Energy Efficiency, Local Entropy Sources and Exergy Analysis in Measuring Orifice Plates: A Computational Fluid Dynamics Approach

Abstract

Accurate flow measurement is crucial for energy efficiency in industrial applications. This study investigates entropy generation in measuring orifice plates under high-pressure conditions (80 bar, 400 °C) using computational fluid dynamics (CFD) in OpenFOAM. Two turbulence models, k-ω SST and Spalart–Allmaras, are employed to analyze compressible steam flow and identify local entropy sources. Building on recent findings, this research explores the hypothesis that the discharge coefficient reflects entropy generation. The orifice plate’s abrupt flow contraction and expansion contribute to significant energy dissipation, affecting exergy efficiency. By quantifying entropy sources through numerical simulations, this study provides insights into optimizing flow metering techniques and reducing irreversibilities. The results show a strong correlation between entropy generation and the discharge coefficient, offering a new approach to improving measurement accuracy. This research supports the advancement of energy-efficient flow measurement methods, aligning with sustainable engineering practices.

1. Introduction

The efficient use of energy has long been of interest to researchers. The issue is gaining increasing attention in previously overlooked areas of energy use and conservation. Although the concept of “energy” is widely known, its exact definition remains controversial and ambiguous [1]. Thinkers such as Leibniz and Descartes attempted to define it as a “quantity of motion” [2]. Another definition suggests that energy is “the first integral of the equations of motion of a physical system”. Defining energy as a quantity of motion becomes meaningless when considering internal energy, which depends on temperature and not on the quantity of motion measured macroscopically. Because of imperfections in the description of the physical world, energy, despite its fundamental role, has not been subject to a coherent definition. However, its various forms have been accurately characterized both mathematically and physically. The lack of a clear definition of energy has fundamental implications for energy analysis. The division of energy into kinetic energy, potential energy, enthalpy and entropy clearly suggests that the lack of a clear definition will lead to further proposals to disambiguate the concept.

Energy is usually considered a scalar quantity that defines the ability of a physical system to do work. During work, energy is transformed in both form and quality. This process results in irreversible losses, which ultimately manifest themselves as heat. The amount of heat produced is described by the second law of thermodynamics, which is related to the degree of order of the physical system. Therefore, the concept of exergy was introduced to provide a more precise description of energy and its use. This concept was initially presented as a type of energy that can be destroyed [3] and was applied to thermal systems operating in steady-state conditions [4]. The resulting energy savings showed that exergy could also be applied to smaller appliances and equipment once they had been broken down and the component parts treated as separate thermal machines [5]. Over time, the concepts of entropy and exergy have evolved to include non-equilibrium processes, and the mathematical apparatus has enabled these concepts to be applied to local analysis during fluid flow [6]. Exergy quantifies the quality of energy by determining how much energy can be converted into useful work under given environmental conditions. This distinction is crucial from a practical point of view because no physical system is completely isolated.

Given the increasing need to consider energy efficiency in new domains, this study focuses on a quantitative analysis of energy flow through measurement components and the portion of energy lost due to the second law of thermodynamics. In this context, the following statement is relevant: “... epochs differ not by what is produced, but by how it is produced and by what means of labour” [7]. This article presents an energy and exergy analysis in an unexplored area related to energy quality losses in measurement equipment used in industrial applications. Specifically, this study examines flow meters based on differential pressure measurements.

Differential pressure flow meters are widely used in industry to measure fluid flow. These devices create a pressure difference by altering the flow cross-section, which results in a velocity change according to Bernoulli’s principle [8]. A commonly used type of flow meter is an orifice plate—a perforated plate containing one or more holes. Numerical and experimental analyses of multi-hole orifice plates are presented in [9]. This research is significant for comparing turbulence models in measurement applications. Another notable study is [10], where the author numerically compares pressure drops and mass flow rates obtained for single-hole and slotted orifice plates. However, the study lacks clarification on whether the analysis was transient or stationary, a crucial factor regarding the applied equations. A comprehensive comparative analysis of multi-hole and single-hole orifice plates can be found in [11], where the authors conducted numerical and experimental investigations, focusing primarily on measurement parameters. Similar topics are explored in [12], though the study is restricted to numerical methods.

Alternative solutions include Venturi tubes and measurement nozzles, each with advantages and limitations. Measurement accuracy depends on proper device calibration. This study analyzes a single-hole orifice plate [13].

The primary objective of this research is to assess the quality (exergy) and quantity of energy lost in an orifice plate. As mentioned, orifice plates have been extensively studied regarding flow phenomena using numerical and experimental methods; however, energy measurement aspects remain outside mainstream research. Exceptions include [14], where the author focuses on turbulence kinetic energy analysis to determine losses in flow measurements. Previous studies have analyzed flow streams concerning heat exchange [15] and entropy generation [16,17]. These studies are closely related to the present research through their methodology for identifying local entropy sources. Similar issues are addressed in [18,19], where the authors examine practical applications in combustion chamber analysis [19] and steam turbines [18]. A detailed entropy generation analysis is presented in [20], which describes entropy sources within various sections of an aircraft engine turbine.

Some analytical studies are reported in [21,22], where the simplest geometries and laminar flows were analyzed in the context of entropy generation. From a computational fluid dynamics perspective, a key limitation of these studies is the absence of three-dimensional models. The significance of incorporating the third dimension will be demonstrated in subsequent sections of this article.

Energy and exergy analysis can be conducted on both global and local scales. This article presents two approaches: a global exergy analysis based on flow parameter values at the inlet and outlet and a local analysis based on identifying local entropy sources.

The author has already conducted a global analysis from a slightly different perspective for transient and compressible phenomena, with results presented in [23,24]. These analyses relied on numerical simulations and Reynolds’ transport theorem for specific energy [25].

Determining the energy and exergy of gases is rarely considered in the literature, particularly for compressible and unsteady flows. Previous studies have simplified geometry using two-dimensional models. While this reduces computational costs, it fails to capture flow phenomena in fluids fully. Furthermore, this research has unique industrial applications. A literature review indicates that analytical models do not accurately represent the flow complexity in measurement nozzles. Additionally, it has been shown that steady-state models lack accuracy and fail to reflect real-world phenomena. Notably, energy analysis has not been the subject of such studies. This work fills a research gap regarding unsteady compressible flows in measurement orifices.

2. Physical and Numerical Model

2.1. Assumptions

The results presented in this paper are derived from numerical calculations performed on a flow model of a measuring orifice. The schematic representation of the orifice is shown in Figure 1. The design and dimensions adhere to the ISO standard [13] for monolithic orifices. The flow conditions replicate those commonly applied in the chemical industry, including a broad range of variable parameters. The working fluid considered is steam, characterized by an average velocity of $\overline{U}=7\mathrm{m}/\mathrm{s}$ and a corresponding mass flow rate of $\dot{m}=16.763\mathrm{kg}/\mathrm{s}$. The nozzle is installed in a pipe with a nominal diameter DN 300, which corresponds to an internal diameter of $D=323.6\mathrm{mm}$. The contraction of the orifice, defined as the pipe diameter to the nozzle throat diameter ratio, equals $\beta =d/D=0.6$. Temperature and pressure are also critical factors in the physical model. Given the application-oriented nature of the simulations, this study assumes high steam pressure of p = 8 × 10⁶ Pa and a temperature of T = 400 °C = 673.16 K.

2.2. Geometry and Numerical Mesh

The orifice shown in Figure 1 is the basis of the numerical flow model geometry. The orifice structure is symmetrical along the pipeline axis in which it is installed. This symmetry was initially utilized in the computational mesh. However, errors during the preliminary simulations prevented further progress with the simplified geometry. Consequently, a fully three-dimensional model was employed, shown in Figure 2.

The numerical mesh has been generated to reflect fundamental entropy and exergy phenomena. One of the key elements is the determination of the dimensionless distance of the first grid cell from the wall $y^{+}=U_{\tau }y/\nu$. This quantity depends on the friction velocity of the fluid at the wall $U_{\tau }$, its viscosity $\nu$ and the distance of the center of the cell element from the wall y. Depending on the scope and accuracy of the analysis, $y^{+}\approx 1$ is used for accurate analyses with no approximations in the behavior at the wall, or $y^{+}=30÷500$ if wall functions can model the behaviour of the fluid at the wall. In the present study, the dimensionless wall distance reached values of y⁺ =3.7 × 10⁻⁵ at the wall of the orifice inlet and maximum values of y⁺ =5 × 10¹ at the inner walls of the orifice, with average values of $y^{+}=0.99÷13.10$. The wall functions are applied to the boundary conditions for the turbulence parameters, i.e., k, $\omega$, ${\nu }_t$.

The computational mesh is conditioned, on the one hand, to maintain the accuracy of the calculations and, on the other hand, not to exceed the limit of using another model to solve the flow equations. This is when, instead of Reynolds averaging, local eddies modeled by Large Eddy Simulation or even Direct Numerical Simulation become relevant in some areas. For the above reasons, the numerical mesh presented is a compromise between computational accuracy and computational cost for conditions used in industry.

2.3. Thermophysical Properties of the Fluid

Thermodynamic models describe variations in fluid properties during thermodynamic gas conversion. Numerical simulations were conducted using the OpenFOAM CFD toolbox [26], where thermodynamic gas models (and other fluid models) are constructed based on a pressure–temperature system. The gas density, a key variable in the governing equations, is determined from these parameters, which is particularly relevant for the considered compressible flow. The working medium is steam, modeled using the ideal gas equation. The thermal conductivity of the fluid is derived from the Prandtl number, while other physical properties are assumed constant at a temperature of T = 400 °C and pressure of p = 8 × 10⁶ Pa as follows [27]:

• Fluid density: $\rho =p/\left(R·T\right)$,
• Fluid thermal conductivity: $\kappa =c_p·\mu /\mathrm{Pr}$,
• Dynamic viscosity: μ = 2.4465 × 10⁻⁵ Pa · s,
• Specific heat capacity at constant pressure: $c_p=2804\mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)$,
• Heat of fusion: $H_f=2.551\mathrm{MJ}/\mathrm{kg}$,
• Prandtl number: $\mathrm{Pr}=1.05355$,
• Specific gas constant: $R=R_U/M$,
• Universal gas constant: $R_U=8.31446\mathrm{J}/\left(\mathrm{mol}·\mathrm{K}\right)$,
• Molar mass: $M=18.0153\mathrm{kg}/\mathrm{kmol}$.

2.4. Governing Equations

The calculations were executed by utilizing the finite volume method, a computational approach that involves the discretization of the equations governing the conservation of mass, momentum and energy. For a single cell within the designated computational domain, the sum of the incoming and outgoing fluxes must be zero under the condition that no net mass is created or destroyed during the flow process. In the case under consideration, the absence of chemical or nuclear reactions ensures that the net mass flux is zero at both the inlet and outlet. Given the transient and compressible nature of the system, the mass balance must take into account the change in density over time and the effect of density on mass flux, as represented by the continuity equation, which establishes the relationship between density change and mass flux over time:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho U_i\right)=0$$

(1)

where t denotes time, $x_i$ is a coordinate and $U_i$ is a velocity vector.

The forces acting on a fluid element in the computational cell result in deformations, which induce stresses in this element. Evaluating these deformations and stresses is possible using the momentum equilibrium (i.e., the momentum conservation law). This equilibrium includes the forces from shear and normal stresses in the fluid element. Mass forces (i.e., gravity) are not included in the equation below due to their negligible influence on the overall phenomena. The contribution of mass forces is influential in situations where significant temperature gradients in the flow are being considered, resulting in a significant influence of convection on the phenomena within the fluid. The effects of mass forces have a strong influence on flows with low Reynolds number values. In flows being considered, where the Reynolds number is of the order of ${10}^6$, neglecting the influence of gravitational forces in gases over short flow distances does not introduce a calculation error. Within the context of the local conditions and the flow under consideration, the aforementioned equation assumes the following form:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho U_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j{U}_i\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left({\tau }_{ij}\right)$$

(2)

where ${\tau }_{ij}$ is the stress tensor that acts on the fluid element and is described by the following equation:

$${\tau }_{ij}={\mu }_{eff}\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial U_k}{\partial x_k}}\right)$$

(3)

where ${\mu }_{eff}$ is effective dynamic viscosity and ${\delta }_{ij}$ is a Kronecker delta (unit tensor). Effective dynamic viscosity is a sum of molecular viscosity $\mu =\rho \nu$ and turbulent dynamic viscosity ${\mu }_t=\rho {\nu }_t$, where $\nu$ and ${\nu }_t$ are kinematic viscosities, molecular and turbulent, respectively.

In order to determine entropy and exergy, it is necessary to consider the energy balance in conjunction with mass and momentum balances. In the case of a computational cell, the energy flux entering the cell is amplified by thermal and mechanical energy fluxes, with the sum of these energy fluxes being equal to the remaining energy flux leaving the cell (see below for a mathematical expression of this relationship in Equation (4)). As with the conservation of momentum, gravitational forces have been neglected. In the computational cell, the incoming energy flux is enhanced by the contributions of thermal and mechanical energy fluxes, indicating that their combined total matches the outgoing energy flux leaving the cell. This relationship is expressed through an energy balance, accounting for both inflows and outflows of energy.

$${\displaystyle \frac{\partial \left(\rho h\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho U_{i}h\right)+{\displaystyle \frac{\partial \left(\rho K\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho U_{i}K\right)-{\displaystyle \frac{\partial p}{\partial t}}=-{\displaystyle \frac{\partial q_i}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_i}}\left({\tau }_{ij}{U}_i\right)$$

(4)

where $h=e+p/\rho$ is specific enthalpy, $e=c_{p}T$—specific internal energy, $K={\displaystyle \frac{1}{2}}{U}_i{U}_i$—specific kinetic energy and $q_i$—heat flux density.

The aforementioned equation (Equation (4)) is employed to balance thermal energy in terms of specific entropy whilst also accounting for the magnitude of the conducted heat flux in the fluid. The value of this flux is determined by an equation representing Fourier’s law:

$$q_i=-{\kappa }_{eff}{\displaystyle \frac{\partial T}{\partial x_i}}$$

(5)

where ${\kappa }_{eff}$ is the effective thermal conductivity of the medium, and it is a sum of fluid thermal conductivity and turbulent thermal conductivity ${\kappa }_{eff}=\kappa +{\kappa }_t$. Whereas fluid thermal conductivity is derived from the Prandtl number, the turbulent thermal conductivity is derived from the turbulent Prandtl number ${\mathrm{Pr}}_t=\rho c_p{\nu }_t/{\kappa }_t$. In the cases studied, the turbulent Prandtl number was assumed constant ${\mathrm{Pr}}_t=0.85$.

The above system of equations was solved numerically using the finite volume method. The equations were discretized using an appropriate discretization scheme. A second-order linear Gaussian scheme was used for the spatial discretization. For the temporal discretization, a second-order Crank–Nicolson scheme [28] was used to handle transient steam flow.

These equations are solved analytically only in the most uncomplicated cases, and numerical methods are required for cases describing industrial problems. An additional complication arises from these equations being valid for laminar flows or flows where the length and time scales follow the Kolmogorov hypothesis. The present case achieves a Reynolds number of Re = 2.7 × 10⁶, and more than 12.48 × 10¹² numerical cells are required to represent the Kolmogorov scale accurately. However, such a number of elements is computationally expensive and would take years to compute. The solution is to use turbulence models that provide relatively reliable calculations in a reasonable time.

Two turbulence models describing turbulent flows were used to perform the calculations:

• Wilcox’s modified k-ω SST 2006 Shear Stress Transport model [29]. This model is one of the two-equation turbulence models, adding to the balance equations presented in Section 2.4, the transport equation for turbulent kinetic energy k and the equation for the specific dissipation rate of the turbulent kinetic energy $\omega$. In index notation, these equations are expressed as follows:

  −

  turbulent kinetic energy k:

  $${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho U_{j}k\right)}{\partial x_j}}=P_k-{\beta }^{*}\rho k\omega +{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\rho \nu +{\sigma }_k\rho {\nu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

  (6)

  −

  turbulent kinetic energy specific dissipation rate $\omega$:

  $${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho U_j\omega \right)}{\partial x_j}}=\alpha {\displaystyle \frac{\omega }{k}}{P}_k-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\sigma }_{\omega }\rho {\nu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]$$

  (7)

  There are experimentally determined constants in the above equations. Their values are as follows: $\alpha ={\displaystyle \frac{13}{25}}$, $\beta ={\beta }_0{f}_{\beta }$, ${\beta }^{*}={\beta }_0^{*}{f}_{{\beta }^{*}}$, ${\sigma }_k=0.6$, ${\sigma }_{\omega }=0.5$, ${\beta }_0=0.0708$, ${\beta }_0^{*}=0.09$.

  The following augmentations are incorporated:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{\beta }& ={\displaystyle \frac{1+70\chi \omega }{1+80\chi \omega }},f_{{\beta }^{*}}=\left\{\begin{array}{cc}1,& {\chi }_k\le 0\\ {\displaystyle \frac{1+680\chi k^2}{1+80{\chi }_k^2}},& {\chi }_k>0\end{array}\right\},{\chi }_{\omega }=\left|{\displaystyle \frac{{\Omega }_{ij}{\Omega }_{jk}{S}_{ki}}{{\left({\beta }_0^{*}\omega \right)}^3}}\right|,\hfill \\ \hfill {\chi }_k& \equiv {\displaystyle \frac{1}{{\omega }^3}}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{\Omega }_{ij}\equiv {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial U_j}{\partial x_i}}-{\displaystyle \frac{\partial U_i}{\partial x_j}}\right),\hfill \\ \hfill & S_{ij}\equiv {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial U_j}{\partial x_i}}+{\displaystyle \frac{\partial U_i}{\partial x_j}}\right),P_k={\nu }_t\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right){\displaystyle \frac{\partial U_i}{\partial x_j}}-{\displaystyle \frac{2}{3}}\rho k{\displaystyle \frac{\partial U_i}{\partial x_j}}.\hfill \end{array}\end{array}$$

  (8)

• Spalart–Allmaras turbulence model [30,31]. This one-equation model adds a turbulent viscous eddy kinetic transport equation to the balance equations listed in Section 2.4. It was used initially in aeronautical areas but has also been positively validated in compressible flows in confined channels such as stream measurements. In index notation, this equation is as follows:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\rho \tilde{\nu }\right)& +{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\tilde{\nu }\right)={\displaystyle \frac{1}{{\sigma }_{{\nu }_t}}}{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho (\nu +\tilde{\nu }){\displaystyle \frac{\partial \tilde{\nu }}{\partial x_j}}\right)+{\displaystyle \frac{C_{b2}}{{\sigma }_{{\nu }_t}}}\rho {\displaystyle \frac{\partial \tilde{\nu }}{\partial x_j}}{\displaystyle \frac{\partial \tilde{\nu }}{\partial x_j}}\hfill \\ \hfill & +C_{b1}\rho \tilde{S}\tilde{\nu }\left(1-f_{t2}\right)-\left(C_{w1}{f}_w-{\displaystyle \frac{C_{b1}}{{\kappa }^2}}{f}_{t2}\right)\rho {\displaystyle \frac{{\tilde{\nu }}^2}{{\tilde{d}}^2}}+f_{t1}\Delta U^2\hfill \end{array}\end{array}$$

  (9)
  In this equation, the additional functions are as follows:

  $$\begin{array}{cc}\hfill {\nu }_t& =\tilde{\nu }{f}_{v1},f_{v1}={\displaystyle \frac{{\chi }^3}{{\chi }^3+C_{v1}^3}},\chi ={\displaystyle \frac{\tilde{\nu }}{\nu }},\tilde{S}\equiv S+{\displaystyle \frac{\tilde{\nu }}{{\kappa }^2{d}^2}}{f}_{v2},f_{v2}=1-{\displaystyle \frac{\chi }{1+\chi f_{v1}}}\hfill \\ \hfill f_w& =g{\left[{\displaystyle \frac{1+C_{w3}^6}{g^6+C_{w3}^6}}\right]}^{1/6},g=r+C_{w2}(r^6-r),r\equiv {\displaystyle \frac{\tilde{\nu }}{\tilde{S}{\kappa }^2{d}^2}}\hfill \\ \hfill f_{t1}& =C_{t1}{g}_{t}exp\left(-C_{t2}{\displaystyle \frac{{\omega }_t^2}{\Delta U^2}}[d^2+g_t^2{d}_t^2]\right),f_{t2}=C_{t3}exp\left(-C_{t4}{\chi }^2\right),S=\sqrt{2{\Omega }_{ij}{\Omega }_{ij}}\hfill \end{array}$$

  where rotation tensor ${\Omega }_{ij}$ is defined by Equation (8). Other constants appearing in Equation (9): ${\sigma }_{{\nu }_t}=2/3$, $C_{b1}=0.1355$, $C_{b2}=0.622$, $C_{w1}={\displaystyle \frac{C_{b1}}{{\kappa }^2}}+{\displaystyle \frac{1+C_{b2}}{{\sigma }_{{\nu }_t}}}$, $C_{w2}=0.3$, $C_{w3}=2$, $C_{v1}=7.1$, $\kappa =0.41$.

2.5. Solver Validation

The above turbulence models were not chosen without reason. Both turbulence models were validated against the model flow in the paper [32]. The values of the velocity profiles obtained with the k-ω SST 2006 model and the Spalart–Allmaras turbulence model were closest to those obtained experimentally. This work was chosen as an experimental validation because of the high velocity of the gas flow and the abrupt change in the channel cross-section. Such conditions are present in the flow through the measuring orifice.

Solver validation was carried out using the velocity profiles obtained from the simulations. They are shown in Figure 3a and show good convergence of the velocity profiles in the k-ω SST 2006 model and relatively good in the standard k-ε model. The local entropy flux values also depend on the value of the viscous dissipation function, which can be well visualized from the stress tensor and the Reynolds stress tensor. In the comparison shown in Figure 3b, the best convergence is with the results for the Spalart–Allmaras model. Worse for the other two. For this reason, the choice of turbulence models was made and fell on the k-ω SST 2006 and Spalart–Allmaras models.

2.6. Boundary Conditions

The boundary and initial conditions determine the phenomena’s nature and the flow characteristics through the orifice. In the case under consideration, these are derived from conditions prevailing in industrial applications. The assumption of an adiabatic wall has been used so that the large unknown in the form of the quality of the pipe wall insulation is not taken into account. Calculations were carried out using boundary conditions for the steady-state and transient cases in both turbulence models. The results obtained using the steady-state conditions were, as it were, the initial conditions for the simulations using the transient flow conditions. These conditions for all the simulations run are presented below, broken down into inlet and outlet conditions:

• Inlet boundary conditions:

  −

  pressure (transient calculations): $n_j{\displaystyle \frac{\partial p}{\partial x_j}}=0$,

  −

  pressure (steady-state calculations): $n_j{\displaystyle \frac{\partial p}{\partial x_j}}=0$

  −

  velocity $U=7\mathrm{m}/\mathrm{s}$ at mass flow rate $\dot{m}=16.763\mathrm{kg}/\mathrm{s}$,

  −

  temperature: $T=673.16\mathrm{K}$,

  −

  turbulent kinetic energy: $k=0.05163\mathrm{J}/\mathrm{kg}$,

  −

  specific turbulence dissipation rate: $\omega =36.01/\mathrm{s}$.

• Outlet boundary conditions:

  −

  pressure (transient calculations): ${\displaystyle \frac{\partial p}{\partial t}}+U_i\left(n_j{\displaystyle \frac{\partial p}{\partial x_j}}\right)={\displaystyle \frac{U_i}{l_0}}\left(p_0-p\right)$

  −

  pressure (steady-state calculations): p = 8 × 10⁶ Pa

  −

  velocity: $n_j{\displaystyle \frac{\partial U_i}{\partial x_j}}=0$

  −

  temperature: $n_j{\displaystyle \frac{\partial T}{\partial x_j}}=0$

  −

  turbulent kinetic energy: $n_j{\displaystyle \frac{\partial k}{\partial x_j}}=0$,

  −

  specific turbulence dissipation rate: $n_j{\displaystyle \frac{\partial \omega }{\partial x_j}}=0$

As observed, most boundary conditions are of Dirichlet type ($\varphi =const$) or Neumann type ($n_j{\displaystyle \frac{\partial \varphi }{\partial x_j}}=0$), where the domain boundary is oriented by the normal vector $n_j$. However, these two conditions can lead to hydraulic wave reflections for compressible transient flows. To mitigate this effect, a Robin-type boundary condition [33] was applied to the pressure. The OpenFOAM toolbox [26,34] incorporates this condition within its wave transmissive setting, where the outlet pressure is determined using the equation described earlier, based on the assumed far field parameters: the distance of the far field from the boundary is $l_0=5\mathrm{m}$, and the far field pressure is p₀ = 8 × 10⁶ Pa.

2.7. Energy

The energy entering a system and any alterations occurring within the designated control volume must be equivalent to the energy departing from the system. In accordance with the principle of energy conservation, energy is neither created nor annihilated, although it has the capacity to transition from one form to another. The rate of energy variation within the control volume is accurately depicted in integral form. At any particular instance and for a defined control volume, the comprehensive energy flow rate consists of both the rate of energy variation and the net energy flux traversing the system boundaries. Mathematically, this correlation is articulated by the First Law of Thermodynamics (FLT) for thermodynamic systems, which aligns with the Reynolds Transport Theorem for specific energy [25]:

$$\dot{E}={\displaystyle \frac{\partial }{\partial t}}{\int }_V\rho \left(e+{\displaystyle \frac{1}{2}}{U}_i{U}_i\right)\mathrm{d}V+{\oint }_A\rho \left(e+{\displaystyle \frac{1}{2}}{U}_i{U}_i\right)U_i{n}_i\mathrm{d}A,$$

(10)

where $\dot{E}$ denotes energy rate, V is control volume and $n_i$ is normal vector to the boundary area A. The last term in Equation (10) $U_i{n}_i\mathrm{d}A$ is volumetric flow rate through infinitesimal boundary area $\mathrm{d}A$. The initial integral in Equation (10) holds no significance in the absence of chemical or nuclear reactions, meaning that internal heat sources are non-existent. The flow of water vapor is classified as adiabatic, and under these assumptions, the First Law of Thermodynamics (FLT) is derived as follows:

$$\dot{E}={\oint }_A\rho \left(e+{\displaystyle \frac{1}{2}}{U}_i{U}_i\right)U_i{n}_i\mathrm{d}A.$$

(11)

Substituting the elementary mass flow rate derived from the law of conservation of mass Equation (1):

$$\mathrm{d}\dot{m}=\rho U_i{n}_i\mathrm{d}A,$$

(12)

and integrating the surface integral Equation (11), one can obtain an energy rate dependent from mass flow rate, which is more convenient for the present study.

The boundary area A is limited to inlets and outlets because of the impermeability of nozzle walls. The values have been area-weighted averaged to predict total energy rates, and the FLT is:

$$\dot{E}=\dot{m}\left(\overline{e}+{\displaystyle \frac{1}{2}}\overline{U_i{U}_i}\right),$$

(13)

2.8. Exergy

Exergy, as previously elucidated, quantitatively signifies the utmost energy that is capable of being transformed into work. The system under examination, namely the nozzle, constitutes an open system wherein no work is executed. The gas will only perform work subsequent to its exit from the system. The lack of work conducted by the gas is a crucial prerequisite for the exergetic analysis of the transition state [35]. This analysis seeks to ascertain the extent of exergy destruction engendered by the passage of the gas solely through the nozzle. By incorporating all input and output streams in a comprehensive evaluation, the quantity of exergy that is destroyed can be quantified. The exergy that is annihilated is directly proportional to the entropy that is produced. The exergy is irreversibly lost, either partially or entirely; this assertion is a fundamental tenet of the Second Law of Thermodynamics. The exergy that is destroyed or the entropy that is generated is accountable for the thermodynamic efficiency of the nozzle falling short of the theoretical expectations [36].

In an open system, the exergy rate associated with a flowing mass constitutes a synthesis of the non-flow exergy and the exergy resultant from the alterations in the parameters of the flowing gas. The exergy flow rate can be categorized into physical, chemical, kinetic and potential exergy, as delineated in [37]. The transit of steam through the nozzle does not entail chemical or nuclear reactions, and the nozzle itself does not experience positional changes. As a result, the potential and kinetic exergy of the system are maintained at a constant state. Only the physical exergy necessitates consideration, which is influenced by variations in pressure and temperature. The physical exergy rate in $\left[\mathrm{W}\right]$ is represented as follows:

$$\dot{X}=\dot{m}\left[(h-h_0)-T_0(s-s_0)\right],$$

(14)

where $\dot{X}$ denotes exergy rate, h is specific enthalpy, s is specific entropy and index ₀ points to environmental (reference) values. Physical exergy can be split apart at temperature-dependent and pressure-dependent ones:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{X}& ={\dot{X}}_T+{\dot{X}}_p\hfill \\ \hfill & {\dot{X}}_T=\dot{m}{\left[(h-h_0)-T_0(s-s_0)\right]}_{p=const}{|}_{T_0}^T\hfill \\ \hfill & {\dot{X}}_p=\dot{m}{\left[(h-h_0)-T_0(s-s_0)\right]}_{T=const}{|}_{p_0}^p.\hfill \end{array}\end{array}$$

(15)

Combining Equation (15) with the equation of state presented in Section 2.3, the physical exergy is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\dot{X}}_T=\dot{m}{c}_p\left(T-T_0-T_{0}ln{\displaystyle \frac{T}{T_0}}\right)\hfill \\ \hfill & {\dot{X}}_p=\dot{m}R{T}_{0}ln{\displaystyle \frac{p}{p_0}}\hfill \\ \hfill \dot{X}& =\dot{m}{c}_p\left(T-T_0-T_{0}ln{\displaystyle \frac{T}{T_0}}\right)+\dot{m}R{T}_{0}ln{\displaystyle \frac{p}{p_0}}.\hfill \end{array}\end{array}$$

(16)

Total physical exergy is calculated according to aforementioned equation at the outlet cross-section of the nozzle. Quantities included in Equation (16) are area-averaged in space to conveniently proceed exergy analysis.

2.9. Entropy Generation

The method presented in the previous Section 2.8 allows the exergy value to be determined from global flow parameters such as mass flow, temperature and pressure. It is also possible to determine the entropy within the flow from values obtained during numerical simulations. The entropy generated during flow through the nozzle is a function of two mechanisms: viscous and/or turbulent dissipation and heat dissipation. Both mechanisms for fluid flows have been described mathematically in the literature [6,38,39].

Entropy sources are mainly functions of gradients: velocity and temperature. The local flow rate values of the generated entropy are as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{s}}_{gen}=& {\dot{s}}_{\tau }+{\dot{s}}_T\hfill \\ \hfill & {\dot{s}}_{\tau }={\displaystyle \frac{{\mu }_{eff}}{T}}\left({\displaystyle \frac{1}{{\mu }_{eff}}}\underline{\mathrm{T}}:\nabla \mathbf{U}\right)={\displaystyle \frac{1}{T}}{\tau }_{ij}{\displaystyle \frac{\partial U_i}{\partial x_j}}\hfill \\ \hfill & {\dot{s}}_T={\displaystyle \frac{k_{eff}}{T^2}}{∥\nabla T∥}^2={\displaystyle \frac{k_{eff}}{T^2}}{\displaystyle \frac{\partial T}{\partial x_i}}{\displaystyle \frac{\partial T}{\partial x_i}}\hfill \end{array}\end{array}$$

(17)

The local values of the generated entropy rates shown in Equation (17) make it possible to determine the total value of generated entropy for the entire considered area volume V:

$${\dot{S}}_{gen,tot}={\int }_V{\dot{s}}_{gen}dV$$

(18)

Relating the value of the total entropy generated rate to the reference temperature value represents the value of the irreversibly destroyed exergy:

$${\dot{X}}_{des}=T_0{\dot{S}}_{gen,tot}$$

(19)

For a properly chosen reference temperature value $T_0$, the absolute values obtained for the global parameters according to Equation (16) should be comparable to the values obtained from the local values calculated according to Equation (19). The reference values of temperature and pressure were $T_0=673.16\mathrm{K}$ and $p_0$ = 8 × 10⁶ Pa, respectively.

The process of simulation and determination of the values of the respective quantities is shown in the diagram in Figure 4. The calculations were performed in parallel for both turbulence models. The common parts are only data preprocessing, comparison and visualization.

3. Results

3.1. Energy and Exergy

This section presents the results obtained from the relationships outlined in Section 2.7 and Section 2.8. Figure 5 illustrates the time evolution of total energy at the outlet of the orifice plate. The mean values of total energy flow rate and their standard deviations for both turbulence models are as follows: for the k-ω SST 2006 model, $\dot{E}$ = 24,582.6 × 10³ W with standard deviation $0.21\%$; for the Spalart–Allmaras model, $\dot{E}$ = 24,582.7 × 10³ W with standard deviation $0.00053\%$.

Determining energy losses using a global exergy analysis reveals significant discrepancies depending on the turbulence model applied. The time evolution of exergy at the orifice plate outlet is shown in Figure 6. The mean values of total exergy flux and their standard deviations for both turbulence models are as follows: for the k-ω SST 2006 model, $\dot{X}=-\mathrm{2,228,661.6}\mathrm{W}$ with standard deviation $7.35\%$; for the Spalart–Allmaras model, $\dot{X}=-\mathrm{849,801.7}\mathrm{W}$ with standard deviation $1.21\%$.

The nature of these variations depends on the turbulence model used. According to the model authors’ concept, the k-ω SST 2006 turbulence model differs from the Spalart–Allmaras model primarily in its treatment of regions with high gradients. Consequently, differences in results were expected. Despite significant fluctuations during the initial phase of the simulation, the Spalart–Allmaras model tends to dampen them, whereas the k-ω SST 2006 model maintains fluctuations in energy flux intensity around the mean level. No suppression of these variations was observed. Additionally, differences in outlet energy values from the nozzle were noted, though they were negligibly small between the models.

Exergy is commonly defined as a positive quantity representing the maximum amount of useful work a system can perform, given its environment (surroundings) in equilibrium. However, there are scenarios in which exergy can be negative. This situation arises when the system is not in equilibrium or when the exergy reference point is defined in a way that results in negative exergy values. The presence of negative exergy may indicate that the system has reduced work potential compared to the reference state. Negative exergy values occur when the system does not favor performing work under the selected reference conditions or when certain system components exhibit properties that hinder work generation. While negative exergy values are relatively rare and may seem counter-intuitive at first, they can provide valuable insights into the thermodynamic properties of a system, particularly in the analysis of complex or non-equilibrium processes.

The cases presented are for the energy analysis of the nozzle itself. Consideration of ambient values is necessary when work is required to deliver the values obtained in the pipeline in the first place. For example, to produce steam at 400 °C, it is necessary to heat the steam, which of course causes losses. Similarly, to achieve a pressure of $8\mathrm{MPa}$, the relevant energy processes must also be taken into account. In the case under consideration, the energy quality losses during the flow through the orifice itself are analyzed. In such a situation, referring to ambient values instead of the initial values prevailing in the pipeline would probably result in exergy loss values at a negligible level, within the limits of the calculation error.

The reference values used to determine exergy were set as constant pipeline conditions: pressure $p_0=8\mathrm{MPa}$ and temperature $T_0$ = 400 °C. Consequently, the computed exergy values indicate negative energy transfer via steam. The negative exergy value represents a reduction in energy quality transferred by steam of less than 0.009% for the k-ω SST 2006 model and 0.003% for the Spalart–Allmaras model. On the other hand, a significant difference between the turbulence models is evident, reaching up to 62% in their mean values. Such a large discrepancy may be mainly due to the model of momentum transport and heat transfer within the fluid. As can be seen in Figure 6, in the Spalart–Allmaras model, the change in the exergy rate values is smoother than in the case of the k-ω SST 2006 model considering the same simulation time. That kind of behavior (smooth change of mean values) does not occur for energy rates (see Figure 5). In energy rate calculations, velocity and internal energy are key quantities (see Equation (13)). Regarding the exergy rate, the key factors (as Equation (16) shows) are enthalpy and entropy, and therefore, directly, temperature and pressure. Internal energy is a linear function of temperature. So, it can be concluded that the main reason for the significant discrepancies is the difference in enthalpy transfer within the fluid. This means that modeling only turbulent viscosity changes (as with the Spalart–Allmaras model) is insufficient to model the exergy rates accurately.

3.2. Entropy Generation

This section presents the results obtained from numerical simulations using the methodology described in Section 2.9. The numerical computations produced a series of images and graphs illustrating parameters related to energy, exergy and entropy for the flow through the measuring orifice. The results can be analyzed both qualitatively and quantitatively. Qualitative analysis is based on visualizations, while quantitative analysis uses graphs and histograms depicting variations in selected parameters throughout the simulation time range of $t=0÷1.5\mathrm{s}$.

Figure 7 shows the local entropy production rate visualization for two turbulence models. Five entropy production rate contour levels in the $50÷250\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ are displayed. The k-ω SST 2006 model shows a significantly larger total entropy production rate region. These visualizations represent a single time instant in the simulation, specifically $t=0.5\mathrm{s}$.

More detailed analyses can be performed using graphs depicting the unsteady characteristics of the flow through the orifice. The visualizations in Figure 7 correspond to the instantaneous values presented in Figure 8.

The entropy production rate consists of two primary components: production due to viscous dissipation, ${\dot{s}}_{\tau }$, and production due to heat exchange within the fluid, ${\dot{s}}_T$. The visualization of entropy production associated with viscous dissipation is shown in Figure 9.

Comparisons between ${\dot{s}}_{\tau }$ and ${\dot{s}}_{gen}$ indicate that the majority of entropy production arises from viscous dissipation. This conclusion is further supported by histogram comparisons of total entropy production rate and entropy production due to viscous dissipation, as illustrated in Figure 10.

Both analyzed turbulence models exhibit minimal differences between total and entropy production rates due to viscous dissipation. These differences are visible in the slightly higher occurrence of large values for the k-ω SST 2006 model and small deviations in mean values for the Spalart–Allmaras model. The mean value for the k-ω SST 2006 model is ${\dot{s}}_{gen}=31.9342\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ with a standard deviation of $0.257\%$. The entropy production rate due to viscous dissipation in this model is ${\dot{s}}_{\tau }=31.9332\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ with a standard deviation of $0.256\%$. The difference in the mean value appears only in the third decimal place. Similarly, for the Spalart–Allmaras model, the mean values are ${\dot{s}}_{gen}=36.9373\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ and ${\dot{s}}_{\tau }=36.9362\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$, with standard deviations of $0.064\%$ for both quantities. The Spalart–Allmaras model yields higher mean values by $15.7\%$ while demonstrating four times lower standard deviation, suggesting that its results are more concentrated around the mean value. This fact indicates that viscous dissipation has the most significant impact on total entropy production.

Entropy production due to heat exchange exhibits a different behavior. Figure 11 provides a visualization of the instantaneous results and contours for ${\dot{s}}_T$.

A completely different entropy field distribution is noticeable. In the Spalart–Allmaras model, entropy production rate field fragmentation is significantly more extensive than in the k-ω SST 2006 model. This difference is also evident in the time-dependent graphs shown in Figure 12.

The temporal evolution of entropy production due to heat exchange is entirely different from that of total entropy generation. Specifically, its values are four to five orders of magnitude lower than those generated by viscous dissipation. The mean values are ${\dot{s}}_T=9.9869\times {10}^{-4}\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ for the k-ω SST 2006 model and ${\dot{s}}_T=10.6283\times {10}^{-4}\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$ for the Spalart–Allmaras model, with standard deviations of $5.76\%$ and $0.76\%$, respectively. The Spalart–Allmaras model shows a higher concentration of results, as evidenced by its seven times lower standard deviation. The entropy production rate values ${\dot{s}}_T$ differed by $15.7\%$ between models, whereas for entropy production due to heat exchange, this difference was only $6.4\%$. The distribution of entropy production rate values due to heat exchange is illustrated in Figure 13.

All obtained mean values are presented in

Table 1. Average values of entropy generated.

entropy rates, $\mathrm{W}/\left({\mathrm{m}}^3\mathrm{K}\right)$	${\dot{s}}_{gen}$	${\dot{s}}_{\tau }$	${\dot{s}}_T$
k-ω SST 2006	31.9342	31.9332	9.9869 × 10−4
$Spalart-Allmaras$	36.9373	36.9362	10.6283 × 10−4
entropy, W/K	${\dot{S}}_{gen,tot}$	${\dot{S}}_{gen,\tau }$	${\dot{S}}_{gen,T}$
k-ω SST 2006	7.6235	7.6233	2.3841 × 10−4
$Spalart-Allmaras$	8.8179	8.8176	2.5372 × 10−4

below, which includes entropy production rates for total, viscous dissipation and heat exchange. The lower part of the table utilizes Equation (18) to determine the total entropy production and its components.

Relative entropy flux values related to viscous dissipation and heat exchange are compared in Figure 14. A linear correlation is observed for viscous dissipation, as indicated by the straight line in Figure 14a, demonstrating a nearly direct proportionality to the total entropy production rate. However, such a correlation is not visible for heat exchange entropy production rate. The Spalart–Allmaras model results also show a higher concentration around the mean value of entropy production rate.

Using the relationship between Equations (18) and (19), we are able to determine the values of the integral exergy flow rate from the results of the local entropy production rates. The values of these rates are shown below in Figure 15. The mean values are ${\dot{X}}_{des}=5.13\mathrm{kW}$ for the k-$omega$ SST 2006 model and ${\dot{X}}_{des}=5.94\mathrm{kW}$ for the Spalart–Allmaras model, with standard deviations of $0.26\%$ and $0.064\%$, respectively.

3.3. Error Estimation

To estimate discretization error, the procedure described in [40] was applied, utilizing Richardson extrapolation [41] and its references.

The most critical quantity for error analysis is the total entropy generation (${\dot{S}}_{gen,tot}$). In the following procedure, index ₀ refers to the mesh used in computations with $n_0$ = 4,287,850 cells, index _c corresponds to a coarser mesh with $n_c$ = 1,033,675 cells and index _f represents a refined mesh with $n_f$ = 12,571,677 cells. The computational domain volume remained the same, with $V_c=V_0=V_f=0.23873{\mathrm{m}}^3$. The estimation procedure requires additional simulations with finer and coarser meshes relative to the base case. The refinement factor, defined as the ratio of mean cell sizes between finer-to-base and base-to-coarser meshes, was $r_{0f}=h_0/h_f\approx 1.43$ and $r_{c0}=h_c/h_0\approx 1.61$. To maintain clarity, the notation $\varphi$ is used to represent the computed result, where $\varphi \equiv {\dot{S}}_{gen,tot}$.

The convergence ratio is given by:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill CR={\displaystyle \frac{{\varphi }_0-{\varphi }_f}{{\varphi }_c-{\varphi }_0}}.\end{array}\end{array}$$

(20)

The observed order of accuracy is computed iteratively using:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{cc}\hfill P& ={\displaystyle \frac{1}{ln\left(r_{0f}\right)}}\left|{\displaystyle \frac{{\varphi }_c-{\varphi }_0}{{\varphi }_0-{\varphi }_f}}\right|+\left|ln\left({\displaystyle \frac{r_{0f}^P-s}{r_{c0}^P-s}}\right)\right|,\hfill \\ \hfill s& =\mathrm{sign}\left({\displaystyle \frac{{\varphi }_0-{\varphi }_c}{{\varphi }_f-{\varphi }_0}}\right),\hfill \end{array}\end{array}\end{array}$$

(21)

where P is the observed order of accuracy. The above equation is solved iteratively due to its implicit form.

Table 2. Error estimation results.

	k-$\mathit{\omega }$ SST 2006	Spalart–Allmaras
${\varphi }_c$	7.04477	8.41465
${\varphi }_0$	7.6235	8.8179
${\varphi }_f$	7.87311	9.04847
$CR$	0.4313	0.5718
convergence	monotonic	monotonic
P	2.82	1.97
$GC{I}_{0f}$	2.27%	3.11%
$GC{I}_{c0}$	3.38%	3.71%

presents the input data and error estimation results.

Following the procedure outlined in [42], the Grid Convergence Index (GCI) was computed using:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill GC{I}_{0f}& ={\displaystyle \frac{1.25}{r_{0f}^P-1}}\left|{\displaystyle \frac{{\varphi }_f-{\varphi }_0}{{\varphi }_f}}\right|\hfill \\ \hfill GC{I}_{c0}& ={\displaystyle \frac{1.25}{r_{c0}^P-1}}\left|{\displaystyle \frac{{\varphi }_0-{\varphi }_c}{{\varphi }_0}}\right|\hfill \end{array}\end{array}$$

(22)

For detailed analysis, the GCI should fall between $1\%$ and $5\%$. These conditions are satisfied for the k-ω SST 2006 model, as both $GCI<5\%$ and $GC{I}_{c0}>GC{I}_{0f}$ are met. In the case of the Spalart–Allmaras model, only the condition $GCI<5\%$ is satisfied.

Based on the above results of discretization error estimation, one might be tempted to determine the predicted values based on Richardson extrapolation. The following relation Equation (23) takes into account the values of the results obtained and the order of accuracy:

$${\varphi }_{ext}=r_{0f}^p{\displaystyle \frac{{\varphi }_f-{\varphi }_0}{r_{0f}^P-1}}$$

(23)

The values obtained for each grid size (provided in Table 2) and the resulting extrapolated values calculated using the Equation (23) are shown collectively in Figure 16. Determining the extrapolated value makes it possible to assess how far the solution obtained is from the correct solution. The extrapolated value is ${\varphi }_{ext,(k-\omega )}=8.01592$ for the k-ω SST 2006 model and ${\varphi }_{ext,(S-A)}=9.27342$ for the Spalart–Allmaras model. These values differ by $4.90\%$ and $4.91\%$, respectively, from the values obtained in the $n_0$ grid calculations. These values confirm the acceptable level of the calculation inaccuracies.

4. Discussion

The entropy generated rate is only $0.003\%$ of the total entropy rate. Such a small value suggests that an analysis of compressible, adiabatic flow may be redundant. At the same time, since the entropy generation rate is partly temperature-dependent (according to Equation (17)), it is necessary to determine the degree of temperature dependence of entropy generation for incompressible adiabatic viscous flows.

Most of the energy carried by the flow is enthalpy, while kinetic energy is relatively minor. The dominant entropy transport mechanism is viscous dissipation. Increasing the temperature gradient significantly affects the entropy rate associated with heat transfer.

A full assessment of the actual energy quality loss is only possible when the value of the heat transfer with the environment of the measurement element is taken into account. In the context of the results presented in this paper, these losses can be significant. This is a significant limitation of the research carried out, but it provides valuable insight into the mechanism of entropy generation in industrial measurements. The average absolute values of the generated entropy $\dot{X}$, due to their reference to the average flow parameters, should be equal to the average values of the destroyed entropy rate ${\dot{X}}_{dex}$. These values vary considerably, not only between each other but also depending on the model used. This suggests a significant lack of precision in the application of global exergetic analysis (which only considers inlet and outlet parameters). In addition, it is not possible to directly compare this method with the literature data.

The dimensionless entropy production ($\psi ={\dot{S}}_{gen}/\left(\dot{m}{c}_p\right)$), in this case, is $1.62\times {10}^{-4}$ for the model k-ω SST 2006 and $1.88\times {10}^{-4}$ for the model Spalart–Allmaras. As this case does not include heat generation from chemical reactions or heat exchange with the environment, these values can be compared to the results in Ref. [21], where flow in channels with different cross-sectional shapes is analyzed. The values given, which are in the range $0.4\times {10}^{-5}$ to $2\times {10}^{-5}$, are lower, but the Reynolds number in that study is also lower, being in the range $\mathrm{Re}=200÷3500$, and there is no heat transfer with the environment. In Ref. [22], where channel heating is considered, the author gives values for water $1.39\times {10}^{-3}$ and for oil $3.10\times {10}^{-2}$, indicating an increase as a function of viscosity. Unfortunately, the Reynolds number is not given, but it is expected to be low. Therefore, the values obtained from the present investigation are reliable in the context of comparison to the studies reported in the literature.

The k-ω SST 2006 model shows higher computational accuracy, while the Spalart–Allmaras model has a lower scatter of results. The choice of turbulence model has a significant impact on the results. Energy transformations using kinetic energy values depend highly on the turbulence kinetic energy model and its dissipation. Since the Spalart–Allmaras model mainly predicts turbulent viscosity, the author is inclined to conclude that this model may be more reliable due to the minor effect of turbulence energy on total flow energy, including entropy and exergy.

The differences in the energy flow rate at the outlet are negligible; however, the spread of the results, represented by the standard deviation, varies significantly. The standard deviation for the k-ω SST 2006 model is almost 400 times larger than for the Spalart–Allmaras model. The energy losses, represented by the destruction of exergy in local entropy sources or exergy related to flow conditions, are small compared to the total energy carried by the flow. Despite exergy losses amounting to only a few thousandths of a percent concerning the total flow rate, further investigation of local energy values and their forms is needed. It remains an open question whether the entropy generated in the flow should be related to the total or kinetic energy. A similar problem concerns the enthalpy of the flow. These values should be the subject of future research.

It can be concluded that the models behave quite differently in global analyses of exergy rates. Due to the greater inertia of the Spalart–Allmaras model in heat transfer within the fluid, it can be suggested that this model is better suited to local analyses.

There are no significant differences in the local analysis of entropy generation rates, but the Spalart–Allmaras model tends to flatten the time series. At the same time, the k-ω SST 2006 model consistently produces oscillatory fluxes in both global and local analyses.

Unfortunately, it is not possible to say which of the models is more reliable. On the other hand, the obtained computational error estimates suggest a higher accuracy of the k-ω SST 2006 model.

Future research will focus on the applicability of the presented energy models to determine the values of quantities that are currently determined from experimental data. The author hypothesizes that quantities such as discharge coefficients or expansion numbers can be reliably determined from numerical simulations using the numerical methods presented in this paper.

However, this requires further research involving a change of medium, consideration of incompressible flow and consideration of heat exchange with the surroundings. The results obtained will allow further research into components and measurement systems not covered by ISO standards.