Optimization of Arrangements of Heat-Storage Bricks in a Regenerative Combustion System by Tree Search

1. Introduction

In this article, we will propose a method for solving an optimization problem on the arrangements of heat-storage ceramic bricks (checkers) in a heat regenerator system.

Usually two fixed-bed heat regenerators are utilized in pairs with a central furnace to form a regenerative combustion system. See Figure 1. In each heat regenerator, checkers are arranged. The regenerative combustion system operates periodically with a two-phase cycle. The checkers arranged in the heat regenerators, on both ends of a regenerative combustion system, are used to absorb heat from the high temperature exhaust gas expelled from the central furnace.

In Phase 1, the cool fresh air flows into the left heat regenerator and is preheated by the checkers arranged there. (The checkers arranged in the left heat regenerator are designed to absorb periodically heat from the high temperature exhaust gas expelled from the central furnace in Phase 2.) The preheated fresh air then flows into the central furnace for combustion with Natural Gas. After combustion with Natural Gas, the high temperature exhaust gas, expelled from the central furnace, then flows into the right heat regenerator and heats up the checkers arranged there. Then Phase 2 starts at the Phase Switch point of time. In Phase 2, the above process is reversed. The cool fresh air flows into the right heat regenerator and is preheated by the checkers arranged there, which already absorbed heat from the high temperature exhaust gas expelled from the central furnace during Phase 1. Then the preheated fresh air flows into the central furnace for combustion with Natural Gas. Finally, the high temperature exhaust gas, expelled from the central furnace, flows into the left heat regenerator and heats up the checkers arranged there. Then Phase 1 starts again at the Phase Switch point of time. A new cycle of operations follows.

Usually there are several different types of heat-storage ceramic bricks that can be arranged in a heat regenerator. So one must find an optimal arrangement of these checkers, of possibly several different types, in a heat regenerator to reach the best performance of the regenerative combustion system with this heat regenerator. Naturally, the objective function for this optimization problem is the long-term Waste Heat Recovery Ratio defined for each arrangement of selected heat-storage ceramic bricks (of possibly different types) in this heat regenerator.

However, the number of possible arrangements of checkers in a heat regenerator could be very huge. Thus finding the optimal arrangement of checkers, of possibly different types, in a heat regenerator could be difficult. In fact, when there are $p$ different types of checkers that can be selected for each of $n$ positions in a fixed-bed heat regenerator, the total number of possible arrangements of checkers at $n$ positions is

$$p^n.$$

Thus, when 5 different types of checkers are available for each of 14 positions in a heat regenerator, the total number of possible arrangements of checkers, of possibly different types, is

$$5^{14}=\mathrm{6,103,515,625}$$

This means that one must evaluate more than 6 billion arrangements to reach an optimal arrangement of checkers at 14 positions in a heat regenerator. To tackle this optimization problem, one must find a effective method for deciding the optimal (most efficient) arrangements of checkers among the huge number of possible arrangements of checkers, of possibly different types, at $n$ positions in a fixed-bed heat regenerator.

Over the past few years, the Artificial Intelligence (AI), based on combination of “Deep Learning” with MCTS, has proved to be very successful in creating computer player for the incredibly complicated board game Go. Motivated by this great progress in AI, we propose, in this article, a simple stochastic tree search method to find the most efficient arrangement of checkers in a heat regenerator.

Though the total number

$$p^n$$

depends exponentially on the number $n$ of positions for checkers in a heat regenerator, our stochastic tree search method usually exhibits only complexity of polynomial growth on the number $n$ of positions for checkers in a heat regenerator.

To speed up the search process for the most efficient arrangements of checkers, we use a simplified system of 1-dimensional partial differential equations to evaluate the long-term Waste Heat Recovery Ratio of each node (arrangement of checkers) considered in our search tree. We demonstrate this search method to find the most efficient arrangements of checkers when the number of positions for checkers in a heat regenerator is 6. Empirical results reveal that our search method is highly efficient for finding the optimal arrangements of checkers in a heat regenerator.

2. Materials and Methods

The “Monte-Carlo Method” was proposed in 1949, [1], to calculate the area of a region, based on Random Sampling, to solve problems in statistical physics. According to the mathematical principle “Law of Large Numbers”, [2,3,4], one should be able to speculate almost correctly the genuine area of a region when the sampling number is large enough. By the way, other probabilistic methods “Stochastic Integrals”, [5,6], based on the Central Limit Theorem, were applied to mathematical finance, [7], in those decades.

2.1. Tree Search by the Monte-Carlo Method

In 1987, B. Abramson applied the “Monte-Carlo Method” to certain “Two-Player Games”： tic-tac-toe, Othello and Chess, [8]. B. Abramson proposed the “Expected-Outcome model”, based on the statistical data of “random game playouts to the end”, as the basis of strategies for a game player. B. Brügmann considered the applications of the “Monte-Carlo Method” to the board game “Go” in 1993, but not seriously, [9]. In 2006, inspired by many predecessors, R. Coulom considered seriously the applications of the “Monte-Carlo Method” to the search tree for the board game “Go”, [10]. See also [11] for the work by Kocsis and Szepesvári. The decision-making algorithm proposed in [10] was considered as “Monte-Carlo Tree Search (MCTS)” by R. Coulom. More modifications and improvements, [12,13,14,15,16,17,18], are made for MCTS after the important contributions by Kocsis and Szepesvári, [11] and by Coulom, [10]. The MCTS algorithm was also applied to certain games, [19,20,21,22,23]. By combining MCTS with Reinforcement Learning based on CNN (Convolutional Neural Networks), the computer Go player AlphaGo, developed by Google DeepMind, defeated Lee, Sedol, one of the most brilliant Go players, in a five game Go match in 2016. See [24,25]. The following three pictures are taken from [26], which can be found on the website of Rémi Coulom.

Figure 2. List of Game Complexity. Taken from [26].

Figure 2. List of Game Complexity. Taken from [26].

Figure 3. Principle of Monte-Carlo Evaluation. Taken from [26].

Figure 3. Principle of Monte-Carlo Evaluation. Taken from [26].

Figure 4. Monte-Carlo Tree Search. Taken from [26].

Figure 4. Monte-Carlo Tree Search. Taken from [26].

Figure 5. The board game Go.

Figure 5. The board game Go.

In 2017, Google DeepMind added a new ingredient “Dirichlet noise” to AlphaGo Zero to encourage further exploration by MCTS, which could be omitted by the earlier versions of AlphaGo because of low prior probabilities. This deepened the MCTS reinforcement learning of AlphaGo Zero. AlphaGo Zero is considered as a genuine AI because AlphaGo Zero is able to explore the board game Go by MCTS without human knowledge (experience), [27]. Applications of Artificial Intelligence with MCTS to other games were developed in [28,29,30,31]. Artificial Intelligence with MCTS was also applied to chemical syntheses of organic compounds, [32]. Application of MCTS to floor plans was considered in [33]. More applications of MCTS are still explored, [34,35]. For a recent review on MCTS, see [36].

2.2. Our Simple Tree Search Method

Motivated by the principles of MCTS, we will introduce a simple Tree Search method to find the optimal (most efficient) arrangements of heat-storage ceramic bricks (checkers) in a heat regenerator, when there are $p$ different types of checkers that can be selected for each of $n$ positions in a fixed-bed heat regenerator. Our search method can be performed easily on an ordinary computer, while a super computer is usually necessary for AI with MCTS.

We begin with the notion of a “Partition” of the set

$$S=\left\{1,\dots ,n\right\}$$

of positions, to be placed by checkers, in a fixed-bed heat regenerator. A Partition $\alpha$ of $S$ is a subset of $S=\left\{1,\dots ,n\right\}$ such that the largest integer $n$ must be included in $\alpha$.

Example 1. For the set $S=\left\{1,\dots ,9\right\}$, $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$ are Partitions of $S$.

Assume that

$$\alpha =\left\{c_1,\dots ,c_m\right\}\mathrm{with}{c}_1<\dots <c_m=n$$

(1)

is a Partition of $S=\left\{1,\dots ,n\right\}$. Using the positive integers $c_1,\dots ,c_m=n$ in $\alpha$, we may create the following decomposition

$$S=I_1\cup \dots \cup I_m$$

(2)

for $S=\left\{1,\dots ,n\right\}$, in which

$$I_1=\left\{1,\dots ,c_1\right\},I_2=\left\{c_1+1,\dots ,c_2\right\}\dots ,I_m=\left\{c_{m-1}+1,\dots ,c_m=n\right\}.$$

(3)

Example 2. For the Partition $\alpha =\left\{4,9\right\}$ of the set $S=\left\{1,\dots ,9\right\}$, we have the following decomposition

$S=I_1\cup I_2$ with $I_1=\left\{1,2,3,4\right\}$ and $I_2=\left\{5,6,7,8,9\right\}$.

For the Partition

$\alpha =\left\{c_1,\dots ,c_m\right\}$ with $c_1<\dots <c_m=n$

of the $S=\left\{1,\dots ,n\right\}$, we will conveniently say that each $I_k$ is an “Interval Component” of $S$ with respect to $\alpha$, because each $I_k$ contains certain consecutive positive integers taken from $S=\left\{1,\dots ,n\right\}$. This phenomenon is clear from the example (Example 2) shown above.

We have the following simple facts about the decomposition (2) of $S=\left\{1,\dots ,n\right\}$.

Fact A. The endpoints of the Interval Components $I_k$ in (2) simply constitute the Partition set $\alpha$ of $S$.

Fact B. The Interval Components of the decomposition (2) of $S=\left\{1,\dots ,n\right\}$, with respect to the Partition $\alpha$, are disjoint.

Fact C. Elements of the Interval Components of the decomposition (2) of $S=\left\{1,\dots ,n\right\}$ are linearly ordered.

There is a Partial Ordering on the collection of all Partitions of $S=\left\{1,\dots ,n\right\}$. For Partitions $\alpha$ and $\beta$ of $S=\left\{1,\dots ,n\right\}$, we say that $\beta$ is finer than $\alpha$ if and only if

$$\alpha \subset \beta .$$

We say that $\beta$ is strictly finer than $\alpha$ if and only if

$$\alpha \subset \beta \mathrm{and}\beta \ne \alpha$$

We will use the notation

$$\alpha \prec \beta$$

to indicate the condition that $\beta$ is strictly finer than $\alpha$.

Example 3. For the Partitions $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$ of $S=\left\{1,\dots ,9\right\}$, we have $\alpha \prec \beta$.

Now we discuss the notion of “qualified arrangements with respect to a Partition”. Assume that $\alpha$ is a Partition of $S=\left\{1,\dots ,n\right\}$ as shown in (1). We say that an arrangement $U$ of checkers on $S=\left\{1,\dots ,n\right\}$ is a “qualified arrangement with respect to the Partition $\alpha$” if and only if the following condition (4) is satisfied.

Definition. We say that an arrangement $U$ of checkers on $S=\left\{1,\dots ,n\right\}$ is a “qualified arrangement with respect to the Partition $\alpha$” if the following condition is satisfied.

For each Interval Component $I_k$ of $S=\left\{1,\dots ,n\right\}$ in the decomposition (2) with respect to the Partition $\alpha$,

$$\mathrm{the}\mathrm{checkers}\mathrm{on}{I}_k,\mathrm{arranged}\mathrm{by}U,\mathrm{are}\mathrm{of}\mathrm{the}same\mathrm{type}$$

(4)

Let ${\Omega }_{\alpha }$ denote the class of all “qualified arrangements with respect to the Partition $\alpha$”. Let $\left|{\Omega }_{\alpha }\right|$ denote the norm (the total number of qualified arrangements with respect to $\alpha$) of ${\Omega }_{\alpha }$. Then we have

$$\left|{\Omega }_{\alpha }\right|=p^m=p^{\left|\alpha \right|}$$

(5)

in which $m=\left|\alpha \right|$ is the norm (number of elements) of $\alpha$, as shown in (1).

Example 4. For the Partition $\alpha =\left\{4,9\right\}$ of $S=\left\{1,\dots ,9\right\}$, the total number of “qualified arrangements with respect to the Partition $\alpha$” is

$$p^2,\mathrm{in}\mathrm{which}\left|\alpha \right|=2.$$

For the Partition $\beta =\left\{4,6,9\right\}$, the total number of “qualified arrangements with respect to the Partition $\beta$” is

$$p^3,\mathrm{in}\mathrm{which}\left|\beta \right|=3$$

However, the number of all possible arrangements of checkers on $S=\left\{1,\dots ,9\right\}$ is $p^9$.

For Partitions $\alpha$ and $\beta$ of $S=\left\{1,\dots ,n\right\}$, we have the following important

Fact D. For Partitions $\alpha$ and $\beta$ of $S=\left\{1,\dots ,n\right\}$ satisfying $\alpha \prec \beta$, we have

$${\Omega }_{\alpha }\subset {\Omega }_{\beta }\mathrm{but}{\Omega }_{\alpha }\ne {\Omega }_{\beta }.$$

(6)

Thus any “qualified arrangement of checkers with respect to the Partition $\alpha$” is naturally a “qualified arrangement of checkers with respect to the Partition $\beta$”. Besides, the class ${\Omega }_{\beta }$ of arrangements is strictly larger than the class ${\Omega }_{\alpha }$ of arrangements.

To initialize a search tree for the optimal (most efficient) arrangements of checkers in a heat regenerator, when $p$ different types of checkers are available for selection at each of $n$ positions in the heat regenerator, we must choose a strictly increasing sequence

$${\alpha }_0\prec {\alpha }_1\prec \dots \prec {\alpha }_q$$

(7)

of Partitions of $S=\left\{1,\dots ,n\right\}$. Let ${\Omega }_{{\alpha }_k}$ denote the class of all “qualified arrangements with respect to the Partition ${\alpha }_k$”. Then, according to Fact D, the sequence of classes

$${\Omega }_{{\alpha }_0}\subset {\Omega }_{{\alpha }_1}\subset \dots \subset {\Omega }_{{\alpha }_q},$$

(8)

of arrangements corresponding to the strictly increasing sequence ${\alpha }_0\prec {\alpha }_1\prec \dots \prec {\alpha }_q$ of Partitions, is naturally strictly increasing.

In our tree search, each arrangement of checkers on $S=\left\{1,\dots ,n\right\}$ will be considered as a “node” in the search tree. At the initial stage ${\alpha }_0$, we evaluate the efficiency of each of the arrangements in ${\Omega }_{{\alpha }_0}$ (“qualified arrangements with respect to the Partition ${\alpha }_0$”) by numerical Simulation. Then we select the top $K$ arrangements in ${\Omega }_{{\alpha }_0}$ as the “mother (root) nodes” in ${\Omega }_{{\alpha }_1}$. These selected “mother (root) nodes” will be used to create “child nodes” (qualified arrangements with respect to the Partition ${\alpha }_1$) in ${\Omega }_{{\alpha }_1}$. This process is the Expansion operator of our algorithm. Then we evaluate the efficiency of each of the child nodes, created by the Expansion operator, in ${\Omega }_{{\alpha }_1}$ by numerical Simulation. See Figure 6 for the phases of our tree search.

In the Expansion phase of Figure 6, The “mother (root) nodes” selected in ${\Omega }_{{\alpha }_k}$ will be used to create “child nodes” in ${\Omega }_{{\alpha }_{k+1}}$.

2.3. The Expansion Operators

We will discuss two Expansion operators, “Gradient Expansion operator” and “Mutation Expansion operator”, for our tree search method.

We consider the following typical case. Assume that $\alpha$ and $\beta$ are Partitions of $S=\left\{1,\dots ,n\right\}$ satisfying

$$\alpha \prec \beta .$$

(9)

Then, by Fact D, we have

$${\Omega }_{\alpha }\subset {\Omega }_{\beta }.$$

Let

$$S=I_1\cup \dots \cup I_{m_{\beta }}$$

(10)

be the decomposition of $S=\left\{1,\dots ,n\right\}$ into a union of Interval Components with respect to the Partition $\beta$. Assume that $U\in {\Omega }_{\alpha }$. Since ${\Omega }_{\alpha }\subset {\Omega }_{\beta }$, we note that, for each Interval Component $I_k$, only checkers of the same type are placed on $I_k$ by the arrangement $U$. We assume that, on the Interval Component $I_k$, the checkers arranged by $U$ are all of the type

$$U_k.$$

(11)

Assume that $V\in {\Omega }_{\beta }$ is a “qualified arrangement with respect to the Partition $\beta$”. Assume that, on the Interval Component $I_k$, the checkers arranged by $V$ are all of the type

$$V_k.$$

(12)

We say that the arrangement $V\in {\Omega }_{\beta }$ of checkers is a “child-node” of $U$ in ${\Omega }_{\beta }$, by the “Gradient Expansion operator”, if the following GE Condition is satisfied.

GE(Gradient Expansion) Condition. There is a sequence $G{E}_{UV}=\left\{\varphi (1),\dots ,\varphi (q)\right\}$ of positive integers taken from $\left\{1,\dots ,m_{\beta }\right\}$, satisfying

$$\varphi (s)+2\le \varphi (s+1),\mathrm{for}\mathrm{all}s=1,\dots ,(q-1),$$

(13)

such that

$$U_{\varphi (s)}\ne U_{\varphi (s)+1},\mathrm{for}\mathrm{all}s=1,\dots ,q.$$

(14)

For each $s=1,\dots ,q$, we have

$$\begin{gathered} (V_{\varphi (s)},V_{\varphi (s)+1})=(U_{\varphi (s)},U_{\varphi (s)})\mathrm{or}(V_{\varphi (s)},V_{\varphi (s)+1})=(U_{\varphi (s)},U_{\varphi (s)+1}) \\ \mathrm{or} \\ (V_{\varphi (s)},V_{\varphi (s)+1})=(U_{\varphi (s)+1},U_{\varphi (s)+1}), \end{gathered}$$

(15)

for the arrangements $U$ and $V$ of checkers on the pair $(I_{\varphi (s)},I_{\varphi (s)+1})$ of adjacent Interval components $I_{\varphi (s)}$ and $I_{\varphi (s)+1}$. Besides, outside these pairs $(I_{\varphi (s)},I_{\varphi (s)+1})$ of adjacent Interval components, $s=1,\dots ,q$, we have

$$V_K=U_k, \mathrm{if} k\mathrm{is}\mathrm{not}\mathrm{in}\mathrm{the}\mathrm{union}\left\{\varphi (1),\dots ,\varphi (q)\right\}\cup \left\{\varphi (1)+1,\dots ,\varphi (q)+1\right\}.$$

(16)

Example 5. For the Partitions $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$ of $S=\left\{1,\dots ,9\right\}$, we have

$$S=\left\{1,2,3,4\right\}\cup \left\{5,6,7,8,9\right\}\mathrm{and}S=\left\{1,2,3,4\right\}\cup \left\{5,6\right\}\cup \left\{7,8,9\right\}.$$

These are the decompositions of $S=\left\{1,\dots ,9\right\}$ into unions of Interval Components respectively with respect to the Partitions $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$. Assume that we have two different types of checkers $A$ and $B$. Then

$$U=(A,A,A,A,B,B,B,B,B)$$

is a qualified arrangement of checkers with respect to the Partition $\alpha =\left\{4,9\right\}$. When $U$ is considered as a qualified arrangement with respect to the Partition $\beta =\left\{4,6,9\right\}$, it has the following child-nodes

$$(A,A,A,A,A,A,B,B,B),(A,A,A,A,B,B,B,B,B),(B,B,B,B,B,B,B,B,B)$$

by the “Gradient Expansion operator”.

This “Gradient Expansion operator” is motivated by the usual Gradient Descent method adopted in Machine Learning. See, for example, [37]. Search through the simple Gradient Descent method usually converges slowly. (To expedite the training process of AI, Stochastic Gradient Descent was introduced. See [37].)

Now we discuss the “Mutation Expansion operator” under the assumptions (9) to (12). We say that the arrangement $V\in {\Omega }_{\beta }$ of checkers is a “child-node” of $U$ in ${\Omega }_{\beta }$, by the “Mutation Expansion operator”, if the following ME Condition is satisfied.

ME(Mutation Expansion) Condition. There is a sequence $G{E}_{UV}=\left\{\varphi (1),\dots ,\varphi (q)\right\}$ of positive integers taken from $\left\{1,\dots ,m_{\beta }\right\}$, satisfying

$$\varphi (s)+2\le \varphi (s+1),\mathrm{for}\mathrm{all}s=1,\dots ,(q-1),$$

(17)

such that

$$U_{\varphi (s)}\ne U_{\varphi (s)+1},,\mathrm{for}\mathrm{all}s=1,\dots ,q.$$

(18)

For each $s=1,\dots ,q$, we have

$$(V_{\varphi (s)},V_{\varphi (s)+1})=(U_{\varphi (s)},U_{\varphi (s)+1})\mathrm{or}(V_{\varphi (s)},V_{\varphi (s)+1})=(U_{\varphi (s)+1},U_{\varphi (s)})$$

(19)

for the arrangements $U$ and $V$ of checkers on the pair $(I_{\varphi (s)},I_{\varphi (s)+1})$ of adjacent Interval components $I_{\varphi (s)}$ and $I_{\varphi (s)+1}$. Besides, outside these pairs $(I_{\varphi (s)},I_{\varphi (s)+1})$ of adjacent Interval components, $s=1,\dots ,q$, we have

$$V_K=U_k, \mathrm{if} k \mathrm{is}not \mathrm{in} \mathrm{the} \mathrm{union}\left\{\varphi (1),\dots ,\varphi (q)\right\}\cup \left\{\varphi (1)+1,\dots ,\varphi (q)+1\right\}$$

(20)

Example 6. For the Partitions $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$ of $S=\left\{1,\dots ,9\right\}$, we have the following decompositions

$$S=\left\{1,2,3,4\right\}\cup \left\{5,6,7,8,9\right\}\mathrm{and}S=\left\{1,2,3,4\right\}\cup \left\{5,6\right\}\cup \left\{7,8,9\right\}$$

of $S=\left\{1,\dots ,9\right\}$ into unions of Interval Components respectively with respect to the Partitions $\alpha =\left\{4,9\right\}$ and $\beta =\left\{4,6,9\right\}$. Assume that we have two different types of checkers $A$ and $B$. Then

$$U=(A,A,A,A,B,B,B,B,B)$$

is a qualified arrangement of checkers with respect to the Partition $\alpha =\left\{4,9\right\}$. When $U$ is considered as a qualified arrangement with respect to the Partition $\beta =\left\{4,6,9\right\}$, it has the following child-nodes

$$(A,A,A,A,B,B,B,B,B)\mathrm{and}(B,B,B,B,A,A,B,B,B)$$

by the “Mutation Expansion operator”.

This “Mutation Expansion operator” is motivated by the Differential Evolution method. See, for example, [38,39,40,41]. Search through this “Mutation Expansion operator” usually lead to rapid convergence. Thus we may use “tree search through the Mutation Expansion operator” to help us to find a satisfactory strictly increasing sequence

$${\alpha }_0\prec {\alpha }_1\prec \dots \prec {\alpha }_q$$

of Partitions of $S=\left\{1,\dots ,n\right\}$. Then we may use this proper strictly increasing sequence of Partitions of $S=\left\{1,\dots ,n\right\}$ to execute a delicate “tree search through the Gradient Expansion operator”, which usually converges slowly.

2.4. The Numerical Simulation Methods

To speed up the evaluation process for an arrangement of checkers in a fixed-bed heat regenerator, our numerical simulations are based on certain 1-dimensional partial differential equations as in [42,43,44]. These 1-dimensional partial differential equations were developed in [45,46] based on the theories of W. Nusselt. Empirical evidence shows that, in the long term, simulation results of the Inlet temperature and the Outlet temperature, by the 1-dimensional partial differential equations, are consistent with that by CFD (Computational Fluid Dynamics) on Ansys Fluent.

Let us consider the two-phase cycle of a fixed-bed heat regenerator, which is placed on the right end of a regenerative combustion system, as shown in Figure 7. In Phase 1, the high temperature exhaust gas, from the central furnace, flows into the right heat regenerator and heats up the checkers arranged there. Then Phase 2 starts at the Phase Switch point of time. In Phase 2, the above process is reversed. The cool fresh air flows into the right heat regenerator and is preheated by the checkers arranged there, which already absorbed heat from the high temperature exhaust gas expelled from the central furnace during Phase 1. Then the preheated fresh air flows into the central furnace for combustion with Natural Gas.

Let $t_{sw}$ denote the switch time of the operation cycle of the fixed-bed heat regenerator shown above. Thus

$$2t_{sw}=t_{sw}+t_{sw}$$

is the period of the operation cycle. When $0\le t\le t_{sw}$, Phase 1 operates. When $t_{sw}\le t\le 2t_{sw}$, Phase 2 operates. To evaluate the efficiency of an arrangement of checkers on a fixed-bed heat regenerator, this operation cycle must be performed periodically until we reach approximately at a steady state. We assume that the fixed-bed heat regenerator is located on the interval $[0,l]$ of the $x$-axis. See Figure 7.

We use the subscript ar to indicate functions related to the fresh air flow. We use the subscript ex to indicate functions related to the exhaust gas flow. We use the subscript s to indicate functions related to the checkers (solid porous medium). Let $n$ be a nonnegative integer. When

$$n\cdot 2t_{sw}+0\le t\le n\cdot 2t_{sw}+t_{sw},$$

we have the following 1-dimensional partial differential equations:

$$\epsilon \cdot {\displaystyle \frac{\partial ({\rho }_{ex}\cdot C_{ex}\cdot T_{ex})}{\partial t}}+{\displaystyle \frac{\partial ({\rho }_{ex}\cdot C_{ex}\cdot u\cdot T_{ex})}{\partial x}}={\displaystyle \frac{\partial }{\partial x}}\left(\epsilon \cdot k_{ex}\cdot {\displaystyle \frac{\partial T_{ex}}{\partial x}}\right)-h\cdot a_s\cdot (T_{ex}-T_s)$$

(21)

and

$$(1-\epsilon )\cdot {\displaystyle \frac{\partial ({\rho }_s\cdot C_s\cdot T_s)}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left[(1-\epsilon )\cdot k_s\cdot {\displaystyle \frac{\partial T_s}{\partial x}}\right]+h\cdot a_s\cdot (T_{ex}-T_s)-q_d$$

(22)

satisfying the following boundary conditions

$${\left[{\displaystyle \frac{\partial T_s}{\partial x}}\right]}_{x=0}=0={\left[{\displaystyle \frac{\partial T_s}{\partial x}}\right]}_{x=l},{\left[{\displaystyle \frac{\partial T_{ex}}{\partial x}}\right]}_{x=l}=0\mathrm{and}{T}_{ex}(0,t)given.$$

(23)

When $n\cdot 2t_{sw}+t_{sw}\le t\le n\cdot 2t_{sw}+2t_{sw}$, we have the following 1-dimensional partial differential equations:

$$\epsilon \cdot {\displaystyle \frac{\partial ({\rho }_{ar}\cdot C_{ar}\cdot T_{ar})}{\partial t}}+{\displaystyle \frac{\partial ({\rho }_{ar}\cdot C_{ar}\cdot [-u]\cdot T_{ar})}{\partial x}}={\displaystyle \frac{\partial }{\partial x}}\left(\epsilon \cdot k_{ar}\cdot {\displaystyle \frac{\partial T_{ar}}{\partial x}}\right)-h\cdot a_s\cdot (T_{ar}-T_s)$$

(24)

and

$$(1-\epsilon )\cdot {\displaystyle \frac{\partial ({\rho }_s\cdot C_s\cdot T_s)}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left[(1-\epsilon )\cdot k_s\cdot {\displaystyle \frac{\partial T_s}{\partial x}}\right]+h\cdot a_s\cdot (T_{ar}-T_s)-q_d$$

(25)

satisfying the following boundary conditions

$${\left[{\displaystyle \frac{\partial T_s}{\partial x}}\right]}_{x=0}=0={\left[{\displaystyle \frac{\partial T_s}{\partial x}}\right]}_{x=l},{\left[{\displaystyle \frac{\partial T_{ar}}{\partial x}}\right]}_{x=0}=0\mathrm{and}{T}_{ar}(l,t)given.$$

(26)

Here $u$ (m/s) is the superficial flow velocity of the gas/air. $\rho$ (kg/m³) and ${\rho }_s$ (kg/m³) are respectively the densities of the gas/air and the solid porous medium (checkers). $C$ (J/kgK) and $C_s$ (J/kgK) are respectively the specific heat capacities of the gas/air and the solid porous medium (checkers). $k$ (W/mK) and $k_s$ (W/mK) are respectively the thermal conductivities of the gas/air and the solid porous medium (checkers). $T$ (K) and $T_s$ (K) are respectively the temperatures of the gas/air and the solid porous medium (checkers). $\epsilon$ is the void fraction (porosity) of the solid porous medium (checkers). $a_s$ (m²) is the specific surface area of the porous ceramic medium (checkers) exposed to gas/air per unit volume. $h$ (w/m²K) is the heat transfer coefficient of the solid porous medium (checkers). $q_d$ is the heat dissipation through the external surface of a fixed-bed heat regenerator. We assume that $q_d=0$.

We have the following relation for the Nusselt number of the fluid (gas/air) flow:

$$Nu={\displaystyle \frac{h\times d}{k}}$$

(27)

in which $d$ is the characteristic length. Let $\mathrm{Re}$ and $\mathrm{Pr}$ denote respectively the Reynolds number and the Prandtl number of the fluid (gas/air) flow. We will use the following classic relation

$$\mathrm{Nu}=1.86\times {\left({\displaystyle \frac{d}{L}}\times \mathrm{Re}\times \mathrm{Pr}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\times {\left({\displaystyle \frac{{\mu }_b}{{\mu }_w}}\right)}^{0.14}$$

(28)

of Sieder and Tate to calculate the Nusselt number. Here ${\mu }_b$ and ${\mu }_w$ are respectively the viscosity of fluid at the mean bulk temperature and the viscosity of fluid at the wall. See page 583 of [47]. Note that

$$M^{0.14}=e^{(0.14)\times \ln M}\approx e^0=1$$

when $\ln M$ is not large. Thus, when performing numerical simulations, we will use the simplified formula

$$\mathrm{Nu}=1.86\times {\left({\displaystyle \frac{d}{L}}\times \mathrm{Re}\times \mathrm{Pr}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

(29)

Let $M_{ex}$ (kg/s) and $M_{ar}$ (kg/s) be respectively the mass flow rates of the exhaust gas flow and of the fresh air flow. Let $C_{ar,i}$ (J/kgK) and $C_{ar,o}$ (J/kgK) be the specific heat capacities of the fresh air respectively at the Inlet of fresh air (into the right heat regenerator) and at the Outlet of fresh air (into the central furnace). Let $T_{ar,i}$ (K) and $T_{ar,o}$ (K) be the temperatures of the fresh air respectively at the Inlet of fresh air (into the right heat regenerator) and at the Outlet of fresh air (into the central furnace). Let $C_{ex,i}$ (J/kgK) be the specific heat capacity of the exhaust gas flow at the Inlet of exhaust gas (into the right heat regenerator). Let $T_{ex,i}$ (K) be the temperature of the exhaust gas flow at the Inlet of exhaust gas (into the right heat regenerator).

We define the Waste Heat Recovery Ratio (%) of an arrangement of checkers in a fixed-bed heat regenerator as follows:

$$E\text{'}={\displaystyle \frac{M_{ar}\cdot (C_{ar,o}\times T_{ar,o}-C_{ar,i}\times T_{ar,i})}{M_{ex}\times C_{ex,i}\times T_{ex,i}}}\times 100.$$

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We use the long term Waste Heat Recovery Ratio to evaluate the efficiency of an arrangement of checkers in a fixed-bed heat regenerator.

To expedite the process of simulations, we use the following polynomials in our simulations.

$$C_{ex}=-7\times {10}^{-8}\times {T_{ex}}^3+0.0002\times {T_{ex}}^2+0.1756\times T_{ex}+1043.4.$$

$$C_{ar}=-5\times {10}^{-20}\times {T_{ar}}^3+0.0002\times {T_{ar}}^2+0.0683\times T_{ar}+976.04.$$

$$k_{ex}=-{10}^{-8}\times {T_{ex}}^2+8\times {10}^{-5}\times T_{ex}+0.0017.$$

$$k_{ar}=-7\times {10}^{-9}\times {T_{ar}}^2+7\times {10}^{-5}\times T_{ar}+0.0078.$$

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These polynomials are derived based on the data provided in the Appendices of [48] at atmospheric pressure. For the densities of the air/gas, we use the following polynomial

$$\rho =7.7352\times {10}^{-13}\times T^4-3.8908\times {10}^{-9}\times T^3+7.2935\times {10}^{-6}\times T^2-0.0063\times T+2.4655$$

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in our simulations. This polynomial is derived based on the data for air provided in the Appendices of [48] at atmospheric pressure.

3. Results

Before starting our optimization process by tree search, we must test the validity of our 1-dimensional partial differential equations, though similar 1-dimensional modeling had been developed before in [42,43,44,45,46]. We test our 1-dimensional modeling for two types of checkers: Cordierite and Mullite with material parameters shown in Table 1. We consider the following arrangement of checkers

$$(\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Mullite},\mathrm{Mullite})$$

at 6 positions in a fixed-bed heat regenerator. We assume that the temperature of exhaust gas, expelled from the central furnace, is 900℃ (1173.15K). We assume that the temperature of fresh air, at the Inlet, is 40℃ (313.15K). We assume that the time of Phase Switch is 30s. We assume that the superficial flow velocity of the exhaust gas is $2.52\mathrm{m}/\mathrm{s}$. We assume that the superficial flow velocity of the fresh air is $0.56\mathrm{m}/\mathrm{s}$.

The comparison of our 1-D simulations with the 3-D CFD simulations on Ansys Fluent for the arrangement of checkers

$$(\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Cordierite},\mathrm{Mullite},\mathrm{Mullite})$$

at 6 positions, in a fixed-bed heat regenerator, is shown in Figure 8.

In Figure 8, the Inlet temperature of exhaust gas (expelled from the central furnace) and the Outlet temperature of fresh air (into the central furnace), based on our 1-D simulations using C++, is shown in black color. In Figure 8, the Inlet temperature of exhaust gas (expelled from the central furnace) and the Outlet temperature of fresh air (into the central furnace), based on 3-D CFD simulations on Ansys Fluent, is shown in red color.

In Figure 8, the Outlet temperature of exhaust gas (expelled from the right heat regenerator) and the Inlet temperature of fresh air (into the right heat regenerator), based on our 1-D simulations using C++, is shown in blue color. In Figure 8, the Outlet temperature of exhaust gas (expelled from the right heat regenerator) and the Inlet temperature of fresh air (into the right heat regenerator), based on 3-D CFD simulations on Ansys Fluent, is shown in green color.

It can be observed that, in the long term, our numerical simulation results, based on the 1-dimensional partial differential equations, are consistent with the simulation results based on CFD (Computational Fluid Dynamics) on Ansys Fluent. See Figure 8.

Now we apply our simple tree search method with Mutation Expansion operator to solve the optimization problem for 3 types of checkers to be arranged at 6 positions in a fixed bed regenerator. The material parameters of these 3 types, A, B and C, of checkers are shown in Table 2. The operating parameters for a regenerative combustion system with two fixed-bed heat regenerators are shown in Table 3.

Thus we have $S=\left\{1,2,3,4,5,6\right\}$. We choose the strictly increasing sequence

$${\alpha }_0=\left\{1,2,3\right\}\prec {\alpha }_1=\left\{1,2,3,4,5,6\right\}$$

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of Partitions of $S=\left\{1,2,3,4,5,6\right\}$. Thus we have the following decomposition

$$S=\left\{1,2\right\}\cup \left\{3,4\right\}\cup \left\{5,6\right\}$$

of $S$ into a union of Interval Components

$$I_1=\left\{1,2\right\},I_2=\left\{3,4\right\},I_3=\left\{5,6\right\}$$

with respect to the Partition ${\alpha }_0=\left\{1,2,3\right\}$.

We select the top 15 arrangements in ${\Omega }_{{\alpha }_0}$ as the “mother (root) nodes” in ${\Omega }_{{\alpha }_1}$ to create “child nodes” (qualified arrangements with respect to the Partition ${\alpha }_1$) in ${\Omega }_{{\alpha }_1}$.

For a complete evaluation process for the total

$$729=3^6$$

possible arrangements of checkers, based on 1-D simulations, it takes 63 hours to finish this evaluation process. The top 3 arrangements are shown in Table 4.

For the optimization process by our simple tree search method, it takes only 6 hours to evaluate

$$42=3^3+15$$

possible arrangements of checkers, based on 1-D simulations. The top 3 arrangements, found by this simple tree search method, are shown in Table 5.

Thus empirical evidence shows that our tree search method is helpful and efficient. An algorithm based on our simple tree search method, with the Mutation Expansion operator, is shown in Figure 9.

4. Discussion

In this article, we propose a tree search method for finding the optimal arrangements of heat-storage ceramic bricks (checkers) in a heat regenerator. Empirical evidence shows that this simple tree search method is efficient and helpful.

This tree search method is motivated by the Monte-Carlo Tree Search (MCTS) algorithm used in Artificial Intelligence, combined with “Deep Learning”, to create computer players for the incredibly complicated board game Go. The combination of “Deep Learning” with MCTS has proved to be very successful in creating powerful computer players for the incredibly complicated board game Go.

The principles and ideas for MCTS are partly adopted and developed in this research. However, there is key difference between these problems. The mathematics behind the incredibly complicated board game Go is still not properly understood by humans. But for the optimization problem of arrangements of heat-storage ceramic bricks (checkers) in a heat regenerator, we do understand that the performance of all the arrangements of checkers is governed by Physics Laws. This might explain why our simple stochastic tree search method seems to exhibit high efficiency.

5. Conclusions

In this article, we propose a tree search method for finding the optimal arrangements of heat-storage ceramic bricks (checkers) in a heat regenerator. Empirical evidence shows that this simple tree search method is efficient and helpful.