# Determination of the average shear rate in a stirred and aerated tank bioreactor

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00449-008-0242-4

- Cite this article as:
- Campesi, A., Cerri, M.O., Hokka, C.O. et al. Bioprocess Biosyst Eng (2009) 32: 241. doi:10.1007/s00449-008-0242-4

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## Abstract

A method for evaluating the average shear rate (\( \dot \gamma _{{\text{av}}} \)) in a stirred and aerated tank bioreactor has been proposed for non-Newtonian fluids. The volumetric oxygen transfer coefficient (*k*_{L}*a*) was chosen as the appropriate characteristic parameter to evaluate the average shear rate (\( \dot \gamma _{{\text{av}}} \)). The correlations for the average shear rate as a function of *N* and rheological properties of the fluid (*K* and *n*) were obtained for two airflow rate conditions (ϕ_{air}). The shear rate values estimated by the proposed methodology lay within the range of the values calculated by classical correlations. The proposed correlations were utilized to predict the \( \dot \gamma _{{\text{av}}} \) during the *Streptomyces clavuligerus* cultivations carried out at 0.5 vvm and four different rotational impeller speeds. The results show that the values of the average shear rate (\( \dot \gamma _{{\text{av}}} \)) varied from 437 to 2,693 s^{−1} by increasing with *N* and flow index (*n*) and decreasing with the fluid consistency index (*K*).

### Keywords

Average shear rateConventional bioreactorMass transferMixingNon-Newtonian fluidRheology### List of symbols

*a*,*b*,*c*parameters of Eq. 10

*d*,*e*,*f*parameters of Eq. 12

- Ce
dissolved oxygen concentration (mmol L

^{−1})- Ce
_{0} dissolved oxygen concentration at

*t*=*t*_{0}(mmol L^{−1})- Ce
_{s} saturation concentration of dissolved oxygen with air (mmol L

^{−1})*d*_{i}impeller diameter (m)

*k*_{e}constant of the oxygen probe (s

^{−1})*k*proportionality constant

*k*_{L}*a*volumetric oxygen transfer coefficient (s

^{−1})*K*consistency index (Pa s

^{n})*n*flow index

*N*rotational impeller speed (rpm)

*Q*volumetric flow rate (m

^{3}s^{−1})*P*power input (W)

*t**U*_{GR}superficial gas velocity in the riser region (m s

^{−1})*V*working volume (m

^{3})

### Greek symbols

- ϕ
_{air} specific air flow rate (vvm)

- \( \dot \gamma \)
shear rate (s

^{−1})- \( \dot \gamma _{{{\text{av}}}} \)
average shear rate (s

^{−1})- \( \dot \gamma _{{\max }} \)
maximum shear rate (s

^{−1})*μ*dynamic viscosity (Pa s)

*μ*_{app}apparent viscosity (Pa s)

*μ*_{L}dynamic viscosity of liquid (Pa s)

- σ
surface tension of fluid (N m

^{−1})*τ*shear stress (Pa)

*τ*_{e}response time (s)