Engineering task and calculation objective
The TKL module calculates filling levels and filling volumes in storage tanks: for horizontal and vertical cylindrical vessels with torispherical heads (Kloepper or Korbbogen type), hemispherical heads or elliptical heads, it determines the liquid volume from the filling level – and conversely the filling level belonging to a target volume. Anyone who needs to calculate tank volume or create a gauge table (level-volume table) for a vessel will find the complete geometry, including the dished ends, represented here.
Inputs are the orientation (horizontal or vertical), head type and head arrangement (convex, concave or mixed), tank outside or inside diameter, shell wall thickness and the length of the cylindrical part. From these, the module delivers the total inside height, the maximum filling volume, the liquid volume and filling fraction at the current level and – via the density of the medium – the total mass of the tank contents.
In practice, this calculation is needed constantly: for gauge tables and level measurements in the tank farm, for inventory determination and custody transfer, for the design of overfill protection (permissible filling fraction) and for weight data in structural and foundation design. Especially for horizontal tanks, the relationship between level and volume is strongly non-linear, and the dished heads contribute a share that cannot be neglected.
Calculation workflow
- Define the vessel geometry: Orientation (horizontal/vertical), head type (torispherical Kloepper or Korbbogen head, hemispherical or elliptical head), head arrangement, outside diameter and shell wall thickness are entered; from these follows the inside diameter. The length of the cylindrical part completes the geometry.
- Calculate overall dimensions and maximum volume: The module determines the total inside height (for vertical orientation) or inside length of the vessel and the maximum filling volume as the sum of the cylindrical part and the two head volumes according to the selected head type.
- Specify the level and determine the partial volume: For the current filling level, the module calculates the liquid volume: for the vertical tank via the cross-sectional area and the filling height in head and cylinder region, for the horizontal tank via the circular segment area of the partly filled cylinder plus the partly filled head caps.
- Report filling fraction and mass: From partial and maximum volume follows the filling fraction; with the density of the medium, the total mass of the tank contents is calculated – the basis for inventory determination, structural design and overfill protection.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Outside diameter | D | mm |
| Length of the cylindrical part | L | mm |
| Wall thickness | s | mm |
| Filling level of the tank | h | mm |
| Knuckle radius | r2 | mm |
| Crown radius | r1 | mm |
| Skirt height | h1 | mm |
| Total liquid volume of the tank | Vges | m³ |
| Inside diameter | d | mm |
| Total outside height of the head | hBod | mm |
| Feed / efflux | V | m³/h |
| 1 | 1 | min |
| 2 | 2 | min |
| 3 | 3 | min |
| 4 | 4 | min |
| 5 | 5 | min |
| 6 | 6 | min |
| Tiefstand | Tiefstand | mm |
| Tiefalarm | Tiefalarm | mm |
| Normalstand | Normalstand | mm |
| Hochalarm | Hochalarm | mm |
| Hochstand | Hochstand | mm |
| 1 | 1 | m³ |
| 2 | 2 | m³ |
Worked example
A horizontal cylindrical storage tank has an inside diameter of 2,000 mm and a cylindrical length of 6 m. The filling level is 600 mm above the lowest point. This worked example determines the liquid volume of the cylindrical part (the contributions of the two torispherical heads are additionally determined by the module; they are neglected in this hand calculation).
Given values
| Inside diameter Di | 2,000 mm (R = 1.0 m) |
| Cylindrical length L | 6 m |
| Filling level h | 600 mm |
Solution
Circular segment area of the partly filled cross-section
A = R² · arccos((R − h)/R) − (R − h) · √(2·R·h − h²)
A = 1.0² · arccos(0.4/1.0) − 0.4 · √(2 · 1.0 · 0.6 − 0.6²)
A = 1.1593 − 0.4 · √0.84 = 1.1593 − 0.4 · 0.9165 = 1.1593 − 0.3666 ≈ 0.7927 m²
Liquid volume in the cylindrical part
V = A · L = 0.7927 · 6 ≈ 4.76 m³
Filling fraction of the cylindrical part
Full cylinder volume: Vfull = π · R² · L = π · 1.0² · 6 ≈ 18.85 m³
Filling fraction: 4.76 / 18.85 ≈ 25.2% – even though the level is at 30% of the diameter. This illustrates the non-linearity of the level-volume relationship in a horizontal tank.
Result
| Segment area A | ≈ 0.793 m² |
| Volume of cylindrical part V | ≈ 4.76 m³ |
| Filling fraction (cylindrical part) | ≈ 25.2% |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why is the level-volume curve non-linear for a horizontal tank?
In a horizontal cylinder, the free liquid surface changes with the filling height: at mid-height (half full) it is largest, and there one centimeter of level corresponds to the greatest volume increment; near the bottom and the top the increment is small. A linear level-to-volume conversion therefore leads to considerable errors – which is why the circular segment calculation, or a gauge table, is needed.
How much do the dished heads contribute to the volume?
That depends on the slenderness of the vessel. A Kloepper (torispherical) head holds around 0.1 · Di³, a Korbbogen (semi-ellipsoidal type) head about 0.13 · Di³, a hemispherical head π/12 · Di³ ≈ 0.26 · Di³. For short, squat vessels, the two heads together can easily account for 10 to 20% of the total volume – neglecting them would be inadmissible for gauge tables or custody transfer purposes.
What does the head arrangement convex/concave mean?
Convex means the head bulges away from the vessel interior (the normal case for both heads). Concave means an inward-dished design, as occurs for example in stacked vessels or special configurations; the head volume then reduces rather than increases the total. The mixed arrangement (convex/concave) covers vessels with different heads at the two ends.
Which diameter is the volume calculated with – outside or inside?
The inside diameter always governs the filling volume. The module converts from outside diameter and shell wall thickness to the inside diameter. For precise inventory determination it must also be considered that internals (heating coils, agitators, baffles) displace volume and that the usable filling fraction is limited by the overfill protection.