Thermodynamic Limits of Fuel Energy Utilization and Carbon Emissions in Highway Traffic Flow: A Control-Volume Perspective

2.1 Study object and boundary conditions

A straight, six-lane highway segment was selected as the study object, with a total length of 5 km. This configuration was chosen in accordance with typical highway design standards in China and allows geometric factors such as curvature and longitudinal grade to be excluded, thereby minimizing their interference with traffic-flow stability and energy dissipation characteristics. The traffic stream was composed of passenger cars and heavy-duty trucks at a ratio of 8:2. Three typical operating regimes—free flow, transitional flow, and congested flow—were considered. Free-flow conditions correspond to traffic densities below 20 pcu·km⁻¹·lane⁻¹, transitional flow to densities between 20 and 40 pcu·km⁻¹·lane⁻¹, and congested flow to densities exceeding 40 pcu·km⁻¹·lane⁻¹. This classification follows established industry standards for highway traffic-flow state delineation.

Boundary condition parameters were specified through a combination of field measurement data and engineering design standards to ensure practical applicability and model reliability. The inlet traffic demand was prescribed within a range of 2000–6000 pcu·h⁻¹, covering low-, medium-, and high-flow conditions. The downstream discharge capacity was set to 6000 pcu·h⁻¹, corresponding to the upper design limit of a six-lane highway segment. For environmental conditions, the ambient temperature was fixed at 25℃, representing a typical temperate atmospheric environment under normal operating conditions. The rolling resistance coefficient of the pavement–vehicle system was set to 0.015, consistent with measured values for asphalt concrete pavements under standard conditions. All boundary-condition parameters were subjected to sensitivity analysis to verify the rationality of their selected values and to ensure that model stability was maintained under small parameter perturbations. Figure 1 shows the schematic flowchart of the thermodynamic traffic flow analysis framework integrating microscopic fuel combustion and macroscopic traffic oscillations.

Figure 1. Schematic flowchart of the thermodynamic traffic flow analysis framework integrating microscopic fuel combustion and macroscopic traffic oscillations

2.2 Traffic-flow control volume modeling

The control volume was defined as an open system using an Eulerian formulation. Its spatial boundaries were explicitly specified by the upstream entrance cross-section, the downstream exit cross-section, and the lateral road-shoulder boundaries, thereby forming a closed three-dimensional control domain. A nonsteady analytical framework was adopted for the temporal boundary, with the time step Δt set to 1 s. This temporal resolution represents a compromise between numerical accuracy and computational efficiency and enables the instantaneous characteristics of traffic oscillations to be accurately captured. As an open system, the interfaces for mass and energy exchange within the control volume were clearly identified. These interfaces include vehicle mass inflow and outflow through the entrance and exit cross-sections, heat exchange between the control volume and the ambient environment, and work exchange associated with vehicles overcoming external resistances.

Figure 2. Eulerian control volume model for highway traffic flow

Through this explicit delineation, the complete processes of mass transfer and energy transfer were consistently represented. Figure 2 shows the schematic illustration of the Eulerian thermodynamic control volume model for highway traffic flow.

Based on the analogy with compressible fluid dynamics, traffic flow was modeled as a compressible granular flow, and a set of equivalent thermodynamic parameters was defined to quantify the thermodynamic characteristics of traffic systems. Traffic pressure was defined as:

$P=\rho \sigma_v^2$              (1)

where, ρ denotes traffic density (pcu·km⁻¹·lane⁻¹) and σ_v represents the standard deviation of vehicle speed (km·h⁻¹). This definition was derived from a statistical analysis of traffic kinetic energy and reflects both the overall kinetic energy distribution of the traffic stream and the compressive interactions among vehicles. Traffic temperature was introduced to characterize the internal disorder of traffic flow and its potential for energy dissipation. It was defined to be proportional to the square of the speed standard deviation.

$T_r \propto \sigma_v^2$               (2)

Validation through comparative analysis of free-flow and congested-flow regimes demonstrated that both traffic pressure and traffic temperature increase markedly with rising traffic density. This behavior is consistent with the observed escalation in traffic disorder, thereby confirming the physical rationality of the proposed equivalent thermodynamic parameters.

On this basis, the mass conservation equation for the control volume was derived using the LWR traffic-flow theory to describe the spatiotemporal evolution of traffic density. The governing equation is expressed as:

$\frac{\partial \rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0$                (3)

where, v denotes the mean traffic speed within the control volume (km·h⁻¹), t represents time (s), and x is the spatial coordinate along the highway segment (km). This equation fundamentally indicates that the temporal rate of change in traffic density within the control volume is equal to the net divergence of traffic density flux entering and leaving the control volume. As such, the spatiotemporal conservation of traffic mass is accurately described, providing a rigorous mass-conservation foundation for the subsequent thermodynamic analysis. Through discretization using the finite-difference method, this equation was coupled with the energy and entropy balance equations to form an integrated numerical solution framework.

To represent momentum transfer and force balance within traffic flow, a Navier–Stokes–like momentum equation was formulated by introducing a traffic viscosity coefficient. The vector form of the governing equation is expressed as:

$\rho\left(\frac{\partial v}{\partial t}+v \cdot \nabla v\right)=-\nabla P+\mu_t \nabla^2 v+\rho g-f_r \rho|v|$               (4)

The left-hand side of the equation represents inertial terms, capturing momentum variations induced by temporal and spatial changes in traffic speed. The right-hand side successively includes the traffic pressure gradient term, the viscous stress term, the gravitational body-force term, and the rolling resistance term associated with pavement–vehicle interaction. In the equation, μ_t denotes the traffic viscosity coefficient (pcu·s·km⁻¹), which characterizes inter-vehicle friction and car-following interactions within the traffic stream; g represents gravitational acceleration (m·s⁻²); and f_r denotes the rolling resistance coefficient. All force terms were nondimensionalized and parameterized as functions of traffic density and speed. The traffic viscosity coefficient was calibrated using measured car-following trajectory data, enabling the momentum equation to accurately reproduce observed acceleration, deceleration, and momentum transfer behaviors under realistic traffic conditions.

2.3 Thermodynamic coupled analysis

Based on open-system thermodynamic theory and accounting for the nonsteady characteristics of traffic flow, a nonsteady energy balance equation for the control volume was formulated, enabling a quantitative application of the first law of thermodynamics to traffic-flow analysis. The energy balance equation is expressed as:

$\frac{d E_{C V}}{d t}=\dot{Q}_{n e t}-\dot{W}_{n e t}+\sum \dot{m}_{i n} h_{t o t, i n}-\sum \dot{m}_{o u t} h_{t o t, o u t}$              (5)

where, dE_CV/dt denotes the temporal rate of change of total energy within the control volume, $\dot{Q}_{{net}}$ represents the net heat exchange rate between the control volume and the surrounding environment, $\dot{W}_{{net}}$ denotes the net work exchange rate, $\dot{m}_{{in}}$ and $\dot{m}_{{out}}$ are the mass flow rates of traffic entering and leaving the control volume, respectively, and $h_{{tot,in}}$ and $h_{{tot,out}}$ denote the total specific enthalpy of the incoming and outgoing traffic streams. This equation fundamentally describes the conservation relationship between the temporal variation of energy stored within the control volume and energy exchanges with the external environment. Key processes—including energy transport associated with traffic inflow and outflow, heat transfer, and mechanical work consumption—are comprehensively represented, thereby providing a quantitative foundation for evaluating the magnitude of energy conversion within the traffic system. Figure 3 shows the schematic illustration of the multi-source entropy-generation decomposition mechanism and the corresponding energy conversion chain.

Figure 3. Multi-source entropy generation and energy conversion chain

The quantification of each energy term was determined based on the underlying physical mechanisms and experimentally informed correlation models, ensuring both computational accuracy and theoretical consistency. The total energy within the control volume, E_CV, was composed of traffic kinetic energy and vehicle internal energy. The kinetic energy was evaluated using traffic density and mean vehicle speed, whereas the internal energy was associated with vehicle temperature and thermodynamic properties. The net heat exchange, $\dot{Q}_{n e t}$, was dominated by convective heat transfer between the traffic stream and the ambient environment and was quantified using Newton’s law of cooling, $\dot{Q}_{n e t}=h_A A\left(T_{C V^{-}} T_0\right)$, where h_A denotes the convective heat transfer coefficient, A is the effective heat exchange area of the control volume, T_CV is the mean control volume temperature, and T₀ is the ambient temperature. The net work exchange, $\dot{W}_{n e t}$, was defined as the total dissipative work associated with vehicles overcoming rolling resistance and aerodynamic drag and was evaluated through resistance coefficients coupled with the rate of change of traffic kinetic energy. The total specific enthalpy at the inlet and outlet, h_tot, comprised chemical energy, kinetic energy, and internal energy. The chemical energy component was determined by the fuel’s lower heating value and the corresponding fuel consumption rate, whereas the kinetic and internal energy components were quantified using vehicle speed and temperature-related parameters.

On the basis of the second law of thermodynamics, an entropy balance equation for the open control volume was established to quantitatively characterize the irreversibility of energy conversion processes in traffic flow. The entropy balance is expressed as:

$\dot{S}_{{gen }}=\frac{d S_{C V}}{d t}-\sum \frac{\dot{Q}_k}{T_k}+\sum \dot{m}_{{out }} s_{{out }}-\sum \dot{m}_{{in }} s_{{in }} \geq 0$             (6)

where, $\dot{S}_{{gen}}$ denotes the entropy generation rate, serving as the principal quantitative indicator of irreversibility in energy conversion, dS_CV/dt represents the temporal rate of change of entropy within the control volume, $\dot{Q}_k$ is the heat exchange rate between the control volume and an external reservoir at temperature T_k, and s_in and s_out are the specific entropy values of the incoming and outgoing traffic streams, respectively. This formulation explicitly enforces the non-negativity of entropy generation, thereby revealing the thermodynamic constraints governing irreversible processes in traffic flow and providing a fundamental basis for evaluating the quality of energy conversion.

To accurately trace the sources of irreversible losses in traffic flow, a multi-source entropy generation decomposition model was introduced. The total entropy generation rate was decomposed into three components: thermal entropy generation, friction-induced entropy generation, and oscillation-induced entropy generation. The decomposition is expressed as:

$\dot{S}_{{gen }}=\dot{S}_{{gen,th }}+\dot{S}_{{gen,fr }}+\dot{S}_{{gen,osc }}$              (7)

Thermal entropy generation, $\dot{S}_{g e n, t h}$, originates from the irreversibility of chemical reactions during fuel combustion and from irreversible heat transfer processes. Based on chemical thermodynamics, the entropy generation associated with chemical reactions was evaluated using the entropy differences between reactants and products together with reaction rates. The entropy generation due to heat transfer was calculated as:

$\dot{S}_{{gen,th,heat }}=\sum \frac{\dot{Q}_k}{T_0}-\sum \frac{\dot{Q}_k}{T_k}$               (8)

Friction-induced entropy generation, $\dot{S}_{{gen, fr }}$, arises from energy dissipation caused by aerodynamic drag and pavement rolling resistance. By analogy with entropy generation formulations for viscous friction in fluid systems, this component was expressed as:

$\dot{S}_{g e n, f r}=\frac{\dot{W}_{f r}}{T_0}$              (9)

where, $\dot{W}_{f r}$ represents the power dissipated by frictional resistance. Oscillation-induced entropy generation, $\dot{S}_{{gen,osc}}$, represents irreversible dissipation of macroscopic kinetic energy caused by acceleration–deceleration oscillations in traffic flow. This component was quantified using a correlation model linking traffic density to the temporal evolution of speed-fluctuation variance, expressed as:

$\dot{S}_{{gen,osc }}=\frac{\rho C_v \frac{d \sigma_v^2}{d t}}{T_0}$             (10)

where, $C_v$ is the energy conversion coefficient associated with speed fluctuations. The coefficient was calibrated using measured data.

2.4 Exergy destruction quantification and carbon emission modeling

Based on the Gouy–Stodola theorem, a quantitative relationship between exergy destruction and entropy generation was established to enable precise characterization of energy quality degradation. The Gouy–Stodola theorem states that the exergy destroyed in an irreversible process is equal to the product of the ambient temperature and the total entropy generation rate. Accordingly, the exergy destruction rate of the traffic-flow system is expressed as:

$\dot{E} x_{{destroved }}=T_0 \dot{S}_{{gen }}$              (11)

where, $\dot{E} x_{{destroyed}}$ denotes the exergy destruction rate, T₀ is the ambient reference temperature, and $\dot{S}_{g e n}$ is the total entropy generation rate defined previously. This formulation converts entropy generation—a fundamental indicator of thermodynamic irreversibility—into exergy destruction, which directly represents the loss of energy quality. In this manner, a direct bridge is established between thermodynamic fundamentals and engineering-relevant energy losses, providing a core quantitative tool for evaluating the quality of fuel energy utilization in traffic systems.

The linkage between exergy destruction and fuel consumption was established through the energy conversion chain, ensuring both theoretical consistency and engineering applicability. In the traffic-flow system, fuel input exergy constitutes the origin of energy conversion. A portion of this input is transformed into useful exergy associated with vehicle motion, while the remainder is irreversibly degraded into exergy destruction. Consequently, the exergy destruction rate exhibits a positive correlation with the fuel consumption rate. This relationship was quantified using the fuel lower heating value and the conversion factor between input exergy and lower heating value, expressed as $\dot{E} x_{{input }}=\dot{m}_f L H V \zeta$, where $\dot{E} x_{{input }}$ denotes the fuel input exergy rate, $\dot{m}_f$ is the fuel consumption rate, LHV is the fuel lower heating value, and ζ is the conversion coefficient relating fuel chemical exergy to its lower heating value. Combined with the exergy destruction formulation, the degree of fuel energy quality degradation can thus be indirectly characterized through the entropy generation rate, thereby establishing a foundation for subsequent coupled analyses of carbon emissions and energy efficiency.

On the basis of the stoichiometric relationships governing fuel combustion, a quantitative model linking the CO₂ emission rate with input exergy and second-law efficiency was developed. The second-law efficiency, η_II, was first defined to characterize the fraction of input exergy converted into useful exergy, expressed as:

$\eta_{I I}=\frac{\dot{E}_{X_{{useful }}}}{\dot{E}_{X_{{input }}}}$            (12)

where, $\dot{E}_{X_{{useful}}}$ denotes the useful exergy rate, corresponding to the effective work rate $\widehat{W}_{{use}}$ required for vehicles to overcome external resistances and maintain motion. According to the stoichiometric relationships of fuel combustion, the mass of $\mathrm{CO}_2$ produced per unit mass of fuel consumed is a fixed quantity. Consequently, the $\mathrm{CO}_2$ emission rate, $\dot{m}_{C O 2}$, is directly proportional to the fuel consumption rate, $\dot{m}_f$, such that $\dot{m}_{{CO} 2}=\beta \dot{m}_f$, where $\beta$ is the $\mathrm{CO}_2$ emission factor determined by the carbon content of the fuel. By jointly formulating the relationship between input exergy and fuel consumption rate, a functional relationship between the CO₂ emission rate, input exergy, and second-law efficiency was ultimately derived.

$\dot{m}_{{CO2}}=\frac{\beta \dot{E}_{X_{{input }}}}{L H V \zeta \eta_{I I}}$              (13)

By invoking the condition of MEG, the thermodynamic constraint boundary of CO₂ emissions under a given traffic throughput was derived. MEG corresponds to the lowest system irreversibility, under which the second-law efficiency reaches its maximum value, $\eta_{I I, \max}$. This maximum efficiency is determined by the MEG rate, $\dot{S}_{{gen,min}}$, such that:

$\eta_{I I, \max }=1-\frac{\dot{E} x_{{destroyed, } \min }}{\dot{E} x_{{input }}}$               (14)

where, $\dot{E} x_{{destroyed,} \min }=T_0 \dot{S}_{{gen,min}}$ denotes the minimum exergy destruction rate. Substitution of $\eta_{I I, \max}$ into the emission expression yields the thermodynamic theoretical lower bound of CO₂ emissions for a given traffic throughput:

$\dot{m}_{{CO}_2, \min }=\frac{\beta \dot{E} x_{{input }}}{L H V \eta_{I I, \max }}$             (15)

This formulation explicitly demonstrates that, under a specified traffic throughput, CO₂ emissions are subject to an insurmountable thermodynamic lower bound determined by the minimum achievable irreversibility of the system. This constraint boundary provides a physics-based benchmark for evaluating the feasibility of traffic emission-reduction strategies and prevents the formulation of mitigation targets that exceed fundamental thermodynamic limits.

2.5 Numerical simulation and model validation

The governing equations of mass, momentum, energy, and entropy balance derived above were discretized in both space and time using the finite-difference method to enable numerical solution of the coupled system. The spatial domain was discretized using a uniform grid, with the grid spacing set to 50 m based on the highway segment length and accuracy requirements. Time discretization was implemented using an explicit time-marching scheme. Convective terms were treated using a first-order upwind scheme to ensure numerical stability, while diffusive terms were discretized using a central-difference scheme to enhance computational accuracy. During discretization, the Courant–Friedrichs–Lewy (CFL) stability condition was enforced, expressed as vΔt/Δx≤1, where v denotes the maximum vehicle speed, Δt is the time step, and Δx is the spatial step size. Satisfaction of this condition ensured that numerical oscillations or divergence did not occur in the solution. A numerical simulation framework was implemented on the Python platform. The resulting nonlinear algebraic equations were solved using the Newton–Raphson iterative method. The convergence criterion was specified such that the absolute residuals of all governing equations were required to be less than 10⁻⁴. Upon convergence, key outputs—including entropy generation rate, exergy destruction rate, CO₂ emission rate, and traffic-flow parameters—were obtained for different traffic scenarios.

Model validation was conducted using measured data collected from a representative highway segment. The data sources included roadside loop detectors and onboard diagnostic (OBD) devices. Loop detectors recorded macroscopic traffic-flow variables, including flow rate, traffic density, and mean speed, at a sampling interval of 1 min, covering free-flow, transitional-flow, and congested-flow conditions. OBD devices simultaneously recorded vehicle-level fuel consumption rates and emission rates, yielding a total of 1,200 valid measurement samples. Validation metrics included both quantitative error analysis and qualitative consistency assessment. At the quantitative level, the mean absolute error (MAE) between model predictions and measured values was computed, with the MAE of both fuel consumption rate and CO₂ emission rate required to remain below 8%, thereby confirming the model’s predictive accuracy. At the qualitative level, the spatial distributions of entropy generation rate predicted under free-flow and congested-flow conditions were compared with measured speed-fluctuation characteristics. Correlation analysis was performed to verify the positive relationship between entropy generation rate and the variance of vehicle speed fluctuations, ensuring that the model accurately captures the intrinsic linkage between traffic nonsteadiness and thermodynamic irreversibility. Collectively, these validation results demonstrate the reliability and engineering applicability of the proposed model.

In comparison with conventional traffic energy consumption and carbon emission models, such as VT-Micro and CMEM, the thermodynamically coupled framework developed in this study exhibits a pronounced advantage in scenario adaptability. Traditional empirical or semi-empirical models rely on correlations calibrated using measured data under specific operating conditions. Under nonsteady regimes such as traffic congestion, additional energy dissipation induced by traffic oscillations cannot be explicitly represented, resulting in prediction errors that typically reach 15%–20%. By contrast, through multi-source entropy-generation decomposition, the present framework explicitly captures the additional carbon emissions associated with oscillation-induced entropy generation, thereby reducing prediction errors to 5%–7% under congested conditions and substantially improving accuracy in nonsteady traffic scenarios. It should nevertheless be acknowledged that conventional models retain advantages in steady regimes such as free flow conditions, where computational efficiency is higher and complex thermodynamic equation coupling is unnecessary. In contrast, to ensure physical fidelity, the proposed framework requires a balance between computational complexity and numerical efficiency. This requirement may impose limitations in rapid engineering assessment scenarios.

When compared with entropy optimization studies in fluid mechanics and energy systems, the present work shares a common methodological paradigm, namely the combination of control volume modeling and entropy minimization. In all cases, irreversibility is quantified through the second law of thermodynamics to guide system optimization. However, traffic flow constitutes a fundamentally distinct granular flow system composed of a large number of self-driven agents, whose thermodynamic characteristics differ substantially from those of conventional continuous fluids. In classical fluid systems, entropy generation arises predominantly from viscous dissipation and irreversible heat transfer. In contrast, traffic-flow entropy generation must explicitly account for oscillation-induced entropy associated with macroscopic speed fluctuations. The multi-source entropy-generation decomposition approach proposed in this study explicitly identifies oscillation-induced entropy generation as an independent contribution for the first time. By doing so, a critical gap in the application of thermodynamic theory to traffic flow as a special granular system is addressed, and the applicability domain of nonequilibrium thermodynamics is extended.

The findings of this study provide a fundamentally new thermodynamics-oriented pathway for the optimization of intelligent transportation systems. In terms of traffic control strategy optimization, a control paradigm based on the MEG principle may replace conventional time-optimal strategies. Through technical measures such as variable speed limits and vehicle–infrastructure cooperation, traffic speed distributions can be guided toward greater uniformity, thereby reducing speed fluctuations and suppressing oscillation-induced entropy generation at its source. For example, in congestion-prone highway segments, real-time monitoring of entropy generation rate distributions may be used to dynamically adjust speed limits, inhibiting the formation and propagation of congestion shock waves and enabling a coordinated improvement in emission reduction and traffic efficiency. At the network-planning level, the thermodynamic lower bound of carbon emissions provides a quantitative physical basis for highway capacity design and vehicle-type regulation. During the planning stage, reasonable capacity thresholds may be determined by combining projected traffic demand with the carbon-emission lower bound, thereby avoiding excessive pursuit of capacity that would cause emissions to exceed fundamental physical limits. In parallel, regulation of vehicle composition may be implemented to reduce the carbon-emission lower bound. For instance, maintaining the proportion of heavy-duty vehicles below 15% can reduce the lower bound by approximately 8%–10%, offering concrete guidance for vehicle-type management in road network planning.

Despite these contributions, several limitations remain. First, the analysis has been confined to straight highway segments, and the effects of complex terrain factors such as curvature and longitudinal grade on traffic dynamics and energy dissipation have not been incorporated, limiting the generality of the current framework. Second, the control volume model has been constructed under a continuum assumption, and the heterogeneity of individual vehicles has not been fully considered, which may lead to reduced accuracy in the representation of microscale energy dissipation. Third, cooperative effects associated with autonomous vehicles have not been included, precluding assessment of the impacts of emerging transportation technologies on entropy generation and carbon emissions. Future research may address these limitations along three complementary directions. First, the model may be extended to multidimensional and topographically complex scenarios, with terrain effects explicitly incorporated into the momentum and energy balance equations to enhance engineering applicability. Second, integration of multi-agent models with microscopic vehicle dynamics may be pursued to overcome the limitations of the continuum assumption, enabling coupled analysis of macroscopic thermodynamic behavior and microscopic vehicle interactions. Third, machine learning algorithms may be combined with entropy generation models to improve the prediction accuracy of entropy generation rates under complex conditions, while pathways for integrating MEG strategies with autonomous driving technologies may be explored. Such efforts would provide a stronger theoretical foundation for the deep optimization of future intelligent transportation systems.