 78.6383s = 5.565584t = 8.9075 8.9075

u(h) = 5.583195

Limits of the Equation

0.00 h 9.77 9.77 h 14.22 14.22 h Top

h = 15.35379 in h = 15.35379 in h = 24 in

V = 58,420 252.90 Gallons V = 54,255 234.87 Gallons V = 102,183 442.35 Gallons

Cos a = Sin a =Acos a = Asin a =

in3 = in3 = in3 =

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Art Montemayor Determining Vessel Volumes June 15, 2003Rev: 0

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The 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, 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 allow rapid and accurate fluid-volume calculations.

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 calculations are defined in the “Variables and Definitions” sidebar. Fig. 1 and Fig. 2 graphically illustrate horizontal tank variables, and Fig. 3 and Fig. 4 graphically illustrate vertical tank variables.

Exact fluid volumes in elliptical horizontal or vertical tanks can be determined by calculating the fluid volumes of appropriate cylindrical horizontal or vertical tanks using the equations described above, and then by adjusting those 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 to the fluid surface. A guppy head is a conical head with its apex level with the top of the cylindrical section of the tank, 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 “segment” of an ellipsoid will work, as is true in the case of a spherical head that can be a spherical segment. For a

Figure 1. Parameters for Horizontal Cylindrical Tanks with Conical, Ellipsoidal, Guppy or Spherical Heads

spherical 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.

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1. Both heads of a tank must be identical. Above diagram is for definition of parameters only.

2. 3. 4. For convex head other than spherical, 0 < a < a , for concave head a < 0

5.6. Ellipsoidal head must be exactly half of an ellipsoid of revolution

7.

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 the fluid 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 other end, calculate fluid volumes of two tanks, one with conical heads and the other with ellipsoidal heads, and average 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:

•••• f > 0.5 for torispherical heads.•• D > 0.•

Variables used in Volumetric Equations and their Definitions

a This is the distance a horizontal tank's heads extend beyond (a>0) or into (a<0) its cylindrical section or the depth the bottom extends below the cylindrical section of a vertical tank. For a horizontal tank with flat heads or a vertical tank with a flat bottom, a = 0.

This is the cross-sectional area of the fluid in a horizontal tank's cylindrical section.

D This is the diameter of the cylindrical section of a horizontal or vertical tank.

These are the height and width, respectively, of the ellipse defining the cross section of the bodyof a horizontal elliptical tank.

These are the major and minor axes, respectively, of the ellipse defining the cross section of the body of a vertical elliptical tank.

f This is the dish-radius parameter for tanks with torispherical heads or bottoms; fD is the dish radius.

h This is the height of fluid in a tank measured from the lowest part of the tank to the fluid surface.

k This is the knuckle-radius parameter for tanks with torispherical heads or bottoms; kD is the knuckle radius.

L This is the length of the cylindrical section of a horizontal tank.

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

L > 0 for a > 0, L > 2|a| for a < 0

0 < h < D.

|a| < R for spherical heads.|a| < L/2 for concave ends.0 < h < 2R for all tanks.

0 < k < 0.5 for torispherical heads.

L > 0.

Af

DH & DW

DA & DB

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R This is the radius of the cylindrical section of a horizontal of vertical tank.

r This is the radius of a spherical head for a horizontal tank or a spherical bottom of a vertical tank.

This is the fluid volume, of fluid depth h, in a horizontal or vertical cylindrical tank.

Horizontal tank equationsThe following 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 and D/2 = R > 0 for all equations).

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

In the horizontal tank 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 square units consistent with the linear units used for R and h. The equation for Af is given by:

Horizontal cylindrical tank examplesTwo examples can be used to check application of the equations.

Find the volumes of fluid, in gallons, in horizontal cylindrical tanks 108 inches [in.] in diameter with cylinder lengths of 156 in., for conical, ellipsoidal, guppy, spherical and “standard” ASME torispherical (f = 1, k = 0.06) heads, each head extending beyond the ends of the cylinder 42 in. (except torispherical), for fluid depths in the tanks of 36 in. (example 1) and 84 in. (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 in., L = 156 in., a = 42 in., h = 36 in., f = 1 and k = 0.06. The fluid volumes are 2,041.19 gallon (gal) for conical heads, 2,380.96 gal for ellipsoidal heads, 1,931.72 gal 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 in., L = 156 in., a = 42 in., h = 84 in., f = 1 and k = 0.06. The fluid volumes are 6,180.54 gal for conical heads, 7,103.45 gal for ellipsoidal heads, 5,954.11 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 input; it can be calculated from f, k and D. For these torispherical head examples, the calculated value is “a” = 18.288 in.

Vertical cylindrical tanksThe 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 Fig. 3 and Fig. 4 for tank configurations and dimension parameters, which also are defined in the “Variables and Definitions” sidebar.

Figure 2. Parameters for Horizontal Cylindrical Tanks with Torispherical Heads

Vf

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A torispherical bottom is an ASME-type bottom defined by a knuckle-radius factor and a dish-radius factor, as shown in Fig. 4. The knuckle radius then will be kD, and the dish radius will be fD. An ellipsoidal bottom must

bottom and R is the radius of the cylindrical section of the tank.

The following parameter ranges must be observed:•••• D > 0.

Figure 3. Parameters for Vertical Cylindrical Tanks with Conical, Ellipsoidal or Spherical Bottoms

Vertical tank equationsThe specific equations for fluid volumes in vertical cylindrical tanks with conical, ellipsoidal, spherical and torispherical bottoms are provided in the Vertical Tank Equations sidebar (use radian angular measure for all trigonometric functions, and D > 0 for all equations).

Figure 4. Parameters for Vertical Cylindrical Tanks with Torispherical Bottoms

be exactly half of an ellipsoid of revolution. For a spherical bottom, |a| < R, where a is the depth of the spherical

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.

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Vertical cylindrical tank examplesTwo examples can be used to check application of the equations for vertical cylindrical tanks; for each example, calculate the fluid volumes for conical, ellipsoidal, spherical and torispherical bottoms.

For example 1, D = 132 in., a = 33 in., h = 24 in., f = 1, and k = 0.06. The fluid volumes are 250.67 gal for a conical bottom, 783.36 gal for an ellipsoidal bottom, 583.60 gal for a spherical bottom and 904.07 gal for a torispherical bottom.

For example 2, D = 132 in., a = 33 in., h = 60 in., f = 1, and k = 0.06. The fluid volumes are 2,251.18 gal for a conical bottom, 2,902.83 gal for an ellipsoidal bottom, 2,658.46 gal for a spherical bottom and 3,036.76 gal for a torispherical bottom.

For a torispherical bottom, parameter "a" is not required input, but can be calculated from the values of f, k, and D. For these examples, the calculated value is a = 22.353 in.

Horizontal and vertical elliptical tanksThe cross-sections of tank bodies of horizontal and vertical tanks with elliptical bodies are ellipses. For this article, a horizontal elliptical tank must be one of two possible configurations, shown in Fig. 5, where the major and minor axes of the elliptical cross-sections are either vertical or horizontal.

The heads of horizontal elliptical tanks and the bottoms of vertical elliptical tanks may be any of those described above for the corresponding cylindrical tanks, with the assumption that the heads and bottoms are "deformed" proportionately to the deformation of the cylindrical body to form the elliptical body.

In certain cases such as those with torispherical heads and bottoms and spherical heads and bottoms, it is necessary to distinguish which elliptical axis defines the head or bottom shape and which axis has been proportionately stretched or compressed from the cylindrical tank shape to form the elliptical tank shape; therefore, this distinction will be made for all cases for the sake of consistency, not necessity.

To calculate the fluid volume in a horizontal elliptical tank with the elliptical body oriented in one of the two orientations shown in Fig. 5 - where the head parameters are defined in the vertical plane through the tank

equations for horizontal cylindrical tanks with the appropriately shaped heads. Multiply the volume found by

Figure 5. Cross-sections of Horizontal Elliptical Tanks

To calculate the fluid volume in a horizontal elliptical tank with the elliptical body oriented in one of the two orientations shown in Fig. 5 - where the head parameters are defined in the horizontal plane through the tank

centerline (plane goes through DH) - calculate the volume of a horizontal cylindrical tank with D = DH using the

DW/DH to get the elliptical tank fluid volume.

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Examples for horizontal elliptical tanks Find the fluid volumes (in gal.) of horizontal elliptical tanks with ellipsoidal, spherical and torispherical heads with

heads, f = 0.8 and k = 0.1 for torispherical heads, fluid height h = 48 in., and head parameters of each tank defined (1) in a horizontal plane through the tank centerline and (2) in a vertical plane through the tank centerline.

For example 1, calculate horizontal cylindrical tank volumes with D = 120 in., L = 156 in., a = 25 in. for ellipsoidal and spherical heads, f = 0.8 and k = 0.1 for torispherical heads, and h = 57.6 in. (48 in. x 120/100), and multiply the volume found by 100/120. For example 2, calculate horizontal cylindrical tank volumes with D = 100 in., L = 156 in., a = 25 in. for ellipsoidal and spherical heads, f = 0.8 and k = 0.1 for torispherical heads, and h = 48, and multiply the volume found by 120/100. The results are summarized in the following table:

The values for "a" in the above torispherical head cases are 27.065 in. for example 1 and 22.554 in. for example 2.

defining the cross-section of the tank body.

To calculate the fluid volume in a vertical elliptical tank, where the bottom parameters are defined in the plane

To calculate the fluid volume in a vertical elliptical tank, where the bottom parameters are defined in the plane

volume.

Examples for vertical elliptical tanksFind the fluid volumes (in gal.) of vertical elliptical tanks with conical, spherical and torispherical bottoms with the

k = 0.2 for the torispherical bottom, fluid height h = 53 in. Head parameters of each tank defined (1) in a plane

calculate vertical cylindrical tank volumes with D = 96 in., a = 34 in. (for conical and spherical bottoms), f = 0.9 and k = 0.2 (for the torispherical bottom), and h = 53 in., and multiply the volume found by 72/96. For example 2, calculate vertical cylindrical tank volumes with D = 72 in., a = 34 in. (for conical and spherical bottoms), f = 0.9 and k = 0.2 (for the torispherical bottom), and h = 53 in., and multiply the volume found by 96/72. The results are summarized in the following table:

centerline (plane goes through DW) - calculate the volume of a horizontal cylindrical tank with D = DW and a fluid

height h' = h(DW/DH) using the equations for horizontal cylindrical tanks with the appropriately shaped heads.

Multiply the volume found by DH/DW to get the desired elliptical tank fluid volume.

the following measurements: DH = 100 in., DW = 120 in., L = 156 in., a = 25 in. for ellipsoidal and spherical

For a vertical elliptical tank, define DA and DB to be the major and minor axes, respectively, of the ellipse

through both the tank centerline and through DA, use D = DA. Use the equations above for a vertical cylindrical

tank with the appropriately shaped bottom. Multiply the volume found by DB/DA to get the elliptical tank volume.

through both the tank centerline and through DB, use D = DB. Use the equations above for a vertical cylindrical

tank with the appropriately shaped bottom. Multiply the volume found by DA/DB to get the desired elliptical tank

following measurements: DA = 96 in., DB = 72 in., a = 34 in. for conical and spherical bottoms, f = 0.9 and

through the tank centerline and DA and (2) in a plane through the tank centerline and DB. For example 1,

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Calculated values for "a" in the torispherical-bottom cases are 25.684 in. and 22.554 in. for examples 1 and 2, respectively. CP

Horizontal Tank Equations

Conical Heads

V f=A f L+(2a R2

3 ) ( K )⋯⋯for 0≤h <R

V f=A f L+(2a R2

3 )(π2 )⋯⋯for h=R

V f=A f L+(2a R2

3 ) (π−K )⋯⋯for R<h≤2 R

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Ellipsoidal Heads

Guppy Heads (Eccentric Cone)

Spherical Heads

1.

2.

3.

4.

For the condition where:

For the condition where:

For the condition where:

For the condition where:

K=cos−1 M+M 3 cosh−1( 1M )−2M √1−M 2⋯⋯where , M=|

R−hR

|

V f=A f L+π a h2 (1− h3 R )

V f=A f L+ 2 a R2

3cos−1(1− h

R )+ 2 a9 R

√2 R h−h2 (2 h−3 R ) (h+R )

h=R and |a|≤R

V f=A f L+π a6

(3 R2+a2)

h=D and |a|≤R

2233

aRa

LAV ff =p

h=0 or a=0 , R , or −R

V f=A f L+π a h2 (1− h3 R )

h≠R , D ; a≠0 , R , −R ; and |a|≥0 .01 D

V f=A f L+ a|a|{2 r3

3 [cos−1 R2−r wR ( w−r )

+cos−1 R2+r wR (w+r )

− zr (2+(R

r )2) cos−1 w

R ]−2(w r2−w3

3 ) tan−1 yz+ 4 w y z

3 }h≠R , D ; a≠0 , R , −R ; and |a|<0 . 01 D

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5.

For the above 5 spherical heads equations:

Torispherical (ASME Flanged & Dished) Heads

For the condition where: h≠R , D ; a≠0 , R , −R ; and |a|<0 . 01 D

V f=A f L+ a|a|[2∫w

R

(r2−x2) tan−1√ R2− x2

r2−R2dx−A f z ]

r=a2+R2

2 |a|………where , a≠0 ; and a=±(r−√r2−R2) + or (− ) for convex or (concave )heads

w=R−h

y=√2 R h−h2

z=√r2−R2

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Jones is a senior process chemist for Stockhausen Louisiana LLC, Garyville, La. Contact him at [email protected].

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