In a rotary dryer used for solids handling, what is the correct expression for the average solids retention time?\n\nUse the following symbols:\nZ = length of the dryer (m)\nρ = apparent solids density (kg/m^3)\nA = dryer internal cross-sectional area (m^2)\nL = mass flow rate of dry solids (kg/s)\nH = volumetric hold-up fraction of solids (dimensionless)

Introduction / Context:
Rotary dryers are widely used in chemical and process industries to reduce moisture content of bulk solids. A key design and operating parameter is the average solids retention time, sometimes called residence time. Knowing how long solids remain in the dryer helps size the unit, set rotational speed, and balance moisture removal with throughput. This question asks for the correct formula based on standard mass and holdup concepts using commonly defined symbols.

Given Data / Assumptions:

• Z is the dryer length in metres.
• ρ (rho) is the apparent bulk density of the solids in kg/m^3.
• A is the internal cross-sectional area available for solids in m^2.
• L is the mass flow rate of dry solids in kg/s.
• H is the volumetric solids hold-up fraction (dimensionless) representing the fraction of the internal volume that is occupied by solids on average.

Concept / Approach:
The average retention time equals the total mass of solids held inside the dryer divided by the mass flow rate of solids. Mass held inside equals bulk density times the actual solids volume inside the shell. That actual volume is the product of the geometric volume (A * Z) and the hold-up fraction H. Therefore, retention time follows directly from a mass balance on the control volume of the drum interior.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: ρ (kg/m^3) * A (m^2) * Z (m) gives kg; multiplying by H (dimensionless) keeps kg. Dividing by L (kg/s) yields seconds, which confirms unit consistency. The relationship also aligns with general residence time formulas used in solids handling equipment (inventory divided by throughput).

Why Other Options Are Wrong:

• (b) Places L in the numerator, which would imply residence time increases with throughput—physically incorrect.
• (c) Omits density ρ, giving wrong units and ignoring mass/volume conversion.
• (d) Puts area A in the denominator, reducing time with larger cross-section, contrary to the inventory principle.

Common Pitfalls:
Confusing H as a mass fraction instead of volumetric fraction, or forgetting to include density to convert volume to mass. Another frequent mistake is to overlook cross-section area, which directly scales inventory in long drums.

Final Answer:
t = (ρ * A * Z * H) / L