Understanding the Calculation of the Volume of a Horizontal Cylindrical Tank

Calculating the volume of a horizontal cylindrical tank is essential for accurate fluid management. This process converts geometric measurements into usable volume data.

This article explores detailed formulas, common values, and real-world applications for precise volume calculations. You will find comprehensive tables and step-by-step examples.

• Calculate the volume of a horizontal cylindrical tank with diameter 2m and length 5m filled to 1.2m depth.
• Determine the volume of liquid in a 3m diameter, 10m long horizontal tank at 2.5m fill height.
• Find the remaining volume in a 1.5m diameter tank, 4m long, filled to 0.8m depth.
• Compute the volume of a horizontal cylindrical tank with diameter 2.5m, length 6m, filled halfway.

Common Dimensions and Volumes of Horizontal Cylindrical Tanks

Mathematical Formulas for Volume Calculation

Calculating the volume of liquid in a horizontal cylindrical tank requires understanding the geometry of the partially filled cylinder. The key variables are:

• D: Diameter of the cylinder (m)
• L: Length of the cylinder (m)
• h: Height of the liquid fill (m)
• r: Radius of the cylinder, r = D/2 (m)

The volume of liquid V in the tank is calculated by finding the area of the circular segment formed by the liquid surface and multiplying by the length of the tank.

Step 1: Calculate the radius

r = D / 2

Step 2: Calculate the central angle θ (in radians) corresponding to the liquid height

θ = 2 × arccos((r – h) / r)

This angle represents the sector of the circle filled with liquid.

Step 3: Calculate the area A of the circular segment

A = (r² / 2) × (θ – sin(θ))

The area A is the cross-sectional area of the liquid in the tank.

Step 4: Calculate the volume V of liquid in the tank

V = A × L

This gives the volume in cubic meters (m³) if all dimensions are in meters.

Summary of the formula in one expression:

V = L × (r² / 2) × (θ – sin(θ)) where θ = 2 × arccos((r – h) / r)

Explanation of Variables and Typical Values

• Diameter (D): Common diameters range from 1 m to 5 m for industrial tanks.
• Length (L): Lengths vary widely, typically 2 m to 12 m depending on storage needs.
• Fill height (h): Varies from 0 (empty) to D (full).
• Radius (r): Half of diameter, critical for calculating the segment area.
• Central angle (θ): Ranges from 0 to 2π radians (0 to 360 degrees), representing the liquid segment.

Note that when h = 0, volume is zero; when h = D, volume equals the full tank volume, which is:

V_full = π × r² × L

Additional Formulas for Related Calculations

Full tank volume

V_full = π × r² × L

Percentage fill of the tank

Fill % = (V / V_full) × 100

Inverse calculation: Find fill height from volume

Finding h given V requires numerical methods since the formula is transcendental. Iterative methods such as the Newton-Raphson algorithm or bisection method are commonly used.

Real-World Application Examples

Example 1: Fuel Storage Tank Volume Calculation

A horizontal cylindrical fuel tank has a diameter of 2 meters and a length of 5 meters. The fuel level is measured at 1.2 meters. Calculate the volume of fuel in the tank.

• Given: D = 2 m, L = 5 m, h = 1.2 m
• Calculate radius: r = 2 / 2 = 1 m
• Calculate central angle θ:

θ = 2 × arccos((1 – 1.2) / 1) = 2 × arccos(-0.2) ≈ 2 × 1.772 = 3.544 radians

• Calculate segment area A:

A = (1² / 2) × (3.544 – sin(3.544)) = 0.5 × (3.544 – (-0.390)) = 0.5 × 3.934 = 1.967 m²

• Calculate volume V:

V = 1.967 × 5 = 9.835 m³

The tank contains approximately 9.835 cubic meters of fuel.

Example 2: Water Tank Monitoring for Industrial Process

An industrial process uses a horizontal cylindrical water tank with diameter 3 meters and length 6 meters. The water level sensor reads 2.5 meters. Determine the volume of water in the tank and the percentage fill.

• Given: D = 3 m, L = 6 m, h = 2.5 m
• Calculate radius: r = 3 / 2 = 1.5 m
• Calculate central angle θ:

θ = 2 × arccos((1.5 – 2.5) / 1.5) = 2 × arccos(-0.6667) ≈ 2 × 2.3005 = 4.601 radians

• Calculate segment area A:

A = (1.5² / 2) × (4.601 – sin(4.601)) = (2.25 / 2) × (4.601 – (-0.993)) = 1.125 × 5.594 = 6.293 m²

• Calculate volume V:

V = 6.293 × 6 = 37.76 m³

• Calculate full tank volume:

V_full = π × 1.5² × 6 = 3.1416 × 2.25 × 6 = 42.41 m³

• Calculate fill percentage:

Fill % = (37.76 / 42.41) × 100 ≈ 89%

The tank is approximately 89% full with 37.76 cubic meters of water.

Additional Considerations and Best Practices

• Measurement Accuracy: Ensure precise measurement of fill height using calibrated sensors or manual gauges.
• Temperature Effects: Account for thermal expansion of liquids and tank materials, especially in fuel storage.
• Tank Shape Variations: For tanks with elliptical or other cross-sections, different formulas apply.
• Regulatory Compliance: Follow standards such as API 650 for tank design and volume calculations.
• Software Tools: Utilize specialized software for complex volume calculations and real-time monitoring.

References and Further Reading

• Engineering Toolbox: Cylindrical Tank Volume
• American Petroleum Institute (API) Standards
• Tank Connection: How to Calculate Tank Volume
• Wermac: Cylindrical Tank Volume Calculator