**1. Introduction**

In the production process of the steel industry, a large amount of waste heat resources, including sinter sensible heat, is generated. The recycling of sinter waste heat resources is one of the effective ways to reduce energy consumption in the sintering process [1,2].

The research on recycling and utilization of sinter waste heat resources has been carried out for a long time, mainly focusing on sinter ring coolers and sinter belt coolers, which have been applied to engineering projects [3,4]. Shi et al. [5] established a one-dimensional unsteady mathematical model to investigate the gas-solid heat transfer process in sinter ring coolers, and studied the influence of the cold air flow rate and trolley movement rate. Zhang et al. [6] developed a three-dimensional model to investigate the effects of several parameters on the heat transfer process of sinter ring coolers based on porous media theory and the local thermal non-equilibrium model. However, the way of recycling sinter waste heat resources in existing ring cooling machines has some inevitable defects, like the high air leakage rate in the cooling system, low air outlet quality, and low waste heat recovery quantity, resulting in a low efficiency. Therefore, a sinter vertical tank has been put forward as a new way for efficient sinter waste heat recovery [7–10].

Different from the unsteady heat transfer process in sinter ring cooling, the gas-solid heat transfer process in a vertical tank is steady. Due to the complexity and instability of gas flow in the sinter bed layer, the analysis of the gas-solid heat transfer process in the sinter vertical tank is still in the theoretical and experimental research stage. Leong et al. [4] studied the effect of sinter layer porosity distribution on flow and temperature fields in a sinter cooler. Liu et al. [11] experimentally studied gas flow characteristics in a vertical tank for sinter waste heat recovery. Feng et al. [12–14] investigated gas flow characteristics and the modification of Ergun's correlation in a vertical tank for sinter waste heat recovery experimentally. The influence of various structural parameters and operating parameters on heat transfer and evaluation of the process is lacking in systemic research and analysis. Kong et al. [15] numerically investigated the heat transfer and flow process in dry quenching furnace by building a one-dimensional mathematical model. The variation of outlet temperatures of circulating gas and coke under different working conditions and different gas-to-material ratios were obtained. To evaluate the recovery performance of waste heat in a system, the quality of waste heat was considered in a recent study. Feng et al. [16] established a steady gas-solid heat transfer model to numerically analyze the effects of different operating parameters on the cooling air outlet exergy in a sinter vertical tank and optimized parameters by the mixed orthogonal experimental method. In a subsequent study, Gao et al. [17] focused on investigating the resistance characteristics of the gas stream passing through the waste heat recovery tank bed layer by building a homemade experimental bench, then applied the weighted comprehensive scoring method to optimize parameters by comprehensively evaluating cooling air outlet exergy and sinter bed layer resistance loss.

Liu et al. [18–24] proposed the concept of a local exergy destruction rate based on convective heat transfer, obtaining the expression of the local exergy destruction rate, which can be used to represent the irreversible loss in the convective heat transfer process, then applied the exergy destruction minimization as the optimization criterion in the exchanged heat transfer tube.

In this paper, to comprehensively evaluate the waste heat recovery process in the sinter vertical tank from two aspects of heat transfer quantity and heat quality, the theory of exergy destruction minimum is applied based on the two-dimensional steady local thermal non-equilibrium two-equation model and porous medium theory.

To optimize parameters and comprehensively evaluate the flow and heat transfer performance, a multi-objective genetic algorithm [25–29] based on the Back Propagation (BP) neural network is applied. The BP neural network is a feedforward neural network trained according to the error back propagation algorithm. Structurally, it has an input layer, a hidden layer, and an output layer. In essence, it adopts the gradient descent method to calculate the minimum value of the square of the network error. In this paper, the BP neural network is used to train the obtained data for the fitting function. In addition, the traditional method for multi-objective optimization, like the weighted comprehensive scoring method, has some defects, because the allocation of each objective weighted value is subjective and there is no standard for it. Therefore, the multi-objective genetic algorithm is applied to the neural network, with the exergy destruction caused by heat transfer and the exergy destruction caused by fluid flow as the two objectives. Then, the most suitable combination of the three parameters is obtained after the iteration and evolution of the population.

#### **2. Numerical Methodology**

#### *2.1. Physical Model*

Different from the unsteady process of a gas-solid cross flow fixed bed in a sinter ring cooler, the gas-solid heat transfer process in a sinter vertical tank is stable from the gas-solid countercurrent moving bed essentially. In this paper, we concentrate on how to recycle waste heat efficiently, so its focus is the heat transfer enhancement between gas and solids. Considering that this work mainly focuses on the heat transfer mechanism in the vertical tank, a relatively miniaturized physical model is established. Figure 1 shows the simplified schematic of the computational domain for waste heat

recovery in a vertical tank. The inner diameter and height of the cooling section are 1 m and 1.8 m, respectively. The sinter particles with high temperatures fall into the vertical tank from the upper entrance slowly, and then exchange heat with the cooling air from the bottom of the tank. In this way, the high-grade air can be utilized later, such as power generation.

**Figure 1.** Simplified physical model of the sinter vertical tank.

#### *2.2. Mathematical Model and Exergy Destruction Minimization*

#### 2.2.1. Mathematical Model

Due to the wide range of sizes and irregular shapes of sinter particles, it is difficult to mathematically describe and numerically investigate the gas solid heat transfer process precisely. In this paper, sinter particles are considered to be spherical particles of equal size and the cooling section of the vertical tank is assumed to be the porous zone. Simultaneously, the radiant heat transfer between/in the sinter particles and the gas is ignored and the wall of the vertical tank is assumed to be insulated. Considering the turbulent flow of the cooling air in the bed layer, the standard turbulent model is adopted. The following equations are employed to describe the gas-solid heat transfer:

Continuity equation:

$$\frac{\partial}{\partial \mathbf{x}\_j}(\rho\_\mathcal{g} \boldsymbol{u}\_i) = 0 \tag{1}$$

Momentum equation:

$$\frac{\partial}{\partial \mathbf{x}\_{j}}(\rho\_{3}u\_{i}u\_{j}) = -\frac{\partial}{\partial \mathbf{x}\_{i}}(p + \frac{2}{3}\rho k) + \frac{\partial}{\partial \mathbf{x}\_{j}}[(\mu + \mu\_{l})(\frac{\partial u\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial u\_{j}}{\partial \mathbf{x}\_{i}} - \frac{2}{3}\delta\_{ij}\frac{\partial u\_{l}}{\partial \mathbf{x}\_{l}})] + \mathcal{S}\_{i} \tag{2}$$

*k* equation:

$$
\rho\_{\mathcal{S}} u\_{\hat{j}} \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_{\hat{j}}} = \frac{\partial}{\partial \mathbf{x}\_{\hat{j}}} \left[ \left( \mu + \frac{\mu\_{t}}{\sigma\_{k}} \right) \frac{\partial}{\partial \mathbf{x}\_{\hat{j}}} \right] + \mu\_{t} \frac{\partial u\_{i}}{\partial \mathbf{x}\_{\hat{j}}} \left( \frac{\partial u\_{j}}{\partial \mathbf{x}\_{i}} + \frac{\partial u\_{i}}{\partial \mathbf{x}\_{\hat{j}}} \right) - \rho\_{\mathcal{S}} \varepsilon \tag{3}
$$

*ε* equation:

$$\rho\_{\mathcal{S}}\mu\_{\hat{j}}\frac{\partial\varepsilon}{\partial\mathbf{x}\_{\hat{j}}} = \frac{\partial}{\partial\mathbf{x}\_{\hat{j}}}[(\mu + \frac{\mu\_{\ell}}{\sigma\_{\mathbf{c}}})\frac{\partial}{\partial\mathbf{x}\_{\hat{j}}}] + \frac{c\_{1}\varepsilon}{k}\mu\_{l}\frac{\partial\mu\_{\hat{i}}}{\partial\mathbf{x}\_{\hat{j}}}(\frac{\partial\mu\_{\hat{i}}}{\partial\mathbf{x}\_{\hat{j}}} + \frac{\partial\mu\_{\hat{j}}}{\partial\mathbf{x}\_{\hat{i}}}) - c\_{2}\rho\_{\mathcal{S}}\frac{\varepsilon^{2}}{k} \tag{4}$$

*Energies* **2019**, *12*, 385

The momentum equation of the porous media has additional momentum source terms, which are the momentum loss essentially consisting of two parts, the viscous loss term and the inertial loss term:

$$S\_{\bar{i}} = - (\frac{\mu}{\alpha} u\_{\bar{i}} + \frac{1}{2} \mathbb{C}\_2 \rho\_{\mathcal{S}} |u| u\_{\bar{i}}) \tag{5}$$

The viscous resistance coefficient and inertial resistance coefficient are calculated by the following two equations, respectively [6]:

$$\frac{1}{a} = \frac{85.4(1 - \varepsilon^2)}{\varepsilon^3 d\_p^{-2}}\tag{6a}$$

$$C\_2 = \frac{0.632(1 - \varepsilon)}{\varepsilon^3 d\_p} \tag{6b}$$

Considering the temperatures between the air and the sinter in the vertical tank are different after the heat exchange and this tends to be stable, the local thermal non-equilibrium model is adopted to calculate the heat transfer process. The cooling air as the gas phase and the sinter as the solid phase have independent energy equations. The two energy equations are shown below [16]:

Gas phase:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho\_{\mathcal{S}} c\_{\mathcal{S}} \boldsymbol{u}\_{\mathcal{S}} T\_{\mathcal{S}}) = \varepsilon \frac{\partial}{\partial \mathbf{x}\_i} (\lambda\_{\mathcal{S}} \frac{\partial T\_{\mathcal{S}}}{\partial \mathbf{x}\_i}) + h\_v (T\_s - T\_{\mathcal{S}}) \tag{7}$$

Solid phase:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho\_s c\_s \mathbf{u}\_s T\_s) = (1 - \varepsilon) \frac{\partial}{\partial \mathbf{x}\_i} (\lambda\_s \frac{\partial T\_s}{\partial \mathbf{x}\_i}) - h\_v (T\_s - T\_\S) \tag{8}$$

where *ρ*, c, and *λ* are the density, specific heat, and thermal conductivity, respectively. Their special values are listed in Table 1. Additionally, *hv* is the volume heat transfer coefficient, which is calculated with the correlation below:

$$h\_v = \frac{6h(1-\varepsilon)}{d\_p} \tag{9}$$

The Nusselt number (*Nu*) is calculated as follows by referring to [16]:

$$Nu = \frac{hd\_p}{\lambda\_\text{\%}} = 0.2 \varepsilon^{0.055} \text{Re}\_p^{0.657} \text{Pr}^{1/3} \tag{10}$$

Then, according to the solid phase energy equation, the velocity of the solid is added to the left side of the equation as a source term, which can achieve the slow falling process of sinter particles in the vertical tank and obtain the goal of steady-state gas-solid heat transfer. The source terms in the momentum equation and energy equation are applied into the solution equations through the user-defined functions, as well as the variations of the physical parameters of the air and sinter with temperature.

#### 2.2.2. Exergy Destruction Minimization

As we know, available potential is a state variable, characterizing the ability to do the work of a fluid. The convective heat transfer process can be comprehensively studied from two aspects of heat transfer quantity and heat quality when the available potential is used for the physical analysis of fluid particles. Additionally, exergy is a process variable, representing the change of the available potential and the maximum ability to do the work of a process. It is described as follows [19]:

$$\mathbf{e}\_x = (\mathbf{h} - T\_0 \mathbf{s}) - (\mathbf{h}\_0 - T\_0 \mathbf{s}\_0) = \mathbf{e} - \mathbf{e}\_0 \tag{11}$$

where *h*<sup>0</sup> and *s*<sup>0</sup> are the enthalpy and entropy in the environmental state, respectively, and *e*<sup>0</sup> is the available potential of the fluid in the environmental state.

*Energies* **2019**, *12*, 385

The change of available potential comes from two parts, the heat flow exergy from the outside and the loss generated during the heat transfer. The local destruction rate represents the loss of input exergy during the heat transfer process, thus it is called exergy destruction. The local exergy destruction rate can be expressed as follows [22]:

$$\varepsilon\_{\rm xd} = T\_0 \frac{\lambda \left(\nabla T\right)^2}{T^2} + \mathcal{U} \cdot \left(\rho \mathcal{U} \cdot \nabla \mathcal{U} - \mu \nabla^2 \mathcal{U}\right) \tag{12}$$

where the first item reacts with the irreversible heat loss. The irreversible source is the local temperature gradient. Additionally, it will never be negative, reflecting the irreversibility of the temperature difference heat transfer process. The next item is shown as Equation (13b). In steady-state flow, it is expressed as the product of the velocity and pressure gradient. Liu et al. [22] considered its physical meaning as the pumping work in the flow process, which includes the change of kinetic energy and viscous loss; that is, mechanical work consumed during the flow. This also meets the understanding in thermodynamics that the mechanical work lost during the flow is all mechanical exergy.

$$\varepsilon\_{\rm xd, \Delta T} = T\_0 \frac{\lambda \left(\nabla T\right)^2}{T^2} \tag{13a}$$

$$e\_{\text{xd},\Delta p} = \mathcal{U} \cdot (\rho \mathcal{U} \cdot \nabla \mathcal{U} - \mu \nabla^2 \mathcal{U}) \tag{13b}$$

The proposed local exergy destruction rate provides the possibility to quantitatively analyse the irreversibility of each point during the process. The total exergy destruction caused by heat transfer and fluid flow can be synthesized:

$$\mathbf{E}\_{xd} = \iiint \mathbf{e}\_{xd}dV = \iiint \left[ T\_0 \frac{\lambda \left(\nabla T\right)^2}{T^2} + \mathcal{U} \cdot \left(\rho \mathcal{U} \cdot \nabla \mathcal{U} - \mu \nabla^2 \mathcal{U}\right) \right] dV \tag{14}$$

According to Equation (14), the total exergy destruction consists of the thermal dissipation owing to a temperature difference and the power consumption owing to a pressure drop, which are adopted to respectively represent the irreversible heat loss and the irreversible pressure loss of the gas-solid heat transfer process.

$$\mathcal{E}\_{\rm xd, \Delta T} = \iiint\limits\_{\Omega} T\_0 \frac{\lambda \left(\nabla T\right)^2}{T^2} dV \tag{15a}$$

$$\mathcal{E}\_{\rm xd, \Delta p} = \iiint \mathcal{U} \cdot (\rho \mathcal{U} \cdot \nabla \mathcal{U} - \mu \nabla^2 \mathcal{U}) dV \tag{15b}$$
