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)?
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AVL = (4/3) * π * HL * (Dt^3 / 8)
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BVL = HL * π * (Dt^2 / 4)
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CVL = HL * π * Dt^2
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DVL = 4 * π * Dt^2
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EVL = π * HL^2 * Dt / 4
Answer
Correct Answer: VL = HL * π * (Dt^2 / 4)
Explanation
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:
Compute base area: A = π * (Dt^2 / 4).Multiply by height HL: VL = A * HL.Therefore, VL = HL * π * (Dt^2 / 4).Check units: if Dt and HL are in meters, VL is in cubic meters.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)