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Data-Driven Dynamic Modeling and Control of a Surface Aeration System

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Institute of Chemical Technology, Matunga, Mumbai-400 019, India, and Chemical Engineering & Process Development Division, National Chemical Laboratory, Pune-411008, India
Cite this: Ind. Eng. Chem. Res. 2007, 46, 25, 8607–8613
Publication Date (Web):May 15, 2007
Copyright © 2007 American Chemical Society

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    In this study we have developed a support vector regression (SVR) based data-driven model for predicting two important design parameters of surface aerators, namely, the volumetric mass transfer coefficient (kLa) and fractional gas hold-up (εG). The dynamical state of the surface aerator system was captured by acquiring pressure fluctuation signals (PFSs) at various design and operating conditions. The most informative features from PFS were extracted using the chaos analysis technique, which includes estimation of Lyapunov exponent, correlation dimensions, and Kolmogorov entropy. At similar conditions the values of kLa and εG were also measured. Two different SVR models for predicting the volumetric mass transfer coefficient (kLa) and overall gas hold-up (εG) as a function of chaotic invariants, design parameters, and operating parameters were developed showing test accuracies of 98.8% and 97.1%, respectively. Such SVM based models for the surface aerator can be potentially useful on a commercial scale for online monitoring and control of desired process output variables.

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     Institute of Chemical Technology.

     National Chemical Laboratory.


     To whom correspondence should be addressed. Tel:  +91-20-5893095. Fax:  +91-20-5893041. E-mail:  [email protected].

    Supporting Information Available

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    List of SVs and corresponding Lagrange multipliers (weights) are available for SVR based kLa and εG models, respectively (PDF). This material is available free of charge via Internet at

    b = bias term

    C = impeller clearance/distance between the impeller and the reactor bottom

    C = cost function

    CC = correlation coefficient

    C(r) = correlation integral

    CD = correlation dimension

    D = impeller diameter

    DT = disc type impeller

    ETrn = RMSE for training set

    ETst = RMSE for test set

    F = high dimensional feature space

    f(x) = regression function

    H = Heaviside function

    HL = liquid height

    k = first positive Lyapunov exponents

    K(xi, xj) = kernel function

    K = Kolmogorov entropy

    kLa= volumetric mass transfer coefficient

    L = lagrangian function (dual form)

    m = embedding dimensions

    Nimp = impeller speed

    N = number of training data points

    Np = number of vector data points constituting attractor

    Nsv = number of support vectors

    P = number of recorded pressure data points

    PBTU = pitched blade turbine up flow

    PBTD = pitched blade turbine down flow

    p(n) = one-dimension pressure data

    r = radius of m-dimension hyper-space

    rps = impeller revolutions per second

    ℛ = input space

    RMSE = root-mean-square error

    T = tank diameter

    xi = ith input vector

    w = weight vector

    yi = target output corresponding to the ith vector

    y(n) = m-dimension pressure vector

    Greek Letters εG = overall gas hold-up

    τ = time delay

    λ = Lyapunov exponent

    φ(xi) = feature space for ith input vector x

    ξi(*) = slack variables

    αi,j(*) = Lagrange multipliers

    ε = precision parameter

    σ = width of radial basis function (RBF) kernel

    γ = 1/2σ2

    ∞ = infinity

    Subscripts G = gas phase

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