Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 313 https://internationalpubls.com Mathematical Analysis of Thermodynamic Model to Predict the Engine Parameters Anbuchezian Ashokan1*, Hariharan Elangandhi2,5, Ravikumar Venkatachalam3, Inbasekaran Alagarasan4, Silambarasan Rajendran5,6,8*, Balu Pandian7 1Department of Civil Engineering, Annapoorana Engineering College, Seeragapadi , Salem- 636308, Tamil Nadu, India. 2,5Department of Mechanical Engineering, Annapoorana Engineering College, Seeragapadi , Salem-636308, Tamil Nadu, India. 3Department of Mechanical Engineering, Sona College of Technology, Salem-636005, Tamil Nadu, India. 4Department of Mechanical Engineering, R P Sarathy Institute of Technology, Poosaripatti, Salem-636305, Tamil Nadu, India. 6Department of Mechanical Engineering, Saveetha School of Engineering,Saveetha Institute of Medical and Technical Sciences,Chennai, Tamil Nadu, India. 7Department of Automobile Engineering, Bharath Institute of Higher Education and Research, Chennai, Tamil Nadu, India . 8Centre for Research Impact and Outcome, Chitkara University Institute of Engineering and Technology, Chitkara University, Rajpura- 140417,Punjab,India. *Corresponding author: simbu2explore@gmail.com Article History: Received: 12-11-2024 Revised:24-12-2024 Accepted:08-01-2025 Abstract: To Predict the thermodynamic performance parameters SI engine two zone thermodynamic simulation model was developed. The Engine performance and thermodynamic parameters was predicted using first order mathematical ordinary differential equations such as peak pressure, burned gas temperature, unburned gas temperature, heat transfer, heat leakage, heat flux and Adiabatic flame temperature. The Fuel is specified by the way of C-H-O-N Values, the equilibrium state of combustion products was determined by olikara and borman method. The model was developed for Air cooled, single cylinder, 4 stroke SI Engine with variable compression ratio of 6-8. Curve fit Co-efficient are used to simulate air and fuel data along with residuals. Thermodynamic parameters are plotted with respect to crank angle. The objective of this work was to study the thermodynamic, performance parameters of SI Engine using various mathematical models. Keywords: Spark ignition engine; Alternate fuels; simulation; Two Zone; Thermodynamic model Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 314 https://internationalpubls.com I. INTRODUCTION The present trend is towards the development of comprehensive 3-D models, which describes the functioning of engines at a very high level of detail and accuracy; however, these require substantial computational power. Also, the need for precise experimental input makes the process significantly complicated and time consuming. There are several instances where theoretical methods, which are based on a limited set of experimental data, are preferred. From these considerations, the need for a simple, fast and accurate engine simulation model is quite evident. A two-zone, Zero-dimensional model was used to simulate the engine operations. The most important assumptions were that, a) The working medium was considered, in general, to be a mixture of 14 species (O2, N2, CO2, H2O, H2, OH, NO, CO, O, H, N, Ar, NO2, HO2) and fuel vapor. b) All 14 species were considered as ideal gases. And c) The alternate fuels are limited to C-H-O-N species. Hence, this paper aims to combine the benefits of various known models to achieve this goal. 2. Zero-Dimensional Thermodynamic Model Formulation The combustion chamber is divided into two zones consisting of unburned gas (mixture of fuel, air and residuals) and burned gas (mixture of 10 product species), each under uniform composition. Following assumptions are considered while developing the model. The pressure at any instant is assumed to be uniform throughout the cylinder. At any instant of time during combustion, the cylinder volume is divided into burned and unburned zones by an infinitesimally thin flame-front with a spherical shape. There is no heat transfer between burned and unburned zones. The burned gases are assumed to be in chemical equilibrium during combustion and for the main expansion stroke while the end of expansion stroke the mixture is assumed to be frozen. The zero-dimensional model includes the formulation of mass and energy balance. II. THERMODYNAMIC MODEL In the present model, a Zero-dimensional combustion model is employed. The combustion chamber is divided into two zones consisting of unburned gas (mixture of fuel, air and residuals) are burned gas (mixture of 14 product species), each under uniform composition. This model assumes that at any instant of time during the combustion, the cylinder volume is divided into burned and unburned zones byu an infinitesimally thin flame-front with a spherical shape. The burned gases are assumed to be in chemical equilibrium during combustion and form the main expansion stroke, while near the end of expansion stroke the mixture is assumed frozen [7], [10], [8]. A wiebe function specifies the fule burn rate and controls the rate at which mixtures from the unburned zone is converted to the burned zone [10]. Mass and energy conservation relations and equations of state from the principle governing equations. Also considering crank angle as the independent variable, we thus form the base of our thermodynamic model. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 315 https://internationalpubls.com A. Mass and Energy Balance The equation of state for an ideal gas is 𝑃𝑉 = π‘šπ‘…π‘‡ (1) The rate of change of mass within any open system is the net flux of mass across the system boundaries. Hence for a control volume enclosing the air-fuel mixture, we have π‘š = βˆ‘π‘šπ‘˜Μ‡ π‘˜ Μ‡ (2) The first law of thermodynamics to an open system yields the energy equation as οΏ½Μ‡οΏ½ = οΏ½Μ‡οΏ½ βˆ’ π‘Š + βˆ‘π‘šπ‘˜Μ‡ π‘˜ β„Žπ‘˜ (3) Equations (2) and (3) can be written as π‘‘π‘š π‘‘πœƒ = βˆ‘ π‘‘π‘šπ‘˜ π‘‘πœƒ π‘˜ (4) 𝑑(π‘šπ‘’) π‘‘πœƒ = 𝑑𝑄 π‘‘πœƒ βˆ’ 𝑝 𝑑𝑉 π‘‘πœƒ + βˆ‘β„Žπ‘˜ π‘‘π‘šπ‘˜Μ‡ π‘‘πœƒ π‘˜ (5) Equation (5) neglects changes of kinetic and potential energy in the control volume. B. Air and Combustion products Data Gordon and McBride [8], proposed the following expressions that wee curve-fitted to the tabulated JANAF Thermchemical tables [11]. 𝐢𝑝 𝑅 = π‘Ž1 + π‘Ž2𝑇 + π‘Ž3𝑇 2 + π‘Ž4𝑇 3 + π‘Ž5𝑇 4 (6) β„Ž 𝑅𝑇 = π‘Ž1 + π‘Ž2 2 𝑇 + π‘Ž3 3 𝑇2 + π‘Ž4 4 𝑇3 + π‘Ž5 5 𝑇4 + π‘Ž6 𝑇 (7) 𝑆 𝑅 = π‘Ž1𝐼𝑛𝑇 + π‘Ž2𝑇 + π‘Ž3 2 𝑇2 + π‘Ž4 3 𝑇3 + π‘Ž5 4 𝑇4 + π‘Ž7 (8) Where 𝑐𝑝 is the specific heat at constant pressure, h is the specific enthalpy and s is the specific entropy. The coefficients π‘Ž1 to π‘Ž7 are calculated over two different temperature ranges: 1)300