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<p>Data-driven surrogates are investigated for the zero-dimensional population balance equation (0D–PBE) of turbulent liquid–liquid dispersions, using the quadrature method of moments (QMOM) as a reference. Five architectures are considered: a multilayer perceptron (MLP), an autoregressive neural network (ARNN), a long short-term memory (LSTM), a source-term surrogate for breakage and coalescence (RHS), and a Neural ordinary differential equation (Neural ODE). Surrogates are trained on thousands of QMOM simulations that span a wide range of initial droplet size, dispersed-phase holdup, and turbulent dissipation rate and are evaluated on 12,000 unseen operating conditions. The MLP, ARNN and LSTM accurately reproduce the decay of low-order moments, the nonlinear evolution of higher orders, and average diameters, with typical relative errors of a few percent and mean relative errors down to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(10^{-3}\)</EquationSource> </InlineEquation>, while delivering speed-ups of about three orders of magnitude relative to the direct QMOM numerical solution. The RHS and Neural ODE models are less data-efficient and exhibit larger errors; the former is also slower than QMOM because it requires an additional ODE integration. The proposed methodology is transferable to other homogeneous PBEs and provides a basis for extensions to spatially resolved CFD–PBE frameworks.</p>

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Deep learning for modeling the evolution of droplet size distribution in liquid–liquid dispersed systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月26日

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  • Mazhar Bayazidi,
  • Daniele Marchisio,
  • Agnese Marcato,
  • Antonio Buffo

摘要

Data-driven surrogates are investigated for the zero-dimensional population balance equation (0D–PBE) of turbulent liquid–liquid dispersions, using the quadrature method of moments (QMOM) as a reference. Five architectures are considered: a multilayer perceptron (MLP), an autoregressive neural network (ARNN), a long short-term memory (LSTM), a source-term surrogate for breakage and coalescence (RHS), and a Neural ordinary differential equation (Neural ODE). Surrogates are trained on thousands of QMOM simulations that span a wide range of initial droplet size, dispersed-phase holdup, and turbulent dissipation rate and are evaluated on 12,000 unseen operating conditions. The MLP, ARNN and LSTM accurately reproduce the decay of low-order moments, the nonlinear evolution of higher orders, and average diameters, with typical relative errors of a few percent and mean relative errors down to \(10^{-3}\) , while delivering speed-ups of about three orders of magnitude relative to the direct QMOM numerical solution. The RHS and Neural ODE models are less data-efficient and exhibit larger errors; the former is also slower than QMOM because it requires an additional ODE integration. The proposed methodology is transferable to other homogeneous PBEs and provides a basis for extensions to spatially resolved CFD–PBE frameworks.