## 2 Simulation Method  
The simulations were carried out using the Ansys Fluent software [42]. There were three different types of circular cross-section channels used. One is a one-meter-long straight channel, and the other is 1.5-m-long. The third channel is a U-shaped one (Figure 1a), with two straight arms of 1 m length, a 0.25 m middle curved part of the outer-wall perimeter, and a 0.05 m wall-wall separation between the two straight parts. All of the channels have the same diameter of 55 mm. The structured mesh of the three channels has approximately 600,000 nodes (Figure 1b), and we tested a mesh with more nodes that would produce nearly identical results. The simulations' boundary conditions are an inlet velocity \(V\) and a constant temperature \(T_{0}=300\) K for the incoming fluid, zero pressure and an open-to-the-air temperature \(T_{1}\) at the outlet, and no-slip on the wall with a constant temperature \(T_{1}\). Two inlet velocities and two temperatures \(T_{1}=273\)/ 283 K were imposed for each channel. The Bingham model describes slurry viscosity, and the Arrhenius model [43] reflects temperature influence:  
\[\left\{\begin{aligned} \eta=\left(\tau_{y}/\dot{\gamma}+b \right)\exp(\frac{280.394}{T}),&|\tau|>\tau_{y}\\ \dot{\gamma}=0,&|\tau|<\tau_{y}\end{aligned}\right. \tag{1}\]  
Figure 1: (**a**) The model of the U-shaped channel and the cross sections within the bend at five different angles: \(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), \(135^{\circ}\), and \(180^{\circ}\). (**b**) Mesh structures of the U-shaped channel, viewed from the inlet and the outlet.  
Here, \(\tau_{y}=7.1\) Pa is the yield stress, \(b=2.3\) Pa-s is the consistency index, and \(T\) is the temperature whose unit is \(K\). The values of \(\tau_{y}\) and \(b\) are referred from [44], in which the rheological properties of coal slurries are systematically measured using a rheometer. The thermal conductivity of the slurry is referred to [37]:  
\[k=k_{0}(k_{0}^{*}+k_{s}^{*}T), \tag{2}\]  
where \(k_{0}=0.08\) W/(mK), \(k_{0}^{*}=0.7753\), and \(k_{s}^{*}=0.00118\) 1/\({}^{\circ}\)C. The temperature-dependent viscosity \(\eta\) and thermal conductivity \(k\) were written into Fluent through a User Defined Function. The Reynolds numbers \(Re=\frac{D\rho V}{b}\), as defined by [39], are approximately 12 and 24, respectively, for the two flow rates, demonstrating that the turbulence can be neglected. Here, \(\rho\) is the density of the fluid and is taken as 1250 kg/m\({}^{3}\) in the simulations.  
The equations to be solved for the incompressible flow of non-Newtonian fluid are the conservation of mass  
\[\frac{\partial u_{i}}{\partial x_{i}}=0, \tag{3}\]  
linear momentum  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial\big{(}\rho u_{i}u_{i} \big{)}}{\partial x_{j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial \tau_{ij}}{\partial x_{j}}, \tag{4}\]  
and thermal energy  
\[\frac{\partial\big{(}\rho c_{p}T\big{)}}{\partial t}+\frac{\partial\big{(} \rho c_{p}u_{j}T\big{)}}{\partial x_{j}}=-\frac{\partial}{\partial x_{j}} \bigg{(}k\frac{\partial T}{\partial x_{j}}\bigg{)}+\tau_{ij}\frac{\partial u_ {i}}{\partial x_{j}}. \tag{5}\]  
\(c_{p}=4182\) J/kg\({}^{\circ}\)C is the specific heat, \(u_{i}\) is the velocity in the \(i\) direction, \(P\) is the fluid pressure, and \(\tau_{ij}=\eta\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{ j}}{\partial x_{i}}\right)\) is the stress tensor.