Q: In a two-phase gas–liquid system with stationary (non-flowing) liquid, express the gas hold-up (ε) in terms of the superficial gas velocity Vs and the bubble rise velocity Vt.

Introduction / Context: Gas hold-up (ε) is the volumetric fraction of gas in a gas–liquid dispersion and is a key parameter for mass transfer, residence time, and hydrodynamics in bubble columns and aerated bioreactors. A simple kinematic relation links ε to the superficial gas velocity (Vs) and the bubble rise velocity (Vt) when the liquid phase is not flowing on average. Given Data / Assumptions: Continuous liquid phase is stationary on average (no net bulk liquid flow). Dispersed gas phase ascends as bubbles with average rise velocity Vt relative to the liquid. Superficial gas velocity Vs equals volumetric gas flux divided by column cross-section. Concept / Approach: The volumetric gas flux Jg equals the product of gas volume fraction and bubble rise speed: Jg = ε * Vt. By definition, Vs = Jg. Therefore, ε = Vs / Vt. This relation captures how more gas flux or slower-rising bubbles (e.g., due to smaller size or higher viscosity) increase hold-up. Step-by-Step Solution: Define gas flux: Jg = volumetric gas flow per area = Vs.Relate phases: Jg = ε * Vt (gas fraction times slip velocity).Equate: Vs = ε * Vt.Solve: ε = Vs / Vt. Verification / Alternative check: Dimensional check: Vs and Vt both have units of length/time, so their ratio is dimensionless as required for ε. Empirically, increased aeration (higher Vs) at fixed bubble rise speed raises ε, consistent with observations. Why Other Options Are Wrong: Forms with Vs ± Vt in numerator/denominator do not follow from the flux balance in a static liquid.ε = (Vs + Vt)/Vs would exceed 1 for typical conditions, which is non-physical for hold-up. Common Pitfalls: Confusing bubble rise velocity relative to liquid (Vt) with superficial gas velocity (Vs); applying this relation when there is significant liquid co-current flow, which requires modified expressions. Final Answer: ε = Vs / Vt

Q: For fully turbulent agitation in a geometrically similar stirred tank, how is the impeller power number (Np) correlated with the impeller Reynolds number (Rei)?

Introduction / Context: The power number Np characterizes energy draw by an impeller and is central to scale-up. Its dependence on Reynolds number Rei changes across flow regimes. Recognizing the turbulent asymptote simplifies power estimation at industrial scales where Rei is large. Given Data / Assumptions: Newtonian fluid, geometric similarity, standard baffling. Turbulent regime with Rei ≫ 10^4. Same impeller type across scales. Concept / Approach: At low Rei (laminar), Np ∝ 1/Rei. In transition, Np varies with Rei. In fully turbulent flow, viscous effects are negligible and Np approaches a constant that depends on impeller and tank geometry. Thus P = Np * ρ * N^3 * Di^5 with Np ≈ constant is used for design and scale-up. Step-by-Step Solution: Identify regime: Rei > 10^4 ⇒ fully turbulent for typical impellers.Apply asymptote: Np → constant K (geometry-dependent).Use power law: P = Np * ρ * N^3 * Di^5 with Np ≈ K.Therefore, Np does not depend on Rei in this regime. Verification / Alternative check: Manufacturer curves show flat Np vs Rei at high Rei for Rushton or pitched-blade impellers, validating constancy. Why Other Options Are Wrong: Np = K * (Rei)^-1 applies to laminar, not turbulent, conditions.Including N^2 * Di with Rei^-1 mixes variables redundantly; not a standard correlation.“Cannot be correlated” is incorrect; the constant-Np correlation is standard. Common Pitfalls: Applying laminar correlations to turbulent scale-up; ignoring the need for geometric similarity and consistent baffling when using constant Np. Final Answer: Np = K (approximately constant, geometry-dependent)

Question 1

Oxygen balance check in a bioreactor: assuming dissolved oxygen is at steady state, if the oxygen transfer rate (OTR) is measured as 23 kg·h^-1 at a given instant, what must the oxygen uptake rate (OUR) be at that same instant?

Accepted Answer

Introduction / Context: Matching oxygen supply with cellular demand is central to aerobic bioprocess control. Two key quantities are the oxygen transfer rate (OTR), driven by aeration and agitation, and the oxygen uptake rate (OUR), dictated by cellular metabolism. At dissolved oxygen (DO) steady state, these two rates are equal by mass balance. Given Data / Assumptions: OTR = 23 kg·h^-1 at the time of interest. DO concentration is momentarily steady (dC_L/dt = 0). No other oxygen sources/sinks besides gas–liquid transfer and cellular uptake. Concept / Approach: The dissolved oxygen mass balance is dC_L/dt = k_La * (C* − C_L) − OUR/V. Multiplying by V gives d(C_L V)/dt = OTR − OUR. At steady state, d(C_L V)/dt = 0, hence OTR = OUR. Therefore, when OTR is 23 kg·h^-1, OUR must also be 23 kg·h^-1 at that instant. Step-by-Step Solution: Write oxygen balance: accumulation = in − out = transfer − uptake.Set accumulation to zero (steady DO): 0 = OTR − OUR.Rearrange: OUR = OTR.Substitute values: OUR = 23 kg·h^-1. Verification / Alternative check: If DO were rising (positive accumulation), OTR > OUR; if falling, OTR < OUR. Only at steady DO are the two exactly equal. Controllers often adjust agitation or airflow to maintain this balance. Why Other Options Are Wrong: Less than or greater than 23 kg·h^-1 would imply changing DO levels, contradicting the steady-state assumption.2.3 kg·h^-1 is a simple scaling distraction with no basis. Common Pitfalls: Confusing OTR capacity with actual transfer; forgetting to check if DO is at steady state before asserting OTR = OUR. Final Answer: 23 kg·h^-1

Question 2

In a two-phase gas–liquid system with stationary (non-flowing) liquid, express the gas hold-up (ε) in terms of the superficial gas velocity Vs and the bubble rise velocity Vt.

Accepted Answer

Introduction / Context: Gas hold-up (ε) is the volumetric fraction of gas in a gas–liquid dispersion and is a key parameter for mass transfer, residence time, and hydrodynamics in bubble columns and aerated bioreactors. A simple kinematic relation links ε to the superficial gas velocity (Vs) and the bubble rise velocity (Vt) when the liquid phase is not flowing on average. Given Data / Assumptions: Continuous liquid phase is stationary on average (no net bulk liquid flow). Dispersed gas phase ascends as bubbles with average rise velocity Vt relative to the liquid. Superficial gas velocity Vs equals volumetric gas flux divided by column cross-section. Concept / Approach: The volumetric gas flux Jg equals the product of gas volume fraction and bubble rise speed: Jg = ε * Vt. By definition, Vs = Jg. Therefore, ε = Vs / Vt. This relation captures how more gas flux or slower-rising bubbles (e.g., due to smaller size or higher viscosity) increase hold-up. Step-by-Step Solution: Define gas flux: Jg = volumetric gas flow per area = Vs.Relate phases: Jg = ε * Vt (gas fraction times slip velocity).Equate: Vs = ε * Vt.Solve: ε = Vs / Vt. Verification / Alternative check: Dimensional check: Vs and Vt both have units of length/time, so their ratio is dimensionless as required for ε. Empirically, increased aeration (higher Vs) at fixed bubble rise speed raises ε, consistent with observations. Why Other Options Are Wrong: Forms with Vs ± Vt in numerator/denominator do not follow from the flux balance in a static liquid.ε = (Vs + Vt)/Vs would exceed 1 for typical conditions, which is non-physical for hold-up. Common Pitfalls: Confusing bubble rise velocity relative to liquid (Vt) with superficial gas velocity (Vs); applying this relation when there is significant liquid co-current flow, which requires modified expressions. Final Answer: ε = Vs / Vt

Question 3

For fully turbulent agitation in a geometrically similar stirred tank, how is the impeller power number (Np) correlated with the impeller Reynolds number (Rei)?

Accepted Answer

Introduction / Context: The power number Np characterizes energy draw by an impeller and is central to scale-up. Its dependence on Reynolds number Rei changes across flow regimes. Recognizing the turbulent asymptote simplifies power estimation at industrial scales where Rei is large. Given Data / Assumptions: Newtonian fluid, geometric similarity, standard baffling. Turbulent regime with Rei ≫ 10^4. Same impeller type across scales. Concept / Approach: At low Rei (laminar), Np ∝ 1/Rei. In transition, Np varies with Rei. In fully turbulent flow, viscous effects are negligible and Np approaches a constant that depends on impeller and tank geometry. Thus P = Np * ρ * N^3 * Di^5 with Np ≈ constant is used for design and scale-up. Step-by-Step Solution: Identify regime: Rei > 10^4 ⇒ fully turbulent for typical impellers.Apply asymptote: Np → constant K (geometry-dependent).Use power law: P = Np * ρ * N^3 * Di^5 with Np ≈ K.Therefore, Np does not depend on Rei in this regime. Verification / Alternative check: Manufacturer curves show flat Np vs Rei at high Rei for Rushton or pitched-blade impellers, validating constancy. Why Other Options Are Wrong: Np = K * (Rei)^-1 applies to laminar, not turbulent, conditions.Including N^2 * Di with Rei^-1 mixes variables redundantly; not a standard correlation.“Cannot be correlated” is incorrect; the constant-Np correlation is standard. Common Pitfalls: Applying laminar correlations to turbulent scale-up; ignoring the need for geometric similarity and consistent baffling when using constant Np. Final Answer: Np = K (approximately constant, geometry-dependent)

Question 4

Considering spherical gas bubbles of different diameters dispersed in a liquid, which bubble size provides the largest interfacial area per unit gas volume (a = surface area/volume)?

Accepted Answer

Introduction / Context: Interfacial area per unit volume is a key determinant of gas–liquid mass transfer (k_La). For a given gas holdup, smaller bubbles create more interface area, increasing transfer rates—vital for aerobic bioprocesses and chemical absorbers. Given Data / Assumptions: Spherical bubbles with diameters 1, 5, 15, and 50 mm. We compare a = surface area / volume for individual bubbles, which scales to interfacial area density for a swarm. Neglect coalescence and deformation effects. Concept / Approach: For a sphere, area A = π * d^2 and volume V = π * d^3 / 6. Thus a = A / V = 6 / d. Therefore, a is inversely proportional to diameter: the smaller the bubble, the greater the interfacial area per unit volume of gas. Step-by-Step Solution: Write a = 6 / d for spherical bubbles.Compare diameters: 1 mm yields a = 6 mm^-1; 5 mm yields 1.2 mm^-1; 15 mm yields 0.4 mm^-1; 50 mm yields 0.12 mm^-1.Identify the maximum: 1 mm bubble provides the largest a.Conclude smallest diameter → largest interfacial area per volume. Verification / Alternative check: Empirical mass-transfer studies show higher k_La with fine-pore spargers that produce small bubbles, consistent with the 6/d relationship. Why Other Options Are Wrong: Larger diameters reduce a due to the inverse proportionality; thus 5, 15, and 50 mm offer progressively less area. Common Pitfalls: Confusing total area at fixed gas flow with area per bubble; overlooking that very small bubbles can coalesce without surfactants or with high ionic strength, changing the actual a achieved. Final Answer: A bubble with a diameter of 1 mm

Question 5

In an unbaffled stirred tank, what typically happens as the impeller speed is increased from low to high values, considering flow pattern and free-surface behavior?

Accepted Answer

Introduction / Context: Baffles suppress swirling and vortex formation. In unbaffled vessels, however, the absence of baffles allows bulk rotation and a vortex at the free surface as speed increases. Understanding this progression helps avoid gas entrainment, poor mixing, and erratic level control. Given Data / Assumptions: Unbaffled cylindrical tank with a free liquid surface. Single impeller operating in a Newtonian liquid. Air entrainment is undesirable at high vortex depth. Concept / Approach: At low speeds, flow is laminar or transitional with gentle circulation. As N increases, tangential velocity rises and the free surface depresses at the center due to radial pressure gradients, eventually forming a stable vortex. The Froude number Fr = N^2 * Di / g indicates the propensity for vortexing; larger Fr favors deeper vortices and potential air entrainment. Step-by-Step Solution: Increase N → stronger circulation and swirling motion.Surface depression grows with centrifugal effects.At sufficient Fr, a well-defined vortex forms, possibly entraining air.Conclusion: circulation precedes vortex; vortex deepens as speed rises. Verification / Alternative check: Visual observations and surface profile measurements show vortex depth scaling with Fr; adding baffles (reducing swirl) markedly reduces vortexing at the same N. Why Other Options Are Wrong: Immediate turbulence after small speed increase is unrealistic; regime depends on Rei and geometry.Ignoring surface effects misses the defining unbaffled behavior.Vortex at low speed with no prior circulation is not typical; circulation occurs at all speeds. Common Pitfalls: Confusing turbulence onset with vortex formation; assuming vortex depth is solely a function of speed rather than Fr and liquid properties. Final Answer: Circulation strengthens with speed; eventually a central vortex forms at the free surface

Question 6

When free-surface vortexing is relevant in stirred bioreactors, which dimensionless number is most useful to characterize the balance of inertial forces to gravity?

Accepted Answer

Introduction / Context: Free-surface behavior in stirred tanks affects gas entrainment and mixing. A central measure for vortex formation is the Froude number, which compares inertial forces generated by rotation to gravitational forces that flatten the surface. Given Data / Assumptions: Unbaffled or partially baffled tank with a free liquid surface. Impeller of diameter Di rotating at speed N. Newtonian liquid under normal gravity g. Concept / Approach: The impeller Froude number is Fr = N^2 * Di / g (or sometimes defined with tip speed squared over g * Di). Higher Fr implies stronger swirling relative to gravity and deeper vortices. Other dimensionless numbers (Stanton, Rayleigh, Bond) are used for different phenomena: Stanton for convective heat/mass transfer coefficients, Rayleigh for buoyancy-driven convection, and Bond for gravity vs surface tension for droplets/bubbles—none directly quantify vortexing due to mechanical agitation. Step-by-Step Solution: Identify phenomenon: vortex depth and stability at the free surface.Select ratio: inertial force (∝ N^2 * Di) to gravitational force (∝ g).Compute Fr: Fr = N^2 * Di / g.Use Fr to compare conditions and predict onset/intensity of vortexing. Verification / Alternative check: Empirical correlations show vortex depth normalized by liquid height increases with Fr, and baffling effectively reduces Fr-relevant swirling, diminishing vortex formation. Why Other Options Are Wrong: Stanton: relates h/(ρ * U * c_p) or mass-transfer analogs; not a vortexing metric.Rayleigh: buoyancy-driven convection in non-agitated systems.Bond: gravity vs surface tension in droplets/bubbles, not tank vortexing. Common Pitfalls: Using Reynolds number to predict vortexing (Re gauges turbulence, not free-surface deformation) or ignoring the role of liquid height and baffling. Final Answer: Froude number

Question 7

Identify the mixing regime for an agitated vessel based on the impeller Reynolds number (Rei). Which range corresponds to fully turbulent flow for typical stirred-tank geometries?

Accepted Answer

Introduction / Context: Determining whether flow is laminar, transitional, or turbulent in a stirred tank guides the choice of correlations for power, mixing time, and mass transfer. The impeller Reynolds number Rei provides the basis for these regime distinctions. Given Data / Assumptions: Newtonian fluid with consistent properties. Standard baffled stirred tanks and common impellers. Rei = ρ * N * Di^2 / μ. Concept / Approach: Rule-of-thumb thresholds place laminar flow at Rei < 10, transitional between about 10 and 10^4, and fully turbulent at Rei > 10^4. In the turbulent regime, correlations often simplify (e.g., constant Np), enabling straightforward scale-up. Step-by-Step Solution: Compute Rei for the process conditions.Compare with thresholds: <10 (laminar), 10–10^4 (transition), >10^4 (turbulent).Select the turbulent range if Rei exceeds 10^4.Apply relevant turbulent correlations thereafter. Verification / Alternative check: Plotting Np vs Rei exhibits a 1/Rei trend in laminar, curvature in transition, and a plateau for Rei > 10^4, confirming the threshold utility. Why Other Options Are Wrong: Rei < 10: laminar regime.10 < Rei < 10^4: transitional, not fully turbulent.“None of these” is incorrect because a standard turbulent threshold exists. Common Pitfalls: Treating thresholds as exact rather than guidelines; neglecting that very high-viscosity media or non-Newtonian behavior shifts practical boundaries. Final Answer: Rei > 10^4

Question 8

Instrumentation check: which device is commonly used to measure the rotational speed (rpm) of a stirrer shaft in laboratory and industrial bioreactors?

Accepted Answer

Introduction / Context: Accurate measurement of impeller speed is fundamental for controlling mixing intensity, oxygen transfer, and shear in stirred tanks. The standard instrument for rotational speed is the tachometer, available in contact and non-contact (optical or laser) forms. Given Data / Assumptions: We need to measure shaft rpm directly or via motor feedback. Laboratory or industrial bioreactor context. Device recognition rather than calibration details. Concept / Approach: A tachometer measures angular speed, typically returning rpm or rps. In bioreactors, speed readouts may come from motor encoders (a form of tachometer) or external handheld devices used for verification. Other instruments target different physical quantities: anemometers measure air velocity, manometers measure pressure differences, and barometers measure atmospheric pressure. Step-by-Step Solution: Identify required variable: rotational speed (rpm).Select instrument class: tachometer/encoder for angular velocity.Confirm alternatives do not measure rotation directly (anemometer, manometer, barometer).Conclude tachometer is the appropriate device. Verification / Alternative check: Most bioreactor controllers display rpm using internal motor tach feedback; periodic verification with an external optical tachometer ensures calibration accuracy. Why Other Options Are Wrong: Anemometer: measures linear air speed, not shaft rpm.Manometer: measures pressure head, not rotation.Barometer: measures atmospheric pressure, not rotation. Common Pitfalls: Confusing motor nameplate frequency (Hz) with actual shaft rpm after gearing; neglecting to account for slip in induction motors without feedback control. Final Answer: Tachometer

Question 9

Axial-flow impellers in bioreactors: which general statement best describes their hydrodynamics and suitability for shear-sensitive bioprocesses?

Accepted Answer

Introduction / Context: Impeller selection strongly influences mixing pattern, shear environment, and mass transfer in bioreactors. Axial-flow impellers (for example, marine propellers and pitched-blade turbines set for down-pumping) primarily push fluid along the impeller axis and are widely used when cultures are shear sensitive, such as mammalian cells or delicate filamentous organisms. Given Data / Assumptions: The question contrasts axial-flow impellers with high-shear, predominantly radial-flow designs. Goal is to identify the statement that captures typical hydrodynamics and process suitability. Context is aerobic, mechanically agitated, baffled tanks. Concept / Approach: Axial-flow devices generate bulk circulation loops with relatively uniform energy dissipation compared with radial-flow Rushton turbines. This lowers peak shear rates, improves suspension and blending, and maintains oxygen transfer by renewing surface area without creating extreme micro-environments that can damage sensitive cells. Step-by-Step Solution: 1) Define axial flow: discharge along the shaft axis (up- or down-pumping), forming top-to-bottom circulation loops.2) Compare shear: axial impellers produce lower peak shear than sharp-edged radial turbines at the same power input.3) Connect to biology: lower shear favors viability and productivity for shear-sensitive cultures.4) Conclude that the best description is “low shear and suitable for shear-sensitive processes.” Verification / Alternative check: Empirical practice pairs axial-flow impellers with large-diameter, low-rpm operation to cap tip speed and shear, while still achieving acceptable kLa via adequate circulation and gas dispersion. Why Other Options Are Wrong: Option A: High shear is characteristic of radial turbines, not axial propellers. Option C: Radial discharge toward the walls describes radial turbines. Option D/E: Axial impellers do not produce only tangential swirl; they create strong axial circulation. Common Pitfalls: Assuming “low shear” means “low mass transfer.” With proper gas rate and baffling, axial impellers can deliver strong circulation and good kLa while protecting cells. Final Answer: Provide low shear conditions and are suited for use in shear-sensitive processes

Question 10

Diagnosing mass transfer limitation from agitation effects: if increasing stirring speed from 300 rpm to 500 rpm doubles the reaction rate, what is the most reasonable conclusion?

Accepted Answer

Introduction / Context: In gas–liquid or liquid–solid bioreactions, overall rate can be limited by intrinsic enzyme/cellular kinetics or by transport of substrates/products through boundary layers. Observing how rate responds to agitation provides a practical diagnostic for transport limitation during process development. Given Data / Assumptions: Stirring speed increase: 300 → 500 rpm. Measured overall reaction rate doubles. Other variables (temperature, composition) unchanged. Concept / Approach: If rate increases substantially with agitation, hydrodynamics—and thus mass transfer coefficients—have become more favorable. When boundary-layer resistance dominates at low rpm, increasing rpm thins boundary layers and enhances interfacial renewal, raising flux and overall rate. Once mass transfer ceases to limit, further rpm increases do little, and rate plateaus at the kinetic limit. Step-by-Step Solution: 1) At 300 rpm, agitation is weaker; film resistance is larger; interfacial area and kL are lower.2) Raising to 500 rpm increases turbulence and circulation, reducing resistance and boosting kLa or kL.3) The observed doubling of rate indicates transport control was significant at 300 rpm.4) Therefore, the logical conclusion is mass transfer limitation at the lower speed. Verification / Alternative check: Plotting rate versus rpm typically shows a rising section (transport-limited) and a plateau (kinetic-limited). The data point lies on the rising section, consistent with transport limitation at 300 rpm. Why Other Options Are Wrong: Option B: If 500 rpm were transport-limited, increasing rpm would still increase rate; the conclusion concerns the lower rpm. Option C: Clear agitation dependence contradicts “no mass transfer.” Option D: Even at 300 rpm, bulk convection exists; diffusion-only is implausible. Option E: Intrinsic kinetics do not change with rpm absent temperature or chemistry changes. Common Pitfalls: Confusing enhanced kLa with altered intrinsic kinetics; agitation changes transport, not enzyme turnover numbers. Final Answer: The reaction was mass transfer limited at 300 rpm

Question 11

Impeller choice for solids suspension: which impeller is preferred when strong vertical currents are needed to keep heavy particles suspended?

Accepted Answer

Introduction / Context: Solid–liquid mixing often requires reliable suspension of dense or large particles to prevent settling and to ensure uniform contact between reactants. The hydrodynamic pattern created by the impeller determines whether solids lift off the bottom and remain in circulation. Given Data / Assumptions: Particles are “heavily” settling and need vertical lift. Tank is baffled and mechanically agitated. Objective is strong axial circulation. Concept / Approach: Axial-flow impellers such as marine propellers generate pronounced top-to-bottom circulation loops that sweep the tank floor, lifting solids into suspension. Radial-flow turbines discharge sideways, which is less effective for vertical lifting unless combined with multiple stages and careful placement. Paddles produce weak flow and are better for gentle blending at low viscosities. Helical ribbons suit very viscous fluids, not typical solids-laden, low-viscosity broths seeking strong vertical currents. Step-by-Step Solution: 1) Identify the needed flow pattern: vertical axial currents to scour the bottom.2) Match impeller: marine propeller (axial) is designed for strong axial throughput.3) Conclude propeller is the preferred choice for suspending heavy particles. Verification / Alternative check: Correlations for just-suspended speed (Njs) show lower Njs for axial impellers than for radial turbines at comparable conditions, confirming better suspension efficiency. Why Other Options Are Wrong: Paddle mixing is weak for suspension; radial turbines emphasize radial jets, not vertical lift; helical ribbons target laminar, viscous mixing and are not optimal for solids suspension in low-viscosity systems; “None” is incorrect because propellers fit the need. Common Pitfalls: Confusing gas–liquid dispersion needs with solids suspension; axial flow is especially beneficial for keeping particles off the bottom. Final Answer: Propeller

Question 12

Which dimensionless number is most directly calculated from impeller diameter and peripheral (tip) speed in an agitated tank?

Accepted Answer

Introduction / Context: Dimensionless numbers help generalize mixing behavior across scales. In stirred tanks, the Froude number relates inertial forces to gravity and is especially relevant when free-surface vortexing or surface waves may occur. It can be expressed using impeller diameter and tip speed (peripheral speed). Given Data / Assumptions: Impeller diameter D and rotational speed N define tip speed u_tip = π*D*N. Gravity g is constant. Liquid properties are otherwise unspecified. Concept / Approach: The Froude number for mixing is often written as Fr = N^2*D/g, which is dimensionally proportional to (u_tip^2)/(g*D). Thus, Fr depends directly on diameter and peripheral speed. By contrast, the Reynolds number Re = ρ*N*D^2/μ requires viscosity and density; the power number is an empirical function of Re (and geometry), not directly computed from tip speed alone. Step-by-Step Solution: 1) Recognize the formula connection: Fr ∝ u_tip^2/(g*D).2) Since u_tip = π*D*N, Fr can be calculated from D and N (or u_tip) with g.3) Therefore, the number “most directly” tied to diameter and peripheral speed is Froude. Verification / Alternative check: Free-surface behavior and vortex depth often correlate with Fr, highlighting its practical link to tip speed and diameter. Why Other Options Are Wrong: Reynolds requires fluid properties; power number is not a stand-alone input calculation; Schmidt relates momentum and mass diffusivity and is unrelated to impeller tip speed. Common Pitfalls: Interchanging Re and Fr in scale-up; they capture different physics (viscous versus gravitational effects). Final Answer: Froude number

Question 13

Cooling jackets with dimples: what is the principal purpose of providing dimples in a bioreactor's cooling jacket?

Accepted Answer

Introduction / Context: Efficient temperature control is critical in bioprocessing. Cooling jackets often include embossed dimples or corrugations. These geometric features are more than cosmetic; they directly affect hydrodynamics and heat-transfer coefficients on the jacket side. Given Data / Assumptions: Flowing coolant (typically water or glycol) circulates in the jacket. Objective is to enhance overall heat-transfer coefficient U. Bioreactor wall is the primary conductive barrier; convection on both sides matters. Concept / Approach: Dimples disturb boundary layers and promote secondary flows, raising turbulence intensity at a given flow rate. Turbulence reduces the thermal boundary layer thickness, increases film heat-transfer coefficient h, and thereby improves U. Longer residence time (option B) is not the driver—excess residence can actually reduce h if velocity drops. Increasing boundary-layer thickness (option C) would hinder heat transfer, not enhance it. Step-by-Step Solution: 1) Add geometric roughness (dimples) → induce local mixing and vortices.2) Higher local turbulence → thinner thermal boundary layer → larger h.3) Net effect: improved jacket-side heat-transfer performance at the same pump power. Verification / Alternative check: Correlations for enhanced surfaces show increased Nusselt numbers for dimpled/corrugated channels versus smooth channels at comparable Reynolds number. Why Other Options Are Wrong: Option B confuses volume hold-up with heat-transfer coefficient; option C proposes the opposite of what is desired; option D cannot be true if B and C are false; option E contradicts the purpose of enhancement. Common Pitfalls: Assuming bigger jacket volume equals better cooling; velocity and turbulence dominate jacket-side performance. Final Answer: Increase the level of turbulence of the cooling water in the jacket

Question 14

Effect of liquid height on oxygen transfer: if higher liquid height increases gas hold-up, where in the reactor will the oxygen transfer rate be higher?

Accepted Answer

Introduction / Context: Oxygen transfer rate (OTR) in aerated bioreactors depends on both the mass-transfer coefficient (kLa) and the driving force (C* − C), where C* is the saturation concentration. Hydrostatic pressure increases with depth, raising C* for oxygen. Therefore, depth-dependent effects can produce spatial variation in OTR. Given Data / Assumptions: Liquid height is increased, raising overall gas hold-up. Aeration and mixing maintain bubbles through the column. Temperature and gas composition are unchanged. Concept / Approach: Henry’s law implies that C* increases with partial pressure; at the base, hydrostatic pressure is higher, so the effective oxygen solubility is higher. Additionally, bubbles entering at the bottom are smaller (due to compression) and have more residence time, often raising local interfacial area a and kLa. Combined, these effects make OTR generally higher near the base than near the surface. Step-by-Step Solution: 1) Recognize depth → higher hydrostatic head → higher C* at the base.2) Bubble compression at depth → smaller effective size → larger a.3) OTR = kLa*(C* − C) → both kLa and driving force tend to be larger at the base.4) Conclude OTR is higher at the base under typical conditions. Verification / Alternative check: Measured dissolved oxygen profiles in tall, sparged reactors often show higher transfer (and lower DO) near the sparger region, consistent with stronger mass-transfer activity at the bottom. Why Other Options Are Wrong: Uniform or “higher throughout” ignores hydrostatic effects; “lower at base” contradicts increased C* and residence time; “lower throughout” misinterprets the role of gas hold-up. Common Pitfalls: Assuming perfect axial mixing eliminates gradients; in practice, gradients remain, especially in tall vessels. Final Answer: Higher at the base of the reactor

Question 15

Depth profile in tall aerated reactors: where are oxygen transfer rates typically the highest?

Accepted Answer

Introduction / Context: Oxygen transfer in tall aerated reactors varies with depth due to hydrostatic pressure and bubble dynamics. Understanding where OTR peaks helps in placing probes, sizing spargers, and interpreting gradients in large-scale fermentations. Given Data / Assumptions: Bottom sparging introduces bubbles near the base. Hydrostatic pressure increases with depth. Mixing is effective but not perfectly uniform over height. Concept / Approach: At greater depth, the saturation concentration C* is elevated, increasing the driving force for transfer. Bubbles at the base are smaller and more numerous due to compression and break-up near the sparger, raising interfacial area a and residence time. Consequently, kLa and OTR are generally largest near the bottom region and decline toward the surface where pressure (and thus C*) is lower and bubbles have enlarged by coalescence. Step-by-Step Solution: 1) Identify key variables: C*, kLa, residence time—all favor higher OTR at depth.2) Recognize bubble evolution: small to larger as they rise; interfacial area decreases upward.3) Combine effects: OTR peaks at the base, then diminishes with height. Verification / Alternative check: Typical DO profiles show lower DO near the base (due to stronger uptake and transfer), consistent with high local OTR. Computational fluid dynamics of gas–liquid systems also predict higher mass-transfer coefficients at depth near the sparger zone. Why Other Options Are Wrong: Option A/C presume mid/surface dominance, which ignores pressure effects; Option D assumes perfect mixing; Option E refers to gas-phase headspace, not liquid-phase mass transfer. Common Pitfalls: Misplacing DO sensors only near the surface and inferring whole-tank behavior; depth matters in scale-up. Final Answer: Highest at the base of the reactor

Question 16

Why does increasing stirrer speed raise oxygen transfer rate (OTR) in an aerated bioreactor?

Accepted Answer

Introduction / Context: Volumetric oxygen transfer, kLa, is central to aerobic bioprocess performance. Agitation affects several contributors to kLa and to the overall OTR, including bubble size distribution, interfacial area, and the liquid-side mass-transfer coefficient. Given Data / Assumptions: Aerated, baffled stirred tank with mechanical agitation. Gas rate held constant while stirrer speed increases. Broth rheology in the low-to-moderate viscosity range. Concept / Approach: Agitation promotes bubble breakup and suppresses coalescence, reducing mean bubble diameter and increasing interfacial area a. Turbulence intensifies micro-mixing and thins boundary layers, raising kL. Stronger circulation renews oxygen-depleted liquid at bubble surfaces and transfers oxygenated liquid away, maintaining driving force (C* − C). These factors act together to raise OTR as speed increases—up to a point where gas flooding, shear damage, or power limits intervene. Step-by-Step Solution: 1) Higher rpm → higher energy dissipation → smaller bubbles → larger a.2) Turbulence increases local gradients and reduces boundary-layer thickness → higher kL.3) Enhanced circulation distributes oxygen rapidly, sustaining driving force and preventing local depletion.4) Net: OTR rises with rpm until other constraints dominate. Verification / Alternative check: Dynamic gassing-out tests show kLa increasing with rpm at fixed aeration; off-gas analysis confirms higher oxygen uptake capacity. Why Other Options Are Wrong: Each individual mechanism contributes; selecting only one understates the system behavior. “None” contradicts established mixing and transfer principles. Common Pitfalls: Overdriving rpm can increase shear and antifoam use, which may reduce kLa—there is a practical optimum. Final Answer: All of the above

Question 17

Heat-transfer enhancement in stirred tanks: which change typically increases heat-transfer performance at the vessel wall or in coils?

Accepted Answer

Introduction / Context: Overall heat-transfer coefficient U in stirred vessels depends on film coefficients on both the process side and the jacket/coil side as well as wall resistance. Agitation affects the process-side film coefficient by changing turbulence near the wall and around internal coils. Given Data / Assumptions: Standard baffled, mechanically agitated tank. Cooling medium flow is adequate. Process fluid is not excessively viscous. Concept / Approach: Increasing stirrer speed intensifies turbulence and wall shear, thinning the thermal boundary layer on the process side and elevating the process-side heat-transfer coefficient h. In contrast, removing baffles increases swirling and can lower effective turbulence near surfaces. A higher coolant inlet temperature reduces the driving temperature difference, decreasing heat flux. Slower agitation reduces h and mixing uniformity. Step-by-Step Solution: 1) Identify U drivers: U depends on h_process, h_jacket, and conduction through the wall.2) Higher rpm → higher h_process via stronger turbulence.3) Therefore, at fixed coolant conditions, increasing rpm increases heat-transfer rate. Verification / Alternative check: Empirical correlations for h around coils and walls scale positively with Reynolds number based on impeller speed and diameter. Why Other Options Are Wrong: Option A reduces driving force; option C harms mixing pattern; option D lowers h; option E risks poor jacket-side performance. Common Pitfalls: Ignoring jacket-side limits; if jacket flow is insufficient, gains from higher rpm may be muted. Final Answer: Increasing the stirrer speed

Question 18

Laminar mixing correlation: for laminar flow around an impeller, how is power number Np correlated with impeller Reynolds number Rei?

Accepted Answer

Introduction / Context: Power consumption in stirred vessels is often expressed using dimensionless groups: power number Np and Reynolds number Rei (based on impeller). In the laminar regime, viscous forces dominate and classical correlations show a simple inverse relationship between Np and Rei for a given geometry. Given Data / Assumptions: Laminar regime (low Rei, typically Rei < 10–100 depending on geometry). Constant impeller and tank geometry. K denotes a geometry-dependent constant. Concept / Approach: By definition, Np = P / (ρ*N^3*Di^5) and Rei = ρ*N*Di^2/μ. In viscous-dominated (laminar) flow, P scales with μ, N, and Di such that Np ∝ 1/Rei. Thus, doubling Rei halves Np, all else equal, until transitional/turbulent regimes are reached where Np becomes nearly constant with Rei for baffled tanks. Step-by-Step Solution: 1) Start from dimensional analysis and laminar scaling: P ∝ μ*N^2*Di^3 (for a given geometry).2) Substitute into Np and collect terms with Rei = ρ*N*Di^2/μ.3) Resulting relation shows Np = K*(Rei)^-1 for laminar mixing. Verification / Alternative check: Mixing handbooks list Np = K/Rei for many impeller types under laminar conditions, with K depending on blade shape and clearance. Why Other Options Are Wrong: Option B/D introduce extra dimensional terms; option C makes Np grow with Rei, opposite to laminar behavior; option E applies to turbulent, not laminar flow. Common Pitfalls: Applying turbulent, constant-Np assumptions to laminar design; ensure regime check via Rei before selecting correlations. Final Answer: Np = K*(Rei)^-1

Question 19

Impeller hydraulics in stirred tanks: Which impeller type generates bulk liquid flow that is predominantly parallel to the agitator shaft (i.e., along the vessel’s axis)?

Accepted Answer

Introduction / Context: Mixing impellers are broadly classified by the dominant direction of the liquid they discharge: axial, radial, or tangential. Correctly identifying which class sends flow parallel to the agitator shaft is essential for selecting equipment for suspension, blending, or gas dispersion in biochemical and chemical reactors. Given Data / Assumptions: The question concerns the principal direction of bulk flow created by an impeller. Axial flow means along the tank’s vertical axis (parallel to the shaft). Radial flow means perpendicular to the shaft, toward the tank wall. Tangential flow largely follows circular motion around the wall. Concept / Approach: Axial-flow impellers (e.g., pitched-blade turbines, hydrofoils/propellers) pump fluid up or down along the vessel axis, creating top-to-bottom circulation loops that are excellent for blending, heat transfer, and solids suspension at moderate power numbers. Radial-flow impellers (e.g., Rushton turbines) eject fluid outward from the blade tips toward the wall, producing strong shear near the impeller for gas dispersion but less axial turnover unless baffles and multiple stages are used. Tangential flow dominates in anchor or helical ribbon mixers used for high-viscosity systems and is not parallel to the shaft. Step-by-Step Solution: Identify the definition: “parallel to the shaft” equals axial direction.Recall common axial devices: pitched-blade at 45°, marine propellers, hydrofoils.Contrast with radial devices: Rushton-type, flat-blade disc turbines.Conclude that only axial-flow impellers meet the stated criterion. Verification / Alternative check: Vendor literature and mixing textbooks plot flow number vs. power number; hydrofoils/propellers are labeled axial-flow with high pumping efficiency and low power number compared to radial turbines. Why Other Options Are Wrong: Radial flow: direction is perpendicular to the shaft. Both (a) and (b): mutually exclusive definitions. None of these: axial is valid. Tangential flow: mainly circumferential, not axial. Common Pitfalls: Assuming pitch angle creates radial flow; pitched blades primarily drive axial circulation despite some radial component. Final Answer: Axial flow

Question 20

Oxygen transfer in salty/sugary media: How do higher concentrations of dissolved salts and sugars typically affect oxygen transfer rate (OTR) in bioreactors, and why?

Accepted Answer

Introduction / Context: Oxygen transfer rate, OTR, is central to aerobic fermentations and is often written as OTR = KLa * (C* - C), where KLa is the overall mass transfer coefficient and C* is the saturation concentration. Media rich in salts or sugars (high osmolarity broths, high Brix feeds) frequently show reduced oxygen transfer performance. Understanding whether this is due to solubility, hydrodynamics, or both is key to troubleshooting scale-up. Given Data / Assumptions: Addition of salts/sugars increases ionic strength and often increases viscosity. Gas type, sparger, and temperature are otherwise unchanged for comparison. We examine qualitative trends rather than exact numerical correlations. Concept / Approach: Two mechanisms lower OTR in such media. First, oxygen solubility (C*) decreases as dissolved solids increase (salting-out), shrinking the driving force (C* - C). Second, higher viscosity and altered surface properties reduce bubble breakup and circulation, lowering KLa by reducing interfacial area (a) and the liquid-side coefficient (KL). Therefore, both a drop in C* and a drop in KLa act in the same direction to depress OTR. Step-by-Step Solution: Write OTR = KLa * (C* - C).Increase salts/sugars → lower C* via salting-out → smaller (C* - C).Increase salts/sugars → higher viscosity, poorer dispersion → lower KLa.Hence, both effects combine to reduce OTR. Verification / Alternative check: Empirical correlations show KLa inversely related to viscosity; oxygen solubility tables demonstrate reduced C* in saline or high-sugar solutions at fixed temperature and pressure. Why Other Options Are Wrong: (a) alone or (b) alone: each is true but incomplete; both occur together. (c): Claims KLa increases while OTR falls—contradicts typical viscosity effects. (d): Dismisses well-documented phenomena. Common Pitfalls: Assuming aeration rate alone fixes OTR; in viscous, salty media, impeller selection and oxygen enrichment may also be required. Final Answer: Both (a) and (b)

Agitation and Aeration Questions