Derivation of the Adjoint Drift Flux Equations for Multiphase Flow

Round 1

Reviewer 1 Report

This paper presents the development of the continuous adjoint approach for multiphase flows modeled by the drift flux equations. The objective function is defined in a generic form (for the reviewer, the physical meaning of this objective is a bit unclear). Development includes the Takacs and Dahl models. No results are presented in this paper, so work cannot be evaluated to this extent.

The frozen turbulence assumption is made. Despite the fact that this might be a source of inaccuracies (the authors mention this but the cited paper -ref. 11 - is one that is also making the turbulence frozen assumption, rather than one that demonstrates the resulting inaccuracies!) in some cases, I am not convinced about the need to talk about this assumption in a paper that derives the math. framework of the adjoint method. I might be wrong but could the same material be presented only for laminar flows?

One last minor question: The omega term in eq. 34 is for correction units. This is not clear to me. There is no need to correct units when the objective function is defines. Later on, in the development of the adjoint method, the presence of the adjoint fields is enough to "absorb" difference in units. So, the question is why do the authors need this omega.

Author Response

We would like to thank the referee for their useful and (we believe) generally supportive comments on our paper. The aim of the paper was to present for the first time the mathematical derivation of the adjoint system for this common multiphase flow set of equations. Results in the context has to refer to the mathematical derivation, the presentation of which we believe to be sufficient for the publication (and we believe the referee agrees on this). The definition of the objective function was slightly unclear; we are interested in maximising the dispersed phase flow through the outlet, whilst the inlet flow rate is fixed; thus the objective function is defined as the mass flow rate of solid at the outlet. The discussion in section 4 has been revised to (hopefully) clarify this.

The issue of frozen turbulence is complex and not something we feel qualified to address in this paper. In the literature there are contributions showing the errors consequent on this assumption (eg. Kavvadias et al 2015) but also work demonstrating that it can be justified (Schramm et al 2018). Working on the laminar system would make the mathematics easier; however we would like to retain the turbulence/frozen turbulence approach here for its more general applicability, to show where the frozen turbulence terms enter the equations and to indicate where in the derivation further work needs to be done. The laminar case can be recovered from these straightforwardly (for example by setting K=0 in equation (1c)). We have included these references to bolster (slightly) the argument about frozen turbulence.

We have removed omega in the derivation in line with the refeeres comments.

Changes to the text have been made in red. We hope that these changes adequately address the referee's comments.

Reviewer 2 Report

This work derives adjoint drift flux equations for multiphase flow.  The paper is well written and can be published.  However, the authors are suggested to provide more background on this topic.  Also, most of the CFD code employs two-fluid model governing equations and have the authors thought about deriving an adjoint method for two-fluid questions?

Author Response

We would like to thank the referee for their comments which we believe are positive. 

To address the issues raised; we have extended the introduction section to provide a little more background on the Adjoint method and its current applications (particularly in the automotive sector). However a complete review of the area would be a large undertaking and the intention was not to do so. 

The drift flux equations were used as we are primarily interested in the optimisation of sedimentation systems. A full two-fluid model would be more generally applicable, however would represent a significantly more complex challenge to perform the mathematical manipulations.

Changes to the manuscript have been highlighted in red. We hope that these meet with the referee's approval.