Technical publication · Fluid mixing

Bioreactor mixing time calculation from multi-probe tracer data

Abstract. This method note defines a reproducible 95% bioreactor mixing-time calculation, prevents a local probe response from being misreported as global homogeneity, and connects the result to Reynolds number, impeller tip speed, power per volume and scale-up review. A synthetic three-probe dataset exposes every calculation.

Acatian EngineeringPublished and reviewed 11 September 202617 min technical publication

Synthetic three-probe tracer response. The bottom probe controls t₉₅ = 50 s; the values are illustrative and contain no customer or experimental Acatian data.

Definition and evidence boundary

Mixing time is a stated homogeneity test, not an equipment label.

Technical question

After a defined tracer pulse, how long does the entire instrumented liquid volume need to enter and remain within a stated band around its final mixed concentration? The answer depends on the homogeneity criterion, injection location, probe positions, tracer, acquisition rate, operating point and fluid properties.

Why several probes

A point sensor records local tracer passage. It can detect an injection shortcut or reach the endpoint before a remote low-circulation region. Pilot-scale experiments and flow-following measurements show why circulation and mixing information are spatially dependent [4].

Evidence classes

Separate measured time series, calibrated vessel inputs and liquid properties from assumed power number or target thresholds. Keep calculated values, uncertainty and engineering judgments in distinct fields. This worked example is entirely synthetic and is suitable for method verification, not process qualification.

Nomenclature

Declare the signal basis and endpoint

SymbolDefinitionUnit or basis

C_j(t)Tracer concentration or calibrated signal at probe j and time tTracer-specific unit

M_j(t)Probe response normalized between pre-pulse and final plateau valuesDimensionless

E(t)Largest absolute deviation |M_j(t) − 1| across all accepted probesDimensionless

t₉₅First time after which E remains at or below 0.05s

NImpeller rotational speeds⁻¹, not rpm

DImpeller diameterm

Normalize each probeM_j(t) = [C_j(t) − C_j,0] / [C_j,∞ − C_j,0]

Estimate C_j,0 and C_j,∞ from stable windows rather than single samples. Normalization makes probes with different offsets comparable, but it cannot repair drift, saturation, nonlinear calibration, chemical reaction, sensor lag or insufficient final equilibration.

Tracer-test workflow

Make the injection, sensing and endpoint reproducible.

1. Fix the test condition

Record vessel and impeller geometry, baffles, working volume, agitation, aeration, pressure, temperature, density, viscosity, surface condition and sensor positions. Use a tracer whose quantity does not materially change density, viscosity, pH or biology. Sodium-chloride conductivity tests have been used to validate CFD mixing predictions in Newtonian and non-Newtonian bioreactor fluids [3].

2. Standardize the pulse

Define injection point, depth, volume, concentration, delivery time and t = 0 trigger. Automated addition can reduce operator-to-operator variability; a 2026 stirred single-use bioreactor study explicitly investigated injection automation and camera-based evaluation to improve repeatability [1].

3. Qualify every sensor

Confirm calibration, time synchronization, sample interval, filtering and response time. A slow pH probe can overstate mixing time; experimental work found that a measured response of about 60 s could still carry roughly 15% probe-related bias [2].

Global strict endpointE(t) = max_j|M_j(t) − 1|; t₉₅ = min{t: E(t′) ≤ 0.05 for every t′ ≥ t}

The persistence clause matters. A probe that briefly crosses the ±5% band and exits again has not met the endpoint. Apply the rule to the recorded interval through a stable final plateau, report the sampling resolution, and repeat the experiment to quantify run-to-run variation.

Synthetic worked example

Three probes yield t₉₅ = 50 s.

Inputs

Use an 8.0 L liquid volume, D = 0.180 m, N = 2.50 s⁻¹ (150 rpm), density ρ = 1,000 kg m⁻³, viscosity μ = 0.001 Pa s and an assumed ungassed power number N_P = 5.0. The normalized top, middle and bottom responses are sampled every 10 s.

Endpoint calculation

At 40 s, the bottom probe is M = 0.92, so E = 0.08 and the criterion fails. At 50 s the three values are 0.99, 1.02 and 0.95; E = 0.05. At 60 s E = 0.03. Because all subsequent values remain inside the band, the strict result is t₉₅ = 50 s.

Reproduce it

Download the seven-point multi-probe CSV. Recalculate each row’s maximum deviation, identify the first persistent pass, and test the method by changing the endpoint from 5% to 10% without altering the raw normalized observations.

Hydrodynamic context

Translate the endpoint into reviewable scale-up quantities

QuantityWorked calculationInterpretation

Mixing timet₉₅ = 50 sCalculated from all three synthetic probe histories

Impeller revolutionsNt₉₅ = 2.50 × 50 = 125Dimensionless mixing time; useful for like-for-like comparison

Tip speedu_tip = πND = π × 2.50 × 0.180 = 1.41 m s⁻¹Mechanical context, not a universal shear metric

Reynolds numberRe = ρND²/μ = 81,000Turbulent on the stated Newtonian property basis

Shaft powerP = N_PρN³D⁵ = 14.76 WScreening estimate based on assumed ungassed N_P

Power per volumeP/V = 14.76/0.008 = 1.85 kW m⁻³Replace with torque or qualified vendor data for decision use

Scale-up interpretation

Keep the mixing criterion inside the complete transport envelope.

Do not preserve one number blindly

Constant t₉₅, P/V, tip speed, k_La or gas flow cannot generally all be maintained when volume and geometry change. A matched mixing time supported successful CHO scale-down in one defined system, but that result does not make equal mixing time universally sufficient [5].

Test the process time scales

Compare t₉₅ with feed-addition duration, acid/base dosing, oxygen uptake, substrate consumption, heat generation and sensitive biological response times. A bulk endpoint can pass while cells still experience recurring local substrate, dissolved-oxygen, pH or carbon-dioxide gradients.

Bracket the operating space

Repeat the experiment across minimum and maximum agitation, gas flow, working volume and representative rheology. Include the gassed state if that is the process state. Store mean, standard deviation, replicate count, sensor correction and excluded runs, then validate any CFD model against the same endpoint definition.

Limitations

Common ways to obtain a precise but wrong mixing time

Failure modeBias or ambiguityRequired control

One probeA local circulation path is treated as global homogeneity.Map representative top, middle, bottom and radial positions.

Uncontrolled injectionPulse duration or placement dominates between-run variation.Fix the injection geometry and synchronize the t = 0 trigger.

Sensor lagThe instrument, rather than hydrodynamics, controls the response.Characterize and correct the complete measurement train.

First crossing onlyAn oscillatory response passes before it remains homogeneous.Use the persistent-band definition over a stable plateau.

Water-only testShear-thinning broth or gas holdup changes circulation and mixing.Test representative property and gassing ranges.

Assumed power numberP/V appears more certain than the impeller and gassed-power evidence.Use torque, calibrated motor data or qualified correlations with stated validity.

Reusable experiment record

Minimum evidence for engineering review

Test definition

Vessel, internals, impellers, baffles, probes and injection coordinates; volume; liquid properties; agitation; gas rate and composition; pressure; temperature; tracer chemistry, volume, concentration and delivery duration.

Signal processing

Raw time series; calibration; clocks; acquisition rate; filtering; response-time test; baseline and plateau windows; normalization equation; homogeneity band; persistence rule; replicate statistics and uncertainty.

Decision context

Scale and operating range; process time scales; P/V, tip speed, Re and gas flow; oxygen, carbon-dioxide, heat and shear limits; CFD acceptance criteria; model version; evidence owner; reviewer and extrapolation boundary.

Primary technical sources

References

Barth, I.; Holbeck, C.; Capitain, C.; Schirmeister, T.; Kampeis, P. “Improving the reproducibility of mixing-time experiments in stirred single-use bioreactors.” Applied Microbiology and Biotechnology 110, 192 (2026). doi:10.1007/s00253-026-13931-w.
Zhang, A.; Tsang, V. L.; Korke-Kshirsagar, R.; Ryll, T. “Effects of pH probe lag on bioreactor mixing time estimation.” Process Biochemistry 49(6), 913–916 (2014). doi:10.1016/j.procbio.2014.03.005.
Bach, C.; Yang, J.; Larsson, H. K.; Stocks, S. M.; Gernaey, K. V.; Albæk, M. O.; Krühne, U. “Evaluation of mixing and mass transfer in a stirred pilot scale bioreactor utilizing CFD.” Chemical Engineering Science 171, 19–26 (2017). doi:10.1016/j.ces.2017.05.001.
Bisgaard, J. et al. “Characterization of mixing performance in bioreactors using flow-following sensor devices.” Chemical Engineering Research and Design 174, 471–485 (2021). doi:10.1016/j.cherd.2021.08.008.
Wiegmann, V.; Gardner, R. A.; Spencer, D. I. R.; Baganz, F. “Equal mixing time enables scale-down and optimization of a CHO cell culture process using a shaken microbioreactor system.” Biotechnology Journal 16(11), e2100360 (2021). doi:10.1002/biot.202100360.

Technical FAQ

Questions engineers ask before using t₉₅

How is bioreactor mixing time calculated from tracer data?

Normalize each probe between its pre-injection and final plateau values, calculate the largest absolute deviation from the final mixed state at every time point, and report t₉₅ only when every probe remains within ±5% thereafter.

Why are several tracer probes needed in a bioreactor?

A single probe measures a local response and can miss slow zones or an injection shortcut. Multiple axial and radial positions support a global homogeneity claim and reveal which location controls the reported mixing time.

Can mixing time alone be used as a bioreactor scale-up criterion?

No. Review mixing time with power per volume, tip speed, Reynolds number, gas flow, oxygen and carbon-dioxide transfer, heat removal, shear exposure, geometry and the rheology expected during the process.