Entropy generation analysis and optimization of cooling systems in industrial and engineering operations

Efficient cooling is crucial for maintaining the reliability and performance of industrial and engineering systems that generate excess heat. Entropy generation analysis, established on the second law of thermodynamics, plays a vital role in identifying inefficiencies within these systems and improving their overall efficiency. This study focuses on a theoretical investigation of the entropy generation, considering the cumulative impact of surface slipperiness, Brownian motion, applied magnetic field, thermophoresis on the nanofluid flow toward a convective heated stretching surface. The governing model partial differential equations undergo a transformation through similarity transformation, resulting in a nonlinear ordinary differential equation. This equation is subsequently solved numerically by employing the Runge–Kutta–Fehlberg integration scheme in conjunction with the shooting method. The obtained results reveal the influence of various parameters on the temperature, velocity, Nusselt number, skin friction, Sherwood number, Bejan number, nanoparticle concentration, and entropy generation. Upon analysis, it was notable that the introduction of a magnetic field, higher Biot numbers, Eckert numbers, and elevated Brownian motion led to an increase in the entropy generation within the system. Conversely, the presence of thermophoresis and reduced surface slipperiness resulted in a decrease in entropy. These results are presented through graphical representations, tables, and quantitative discussions, providing valuable insights for optimizing the cooling and performance of industrial and engineering systems.

nanofluid flow on a rotating disk.Berrehal et al. 16 investigated the impact of nanoparticle shape on entropy generation in a theoretical model of magnetohydrodynamics nanofluid flow over a flat sheet with heat source/sink.Jawad et al. 17 numerically analyzed entropy generation, considering the mutual impact of Dufour, Soret, thermal conductivity, and thermal radiation in the magnetohydrodynamics nanofluid flow over a slippery, porous, and flat sheet.By incorporating chemical reaction and Arrhenius activation energy, Oyelakin et al. 18 performed an optimization study on the entropy generation of Casson nanofluid flow over an unsteady stagnation point on a flat surface.Considering the space-dependent and exponential nonlinear thermal radiation, Yusuf et al. 19 examined the entropy generation in the magnetohydrodynamics flow of Casson fluid over a curved stretching sheet.Ullah et al. 20 scrutinized entropy generation in the magnetohydrodynamic hybrid nanofluid flow over a curved stretching surface, focusing on nanoparticle shape properties.
Examining the existing body of literature, it becomes evident that there is a gap in our understanding of entropy generation and the optimization of cooling systems in industrial and engineering operations.This gap specifically relates to the absence of a comprehensive model that considers the collective impact of surface slipperiness, Brownian motion, applied magnetic fields, and thermophoresis on the flow of nanofluids towards a convective heated stretching surface.The objective of this paper is to investigate how the combine effects of surface slipperiness, Brownin motion, applied magnetic fields, and thermophoresis can be analysed to minimize entropy generation and optimize cooling.In the subsequent sections, we will detail the formulation of the model problem, apply appropriate similarity transformation, solve using an appropriate numerical approach, and then present and discuss the relevant findings.

| MODEL FORMULATION
Considering the model problem depicted in Figure 1, we assume that the flow is incompressible, two-dimensional, steady, slippery, and convectively heated.These assumptions lead to the modified governing equations, which are expressed as follows [21][22][23] : (2) (3) with the boundary condition, when , , , .
w w f ff (6)   In Equations ( 1)-( 6), the variables are defined as follows: u and v represent the velocity of components, x and y are the Cartesian coordinates, ∞ U represents the free stream velocity, ∞ T represents the free stream temperature, ∞ C represents the concentration at the free stream, U w represents the velocity at the surface, T f represents the temperature from the heat source, B 0 We introduce the following variables and parameters: η as the dimensionless surface length, a and b the positive real numbers, Re Reynolds number, θ the dimensionless temperature, ϕ the dimensionless concentration.
By substituting the non-dimensionless variables in Equation ( 8) into Equations ( 1)-( 6), we obtained the following expressions, (10) subjected to the boundary condition, The parameters in Equations ( 9)-( 13) are defined as follows: The parameter in Equation ( 14  The specific physical properties of interests are the coefficient of skin friction (C f ), the coefficient of Nusselt number (Nu), the coefficient of Sherwood number (Sh), and the Bejan number (Be).Their expressions are defined as follows: The heat flux (q w ), shear stress (t w ), and mass flux of nanoparticles (b w ) at the surface is given by, respectively, By substituting the similarity variables from Equation (8) into Equation (15), we obtain the following expression: with,  (18)

| RESULTS AND DISCUSSION
To authenticate the theoretical calculations, we matched the values of F″(0) for stretching surface with findings reported by Wang, 24 Ishak, 25 and Tshivhi and Makinde. 26The comparison was performed under the conditions where Ha β = = 0 and λ varies, as presented in Table 1 below.The results demonstrated a favorable agreement between our findings and the previously reported.
The set of Equation (19) with their corresponding boundary conditions specified in Equation (20) were numerically solved using the MAPLE software, which employs the Runge-Kutta-Fehlberg known for its high accuracy.The obtained results, illustrating the influence of magnetic field (Ha), surface slipperiness (β), Eckert number (Ec), Biot number (Bi), thermophoresis (Nt), and Brownian motion (Nb) on the temperature, velocity, Nusselt number, skin friction, Sherwood number, Bejan number, nanoparticle concentration, and entropy generation, are presented graphically in Figures 2-18.These figures provide a visual representation of the effects of the mentioned parameters on the various aspects of the system.

| Temperature profile
Figures 3-5 illustrate the influence of various parameters, including the Ha, β, Bi, Ec, Nt, and Nb, on the temperature profile of a heated stretching surface.These figures revel that the surface temperature is notably higher than that of the free stream due to convection heating, with a gradual temperature decrease away from the surface.Enhance values of Ha, Ec, Bi, Nt, and Nb correspond to increased surface temperature.The magnetic field's effect is attributed to the Lorentz force, impacting heat transfer, while a high Eckert number signifies a dominance of kinetic energy over internal energy in the fluid.A high Boit number suggest significant joule heating relative to heat conduction.Thermophoresis leads to particle migration in response to temperature variations, affecting heat transfer.Brownian motion, typical in colloidal systems, increases particle dispresion and random motion.Conversely, rising values β of decrease temperature due to surface slipperiness.

| Concentration profile
Figures 6 and 7 illustrate how the parameters of Nt and Nb affect the concentration profiles on the heated stretching surface.These figures demonstrate that the surface concentration of nanoparticles surpasses that of the free stream, adhering to the specified boundary conditions.As Nt values increase, the surface nanoparticle concentration rises due to thermophoresis induced selective particle migration towards temperature gradients, influencing fluid concentration.Conversely, higher Nb values result in a reduction of nanoparticle concentration.In the case of low Brownian motion, typically associated with large particles or limited particle dispersion, mixing and diffusion are impede, leading to faster settling and uneven concentration gradients, making it challenging to achieve a uniform particle distribution in the fluid.

| Skin friction
Figure 8 portrays the analysis of skin friction between the heated stretching surface and the nanofluid, considering the influence of parameters β and Ha.It is evident that an rise in the parameter values of Ha leads to an increase in the skin friction.This can be attributed to the presence of the imposed magnetic field, which enhances the interaction between the nanofluid and the surface, leading to increased friction.On the other hand, an increase in the values of β and λ leads to a decline in the skin friction.This decrease is due to surface slipperiness, which reduces the resistance to fluid flow and consequently decreases the skin friction.

| Nusselt number
The Nusselt number is important in determining the heat transfer between the moving nanofluid and the stretching surface.The influence of parameters Ha, β, Nt, Ec, Nb, and Bi on the Nusselt number is illustrated in Figures 9 and 10.It is observed that the Nusselt number number increases with higher values of β and Bi.This increase is attributed to the surface slipperiness, which promotes enhanced heat transfer.Furthermore, the Nusselt number demonstrates a decrease with rising in the values of Ha, Nt, Ec, and Nb.This decrease is influenced by the availability of joule heating, thermophoresis distribution, and internal heat transfer of the nanoparticles, as well as Brownian motion, which collectively result in reduced heat transfer efficiency and a lower Nusselt number.

| Sherwood number
The Sherwood number represents the mass transfer between the nanofluid and the surface.The impact of parameters Ha, β, Nb, Ec, Nt, and Bi on the Sherwood number is depicted in Figures 11 and 12.The computed Sherwood number values indicate that the rise in the values of Ha, Ec, and Nb results to the raise in the Sherwood number.This increase is because presence of Brownian motion, joule heating, and internal heat transfer within the nanofluid, which collectively enhance the mass transfer between the surface and the nanofluid.Conversely, the upsurge in the values of β, Bi, and Nt results in a drop in the Sherwood number.This decrease is due to the slipperiness of the surface, heat transfer, and the impact of thermophoresis, all of which contribute to a reduction in the mass transfer rate.

| Entropy generation rate
The entropy generation rate serves as a measure of the disorder generated during irreversible processes in nanofluid flow.The influence of parameter values Ha, β, Bi, Ec, Nt, and Nb on the

| Bejan number
The Bejan number, ranging from 0 to 1, provides insights into the dominant factors controlling entropy generation.When the Bejan number falls between 0 and 0.5, it reveals that the entropy generation is primarily governed by magnetic field irreversibility and nanofluid friction.Conversely, if the Bejan number exceeds 0.5, the entropy generation is predominantly influenced by heat transfer irreversibility and Brownian motion.The profiles of the Bejan number are depicted in Figures 16-18.The rise in the values of β, Ec, and Nb will increase the Bejan number, signifying that entropy generation is led by Brownian motion and heat transfer irreversibility.Conversely, rising in values of Ha, Bi, and Nt results in a decrease in the Bejan number, indicating that the entropy generation rate is primarily influenced by nanofluid friction and magnetic field irreversibility.The study aimed to investigate the cumulative impact of surface slipperiness, Brownian motion, applied magnetic field, thermophoresis on the nanofluid flow toward a convective heated stretching surface.The solution of model was obtained using the Runge-Kutta-Fehlberg integration scheme and shooting method.The findings highlight the following key points: • The width of the velocity boundary layer reduces as the values of β and Ha increase.
• The nanofluid temperature increases with higher values of Ha, Ec, Bi, Nt, and Nb, but declines with increasing β.These findings provide valuable insights into the entropy generation and optimization of cooling systems in industrial and engineering operations.They contribute to our understanding of the combined impacts of surface slipperiness, applied magnetic field, thermophoresis, and Brownian motion on the flow characteristics, heat transfer, mass transfer, and entropy generation in nanofluid systems.

T A B L E 1 F I G U R E 2
Comparison of the results.Impacts of Ha and β on the velocity profile of the model.

F I G U R E 3
Effects of Ha and β on the temperature graph of the model.represents the constant applied magnetic field, T represents the temperature, C p represents the fluid specific heat capacity, τ represents the heat capacity of the fluid, D B represents the coefficient of Brownian diffusion, C represents the fluid concentration, D T represents the thermophoretic diffusion coefficient, E g represents the entropy generation rate, μ represents the fluid dynamic viscosity, ρ represents the density of the fluid, σ represents the electrical conductivity of the base fluid, k represents the fluid thermal conductivity, α represents the fluid thermal diffusion, ν f represents the fluid kinematic viscosity, L represents the coefficient of slip length, R represents the nanoparticle diffusion parameter, k f represents the thermal conductivity of the fluid, and h f represents the heat coefficient.Furthermore, the stream function ψ is incorporated into Equation (1) as follows:

F I G U R E 4
Impacts of Ec and Bi on the temperature graph of the model.

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Impacts of Nt and Nb on the temperature graph of the model.TSHIVHI | 739 ) are defined as follows: Ec represents the Eckert number, Nb represents the Brownian motion parameter, Ha represents the applied magnetics field parameter, Nt represents the thermophoresis parameter, Sc represents the Soret number, λ represents the stretching surface parameter, Ns represents the dimensionless entropy generation, β represents the slip parameter of the surface, and Bi represents the Biot number, Pr represents the Prandtl number.

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I G U R E 6 Impact of Nb on the concentration profile.

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Impacts of Ha and β on the skin friction.

F I G U R E 9
Impacts of Ha and β on the skin friction.TSHIVHI | 743 4.1 | Velocity profile

Figure 2
Figure 2 depicts the influence of β and Ha on the velocity profile.It is evident that the velocity at the surface adheres to the specified boundary conditions by being less than the free stream velocity.With increasing values of β, the surface velocity increases due to enhanced slipperiness.Additionally, heightened magnetic field strength within the nanofluid's boundary layer induces an increase in fluid velocity, a noteworthy occurrence in magnetohydrodynamics with practical significance for various industries.The augmentation of surface slipperiness leads to levated fluid velocity in nanofluid flow, reducing surface friction and potentially improving flow rate and heat transfer characteristics, making it advantageous for numerous industrial and engineering applications.

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I G U R E 10 Impact of Ha and β on the skin friction.

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I G U R E 11 Impact of Ha and β on the skin friction.TSHIVHI | 745

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I G U R E 12 Impact of Ha and β on the skin friction.

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I G U R E 13 Impacts of Ha and β on the skin friction.TSHIVHI | 747entropy generation rate is shown in Figures13-15.It is evident that rising in the values of Ha, Bi, Ec, and Nb lead to an increase in the entropy generation rate.This increase can be attributed to factors such as Joule heating of the stretching surface, viscous dissipation, internal heat transfer within the nanofluid, and Brownian motion.Conversely, the rise in the values of β and Nt will decrease the entropy generation rate.This decrease is a consequence of the surface slipperiness and the distribution of thermophoresis.

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I G U R E 14 Impacts of Ha and β on the skin friction.F I G U R E 15 Impacts of Ha and β on the skin friction.F I G U R E 16 Effects of Ha and β on the skin friction.TSHIVHI | 749 F I G U R E 17 Effects of Ha and β on the skin friction.F I G U R E 18 Effects of Ha and β on the skin friction.
velocity components along the x y ( , ) directions (m/s) U w fluid velocity at the surface (m/s) ∞ U fluid velocity at the free stream (m/s) x Cartesian coordinate along the surface (m) y Cartesian coordinate perpendicular to the surface (m) μ fluid dynamic viscosity (kg m −1 s −1 ) ρ density of the fluid (kg/m 3 ) σ electrical conductivity of the base fluid (S/m) τ heat capacity of the fluid ν f fluid kinematic viscosity (m 2 /s) ϕ dimensionless concentration ψ stream function Ω Bejan number • The concentration of the nanofluid increases with rises in values of Nt, while it declines with higher values of Nb. • The skin friction rises with increase in value of Ha, but reduces with increasing in β, indicating a reduction in nanofluid friction.• The Nusselt number rises with increase in values of β and Bi, but lessens with Ha, Nt, Ec, and Nb.• The Sherwood number rises with increase in values of Ha and Ec, but declines with β and Bi.• The entropy generation rate increases with higher values of Ha, Bi, Ec, and Nb, but decreases with β and Nt.• The Bejan number increases with higher values of β, Bi, and Nb, but decreases with Ha, Ec, and Nt.