Dimensional analysis
The key dimensionless flow parameters, i.e., the coefficient of restitution, the friction coefficient, the dimensionless stresses, the particle volume fraction, and the dimensionless shear rate have been well documented by Chialvo et al. [11], and Mohan et al. [13]. In agreement with Chialvo et al. [11] we define a scaled shear rate as
$$\begin{aligned} \dot{\gamma } = \gamma \, d^{3/2}_\mathrm{p} / \sqrt{k_\mathrm{n}/\rho _\mathrm{p}}. \end{aligned}$$
(16)
According to the experiments by Nordstrom et al. [32], and recent simulations [10, 33, 34], simple expressions that correlate the above defined dimensionless quantities can be found. Specifically, the scaled stress-shear rate data can be collapsed into curves by appropriately adjusting the critical volume fraction (\(\phi _\mathrm{c}\)), as well as a set of exponents as shown below. This scaling will be an inspiration for the re-scaling of the conductive heat flux in the present contribution (see Sect. 4). Thus, of key relevance are the three resulting stress regimes (here illustrated by the pressure), which are characterized by the following expressions:
$$\begin{aligned}&p_\mathrm{inert} \sim |\phi - \phi _\mathrm{c}|^{-2} \end{aligned}$$
(17)
$$\begin{aligned}&p_\mathrm{QS} \sim |\phi - \phi _\mathrm{c}|^{2/3} \end{aligned}$$
(18)
$$\begin{aligned}&p_\mathrm{int} \sim |\phi - \phi _\mathrm{c}|^{0}. \end{aligned}$$
(19)
Previous work only considered the conductive and convective heat transport rate in a sheared, but uncooled, particle bed (see, e.g., [17]). As pointed out by Mohan et al. [13], the Peclet number is the most influential parameter for heat transport in granular materials and reads:
$$\begin{aligned} Pe = \dfrac{(d_\mathrm{p}/2)^2}{\lambda _\mathrm{p}/(\rho _\mathrm{c} \, c_\mathrm{p})} \, \gamma \end{aligned}$$
(20)
Clearly, Pe characterizes the relative rates of convective and conductive heat transport within the particles in the absence of heat transfer to the ambient fluid. In the present work, however, we consider exactly this heat transfer rate to the ambient fluid, which is characterized by the heat transfer coefficient \(\alpha \). Therefore, we must introduce a new dimensionless quantity, which we choose to be the Biot number defined as:
$$\begin{aligned} Bi = \dfrac{\alpha \, d_\mathrm{p}}{\lambda _\mathrm{p}} \end{aligned}$$
(21)
The Biot number can be interpreted as the ratio of (i) the intra-particle and (ii) the external resistance to heat transport. Thus, for high Biot numbers we expect intra-particle temperature profiles to be significant. We will see that also the Peclet number affects intra-particle temperature profiles, since conductive transport between particles (caused by particle-particle collisions) leads to a heat up of the outer shell of the particles.
As an alternative to the Peclet number, one could define a dimensionless convective heat transport rate by using the transferred heat flux (to the ambient fluid) as the reference, i.e.,
$$\begin{aligned} {\varPi }= \dfrac{Pe}{Bi} = \dfrac{\rho _\mathrm{p} \, c_\mathrm{p} \, d_\mathrm{p} \, \gamma }{\alpha } \end{aligned}$$
(22)
\({\varPi }\) can be interpreted as a scaled length over which convective transport is able to balance the cooling (or heating) caused by the ambient fluid. This dimensionless number is useful in situations in which one wants to probe the effect of the the particles’ heat conductivity on the results: for fixed values of \({\varPi }\) one can illustrate the effect of heat conduction within the particles by varying the Biot number. Note, that for fixed Pe a variation of Bi does not lead to such a transparent illustration, since also Pe is a function of the particles’ heat conductivity, and both dimensionless numbers are zero in the limit of infinitely fast heat conduction within the particles. Since \({\varPi }\) is proportional to Pe and Bi, it is straight forward to interpret our results also in light of this new dimensionless quantity. We have already attempted this in most parts of the present contribution.
Effect of heat conduction within the particle
The goal of this Section is to theoretically investigate the cooling of a sphere under different conditions, and justify the assumption of spherical symmetry made in the current work. As pointed out by Schmidt et al. [35], as well as Oschmann et al. [23], the computational cost for a fully resolved particle is enormous: calculation time and data storage increase by a factor of about 1000 when using three-dimensionally resolved simulations. Fortunately, a one-dimensional discretization (as followed in the current work) is often an excellent approximation (in case of a sphere) when considering heat exchange in a certain Reynolds number range (see Nikrityuk et al. [36]). To support this finding, we study the transient cooling of a sphere, and consider different Reynolds numbers. We then compare the temperature change over the radius with the data provided by Nikrityuk et al. [36]. The latter performed a fully three-dimensional simulation, including a resolved flow and temperature profile simulation in the fluid surrounding the particle. For the simulation details we refer to Nikrityuk et al. [36]. We have summarized the key dimensionless parameters in the caption of Fig. 2.
Table 3 Boundary conditions used for the cooled sphere simulations
In general, an excellent agreement between the DNS data and simulations performed with ParScale over a wide range of dimensionless times, i.e., Fo numbers, can be observed. This is especially true for large Re numbers, where only marginal differences between DNS data and predictions by ParScale are observed. As already pointed out by Nikrityuk et al. [36], the maximum difference between a fully three-dimensional simulation and a one-dimensional simulation is between 2\(\%\) and \(\approx 5\%\). To support this finding, we investigated conjugate heat transfer associated with the flow around a single sphere under significant internal heat transport limitation (see Fig. 3 for details). We listed all boundary conditions in Table 3 and governing equations in Table 1. As can be seen, the temperature profiles inside the particles approach spherical symmetry for higher Reynolds numbers. The reason for this behaviour is that the the temperature distribution in the ambient fluid becomes more uniform, since the rate of convective heat transport in the fluid increases with increasing Reynolds number. At low Reynolds numbers (see panel a of Fig. 3) a certain temperature gradient in the fluid in the main flow direction is observed. This temperature gradient leads to a shift of the maximum intra-particle temperature towards the rear stagnation point, causing a significant temperature gradient in the azimuthal direction. Fortunately, most industrial applications are concerned with improving external transport limitations, and hence operate at Reynolds numbers larger than unity. Also, a closer inspection of the global temperature difference in the left panel of Fig. 3 indicates a temperature fluctuation of about 0.17 dimensionless units. This is because \(Bi<1\), i.e., the external heat exchange rate is typically much smaller than the heat conduction rate in the particle in case of low Reynolds number flows. In summary, the results in Figs. 2 and 3 support our assumption of a spherically-symmetric temperature profile within particles for most gas-particle systems. Of course, this statement only holds for flows involving single spherical particles, and cannot be easily generalized to non-spherical particles, or dense particle ensembles. Thus, we currently cannot provide a rigorous justification for spherical symmetry in dense granular flows where particle-particle heat conduction occurs. Fortunately, this mode of thermal transport (i.e., heat conduction through the contact areas) is typically small: an analysis of the amount of heat exchanged due to conduction (denoted as \(\mathbf {Q}^\mathrm{cond}\)) and transfer to the surrounding fluid (i.e., \(Q^\mathrm{trans}\)) on a per particle basis helps to support this argument. Specifically, one finds that the ratio of these heat exchange rates is (based on the expressions for the heat fluxes introduced in Eqs. (10) and (11):
$$\begin{aligned} \dfrac{\mathbf {Q}^\mathrm{cond}}{Q^\mathrm{trans}} = \dfrac{1}{Bi} \, \dfrac{2 \epsilon }{\pi } \, \dfrac{\langle \bigtriangleup T_\mathrm{co} \rangle }{T_\mathrm{i} - T_\mathrm{f,i}} \end{aligned}$$
(23)
Here we have introduced \(\epsilon \) as a parameter that characterizes a typical overlap area, which is typically much smaller than unity. Also, we have denoted with \(\langle \bigtriangleup T_\mathrm{co} \rangle \) a mean temperature difference of the contacting particles. If we now accept that the term \(\langle \bigtriangleup T_\mathrm{co} \rangle / T_\mathrm{i} - T_\mathrm{f,i}\) is of order unity, we find
$$\begin{aligned} \dfrac{\mathbf {Q}^\mathrm{cond}}{Q^\mathrm{trans}} = \dfrac{\epsilon }{Bi}. \end{aligned}$$
(24)
Thus, the relative importance of the conductive heat flux decreases with increasing Biot number. It is now obvious that the heat exchange rate due to particle-particle contacts will not impact the shape of intra-particle temperature profiles, since the latter are significant only for \(Bi>1\). Thus, one can readily transfer the findings related to intra-particle temperature profiles in a single particle to a cooled bed of particles in case one considers Biot numbers of order unity or larger.
Analytical solution for heat conduction in a cooled particle bed
In the following we present an analytical approach to calculate the temperature distribution in a sheared bed with a constant ambient fluid temperature \(T_\mathrm{f}\), and study the limiting cases with respect to the Biot number. To the best of our knowledge no analytical solution for the mean temperature profile in a cooled bed of moving particles exists. To approach this issue, we first calculate the mean particle temperature distribution inside a static bed of cooled particles. Therefore, we use the particle heat conductivity \(\lambda _\mathrm{p}\), as well as simple geometrical factors that describe the morphology of the particle bed. Specifically, we assume that the bed has an average effective cross-sectional area A, and an average effective cross-sectional perimeter U (see Fig. 4 that also illustrates their local counterparts). We note that the calculation of the effective area and perimeter cannot be done by taking a simple arithmetic mean of their local counterparts. This is due the fact that in the governing equation for heat conduction in the fin (discussed in the next paragraph) the cross-sectional area A comes into play in a non-linear fashion. Consequently, A will be also impacted by the average particle-particle contact area, which depends on \(\phi _\mathrm{p}\), and hence the applied stress. Fortunately, in what follows it is not necessary to provide an expression for the effective cross-sectional area and perimeter. We will see that considering these effective quantities is sufficient to explain the qualitative features of the mean particle temperature distribution.
In case we accept the above simplification, we can reduce the problem of heat conduction inside a static bed to the problem of heat transport in and around a convectively-cooled fin (see Fig. 4, left panel). The steady-state solution of this classical heat transport problem is straightforward, and relies on the following differential heat balance:
$$\begin{aligned} \lambda _\mathrm{p} \, \dfrac{\text {d}^2 T}{\text {d}y^2} - \dfrac{\alpha \, U}{A} (T-T_\mathrm{f}) = 0 \end{aligned}$$
(25)
Note that the above expression only considers thermal transport through the particles, and hence avoids complexities arising due to, e.g., heat conduction via the ambient fluid as mentioned in Kuipers et al. [24]. We feel that such complexities should be incorporated by a separate model for thermal transport in the surrounding fluid, and hence can be added in the future.
Normalization of the above heat balance equation by introducing
$$\begin{aligned}&y^{*} =\dfrac{y}{d_\mathrm{p}} \end{aligned}$$
(26)
$$\begin{aligned}&\theta = T - T_\mathrm{f} \end{aligned}$$
(27)
$$\begin{aligned}&\text {d} \theta = \text {d} T \end{aligned}$$
(28)
$$\begin{aligned}&\text {d}^2 \theta = \text {d}^2 T \end{aligned}$$
(29)
leads to
$$\begin{aligned} \dfrac{\text {d}^2 \theta }{\text {d}y^{*\,2}} = \underbrace{\dfrac{d_\mathrm{p} \, \alpha }{\lambda _\mathrm{p}}}_\mathrm{Bi} \, \underbrace{\dfrac{d_\mathrm{p} \, U}{A}}_{\mathrm{g}^{*} (Pe, \phi _\mathrm{p},\dot{\gamma })} \, \theta \end{aligned}$$
(30)
The first term on the right hand side of the above equation is equal to the Biot number. The second term, i.e., (\(g^{*}\)), represents a dimensionless metric for the bed morphology. The latter fluctuates in case the bed of particles is sheared (see our discussion below). However, it is reasonable to assume that this is irrelevant for the steady-state solution to a first approximation.
We next consider thermal transport in a sheared bed. Clearly, the rate of heat transport through the bed will be increased due to the motion of particles in such a case. This will not change the structure of the above differential equation as long as we assume that particle motion in the gradient direction leads to a diffusive transport of thermal energy. In other words, we assume that the additional heat flux caused by particle motion linearly scales with the temperature gradient. Consequently, only the magnitude of the heat conductivity will change, and one could simply introduce an effective heat conductivity of the sheared particle bed, i.e., \(\lambda _\mathrm{eff}\) in Eq. 30. This effective heat conductivity will be some function of \(\phi _\mathrm{p}\), \(\dot{\gamma }\) and the particle parameters. Naturally, one would express \(\lambda _\mathrm{eff} / \lambda _\mathrm{p}\) as some function of the Peclet number, the particle concentration and the dimensionless shear rate. Hence, one can easily split off the particles’ heat conductivity from \(\lambda _\mathrm{eff}\) and remain the general structure of Eq. 30. Thus, we simply lump the effect of particle flow, i.e., the convective heat transport due to random particle motion, and the consequent change of A into the (unknown) function \(g^{*}(Pe, \phi _\mathrm{p}, \dot{\gamma })\). Clearly, \(g^{*}\) will decrease with increasing shear rate and increasing Peclet number (i.e., softer particles moving faster), since the effective heat conductivity and the area available for conduction will increase with increasing speed of shearing (or increasing particle softness). While \(g^{*}\) lumps a large number of phyical effects, we have not attempted to decouple these effects in the present contribution. This is done since such a decoupling is not essential to understand the qualitative features of the particle temperature profiles discussed in the next paragraphs. Also, we note in passing that in the following, we have set \(T_\mathrm{f}\) to be constant as indicated in Fig. 4.
We now use the following Ansatz for the solution of the temperature profile:
$$\begin{aligned} \theta = c_\mathrm{1 e^\mathrm{(Bi \, g^{*})y^{*}}} + c_\mathrm{2 e^{-(\mathrm Bi \, g^{*})y^{*}}} \end{aligned}$$
(31)
with the boundary conditions
$$\begin{aligned}&(1)&\, \, \, \, \text {for} \, y^{*}= 0: \theta = \theta _\mathrm{0} = T_\mathrm{bot} - T_\mathrm{f} \end{aligned}$$
(32)
$$\begin{aligned}&(2)&\, \, \, \, \text {for} \, y^{*}= H / d_\mathrm{p} = H^{*}: \theta = \theta _\mathrm{H} = T_\mathrm{top} - T_\mathrm{f} \end{aligned}$$
(33)
This leads us to the following constants:
$$\begin{aligned} c_\mathrm{1}= & {} \theta _\mathrm{0} - \dfrac{\theta _\mathrm{0} \, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^\mathrm{(Bi \, g^{*})H^{*}} - e^{-\mathrm (Bi \, g^{*})H^{*}}} \end{aligned}$$
(34)
$$\begin{aligned} c_\mathrm{2}= & {} \dfrac{\theta _\mathrm{0}\, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^\mathrm{(Bi \, g^{*})H^{*}} - e^{-(\mathrm Bi \, g^{*})H^{*}}}. \end{aligned}$$
(35)
Combining these constants with Eq. (31) one obtains the following mean particle temperature distribution:
$$\begin{aligned} \theta= & {} \left( \theta _\mathrm{0} - \dfrac{\theta _\mathrm{0}\, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^\mathrm{(Bi \, g^{*})H^{*}} - e^{-(\mathrm Bi \, g^{*})H^{*}}} \right) e^\mathrm{(Bi \, g^{*})y^{*}}\nonumber \\&+ \left( \dfrac{\theta _\mathrm{0}\, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^\mathrm{(Bi \, g^{*})H^{*}} - e^{-(\mathrm Bi \, g^{*})H^{*}}} \right) \text {e}^{-(\mathrm Bi \, g^{*})y^{*}} \end{aligned}$$
(36)
This equation does not allow us to explicitly calculate the effect of the shear rate on the resulting temperature profile. However, it provides us with an analytical solution to study certain limiting cases, namely sheared beds under the influence of high and low thermal heat exchange rates to the ambient fluid. Also, this analytical solution allows us to draw qualitative conclusions on the effect of the particle shear rate as mentioned above: faster shearing (i.e., increasing \({\varPi }\) or Pe) will lead to a more uniform temperature profile, since convective heat transport (due to random particle motion) becomes the dominating mode of thermal transport compared to heat transfer to the ambient fluid.
We note in passing that after calculating the temperature gradient from the above temperature profile, as well as integrating over the bed height, the mean heat flux based on the above analytical solution bed can be calculated as:
$$\begin{aligned} \dfrac{\partial \theta }{\partial y^{*} } = \theta '= & {} \left( \theta _\mathrm{0} - \dfrac{\theta _\mathrm{0}\, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^{(\mathrm Bi \, g^{*})H^{*}} - e^{-\mathrm (Bi \, g^{*})H^{*}}} \right) \, (\mathrm Bi \, g^{*}) \, e^\mathrm{(Bi \, g^{*})y^{*}} \nonumber \\&- \left( \dfrac{\theta _\mathrm{0}\, e^\mathrm{(Bi \, g^{*})H^{*}} - \theta _\mathrm{H}}{e^\mathrm{(Bi \, g^{*})H^{*}} - e^\mathrm{-(Bi \, g^{*})H^{*}}} \right) \, (\mathrm Bi \, g^{*}) \, e^{-(\mathrm Bi \, g^{*})y^{*}}\end{aligned}$$
(37)
$$\begin{aligned} \overline{q^\mathrm{y,analyt}}^{*}= & {} \frac{\lambda _\mathrm{eff} }{\lambda _\mathrm{p}} \end{aligned}$$
(38)
This heat flux has been normalized with the reference flux (i.e., \(q_\mathrm{s}\)), and requires the knowledge of \(\lambda _\mathrm{eff}/ \lambda _\mathrm{p}\) or, alternatively, \(g^{*}\). Unfortunately, the latter quantities are unknown, and hence must be modelled (e.g., via models for the conductive and convective fluxes developed in the present contribution). In the following we study certain limiting cases, and only compare the qualitative features of the temperature profiles inside the sheared bed between the simulation and the above analytical solution. Note, that \(g^{*}\) has been adjusted in the following such that the difference between analytical solution and simulation data is a minimum. Thus, the following comparison is only helpful to judge whether the shape of the temperature profiles can be predicted. Models for the effective heat flux (due to conduction and convection) are proposed in Sect. 4, which can be used to compute the effective heat conductivity in a straight forward manner.
Small Biot number limit
The first limiting case we study is a sheared bed without any heat transfer to the ambient fluid. This is in line with the work of Rognon et al. [16] and Mohan et al. [13]. Therefore, it is clear that for a steady-state situation the temperature profile is linear, as seen in Fig. 5. A very good agreement can be observed from the above figure. Volume-averaged and surface temperature are collapsing in one point since the Biot number is small (and in the limit of thermally insulated particles, i.e., no heat exchange with the ambient fluid, \(Bi=0\)). Due to the low Biot number the solution is independent of the fluid temperature, the dimensionless shear rate and the volume fraction. Unfortunately, this solution does not lead to a deeper understanding of the total fluxes inside the bed and the relative contributions of convective and conductive transport rates. Thus, in the following we study a set of dimensionless parameters for which the temperature distribution is mainly governed by the heat transferred to the ambient fluid, and hence might help us to better understand the associated heat fluxes.
High Biot number limit
The second limiting case is a sheared bed which is strongly cooled by the ambient fluid, i.e., heat transfer to the ambient fluid is fast. It is obvious that in this limiting case the temperature of the ambient fluid is of key importance. Also, we expect that for \({\varPi }\) approaching zero (at finite Bi) the convective and conductive flux will vanish, since the particle temperature approaches the fluid temperature. To illustrate this, we study high Biot number flows to highlight the mean particle temperature profiles in a sheared particle bed. For the shown set of dimensionless parameters a clear difference to Fig. 5 can be observed. The heat transfer to the ambient fluid leads to a fast relaxation of the bed temperature to the fluid temperature. Also, for even higher Biot numbers we observe a steeper gradient near the top and bottom of the bed (data not shown). This is explained by the corresponding decrease of \({\varPi }=Pe/Bi\), i.e., the dimensionless distance over which convective heat transport outbalances heat transfer to the ambient fluid decreases. In contrast, this gradient decreases with increasing Peclet number: fast shearing in combination with a low thermal conductivity of the particles smoothens out temperature gradients.
Another key observation is that in case of low \({\varPi }\)-number flows, the temperature gradient (and hence the thermal fluxes through the particle bed) is no longer constant over the simulation domain. Consequently, the size of the considered domain may affect the predicted mean thermal fluxes in that domain. Hence, in what follows it is essential to consider the size of the simulation domain for a correct interpretation of the results. Specifically, we now explore a wider range of dimensionless parameters to establish a more quantitative understanding of thermal fluxes through sheared particle beds (Fig. 6).