Heat and Mass Transfer

, Volume 54, Issue 10, pp 2975–2986 | Cite as

Reliable prediction of heat transfer coefficient in three-phase bubble column reactor via adaptive neuro-fuzzy inference system and regularization network

  • A. Garmroodi Asil
  • A. Nakhaei Pour
  • Sh. Mirzaei


In the present article, generalization performances of regularization network (RN) and optimize adaptive neuro-fuzzy inference system (ANFIS) are compared with a conventional software for prediction of heat transfer coefficient (HTC) as a function of superficial gas velocity (5–25 cm/s) and solid fraction (0–40 wt%) at different axial and radial locations. The networks were trained by resorting several sets of experimental data collected from a specific system of air/hydrocarbon liquid phase/silica particle in a slurry bubble column reactor (SBCR). A special convection HTC measurement probe was manufactured and positioned in an axial distance of 40 and 130 cm above the sparger at center and near the wall of SBCR. The simulation results show that both in-house RN and optimized ANFIS due to powerful noise filtering capabilities provide superior performances compared to the conventional software of MATLAB ANFIS and ANN toolbox. For the case of 40 and 130 cm axial distance from center of sparger, at constant superficial gas velocity of 25 cm/s, adding 40 wt% silica particles to liquid phase leads to about 66% and 69% increasing in HTC respectively. The HTC in the column center for all the cases studied are about 9–14% larger than those near the wall region.



Unit vector


Green’s matrix


Smoother matrix


Identity matrix


Number of neurons

\( \underline{w} \)

Synaptic weight vector

\( \underline{x} \)

Input vector

\( \underline{y} \)

Real response values


Membership function


Input variable

a, b, c

Premise parameters

p, q, r

Consequent parameters


Membership grade


Firing strengths

A, B

Linguistic label


Surface temperature


Bulk fluid temperature


Heat transfer coefficient

Greek letters


Regularization parameter

\( {\underline{w}}_{\lambda } \)

Linear synaptic weight


Isotropic spread



Neuro-fuzzy inference system


Heat transfer coefficient


Regularization network


Slurry bubble column reactor


Artificial neural network


Root-mean-square error



The authors wish to acknowledge the financial support granted by Ferdowsi University of Mashhad.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • A. Garmroodi Asil
    • 1
  • A. Nakhaei Pour
    • 2
  • Sh. Mirzaei
    • 3
  1. 1.Chemical Engineering Department, Faculty of EngineeringUniversity of BojnordBojnordIran
  2. 2.Chemistry Department, Faculty of ScienceFerdowsi University of MashhadMashhadIran
  3. 3.Chemical Engineering Department, Faculty of EngineeringFerdowsi University of MashhadMashhadIran

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