Session E01: Focus Session: Understanding Thermal Transport in Flows of Dense Suspensions (3:10pm - 3:55pm CST)

Two fluid modeling of heat transfer in dense suspension flows in Couette cells

We propose a two fluid model (TFM) to capture thermal transport coupled to particle migration in shear flows of suspensions in the non-Brownian regime. Specifically, we introduce a closure relation for the inter-phase heat transfer coefficient as $K_h(\dot\gamma,\phi) = K_{h,0}[1 + \beta\phi(\|\dot{\boldsymbol{\gamma}}_p\|d_p^2/\alpha_p)^m]$, where $\phi$, $d_p$, $\dot{\boldsymbol{\gamma}}_p$ and $\alpha_p$ are the particle volume fraction, diameter, rate-of-shear-strain tensor and thermal diffusivity, respectively. Importantly, we capture shear-induced effects by using the full tensor $\dot{\boldsymbol{\gamma}}_p$, which is made possible by use of a TFM. We successfully calibrate $\beta$ and $m$ (and, thus, the TFM) by comparing to experiments in a Couette cell. Next, we perform a parametric study to understand how radial shear-induced migration influences the thermal transport performance in this system for different shear rates set by the rotation of the inner cylinder $\Omega_{in}$, $d_p$ and bulk volume fraction $\phi|_{t=0}$. Compared to a clear fluid, suspensions enhance thermal transport, and our computational model identifies the combinations of $d_p/(R_{out}-R_{in})$, $Pe_p = \Omega_{in} d_p^2/\alpha_p$, and $\phi|_{t=0}$ that maximize transport and/or efficiency.   

Regimes of heat transfer in particle suspensions

We present results of interface-resolved simulations of heat transfer in suspensions of finite-size neutrally-buoyant spherical particles for solid volume fractions up to 35\% and bulk Reynolds numbers from 500 to 5600. An Immersed Boundary-Volume of Fluid method is used to solve the energy equation in the fluid and solid phase. We relate the heat transfer to the regimes of particle motion previously identified, i.e. a viscous regime at low volume fractions and low Reynolds number, particle-laden turbulence at high Reynolds and moderate volume fraction and particulate regime at high volume fractions. We show that in the viscous dominated regime, the heat transfer is mainly due to thermal diffusion with enhancement due to the particle-induced fluctuations. In the turbulent-like regime, we observe the largest enhancement of the global heat transfer, dominated by the turbulent heat flux. In the particulate shear-thickening regime, however, the heat transfer enhancement decreases as mixing is quenched by the particle migration towards the channel core. As a result, a compact loosely-packed core region forms and the contribution of thermal diffusion to the total heat transfer becomes significant once again.   

Direct numerical simulations of heat transfer in liquid-solid circulating fluidized beds

Controlling heat and mass transfer in particulate suspensions has many important applications such as packed and fluidized bed reactors and industrial dryers. In this work, we study the effect of particle volume fraction and Galileo number on the heat transfer within suspensions of rigid spherical particles in liquid-solid circulating fluidized beds (L-S CFBs). To this purpose, particle-resolved direct numerical simulations (PR-DNS) are performed, using the immersed boundary method (IBM) to account for the solid-fluid interactions and a volume of fluid (VoF) approach to resolve the temperature equation both inside and outside the particles. A vertical box is considered as numerical domain, where hot particles are introduced at the bottom of the box in the presence of a cold inflow, cooling down during their rise to the top of the box. Different Galileo numbers are simulated at various particle volume fractions up to $10\%$. The average particle velocity and temperature are monitored, aiming to maximize the heat transfer while also accelerating the process. Our results indicate an optimum Galileo number where the efficiency of the heat transfer process is maximum. Detailed statistics of the fluid and particle phase will be presented at the conference.   

Non local heat transfer in dense suspensions with Stefan effect and thin surface layers

Suspension flows are pivotal in a large number of industrial processes due to their good heat and mass transfer properties. While a large amount of literature has been devoted to the study of heat transfer in such flows, the realm of pre-asymptotic conjugate transfer (where intra-particle transfer and non local effects are dominant) is still relatively unexplored. Particle-tracking techniques are often preferred to rigorous multiscale modelling to account for the thermal history of particles and particle clouds. In this work, a novel method to describe pre-asymptotic conjugate transfer between suspended particles and a suspending fluid is presented. This method is based on the Generalised Multi-Rate Transfer (GMRT), a formal multiscale method that describes temporal non locality (memory effects) using a set of local equations. The GMRT is extended to account for the effect of fluid displacement due to surface reactions (Stefan or "blowing" effect) and thin layers covering the particle surface. It is shown that the method, first developed for flows over immobile regions, can be extended to suspended solids, and numerical results are presented for the case of spherical particles.   

Rayleigh-Benard convection in non-colloidal suspensions

This study explores the Rayleigh-Benard convection in suspensions of neutrally-buoyant, non-colloidal suspensions confined between horizontal walls. A constitutive diffusion equation is used to model the dynamics of the particles suspended in a viscous fluid. We employ a simple model for the effective thermal diffusivity of suspensions that considers the thermal diffusivity increasing linearly with the thermal Peclet number (\textit{Pe}) and the particle volume fraction ($\phi )$. We perform both linear stability analysis (LSA) and direct numerical simulation (DNS) for various bulk particle volume fractions ($\phi_{b})$ ranging from 0 to 0.3. The critical Rayleigh number (\textit{Ra}) grows gradually by increasing $\phi_{b}$ from the critical value for a pure fluid, whereas the critical wavenumber ($k_{c})$ remains constant at 3.12. The transition from the conduction state is subcritical (or hysteretic) and the heat transfer rate in dense suspensions is significantly enhanced by the convective flow for small \textit{Ra} close to the critical \textit{Ra}. We also found a power-law increase of the Nusselt number (\textit{Nu}) with \textit{Ra} where the scaling exponent $b$ decreases with $\phi_{b}$   

Stochastic Functional Expansion for Identifying the Effective Heat Conductivity Coefficient of Polydisperse Suspension

We consider a random two-phase medium which represents a matrix containing an array of non-overlapping spherical inclusions with random radii. A statistical theory of transport phenomena in the medium is constructed by means of the functional (Volterra-Wiener) series approach for identifying the effective heat conductivity of a polydisperse spherical suspension. An approximate model based on power-series expansion of the kernels with respect to the volume fraction is developed. The functional series for the temperature is rendered virial in the sense that its truncation after the $p$-tuple term asymptotically correct to the order $\gamma^p$ where $\gamma$ is the mean number of spheres per unit volume -- also proportional to the volume fraction. The case $p=2$ is considered in detail and the needed kernels of the functional series are found to the order $\gamma^2$. The truncated Volterra-Wiener expansion is applied consistently to derive the equations for the kernels and their contributions to the overall (effective) modulus are identified. In this way, not only the effective conductivity, but also all needed correlation functions can be expressed in closed form, correct to the said order.