Fired Heaters Optimization by Estimating Real-Time Combustion Products Using Numerical Methods

Abstract

Fired heaters upstream of distillation towers, despite their optimal thermal efficiency, often suffer from performance decline due to fluctuations in fuel composition and unpredictable operational parameters. These heaters have high energy consumption, as fuel properties vary depending on the source of the crude oil. This study aims to optimize the combustion process of a three-gas mixture, mainly refinery gas, by incorporating more stable fuels such as natural gas and liquefied petroleum gas (LPG) to improve energy efficiency and reduce LPG consumption. Using real-time gas chromatography-mass spectrometry (GC-MS) data, we accurately calculate the mass fractions of individual compounds, allowing for more precise burner flow rate determinations. Thermochemical data are used to calculate equilibrium constants as a function of temperature, with the least squares method, while the Newton–Raphson method solves the resulting nonlinear equations. Four key variables ($X_4,X_6,X_8,$ and $X_{11}$), representing $H_2,CO,O_2,$ and $N_2$, respectively, are defined, and a Jacobian matrix is constructed to ensure convergence within a tolerance of 1 ×${10}^{-6}$ over a maximum of 200 iterations, implemented via Python 3.10.4 and the scipy.optimize library. The optimization resulted in a reduction in LPG consumption by over 50%. By tailoring the fuel supply to the specific thermal needs of each processing unit, we achieved substantial energy savings. For instance, furnaces in the hydrocracking unit, which handle cleaner subproducts and benefit from hydrogen’s adiabatic reactions, require much less energy than those in the primary distillation unit, where high-impurity crude oil is processed.

1. Introduction

In a wide range of refinery configurations, heaters are strategically positioned upstream of the crude distillation tower to ensure that the incoming crude oil is heated to a specific, required temperature. This temperature regulation is imperative for optimal operation. In instances where the crude oil does not attain this essential temperature, it becomes necessary to either increase the combustion of fuel gas or adjust the calorific value of the gas mixture. This adjustment is essential to achieve the desired process temperature, underscoring the importance of thermal management and energy efficiency in refinery operations. Fired heaters represent the primary energy consumers in both the refining and petrochemical sectors, with approximately 55% of total energy consumption being attributed to their function as heat exchangers [1]. Although most were designed for thermal efficiencies of up to 80%, current operating efficiencies often fall short of expectations [2]. This is thought to result from the inconsistent operation of the furnaces in line with their design conditions.

Hydrocarbons have played a crucial role in the global economy for centuries, acting as key drivers of industrial development [3,4,5,6]. Hydrocarbons in the gaseous state serve as a common fuel burned in this equipment [7,8]. However, exclusive reliance on one type of gas is not feasible due to source limitations. This leads to increased costs and insufficient combustion efficiency. Currently, a mixture of natural gas (NG) and refinery gas (RG) is used as the primary fuel source in industrial applications [9]. RG is usually recovered from various processes, including cracking, desulphurization, and catalytic reforming units [10]. Liquefied petroleum gas (LPG), a by-product of crude oil refining, is occasionally used as a third gas due to its high propane and butane content, which increases the calorific value of the mixture. This is particularly useful when the mixture of NG and RG has a significantly low calorific value or when considering the fluctuating costs of these fuels in the market [11]. These procedures generate a diverse matrix of fuel blends, resulting in a wide range of compositions suitable for combustion processes [12]. Highlighting that NG and LPG are relatively more environmentally friendly, their combustion produces lower pollutant emissions into the atmosphere [13].

The optimization of thermal efficiency in fired heaters can be achieved through different strategies [14,15,16]. Thorat and Garg, both emphasize the importance of optimal design and application of heat tracing and management systems, which can significantly reduce energy consumption [2,17]. Masoumi further underscores the potential for improvement in refinery furnaces’ efficiency through mathematical modeling, particularly by considering ambient and operational conditions [18]. These studies collectively highlight the need for a holistic approach to addressing the challenges of energy usage in petrochemical refineries. It has also been emphasized that net-zero global carbon dioxide $C{O}_2$ emissions need to be reached to achieve this goal by 2050 [19]. Thus, reducing greenhouse gas emissions, especially $C{O}_2$, is one of the main global challenges to achieving a more sustainable future. Combustion furnaces within oil refineries generally account for over 65% of $C{O}_2$ emissions [20]. Single-burner research furnaces have undergone testing, revealing staged combustion air injection and flue gas recirculation as the most promising combustion modifications to decrease $NO$ emissions from the gas-fired process heater. The modifications have a potential for up to a 67% reduction [21].

Research also highlights the significance of fuel composition, particularly hydrogen enrichment, in enhancing combustion efficiency. Additionally, experts are developing simulation methods for optimizing gaseous fuel mixtures [22,23,24,25,26]. Saifullin’s specialized techniques have significantly improved combustion efficiency in thermal power plants that use variable fuel compositions [27]. Some researchers have studied real-time optimization [28]. However, there is still a gap in the real-time optimization of fuel mixtures, particularly for RG, when responding to their changing compositions. Gas chromatography-mass spectrometry (GC-MS) data play a vital role in comprehending RG behavior. In an industrial context, it is important to maintain a constant calorific value, which is critical for cost management as outlined by Cote [29]. The imperative to maintain a consistent calorific value drives many end-users to monitor and regulate their flue gas quality [30]. Chomiak observes that the utilization of gas with higher calorific value can minimize expenses linked to fuel consumption, thereby affecting the profitability per cubic meter of refined crude oil [31,32].

This study aims to bridge the knowledge gap by developing a method to dynamically optimize the combustion of a three-gas mixture. We utilize GC-MS data to analyze RG in real time, determining the mass fraction of individual compounds. These data enable the selection of optimal flow rate combinations for burners, ensuring efficient combustion. Equilibrium constants are also calculated as a function of temperature by applying the Newton–Raphson method in Python to solve the resulting nonlinear equations [33]. The research extends beyond theoretical analysis, incorporating simulations validated with in situ chromatography data. The novel aspect of this research lies in its real-time approach to optimizing fuel mixtures, examining the inherent variability of RG. This approach is expected to contribute to a significant reduction in LPG consumption, thereby offering substantial economic benefits to oil companies. Additionally, the study contributes to the broader goal of enhancing the overall efficiency and performance of furnaces in the petrochemical industry.

The paper is structured as follows: Section 2 details the setup of chemical reactions involved in combustion and the dynamic model of the gas mixture. Section 2.2 discusses the reduction in equations from twelve variables to four, which is essential for accurately computing combustion product mass fractions. Section 3 presents the method’s numerical performance, supported by simulations and real-time chromatography data. Finally, Section 4 offers conclusions, highlighting the study’s contributions to efficient energy management in fired heaters.

2. Methodology and Simulation Model

2.1. Composition and Properties of Gas Mixture

This research addresses the combustion of a three-gas mixture, with a particular focus on how the varying composition of RG (shown as a dashed yellow line in Figure 1) affects the process, alongside NG and a standard gas mixture (depicted by the green and red lines, respectively).

One of the key properties of gases is their ability to mix uniformly with each other, resulting in a solution where each gas component can be analyzed independently while maintaining the same temperature and volume within the mixture [34]. Understanding gas mixtures becomes more straightforward with knowledge of their composition, which we investigated through a review of statistical reports from approximately 250 chromatography analyses of RG streams collected every 24 h.

Table 1. Selecting gases for the numerical model (MF = Mole Fraction).

Fuels	${\mathit{RG}}_{29}$	${\mathit{RG}}_{87}$	${\mathit{RG}}_{\mathbf{176}}$	$\mathit{NG}$	$\mathit{LPG}$
Comp.	MF	MF	MF	MF	MF
$H_2$	0.16983526	0.28839122	0.15382659	0.0	0.0
$C{O}_2$	0.00355668	0.0018357	0.00180876	0.0	0.0
$C{H}_4$	0.50036226	0.49431918	0.63173331	0.9831	0.0
$C_2{H}_6$	0.1371265	0.04462025	0.11880152	0.00258124	0.9838
$C_2{H}_4$	0.05565108	0.03992771	0.04821289	0.0	0.0
$C_3{H}_8$	0.0549727	0.0147482	0.00479593	0.002	0.0054
propylene	0.02094179	0.00305931	0.00649243	0.0	0.0098
$H_{2}S$	0.0	0.0	0.00327537	0.0	0.001
$C_2{H}_2$	7.54 $\times {10}^{-6}$	0.0	6.62 $\times {10}^{-6}$	0.0	0.0
isobutane	0.00558021	0.004937739	0.00230704	0.0	0.0
propadiene	0.0	0.0	0.0	0.0	0.0
n-butane	0.00679208	0.00343208	0.00306676	0.0	0.0
$O_2$	0.0004077	0.0153073	0.00022166	0.001	0.0
trans-2-butene	0.0032849	0.0023618	0.0013005	0.0	0.0
$N_2$	0.0212296	0.080129	0.0109625	0.011	0.0
1-butene	0.0032146	0.00032	0.0010858	0.0	0.0
isobutene	0.0042796	0.0008848	0.0009842	0.0	0.0
2-butene	0.0022651	0.0015048	1.32 $\times {10}^{-5}$	0.0	0.0
isopentene	0.0020144	0.000221	0.0022329	0.0002048	0.0
n-pentane	0.00162679	5.559 $\times {10}^{-5}$	0.0021111	6.879 $\times {10}^{-5}$	0.0
1,3-butadiene	0.0001166	0.0	4.06 $\times {10}^{-5}$	0.0	0.0
$CO$	0.00653259	0.0039111	0.0067198	0.0	0.0
Hexans+	0.0002015	3.27 $\times {10}^{-5}$	0.0	3.03 $\times {10}^{-5}$	0.0
$\sum X_i\equiv 1$	0.9999994	0.99999949	0.99999948	0.99998514	1.0

outlines the composition of three specific refinery gases at 29, 87 and 176 days, thus: ${\mathit{RG}}_{\mathbf{29}}$, ${\mathit{RG}}_{\mathbf{87}}$, and ${\mathit{RG}}_{\mathbf{176}}$ (See Figure 1), which represent a range of refinery gas compositions with varying calorific values and sulfur content. These gases were selected due to their distinct low heating value (LHV), thus, the minimum LHV ${\mathit{RG}}_{\mathbf{29}}$, the maximum LHV ${\mathit{RG}}_{\mathbf{87}}$, and the last one with a typical sulfur content in ${\mathit{RG}}_{\mathbf{176}}$. In addition, the composition of natural gas (NG) is predominantly methane (98%), while LPG is primarily composed of butane (98%), making both ideal for numerical modeling purposes due to their stable and predictable compositions. Our code is developed to flexibly process chromatographic data for these gases across various process parameters.

For the calculation of the thermodynamic properties of the air–fuel mixture, the engineering equations solver (EES) software (https://fchartsoftware.com/ees/, accessed on 19 September 2024) was used, where NASA’s ideal gas data were taken. These data consist of specific heat, specific enthalpy and density, among others, at standard pressure as a function of temperature. Another advantage is the ease of numerically solving thousands of coupled non-linear algebraic and differential equations.

But first, as we observe in Table 1, none of the refinery gas samples yield the sum of the volumetric composition equal to 1 as the other two gases, so it must be readjusted to give exactly 1. Implementing them guarantees the precision of the solution for a great variety of systems, in a wide range of conditions with a high degree of efficiency and reliability.

2.2. Chemical Reactions in the Combustion Process

Considering a fuel with a composition of $C,H,O,N$ and S, mixed with air at an equivalence ratio denoted as $\varphi$, and examining its reaction in the framework of equilibrium thermodynamics applied to the system, we observe the formation of products at a temperature set as T and a pressure set as P [35]. Under conditions where high temperatures induce dissociation, up to 12 combustion products can be generated [36].

The entire representation of the chemical reaction can be formulated as follows,

$$\begin{array}{l}{X}_{14}[C_{n1}+H_{m1}+O_{l1}+N_{k1}+S_{j1}]+X_{15}[C_{n2}+H_{m2}+O_{l2}+N_{k2}+S_{j2}]\\ +X_{16}[C_{n3}+H_{m3}+O_{l3}+N_{k3}+S_{j3}]+\left[{\displaystyle \frac{(n+j+\frac{m}{4}-\frac{l}{2})}{\varphi }}\right](O_2+3.76N_2)\Rightarrow \\ X_{1}H+X_{2}O+X_{3}N+X_4{H}_2+X_{5}OH+X_{6}CO+X_{7}NO\\ +X_8{O}_2+X_9{H}_{2}O+X_{10}C{O}_2+X_{11}{N}_2+X_{12}S{O}_2\end{array}$$

(1)

where $j,k,l,m$ and n are the quantities of $S,N,O,H$ and C, respectively, for the gases $1=X_{14},2=X_{15}$, and $3=X_{16}$. In this context, $X_{14}$, $X_{15}$, and $X_{16}$ represent the quantities of refinery gas, natural gas, and liquefied petroleum gas, respectively, constituting the entirety of the fuel blend. The values ranging from $X_1$ to $X_{12}$ are the mole fraction of the combustion products. For the sake of simplicity, we treat air as a mixture consisting solely of $O_2$ and $3.76N_2$, disregarding other components. The quantity of air utilized in the combustion process depends directly on the fuel, and the concentration of air is defined by $\left[(n+j+m/4-l/2)/\varphi \right](O_2+3.76N_2)$, where, $j=X_{14}j1+X_{15}j2+X_{16}j3$; $k=X_{14}k1+X_{15}k2+X_{16}k3$; $l=X_{14}l1+X_{15}l2+X_{16}l3$; $m=X_{14}m1+X_{15}m2+X_{16}m3$ and $n=X_{14}n1+X_{15}n2+X_{16}n3$.

The equivalence ratio between air and fuel, denoted as $\varphi$, is defined as follows:

$$\varphi =\left\{\begin{array}{cc}>1,& \mathrm{for}\mathrm{fuel-}\mathrm{rich}\mathrm{mixtures},\\ =1,& \mathrm{for}\mathrm{a}\mathrm{stoichiometric}\mathrm{mixture},\\ <1,& \mathrm{for}\mathrm{mixtures}\mathrm{that}\mathrm{are}\mathrm{fuel-}\mathrm{lean}.\end{array}\right.$$

The excess air coefficient is another commonly used parameter for characterizing the stoichiometry of the fuel-air mixture, and it is related to the equivalence ratio as $(1-\varphi )/\varphi \times 100\%$. By simplifying the expression, we used $X_{13}$ as the sum of all the reactants, $r_0=(n+j+m/4-l/2)/\varphi$, $r_1=l/2+r_0$, and $r_2=k/2+3.76r_0$. Therefore, we arrive at the following equation:

$$\begin{array}{l}{X}_{13}{(}_{n}C{+}_{m}H{+}_{r_1}O{+}_{r_2}N{+}_{j}S)\Rightarrow \\ X_{1}H+X_{2}O+X_{3}N+X_4{H}_2+X_{5}OH+X_{6}CO+X_{7}NO+X_8{O}_2+X_9{H}_{2}O+X_{10}C{O}_2+X_{11}{N}_2+X_{12}S{O}_2\end{array}$$

(2)

This represents the atom balance for the general equation.

$$C:n{X}_{13}=X_6+X_{10}$$

(3)

$$H:m{X}_{13}=X_1+2X_4+X_5+2X_9$$

(4)

$$O:2r_1{X}_{13}=X_2+X_5+X_6+X_7+2X_8+X_9+2X_{10}+2X_{12}$$

(5)

$$N:2r_2{X}_{13}=X_3+X_7+2X_{11}$$

(6)

$$S:j{X}_{13}=X_{12}$$

(7)

In addition, a condition is introduced in this system which requires that the total sum of the mole fractions of all products be equal to one mole. Therefore, this implies:

$$\sum _{i=1}^{12}{X}_i=1$$

(8)

To solve the system of six equations (from Equations (3)–(8)) with 13 unknowns, an additional set of seven equations is required (from Equations (10)–(16)). These equations are sourced from the equilibrium reactions.

Gas Equilibrium Constants

The equilibrium constant ($K_p$) is determined by the ratio of the molar concentrations (mol/L) of reactants and products in a chemical reaction. Its value is temperature-dependent and must always be specified [37]. We begin by examining the general chemical reaction to determine the equilibrium constants. The equation that defines equilibrium constants as a function of partial pressures for a given combustion reaction is as follows:

$$K_p={\displaystyle \frac{{\Pi }_i{(P_i/P)}^{{\mu }_i}}{{\Pi }_j{(P_j/P)}^{{\mu }_j}}}$$

(9)

where the partial pressures of flue gases are represented by $P_i/P=X_i$ and the respective stoichiometric coefficients by ${\mu }_i$. The seven missing equations to solve the system correspond to the seven chemical reactions associated with natural gas, according to Olikara as follows [34].

$$\begin{array}{ll}{\displaystyle \frac{1}{2}}{H}_2\Rightarrow H& K_{p1}={\displaystyle \frac{X_1}{X_4^{\frac{1}{2}}}}\end{array}$$

(10)

$$\begin{array}{ll}{\displaystyle \frac{1}{2}}{O}_2\Rightarrow O& K_{p2}={\displaystyle \frac{X_2}{X_8^{\frac{1}{2}}}}\end{array}$$

(11)

$$\begin{array}{ll}{\displaystyle \frac{1}{2}}{N}_2\Rightarrow N& K_{p3}={\displaystyle \frac{X_3}{X_{11}^{\frac{1}{2}}}}\end{array}$$

(12)

$$\begin{array}{ll}{\displaystyle \frac{1}{2}}{O}_2+{\displaystyle \frac{1}{2}}{H}_2\Rightarrow OH& K_{p5}={\displaystyle \frac{X_5}{X_4^{1/2}{X}_8^{1/2}}}\end{array}$$

(13)

$$\begin{array}{ll}{\displaystyle \frac{1}{2}}{O}_2+{\displaystyle \frac{1}{2}}{N}_2\Rightarrow NO& K_{p7}={\displaystyle \frac{X_7}{X_8^{1/2}{X}_{11}^{1/2}}}\end{array}$$

(14)

$$\begin{array}{ll}{H}_2+{\displaystyle \frac{1}{2}}{O}_2\Rightarrow H_{2}O& K_{p9}={\displaystyle \frac{X_9}{X_4{X}_8^{1/2}}}\end{array}$$

(15)

$$\begin{array}{ll}CO+{\displaystyle \frac{1}{2}}{O}_2\Rightarrow C{O}_2& K_{p10}={\displaystyle \frac{X_{10}}{X_6{X}_8^{1/2}}}\end{array}$$

(16)

Applying Equation (9), the equilibrium constants data were taken from JANAF Thermochemical Tables [34], where ${log}_{10}{K}_p$ formation for all species are tabulated as functions of the absolute temperature (K). Theoretical investigations [38] propose a functional relationship of this nature to compute the $K_p$ as follows:

$$log{K}_p=Aln{T}_A+{\displaystyle \frac{B}{T_A}}+C+D{T}_A+E{T}_A^2$$

(17)

where a transformed temperature $T_A$ defined as 0.005 $T/9$ (T is in °R) was used for fitting. The constants $A,B,C,D$ and E are listed in

Table 2. Constant coefficients of Reactions.

Equation	A	B	C	D	E
(10)	$0.432168$	−0.112464 ×${10}^2$	0.267269 × ${10}^1$	−0.7457 ×${10}^{-1}$	0.242484 × ${10}^{-2}$
(11)	$0.310805$	−0.129540 ×${10}^2$	0.321779 × ${10}^1$	−0.7383 ×${10}^{-1}$	0.344645 × ${10}^{-2}$
(12)	$0.389716$	−0.245828 ×${10}^2$	0.314505 × ${10}^1$	−0.9637 ×${10}^{-1}$	0.585643 × ${10}^{-2}$
(13)	$-0.141784$	−0.213308 ×${10}^1$	$0.853461$	0.355015 × ${10}^{-1}$	−0.3102 ×${10}^{-2}$
(14)	0.150879 × ${10}^{-1}$	−0.470959 ×${10}^1$	$0.646096$	0.272805 × ${10}^{-2}$	−0.1544 ×${10}^{-2}$
(15)	$-0.752364$	0.124210 × ${10}^2$	−0.260286 ×${10}^1$	$0.259556$	−0.1626 ×${10}^{-1}$
(16)	−0.4153 ×${10}^{-2}$	0.148627 × ${10}^2$	−0.475746 ×${10}^1$	$0.124699$	−0.9002 ×${10}^{-2}$

. This model was used to fit tabulated data through a least squares fitting method. To strike a balance between precision and temperature range, we chose the 1500 to 3000 K (2700 to 5400 °R) range for studying combustion purposes [39]. The adiabatic flame temperature, representing the maximum possible temperature without heat exchange with the surroundings, is crucial for optimizing combustion. Understanding this temperature helps minimize harmful emissions, maximize efficiency, and determine the ideal air–fuel mixture and fuel blends. Figure 2 illustrates a case of combustion wherein the adiabatic flame would attain a peak temperature of approximately 2000 K (3600 °R). Consequently, $K_p$ values are computed simultaneously for each of the reactions.

The log $K_p$ values computed from the equations were compared with the original data, showing deviations of less than 0.0009, these deviations are considered negligible. In the process of data collection, we observe the following expressions, where the values of the variables are influenced by specific constants: $X_1=K_{p1}\sqrt{X_4}$, $X_2=K_{p2}\sqrt{X_8}$, $X_3=K_{p3}\sqrt{X_{11}}$, $X_5=K_{p5}\sqrt{X_4{X}_8}$, $X_7=K_{p7}\sqrt{X_8{X}_{11}}$, $X_9=K_{p9}{X}_4\sqrt{X_8}$, $X_{10}=K_{p10}{X}_6\sqrt{X_8}$. When expressing the molar fractions using the equilibrium constants in Equations (3)–(7), we decrease the number of constraints, resulting in a fresh system of four nonlinear equations with four unknowns. This means that every fraction is depending only on $X_4$, $X_6$, $X_8$, and $X_{11}$ ($H_2$, $CO$, $O_2$, and $N_2$) exclusively as follows:

$$K_{p1}\sqrt{X_4}+2X_4+K_{p5}\sqrt{X_4{X}_8}+2K_{p9}{X}_4\sqrt{X_8}-d_1(X_6+K_{p10}{X}_6\sqrt{X_8})=0$$

(18)

$$K_{p2}\sqrt{X_8}+K_{p5}\sqrt{X_4{X}_8}+X_6+K_{p7}\sqrt{X_8{X}_{11}}+2X_8+K_{p9}{X}_4\sqrt{X_8}+AA=0$$

(19)

where $AA=2K_{p10}{X}_6\sqrt{X_8}-d_2(X_6+K_{p10}{X}_6\sqrt{X_8})$

$$K_{p3}\sqrt{X_{11}}+K_{p7}\sqrt{X_8{X}_{11}}+2X_{11}-d_3(X_6+K_{p10}{X}_6\sqrt{X_8})=0$$

(20)

$$K_{p1}\sqrt{X_4}+K_{p2}\sqrt{X_8}+K_{p3}\sqrt{X_{11}}+X_4+K_{p5}\sqrt{X_4{X}_8}+X_6+K_{p7}\sqrt{X_8{X}_{11}}+BB=0$$

(21)

where $BB=X_8+K_{p9}{X}_4\sqrt{X_8}+K_{p10}{X}_6\sqrt{X_8}+X_{11}+d_4(X_6+K_{p10}{X}_6\sqrt{X_8})-1$, where $d_1=m/n$, $d_2=2r_0/n$, $d_3=2r_1/n$, and $d_4=r_2/n$. This set of four interrelated non-linear equations can be expressed as a function involving four variables: $f_i(X_4,X_6,X_8,X_{11})$, here $i=1,2,3,4$.

To linearize Equations (18)–(21), a Taylor series expansion is applied, yielding the following generalized expression.

$$f_i+{\displaystyle \frac{\partial f_i}{\partial X_4}}\Delta X_4+{\displaystyle \frac{\partial f_i}{\partial X_6}}\Delta X_6+{\displaystyle \frac{\partial f_i}{\partial X_8}}\Delta X_8+{\displaystyle \frac{\partial f_i}{\partial X_{11}}}\Delta X_{11}=0$$

(22)

2.3. Numerical Modeling of the Combustion Process

Using real-time gas chromatography–mass spectrometry (GC-MS) data, we accurately calculate the mole fractions of individual compounds, allowing for more precise burner flow rate determinations. Thermochemical data are used to calculate equilibrium constants as a function of temperature, with the least squares method, while the Newton–Raphson method solves the resulting nonlinear equations. The system of four nonlinear equations is solved using a $4\times 4$ Jacobian matrix of first-order derivatives to ensure convergence within a tolerance of $1\times {10}^{-6}$ over a maximum of 200 iterations (See Equation (23)). The optimization method is implemented in Python using the scipy.optimize library and the newton() function.

$$\left[\begin{array}{cccc}{\displaystyle \frac{\partial f_1}{\partial X_4}}& {\displaystyle \frac{\partial f_1}{\partial X_6}}& {\displaystyle \frac{\partial f_1}{\partial X_8}}& {\displaystyle \frac{\partial f_1}{\partial X_{11}}}\\ {\displaystyle \frac{\partial f_2}{\partial X_4}}& {\displaystyle \frac{\partial f_2}{\partial X_6}}& {\displaystyle \frac{\partial f_2}{\partial X_8}}& {\displaystyle \frac{\partial f_2}{\partial X_{11}}}\\ {\displaystyle \frac{\partial f_3}{\partial X_4}}& {\displaystyle \frac{\partial f_3}{\partial X_6}}& {\displaystyle \frac{\partial f_3}{\partial X_8}}& {\displaystyle \frac{\partial f_3}{\partial X_{11}}}\\ {\displaystyle \frac{\partial f_4}{\partial X_4}}& {\displaystyle \frac{\partial f_4}{\partial X_6}}& {\displaystyle \frac{\partial f_4}{\partial X_8}}& {\displaystyle \frac{\partial f_4}{\partial X_{11}}}\end{array}\right]\left[\begin{array}{c}\Delta X_4\\ \Delta X_6\\ \Delta X_8\\ \Delta X_{11}\end{array}\right]=\left[\begin{array}{c}-f_1\\ -f_2\\ -f_3\\ -f_4\end{array}\right]$$

(23)

The matrix equation in the expanded form is as follows:

$$\left[\begin{array}{c}\\ {\displaystyle \frac{k_{p1}}{2\sqrt{X_4}}}+{\displaystyle \frac{k_{p5}\sqrt{X_8}}{2\sqrt{X_4}}}+2k_{p9}\sqrt{X_8}+2& -d_1{k}_{p10}\sqrt{X_8}-d_1& -{\displaystyle \frac{d_1{k}_{p10}{X}_6}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p5}\sqrt{X_4}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p9}{X}_4}{\sqrt{X_8}}}& 0\\ {\displaystyle \frac{k_{p5}\sqrt{X_8}}{2\sqrt{X_4}}}+k_{p9}\sqrt{X_8}& d_5+d_6{k}_{p10}\sqrt{X_8}& {\displaystyle \frac{d_6{k}_{p10}{X}_6}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p2}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p5}\sqrt{X_4}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p7}\sqrt{X_{11}}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p9}{X}_4}{2\sqrt{X_8}}}+2& {\displaystyle \frac{k_{p7}\sqrt{X_8}}{2\sqrt{X_{11}}}}\\ 0& -d_3{k}_{p10}\sqrt{X_8}-d_3& -{\displaystyle \frac{d_3{k}_{p10}{X}_6}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p7}\sqrt{X_{11}}}{2\sqrt{X_8}}}& {\displaystyle \frac{k_{p3}}{2\sqrt{X_{11}}}}+{\displaystyle \frac{k_{p7}\sqrt{X_8}}{2\sqrt{X_{11}}}}+2\\ {\displaystyle \frac{k_{p1}}{2\sqrt{X_4}}}+{\displaystyle \frac{k_{p5}\sqrt{X_8}}{2\sqrt{X_4}}}+k_{p9}\sqrt{X_8}+1& d_7{k}_{p10}\sqrt{X_8}+d_7& {\displaystyle \frac{d_7{k}_{p10}{X}_6}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p2}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p5}\sqrt{X_4}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p7}\sqrt{X_{11}}}{2\sqrt{X_8}}}+{\displaystyle \frac{k_{p9}{X}_4}{2\sqrt{X_8}}}+1& {\displaystyle \frac{k_{p3}}{2\sqrt{X_{11}}}}+{\displaystyle \frac{k_{p7}\sqrt{X_8}}{2\sqrt{X_{11}}}}\\ \end{array}\right]$$

$$\left[\begin{array}{c}\Delta X_4\\ \Delta X_6\\ \Delta X_8\\ \Delta X_{11}\end{array}\right]=\left[\begin{array}{cc}& -(X_1+2X_4+X_5+2X_9)+d_1(X_6+X_{10})\\ & -(X_2+X_5+X_6+X_7+2X_8+X_9+2X_{10})-d_2(X_6+X_{10})\\ & -(X_3+X_7+2X_{11})+d_3(X_6+X_{10})\\ & 1-\left({\sum }_{i=1}^{11}{X}_i\right)-d_4(X_6+X_{10})\end{array}\right]$$

Initial Value Estimation

To estimate initial values, we focus on the products generated during complete combustion. This involves narrowing the scope from the original 12 products to focus on only four key ones, specifically $H_{2}O$, $C{O}_2$, $N_2$, and $S{O}_2$. Additionally, we consider the products $H_2$, $CO$, $O_2$, and $N_2$, which correspond to the fractions $X_4$, $X_6$, $X_8$, and $X_{11}$, respectively. This approach establishes the following products within the combustion equation:

$$X_{13}{(}_{n}C{+}_{m}H{+}_{r}O{+}_{r^{\prime }}N{+}_{j}S)\Rightarrow X_4{H}_2+X_{6}CO+X_8{O}_2+X_9{H}_{2}O+CC$$

(24)

where $CC=X_{10}C{O}_2+X_{11}{N}_2+X_{12}S{O}_2$.

When performing the elemental balance for each constituent of the fuel in this particular case, we have $C:n{X}_{13}=X_6+X_{10},H:m{X}_{13}=2X_4+2X_9,O:2r_1{X}_{13}=X_6+2X_8+X_9+2X_{10}+2X_{12},N:2r_2{X}_{13}=2X_{11},S:j{X}_{13}=X_{12}$.

By substituting the fractions using the constants from Section 2.2, we obtain the following:

$$X_6={\displaystyle \frac{n{C}_{10}{X}_{13}}{C_{10}+\sqrt{X_8}}}$$

(25)

$$X_4={\displaystyle \frac{\frac{1}{2}m{C}_5{X}_{13}}{C_5+\sqrt{X_8}}}$$

(26)

$$X_{11}=r_2{X}_{13}$$

(27)

$$X_{12}=j{X}_{13}$$

(28)

$$0={\displaystyle \frac{n(C_{10}+2\sqrt{X_8})}{C_{10}+\sqrt{X_8}}}+{\displaystyle \frac{\frac{1}{2}m\sqrt{X_8}}{C_5+\sqrt{X_8}}}+{\displaystyle \frac{2X_8}{X_{13}}}+2j-2r_1$$

(29)

The quantity of $X_{13}$ can be accurately estimated to ensure that the sum of the molar fractions equals one.

$$\begin{array}{l}{X}_4+X_6+{\displaystyle \sum _{i=8}^{12}}{X}_i=1\end{array}$$

(30)

When the air–fuel equivalence ratio is less than or equal to 1, we can obtain a reliable estimate of $X_{13}$ through complete combustion.

$$\begin{array}{l}{X}_{13}={\displaystyle \frac{1}{\frac{m}{4}+r_1+2r_2}}\end{array}$$

(31)

Conversely, when the air–fuel equivalence ratio exceeds 1, a dependable estimation of $X_{13}$ can be derived from incomplete combustion.

$$\begin{array}{l}{X}_{13}={\displaystyle \frac{1}{n+\frac{m}{2}+r_2+j}}\end{array}$$

(32)

By replacing the estimated value of $X_{13}$ into Equation (29), we obtain $X_8$. Once Equation (29) is solved, the remaining unknowns can be readily determined through substitution into Equations (25)–(28). This approximation is used as the initial value for the model.

$$\begin{array}{l}[X_4^{\left(1\right)},X_6^{\left(1\right)},X_8^{\left(1\right)},X_{11}^{\left(1\right)}]\end{array}$$

(33)

The previous vector approximates closely to the solution vector:

$$\begin{array}{l}[X_4^{(\ast )},X_6^{(\ast )},X_8^{(\ast )},X_{11}^{(\ast )}]\end{array}$$

(34)

At each iteration, the updated vector is used to compute the partial derivatives and evaluate the functions until the two criteria of convergences (200 iterations or tolerance of $1\times {10}^{-6}$).

3. Results

3.1. Numerical Model Validation

In the research, the numerical model of the combustion process, and its validation through operating data, considered crucial parameters such as LHV, density, adiabatic flame temperature and $CO$ emissions. The model postulates that emissions of $C{O}_2$ and other greenhouse gases are inextricably linked to $\varphi$. The algorithm begins with $\varphi =1$, which represents a stoichiometric mixture. It then optimizes between stoichiometric and fuel-rich mixtures ($\varphi \le 1$) with the objective of reducing emissions while maximizing energy efficiency and ensuring regulatory compliance. In the case of ideal combustion, $C{O}_2$ emissions are at their highest, while $CO$ emissions are minimized, as illustrated in Figure 3.

Accordingly, the model’s assumptions are constrained within a range of $\varphi =1$ to $\varphi =1.5$, where excess air is introduced to ensure more complete combustion, thereby balancing $C{O}_2$ production with lower levels of other harmful emissions, such as $CO$.

As shown in Figure 4, the density information for each chromatographic sample clearly shows consistent alignment between the model results (blue circles) and the data reported (red circles), thus affirming the reliability of our thermodynamic property calculations. The error is below 0.014% among the 247 samples, providing substantial validation of the computational model proposed in this research.

On the other hand, when it comes to the computed calorific values of each specific chromatographic sample, a disparity becomes evident when compared to the reported data in Figure 5. This examination indicates that this difference can be attributed to the temperature conditions during their laboratory analysis, which plays a pivotal role. It is worth noting that each sample may exhibit a temperature falling within the range of 90 °C to 120 °C. If the exact temperatures employed for this purpose were used, this validation would likely yield results much closer to the reported values.

3.2. Approaches for Achieving a 50%+ Reduction in LPG Use

The algorithm for determining optimal burner flow rates is based on the principle that, regardless of the volume of crude oil processed at any given time (e.g., 100,000 barrels per day), approximately 70% of the refinery’s furnace fuel demand is supplied by recovered refinery gas. This volume, in conjunction with its physicochemical properties, which are obtained through chromatographic analysis, serves as a fixed parameter at the outset of the optimization process. The objective is to achieve an LHV as close as possible to 1000 BTU/ft³, in some cases without the need to incorporate LPG into the fuel mixture. Initially, the physicochemical characterization of gases used for combustion included determining certain properties.

Table 3. Physicochemical properties of gases.

Gas	Mass	Density	Density	LHV
	gmol	kg/m3	lb/ft3	BTU/ft3
$R{G}_{29}$	15.48	0.6558	0.04094	743.52
$R{G}_{87}$	20.06	0.8513	0.05314	1071.28
$R{G}_{176}$	17.38	0.7371	0.04601	952.62
$NG$	16.29	0.6907	0.04311	903.98
$LPG$	44.23	1.9040	0.11886	2225.47

presents the initial calculations of these properties such as molecular mass, density, and LHV.

NG is the usual source for the rest unless there is a significant price difference or LHV. In such cases, we introduce LPG as a third component to ensure optimal combustion [40]. Under these conditions, the numerical model of the mixing of the three gases must take into account the following considerations:

• Calculate the mass and density of the gas mixture.
• The LHV of the mixture must be $\ge 1000{\mathrm{Btu}/\mathrm{ft}}^3$.
• Obtain the adiabatic flame temperature.
• Reducing $C{O}_2$ emissions and minimizing excess air.

Once these conditions are satisfied, the resulting mixture is optimized for combustion.

Table 4. Properties of modeled mixtures.

Mix	RG-NG-LPG	Mass	Density	LHV
	%-%-%	gmol	lb/ft3	Btu/ft3
$M_{R{G}_{29}}$	70-14-16	20.19	0.0537	1003.09
$M_{R{G}_{87}}$	70-30-00	18.92	0.0501	1021.09
$M_{R{G}_{176}}$	70-25-05	18.45	0.0489	1004.10

presents the blend that meets these optimization criteria.

For the mixture $M_{R{G}_{29}}$ including the RG gas containing the lowest calorific value, the mixture simulation resulted in the following flow ratio: 70% RG, 14% NG and 16% LPG (see Table 4). It was necessary to use LPG gas due to the low calorific value contained in the refinery gas sample. In flow terms, this indicates that when the refinery gas has a calorific value greater than 1050 BTU/ft³, it is not necessary to add LPG to the mixture. In such cases, only 70% RG and 30% NG are kept in the mixture.

As we can see in Figure 6, when applying the Newton–Raphson method to the initial values for the $M_{R{G}_{29}}$ mixture, the initial values converge after 70 iterations, although the blue line stops acquiring negative values and the line attempts to stabilize. Consequently, these values are replaced in the other combustion products to compute the total composition of the molecule.

The lines displayed in Figure 7 reveal that gradual changes occur after each iteration. During the initial 50 iterations, the lines extend towards the bottom of the graph in search of negative values, which is illogical as they calculate mass or volume fractions. However, they eventually stabilize, extending horizontally with a straight line after 60 iterations, indicating accurate convergence of the method. The results show that the dynamic behavior was accurate in both cases. After solving the numerical method for the three different mixture combinations and obtaining the real values of $X_4$, $X_6$, $X_8$, and $X_{10}$, these values are automatically substituted into Equations (11) to (16) to determine the exact values for the remaining combustion products. This process completes the calculation of all 12 variables. As shown in

Table 5. Combustion products, Numerical mixtures results $M_{R{G}_{29}}$, $M_{R{G}_{87}}$, $M_{R{G}_{176}}$.

	Comb.	${\mathit{M}}_{{\mathit{RG}}_{29}}$	${\mathit{M}}_{{\mathit{RG}}_{87}}$	${\mathit{M}}_{{\mathit{RG}}_{176}}$
${\mathit{X}}_{\mathit{i}}$	Prod.	MF	MF	MF
$X_1$	H	$4.441\times {10}^{-4}$	$3.068\times {10}^{-4}$	$4.200\times {10}^{-4}$
$X_2$	O	$1.222\times {10}^{-6}$	$8.294\times {10}^{-7}$	$1.090\times {10}^{-6}$
$X_3$	N	$9.137\times {10}^{-10}$	$7.540\times {10}^{-10}$	$9.574\times {10}^{-10}$
$X_4$	$H_2$	$4.616\times {10}^{-2}$	$1.513\times {10}^{-2}$	$4.019\times {10}^{-2}$
$X_5$	$OH$	$1.529\times {10}^{-4}$	$1.988\times {10}^{-4}$	$1.238\times {10}^{-4}$
$X_6$	$CO$	$5.841\times {10}^{-2}$	$6.150\times {10}^{-2}$	$5.742\times {10}^{-2}$
$X_7$	$NO$	$2.739\times {10}^{-5}$	$1.818\times {10}^{-5}$	$2.461\times {10}^{-5}$
$X_8$	$O_2$	$1.439\times {10}^{-6}$	$3.496\times {10}^{-7}$	$1.034\times {10}^{-6}$
$X_9$	$H_{2}O$	$1.022\times {10}^{-1}$	$2.133\times {10}^{-1}$	$7.422\times {10}^{-2}$
$X_{10}$	$C{O}_2$	$4.302\times {10}^{-2}$	$6.817\times {10}^{-2}$	$3.564\times {10}^{-2}$
$X_{11}$	$N_2$	$7.496\times {10}^{-1}$	$6.414\times {10}^{-1}$	$7.913\times {10}^{-1}$
$X_{12}$	$S{O}_2$	$0.000\times {10}^{+0}$	$0.000\times {10}^{+0}$	$6.621\times {10}^{-4}$
	SUM	$1.00$	$1.00$	$1.00$

, the sum of the molar fractions confirms that they total to one, as expected.

4. Conclusions

This study offers a significant advance in the optimization of combustion processes in fired heaters by integrating real-time analysis of refinery gas composition. This research aims to establish a foundation for the integration of tools such as AI with immediate control in equipment that regulates the flow rates of a three-gas mixture, focusing on the variable composition of RG along with NG and LPG. This approach effectively improves combustion efficiency and reduces fuel consumption, in particular, by reducing LPG use by more than 50%, which offers significant economic benefits. The proposed improvement has the potential to reduce LPG consumption by over 50% due to the current uniform distribution of a single fuel mix (a blend of three gases) to all furnaces across the refinery, without consideration of their specific heating requirements. By customizing the fuel mix for each furnace based on its heating requirements, substantial savings are possible. For instance, furnaces in the hydrocracking unit require less energy compared to those in the primary distillation unit, which handles unrefined crude and demands higher temperatures. Tailoring the fuel supply to match these requirements improves efficiency and can lead to a reduction in LPG consumption of over 50%, particularly when NG prices are relatively higher than LPG costs.

The application of GC-MS data in real-time allowed for the accurate calculation of individual compounds’ mass fractions. This innovation in monitoring and adjusting the composition of the fuel mixture ensures a more consistent and efficient combustion process. The successful implementation of the Newton–Raphson method in Python to solve the non-linear equations derived from the study demonstrates the practical utility of the approach.

Furthermore, this research focus on reducing greenhouse gas emissions conforms to worldwide sustainability targets. The numerical model helps to reduce operational costs and contributes to improved efficiency and performance within petrochemical furnace systems. This research represents a noteworthy advancement in achieving more sustainable and efficient energy management within industrial processes. Additionally, it enables the storage of mass flow data on pollutant gas emissions to report these metrics in annual projections for reducing $C{O}_2$.