Scaling of heat transfer and temperature distribution in granular flows in rotating drums

Accurate prediction of the time required to heat up granular materials to a target temperature is crucial for several processes. However, we do not have quantitative models to predict the average temperature or the temperature distribution of the particles. Here, we computationally investigate the scaling of heat transfer in granular flows in rotating drums. Based on our simulations, which include a wide range of system and material properties, we identify the appropriate characteristic time that is used to derive equations that predict the particles' average temperature and the particles' temperature distribution.

Figure 1

Snapshots from a rotating drum simulation: (a) $\varphi$ = 0.05, low $\varphi$ case—particles heat up like a solid body, (b) $\varphi$ = 1.2, intermediate $\varphi$ case—particles heat up with a cool inner core, and (c) $\varphi$ = 35, high $\varphi$ case—particles heat up relatively uniformly. Particles are colored based on their temperature (red = $T_w$ and blue = $T_o$). $\omega$ is the speed of rotation, $\theta$ is the fill angle, and ${\tau }_c=\frac{\theta }{\omega }$ is the time a particle spends in contact with the drum wall. (d) Computation of the thermal time parameter $\tau$ from DEM simulations. $\mathrm{\Delta }=T_w-\overline{T}$, and ${\mathrm{\Delta }}_o=T_w-T_o$. The results from the DEM simulations ($•$) are consistent with the lumped parameter formulation ($-$) [Eq. (1)].

Figure 2

(a) $\frac{\tau }{{\tau }_c}$ vs $\varphi$ for various $D$. For each $D, \omega$ is varied between 0.1 and 60 rpm and $\frac{\tau }{{\tau }_c}=a{\varphi }^{0.88}$. $a$ = 7.2 for $D=15$ ($\circ$), $a$ = 19.2 for $D=30$ ($\diamond$), $a$ = 22.4 for $D=40$ ($\times$), $a$ = 47.1 for $D=60$ ($▹$), $a$ = 62.6 for $D=90$ ($\square$), and $a$ = 125.8 for $D=120$ cm (+). ($F_L$ = 15%, $k=3000$ W/mK, $C_p$ = 880 J/kgK, and $\overline{d}$ = 4 mm). (b) Same data as in (a) but $\frac{\tau }{{\tau }_c}$ is normalized by $A^{\frac{2}{3}}$ and the data collapses into a single equation. (c) $\frac{\tau }{{\tau }_c}$ vs $\varphi$ plot for several values of $D$ and properties listed in the Supplemental Material [20]. The values of $\varphi$ range over four orders of magnitude. (d) Same data as in (c) but normalized by ${\left(\frac{A}{A_p}\right)}^{\frac{2}{3}}$ and experimental data for expanded clay ($▸$), glass beads ($⧫$), and steel balls ($★$) [7]. The line represents Eq. (3).

Figure 3

Evolution of temperature of individual particles for (a) $\varphi$ = 0.03 (low $\varphi$ case) and (b) $\varphi$ = 35 (high $\varphi$ case). (c) Probability distribution function of the particles' temperature, $f\left(T\right)$, for $\varphi$ = 0.03 (blue), 3.5 (black), and 35 (red) when $\overline{T}=\frac{T_w+T_o}{2}$. (d) Temperature of individual particles vs size of particles for $\varphi$ = 35 at $t=0$ ($\overline{T}=T_o$), $t=80.0$, and $t=290.0$ s.

Figure 4

(a) Evolution of ${\sigma }_T^{*}$ for $\varphi$ =0.05, 1.2, and 35.0 as a function of normalized time $\frac{t}{\tau }$. (b) $\gamma$ vs $\varphi$ for various values of $\varphi$; $\gamma \sim {\varphi }^{-2/5}$. (c) $f\left(T\right)$ for $\varphi$ = 0.05 and $\tau =2.0\text{s}$. (d) $f\left(T\right)$ for $\varphi$ = 35 and $\tau =115\text{s}$.