(38e) A Two-Fluid Sub-Grid Analysis of Heat Transfer in Gas-Particle Fluidized Flows

A variety of appliances in chemical and energy industries trust on fluidized gas-particle flows to deal with heterogeneous and catalytic chemical reactions, where heat transfer frequently plays a significant role. An approach to large-scale simulation in this complex heterogeneous multiscale environment is by filtered two-fluid conservative formulations applied over coarse grids, where effects of dissipated heterogeneities at scales smaller than grid size must be recovered through additional closure modeling. As for heat transfer, additional closure modeling is required for effective interphase heat transfer coefficients, for filtered thermal conductivities of solid phases, and for residual thermal conductivities of solid and gas phases. A way for deriving effective, filtered and residual data for sub-grid modeling is by means of highly resolved simulations with microscopic two-fluid formulations, a now widely practiced approach mostly introduced and extensively developed by Professor Sankaran Sundaresan of Princeton and his co-workers. This approach has been followed by researchers worldwide for proposing a multiplicity of sub-grid closure models, including hydrodynamic sub-grid closures, heat transfer sub-grid closures, and also mass transfer/reaction rate sub-grid closures in chemically reacting flows. As for heat and mass transfer, literature shows that by capturing meso-scale hydrodynamic heterogeneities, the highly resolved simulations also capture associated heat and mass transfer heterogeneities. It follows that independent variables that suit hydrodynamic sub-grid correlation should also suit heat and mass transfer sub-grid correlation (for instance, meso-scale markers such as filtered solid volume fraction and filtered slip velocity, which are relevant to hydrodynamic sub-grid modeling, should also be relevant to heat and mass transfer sub-grid modeling).

In the present work, filtered results from highly resolved simulations with microscopic two-fluid modeling were used to investigate scale and heterogeneity related effects over sub-grid heat transfer coefficients in gas-particle fluidized flows. Following the analysis that was performed, new sub-grid models were proposed for those coefficients. Cyclic boundaries were applied, and a heat source/sink enforcement procedure was implemented to deal with statistical steady state energy balance. A variety of micro-scale conditions was accounted for (through a range of particle Froude numbers, named micro-scale marker), combined with a variety of macro-scale conditions (enforced through ranges of domain average solid volume fractions and gas Reynolds numbers, named macro-scale markers). The composition of the underlying micro and macro-scale conditions provided for a variety of meso-scale heterogeneity patterns detected by means of filtering related parameters (filtered solid volume fraction, filtered slip velocity and filter size, named meso-scale markers).

The results showed that the overall behaviors of the concerning effective, filtered and residual sub-grid heat transfer coefficients resulted quite similar to previously analyzed effective, filtered and residual hydrodynamic coefficients. With that in mind, the focus of analysis was then turned to the behaviors of the various sub-grid heat transfer coefficients as for their response to hydrodynamic heterogeneity as affected by the various micro, meso and macro-scale parameters of concern. In general, it was expected that the various sub-grid heat transfer coefficients would be diminished at increasing heterogeneity. As expected, the interphase sub-grid effective heat transfer coefficient resulted always inversely related to the heterogeneity degree, as evaluated through the sub-grid effective drag coefficient. The filtered and residual thermal conductivities, otherwise, responded erratically to heterogeneity, suggesting that something more besides heterogeneity, or maybe a more refined accounting of heterogeneity, should be required for their correlation.

Considering the observed similitude of behavior between sub-grid heat transfer and hydrodynamic coefficients, the former were here correlated just like the later have been in previous research. So, just like previously done for the interphase effective drag coefficient, the interphase effective heat transfer coefficient was correlated to particle Froude number, filter size, filtered solid volume fraction and filtered slip velocity. Just like previously done for filtered stresses coefficients, the filtered thermal conductivity of the solid phase was correlated to particle Froude number, filtered solid volume fraction and filtered slip velocity. And, just like previously done for residual stresses coefficients, the residual thermal conductivities were correlated to filter size, filtered solid volume fraction and filtered slip velocity. The new sub-grid models that were proposed stand for a significant range of conditions, for particulates ranging from fine to coarse, for flows ranging from dilute to dense, and from low velocity suspensions up to pneumatic transport.