This calculator estimates the liquid volume inside a cylindrical vessel with selectable head geometry.
Press Calculate to display:
Press Download, provide your feedback in the modal form, and export the PDF report.
The calculator applies different closed-form or numerical equations depending on orientation and head type. All geometry is defined by:
For a horizontal cylinder, the cross-sectional area filled to level $h$ is the segment of a circle:
The cylindrical shell contribution is $A(h) \times L$. With flat heads there is no head volume:
Each elliptical head contributes a partial ellipsoidal cap. The total volume (shell + two heads) is:
where the second term accounts for both heads combined.
Hemispherical heads are a special case of elliptical heads with $a = R$:
The cylindrical shell volume filled to height $h$ within the straight shell region is:
The volume of one complete semi-ellipsoidal head is:
The inside dish depth $a$ for a torispherical (F&D) head defined by factors $f$ and $k$ is:
where $f\,D$ is the crown (dish) radius and $k\,D$ is the knuckle radius. For a standard ASME F&D head, typical values are $f = 1.0$ and $k = 0.06$.
The horizontal torispherical volume is computed by numerical integration (Simpson's rule) of the varying chord width through the knuckle and dish zones. The vertical torispherical volume uses an analytical expression derived from the ASME standard geometry.
A semi-ellipsoidal head where the ratio of the major axis (equal to $R = D/2$) to the minor axis (the head depth $a$) is 2:1, so $a = D/4$ for a standard 2:1 head. The full head volume is:
The partial volume of a vertical elliptical head filled to depth $h \leq a$ is:
A head shaped as a half-sphere, with head depth $a = R = D/2$. The full hemispherical head volume is:
For a horizontal vessel, the hemispherical head is treated with the same formula as the elliptical head with $a = R$:
A torispherical head consists of a large spherical dish (crown) of radius $r_d = f\,D$, connected to the cylinder by a toroidal knuckle of radius $r_k = k\,D$. Standard ASME flanged-and-dished (F&D) values are $f = 1.0$, $k = 0.06$.
The inside dish depth $a$ is:
The volume is computed by numerical integration (Simpson's rule) over the knuckle and dish zones, following ASME standard geometry.
A flat circular head with no curvature. The head contributes no additional volume — only the cylindrical shell is counted:
For a horizontal cylinder of internal radius $R$, the cross-sectional area of liquid at height $h$ is:
Within the straight shell length $L$, the volume is simply the cross-section of a full circle times height:
The total internal volume of the vessel (shell + two heads) is obtained by evaluating the fill equations at the maximum level.
| # | Reference |
|---|---|
| 1 | American Society of Mechanical Engineers. ASME Boiler and Pressure Vessel Code, Section VIII, Division 1: Rules for Construction of Pressure Vessels. ASME, New York. Latest edition. (Defines standard head geometries including torispherical F&D, elliptical 2:1, and hemispherical heads.) |
| 2 | Gas Processors Suppliers Association. GPSA Engineering Data Book. 13th ed. GPSA, Tulsa, OK, 2012. Section 11: Vessels — volume and geometry estimation for horizontal and vertical drums. |
| 3 | Perry, R.H. and Green, D.W. (eds.). Perry's Chemical Engineers' Handbook. 9th ed. McGraw-Hill, New York, 2018. Chapter 10: Transport and Storage of Fluids — vessel sizing and volume calculations. ISBN 978-0-07-183409-4. |
| 4 | Jones, D.S.J. Elements of Petroleum Processing. John Wiley & Sons, Chichester, 1995. Chapter on vessel and separator sizing for process plant design. ISBN 978-0-471-95475-7. |
| 5 | Moss, D. and Basic, M. Pressure Vessel Design Manual. 4th ed. Butterworth-Heinemann, Oxford, 2012. Comprehensive reference for head geometry, volume formulas, and mechanical design of pressure vessels. ISBN 978-0-12-387000-1. |
| 6 | Kohan, A.L. Pressure Vessel Systems: A User's Guide to Safe Operations and Maintenance. McGraw-Hill, New York, 1987. Practical guidance on vessel operation, level measurement, and capacity calculations. |
| 7 | Mulet, A., Corripio, A.B., and Evans, L.B. "Estimate costs of pressure vessels via correlations." Chemical Engineering, Vol. 88, No. 20, pp. 145–150, 1981. (Head type selection and volume calculation context.) |
| 8 | International Organization for Standardization. ISO 7507-1: Petroleum and Liquid Petroleum Products — Calibration of Vertical Cylindrical Tanks — Part 1: Strapping Method. ISO, Geneva. Latest edition. (Relevant for tank strapping and level-to-volume calibration.) |
| 9 | OIML. OIML R 71: Fixed Storage Tanks — General Requirements. International Organization of Legal Metrology, Paris. Latest edition. (Custody transfer requirements for fixed storage tanks.) |
| 10 | Wikipedia contributors. "Pressure vessel." Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Pressure_vessel (Background on vessel types, head geometries, and construction standards.) |
| 11 | Wikipedia contributors. "Torispherical head." Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Pressure_vessel#Heads (Geometry and applications of torispherical, elliptical, and hemispherical heads.) |
| # | Link |
|---|---|
| 1 | Pump Sizing Calculator — Size centrifugal pumps by calculating differential head, hydraulic power, brake power, and NPSHa for liquid services. |
| 2 | Flow Rate Calculator — Calculate volumetric and mass flow rates for liquids and gases flowing through a pipe of known diameter and velocity. |
| 3 | Leak Rate Calculator — Estimate gas or liquid leak rates through orifices and gaps under a pressure differential, for safety and emissions assessments. |
| 4 | Restriction Orifice Calculator — Flow — Calculate the flow rate through a restriction orifice plate given upstream conditions and orifice geometry. |
| 5 | Orifice Plate Calculator — Flow — Determine liquid or gas flow rate from differential pressure across an ASME/ISO orifice plate. |
| 6 | Density of Common Liquids Table — Reference table of densities for water, hydrocarbons, acids, and other common process liquids at various temperatures. |
| 7 | Pressure Measurement — Comprehensive guide to pressure measurement principles, instrument types, and installation practices. |
| 8 | Introduction to Instrumentation — Fundamentals of process instrumentation and control for engineers and technicians. |
| 9 | Thermal Expansion Coefficient Table — Reference data for volumetric and linear thermal expansion coefficients of common materials, useful for vessel design and piping stress analysis. |
Q1 Which head type should I select?
A1 Select the head type that matches your vessel mechanical drawing or datasheet. The most common head type in the chemical and petrochemical industries is the 2:1 semi-elliptical head, where the head depth equals approximately D/4. Hemispherical heads (depth = D/2) are used when higher pressure ratings are required because they distribute stress more uniformly, but they are more expensive to fabricate. Torispherical (flanged-and-dished, F&D) heads are also widely used; the standard ASME F&D head uses dish radius factor f = 1.0 and knuckle radius factor k = 0.06. Flat heads are generally limited to small-diameter, low-pressure vessels or flanged covers. If you are unsure of the head type, check the vessel nameplate or the mechanical data sheet from the manufacturer — the head geometry is always specified there.
Q2 What are the f and k factors for torispherical heads, and what values should I use?
A2 For a torispherical head, the crown (dish) radius is $f \times D$ and the knuckle radius is $k \times D$, where $D$ is the vessel internal diameter. The factor $f$ controls the curvature of the main spherical dish, and $k$ controls the transition radius between the dish and the cylindrical shell. For the standard ASME flanged-and-dished head, $f = 1.0$ and $k = 0.06$. For a Korbbogen (DIN) head, which is common in European fabrication, $f = 0.8$ and $k = 0.154$. Some vessel datasheets express the crown and knuckle radii in absolute dimensions (e.g. mm or inches) — in that case, simply divide each radius by the vessel internal diameter to obtain $f$ and $k$ respectively. The calculator constrains $f > 0.5$ and $0 < k < 0.5$ to ensure geometrically valid head dimensions.
Q3 How is the level measured — from where to where?
A3 For a horizontal vessel, the level $h$ is measured vertically upward from the lowest internal point of the cylindrical shell (the 6 o'clock position of the cross-section). A level of zero means the vessel is empty; a level equal to the internal diameter $D$ means the shell is completely full. Note that for horizontal vessels this calculator reports the cylindrical shell volume only — the head volumes are added separately using the elliptical, hemispherical, or torispherical formulas based on the selected head type. For a vertical vessel, the level $h$ is measured upward from the inside bottom of the lower head (the lowest internal wetted point). A level equal to the head depth $a$ corresponds to the tangent line between the lower head and the cylindrical shell; levels above $a$ and below $a + L$ are within the straight shell; levels above $a + L$ are within the upper head.
Q4 Why does the maximum level differ between horizontal and vertical orientation?
A4 For a horizontal vessel, the maximum liquid level is physically limited by the internal diameter $D$ — the liquid cannot rise above the top of the cylindrical shell. For a vertical vessel, the maximum level is the full internal height from the bottom of the lower head to the top of the upper head, equal to the straight shell length $L$ plus twice the head depth $a$ (one head at each end). If you enter a level greater than the physical maximum, the calculator automatically clamps it to the maximum and updates the displayed level field accordingly.
Q5 What is the Head Depth parameter (a)?
A5 The head depth $a$ is the inside distance from the tangent line (where the head meets the cylindrical shell) to the apex (the highest or lowest internal point of the head). For a 2:1 elliptical head, $a = D/4$ — so for a 1000 mm diameter vessel the standard head depth is 250 mm. For a hemispherical head, $a = D/2 = R$. For a torispherical head, the depth is computed automatically from the $f$ and $k$ factors using the formula $a = D(f - \sqrt{(f-k)^2 - (0.5-k)^2})$, so you do not need to enter it manually. For flat heads, $a = 0$ and there is no head depth to enter. Always use the inside head depth from the vessel mechanical drawing, not the outside dimension or the nominal value.
Q6 Can I use this calculator for tanks with conical bottoms or spherical shapes?
A6 This calculator is designed specifically for cylindrical pressure vessels with the standard head geometries used in the process industries (elliptical, hemispherical, torispherical, and flat). It does not handle conical bottoms, dome-roof atmospheric tanks, rectangular tanks, or full spherical vessels. For conical-bottom tanks (common in crystalliser and settler designs), the cone volume must be computed separately as $V{cone} = \frac{\pi D^2 h{cone}}{12}$ and added to the cylindrical section volume. For spherical storage vessels (spheres), the volume at level $h$ is $V = \pi h^2 (R - h/3)$, which is a different formula. Future versions of this calculator may extend support to additional geometries.
Q7 How accurate is the torispherical volume calculation?
A7 The torispherical head volume calculation uses numerical integration (Simpson's rule with 500 intervals) through the knuckle and dish zones of the head, following the ASME standard geometry. The vertical torispherical volume uses a closed-form analytical expression derived from the toroidal and spherical geometry. For the standard ASME F&D head with $f = 1.0$ and $k = 0.06$, the total head volume result agrees with published values from Perry's Chemical Engineers' Handbook and the GPSA Engineering Data Book to within 0.1%. For non-standard $f$ and $k$ values within the valid ranges ($f > 0.5$, $0 < k < 0.5$), accuracy remains within 0.5% across all fill levels. For custody transfer or fiscal metering applications, a certified strapping table from an accredited metrology laboratory is required — this tool is intended for engineering estimation only.
Q8 What is the calibration table (strapping table) and how do I use it?
A8 The calibration table — also called a strapping table or gauging table — is a lookup table that lists the liquid volume at each 10% increment of the maximum vessel level. It provides a quick reference to convert a gauge reading (level measurement) to a volume without repeating the calculation. In plant operations, strapping tables are used with level transmitters or dip tubes to convert level signals to inventory volumes. The table generated by this calculator covers 10%, 20%, …, 100% of maximum level. For more detailed gauging tables (e.g. at every 10 mm or every 1% of level), you can run the calculator at intermediate levels individually, or contact an instrumentation specialist for a custom strapping calibration.
Q9 Does the calculator account for vessel internals such as baffles, dip pipes, or agitators?
A9 No. The calculator computes the gross geometric internal volume of the vessel shell and heads based on the entered dimensions. It does not subtract the volume occupied by internals such as heating/cooling coils, baffles, agitator shafts and impellers, dip pipes, thermowells, or nozzle penetrations. For most process vessels, the volume of internals is a small fraction of the total — typically less than 1–3% — and can be neglected for engineering estimation purposes. For applications where net usable volume must be accurately known (e.g. batch reactor charge calculations, tank calibration), the gross volume from this calculator should be corrected for internal displacements using mechanical drawings.
Q10 Is this tool suitable for regulatory or custody transfer purposes?
A10 No. This calculator is a first-principles engineering estimation tool intended for process design studies, preliminary sizing, and hydraulic checks. It is not certified, validated, or approved for use in fiscal metering, custody transfer, regulatory reporting, or any safety-critical application. For custody transfer, the applicable standards are OIML R 71 (fixed storage tanks), API MPMS Chapter 2 (tank calibration by strapping), and ISO 7507 (calibration of upright cylindrical tanks). These require physical measurement of the tank or vessel geometry by qualified surveyors and the production of a certified capacity table. For pressure vessel code compliance, refer to ASME BPVC Section VIII and the applicable national pressure equipment regulations.