Vertical Dish End Tank Volume Calculator

This calculator works out the liquid volume of a vertical cylindrical tank with dished (curved) ends, which are common in pressure vessels and storage tanks. You enter the tank’s diameter, the straight side height, the dish depth (the height of the curved end), and the liquid depth, and it returns the volume of liquid in the tank. It also gives the total tank volume and the percentage filled, so you can quickly gauge capacity at any level.

Formula

The total volume of a vertical tank with two identical dished ends is the sum of the cylindrical middle section and the two end caps. The formula for the volume of a dished head (often called a torispherical or ellipsoidal head) is:

V_head = (π × k × (3d² + 4k²)) / 24

where:

• d = tank diameter (same units as volume)
• k = dish depth (the height of the curved part of each end)

For the cylindrical section, the volume is simply the cross‑sectional area times the height:

V_cyl = (π/4) × d² × h

where h is the straight side height (the part between the two dished ends).

So the total tank volume is:

V_total = (π/4) × d² × h + 2 × V_head

For a partially filled tank, the liquid volume depends on whether the liquid level is within the bottom dish, the cylindrical section, or the top dish. The calculator handles all three cases by splitting the calculation into the bottom dish, the cylindrical part, and the top dish.

Worked Example

Using the default values from the calculator: diameter = 2.0 m, straight side height = 3.0 m, dish depth = 0.25 m, and liquid depth = 2.5 m.

Step 1: Calculate the volume of the cylindrical section (since the liquid depth 2.5 m is greater than the straight side height? Actually, the straight side height is 3.0 m, so the liquid is entirely within the cylindrical section, not reaching the top dish. The bottom dish is fully filled.)

First, the bottom dish volume:

V_dish = (π × 0.25 × (3×(2.0)² + 4×(0.25)²)) / 24 = (π × 0.25 × (12 + 0.25)) / 24 = (π × 0.25 × 12.25) / 24 = (π × 3.0625) / 24 ≈ 0.4007 m³

Step 2: The liquid fills the bottom dish completely (0.4007 m³) and then fills the cylindrical section up to a height of (2.5 – 0.25) = 2.25 m. The cylindrical volume is:

V_cyl = (π/4) × (2.0)² × 2.25 = (π/4) × 4 × 2.25 = π × 2.25 ≈ 7.0686 m³

Step 3: Total liquid volume = 0.4007 + 7.0686 ≈ 7.47 m³ (rounded to two decimals). The calculator shows 7.47 m³, which matches.

Total tank volume = cylindrical section (π/4 × 4 × 3 = 3π ≈ 9.4248) + two dish volumes (2 × 0.4007 = 0.8014) = 10.2262 m³, which rounds to 10.23 m³. The fill percentage = 7.47 / 10.23 × 100 ≈ 73.0%.

When to Use This Calculator

Vertical tanks with dished ends are used in many industries: chemical storage, food processing, pharmaceuticals, and water treatment. They are preferred for high-pressure applications because the curved ends distribute stress more evenly than flat ends. If you need to know how much liquid is in such a tank at a given level—for inventory, mixing, or safety—this calculator gives a quick answer without having to climb a ladder with a tape measure.

The calculator also helps when you are designing a new tank: you can experiment with different diameters, heights, and dish depths to see how the total volume changes. It is also useful for estimating the volume of a partially filled tank during a transfer or when checking the accuracy of a level sensor.

How to Choose the Dimensions

When selecting a tank, you usually start with the required total volume. The total volume is the sum of the cylindrical section and the two dished ends. The dish depth (k) is typically a fraction of the diameter—commonly 10% to 25% for torispherical heads, or about 25% for ellipsoidal heads. For example, a 2-meter diameter tank might have a dish depth of 0.2 to 0.5 meters.

If you know the total volume you need, you can rearrange the formula to solve for the straight side height:

h = (V_total – 2 × V_head) / (π/4 × d²)

Just remember that the dish depth adds to the overall height of the tank, so the total height of the tank is h + 2k.

Common Mistakes and How to Avoid Them

• Mixing units: All dimensions must be in the same unit. If you use meters for diameter, use meters for height and dish depth. If you use feet, the volume will be in cubic feet.
• Forgetting the top dish: When the liquid level is above the straight side, the top dish also contributes to the liquid volume. The calculator handles this, but if you are doing a manual calculation, remember to add the partial top dish volume.
• Using the wrong dish volume formula: The formula used here is for a standard torispherical or ellipsoidal head. If your tank has a different shape (e.g., hemispherical), the volume will be different. For a hemispherical head, the volume is (2/3)πr³, which is larger than the torispherical formula.
• Rounding too early: In the worked example, we rounded intermediate values to four decimal places, which is fine. But if you round to two decimals at each step, you might get a slightly different final answer. Keep extra digits until the end.

Limitations of the Formula

This formula assumes the dish is a perfect torispherical or ellipsoidal shape, which is a good approximation for most commercially manufactured heads. However, the actual internal volume may vary slightly due to manufacturing tolerances. Also, the calculator does not account for the thickness of the tank walls—it calculates the internal volume based on the dimensions you enter, so if you enter the outside diameter, the volume will be slightly larger than the actual internal volume.

For tanks with unusual geometries (e.g., conical bottoms, or heads that are not standard), this calculator will not be accurate. In that case, you would need a more detailed geometric model or a 3D CAD program.

Frequently Asked Questions

What is a dished end? A dished end is the curved cap that closes the end of a cylindrical tank. It can be torispherical (a shallow dome with a knuckle radius), ellipsoidal (a more elongated curve), or hemispherical (a half-sphere). The dish depth is the height of the curved part, not including the straight flange that may be welded to the shell.

How do I measure the dish depth? The dish depth is the distance from the tangent line (where the curve meets the cylindrical shell) to the center of the head. If the head has a straight flange (a short cylindrical section), that is not part of the dish depth—it is part of the straight side.

Can I use this for horizontal tanks? No, this calculator is specifically for vertical tanks. Horizontal tanks have a different formula for partial volume, because the cross-section of the liquid is a segment of a circle, not a full circle.

What if the liquid level is above the cylindrical section? The calculator automatically adds the volume of the top dish, but only the portion that is filled. The formula for the partial top dish is more complex, but the calculator handles it correctly.

Why is the total volume not simply πr²×height? Because the dished ends add volume beyond the cylindrical section. The total height of the tank includes the dish depths, but the cylindrical volume only uses the straight side height. The dish volume is extra.

How accurate is this calculator? It is accurate to the geometry you enter. The main source of error is if you enter dimensions that do not match the actual tank, or if the head shape deviates from the assumed torispherical/ellipsoidal profile.

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in d | Tank diameter | m | 2.0 | 0
in h | Straight side height | m | 3.0 | 0
in k | Dish depth (each end) | m | 0.25 | 0
in l | Liquid depth (from bottom) | m | 2.5 | 0
out vol | Liquid volume | m³ | (pi/4)*d^2*min(l,h) + (pi*k*(3*d^2+4*k^2))/24 + (pi*(l-h)*((d/2)^2+((l-h)-k)^2))/2 | 2
out tot | Total tank volume | m³ | (pi/4)*d^2*h + (pi*k*(3*d^2+4*k^2))/12 | 2
out pct | Fill percentage | % | vol/tot*100 | 1