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Mathematical modeling of mixing in a horizontal rotating tubular bioreactor: “Simple flow” model

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Abstract

The construction of the horizontal rotating tubular bioreactor (HRTB) represents a combination of a “thin-layer” bioreactor and a “biodisc” reactor. The bioreactor was made of a plastic tube whose interior was divided by the O-ring shaped partition walls. For the investigation of mixing properties in HRTB the temperature step method was applied. The temperature change in the bioreactor as a response to a temperature step in the inlet flow was monitored by six Pt-100 sensors (t 90 response time 0.08 s and resolution 0.002 °C) which were connected with an interface unit and personal computer. Mixing properties of the bioreactor were modeled using the modified “tank in series” concept which divided the bioreactor into ideally mixed compartments. A mathematical mixing model with “simple flow” was developed according to the physical model of the compartments network and corresponding heat balances. Numerical integration of an established set of differential equations was done by the Runge-Kutt-Fehlberg method. The final mathematical model with “simple flow” contained four adjustable parameters (N1,Ni, F cr andF p ) and five fixed parameters.

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Abbreviations

A u m2 :

inner surface of bioreactor's wall

A ui m2 :

i-th part of inner surface of bioreactor's wall

A v m2 :

outlet surface of bioreactor's wall

A vi m2 :

i-th part of outlet surface of bioreactor's wall

C p kJ kg–1 K–1 :

heat capacity of liquid

C pr kJ kg–1 K–1 :

heat capacity of bioreactor's wall

D h–1 :

dilution rate

E °C °C–1 h–1 :

error of mathematical model

F cr dm3s–1 :

circulation flow in the model

F p dm3 s–1 :

back flow in the model

F t dm3s–1 :

inlet flow in the bioreactor

I °C:

intensity of temperature step, the difference in temperature between the temperature of the inlet liquid flow and the temperature of liquid in the bioreactor before the temperature step

K1 Wm–2K–1 :

heat transfer coefficient between the liquid and bioreactor's wall

K2 Wm–2K–1 :

heat transfer coefficient between the bioreactor's wall and air

m s kg:

mass of bioreactor's wall

L m:

length of bioreactor

L k m:

wetted perimeter of bioreactor

n min–1 :

rotational speed of bioreactor

n s :

number of temperature sensors

N1:

number of cascades

Ni:

number of compartments inside the cascade

Nu:

Nusselt number

Pr :

Prandtl number

r u m:

inner diameter of bioreactor

r v m:

outside diameter of bioreactor

Re :

Reynolds number

s(t) :

step function

t s:

time

T °C:

temperature

T c °C:

calculated temperature

T m °C:

measured temperature

T N1,Ni °C:

temperature of liquid in a defined compartment inside cascade

T N1,S °C:

temperature of defined part of bioreactor's wall

T S °C:

temperature of bioreactor's wall

T v °C:

temperature of liquid in bioreactor

T z °C:

temperature of surrounding air

V t dm3 :

volume of liquid in the bioreactor

λ kJm–1s–1 K–1 :

thermal conductivity of liquid in the bioreactor

ϱ kgm–3 :

density of liquid in the bioreactor

ν m2s–1 :

kinematic viscosity of liquid in the bioreactor

B :

\(\frac{{F_t N1Ni}}{{V_t }}\)

C :

\(\frac{{F_{cr} N1Ni}}{{V_t }}\)

D :

\(\frac{{F_p N1Ni}}{{V_t }}\)

E :

B+C+D

G1 :

\(K1\frac{{2r_u \Pi LNi}}{{V_t }}\)

G2 :

\(K1\frac{{2r_u \Pi LNi}}{{V_t }} + K2\frac{{2r_v \Pi LNi}}{{V_t }}\)

G3 :

\(K2\frac{{2r_v \Pi LNi}}{{V_t }}\)

A ui :

\(\frac{{2r_u \Pi LNi}}{{V_t }}\)

A vi :

\(\frac{{2r_v \Pi LNi}}{{V_t }}\)

Q 1 :

\(K1\frac{{A_{ui} }}{{m_s c_{pr} }}\)

Q 2 :

\(K1\frac{{A_{ui} }}{{m_s c_{pr} }} + K2\frac{{A_{vi} }}{{m_s c_{pr} }}\)

Q 3 :

\(K2\frac{{A_{vi} }}{{m_s c_{pr} }}\)

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Authors and Affiliations

  1. Department of Biochemical Engineering, Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6/IV, 41000, Zagreb, Croatia

    B. Šantek, P. Horvat, S. Novak & V. Marić

  2. Institute of Biotechnology, TU Graz, Petersgasse 12, 8010, Graz, Austria

    B. Mayr & A. Moser

Authors
  1. B. Šantek
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  2. P. Horvat
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  3. S. Novak
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  4. B. Mayr
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  5. A. Moser
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  6. V. Marić
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Šantek, B., Horvat, P., Novak, S. et al. Mathematical modeling of mixing in a horizontal rotating tubular bioreactor: “Simple flow” model. Bioprocess Engineering 14, 195–204 (1996). https://doi.org/10.1007/BF01464734

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