Natural Gas and Biogas Mixtures in Smart Cities: A Mathematical Model of Its Proposal for Use with Biogas Produced by Biomass Plants and Mixture Density Control According to the Biogas Composition

Abstract

This article presents a proposal for blending natural gas and biogas with a control system with feedback to ensure a constant mixture density. To achieve this, we propose the following: a mathematical model to determine the gas density based on its composition; a control system whose main components are a gas mixer, valves, and a natural gas storage tank to regulate the biogas density, where its inputs are gases from biomass plants and the natural gas grid; mathematical models to calculate the volume of natural gas required in the storage tank. It is assumed that the composition at the outlet of the biogas plants is measured and that there are no losses of any kind; a case study simulation is then performed. All models consider random variation in gas composition over time. The main results are as follows: (a) reduced natural gas consumption, the promotion of biogas production and use and of mixtures of lower methane compared to natural gas, and the facilitation of the pumping of the gas mixtures; (b) all the biogas produced is used; (c) different piping, sources, storage tanks, consumers, and mixer schemes, considering the concepts of cities, microgrids, smart grids, and smart cities.

1. Introduction

The topic of interest is the use of natural gas (NG), biogas (BG), and a natural gas–biogas mix (NG–BG mix) to be used by customers as residential, commercial, and/or industrial fuel in cities and smart cities. The importance of biogas in cities and smart cities has previously been reported. For example, (a) biogas is part of the proposal for sustainable and smart solutions for waste management in rapidly urbanizing African cities, where organic waste is converted into biogas, organics are partially dewatered or pre-treated for composting or biogas production, and organic fractions are converted to biogas through anaerobic digestion or briquettes for combustion-based energy [1]; (b) biogas is an energy resource in the energy supply chain of smart city ecosystems, e.g., in Oslo, where the city uses biogas for urban transport (i.e., the electrification of public transport), reducing the dependence on fossil fuels, with biogas produced from organic waste as a renewable energy source to further diversify its clean energy portfolio [2]; (c) biogas can be used in combination with other energy sources (solar PV, on-shore and off-shore wind, ocean, hydro, biomass, geothermal, nuclear, natural gas, concentrated solar thermal, biomass/biogas, fuel oil, and coal) for sustainability assessments of energy efficiency measures and energy supply and storage technologies [3]; (d) biogas is used in futuristic waste management strategies to reduce emissions and assess municipal solid waste in smart cities [4,5,6], smart energy management in smart cities [7], thermal energy storages in future smart energy systems [8], and integrated smart city models [9].

In general, according to a report published by the IEA in 2024 [10], worldwide, cities account for 75% of global energy consumption (they have the highest energy demand for transport, industry, and buildings), emit 70% (approximately 29 billion tonnes of $C{O}_2$ in 2023, and in 2015, their share was 62%) of global greenhouse gas emissions (making urban residents more exposed to unhealthy levels of pollution that mostly result from the use of fossil fuels such as natural gas, which are responsible for some 5 million premature deaths each year), and account for 56% of the Earth’s 8 billion people at a growth rate of about 400 million people between 2015 and 2020 (with more than 90% of this growth occurring in cities in emerging markets and developing economies). However, cities present unique opportunities to create economies of scale and lower prices for clean energy technologies, as between 2024 and 2050, the urban population is estimated to increase from 56% to around 70%. In addition, urban areas contribute more than 80% of the global GDP.

Until the onset of the COVID-19 pandemic, $C{O}_2$ emissions from NG use had a growth of 2.6% per year from 2009 to 2018, despite the fact that at that time, there was already growing concern about the negative impacts of NG. Despite this scenario, investments in infrastructure continued to be made with the argument that NG was a more environmentally friendly alternative (compared to coal and oil) concerning the current energy transition to renewable energy sources, meaning that today, there are almost 500 GW of gas-fired power plants in planning or under construction as a result of the massive expansion of NG infrastructure [11].

According to the most recent data published by the IEA, from 1990 to 2022, the percentage share of natural gas in the total energy supply (TES) has increased from 19.1112% (364,190,600 TJ) in 1990 to 23.1277% (622,184,228 TJ) in 2022, whereas the biofuels and waste (BW) in the percentage share of the TES went from 9.5782% in 1990 to 8.7567% in 2022 (see Figure 1); however, the amount of energy increased from 34,882,742 TJ in 1990 to 54,482,802 TJ in 2022, representing an increase in energy of 56.1884% by 2022 compared to 1990 [12]. The substantial difference is that natural gas is a mass that is extracted from the interior of the Earth, which, once used, does not return to the cavity that housed it for millions of years, while biomass is a resource that circulates on the surface of the Earth, maintaining the same amount of carbon mass. Therefore, the increase in the world’s population implies a greater amount of organic resources that need to be produced from daily activities, and biogas can be generated to reduce the amount of natural gas consumed.

Global gas demand growth was expected to slow to around 2.7% in 2024 (115 billion cubic metres, equivalent to around 4 EJ) [13] and 1.5% in 2025 due to a combination of initially tight market conditions and heightened macroeconomic uncertainties. Low-emission gases (including biomethane, low-emission hydrogen, and e-methane6) can play an important role in decarbonising gas supply chains and the broader energy system. The European Union’s Union Database for Biofuels (UDB) was launched in January 2024 and was recently expanded to include gaseous fuels, including biomethane [14].

For several years now, biomass has been considered a key component in the energy management and planning of smart cities [15], where advanced technologies and systems are considered to perform energy management, such as simulation software in the design and operation phases, with demand-side management (DSM) strategies including energy-efficient technologies and smart metering [16]. To ensure that its products contribute to the concept of sustainable smart cities (SSCs) whose target is decarbonisation by optimising energy consumption through the emerging capabilities of technology [17], biogas needs to be mixed with natural gas, forming a gaseous mixture, which leads to a consequent reduction in the consumption of natural gas from natural gas wells.

Similar studies include [18], which reviews biogas plants, the need for compressed biogas (CBG), and desulfurisation to enable its use in India, which has 12 commercial CBG plants with a total CBG output capacity of 18,461.7 tonnes per year through a sustainable alternative towards affordable transportation (SATAT), launched in 2018 with a proposal for four different models for use of compressed biogas in the CGD sector. However, the pressure swing adsorption (PSA) technique is only used in large biogas systems in India, and there are other technologies that can be costly if a large number of medium- and low-capacity CBG plants are to be implemented in increasingly populated urban environments. Concerning biogas combustion, improvements have been made, such as those mentioned in [19], detailing that biogas combustion can be improved with a content of up to 40% using inhomogeneous partially premixed stratified (IPPS) technology.

In reference to biomass, despite the COVID-19 pandemic, in 2019, the second-highest level of annual additions on record was achieved with an 8.5 GW increase in the global electricity generation capacity using biomass. Of this total, China accounted for 60% using mostly waste-to-energy projects. China is the largest market for biomass projects, followed by Japan, which accounts for one-tenth of the Chinese market. China, Japan, Brazil, the United Kingdom, and four other countries accounted for 90% of the newly installed capacity in 2019. During the COVID-19 pandemic, the markets of China, Japan, Brazil, and the UK were the most affected; however, post-pandemic, no disruptions in the supply of biomass fuels (including wood chips and pellets) to existing projects were observed due to ongoing forestry activity and operational ports [20].

The world’s BG electricity generation capacity trends have been increasing despite the COVID pandemic, from 19,320 MW in 2019 to 21,385 MW in 2023, with an average annual growth of 516.25 MW (an annual percentage growth of 2.6721%). This is similar to renewable municipal waste with an increase from 14,310 MW in 2019 to 21,333 MW in 2023 with an average annual growth of 1755.75 MW (an annual percentage growth of 12.2693%) and that of solid biofuels with a growth from 87,321 MW in 2019 to 103,358 MW in 2023, representing an average annual growth of 4009.25 MW (an annual percentage growth of 4.5913%) [21].

In 2023, the use of BG-biomethane (BG-BM) compared to NG in buildings (NG-B) represented 4.45%. There are three projection scenarios of the percentage use of BG-BM in comparison to NG-B: the stated policies scenario (STEPS) will represent 6.84% in 2030, 10.14% in 2040, and 27.37% in 2050; in the announced pledges scenario (APS), it will be 14.10% in 2030, 25.42% in 2040, and 80.14% in 2050; in the net-zero emissions by 2050 (NZE) scenario, it will be 31.05% in 2030 and 80.13% in 2040, with BG-BM use od 346 billon cubic metres (bcm) in comparison with NG-B use of 1 bcm. This excess implies that it could supply approximately 338 bcm of NG destined for industry [22].

However, the state-of-the-art indicates that there are no studies related to biogas–natural gas mixes or to gas quality control, and local BG production does not ensure that the percentage of methane is high and constant, as it depends on environmental conditions, as well as the materials used for BG production. Therefore, in this paper, the following proposal is modeled and simulated: the implementation of an NG–BG mixed installation with BG composition measurement produced locally, and the calculation of the density. With it, a control system is proposed to ensure that the values of the NG–BG mix density supplied to customers are constant; therefore, the control system will compensate for a decrease in the BG’s quality by injecting a percentage of NG stored in storage tanks or as part of a continuous pipeline supply that exists in many cities. This proposal has the potential to optimise the use of BG and NG with a consequent reduction in NG consumption in local BG production environments that are in accordance with the sustainable development goals [23] relating to access to modern energy in cities and human settlements that are resilient and sustainable. With patterns of consumption and production that are sustainable, this proposal is an urgent action to that can be used to combat climate change and its impacts; additionally, it is compatible with the current concepts of microgrids (MG), smart grids (SG), smart cities (SC), and/or distributed generation (DG).

2. Mathematical Model

2.1. Methodological Framework

The overall modelling approach consists of the following steps: (a) first, a mathematical model is presented that determines the density of natural gas based on its composition; (b) second, a control system scheme is proposed to ensure a NG–BG mix flux with constant density from two gases with different densities; (c) third, a mathematical model is presented to estimate the volume of natural gas needed for the NG–BG mix to have a constant density, which is the objective of the control system; (d) fourth, study scenarios are presented consisting of topologies made up of various biogas sources, one natural gas source, mixers, and demands (pipelines to consumers), which have been proposed based on the criterion that they be implemented in cities and/or microgrids, smart grids, and smart cities that have or would have many biomass plants producing biogas; (e) finally, a simulation scenario is presented in which the simulation configuration assumes the random behaviour for the composition and percentages of biogas components, the natural gas source of constant quality, and the final target density of the mixture that must be kept constant, and the volume of the natural gas tank that regulates the resulting density of the mixture is determined.

The control strategy consists of the following: (a) the composition of biogas and its production volume from the biogas plant, and, with this, the density of the biogas is determined; (b) the biogas is mixed with natural gas in a mixer; (c) at the outlet of the mixer, measurements are taken to determine the density of the resulting mixture and they are compared with a predetermined desired value. If there is a difference between the outlet density and the desired density, a controller activates the valve at the outlet of the natural gas tank to increase or decrease the flow of natural gas, thus achieving the desired density value (negative feedback).

The main assumptions considered are as follows: (a) mixtures of natural gas and biogas are ideal, given that this paper presents an initial proposal; (b) there are no losses in pressure, volume, or similar, either in the pipes, the mixer, or the natural gas tank; (c) the simulation is stationary because, as presented below, the time between readings is greater than 1 min, which is sufficient to rule out the need for transient process studies.

2.2. Density of NG According to Its Composition

NG comes from reservoirs where the reservoir energy that drives petroleum fluids towards wells is directly related to the prevailing reservoir pressure. The pressure, temperature, and composition of the hydrocarbon fluid are the parameters that determine whether the fluid will initially exist in a single phase (oil or gas) or in two phases (oil with a gas cap). It is common practice to determine the reservoir pressure and temperature at discovery and to conduct pressure surveys periodically or continuously at various wells during the life cycle of the reservoir. Most reservoir engineering studies, including reservoir simulation, require knowledge of the reservoir’s pressure response as a function of time and location during production and shutdown of the wells. A robust pressure monitoring program may readily point to reservoir drive mechanisms, the effectiveness of fluid injection, and suspected heterogeneities in the rock, among other factors. Modern reservoir monitoring practices employ downhole sensors that continuously provide data [24].

The value of NG’s density based on its composition is described in a mathematical model based on [25] in which NG is a mixture of n components $c_i$ that are extracted from the Earth’s crust at a reservoir temperature of $T_r$ and a reservoir pressure of $p_r$. This NG’s composition can be determined using techniques such as gas chromatography [26,27]. In addition, it is important to mention that [25] is a very useful bibliographic reference for calculating gas density in the natural gas exploration and exploitation industry, drawing on experimentation, real cases, and industrial experience.

According to [25], each component i has a composition mole fraction of $cm{f}_i$, from which its molecular weight $W_{t_i}$, critical pressure $P_{c_i}$, and critical temperature $T_{c_i}$ can be found. The sum of $cm{f}_i$ of all the NG components is equal to 1. Then, the molecular weight $W_{t_i}$ is calculated using Equation (1), the pseudo-critical pressure $p_{pc}$ is calculated using Equation (2), and the pseudo-critical temperature is calculated $T_{pc}$ using Equation (3). For $M_W$, the specific gravity ${\gamma }_g$ is calculated using Equation (4) [25].

$$M_W=\sum _{i=1}^n{W}_{t_i}·cm{f}_i,$$

(1)

$$p_{pc}=\sum _{i=1}^n{P}_{c_i}·cm{f}_i,$$

(2)

$$T_{pc}=\sum _{i=1}^n{T}_{c_i}·cm{f}_i,$$

(3)

$${\gamma }_g={\displaystyle \frac{M_W}{28.97}}.$$

(4)

Then, using $p_{pc}$, $T_{pc}$, $T_r$, and $p_r$, the pseudo-relative pressure $p_{pr}$ and pseudo-relative temperature $T_{pr}$ are calculated using Equations (5) and (6), respectively [25].

$$p_{pr}={\displaystyle \frac{p_r}{p_{pc}}},$$

(5)

$$T_{pr}={\displaystyle \frac{T_r+460}{T_{pc}}}.$$

(6)

The next step is to determine the value of the gas deviation factor z using the Dranchuk and Abou-Kassem equation of state, which is valid for $T_{pr}=1.0$ and $p_{pr}>1.0$. The formulation of the Dranchuk and Abou-Kassem equation of state by determination of z is shown in Equations (7)–(12), where the value of the constants are $A_1=0.3265$, $A_2=-1.0700$, $A_3=-0.5339$, $A_4=0.01569$, $A_5=-0.05165$, $A_6=0.5475$, $A_7=-0.7361$, $A_8=0.1844$, $A_9=0.1056$, $A_{10}=0.6134$, and $A_{11}=0.7210$ [25].

$$z=1+c_{1\left(T_{pr}\right)}{\rho }_r+c_{2\left(T_{pr}\right)}{\rho }_r^2-c_{3\left(T_{pr}\right)}{\rho }_r^3+c_{4\left({\rho }_r,T_{pr}\right)},$$

(7)

$${\rho }_r=0.27{\displaystyle \frac{p_{pr}}{z·T_{pr}}},$$

(8)

$$c_{1\left(T_{pr}\right)}=A_1+{\displaystyle \frac{A_2}{T_{pr}}}+{\displaystyle \frac{A_3}{T_{pr}^3}}+{\displaystyle \frac{A_4}{T_{pr}^4}}+{\displaystyle \frac{A_5}{T_{pr}^5}},$$

(9)

$$c_{2\left(T_{pr}\right)}=A_6+{\displaystyle \frac{A_7}{T_{pr}}}+{\displaystyle \frac{A_8}{T_{pr}^2}},$$

(10)

$$c_{3\left(T_{pr}\right)}=A_9\left({\displaystyle \frac{A_7}{T_{pr}}}+{\displaystyle \frac{A_8}{T_{pr}^2}}\right),$$

(11)

$$c_{4\left({\rho }_r,T_{pr}\right)}=A_{10}\left(1+A_{11}·{\rho }_r^2\right){\displaystyle \frac{{\rho }_r^2}{T_{pr}}}·exp\left(-A_{11}·{\rho }_r^2\right).$$

(12)

Then, the density of the reservoir’s gas ${\rho }_g$ is calculated using Equation (13), where $R^{\prime }$ is the gas constant equal to 10.73 [25].

$${\rho }_g={\displaystyle \frac{28.97·{\gamma }_g{p}_r}{z·R^{\prime }·T_r}}.$$

(13)

In this paper, z is calculated using the command solve in MATLAB [28] to solve Equation (7). Figure 2 shows the algorithm used, which has been implemented in MATLAB R2025a (25.1.0.2946757) 21 April 2025 [28], and the results are presented in Section 3.

2.3. Control System for the Constant Density of the NG–BG Mix: A Descriptive Case of One BG Source and One NG Tank (NGST)

Control systems in a multidisciplinary field cover many areas in engineering, sciences, and the human body. Control means to regulate, direct, command, or govern. A system is a set of parts, components, machines, equipment, etc., that work together in order to fulfil a function or reach an objective. Usual examples of systems are the circulatory system, which consists of the heart, veins, and arteries that carry oxygen and food to the cells and collect waste from cellular metabolism. Therefore, a control system is the arrangement of parts, components, and/or machines organised in such a way that they seek to provide the desired response, which they achieve by ordering, regulating, measuring, directing, or governing their parts/components or another system [29].

The proposal of a control system scheme to ensure an NG–BG mix flux with density that is constant from two gases with different densities is shown in Figure 3. This demonstrates a system that operates at a constant pressure. In the event of pressure variations, they can be regulated by means of a pressure regulator or control valve según [30]. So, the objective is to keep the gas (NG–BG mix) density ${\rho }_{go}$ constant, which is the output of the control system (see Figure 3). For this, the measurement of the input gas density ${\rho }_{gi}$ (from biomass plant or similar) and ${\rho }_{go}$ are input data for the control system. Then, the readings that are obtained serve as feedback to the control system controller (CSC) that processes the information and acts on the valve of the NG storage tank (NGST) $v_d$, which contains NG with constant composition (${\rho }_d$ constant) over time, and that is mixed with the NG of the reservoir to obtain a gas mixture (NG–BG mix) with constant density. The quantities of inlet mass flow $m_{gi}$, outlet mass flow $m_{go}$, and the mass flow of gas $m_d$ dispatched by the control system from the NGST are analysed in Chapter III. M is the mixer that performs the mixing of gases to obtain a constant density of the mix. NGST is refuelled through a filling pipe from the natural gas distribution network that transports natural gas from extraction fields through gas pipelines and natural gas processing plants, and its volume can be determined based on demand and supply autonomy as a contingency in the event of a natural gas supply interruption to the NGST. Real data are necessary for this, and we recommend conducting this type of study in future research. There are two possible scenarios here: (a) a low-volume NGST, sufficient to compensate for sudden (fluctuations) and random natural gas requirements, and (b) a higher-volume NGST, sufficient to have autonomy to supply demand based on the time required to restore its refill supply in the event of a contingency, which would be part of the response to emergencies and disasters.

The feedback (as detailed in Figure 3) would be used to decrease the system’s sensitivity to biomass plant variations, enable the adjustment of the system’s transient response, reject disturbances, and reduce steady-state tracking errors [31].

To determine the NG’s volume $V_d$ in Figure 3, mathematical models based on the principles of mass conservation are proposed. Thus, for this first case of one BG source and one NG tank (NGST), the NG’s volume $V_d$ calculation that comes out of the NGST to the mixer is considered using the mass of the incoming gas to be regulated ${\rho }_{gi}{V}_{gi}$ and the mass of gas ${\rho }_d{V}_d$ that regulates the density of the mixture (see Figure 3), as shown in the Equations (14) and (15), which serve as the basis for more complex configurations described and studied in the paragraphs and sections below. The proposed mathematical models consider the model that determines the gas density presented in Section 2.2.

$${\rho }_{go}\left(V_{gi}+V_d\right)={\rho }_{gi}{V}_{gi}+{\rho }_d{V}_d,$$

(14)

$$V_d=V_{gi}{\displaystyle \frac{{\rho }_{go}-{\rho }_{gi}}{{\rho }_d-{\rho }_{go}}}.$$

(15)

2.4. Case of Two BG Sources and One NG Tank

According to Figure 4 and the law of the conservation of mass, the ratio of incoming to outgoing mass is shown by Equation (16):

$${\dot{m}}_f={\dot{m}}_1+{\dot{m}}_2+{\dot{m}}_d.$$

(16)

Therefore, given that $\dot{m}=\rho \dot{V}$, where $\dot{m}$ is the mass flow and $\dot{V}$ is the volumetric flow or flow rate, the relationship between the flow rate and densities of the NG tank (${\dot{V}}_d$ and ${\rho }_d$) and the flow rates of BG sources 1 (${\dot{V}}_1$ and ${\rho }_1$) and 2 (${\dot{V}}_2$ and ${\rho }_2$), including the NG–BG mix (${\dot{V}}_f$, ${\rho }_f$) and considering that ${\dot{V}}_f={\dot{V}}_1+{\dot{V}}_2+{\dot{V}}_d$, is shown in Equations (17)–(23).

$${\rho }_f{\dot{V}}_f={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+{\rho }_d{\dot{V}}_d,$$

(17)

$${\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2+{\dot{V}}_d\right]={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+{\rho }_d{\dot{V}}_d,$$

(18)

$${\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2\right]+{\rho }_f{\dot{V}}_d={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+{\rho }_d{\dot{V}}_d,$$

(19)

$${\rho }_f{\dot{V}}_d-{\rho }_d{\dot{V}}_d={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2-{\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2\right],$$

(20)

$${\dot{V}}_d\left[{\rho }_f-{\rho }_d\right]=\left[{\rho }_1-{\rho }_f\right]{\dot{V}}_1+\left[{\rho }_2-{\rho }_f\right]{\dot{V}}_2,$$

(21)

$${\dot{V}}_d={\displaystyle \frac{\left[{\rho }_1-{\rho }_f\right]{\dot{V}}_1+\left[{\rho }_2-{\rho }_f\right]{\dot{V}}_2}{{\rho }_f-{\rho }_d}},$$

(22)

$${\dot{V}}_d=\left[{\displaystyle \frac{{\rho }_1-{\rho }_f}{{\rho }_f-{\rho }_d}}\right]{\dot{V}}_1+\left[{\displaystyle \frac{{\rho }_2-{\rho }_f}{{\rho }_f-{\rho }_d}}\right]{\dot{V}}_2.$$

(23)

2.5. Case of n BG Sources and One NG Tank

In this case, as shown in Figure 5, it is assumed that there is one mixer that has multiple inputs and one output; specifically, for the BG coming from n biomass sources, this involves one NG tank that will act as a density compensator and one output, e.g., that of the NG–BG mix. This is the generalisation of the previous case in which the law of conservation of mass is shown in Equation (24).

$${\dot{m}}_f={\dot{m}}_1+{\dot{m}}_2+\cdots +{\dot{m}}_n+{\dot{m}}_d.$$

(24)

Then, considering that ${\dot{V}}_f={\dot{V}}_1+{\dot{V}}_2+\cdots +{\dot{V}}_n+{\dot{V}}_d$, according to Equations (25)–(31), the relationship between the flow rate of the NG tank needed to regulate the density of the NG–BG mix as a function of the flow rates and densities of BG input is determined (${\dot{V}}_1$, ${\rho }_1$, ${\dot{V}}_2$, ${\rho }_2$, … , ${\dot{V}}_n$, ${\rho }_n$, ${\dot{V}}_f$, ${\rho }_f$).

$${\rho }_f{\dot{V}}_f={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+\cdots +{\rho }_n{\dot{V}}_n+{\rho }_d{\dot{V}}_d,$$

(25)

$${\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2+\cdots +{\dot{V}}_n+{\dot{V}}_d\right]={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+\cdots +{\rho }_n{\dot{V}}_n+{\rho }_d{\dot{V}}_d,$$

(26)

$${\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2+\cdots +{\dot{V}}_n\right]+{\rho }_f{\dot{V}}_d={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+\cdots +{\rho }_n{\dot{V}}_n+{\rho }_d{\dot{V}}_d,$$

(27)

$${\rho }_f{\dot{V}}_d-{\rho }_d{\dot{V}}_d={\rho }_1{\dot{V}}_1+{\rho }_2{\dot{V}}_2+\cdots +{\rho }_n{\dot{V}}_n-{\rho }_f\left[{\dot{V}}_1+{\dot{V}}_2+\cdots +{\dot{V}}_n\right],$$

(28)

$$\left[{\rho }_f-{\rho }_d\right]{\dot{V}}_d=\left[{\rho }_1-{\rho }_f\right]{\dot{V}}_1+\left[{\rho }_2-{\rho }_f\right]{\dot{V}}_2+\cdots +\left[{\rho }_n-{\rho }_f\right]{\dot{V}}_n,$$

(29)

$${\dot{V}}_d={\displaystyle \frac{\left[{\rho }_1-{\rho }_f\right]{\dot{V}}_1+\left[{\rho }_2-{\rho }_f\right]{\dot{V}}_2+\cdots +\left[{\rho }_n-{\rho }_f\right]{\dot{V}}_n}{\left[{\rho }_f-{\rho }_d\right]}},$$

(30)

$${\dot{V}}_d=\left[{\displaystyle \frac{{\rho }_1-{\rho }_f}{{\rho }_f-{\rho }_d}}\right]{\dot{V}}_1+\left[{\displaystyle \frac{{\rho }_2-{\rho }_f}{{\rho }_f-{\rho }_d}}\right]{\dot{V}}_2+\cdots +\left[{\displaystyle \frac{{\rho }_n-{\rho }_f}{{\rho }_f-{\rho }_d}}\right]{\dot{V}}_n.$$

(31)

2.6. Case of Two Mixers and Two NG Tanks (One to Each Mixer), Where Each Mixer Has n and m BG Sources and the Mixture from the FIRST mixer Is Also Input to the Second Mixer: NG Tanks Contain NG of Equal Composition

The present case study is shown schematically in Figure 6, where given that the NG in tanks 1 and 2 are of the same nature, and considering the previous results, the flow rate of tank 1 ${\dot{V}}_{d_1}$ to compensate for density is determined by Equation (32):

$${\dot{V}}_{d_1}=\left[{\displaystyle \frac{{\rho }_{1\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_{d_1}}}\right]{\dot{V}}_{1\left(d_1\right)}+\left[{\displaystyle \frac{{\rho }_{2\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_{d_1}}}\right]{\dot{V}}_{2\left(d_1\right)}+\cdots +\left[{\displaystyle \frac{{\rho }_{n\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_{d_1}}}\right]{\dot{V}}_{n\left(d_1\right)}.$$

(32)

Similarly, the flow rate of tank 2 ${\dot{V}}_{d_2}$ is determined by Equation (33):

$${\dot{V}}_{d_2}=\left[{\displaystyle \frac{{\rho }_{f_1}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_{d_2}}}\right]{\dot{V}}_{f_1\left(d_2\right)}+\left[{\displaystyle \frac{{\rho }_{1\left(d_2\right)}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_{d_2}}}\right]{\dot{V}}_{1\left(d_2\right)}+\cdots +\left[{\displaystyle \frac{{\rho }_{n\left(d_2\right)}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_{d_2}}}\right]{\dot{V}}_{m\left(d_2\right)}.$$

(33)

Assuming that both tank 1 and tank 2 are supplied by the same NG source, then ${\rho }_{1\left(d_1\right)}={\rho }_{1\left(d_2\right)}={\rho }_d$; therefore, the equations are expressed as Equations (34) and (35):

$${\dot{V}}_{d_1}=\left[{\displaystyle \frac{{\rho }_{1\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_d}}\right]{\dot{V}}_{1\left(d_1\right)}+\left[{\displaystyle \frac{{\rho }_{2\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_d}}\right]{\dot{V}}_{2\left(d_1\right)}+\cdots +\left[{\displaystyle \frac{{\rho }_{n\left(d_1\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_d}}\right]{\dot{V}}_{n\left(d_1\right)},$$

(34)

$${\dot{V}}_{d_2}=\left[{\displaystyle \frac{{\rho }_{f_1}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_d}}\right]{\dot{V}}_{f_1\left(d_2\right)}+\left[{\displaystyle \frac{{\rho }_{1\left(d_2\right)}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_d}}\right]{\dot{V}}_{1\left(d_2\right)}+\cdots +\left[{\displaystyle \frac{{\rho }_{n\left(d_2\right)}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_d}}\right]{\dot{V}}_{m\left(d_2\right)}.$$

(35)

Equations (34) and (35) can be expressed using summations, as shown in Equations (36) and (37):

$${\dot{V}}_{d_1}=\sum _{i=1}^n\left[{\displaystyle \frac{{\rho }_{i\left(d_i\right)}-{\rho }_{f_1}}{{\rho }_{f_1}-{\rho }_d}}\right]{\dot{V}}_{i\left(d_1\right)},$$

$${\dot{V}}_{d_1}={\displaystyle \frac{1}{{\rho }_{f_1}-{\rho }_d}}\left[\sum _{i=1}^n\left[{\rho }_{i\left(d_1\right)}-{\rho }_{f_1}\right]{\dot{V}}_{i\left(d_1\right)}\right],$$

(36)

$${\dot{V}}_{d_2}=\left[{\displaystyle \frac{{\rho }_{f_1}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_d}}\right]{\dot{V}}_{f_1\left(d_2\right)}+\sum _{j=1}^m\left[{\displaystyle \frac{{\rho }_{j\left(d_2\right)}-{\rho }_{f_2}}{{\rho }_{f_2}-{\rho }_d}}\right]{\dot{V}}_{j\left(d_2\right)},$$

$${\dot{V}}_{d_2}={\displaystyle \frac{1}{{\rho }_{f_2}-{\rho }_d}}\left[\left[{\rho }_{f_1}-{\rho }_{f_2}\right]{\dot{V}}_{f_1\left(d_2\right)}+\sum _{j=1}^m\left[{\rho }_{j\left(d_2\right)}-{\rho }_{f_2}\right]{\dot{V}}_{j\left(d_2\right)}\right].$$

(37)

2.7. Case of an Arrangement of k Tanks and k Mixers, with the Last Mixer Receiving the Input from $(k-1)$ Mixers, Having Its Own BG Sources Connected: All NG Tanks Have the Same Density

Figure 7 shows the scheme for this case in which, as has been studied in previous items, mixers 1 and 2 can be expressed by Equations (38) and (39), and generally speaking, the flow rate of any mixer k is determined by Equation (40):

$${\dot{V}}_{d_1}={\displaystyle \frac{1}{{\rho }_{f_1}-{\rho }_d}}\left[\sum _{i=1}^{n\left(d_1\right)}\left[{\rho }_{i\left(d_1\right)}-{\rho }_{f_1}\right]{\dot{V}}_{i\left(d_1\right)}\right],$$

(38)

$${\dot{V}}_{d_2}={\displaystyle \frac{1}{{\rho }_{f_2}-{\rho }_d}}\left[\sum _{i=1}^{n\left(d_2\right)}\left[{\rho }_{i\left(d_2\right)}-{\rho }_{f_2}\right]{\dot{V}}_{i\left(d_2\right)}\right],$$

(39)

$${\dot{V}}_{d_{k-1}}={\displaystyle \frac{1}{{\rho }_{f_{k-1}}-{\rho }_d}}\left[\sum _{i=1}^{n\left(d_{k-1}\right)}\left[{\rho }_{i\left(d_{k-1}\right)}-{\rho }_{f_{k-1}}\right]{\dot{V}}_{i\left(d_{k-1}\right)}\right].$$

(40)

Then, the NG flow rate of the tank associated with the mixer k is determined by Equation (41):

$${\dot{V}}_{d_{k-1}}={\displaystyle \frac{1}{{\rho }_{f_k}-{\rho }_d}}\left[\sum _{j=1}^{k-1}\left[{\rho }_{f_j}-{\rho }_{f_k}\right]{\dot{V}}_{j\left(d_k\right)}+\sum _{i=1}^n\left[{\rho }_{i\left(d_k\right)}-{\rho }_{f_k}\right]{\dot{V}}_{i\left(d_k\right)}\right].$$

(41)

Once the flow rate of the tank associated with each mixer is determined, Equation (42) determines the total flow rate of NG to be delivered from all the tanks:

$${\dot{V}}_d={\dot{V}}_{d_1}+{\dot{V}}_{d_2}+\cdots +{\dot{V}}_{d_{k-1}}+{\dot{V}}_k.$$

(42)

The sum of the first (k-1) terms ${\dot{V}}_{d_1\cdots d_{k-1}}$ of Equation (42) (see Equation (43)) can be summarised in a summation that encompasses the production of individual biodigesters, as shown in Equation (44):

$${\dot{V}}_{d_1\cdots d_{k-1}}={\dot{V}}_{d_1}+{\dot{V}}_{d_2}+\cdots +{\dot{V}}_{d_{k-1}},$$

(43)

$${\dot{V}}_{d_1\cdots d_{k-1}}=\sum _{q=1}^{k-1}\left[{\displaystyle \frac{1}{{\rho }_{f_q}-{\rho }_d}}\left[\sum _{i=1}^{n\left(d_q\right)}\left[{\rho }_{i\left(d_q\right)}-{\rho }_{f_q}\right]{\dot{V}}_{i\left(d_q\right)}\right]\right].$$

(44)

Therefore, Equation (42) is transformed into Equation (45), which provides the amount of NG used to regulate the density of BG from different sources:

$${\dot{V}}_d={\dot{V}}_{d_1\cdots d_{k-1}}+{\dot{V}}_k.$$

(45)

2.8. Effect of Density Change on the Transport of the NG–BG Mix

The behaviour of the liquids that flow through the pipe is known; their density remains constant due to a weak dependence of this property on pressure, which simplifies the equations to describe the phenomena. However, this assumption is not valid when the fluid is compressible and the decrease in pressure due to frictional head losses causes a decrease in the gas density [32].

On the other hand, the variation in gas density in the compressors affects the turbocharger pressure ratio, compressor head, system resistance, and turbocharger flow and power. Any change in $W_g$, T, k, or z will change the ratio of the pressure produced. A gas being a compressible substance leads to the velocity of the gas changing with the density of the gas. The effect on the resistance of the system is affected by a change in density that causes a variation in the friction drop in pipes, fittings, and vessels. The effect on turbocharger flow causes changes in the operating point of each stage, which, depending on the selected impeller, can lead to adverse effects on the performance of a dynamic compressor, such as surges, flow changes, elevated temperatures, vibrations, etc. The power consumed by the dynamic compressor is directly proportional to the gas density, and it is until before the region of throttling flow or the stonewall of the performance curve. In the throttling flow region, the head produced by the compressor approaches zero because the gas velocity is equal to the sonic velocity. Finally, all these effects cause a mismatch in the compressor stage, which can cause significant mechanical damage to the compressor train [33].

3. Simulations and Analysis

Simulations have been conducted according to following criteria:

1.

It is assumed that the composition of BG will be methane $C{H}_4$, ethane $C_2{H}_6$, propane $C_3{H}_8$, butane $C_4{H}_{10}$, pentane $C_5{H}_{12}$, and hexane $C_6{H}_{14}$, which mostly make up a typical NG composition, as can be seen in [34] and the research criteria of [35].

2.

Using the combination of mixture components of [34], an experiment with randomly generated data is developed with methane $C{H}_4$ between 80–100% (by volume) in accordance with [19,36,37,38], ethane $C_2{H}_6$ between 0–15% (by volume), and propane $C_3{H}_8$ between 0–20% (by volume). In this paper, in order to verify the operation of the control system and the quantification of random variables, Equations (46)–(51) are used with the command rand() in MATLAB, and m is the state number. The definition of the state number is the shortest possible time between reliable measurements, which may be a few seconds or a few minutes; therefore, there are no sudden variations in the composition of biogas between states. This is ensured both because it is possible to design an adequate measurement system and because of the design of biodigesters or similar equipment that process biomass (in future experiments, this will depend on the measurement system, equipment, sensor quality, and signal processing). These percentages are assumed considering that BG undergoes a purification and concentration process, given that its reported values in production are $C{H}_4$ 51.5%, $C{O}_2$ 38.9%, $O_2$ 1.0%, $N_2$ 8.9%, $H_{2}S$ 350 ppm, and water 17.2 mg/L [39].

$$C{H}_4=0.80+0.2\times rand(1,m),$$

(46)

$$C_2{H}_6=0.07\times rand(1,m),$$

(47)

$$C_3{H}_8=0.05\times rand(1,m),$$

(48)

$$C_4{H}_{10}=0.05\times rand(1,m),$$

(49)

$$C_5{H}_{12}=0.02\times rand(1,m),$$

(50)

$$C_6{H}_{14}=0.01\times rand(1,m).$$

(51)

3.

The properties of the BG components used are shown in

Table 1. Physical properties of paraffin hydrocarbons and other compounds.

Compound	${\mathit{W}}_{\mathit{g}}$	${\mathit{p}}_{\mathit{c}}$	${\mathit{T}}_{\mathit{c}}$
Methane	16.04	673.1	343.2
Ethane	30.07	708.3	549.9
Propane	44.09	617.4	666.0
Butane	58.12	529.1	734.6
Pentane	72.15	489.8	846.2
Hexane	86.17	440.1	914.2

, where $p_c$ is in psia, and $T_c$ is in °R.

4.

The composition of NG in NGST will be of high quality in methane, as shown in [40,41,42], and it is considered that ${\rho }_d=$ 0.04 lb/cu.ft. at 68 °F and 14.7 psia.

3.1. Normalisation of NG Composition

Since Equations (46)–(51) generate random numbers, the sum can exceed a value of 1; therefore, it is necessary to perform normalisation in order to keep the total sum of all the percentages of the NG components equal to 1. An example of normalised and non-normalised methane is shown in Figure 8 and is considered m = 30, where m is the number of states, with each state representing a certain period of time in which the characteristics of the gases remain constant. Consequently, the density of the mixture also remains constant; that is, the duration of each state depends on the variation in the composition of the BG during its production, which may vary with changes in biomass and environmental conditions over the time [43,44,45,46,47]. The horizontal axis of the figures indicates the state number; each state has its own characteristics.

It is assumed that $P_r$ and $T_r$ are equal to 3250 psi and 213 °F, respectively, which can be modified for other study cases. So, with the normalised composition of BG for each of the m states, $M_W$, $p_{pc}$, $T_{pc}$, $p_{pr}$, and $T_{pr}$ are determined. Then, z (see Figure 9) and ${\rho }_g$ (see Figure 10, graphed together with line of ${\rho }_{go}=8$ lb/cu.ft. and ${\rho }_d=7.62$ lb/cu.ft., both have been assumed for this paper, and $V_{gi}=1$ cu.ft. under the criterion of $V_d$ per unit of $V_{gi}$) are calculated. Also, the NG supply volume (see Figure 11) and cumulative volume of NG from NGST (see Figure 12) are shown.

3.2. Volume of NG in the NGST

The calculation determines the discharge of pure NG from NGST and its accumulation. NGST needs to be reloaded after a certain time. The local production of NG (for example, in biodigesters located within metropolis) allows for a reduction in the NG coming from subsoil.

4. Discussion

The limitations of the model stem from the fact that (a) there is no similar experience of mixing natural gas and biogas to maintain a constant mixture density that has been implemented and tested in a laboratory, cities, or similar settings; (b) there is a lack of detailed data for short time intervals (seconds or minutes) from measurements in biogas plants, and to compensate for this, the random behaviour of the composition must be assumed based on the references consulted. The strength of the model is that the mathematical model for calculating gas density based on its composition is widely used in the oil and natural gas industry.

During the development of the state-of-the-art, it was found that there are no studies related to the mixture of biogas and natural gas or to gas quality control. Similar studies that do exist seek to achieve or pursue a high concentration of methane. In contrast, this paper proposes a mixture of natural gas and biogas with lower percentages of methane (compared to what has been reported), which implies a reduction in the equipment needed in biogas plants, which, in principle, would ideally only require equipment for desulfurisation and gas pumping.

The calculation of z, considered an interactive and laborious process, has been developed and tested for a scenario of states that evolve in ordinary time. This is a topic of interest to answer in future research about the duration of each state; however, it must be changed to meet varying demands during the day and remain quasi-constant during the night. A reduction in the consumption of NG from the subsoil is possible from production at the ground surface level and locally using biodigesters, which is very possible given that there is an increasing world population and this also fits into the concept of smart cities for the production, use, and management of heat (in heating systems or similar) and electricity (in micro turbines or similar). In this regard, it can be extended to concepts such as distributed generation, microgrids, and smart grids. For the practical implementation of the system, it is enough to obtain the pressure and temperature characteristics of the NG source and its composition using measurement systems with real-time data reading, capture, and processing with an adequate sensor–computer–machine–human interface, which could be a solution to an important problem in this type of facility.

One can also consider the multi-MG scenario for large urban areas/large populations. In this case, MGs can also have NG surpluses and shortages since a microgrid is fed by other micro-sources and the utility network. Then, not only can the BG production of the MG be for internal consumption, but it can also be dumped into the NG–BG mix transport network and be taken to other MGs. Given that each MG can handle up to 10 MW at its point of common coupling, it should be taken into account that it will be usual that BG production will be less than the amount needed to meet the demand of the MG or multiple MGs; this will be a function of how much population can be engaged and, based on that, a study scenario or scenarios can be developed.

5. Conclusions

The key findings are as follows: (a) a mixture of natural gas and biogas has been proposed in such a way that the density of the resulting mixture remains constant, which facilitates the pumping of the gas mixture between mixers and/or to consumers; (b) a theoretical calculation of the density of natural gas mixed with biogas to obtain a mixture of constant density has been shown; (c) a control system with feedback and control over valves has been proposed to regulate the flow of natural gas and to take advantage of all the biogas produced; (d) different piping and mixer schemes have been proposed considering the reality, future, and concept of cities, microgrids, smart grids, and smart cities where it will be of massive use to biogas plants.

It is possible to reflect on the scope considering that in urban and similar environments where there is a large production of organic waste, the mixture of NG and BG is seen as an alternative for the reduction in greenhouse gases; even the implementation of the micro-production of BG in homes, businesses, and residential areas, and its mixture with the normal supply of NG to the customer, is interesting. The resulting density of the gas mixture has a controlled intermediate value between the density of NG and that of BG. With a potential for massive use in cities around the world, it has the advantage of being able to control the density of the resulting gas (NG–BG mix), and the compressors (and other components and machines in the gas transport pipeline systems) will not have mechanical problems due to sudden variations in the density of the gas they drive and transport. So, another finding is that according to the schemes presented in this paper, the location of biodigesters should be close to the pipeline that supplies NG to users (families, businesses, industries, MGs, etc.).

Finally, an investigation was made into how to compensate for the changing BG density with NG, which has constant properties over time. For this purpose, a mathematical density model was developed, and a feedback control system was proposed that seeks to achieve an intermediate density for the mixture. Ideally, this density will be constant and between the densities of NG and BG. Diagrams of different scenarios linked to a vision of implementation topologies based on tree-shaped mixers were presented. A case study has been conducted. For all scenarios, the respective equations for mixture flow rate, BG source flow rate, and NG flow rate in the different tanks were presented. These, when added together, provided the total value of NG used for planning the purchase and storage of NG to meet demand. Data records made in real-life implementations can be used to characterise the performance of BG sources and the operation of different NG–BG mix distributions and mixing schemes.