Changes in the Operating Conditions of Distribution Gas Networks as a Function of Altitude Conditions and the Proportion of Hydrogen in Transported Natural Gas

Abstract

The article presents a comparison between the pressure conditions of a real low-pressure gas network and the results of hydraulic calculations obtained using various simulation programs and empirical equations. The calculations were performed using specialized gas network analysis software: STANET (ver 10.0.26), SimNet SSGas 7, and SONET. Additionally, the simulation results were compared with calculations based on the empirical Darcy–Weisbach and Renouard equations. In the first part of the analysis, two calculation models were compared. In one model, the geodetic elevation of individual network nodes was included (elevation-aware model), while in the second, calculations were performed without considering node elevation (flat model). For low-pressure gas networks, accounting for elevation is critical due to the presence of the pressure recovery phenomenon, which does not occur in medium- and high-pressure networks. Furthermore, considering the growing need to increase the share of renewable energy, the study also examined the network’s operating conditions when using natural gas–hydrogen mixtures. The following hydrogen concentrations were considered: 2.5%, 5.0%, 10.0%, 20.0%, and 50.0%. The results confirm the importance of incorporating elevation data in the modeling of low-pressure gas networks. This is supported by the small differences between calculated results and actual pressure measurements taken from the operating network. Moreover, increasing the hydrogen content in the mixture intensifies the pressure recovery effect. The hydraulic results obtained using different computational tools were consistent and showed only minor discrepancies.

1. Introduction

Accurate hydraulic calculations of distribution networks, particularly in low-pressure systems, are crucial due to the limited available pressure drops. The fundamental aspects of such calculations, as presented in standard textbooks [1,2], are rarely addressed in subsequent research topics. When research does focus on distribution networks, it typically involves modifications of previously established computational methods [3,4,5], comparison of results obtained using various empirical formulas [6], or the introduction of numerical methods [7]. Even in publications focused on gas installation calculations, this issue is almost entirely marginalized [8].

Pressure recovery—as a phenomenon related, on the one hand, to the density of the medium and, on the other, to significant elevation changes—is primarily utilized in gas installations, as described in available textbooks [9,10]. However, this issue is often approached uncritically in relation to gas networks, where, by definition, gas flow conditions differ fundamentally from those in gas installations or laboratory settings.

In gas pipelines, thermodynamic conditions change along the gas flow, which affects the parameters (properties) of the gas. This phenomenon becomes particularly noticeable in large-scale gas networks characterized by significant pressure variations. It should be noted that the difference between measurement pressure and absolute pressure is a function of the elevation of the measurement points above sea level, implying that these parameters must be considered.

By definition, pressure recovery is possible when two equally essential conditions are met:

• There is a difference between the density of the distributed medium and the density of the surrounding air;
• There is an elevation difference between the entry point and the offtake point.

Natural gas, having a lower density than air, tends to rise naturally due to this property. This effect is well documented in gas installations, particularly in multi-story buildings where there is a substantial height difference between the lowest and highest gas consumption points. While recovery in internal installations is a well-documented phenomenon in the literature, its occurrence in gas networks is less obvious. It is therefore often overlooked in gas flow analyses in pipelines.

Additionally, the physical and chemical properties of the distributed gas mixture may vary over time, primarily due to the introduction of admixtures, such as hydrogen, into the natural gas stream. Since hydrogen is the lightest gas found in the environment, its presence decreases the density of the resulting gas mixture, potentially enhancing the pressure recovery effect.

Currently, considerable attention is devoted to the replacement of pure natural gas with various blends, including mixtures with hydrogen [11,12], biogas [13], or LNG [14,15]. Studies also explore the potential of covering part of the gas demand with other gas mixtures [16], their use for electricity generation [17], and the importance of monitoring hydrogen concentration fluctuations, which is critical for the proper operation of gas appliances [18].

However, during calculations involving elevation effects, the use of such gas mixtures is largely neglected, with only a few exceptions [19]. While this simplification may be acceptable in medium-pressure networks, in low-pressure networks it can lead to significant discrepancies between calculated and measured pressure values in real systems.

Pressure losses in gas pipelines are influenced by pipe roughness and hydraulic resistance due to friction along straight sections, as well as by resistance from welds and joints between pipeline segments and from network nodes [20].

This phenomenon is relevant primarily in low-pressure gas networks where available pressure drops are proportionally small. However, due to the topology of such networks—which typically consist of many short pipeline segments and a high number of fittings and valves—additional losses are introduced through joint and node resistance. In higher pressure ranges, pressure recovery is generally of minor importance due to the considerable discrepancy between the recoverable pressure and the available pressure drop, as well as due to the higher absolute gas density under such pressure conditions.

Due to ongoing efforts to reduce the environmental impact of fossil fuel usage [21]—for example, by incorporating hydrogen as an energy carrier produced and utilized through Power-to-Gas technologies—it is necessary to thoroughly analyze the feasibility of delivering natural gas–hydrogen mixtures to residential consumers, particularly within low-pressure gas networks [22]. This need is often reinforced by the technical limitations of such networks, which frequently cannot be upgraded to operate at medium-pressure levels due to their current condition. For this reason, the issue of pressure recovery in systems distributing natural gas–hydrogen mixtures is a relatively new topic, and there is currently no dedicated literature that comprehensively addresses it.

To verify these hypotheses, simulations were conducted using specialized software for gas network simulation and calculations, incorporating empirical Renouard and Darcy–Weisbach equations for a section of a low-pressure gas network. The objective of these operations was to investigate how elevation differences affect pressure recovery. The simulation results were compared with actual values measured using pressure loggers. In the second part of the experiment, the selected simulation models and calculation methods were applied to assess the influence of hydrogen admixture on pressure recovery—both for the network model including elevation coordinates and for the flat model with no elevation data considered.

2. Materials and Methods

2.1. Low-Pressure Distribution Network

The analysis of the impact of elevation conditions on pressure drop was conducted on a selected segment of a low-pressure gas distribution network. The analyzed network is supplied from a single pressure source, denoted as point C. The elevation coordinate (Z) of this source is 227.0 m, making it the lowest point in the network segment under study. Two gas consumption points were also considered:

• Point A—the point with the highest elevation difference relative to the pressure source, located at 265.0 m above sea level, resulting in a height difference of 38.0 m. This point is approximately 1100 m away from the gas station along the gas flow path.
• Point B—the most distant consumption point from the pressure source, located at an elevation of 255.0 m and approximately 1400 m from the gas station.

The analyzed section of the network consists primarily of steel pipes, with a smaller portion made of polyethylene, with nominal diameters ranging from DN40 to DN250. The system supplies natural gas to consumers in a residential district of Kraków, primarily composed of multi-family buildings. Consequently, the characteristic tariff structure (

Table 1. Characteristics of tariff groups [23].

Tariff Group	Contracted Power b 1 [kWh/h]	Annual Contract Volume a [m3/year]
W-1 (1.1, 1.2, 1.12T)	b ≤ 110	a ≤ 300
W-2 (2.1, 2.2, 2.12T)	b ≤ 110	300 < a ≤ 1 200
W-3 (3.6, 3.9, 3.12T)	b ≤ 110	1 200 < a ≤ 8 000

¹ 110 kWh = 10 m³.

) reflects a predominance of residential consumers classified under tariff groups W-1 and W-2, with a relatively small number of customers in other tariff groups.

A total of 3000 actual gas consumers (gas off-take points) connected to the analyzed network were included in the calculations, which, based on the adopted load profiles (reflecting the non-uniformity of gas consumption), resulted in a source flow rate of approximately 93 m³/h. The pressure at the source (a medium-pressure reduction and metering station) was set at 2.600 kPa. Figure 1 presents a schematic of the network overlaid on a contour map, with the pipe types and key points marked.

The gas network was calibrated, which means that the actual conditions from a specific day and time were reproduced as a result of the simulation.

2.2. Pressure Recovery Calculation

As part of the study on pressure recovery in a gas network, theoretical, static simulation calculations were conducted using three software tools: SimNet SSGas 7, SONET, and STANET (ver. 10). In addition, theoretical calculations were performed using the Darcy-Weisbach equation and Renouard’s equation. It is also worth noting that, when analyzing the operation of low-pressure gas networks, there is no need to account for the time-dependent behavior of pressure variations. This is due to the relatively short distances between the supply source and end users, the low operating pressures within the network, and the very short time required to reach a steady state [20]. For this reason, steady-state calculations were selected for the analysis.

SimNet (Fluid System Sp. z o.o.) is a suite of software applications used for performing static or dynamic simulations of fluid networks (gas, district heating, and water) under defined operating conditions. Static simulations for low-, medium-, elevated-medium-, and high-pressure gas networks are handled by the SimNet SSGas module. Thanks to an advanced computational engine, it imposes no limitations on the number of supply sources, demand nodes, pipeline sections, or valves operating in either open or closed states. Nonlinear elements, such as compressors and regulators, can operate within specified ranges of inlet pressures, outlet pressures, compression ratios, pressure drops, or flow rates. The program does not rely on the physicochemical parameters of the flowing gas, but rather on the characteristics of nodes and linear elements. Operational parameters and spatial coordinates (x, y, z) of network components are imported from databases and online maps, depending on the user’s license. The application integrates Geographic Information System (GIS) technology to provide a clear visualization of simulation results and network elements, and it enables data storage [24].

SimNet uses the following equation to describe the pressure drop:

$$p_1-p_2=\lambda (z,\lambda ,\varrho ,\mu ,T)·L·D_w^b·Q_n^a$$

(1)

SONET (Sygnity S.A.) is a GIS-based application designed for the visualization, analysis, and editing of spatial data [25]. The gas network calculation module is the embedded GASNET application [26]. In user-generated calculation schemes, the locations of fittings, nodes, and pipeline sections, along with data on consumer numbers, tariffs, pipe diameters, and other relevant details, are automatically loaded from databases when the computational model is generated. For simulation purposes, the software also allows for manual editing of network operating parameters, topology, and the selection of computational formulas [27]. Additionally, users can define basic properties of the gas flowing in the network, such as temperature, relative density, and calorific value [26]. In this study, static simulations were conducted using a formula developed in accordance with the Polish standard PN-34034 [26,28]:

$$p_1-p_2=p_1·\left(1-{\displaystyle \frac{w_1{T}_2}{w_2{T}_1}}\right)$$

(2)

STANET is a program designed for performing calculations in networked systems, not limited to gas networks, but also applicable to water supply, sewage, heating, and even electrical grids [29]. In addition to static simulations, it supports quasi-dynamic calculations (i.e., sequences of steady-state computations). It enables the creation, modification, and calculation of networks comprising up to 2,000,000,000 elements, while continuously verifying project completeness. The software supports the calculation of additional parameters such as gas temperature, heat loss to the environment, elevation-related pressure losses, and also optimizes pipeline routes and diameter selection. Like the programs mentioned above, it integrates with GIS platforms and is compatible with most commercial online mapping services. However, it provides a significant advantage over competing tools by allowing for the definition or computation of all physicochemical properties of the transported gas, as well as the modification of all coordinates (X, Y, Z) of selected network nodes [30].

In STANET, the pressure drop is calculated using one of two empirical formulas, depending on the gas flow conditions present in the pipe being analyzed [31]:

• turbulent conditions:

$$p_1-p_2={\displaystyle \frac{16p_n{\varrho }_{n}zT{Q}_n\left|Q_n\right|}{{\pi }^2\left(p_1+p_2\right)T_n{D}_w^4}}\left({\displaystyle \frac{L}{D_w}}\lambda +\zeta \right)-{\displaystyle \frac{{\varrho }_1+{\varrho }_2}{2}}g\left(h_1-h_2\right)$$

(3)

• laminar conditions:

$$p_1-p_2={\displaystyle \frac{128\eta LzT{p}_n{Q}_n}{\left(p_1+p_2\right)\pi T_n{D}_w^4}}-{\displaystyle \frac{{\varrho }_1+{\varrho }_2}{2}}g(h_1-h_2)$$

(4)

Each of the software tools used in this study has its own graphical user interface and method of presenting results. Consequently, graphical representations of the outcomes may differ between the software packages, which can affect the ease of interpreting the obtained results.

Numerous researchers have formulated theoretical and empirical equations since the inception of gas network flow modeling. Their applicability is typically limited by flow characteristics (defined by the Reynolds number) or the range of pipe internal diameters. Pressure range generally plays a secondary role, especially in low-pressure gas pipelines, where equations are typically formulated as a function of the Reynolds number, but without accounting for the fifth flow regime [20].

Two empirical equations were selected for the calculations:

• The Darcy–Weisbach equation:

$$p_1-p_2=\lambda ·{\displaystyle \frac{\varrho w^{2}L}{2D_w}}+\varrho g\left(h_1-h_2\right)$$

(5)

• The Renouard equation:

$$p_1-p_2=2557.07dL{\displaystyle \frac{Q_n^{1.82}}{D_w^{4.82}}}$$

(6)

The Renouard and Darcy–Weisbach equations selected for this study are two independent formulas used to describe flow resistance in pipelines. As can be observed, the Renouard equation does not account for the influence of elevation difference between the inlet and outlet nodes.

The friction factor in the Darcy–Weisbach equation (Equation (5)) is calculated using different formulas depending on the flow conditions present in the pipeline segment under analysis. This coefficient primarily depends on the Reynolds number but is also influenced by the internal wall roughness of the pipe [32].

In engineering practice, pressure recovery in internal gas installations is commonly calculated using the following formula [1], which is also applied in the present study:

$$∆p_h=g·∆h·\left(\rho -{\rho }_p\right)$$

(7)

2.3. Physicochemical Properties of Pure Natural Gas and Natural Gas–Hydrogen Blends

The analyzed segment of the gas distribution network is supplied with natural gas, the composition of which is presented in

Table 2. Composition of natural gas and selected compositions of exit gas mixtures with varying hydrogen content. Own calculations based on [33].

Elements			Input Gas	Hydrogen Content in the Exit Gas
				2.5%	5.0%	10.0%	25.0%	50.0%
Component	methane	% mol.	97.8780	95.4311	92.9841	88.0902	73.4085	48.9390
	ethane	% mol.	0.9100	0.8873	0.8645	0.8190	0.6825	0.4550
	propane	% mol.	0.1670	0.1628	0.1587	0.1530	0.1253	0.0835
	i-butane	% mol.	0.0430	0.0419	0.0409	0.0387	0.0323	0.0215
	n-butane	% mol.	0.0280	0.0273	0.0266	0.0252	0.0210	0.0140
	i-pentane	% mol.	0.0180	0.0176	0.0171	0.0162	0.0135	0.0090
	n-pentane	% mol.	0.0060	0.0059	0.0057	0.0054	0.0045	0.0030
	heavier hydrocarbons	% mol.	0.0170	0.0166	0.0162	0.0153	0.0128	0.0085
	nitrogen	% mol.	0.6800	0.6630	0.6460	0.6120	0.5100	0.3400
	carbon dioxide	% mol.	0.2530	0.2467	0.2404	0.2277	0.1898	0.1265
	hydrogen	% mol.	0.0000	2.5000	5.0000	10.0000	25.0000	50.0000
Parameters	absolute density	kg/m3	0.7350	0.7188	0.7027	0.6704	0.5737	0.4124
	relative density	—	0.5684	0.5559	0.5435	0.5185	0.4437	0.3189
	gross caloric value	kWh/m3	11.102	10.913	10.724	10.345	9.211	7.321
		MJ/m3	39.97	39.29	38.605	37.244	33.160	26.356
	net caloric value	kWh/m3	10.009	9.834	9.658	9.308	8.256	6.502
		MJ/m3	36.03	35.402	34.77	33.508	29.722	23.407
	higher Wobbe index	kWh/m3	14.73	14.64	14.55	14.367	13.829	12.963
		MJ/m3	53.01	52.69	52.37	51.722	49.784	46.666
	lower Wobbe index	kWh/m3	13.28	13.19	13.10	12.926	12.394	11.513
		MJ/m3	47.79	47.48	47.17	46.534	44.618	41.447
	dynamic viscosity	μPa·s	10.523	10.500	10.476	10.426	10.286	9.874
	kinematic viscosity	μm2/s	14.318	14.607	14.909	15.551	17.878	23.944
	summation coefficient	–	0.0497	0.0486	0.0474	0.0452	0.0383	0.0269
	compressibility factor	–	0.9975	0.9976	0.9977	0.9980	0.9985	0.9993

[33]. The same table also provides calculated values of key parameters of the multicomponent gas under normal conditions (pressure: 101.325 kPa, temperature: 273.15 K), depending on its composition. These parameters were determined using the following equations [20]:

• additive parameters: density, gross/net calorific value, summation coefficient (the symbol K denotes one of the listed parameters):

$$K={\sum }_{i=1}^n{x}_i·K_i$$

(8)

• Wobbe index:

$$W={\displaystyle \frac{H}{\sqrt{d}}}$$

(9)

• summation coefficient (used in the calculation of the compressibility factor z):

$$\sqrt{b}=\sqrt{1-z}$$

(10)

• compressibility factor (calculated based on the summation coefficient):

$$z=1-{\left(\sqrt{b}\right)}^2$$

(11)

• dynamic viscosity:

$${\mu }_n={\displaystyle \frac{\sum _{i=1}^n{\mu }_{ni}·x_i·\sqrt{M_i·T_{ki}}}{\sum _{i=1}^n{x}_i·\sqrt{M_i·T_{ki}}}}$$

(12)

• kinematic viscosity:

$$\nu ={\displaystyle \frac{\mu }{\varrho }}$$

(13)

For the Wobbe index (Equation (9)), similar to calorific value, one distinguishes between the lower Wobbe index (based on the lower heating value) and the upper Wobbe index (based on the higher heating value).

The impact of increasing hydrogen content in natural gas–hydrogen mixtures on fundamental parameters—such as absolute and relative density, as well as volumetric gross and net calorific values—is well documented. However, changes in the viscosity of such gas mixtures are far less frequently addressed in the literature, even though viscosity plays a critical role in hydraulic calculations. Therefore, Figure 2 presents the changes in both dynamic and kinematic viscosity as a function of the increasing hydrogen fraction in the natural gas mixture. All values are expressed under normal conditions.

Additionally, due to the pressure values prevailing in the analyzed network (approximately 104 kPa, only slightly different from the standard conditions of 101.325 kPa), variations in the compressibility factor were not considered. This simplification was adopted because the compressibility factor values remain close to unity under the given conditions (see Table 2). Furthermore, even with increasing hydrogen content in the mixture, it is not necessary to account for compressibility, as the lower overall density allows the mixture to be treated as an ideal gas with even smaller error. However, it should be noted that at higher pressures, the compressibility factor should be calculated using, for example, the method described in ISO 12,213 [34]. In the case of hydrogen contents exceeding 10%, the method described in the 2021 DVGW report [35] is recommended.

3. Results

The following calculation scenarios were analyzed:

• calculations for actual flow distributions in the real-world network, taking elevation into account,
• calculations for actual flow distributions in a flat model of the network (neglecting elevation),
• calculations analyzing the effect of hydrogen admixture in natural gas, for both the elevation-aware network model and the flat model.

All of the above scenarios were computed using the Darcy-Weisbach and Renouard equations as well as the STANET software. Additionally, in selected cases, SimNet and SONET were also used.

Additionally, in the first case, a comparison was made between the simulation results and actual pressure values measured at selected points in the gas network.

3.1. Actual Measurement Data

To compare the simulation results with real-world data, pressure loggers were installed at key locations in the analyzed network. The measurement data collected over several days confirmed the presence of pressure recovery in the low-pressure gas network, associated with elevation differences between the pressure source and the consumption points.

Figure 3 illustrates a schematic of the network, showing the locations of the pressure loggers. The diagram also includes simulated pressure values at the nodes and the gas flow rates in individual pipeline sections calculated using the STANET software for one of the assumed operating conditions. Missing values of pressure and flow in some parts of the diagram result from the software’s display function, which automatically adjusts the placement of text fields to prevent overlapping.

Table 3. Pressure readings at measurement points.

Measurement Point	Supply Pressure [kPa]	2.5	2.6	2.7
NS-44	Measured pressure [kPa]	2.97 ± 0.12	2.99 ± 0.12	3.00 ± 0.12
NS-17C		3.18 ± 0.10	3.21 ± 0.12	3.20 ± 0.12
NS-10B		2.84 ± 0.17	2.92 ± 0.13	2.89 ± 0.16

presents the average pressure values determined based on data collected from pressure loggers installed at various points within the analyzed network. For each measurement point, three pressure values are shown, corresponding to three different supply pressure levels at the outlet of the gas station.

3.2. Simulation Calculations

The low-pressure network was modeled in all three previously mentioned software tools using two variants: one that included elevation coordinates, and a flat model that neglected the elevation of individual nodes.

3.2.1. SimNet

A series of static simulations of the gas network demonstrates that the elevation difference between the supply point and the consumption points has a significant impact on the calculated pressure values. According to the results, the lowest pressure within the network corresponds to the pressure at the supply node. Despite the presence of flow resistance, the gravitational forces—resulting from the difference in density between the gas and the surrounding air—are sufficiently strong not only to offset the linear pressure losses fully, but also to cause pressure recovery at locations situated higher than the supply point. As a result, the lowest pressure value determined in the simulation was 2.60 kPa, corresponding to the supply pressure. The calculated pressures at strategic points A and B were 2.74 kPa and 2.71 kPa, respectively (Figure 4).

To better visualize the pressure recovery observed in the hydraulic simulations, a color-coding scheme was employed in the presentation of the results, where red indicated the lowest pressures and green the highest. This representation clearly shows that the lowest pressure occurs at the medium-pressure reduction and metering station, located at the lowest elevation in the network. In contrast, the highest pressures appear at the far ends of the network, where the consumption points are situated significantly higher than the supply source.

To assess the impact of the elevation of consumption points relative to the supply source, a set of simulation calculations was performed under the assumption of no elevation differences between network nodes. For these simulations, the elevation coordinates of all nodes were set to zero.

Assuming identical network operating parameters as in the baseline scenario, the results clearly show that no pressure recovery occurs when there is no elevation difference between the supply point and the consumption points. Due to the relatively low gas flow rate and the large internal diameters of the pipelines, the calculated results indicate that there is no significant pressure drop. The lowest calculated pressure value under these conditions was 2.59 kPa at point A (Figure 5).

Despite the density difference between the transported gas and ambient air, no pressure recovery was observed because of the absence of elevation variation within the network.

The only noticeable pressure drop under the assumed network operating conditions occurs along the pipeline segment between the pressure source and point A. In Figure 5, the displayed pressure values may give the impression that no pressure drop is present. This is due to the limited resolution of the schematic visualization and the rounding of the presented results.

The pressure level in a gas network is influenced not only by pressure recovery but primarily by flow resistance, which occurs as the medium travels through the pipeline. An increase in the gas flow rate leads to a higher linear flow velocity, which in turn causes greater pressure drops along individual segments of the network. These losses may, in some cases, offset the pressure recovery resulting from elevation differences between the supply source and the point of consumption.

3.2.2. SONET/GASNET

To confirm the possibility of pressure recovery in low-pressure gas networks, additional parallel calculations were performed using the SONET software. The same gas network previously analyzed in SimNet was used for this purpose. As in the previous case, both baseline calculations and simulations with elevation coordinates set to zero were performed.

The hydraulic simulations conducted under conditions identical to those in the previous scenario also demonstrated the occurrence of pressure recovery. Due to the relatively low gas flow rate, the calculated pressure values at the network’s ends were as follows: 2.74 kPa at point A and 2.72 kPa at point B, with a supply pressure of 2.60 kPa. The resulting pressure distribution within the network, as determined by the simulation results, is presented in Figure 6.

Unfortunately, the method of presenting the calculation results is somewhat limited, as it does not allow for precise visualization of the obtained values. However, the key elements relevant to the discussed issue—namely the presentation of the network topology and a simplified representation of pressure variations—are sufficiently clear. Other data displayed in the program window are not essential for the considerations presented in this study.

3.2.3. STANET

Another software tool used in the calculations was STANET. The initial assumptions and boundary conditions were analogous to those applied in the previously discussed programs.

For the network model that included elevation differences between individual nodes, pressure recovery was observed. With a supply pressure of 2.600 kPa, the calculated pressure values at the terminal points of the network were: 2.778 kPa at point A and 2.735 kPa at point B. In contrast, for the flat model, where elevation was not considered, no pressure recovery was observed, and the resulting pressures at the terminal points were 2.589 kPa at point A and 2.595 kPa at point B.

Figure 7 presents the STANET simulation results for the elevation-aware model, while Figure 8 shows the results obtained when elevation effects were excluded from the calculations.

The figures illustrate the methods used to present calculation results for different elements of the network. For nodes, the computed pressure is shown both as a numerical value and as a color corresponding to a specific value range. For pipeline segments, the selected visualization includes the numerical value of the gas flow rate, a color indicating the linear flow velocity, and a line thickness representing the magnitude of the flow rate.

3.3. Calculations Using Empirical Equations

Simulation data, including pipeline length, pipe types, flow rates, and elevation coordinates, were used to perform calculations based on the Renouard and Darcy–Weisbach equations. Calculations were carried out for a network supplied with pure natural gas as well as hydrogen-enriched mixtures, with gas properties specified in Table 2. Two elevation variants were considered: a network model including elevation coordinates (Figure 9) and a flat network (Figure 10).

For both the theoretical Darcy–Weisbach equation (Equation (5)) and the empirical Renouard equation (Equation (6)), identical input data were used: geometric parameters of the pipes (internal diameter, roughness, and length), as well as gas flow rates at individual nodes. The supply pressure was also maintained at the same level as in previous simulations (2.600 kPa).

The calculated pressure values at the terminal nodes of the network were as follows:

For the elevation-aware model:

• Point A: 2.829 kPa (Darcy-Weisbach), 2.778 kPa (Renouard);
• Point B: 2.791 kPa (Darcy-Weisbach), 2.747 kPa (Renouard).

For the flat model (neglecting node elevation):

• Point A: 2.585 kPa (Darcy-Weisbach), 2.588 kPa (Renouard);
• Point B: 2.590 kPa (Darcy-Weisbach), 2.595 kPa (Renouard).

Figure 9 shows the pressure profiles along both main supply paths (AC and BC), including elevation differences. Figure 10 presents the corresponding pressure profiles for the flat network model, excluding elevation effects.

Slight differences can be observed in the obtained calculation results. The greater pressure drop along the AC flow path, as shown in Figure 10, despite its shorter length, results from a higher gas flow rate (or slightly larger internal diameters). A more critical conclusion emerges from a comparison with Figure 9, where a more pronounced pressure recovery is visible. This is attributed to the higher elevation of the terminal node in that part of the network.

The discrepancies between the results obtained using the two equations arise primarily from the assumptions made in the formulation of the equations themselves. The Darcy–Weisbach equation allows for the inclusion of the actual physical properties of natural gas, such as density and viscosity. In contrast, the Renouard equation, being empirical in nature, is based on simplifications and constants without directly accounting for physical variables. As a result, it can be observed that the Renouard equation may, in some cases, lead to an underestimation or overestimation of pressure drops compared to results obtained by other methods.

4. Discussion

This section presents a comparison of the results obtained using different tools, namely, simulation software and analytical formulas. The comparison was conducted based on a real-world example of a low-pressure gas network. In additional scenarios, a network with identical topology, consumer locations, and energy demand profiles was used. In the first additional case, the network was supplied with a significantly higher pressure of 250 kPa. In the second, the impact of adding varying amounts of hydrogen to the gas mixture on the operating conditions of the low-pressure network was analyzed.

Figure 11 presents a comparison between the simulation results and the measured pressure values recorded by devices installed at selected points in the network.

The pressure values shown in Figure 11, both measured and calculated, were obtained under identical gas consumption rates for each consumer. A slight discrepancy is observed between the measured and simulated values, with a maximum difference of 80 Pa. Given that the measured pressures are approximately 2.70 kPa, this corresponds to a relative error of less than 3.0%. When comparing the results presented in Table 3, the calculated values fall within the confidence intervals for each of the measured data points.

4.1. Low-Pressure Network

For the low-pressure network, two scenarios were analyzed: a flat model and a model that accounts for the elevation of individual nodes.

Figure 12 and Figure 13 illustrate the pressure variations along the main supply path (CA) for both the flat network and the model that includes node elevations.

The impact of elevation coordinates on pressure recovery in the low-pressure network was also investigated using simulation software.

The initial simulation replicated the scenarios previously analyzed in SimNet and SONET, evaluating pressure recovery for pure natural gas under both a flat network model and a model that included elevation of individual nodes.

A comparison of the results obtained is presented in Figure 14.

The differences observed in the results are primarily attributed to simulation scenarios that incorporate elevation changes, stemming from the specific calculation algorithms employed in each software. From a physical standpoint, the phenomenon of pressure recovery is related to the density of natural gas and the system of forces acting along the analyzed gas pipeline route. Under static operating conditions of a low-pressure network, the influence of topology may lead to pressure increases in nodes located at higher elevations. At low gas densities, elevation-related differences may be comparable to or even exceed pressure losses. In such cases, local pressure increases can be observed despite no actual increase in the gas energy.

Based on the conducted simulations, it can be inferred that pressure recovery is achievable in low-pressure gas networks, provided that a sufficient elevation difference exists between the supply point and the point of gas consumption.

The results obtained from all three software tools, each employing different computational algorithms, are nearly identical. Minor discrepancies can be observed when comparing pressure values along the paths to points A and B. When the network is calibrated correctly, these differences are minimal, and the calculated pressures at the terminal nodes are practically equal. Slightly different results appear in the flat model, where elevation coordinates are set to zero. In this case, pressure values computed using SONET are somewhat lower than those obtained from SimNet and STANET.

A comparison of the calculated values is presented in

Table 4. Pressure at key nodes—low-pressure network, kPa.

Calculation	Supply Node	Flat Model		Elevation Model
	C	A	B	A	B
Darcy-Weisbach	2.600	2.585	2.590	2.829	2.791
Renouard	2.600	2.588	2.595	2.778	2.747
STANET	2.600	2.589	2.595	2.788	2.735
SimNet	2.600	2.590	2.600	2.740	2.710
SONET	2.600	2.587	2.594	2.740	2.720
Measurement	2.600	—	—	2.788	—

.

An analysis of the obtained results reveals that the calculated values are highly consistent, regardless of the computational method used. When comparing the calculated pressures with the measured values, the maximum discrepancies are less than 50 Pa, which corresponds to less than 2% of the measured pressure.

4.2. Medium-Pressure Network

In the first computational scenario, a significant modification was introduced by increasing the supply pressure to 250.0 kPa (2.50 bar). As a result, the originally low-pressure network becomes a medium-pressure distribution network.

The supply pressure was set at 250.0 kPa. Assuming the same load as used during the calibration of the low-pressure network (93 m³/h), the lowest calculated pressure was 249.48 kPa. No pressure values higher than the supply pressure were observed at any consumption point, indicating that pressure recovery does not occur when the system operates under medium-pressure conditions.

Figure 15 presents the pressure distribution at individual nodes as calculated using SimNet, while Figure 16 shows the corresponding results from STANET.

Table 5. Calculated pressures at key nodes—medium-pressure network (simulation tools and empirical equations), kPa.

Calculation	Supply Node	Flat Model		Elevation Model
	C	A	B	A	B
STANET	250.000	249.997	249.999	249.538	249.660
SimNet	250.000	249.991	249.996	249.478	249.589
Darcy-Weisbach	250.000	249.989	249.993	249.559	249.619
Renouard	250.000	249.970	249.987	249.523	249.574

presents the calculated pressure values at the terminal nodes of the network (points A and B) obtained using the SimNet and STANET software, for both the flat model and the model that accounts for node elevations.

Figure 17 shows the pressure variations along the gas flow path from the supply node (C) to node (B), as determined using both empirical equations and simulation software.

Based on the presented results, it can be concluded that in medium-pressure gas networks, the phenomenon of pressure recovery does not occur. Furthermore, analysis of the values shown in Table 5 reveals a trend opposite to that observed in low-pressure networks. In the model that includes node elevation, the pressure drop is greater than in the flat model.

This effect is caused by the higher pressure levels in the gas network, which influence the absolute gas density. At low pressures, natural gas is lighter than air (see Table 2); however, as the pressure increases, the gas density also increases. At a certain pressure threshold, natural gas becomes denser than the surrounding ambient air (Figure 18).

The simulation analyses demonstrated that, after increasing the supply pressure of the gas network, pressure recovery does not occur, even in the presence of a significant elevation difference.

4.3. Mixture of Natural Gas with Hydrogen

The considerations presented in this section are hypothetical, due to existing regulatory constraints such as the Regulation of the Minister of Economy of 2 July 2010, which limits permissible changes in the composition of natural gas transported through gas networks in Poland [36], as well as the Transmission Network Code defined by the Transmission System Operator [37]. Similar restrictions also arise from Polish standards that specify the required quality of gas transported through the gas infrastructure [38,39]. Additionally, it is worth noting that mixtures containing more than 10% hydrogen (H₂) fail to meet a key parameter required for replacing natural gas: the Wobbe index. Maintaining the Wobbe index within the regulated range specified in the legislation ensures combustion conditions that do not pose a threat to human health or the environment, while also ensuring the safe operation of gas appliances, flame stability, and complete combustion [40].

The purpose of these simulations and calculations is to illustrate the changes in physicochemical and hydraulic parameters that occur when natural gas is blended with hydrogen. The fundamental assumption is that hydrogen is perfectly mixed with natural gas—the resulting gas mixture is treated mas a homogeneous fluid. This simplification is justified by the fact that the STANET software and the empirical equations used in the analysis do not account for dynamic stratification or local turbulent phenomena. Such an approach is consistent with typical engineering practice applied in the design and operation of gas networks.

An increase in the hydrogen content in the gas mixture reduces the absolute density of the blend, which in turn leads to greater pressure recovery resulting from elevation differences between nodes in the network. In contrast, increasing hydrogen content in a flat network model shows the opposite trend: the more hydrogen injected, the greater the pressure losses at the network’s endpoint.

These relationships can be explained by analyzing the pressure drop equations used in the STANET program. When elevation coordinates are set to zero, the gravitational term, which accounts for the actual gas density changing based on physical conditions, is eliminated. As a result, the model no longer reflects the elevation-induced pressure effects. The remaining terms in the equation, based on gas density under standard conditions, become insensitive to changes in the gas’s physicochemical properties under actual network conditions.

In this case, pipeline parameters—such as diameter, length, and roughness—become the dominant factors influencing pressure losses through their effect on friction coefficients. This explains the consistent pressure trends observed in flat-network simulations, where pressure recovery does not occur and the final pressure value becomes directly proportional to the gas density under standard conditions.

An additional constraint is the assumption of a constant amount of energy that must be delivered to all consumers. As the hydrogen fraction increases, the heating value of the gas mixture decreases, due to the significantly lower heating value of pure hydrogen. Thus, to maintain the same delivered energy, a higher volumetric flow rate is required.

Superimposed on these two key parameters are changes in gas viscosity coefficients. Therefore, it is not possible to define the impact of increasing hydrogen content on pressure conditions within a gas distribution network unequivocally or arbitrarily.

Figure 19, Figure 20 and Figure 21 present pressure variation profiles along the flow path toward node A, obtained using different computational tools.

In the analyzed system, it can be observed that an increase in hydrogen content leads to an increase in pressure recovery. This indicates the dominant influence of elevation on the calculated pressure values, which outweighs the adverse effect of the increased gas flow rate (required to deliver a constant amount of energy to the consumer). However, this relationship should not be considered universal.

The pressure plots comparing hydrogen–natural gas mixtures at various blending ratios, evaluated under both elevation scenarios, along with the equation-based analysis, further demonstrate the importance of accounting for elevation when calculating pressure variations in a low-pressure pipeline. As the hydrogen content in the mixture increases, the pressure difference at point A grows—from 0.17 kPa for pure natural gas to nearly 0.28 kPa for a 50/50 natural gas–hydrogen blend.

Regardless of the gas composition and computational method used, in the flat model, no pressure recovery is observed; only a pressure drop is observed, which increases in magnitude as the hydrogen fraction increases (Figure 20). This trend is expected and aligns with the simulation results. The differences between the pressures at point A calculated using both equations do not exceed 7 Pa.

When elevation (Z coordinate) is included (Figure 21), the pressure recovery calculated using Renouard’s equation is directly proportional to the hydrogen content in the mixture. In contrast, when using the Darcy-Weisbach equation, an inversely proportional trend is observed, matching the pattern obtained for the flat network.

This inverse relationship between pressure recovery and hydrogen content contradicts the general assumption that a lower-density gas should yield greater pressure recovery. On the other hand, in the Darcy-Weisbach equation, the coefficient associated with flow resistance is calculated based on the Reynolds number, which depends, among other factors, on the dynamic viscosity and absolute density of the gas. The dynamic viscosity of the analyzed gases decreases as the hydrogen content in the natural gas mixture increases, which results in the “deviation” observed in Figure 21. In contrast, the Renouard equation used in the calculations is a simplified empirical formula. The only variable that can be modified—and that depends on the physicochemical properties of the analyzed gas—is the relative density, which, as previously mentioned, decreases proportionally with the increase in H₂ content. This leads to an apparent increase in pressure recovery with increasing hydrogen content, which—as it turns out—is a misleading phenomenon.

5. Conclusions

Based on the analyses conducted using three simulation programs and data from pressure loggers, it has been demonstrated that under appropriate conditions, pressure recovery occurs in low-pressure gas networks. For this phenomenon to arise, two conditions must be met: a difference in elevation between the pressure source and the consumption point, and a density difference between the gas in the pipeline and the surrounding air.

The following conclusions can be drawn from the study:

• In low-pressure networks, pressure recovery does occur. Under conditions of low linear pressure losses resulting from gas flow, this can lead to increased pressure in sections of the network located farther from the supply node. A necessary condition is an appropriate network configuration, specifically an increase in node elevation with increasing distance from the supply point.
• In medium-pressure networks, pressure recovery does not occur, even when node elevations increase significantly. Moreover, comparing results from the flat model and the elevation-aware model reveals that pressure drops are noticeably greater in the elevation model. This is due to the much higher actual gas density under typical operating conditions in medium-pressure networks, where the gas becomes heavier than the surrounding air. As a result, one of the key conditions for pressure recovery is no longer met.
• Blending hydrogen with natural gas further reduces the density of the gas mixture, which enhances pressure recovery in low-pressure networks.

The most accurate and conclusive evidence of pressure recovery was obtained by comparing simulation results with measured data from the same network system. The installation of pressure loggers not only confirmed the occurrence of pressure recovery in the low-pressure network but also revealed that the measured recovery was greater than that predicted by simulation.

These simulations demonstrate that neglecting elevation in the modeling of low-pressure networks can be a significant source of error. This simplification may lead to the inaccurate determination of pressure at individual nodes, resulting in misinterpretation of system behavior and potentially incorrect operational decisions.