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Volume of a spherical tank : step by step calculation guide and calculator

How to calculate the volume of a spherical tank

What is the volume of a sphere ?

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1. Spherical tank

2. STEP 1 : Calculate the angle α

3. STEP 2 : Calculate the radius r1
4. STEP 3 : Calculate the volume of the spherical cap at height h
5. STEP 4 : Sphere volume calculation (full)
6. STEP 5 : Calculate the filled volume of the spherical tank at height h
7. Spherical tank volume calculator Excel

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1. Spherical tank

Spherical tanks are used in process industries for large storages of liquified gases such as ammonia, LPG, butadiene, propylene. Spherical vessels are particularly interesting for high pressure storage as the spherical shape is more advantgeous than cyclindrical ones to resist pressure (since pressure is evenly distributed) while it also reduces the surface of the tank relative to the volume compared to other tank designs.

Although spherical vessels must be properly insulated, the reduction of the surface is key to limit heat transfer to the stored liquid.

With :
R = tank radius (m)
h = actual height of liquid in the tank (m)
r1 = radius at the surface of the liquid level in the tank (m)
α = angle from center of sphere to liquid level

This page is explaining how to calculate the volume in the tank as a function of the level of liquid h. To calculate the partial volume in the tank it is necessary to know the geometrical characteristics of the sphere (its diameter or radius) as well as the height of liquid from the bottom of the tank.

2. STEP 1 : Calculate the angle α

The angle α can be calculated thanks to the following formula :

cos α = (1-h/R)

3. STEP 2 : Calculate the radius r1

The radius r1 which is radius of the pool of liquid can be calculated thanks to the following formula :

r1 = (R-h) tan α

4. STEP 3 : Calculate the volume of the spherical cap at height h

The volume of the spherical cap can be calculated with the following formula :

V_{spherical_cap} = 1/6*π*h*(3*r₁²+h²)

5. STEP 4 : Sphere volume calculation (full)

To be noted that the formula above allows to find out the well known formula for the volume of a sphere : considering that full r1=0 (level reaches the top of the tank) and that h = 2R, then :

V_{sphere_full} = 1/6*π*2R*(0+(2R)²) = 1/6*π*8*R³ = 4/3*π*R³

6. STEP 5 : Calculate the filled volume of the spherical tank at height h

With respect to the definition of the volume of the spherical cap, there are different cases to consider :

• h < R : the filled volume is simply the volume of the spherical cap that can be calculated directly by the formula given above :

• h > R : the spherical cap is at the top of the tank is therefore empty, the volume of liquid in the tank can then be calculated by substracting the volume of the spherical cap to the total volume of the sphere. The top spherical part has an height equal to 2R-h

• h = R : this is simply 50% of the total volume of the spherical tank

7. Spherical tank volume calculator Excel

You can download a calculator to compute the volume of a partially filled spherical tank : Spherical tank volume calculator Excel.

Warning : this calculator is provided to illustrate the concepts mentioned in this webpage, it is not intended for detail design. It is not a commercial product, no guarantee is given on the results. Please consult a reputable designer for all detail design you may need.

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