Agitation in laminar regime: during liquid mixing, which variable does not make the power consumption proportional when flow is strictly laminar and impeller geometry is fixed?

Introduction / Context:
Power draw in mechanically agitated vessels depends on flow regime. In the laminar regime (low Reynolds number), viscous forces dominate and the classical turbulent correlations (constant Power number) do not apply. Understanding which variables control power under laminar flow is essential for sizing lab mixers, dosing tanks, and high-viscosity blending operations without over- or underpowering the drive.

Given Data / Assumptions:

• Flow is strictly laminar (Re << 10).
• Impeller geometry and tank dimensions are fixed (geometric similarity).
• Steady, single-phase liquid mixing; no gas dispersion or solids suspension.

Concept / Approach:
In laminar mixing, the Power number, Po = P / (rho * N^3 * D^5), varies inversely with Reynolds number, Re = rho * N * D^2 / mu. For many laminar impellers, Po = K / Re, where K is a geometry constant. Rearranging gives P = K * mu * N^2 * D^3. Thus, P is directly proportional to viscosity (mu), to the square of rotational speed (N^2), and to the cube of impeller diameter (D^3). Notably, density (rho) cancels from the laminar result; hence, power is not proportional to density in this regime.

Step-by-Step Solution:
Start with Po = P / (rho * N^3 * D^5).Use Po = K / Re and Re = rho * N * D^2 / mu.Therefore, P / (rho * N^3 * D^5) = K * mu / (rho * N * D^2).Solve for P: P = K * mu * N^2 * D^3 (no rho term remains).

Verification / Alternative check:
Practical scale-ups for viscous laminar systems hold N^2 * D^3 * mu constant to preserve shear fields and power density, further confirming the proportionalities above and the independence from density.

Why Other Options Are Wrong:
Viscosity: Appears linearly in P = K * mu * N^2 * D^3; power is proportional to mu.Impeller diameter cubed: Power scales with D^3 in laminar mixing.Rotational speed squared: Power scales with N^2 in laminar mixing.

Common Pitfalls:
Confusing turbulent and laminar correlations; using a constant Power number at low Re leads to major sizing errors. Also, overlooking temperature effects on viscosity can cause large power deviations in laminar service.

Final Answer:
density of the liquid