Thermal analysis of extended surfaces using deep neural networks

Nomenclature

a

cross-section area of the fin (m²)

B 0

magnetic field intensity (T)

c p

specific heat capacity (J/kg K)

F

temperature (K)

F ∞

temperature (ambient) (K)

g

acceleration due to gravity (m/s²)

H

magnetic parameter

ι

internal heat generation (W/m³)

J e

conduction intensity

k

thermal conductivity (W/m K)

k a

thermal conductivity (ambient)

K

permeability (m²)

L

length of the fin (m)

M

thermo-geometric parameter

n

multi-boiling heat parameter

N F

temperature ratio

N P

porosity parameter

N R

thermal radiation parameter

P

perimeter of the fin (m)

Q

heat generation parameter

u

velocity in the porous fin (m/s)

V

velocity vector (m/s)

w

width of the fin (m)

x

Cartesian coordinate

Z r

fin efficiency rate

Greek symbols

α 0 , α 1

thermal coefficient (1/K)

β

thermal expansion coefficient (1/K)

η

heat transfer coefficient (W/m² K)

η b

heat transfer coefficient (base)

ε

emissivity coefficient

ρ

density of the fin (kg/m³)

ν

kinematic viscosity (m²/s)

ϕ 0 , ϕ 1

temperature ratio

σ

electrical conductivity (S/m)

σ ∗

Stefan-Boltzmann constant

ξ

thermal coefficient (1/K)

ξ ∗

temperature ratio

ζ

dimensionless length

Γ

dimensionless temperature

Subscripts

∞

ambient condition

w

reference (base)

1 Introduction

Understanding the behaviour of a system that is influenced by convective, magnetic, and radiative heat transfer conditions is a long-standing and crucial aspect in the field of thermal transport. This subject has considerable significance in various industries, including aerospace, construction, energy, and manufacturing, thus underscoring its broad and practical utility. One example of such a system is the fin. The relevance of fins is evident in our daily lives, influencing cooling processes in refrigerators, the radiator systems in cars, the heating elements in geysers, the thermal fins used in heat dissipation in our computers, and numerous other applications around us. In these applications, fins are crucial to enhancing heat transfer, improving efficiency, and preventing overheating. Fins, or extended surfaces, improve heat transfer efficiency from surfaces that typically undergo cooling via natural or forced convection by gases or fluids. Heat is conducted from the heat source to the fin’s surface through thermal conduction. Heat is usually transferred by radiation, convection, and conduction. On the other hand, heat from the surrounding medium, the magnetic element, and the material porosity can all contribute to thermal dissipation. Fins have three fundamental characteristics: (i) they are metallic; (ii) they have a variety of structures; and (iii) the fin’s length is significantly longer than its diameter [1].

The mechanisms of fins have been the subject of numerous investigations; one noteworthy study is by Gorla and Bakier [2], which examined heat dissipation due to radiation and conduction in a rectangular porous fin. The study used the Darcy model to simplify the momentum equation. The fin system studied by Gorla and Bakier [2] takes into account conduction and radiation-induced thermal equilibrium while ignoring internal heat generation and the role of magnetic fields in heat enhancement. The significance of porosity in fins in a parallel-surface channel was studied by Kiwan and Al-Nimr [3]. Numerical analyses of the heat enhancement resulting from a porous fin were carried out in the study. A 100% decrease in the fin material’s weight was noted, and it was concluded that the fin material has the same performance as a solid fin. The effect of the thickness of a porous fin on the thermal system was examined in the study. Hatami and Ganji [4] investigated the effect of geometrical and material properties in the design of porous fins using the least squares approach. The results show that fins made of aluminum have a high rate of heat transmission, which makes them appropriate for extended surface design. Gireesha et al. [5] investigated the effects of stretching and shrinking on the temperature distribution of a wet longitudinal fin. According to the analysis, the negative impacts of motion and internal heat generation on the fin’s heat transfer rate can be lessened by putting in place a shrinking mechanism on the extended surface. Other studies in this direction include the work of Atouei et al. [6], Nicholls et al. [7], Aziz and Lopez [8], Razani and Ahmadi [9], Buonomo et al. [10], and the references therein.

Research into extended surfaces has made a substantial contribution to understanding the accuracy and efficiency of semi-analytical and numerical discretization methods. These studies are motivated by the complex dynamical equation (a second-order nonlinear ordinary differential equation) that governs heat transfer in extended surfaces. These equations often do not have simple closed-form solutions, especially when they are nonlinear. For instance, Aderogba et al. [11] used the generalized fins model to evaluate the performance of the nonstandard finite difference numerical discretization technique. It was demonstrated that the numerical approach maintained the dynamics of the differential equation with good accuracy. The hybridization of the differential transform and Pade approximation method was used by Sowmya et al. [12] to investigate heat transfer in rectangular fins under the influence of a magnetic field. Kasali et al. [13] used the spectral-based local linearization method introduced by Motsa [14] to investigate the effects of the multi-boiling parameter on linear and nonlinear thermal conductivity on thermal flux on an extended surface. The study also assessed the accuracy of the spectral-based method. Akindeinde [15] investigated the efficiency and accuracy of the Parker-Sochacki semi-analytical method using the dynamical nonlinear fin model as a case study. Sobamowo [16] studied porous wet fins subjected to magnetic body forces and thermal radiation using the conventional finite difference approach. To predict heat dissipation in an extended surface subjected to conductive, convective, and radiative conditions, Zhanga and Li [17] presented a novel Taylor approximate approach. The study evaluated the accuracy of the method by contrasting their approach with explicit solutions in limiting cases, such as those without a radiation effect. Roy et al. [18] examined the model of an accelerating triangular fin and the performance of a modified decomposition method.

Machine learning has revolutionized various fields, including computer vision, natural language processing, and speech recognition, among many others. Recently, researchers have focused on training computer systems to process data and information in a manner similar to the human brain. This technique, known as the neural networks method, is a part of a broader category of machine learning processes called deep learning. Lagaris et al. [19] initially proposed solving an ordinary differential equation using neural networks. The underlying principle involves training a neural network framework to meet the conditions stipulated by a differential equation. Waseem et al. [20] analyzed the temperature distribution in a porous fin through artificial neural networks coupled with the cuckoo algorithm. Tan et al. [21] used the neural network method to analyze heat efficiency in a compact exchanger. Goud et al. [22] used a Levenberg-Marquardt neural network technique to study heat transfer in a trapezoidal-shaped fin. They took into account assumptions on heat conductivity that depart from traditional Fourier principles. Kumar et al. [23] investigated the heat efficiency mechanism in a wavy wall with an artificial neural network. Shafiq et al. [24] used the artificial neural network method to estimate key parameters related to engineering physical quantities in a bioconvection nanofluid flow over a stretching surface containing gyrotactic microorganisms. In other related studies, Shafiq et al. [25] used artificial neural networks to forecast the potential number of deaths from the COVID-19 pandemic in Italy. In addition, Shafiq et al. [26] investigated the effects of Soret and Dufour convective heat transfer using an accelerating needle as the focal point, with the simulations conducted through artificial neural networks.

Notably, none of the many studies published in the literature have examined the application of deep learning methods to the analysis of heat dissipation and temperature distribution in a rectangular porous fin subjected to radiation, magnetic effects, and quadratic-type thermal conductivity. To this end, this study examines heat dissipation in a porous extended surface (fin) exposed to radiation, convection, and conductivity in a magnetic environment. The dynamical system is analyzed using a deep neural network (DNN) and the physical implications of the findings are reported. The spectral-based local linearization method, introduced by Motsa [14], is used to validate the accuracy of the result obtained using the deep learning approach.

2 Analysis of the mathematical model

This section begins by developing the mathematical model. In the context of a porous fin, the following dynamical cumulative heat addition equation describes the equilibrium of thermal transport [13,27]:

Eq. (1) represents the thermal energy balance within a closed system. Heat conduction into the base of the fin element at position x is the sum of heat conduction into the element at the fin tip ( x + δ x ) , heat convection from the element, heat radiation from the element, heat transfer due to the magnetic field, internal heat generation within the element, and the contribution from the porous term. The porous media is homogeneous and saturates with a single-phase fluid. The study focuses on a convective-conductive-radiative porous fin with length L . The fin is exposed on both faces to convection with a temperature F ∞ and subjected to a uniform magnetic field, as illustrated in Figure 1. The longitudinal fin is constrained in length with an insulated tip, creating an adiabatic system where no heat is transmitted from the tip. The temperature at the base of the fin is denoted as F = F w and at the tip takes the form of a Neumann boundary condition, d F d x = 0 . The interaction between the porous medium and the single-phase fluid is formulated using the Darcy model. Body force due to the magnetic field is experienced in the y -direction. It is assumed that the radiative heat flux behaves similarly to an optically thick gas within the porous medium. The effect of the electrical field due to polarization is considered to be negligible, i.e., E = 0 . If this phenomenon takes place in a steady-state condition, then Eq. (1) becomes [13]:

Figure 1

Geometrical configuration of the fin.

Here, u ( x ) represents the movement of the buoyancy-driven flow at any location x , and J e is the intensity of the conduction responsible for the thermal transport due to electric current, given, respectively, as [1]

By substituting Eq. (3) into Eq. (2), we obtain

where δ a = P δ x .

According to Ohm’s law, the current density or conduction intensity is obtained as J e = σ ( E + V × B ) , where E denotes the electric field, V is the charge carrier velocity, which is accounted for using the model u = g β K ν ( F − F ∞ ) , since porosity is considered, σ is the electrical conductivity, and B is the magnetic force. Thus, if the electric field is negligible, that is, E = 0 ;  we have

Here, V = ( u , v ) , Eq. (5) indicates that J e = σ ∣ u ∣ ∣ B ∣ sin θ n ˆ , and thus,

Note that the velocity of the charged particle is perpendicular to the magnetic field, and thus, sin 2 θ = 1 . Hence [1],

Upon division by δ x , Eq. (4) reduces to

where J e × J e σ = σ B 0 2 u ( x ) 2 . Following Fourier’s law of heat conduction [28], heat conduction taking place within the fin can be mathematically described as [4]:

Substituting Eq. (9) into Eq. (8), we have

and using the relation given in Singha et al. [29], Oguntala and Abd-Alhameed [30], and Sobamowo et al. [31], we have

After some simplifications, Eq. (10) becomes

with associated boundary conditions given as [2]:

We further introduce the following dimensionless variables and parameters:

By incorporating the nondimensional variable in Eq. (14) into Eqs (12) and (13), we obtain the following single variable differential equation:

where K 1 ( Γ ) = 1 + ϕ 0 Γ + ϕ 1 Γ 2 and its associated insulated boundary conditions [2]

3 Method of solution

In this section, we introduce the implementation of a DNN to solve Eq. (15) subject to the associated boundary conditions (16). Subsequently, we derive the algorithm and describe the implementation of the spectral local linearization method (SLLM) to solve the differential equation.

3.1 DNNs approximation of the model solution

This study uses a deep and fully connected feedforward artificial neural networks framework to solve the nonlinear differential Eq. (15) with the associated boundary conditions (16). Suppose the differential Eq. (15) is defined as follows:

(18) G ζ , Γ ( ζ ) , d Γ ( ζ ) d ζ , d 2 Γ ( ζ ) d ζ 2 = 0 , ζ ∈ [ 0 , 1 ] ,

subject to the conditions in Eq. (16), where G is a nonlinear operator of Γ ( ζ ) and its derivatives. The collocation technique is used to discretize the domain and boundary of the problem into a finite number of discrete points. In this study, we use the set of Chebyshev-Gauss-Lobatto points,

ζ ˜ j = − cos π j M ζ , j = 0 , … , M ζ ,

mapped onto ζ ∈ [ 0 , 1 ] through the linear mapping, ζ = 1 2 ( ζ ˜ + 1 ) . The nonlinear differential Eq. (18) is then transformed into its discrete equivalent

(19) G ζ j , Γ ( ζ j ) , d Γ ( ζ j ) d ζ , d 2 Γ ( ζ j ) d ζ 2 = 0 , ζ j ∈ 1 2 ( ζ ˜ + 1 ) ≡ [ 0 , 1 ] ,

subject to the conditions

(20) Γ ( ζ 0 ) = 1 and d Γ ( ζ M ζ ) d ζ = 0 .

In this study, we consider the ansatz, Γ t ( ζ ; w , b ) , which satisfies the boundary conditions (20) for the solution of the differential Eq. (19). Here, w and b are the weights and biases, respectively, of the DNNs framework. Therefore, the problem of approximating the solution of the differential equation then becomes the unconstrained minimization problem,

(21) min w , b ∑ ζ j ∈ [ 0 , 1 ] G ζ j , Γ t ( ζ j ; w , b ) , d Γ t ( ζ j ; w , b ) d ζ , d 2 Γ t ( ζ j ; w , b ) d ζ 2 2 .

In the proposed technique, the trial solution, Γ t ( ζ ; w , b ) , is defined as follows:

(22) Γ t ( ζ ; w , b ) = 1 + ζ N ( ζ ; w , b ) − N ( 1 ; w , b ) − d N ( 1 ; w , b ) d ζ .

The trial solution, Γ t ( ζ ; w , b ) , depends on the output, N ( ζ ; w , b ) , of the neural networks framework and is chosen to satisfy the boundary conditions, (16), of the differential model. N ( ζ ; w , b ) is the single output of a feedforward DNN architecture fed with a single input vector, ζ j , with parameters, w and b , which are the weights and biases of the neural network architecture.

The neural network architecture is a L + 1 fully connected layers, as shown in Figure 2, which consist of the first layer, which is the input through which the input vector ζ j is fed, the last layer, which is the output layer and L − 1 hidden layers. Activation functions are introduced in the hidden layers to introduce nonlinearity to the neural network architecture. In this study, the sigmoid activation function, ϱ ( q ) = ( 1 + e − q ) − 1 , is used; however, other functions such as the hyperbolic tangent, rectified linear units and exponential functions can be used. Each neuron in each hidden layer and the output layer has a bias, and the neurons in layers l − 1 and l are connected by the weights. The weights which connect neuron k in layer l − 1 to neuron i in the l th layer are represented by w i k l , and the bias associated with neuron i in layer l is denoted by b i l .

Figure 2

Schematic diagram of a fully connected DNN.

The neural network framework is expressed mathematically as follows:

(23) q 0 = ζ , the input layer q l = ϱ l ( w l q l − 1 + b l ) , 1 < l < L − 1 , the hidden layers q L = w L q L − 1 + b L , the output layer ,

where the result of the output layer, q L = N ( ζ ; w , b ) , is used in the trial solution (22). To find the weights and biases in the layers, we minimize the residual function

(24) ℛ ( w , b ) = ∑ j = 0 M ζ G ζ j , Γ t ( ζ j ; w , b ) , d Γ t ( ζ j ; w , b ) d ζ , d 2 Γ t ( ζ j ; w , b ) d ζ 2 2 ,

using gradient descent optimization with adaptive moment estimation. Lagaris et al. [19] showed how to obtain the automatic partial derivatives of the residual function with respect to the weights and biases. However, in this study, we used the GradientTape sub-module of Python’s TensorFlow package. The weights and biases are then updated through backward propagation. The neural network is completely implemented with Python programming language using the TensorFlow and NumPy packages. The algorithm for the neural networks approximation of the solution of Eq. (15) is given in Algorithm 1. Figure 3 shows the training loss curve of the neural network architecture. The figure depicts the plot of the value of the loss function (24) against the number of training or epochs. The loss function measures how well the trial function (22), which is dependent on the neural network framework (23), approximates the solution of the differential Eq. (15). During the training process, the network adjusts its parameters, viz. weights and biases, iteratively to minimize the loss and improve the predictability of the trial solution as an approximate solution of Eq. (15). We note here that the weights and biases are initialized using normal distributions centred at zero with standard deviations of 0.5.

Figure 3

Training loss curve of the neural networks algorithm with 6 hidden layers and 50 neurons in each hidden layer.

Algorithm 1. Step-by-step algorithm for solving Eq. (15) using a fully connected deep neural network.
1:	procedure DNN approximation of the solution of Eq. (15)
2:	Define the variable q 0 = ζ as input.
3:	Define the number of hidden layers and number of neurons in each layer.
4:	Initialize the weights and biases for each layer and set them as trainable parameters.
5:	Define the neural networks architecture:
6:	for l = 1 to L do
7:	q l = ϱ l ( w l q l − 1 + b l )
8:	end for
9:	Define the trial solution: Γ t ( ζ ; w , b ) = 1 + ζ ( N ( ζ ; w , b ) − N ( 1 ; w , b ) − N ′ ( 1 ; w , b ) ) .
10:	Define the loss function ℛ ( w , b ) in Eq. (24) in terms of the trial function.
11:	Define the model parameters, ϕ 0 , ϕ 1 , M , n , ε , N R , N F , H , N P , Q , ξ ∗ as untrainable parameters.
12:	Train the neural network model to optimize the weights and biases by minimizing the residual function.
13:	end procedure

4 Results and discussion

Insights into the thermal dynamics of extended surfaces are provided through parametric analysis of some key parameters in Eq. (15) using the DNN method. A comparison of the results obtained by DNN and SLLM is made to validate the results. If not specified otherwise, the analysis of the results was carried out using the following parameter values: M = 1.0 , N R = 0.10 , N F = 0.10 , n = 1.0 , ε = 1.0 , ϕ 0 = 0.10 , ϕ 1 = 0.30 , N P = 1.0 , H = 1.0 , Q = 0.2 , and ξ ∗ = 1.0 . The figures presented in this section depict the output obtained from using the DNN to learn the mathematical model in Eq. (15) with its conditions given in Eq. (16).

Figure 4 depicts the effect of linear and nonlinear thermal conductivity variations on the dynamics of the extended surface. Specifically, the figure illustrates how temperature distribution along the surface is influenced by the variations in the linearity and nonlinearity of the thermal conductivity. The distinction between linear and nonlinear thermal conductivity variations is significant for temperature distribution along extended surfaces. The linear relationship suggests that the thermal conductivity of the fin changes uniformly with temperature. On the other hand, the nonlinear thermal conductivity variation has a much more complex, nonuniform or even nonmonotonic relationship with the temperature. It is apparent from Figure 4 that the linear variation of thermal conductivity results in a more uniform temperature distribution, particularly in the vicinity of the fin’s base. This uniform temperature distribution at the fin’s base can be attributed to the consistent and predictable changes in thermal conductivity along the surface. In contrast, the nonlinear variation introduces deviations in temperature distribution along the surface, leading to nonuniform temperature profiles. This nonuniformity may arise due to nonlinear effects such as phase transitions, variations in the material property of the fin or localized heat sources, which can disrupt the thermal equilibrium along the surface and lead to nonuniform temperature profiles, which may sometimes be necessary for the design of fins. Regardless of whether thermal conductivity variations are linear or nonlinear, the overall trend in temperature distribution remains consistent: as both parameters increase, the absolute temperature along the extended surface increases. This is because higher thermal conductivity values enable more efficient heat transfer, leading to higher temperatures along the surface.

Figure 4

Effect of (a) linear, ϕ 0 , (b) nonlinear, ϕ 1 thermal conductivity on temperature distribution on the fin.

The impact of both radiation and internal heat generation on the temperature of the fin are shown in Figure 5. Increasing the thermal radiation parameter reduces the temperature observed on the extended surface. This phenomenon arises because enhanced thermal radiation facilitates heat transfer away from the surface. A significant decrease in the temperature of the fin at its tip is experienced as the fin generates more internal heat. The increased internal heat generation parameter leads to a higher temperature change at the tip of the extended surface. This significant temperature gradient on the fin causes heat to be transferred more rapidly from the base of the fin to the tip. Consequently, the higher the value of the heat generation parameter, the lesser the temperature value at the tip of the extended surface since heat is effectively conducted and convected away from that region.

Figure 5

Effect of (a) radiation parameter, N R , and (b) heat generation parameter, Q on the temperature distribution of the extended surface.

Figure 6 shows the effect of the multi-boiling and magnetic field strength parameters on the fin. The choice of the values of the multi-boiling parameters delineates distinct heat transfer regimes occurring on the extended surface. By varying this parameter, we can simulate and analyze the heat transfer behaviour under different boiling conditions, providing insights into the dynamics and thermal performance of the extended surface. When n = − 1 4 , it indicates laminar film boiling, where a thin film of vapour forms and envelops the fin surface. The temperature distribution in this regime is relatively low due to the insulating effect of the vapour film. On the other hand, n = 1 4 corresponds to natural convection, where heat transfer occurs primarily through fluid motion driven by density variations caused by temperature gradients. Finally, n = 2 represents nucleate boiling, where discrete vapour bubbles form on the extended surface and detach to facilitate heat transfer. This regime is associated with high heat transfer coefficients due to the effective heat removal by vaporization and bubble motion as captured in Figure 6. Increasing the magnetic field parameter has a notable effect on the temperature distribution at the tip of the extended surface. As the parametric value of the magnetic field parameter increases, the temperature at the tip decreases. This dynamic is primarily driven by the amplified influence of the magnetic field due to a rise in the Hartman number. The Hartman number characterizes the ratio of magnetic force to viscous force and measures the strength of the magnetic field relative to fluid motion. An intensified magnetic field exerts a damping effect on the convective heat transfer process. Essentially, the magnetic field inhibits the movement of fluid particles, reducing their ability to carry heat away from the surface efficiently. Consequently, as the magnetic field becomes stronger, the convective heat transfer rate decreases, reducing overall heat transfer on the extended surface.

Figure 6

Effect of (a) multi-boiling, n , and (b) magnetic field strength, H , parameters on the fin’s temperature distribution.

The efficiency rate of the extended surface, as evaluated using both the DNN approach and spectral-based method, is presented in Table 1. Under conditions where there is no internal heat generation and the thermal heat conductivity of the extended surface is constant, it is observed that increasing the thermo-geometric, magnetic field, and porosity parameters results in a decrease in the fin efficiency rate. It was also found that variable thermal conductivity enhances the efficiency of the extended surface. This result suggests that if a fin is designed with the ability to adjust its thermal conductivity in response to changing conditions dynamically, it can improve its heat transfer performance. This underscores the importance of material selection and highlights the potential benefits of using materials with adaptable thermal properties in heat transfer applications. However, the efficiency of the extended surface is suppressed in designs characterized by an apparent increase in heat. This implies that factors such as elevated ambient temperature or enhanced heat transfer rates negatively impact the efficiency of the extended surface.

Table 1 also validates the accuracy and reliability of the methodology used in this study. A comparative analysis shows that the outputs generated by the neural network closely align with those derived through the spectral-based method. The deviation between the two sets of outputs remains within 1 0 − 4 , underscoring the relatively high degree of precision exhibited by the DNN outputs. This result highlights the robustness and efficacy of the DNN in producing solutions consistent with those obtained through established traditional numerical methods such as spectral methods. By demonstrating its ability to closely replicate the results of conventional numerical methods, the neural network technique establishes itself as a reliable predictive modelling and analysis tool. This validation not only bolsters confidence in the methodology adopted for this study but also underscores the potential of neural network-based approaches in various scientific and engineering applications. Its ability to accurately capture complex relationships and patterns makes this approach valuable for scientific research, particularly in seeking efficient and reliable solutions to a wide range of differential models.

5 Concluding remarks

The primary objective of this study is to develop a deep learning algorithm for approximating the solution of a differential equation that describes the intricate dynamics of heat transfer on extended surfaces. The study also seeks to elucidate the physical implications of some physical parameters arising from the differential model. A DNN architecture was designed to approximate the solution of the nonlinear differential model to achieve these objectives. In order to measure the accuracy and reliability of the neural network technique, a spectral-based numerical method was used to benchmark the result of the neural network technique. The results obtained were accurate, validating the efficiency of the proposed method as a reliable tool for solving and analyzing nonlinear differential models, especially those describing heat transfer in extended surfaces. The finding of this study provides several key insights regarding the thermal behaviour of extended surfaces. Notably, it was observed that variations in the thermal radiation parameter exert a comparable influence on reducing the fin temperature at the tip when compared to changes in the internal heat generation parameters. The study further showed that using materials with varying thermal conductivity can enhance the efficiency of an extended surface, thus optimizing the heat transfer process. Overall, these findings are significant in designing, managing and optimizing extended surfaces.