Multiple Impeller Spacing Calculation

Reference ID: MET-D65B | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The multiple‑impeller spacing calculation determines the axial arrangement of several agitator impellers within a cylindrical mixing tank. Proper spacing ensures that each impeller operates in a hydrodynamically independent region, avoids mechanical interference, and provides adequate power transfer to the fluid. This analysis is routinely applied in the design of large‑scale stirred‑tank reactors, fermenters, and chemical‑process vessels where high‑shear mixing and uniform residence time are required.

Methodology & Formulas

1. Reynolds number (turbulent regime check)

\[ Re = \frac{\rho \, N \, d^{2}}{\mu} \]

2. Froude number (vortex formation check)

\[ Fr = \frac{N^{2} \, d}{g} \]

3. Spacing factor (α) enforcement

The user‑provided spacing factor α_input is limited to a minimum value α_min to prevent impeller overlap:

\[ \alpha = \max\!\left(\alpha_{\text{input}},\; \alpha_{\text{min}}\right) \]

4. Centre‑to‑centre spacing (S)

\[ S = \alpha \, d \]

5. Axial positions of impeller centres (z_i)

\[ z_{i}= C_{b} + \frac{d}{2} + i\,S \qquad i = 0,1,\dots ,N_{\text{imp}}-1 \]

6. Stack height (H_stack)

\[ H_{\text{stack}} = d + (N_{\text{imp}}-1)\,S \]

7. Axial coverage of the tank (η)

\[ \eta = \frac{H_{\text{stack}}}{H} \]

8. Power per impeller (P_imp) using the Rushton correlation

\[ P_{\text{imp}} = P_{0}\,\rho\,N^{3}\,d^{5} \]

9. Total shaft power (P_total)

\[ P_{\text{total}} = N_{\text{imp}} \, P_{\text{imp}} \]

Validity Checks

Parameter	Expression	Limit / Criterion	Action / Note
Reynolds number	\(Re\)	\(Re \ge Re_{\text{crit}}\)	Warn if \(Re < Re_{\text{crit}}\) – turbulent correlation may be invalid.
Froude number	\(Fr\)	\(Fr \le Fr_{\text{crit}}\)	Note if \(Fr > Fr_{\text{crit}}\) – vortex formation may require baffles.
Spacing factor (α)	\(\alpha_{\text{input}}\)	\(\alpha_{\text{min}} \le \alpha_{\text{input}} \le \alpha_{\text{max}}\)	Adjust to \(\alpha\) if outside range to avoid overlap.
Axial coverage	\(\eta\)	\(\eta \ge \eta_{\text{min}}\)	Warn if \(\eta < \eta_{\text{min}}\) – consider larger spacing or more impellers.
Top clearance	\(C_{\text{top}} = H - \bigl(C_{b}+H_{\text{stack}}\bigr)\)	\(C_{\text{top}} \ge C_{\text{top,min}}\)	Warn if \(C_{\text{top}} < C_{\text{top,min}}\) – increase tank height or reduce impeller count/size.

The optimal spacing is found by balancing hydraulic performance and mechanical constraints. Follow these steps:

Calculate the theoretical head rise for each stage using the pump affinity laws.
Determine the flow‑induced pressure pulsation frequency (typically blade‑pass frequency).
Apply the empirical spacing rule of thumb: Spacing (mm) ≈ 0.5 × (Impeller Diameter in mm). Adjust based on CFD results.
Run a 1‑D stage interaction model to verify that the pressure recovery between stages does not exceed 0.2 MPa, which could cause cavitation.
Validate the chosen spacing with a transient CFD simulation or a physical prototype test.

Efficiency: Too close a spacing increases hydraulic interference, raising losses and reducing overall efficiency. Too large a spacing adds unnecessary shaft length and frictional losses.
Net Positive Suction Head (NPSH): Inadequate spacing can cause pressure troughs between stages, increasing the required NPSH and raising the risk of cavitation.
Mechanical reliability: Excessive spacing may lead to shaft deflection and vibration, while insufficient spacing can cause impeller‑impeller contact under transient loads.

Standard charts provide a good starting point for conventional designs, but custom calculations are recommended when:

The pump operates at extreme speeds (> 10,000 rpm) or very low flow rates.
Non‑standard impeller diameters or blade counts are used.
Fluid properties differ significantly from water (high viscosity, gas‑laden, or corrosive fluids).
Space constraints force the pump into a compact arrangement.

In these cases, perform a detailed CFD or 1‑D stage interaction analysis to fine‑tune the spacing.

Apply a minimum mechanical clearance of 0.5 mm for stainless‑steel impellers and 0.75 mm for cast‑iron impellers to accommodate thermal expansion.
Include a dynamic clearance factor of 10 % of the calculated static clearance to account for shaft deflection and vibration.
For high‑speed applications (> 12,000 rpm), increase the total clearance by an additional 0.2 mm as a precaution against blade‑to‑blade contact.
Verify all clearances against the pump manufacturer’s recommended limits and relevant industry standards (e.g., API 610).

Worked Example – Multiple Impeller Spacing Calculation

A marine-biology research facility is retrofitting a 2 m diameter flat-bottom tank to grow micro-algae in brackish water. Three down-pumping pitched-blade impellers are to be mounted on a single shaft to give uniform axial circulation while avoiding surface vortexing. Determine the vertical spacing between impellers and confirm that the lowest impeller still clears the vessel bottom by the required 10 % of tank diameter.

Knowns

Tank diameter, D = 2.0 m
Liquid height, H = 6.0 m
Number of impellers, num_imp = 3
Impeller diameter, d = 1.0 m
Rotational speed, N = 2.0 rps
Fluid density, ρ = 1030 kg m⁻³
Fluid viscosity, μ = 0.0012 Pa s
Bottom clearance factor, C_b = 0.1 (10 % of D)
Empirical spacing factor, α = 1.0 (from vendor chart for pitched blades)

Step-by-step calculation

Calculate the impeller Reynolds number to confirm turbulent regime:
\[ Re = \frac{\rho N d^{2}}{\mu} = \frac{1030 \times 2.0 \times 1.0^{2}}{0.0012} = 1.717 \times 10^{6} \] Since Re ≫ 10,000, flow is fully turbulent.
Determine the minimum bottom clearance:
\[ z_{1} = C_{b} D = 0.1 \times 2.0 = 0.2 \text{ m} \]
Compute the recommended vertical spacing between successive impellers:
\[ S = \alpha d = 1.0 \times 1.0 = 1.0 \text{ m} \]
Locate the axial position of each impeller starting from the bottom:
\[ \begin{align*} z_{1} &= 0.2 \text{ m (from tank floor)} \\ z_{2} &= z_{1} + S = 0.2 + 1.0 = 1.2 \text{ m} \\ z_{3} &= z_{2} + S = 1.2 + 1.0 = 2.2 \text{ m} \end{align*} \]
Check the submergence of the top impeller below the liquid surface:
\[ \text{Top clearance} = H - z_{3} - \frac{d}{2} = 6.0 - 2.2 - 0.5 = 3.3 \text{ m} \] This exceeds the rule-of-thumb minimum of one impeller diameter, so surface vortexing is unlikely.
Verify total impeller “stack height”:
\[ \text{Stack height} = z_{3} - z_{1} + d = 2.2 - 0.2 + 1.0 = 3.0 \text{ m} \] This is 50 % of the liquid height, satisfying the vendor guideline for axial flow coverage.

Final Answer

Mount the three impellers with 1.0 m vertical spacing. The lowest impeller is 0.2 m above the tank floor and the top impeller is 3.3 m below the liquid surface, giving adequate clearance and coverage for the specified duty.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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