In this part, we have now carried out the designed FFBPNN-BFGS Qasi Newton algorithm to review the speed of heat transfer in annular fin by various totally different bodily parameters. The options obtained by the designed algorithm are in contrast with some not too long ago used strategies akin to Finite distinction technique, Finite factor technique and homotopy perturbation method29. Table 2 presents values of the heat transfer price on the tip of the annular fin achieved by totally different strategies (FDM, FEM, HPM, RK-4, FFBPNN-BFGS QN) for 3 totally different instances ((gamma = -0.2, 0.0) and 0.2) with mounted values of the opposite parameters akin to (N_c = 0.5, m_1 = 1.0, N_r = 0.1225, E_G = 0.1, theta _S = theta _S= 0.5). In every case the FFBPNN-BFGS Qasi Newton algorithm achieves the worth which could be very near the analytical options offered by the RK-4 technique. The proportion error between the analytical options and designed technique options for every case are (0.08%), (0.01%) and (0.004%) respectively. The time (seconds) taken by the algorithm to calculate options for these instances are proven in Fig. 3. Furthermore, the graph of absolute errors for 2 totally different instances (N_c = 0.5, m_1 = 1.0, gamma = -0.2, N_r = 0.1225, G =0.025, E_G = 0.1, theta _S = theta _S= 0.4) and (N_c = 0.4472, m_1 = 1.0, gamma = -0.2, N_r = 0.0294, G = 0.016, E_G = 0.1, theta _S = theta _S= 0.5) for 2001 knowledge factors (in [0,1]) are proven in Fig. 4. The minimal values for every case are (2.37times 10^{-11}) and (2.89times 10^{-10}) whereas the imply values for every case lies between (10^{-6}) to (10^{-8}) and (10^{-6}) to (10^{-7}) respectively. The FFBPNN-BFGS QN algorithm demonstrates an distinctive stage of accuracy. This demonstrates the algorithm’s potential as a useful gizmo for analyzing heat transfer and different phenomena the place exact numerical approximations are essential, akin to when making an attempt to foretell the habits of a system.
Table 2 Heat transfer values on the tip of the annular fin: a comparative evaluation of FDM, FEM, HPM, and FFBPNN-BFGS-Quasi Newton technique.Figure 3Time taken by the design approach for 3 totally different instances ((gamma = -0.2, 0.0) and 0.2) with mounted values of the opposite parameters akin to (N_c = 0.5, m_1 = 1.0, N_r = 0.1225, E_G = 0.1, theta _S = theta _S= 0.5). Green strains exhibits the imply time (0.0233 s).Figure 4(a, b) Absolute error evaluation: analytical options and FFBPNN-BFGS Quasi-Newton algorithm.Figure 5Investigating the affect of thermo-geometric parameter ((N_c)) and exponent of variable convective heat transfer coefficient ((m_1)) on dimensionless temperature distribution.In addition, we examine in element the impact of varied variables on the dimensionless heat transfer of annular fins. Specifically, we centered on understanding the results of a change in floor emissivity, heat transfer coefficient exponent, heat manufacturing, thermal conductivity variation, and coefficients between conduction and radiation processes. Figure 5 illustrates the affect of the thermo-geometric parameter ((N_c)) and the exponent of the convective heat transfer coefficient ((m_1)) on the temperature subject. It was discovered that (N_c) had an affect on native temperatures, with a decrease worth of (N_c) akin to a sooner price of conductive heat transfer and, thus, increased native temperatures. This relationship was attributed to the inverse correlation between Nc and thermal conductivity. The materials’s thermal conductivity is enhanced as (N_c) drops, bettering the conductive heat transfer contained in the annular fin. Heat conduction via the fin elevated in consequence, resulting in larger temperatures in the neighborhood. Moving on to the variations in (m_1), which represents totally different heat transfer modes with chosen values of (m_1) = -0.25, 0.25, and 0.33 representing laminar movie condensation, laminar pure convection, and turbulent pure convection, respectively. It was noticed that increased values of (m_1) had been related with increased native temperatures throughout all (N_c) values. This phenomenon will be understood by inspecting the bodily mechanisms associated to the convective heat transfer coefficient. In common, the next worth of (m_1) indicated extra intense convection, whereby the fluid circulation exerted a larger affect on heat transfer. This intensified convection facilitated an elevated transfer of heat from the fin floor to the encompassing fluid, resulting in increased temperatures in the native area.Figure 6Exploring the affect of (N_r) and exponent of variable emissivity ((m_2)) on dimensionless temperature distribution.Figure 6 illustrates the affect of the conductive radiative parameter ((N_r)) and the exponent of the variable emissivity parameter ((m_2)) on the temperature subject. The findings present that (N_r) considerably impacts the temperature distribution. The native temperature distribution diminishes as (N_r) rises. With increased values of (N_r), the floor loses extra heat, which explains this phenomenon. The fin loses radiative heat extra considerably when exponent of the variable radiative parameter rises, which lowers the close by space’s temperature. Additionally, we observed that when (m_2) rises, the native temperature drops. This will be defined by the truth that floor emissivity will increase collectively with (m_2). The native temperature decreases in consequence of sooner power emission brought on by increased floor emissivity. As the radiated power can’t be extra in the low-temperature zone than on the base of the fin, the worth of (m_2) should be adverse. The power emitted from the fin floor behaves as anticipated as a result of to this bodily restriction.Figure 7Exploring the connection between two parameters: the heat technology parameter and the coefficient describing the variation of heat technology, and their affect on the distribution of dimensionless temperature.In Fig. 7, we current the temperature distributions obtained for various values of the heat technology parameter (G) and the coefficient describing the variation of heat technology ((E_G)). In explicit underneath steady-state circumstances, increased values of the heat technology parameter (G) and the coefficient describing the variation of heat technology ((E_G)) trigger a rise in the native temperature subject throughout the annular fin as a result of elevated dissipation of heat to the encompassing setting. Indicated by a rise in the heat technology parameter (G), the annular fin is producing extra heat internally. A larger inside temperature of the fin outcomes from this enhanced heat output. Due to the elevated temperature differential between the fin and its surrounds, heat transmission from the fin to the environment is accelerated. As extra heat is transferred from the fin to the environment, the native temperature subject rises in consequence. Similar to the way it exhibits that the speed of heat manufacturing throughout the fin is fluctuating extra shortly because the coefficient representing the variance of heat technology ((E_G)) will increase. The heat transmission course of is made way more intense by this distinction in heat output. Temperature modifications are introduced on by modifications in heat manufacturing contained in the fin, which happen extra typically. As a end result of the system adjusting to the variations in heat technology, the native temperature subject throughout the fin rises.Figure 8Examining the variations in (theta _a) and (theta _s) on the distribution of dimensionless temperature.In order to regulate a system’s or system’s temperature and ensure it stays under the utmost permissible temperature, it’s customary apply to make use of a heat sink. By facilitating the transmission of heat from the item to the encompassing space, heat sinks are supposed to enhance heat dissipation and keep away from overheating. Figure 8 exhibits the plots of the nondimensional temperature subject for varied values of the convective sink temperature ((theta _a)) and the radiative sink temperature ((theta _s)) alongside the radial path. An improve in a and s in both state of affairs causes the encompassing temperature to rise. The affect of (theta _a) and (theta _s) on the heat transfer processes, notably convective and radiative heat transfer, is the trigger of this temperature improve. The distinction in temperature between the system and the sink decreases as (theta _a) or (theta _s) rises. As a end result, there’s a discount in the speed of heat loss through convective and radiative processes. A larger native temperature outcomes from extra heat being held contained in the system in consequence of the lower in heat loss. The analysis additionally exhibits that the radiative sink temperature ((theta _s)) has a far much less affect than the convective sink temperature ((theta _a)). This discovering implies that the convective heat transfer mechanism has the next affect on the system’s potential to control temperature than the radiative heat transfer technique. To put it one other means, modifications to a have a extra important affect on the native temperature subject.Performance evaluationFigure 9(a, b) Assessments of the convergence of performance worth in phrases of imply sq. error.In this part, we current a complete performance evaluation of the designed algorithm, showcasing its effectiveness in calculating the options for the heat transfer drawback of annular fin. The performance plots of the designed approach for various instances generated by the nntool present precious insights into the algorithm’s capabilities and its potential to approximate the specified outcomes. The performance plots proven in Fig. 9 provide a visible illustration of imply sq. error between the focused and predicted values to evaluate the algorithm’s performance throughout the coaching course of. With a greatest validation performance of (4.4014times 10^{-09}) and (3.2749times 10^{-09}) at epoch 997 and 921 respectively, the algorithm performs exceptionally nicely. This means that the algorithm’s predictions intently match the specified values, resulting in a bit distinction between the anticipated and precise outcomes. Such a formidable end result demonstrates the algorithm’s wonderful accuracy and dependability in figuring out options to the annular fin heat transfer drawback.Figure 10(a, b) Histogram matches with regular distribution curves for the performance worth achieved throughout the multipe executions of the designed approach.Furthermore, the designed FFBPNN-BFGS Qasi Newton technique is executed for 30 unbiased runs to increase the a complete performance evaluation. The greatest performance values obtained from these runs had been collected and analyzed. To present insights into the distribution and traits of these performance values, histograms with corresponding matches had been constructed. The histogram represents the frequency distribution of the values, permitting us to research the distribution sample and establish any underlying traits or patterns as proven in Fig. 10. It is fascinating to notice that almost all of the performance values look like concentrated in the center vary of the x-axis, with fewer occurrences on the extremes. This means that the algorithm’s performance is usually constant and achieves comparatively good outcomes, as indicated by the focus of values across the center vary. In every case, the histogram match revealed a distribution of performance values. The minimal performance values noticed was (5.03641times 10^{-09}), (8.73119times 10^{-09}) whereas the imply and median values had been (1.92607times 10^{-08}), (8.73119times 10^{-09}) and (1.75305times {-08}), (3.40955times 10^{-08}), respectively. This signifies that the algorithm exhibited good performance with variations throughout the runs.Figure 11(a, b) Exploring regression performance: evaluating predicted and precise values.Finally, the regression plots are proven via Fig. 11 to check the focused (anticipated values) and the anticipated values generated by the designed approach. It will be noticed that values of regression ((R^2)) are precisely 1 that exhibits the well-fitted regression highlighting a decent clustering of factors alongside a diagonal line indicating the sturdy correlation between the goal and predicted values.
https://www.nature.com/articles/s41598-024-58595-6
Machine learning-based prediction of heat transfer performance in annular fins with functionally graded materials
