Introduction & Context
Mixing endpoint detection is a critical process engineering technique used to determine the precise moment when a batch mixture achieves homogeneity. In industrial stirred-tank reactors, monitoring the power draw of the impeller provides a non-invasive, real-time method to track the progression of blending. As components with different physical properties (specifically density and viscosity) are combined, the resistance against the impeller changes. Once the mixture reaches a uniform state, the power draw stabilizes, signaling the completion of the unit operation. This method is widely applied in chemical, pharmaceutical, and food processing industries to optimize batch times, reduce energy consumption, and ensure product consistency.
Methodology & Formulas
The calculation relies on the relationship between fluid dynamics and mechanical power consumption in a fully baffled stirred tank. The following formulas define the system state:
The Reynolds number, which characterizes the flow regime, is calculated as:
\[ Re = \frac{\rho \cdot N \cdot d^{2}}{\mu} \]
The power draw required to maintain a constant impeller speed is determined by the power number, which remains constant in the turbulent regime:
\[ P = Po \cdot \rho \cdot N^{3} \cdot d^{5} \]
When monitoring the mechanical load on the drive shaft, the torque is derived from the power draw and the rotational speed:
\[ M = \frac{P}{2 \cdot \pi \cdot N} \]
The change in power draw between the initial state and the final homogeneous mixture is defined as:
\[ \Delta P = P_{final} - P_{initial} \]
| Parameter |
Condition / Threshold |
Significance |
| Flow Regime |
\( Re > 10{,}000 \) |
Ensures turbulent flow and constant Power Number (\(Po\)). |
| Geometry |
\( H = T \) |
Standardized tank geometry for valid \(Po\) correlation. |
| Endpoint Detection |
\( \dfrac{dP}{dt} < \text{threshold} \) |
Signal stabilization indicates mixture homogeneity. |
Worked Example: Mixing Endpoint Detection via Power Draw
A batch mixing operation uses a fully baffled stirred tank with a Rushton turbine. The tank diameter is 0.9 m and liquid height is 0.9 m (\(H = T\)). The impeller diameter is 0.3 m and rotates at 5.0 s⁻¹ (300 rpm). The initial liquid has a density of 1000.0 kg/m³ and viscosity of 0.001 Pa·s. After blending with a second liquid, the final mixture has a density of 1100.0 kg/m³ and viscosity of 0.002 Pa·s. The power number for the Rushton turbine in the turbulent regime is 5.0. The endpoint of mixing is detected when the power draw plateaus at the theoretical final value.
Knowns
- Tank diameter \(T = 0.9\ \text{m}\)
- Liquid height \(H = 0.9\ \text{m}\)
- Impeller diameter \(d = 0.3\ \text{m}\)
- Impeller speed \(N = 5.0\ \text{s}^{-1}\)
- Power number \(Po = 5.0\)
- Initial density \(\rho_{A} = 1000.0\ \text{kg/m}^{3}\)
- Initial viscosity \(\mu_{A} = 0.001\ \text{Pa·s}\)
- Final density \(\rho_{mix} = 1100.0\ \text{kg/m}^{3}\)
- Final viscosity \(\mu_{mix} = 0.002\ \text{Pa·s}\)
Step-by-Step Calculation
- Calculate initial Reynolds number to confirm turbulent regime:
\[
Re_{A} = \frac{\rho_{A} \cdot N \cdot d^{2}}{\mu_{A}} = \frac{1000.0 \times 5.0 \times (0.3)^{2}}{0.001} = 450{,}000
\]
Since \(Re_{A} > 10{,}000\), the flow is turbulent and the power number is constant.
- Calculate final Reynolds number:
\[
Re_{mix} = \frac{\rho_{mix} \cdot N \cdot d^{2}}{\mu_{mix}} = \frac{1100.0 \times 5.0 \times (0.3)^{2}}{0.002} = 247{,}500
\]
Again \(Re_{mix} > 10{,}000\), confirming turbulent regime throughout.
- Select power number: For a Rushton turbine at \(Re > 10^{4}\), \(Po = 5.0\).
- Compute initial power draw:
\[
P_{init} = Po \cdot \rho_{A} \cdot N^{3} \cdot d^{5} = 5.0 \times 1000.0 \times (5.0)^{3} \times (0.3)^{5} = 1518.75\ \text{W}
\]
- Compute final power draw (fully mixed):
\[
P_{final} = Po \cdot \rho_{mix} \cdot N^{3} \cdot d^{5} = 5.0 \times 1100.0 \times (5.0)^{3} \times (0.3)^{5} = 1670.625\ \text{W}
\]
- Compute initial torque:
\[
M_{init} = \frac{P_{init}}{2 \cdot \pi \cdot N} = \frac{1518.75}{2 \pi \times 5.0} = 48.343\ \text{N·m}
\]
- Compute final torque:
\[
M_{final} = \frac{P_{final}}{2 \cdot \pi \cdot N} = \frac{1670.625}{2 \pi \times 5.0} = 53.178\ \text{N·m}
\]
- Expected power increase:
\[
\Delta P = P_{final} - P_{init} = 1670.625 - 1518.75 = 151.875\ \text{W}
\]
- Endpoint detection criterion: Monitor the real-time power draw. The endpoint is reached when the measured power stabilizes at \(P_{final} = 1670.625\ \text{W}\) within ±2% for at least 10 seconds, or when the time-derivative of power falls below 0.1% of \(P_{final}\) per second.
Final Answer
The power draw plateaus at 1670.625 W when the mixture is homogeneous. The corresponding torque is 53.178 N·m. The expected increase from the initial state is 151.875 W. This change is easily detectable with standard wattmeters or torque sensors, confirming that the mixing endpoint can be reliably identified via power draw monitoring.
