Tank Dish Volume Calculator: How It Works

A tank dish volume calculator works out the volume of a dished (curved) tank end, such as the heads on a pressure vessel or storage tank. You enter the dish’s diameter, crown radius, and knuckle radius (or the dish depth), and it returns the volume in gallons or liters. This is useful for estimating the total capacity of a tank with dished ends, since the heads hold a significant amount of liquid beyond the cylindrical section.

Formula

The calculator uses the formula for the volume of a spherical cap, because a dished tank end is essentially a portion of a sphere. The formula is:

V = (π × h² × (3r – h)) / 3

Where:

• V = volume of the dish (cubic inches, then converted to gallons or liters)
• h = dish depth (the height of the cap, in inches)
• r = radius of the dish’s base (half the diameter, in inches)
• π ≈ 3.14159

To get gallons, divide the cubic inches by 231 (since 1 US gallon = 231 cubic inches). To get liters, multiply cubic inches by 0.0163871 (since 1 cubic inch = 0.0163871 liters).

Worked Example

Using the default values from the calculator: diameter = 48 inches, depth = 12 inches.

1. Radius r = 48 / 2 = 24 inches.
2. h² = 12² = 144.
3. 3r – h = 3 × 24 – 12 = 72 – 12 = 60.
4. Multiply: 144 × 60 = 8640.
5. Multiply by π: 8640 × 3.14159 ≈ 27143.3 cubic inches.
6. Divide by 3: 27143.3 / 3 ≈ 9047.77 cubic inches.
7. Convert to gallons: 9047.77 / 231 ≈ 39.17 US gallons.
8. Convert to liters: 9047.77 × 0.0163871 ≈ 148.26 liters.

The calculator shows 39.17 gallons and 148.26 liters, matching these steps.

When to Use This Calculator

Use this calculator when you need to know the volume of a single dished end, for example:

• Estimating the total capacity of a horizontal or vertical tank with dished heads.
• Calculating the volume of liquid held in the curved ends of a pressure vessel.
• Comparing different dish depths for a given diameter to see how much extra volume they add.
• Working out the volume of a spherical cap for any other engineering or DIY project.

Note that this formula gives the volume of the dish itself, not the cylindrical section. To get total tank volume, add the volume of the cylinder (π × r² × length) plus both dish ends.

How to Choose the Right Dish Dimensions

Dished tank ends come in several standard shapes, and the depth (h) depends on the type:

• Ellipsoidal (2:1): Depth is about 1/4 of the diameter. For a 48-inch dish, h ≈ 12 inches. This is the most common for pressure vessels.
• Torispherical (ASME F&D): Depth is about 1/5 to 1/6 of the diameter, depending on the knuckle radius. For a 48-inch dish, h might be 8–10 inches.
• Hemispherical: Depth equals half the diameter (h = r). For a 48-inch dish, h = 24 inches. This gives the largest volume but uses more material.

If you don’t know the depth, measure it directly on the dish. If you only know the type, use the standard ratio to estimate h. The calculator’s default (48-inch diameter, 12-inch depth) corresponds to a 2:1 ellipsoidal head.

Common Mistakes and How to Avoid Them

• Using diameter instead of radius: The formula uses radius (half the diameter). Forgetting to divide by 2 will overestimate volume by 8 times.
• Mixing units: If you measure in inches, the volume is in cubic inches. Convert to gallons or liters correctly. If you measure in millimeters, convert to inches first, or the result will be off.
• Confusing depth with height: The dish depth is the distance from the base plane to the top of the curve, not the overall height of the tank.
• Forgetting to add both ends: A tank usually has two dished ends. Multiply the single-dish volume by 2 for total head volume.
• Using the wrong π: Use 3.14159 or the π button on your calculator, not 3.14, for accuracy.

Limitations of the Formula

The spherical cap formula assumes the dish is a perfect sphere segment. Real dished heads (ellipsoidal, torispherical) are not exactly spherical, so the calculated volume is an approximation. For most practical purposes, the error is small (typically under 2% for standard heads). However, for precise engineering calculations, you should use the exact geometry of the head type or consult the manufacturer’s data.

Also, the formula gives the internal volume of the dish if you use internal dimensions. If you use external dimensions, you get the external volume, which includes the metal thickness. For liquid capacity, use internal dimensions.

Frequently Asked Questions

Q: What is a tank dish?
A: A tank dish is the curved end of a cylindrical tank or pressure vessel. It is shaped like a shallow bowl or cap and is welded or bolted to the cylinder to close it. Common types are ellipsoidal, torispherical, and hemispherical.

Q: How do I measure the depth of a dish?
A: Place a straightedge across the rim of the dish (the flat circular opening). Measure from the straightedge down to the deepest point of the curve. That is the depth (h). Make sure the dish is lying flat on its rim.

Q: Can I use this for a hemispherical head?
A: Yes. For a hemisphere, the depth equals the radius (h = d/2). Enter the diameter and depth accordingly, and the calculator will give the correct volume, which is half the volume of a full sphere.

Q: How do I find the total volume of a tank with two dished ends?
A: Calculate the volume of the cylindrical section using V_cyl = π × r² × length (in cubic inches). Then calculate the volume of one dish using this calculator, multiply by 2, and add it to the cylinder volume. Convert the total to gallons or liters.

Q: Does this calculator work for metric measurements?
A: The calculator uses inches and outputs gallons and liters. If you have metric dimensions, convert them to inches first (1 inch = 25.4 mm). Alternatively, you can use the formula with centimeters and convert the result to liters (1 liter = 1000 cm³).

Q: Why is the dish volume important?
A: The dished ends hold a significant portion of a tank’s total volume, especially in smaller tanks. Knowing the exact volume helps with filling, chemical dosing, and capacity planning. It also affects the weight and cost of the tank.

Related

• Round Tank Volume Calculator: Gallons & Liters
• Septic Tank Volume Calculator: How It Works
• Sphere Tank Volume Calculator: How to Find Capacity

in d | Dish diameter | in | 48 | 0
in h | Dish depth | in | 12 | 0
out vol_gal | Volume (US gallons) | gal | (pi * h^2 * (3 * (d/2) - h) / 3) / 231 | 2
out vol_l | Volume (liters) | L | (pi * h^2 * (3 * (d/2) - h) / 3) * 0.0163871 | 2