Cylindrical Reactor Geometry — If the liquid height is HL and the tank diameter is Dt, which expression correctly gives the liquid volume VL (assume a right circular cylinder)?

Introduction:
Accurate reactor volume calculations are crucial for material balances, dosing, and scale-up. For a cylindrical, baffled mixing tank, the liquid volume depends on the cross-sectional area of the tank and the filled height.

Given Data / Assumptions:

• Tank is a right circular cylinder with internal diameter Dt.
• Liquid height is HL (no dished heads considered).
• No internal structures displacing significant volume for this calculation.

Concept / Approach:

The volume of a cylinder is base area times height. The base area is the area of a circle with diameter Dt: A = π * (Dt/2)^2 = π * Dt^2 / 4. Multiplying by the liquid height gives the total liquid volume VL. This expression is widely used for instantaneous inventory calculations in batch and fed-batch operations.

Step-by-Step Solution:

Verification / Alternative check:

Compare with known volumes (for example, Dt = 1 m, HL = 1 m ⇒ VL ≈ 0.785 m^3), consistent with geometric expectations.

Why Other Options Are Wrong:

A mixes spherical and cylindrical forms; C misses the factor 1/4; D lacks height and thus cannot be a volume; E is not the standard cylinder formula.

Common Pitfalls:

Forgetting to subtract dished-head or internal volumes when high accuracy is required; always confirm internal geometry if precision is needed.

Final Answer:

VL = HL * π * (Dt^2 / 4)