Experimental investigation and ANN-based prediction of average Nusselt number for a flat surface subjected to double inclined air jets

Abstract

This study explores the thermal behavior of double-inclined circular air jets impinging on a flat plate. Using a jet diameter (D) of 9.5 mm, experimental analyses are carried out to examine the impact of jet inclination angles (ranging from 0° to 60°), spacing between jets (2D–8D), Reynolds numbers (10,000–40,000), and the distance from the nozzle to plate (2D–8D) on the heat transfer over the impingement plate. An Artificial Neural Network (ANN) is constructed using the experimental dataset to estimate the average Nusselt number based on these parameters. Thermal infrared imaging is employed to capture surface temperature distributions during the experiments. The results indicate that the most effective jet inclination angles, generally between 10° and 20°, enhance heat transfer by optimizing both direct impingement and interaction between jets. Optimal jet spacing, found to be in the range of 3D to 4D, contributes to improved thermal performance with reduced interference and a more uniform temperature field. The ADAM optimizer is used to train the ANN design, which has 16 hidden layers with 24 neurons each. The model employs linear activation function in the output layer, ReLU activation functions in the hidden layers, and a feedforward structure combined with backpropagation to iteratively refine weights and biases. The resulting model demonstrates high prediction accuracy, with R² and r values approximately 1 and low error values: MSE = 6.265, MAPE = 0.796%, MSLE = 9.82e-05, and log-cosh loss = 1.419.

Introduction

Air jet impingement, which utilizes a high-speed airflow directed at a targeted surface to remove localized heat, has emerged as an effective cooling approach for applications involving intense heat flux. The high-velocity turbulent flow created by the jets significantly thins both the thermal and velocity boundary layers, enhancing convective heat transfer¹. Over the years, considerable effort has been aimed to enhance the thermal efficiency of jet impingement systems, covering both single and multiple jet configurations^2,3. Studies have also explored angled impingement jets from circular and planar slot nozzles⁴. A broad overview of the progress in jet impingement techniques—ranging from passive enhancements to active control methods—has been presented in recent literature⁵.

Numerous factors impact the effectiveness of heat transfer in jet impingement systems, such as the Reynolds number (Re), degree of jet confinement⁶, nozzle geometry⁷, jet inclination angle, inlet air temperature, surface texture of the impingement area⁸, jet arrangement⁹, and different nozzle configurations. Inclined jets are commonly employed in a wide range of thermal management systems due to their considerable impact on the Nusselt number distribution across the impingement plate. As the angle of the jet deviates from perpendicular, the distribution pattern becomes increasingly complex. This complexity stems from the intensified turbulence generated by the oblique flow, which disrupts boundary layers more aggressively at higher inclination angles. In real-world cooling scenarios—particularly where spatial constraints are a concern—jets are frequently angled toward the surface. A typical application can be found in gas turbine blade cooling, where arrays of inclined jets are used to direct cooling air onto the curved leading-edge surfaces¹⁰.

A vast array of traditional performance studies regarding jet impingement problems often necessitates extensive datasets, leading to a heavy investment of time and effort. Most traditional performances of inclined jet impingement incorporate numerous datasets based on experimental work that take a significant amount of time and effort. Conducting experiments for inclined jet cooling is often labor-intensive and time-consuming due to the involvement of numerous influencing parameters, including both independent and interacting variables that affect thermal performance. To overcome these challenges, researchers have adopted computational methods for analyzing heat transfer behavior. These methods utilize various algorithms to solve intricate systems of differential equations, often requiring significant computational resources. Through simulation, it becomes possible to explore a wide range of configurations and enhance the design of jet impingement systems for better efficiency and effectiveness. This computational approach not only reduces the reliance on extensive physical testing but also provides insightful information on the complex fluid and thermal dynamics involved. As a result, it has become a key enabler for technological progress and innovation in jet cooling applications¹¹. The use of such algorithms allows for detailed problem-solving despite the high computational demands.

Thus, attention has been given by researchers to utilizing artificial neural networks (ANN) as a universal tool for analyzing inclined jet impingement and extracting crucial information patterns from multifaceted data networks. A wide array of practical software tools utilizing ANNs have been created¹². ANNs are capable of efficiently and precisely estimating the outcomes of complex, non-linear physical systems by processing input data and performing interpolation. This can be achieved without relying on detailed mathematical models or conducting extensive experimental procedures¹³.

ANNs possess the unique capability to predict the desired result of any complex functional system without requiring extensive experimentation or explicit mathematical representations, thanks to their ability to process information and interpolate data rapidly and accurately. ANN techniques are increasingly being adopted across various scientific and engineering disciplines because of their ease of use, independence from derivative information, computational efficiency, robustness, and capability to solve complex and nonlinear problems at high speed¹⁴.

Numerous studies have been carried out to comprehend the thermal performance of inclined air jets. For instance, an experimental study was conducted by Talapati et al.¹⁵ to examine the influence of circular air jet inclination on heat transfer behavior over an impingement plate. Utilizing infrared thermography alongside the thin foil method, they tested inclination angles of 45°, 60°, and 75°, with Re varied from 12,000 to 28,000 and nozzle-to-surface separations from 1 to 6D. Their results indicated that lower inclination angles promoted more uniform thermal distribution. In a different study, Attalla et al.¹⁶ examined the thermal effects of dual inclined circular jets at angles of 0°, 10°, 20°, 30°, 45°, and 60°, operating within a Re range of 10,000 to 40,000. The study varied both the jet-to-plate distance and the spacing between the jets from 2 to 8D. Their findings highlighted that optimal average heat transfer occurred at inclination angles between 10° and 20°.

Multiple studies have utilized ANNs to model, optimize, and predict the key factors influencing jet impingement heat transfer. These efforts predominantly focused on variations in nozzle design and fluid operating parameters. Typically, ANN models use geometrical configurations and flow conditions as input features¹⁷. As the dimensionality of input and output layers increases, the relationship between these variables becomes more intricate. Enhancing model accuracy often involves experimenting with different training methods and activation functions¹⁸. For example, Fawaz et al.¹⁹ applied an ANN to determine both average and stagnation Nusselt numbers in a setup where a helically coiled air jet impinged on an impingement plate. Their study spanned Re from 5000 to 30,000, nozzle-to-plate distances between 2 and 8D, and helical angles of 0°, 20°, 30°, 40°, and 60°. The ANN architecture used a three-layer configuration with hidden neurons structured as 14–10–8, yielding mean absolute percentage errors of 2.35% for the average and 2.52% for the stagnation Nusselt numbers. Likewise, Jahromi and Kowsary²⁰ proposed three highly accurate backpropagation-based ANN surrogate models to assess compression power, average Nusselt number, and thermal uniformity in jet arrays for turbine blade cooling. They employed the NSGA-II algorithm for optimization, while TOPSIS and LINMAP multi-criteria decision-making tools guided the selection of optimal designs. Sensitivity analysis revealed that both compression power and average Nusselt number were significantly affected by the Re. In another study, Sánchez-Chica²¹ combined ANN modeling with a genetic algorithm to refine the performance of an air jet impingement cooling system, taking blower characteristics into account to approach peak system efficiency. Additionally, Kim et al.²² introduced an ANN-based surrogate model for predicting pumping power and thermal resistance in a micro-post jet impingement setup. Their findings indicated that a reduction in the number of jets led to an increase in the jet exit velocity, which strengthened forced convection and consequently lowered thermal resistance.

This research offers a significant advancement in heat transfer by deepening the understanding of double-inclined air jet impingement—a technique crucial for improving localized heating and cooling performance. An extensive experimental study is conducted to evaluate the thermal behavior of double-inclined air jets striking a flat surface, with particular attention given to the interaction among key geometrical and flow-related parameters. These include jet inclination angle (θ), spacing between jets (S/D), Reynolds number (Re), and nozzle-to-surface distance (H/D). The results provide important insights into how these variables influence convective heat transfer, as reflected by the average Nusselt number (Nu), a key nondimensional indicator of thermal transfer efficiency. By systematically adjusting each parameter and examining the resulting heat transfer patterns, the study compiles a rich dataset capturing the intricate dynamics of various jet configurations. Furthermore, an advanced ANN model is developed using the present experimental data to accurately predict the average Nusselt number based on Re, θ, S/D and H/D. This model highlights the practical application of machine learning in thermal analysis, offering a powerful tool for optimizing jet impingement setups. Together, the experimental findings and ANN-based predictive model contribute significantly to the advancement of modern cooling technologies, offering valuable guidance for engineers and researchers across industries such as electronics cooling, aerospace thermal control, and metallurgical processing.

Experimental setup and method

Experimental setup

Figure 1 displays both the schematic layout and a real photograph of the experimental apparatus utilized in this research. Compressed air is supplied through a controlled system, managed by a sequence of valves to regulate flow. Before reaching the orifice-type flow meter, the air stream passes through a filtration unit and a pressure regulator to ensure it remains clean and maintains consistent pressure. Fine adjustment of the air mass flow rate is achieved using a precision needle valve positioned upstream of the digital flow meter. The air then enters a plenum chamber, which distributes the flow toward a pair of inclined jets. To reduce turbulence and achieve uniform flow distribution, a mesh screen is installed inside the plenum. The nozzles, fabricated from stainless steel, feature an internal diameter of 9.5 mm and a length-to-diameter (L/D) ratio of 40, which helps establish a fully developed velocity profile at the nozzle exit^23,24.

Fig. 1

Experimental setup: (a) schematic diagram illustrating the key components, (b) real photograph of the complete setup. 1. Air compressor 2. Pressure regulator 3. Air filter 4. Ball valve 5. Digital flow meter 6. Plenum chamber 7. Assembly of nozzles 8. Impinging flat plate 9. Heating system 10. Electrical cables 11. Thermal camera 12. PC with data acquisition unit 13. Channel frame.

The nozzles are installed on a high-precision positioning stage that allows for fine-tuned adjustments. It provides accurate control in both the horizontal and vertical directions, with a positioning resolution of 0.01 mm. The jet inclination can be modified relative to the Y-axis in 0.1° increments, where an angle of 0° represents jets directed perpendicularly to the surface. This setup offers precise manipulation of the nozzle-to-target distance, jet inclination angles, and the spacing between adjacent nozzles.

The impingement surface consists of an Inconel metal plate measuring 325 mm × 525 mm × 0.5 mm. To ensure mechanical stability, the plate is firmly held between two copper bus bars, with approximately 8 mm of each edge secured. Due to its small thickness, the plate exhibits very limited lateral heat conduction, resulting in a nearly uniform heat flux distribution. For improved thermal imaging accuracy, the bottom surface of the plate is coated with a thin layer of ACRYLIC AG:525 paint to increase its emissivity. This treatment allows the surface to approximate blackbody behavior, with an emissivity close to 0.95, which is crucial for reliable infrared temperature measurements^23,24,25.

The heating of the plate is achieved using a DC power supply integrated with a voltage stabilizer to maintain consistent output. To verify uniform thermal conditions at the nozzle exits, jet temperatures are measured using a hot-wire thermo-anemometer (CFM Thermo-Anemometer, Model 407119A). Thermal imaging of the impingement surface is conducted with an infrared (IR) camera (TROTEC IC120LV), which is installed beneath the heated plate, directly opposite the jets. The IR camera records thermal data at a resolution of 525 × 325 pixels, with each pixel representing an area of 0.35 mm on the plate. A schematic view labeled “View A” illustrates the geometric layout of the double-inclined jet system and the spatial arrangement of its components.

Procedure and data reduction

The temperature profile across the impingement surface is acquired by analyzing digitized infrared (IR) thermal images, which are processed using IC-Report Standard software. For every test configuration, five thermal images are captured, and their data is averaged to determine the surface temperature. Based on these temperature measurements, the heat transfer coefficient is computed using the subsequent formula:

$$h=\frac{{Q}_{conv}}{A\left({T}_{w}-{T}_{jet}\right)}$$

(1)

The local surface area (A) represents the combined area of two adjacent cells, where each cell is associated with a region affected by the adhesive electrical resistance. Additionally, the convective heat transfer rate (${Q}_{conv}$) is computed using the following expression:

$${Q}_{conv}={Q}_{elec}-{Q}_{loss}$$

(2)

The local electrical power (${Q}_{elec}$) dissipated by the Joule effect can typically be defined as:

$${Q}_{elec}=VI$$

(3)

Nevertheless, the loss in heat transfer (${Q}_{loss}$) is evaluated using the following expression:

$${Q}_{loss}={Q}_{rad}+{Q}_{nat}$$

(4)

The radiation heat transfer, denoted as ${Q}_{rad}$, is determined using the following equation:

$${Q}_{rad}=\left({\varepsilon }_{1}+{\varepsilon }_{2}\right)A\sigma \left({{T}_{w}}^{4}-{{T}_{\infty }}^{4}\right)$$

(5)

where ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ are emissivity of both sheet sides.

The IR camera is set at a distance of one meter and oriented perpendicularly to the heated surface to capture a projected image onto the curved surface. To improve the surface emissivity and enhance the accuracy of the IR images, the outer layer of the heated surface is covered with a thin layer of paint that possesses high emissivity. The heated plate is treated as a black surface with an emissivity ${\varepsilon }_{1}$ ranging from 0.95 to 0.97, while the emissivity of the internal unpainted surface (${\varepsilon }_{2}$) is approximately 0.35. The measurements assess the heat transfer through radiation, estimating it to be between 2 and 9% of the total heat within the analyzed range of Reynolds number.

The natural convective heat transfer (${Q}_{nat}$) is negligible in comparison to forced convection, as noted in²⁴. The heat dissipation due to the natural convection is correlated according to the following equation.

$${Q}_{nat}=A{h}_{\infty }\left({T}_{w}-{T}_{\infty }\right)$$

(6)

where ${T}_{w}$ is the wall surface temperature, ${h}_{\infty }$ represents the natural convection heat transfer coefficient, and ${T}_{\infty }$ is the ambient temperature.

The electrical power supplied generates relatively low surface temperature, resulting in natural convective heat dissipation that accounts for less than 3% of the total power. This percentage is regarded as a loss in the total heat dissipated.

The impinging flat plate, constructed from Inconel metal recognized for its excellent thermal conductivity, is considered a heat source that generates consistent heat throughout every area of the sheet, which possesses minimal thickness, making the heat conduction insignificant.

Non-dimensional parameters, such as the Re and the average Nusselt number (Nu), are used in this study to express the results. The formula for the Re number is:

$$Re=\frac{uD}{\upsilon }$$

(7)

The Nu for the flat surface is determined using the subsequent formula:

$$Nu=\frac{hD}{{k}_{a}}$$

(8)

The air thermal conductivity (${k}_{a}$) is calculated based on the air temperature at the nozzle exit.

Uncertainty of experimental measurements

In the present study, careful attention and strict precautions will be considered during the experimental process. It is necessary to assess the measurement errors in the instruments utilized in the experimental work, in order to determine the potential uncertainty in all measurements conducted throughout the experiments. The uncertainties of all measured quantities used to estimate errors in the obtained results were determined using Taylor error analysis²⁶, as detailed below:

$$\delta R=\sqrt{{\left(\frac{\partial R}{\partial {X}_{1}}\delta {X}_{1}\right)}^{2}+{\left(\frac{\partial R}{\partial {X}_{2}}\delta {X}_{2}\right)}^{2}+\dots +{\left(\frac{\partial R}{\partial {X}_{n}}\delta {X}_{n}\right)}^{2}}$$

(9)

The uncertainties for the measured quantities, such as air flow rate, current, voltage, nozzle-to-plate distance, jet spacing, and air jets inclination angle are approximated and listed in Table 1.

Table 1 Uncertainty of measured parameters.

From Eq. 1 and 7, The average Nusselt number, Nu, was computed using the relation:

$$Nu=\frac{{Q}_{conv}.D}{{k}_{a}.A\left({T}_{w}-{T}_{jet}\right)}$$

(10)

To compute the combined uncertainty of Nu, the partial derivatives with respect to each contributing parameter were taken, and the overall uncertainty was calculated using:

$$\delta Nu=\sqrt{{\left(\frac{\partial Nu}{\partial {Q}_{conv}}\delta {Q}_{conv}\right)}^{2}+{\left(\frac{\partial Nu}{\partial D}\delta D\right)}^{2}+{\left(\frac{\partial Nu}{\partial {k}_{a}}\delta {k}_{a}\right)}^{2}+{\left(\frac{\partial Nu}{\partial A}\delta A\right)}^{2}+{\left(\frac{\partial Nu}{\partial {T}_{w}}\delta {T}_{w}\right)}^{2}+{\left(\frac{\partial Nu}{\partial {T}_{s}}\delta {T}_{s}\right)}^{2}}$$

(11)

The estimated uncertainties of the individual variables (air flow rate, voltage, current, temperatures, and geometrical dimensions) were listed in Table 1. Using these, the calculated relative uncertainty in the average Nusselt number was found to range between ± 4.2% and ± 5.8%, depending on the test conditions.

Artificial neural network methodology

The Artificial Neural Network (ANN) used in this study is based on Multilayer Perceptron (MLP) architecture. An MLP is a type of feedforward neural network that consists of an input layer, one or more hidden layers, and an output layer. Each layer is composed of interconnected processing elements called neurons, where each neuron receives a weighted sum of inputs from the previous layer and applies an activation function to produce an output. The MLP structure is capable of modeling complex, nonlinear relationships between inputs and outputs, making it suitable for engineering applications involving high-dimensional, nonlinear data. In this study, the MLP model was designed with four input variables (jet inclination angle, jet spacing, Reynolds number, and nozzle-to-plate distance), sixteen hidden layers with 24 neurons each (using ReLU activation functions), and one output neuron representing the average Nusselt number (using a linear activation function). The network was trained using the backpropagation algorithm in conjunction with the ADAM optimizer.

Data gathering and processing

The dataset employed for training the ANN model was obtained from experimental measurements, incorporating input features such as jet inclination angle (θ), jet spacing (S/D), Reynolds number (Re), and nozzle-to-plate distance (H/D). The output variable was the average Nusselt number (Nu). To promote effective learning and maintain consistent feature scaling, MinMax normalization was applied, transforming all input parameters to a range between 0 and 1. Subsequently, three subsets of the dataset were created: 70% for training, 15% for validation, and 15% for testing, following a conventional split strategy to support model training and accuracy assessment.

Artificial neural network architecture

Figure 2 presents the schematic layout of the ANN architecture utilized in the current investigation. The model aims to predict Nu, a key indicator of convective heat transfer in fluid systems. It receives four input parameters associated with jet behavior and thermal performance: jet inclination angle (θ), jet spacing (S/D), Reynolds number (Re), and nozzle-to-plate distance (H/D). The ANN is implemented as a multilayer perceptron (MLP) configured for regression tasks, employing the backpropagation algorithm for training.

Fig. 2

Schematic diagram of the present ANN model architecture.

The input layer consists of four nodes corresponding to the input variables. Each of these inputs represents a feature in the experimental dataset. Mathematically, the input vector x is:

$$x={\left[{x}_{1},{x}_{2},{x}_{3},{x}_{4}\right]}^{T}$$

(12)

These input values are transmitted to the hidden layers, where they are transformed through associated weights and biases. The neural network consists of 16 hidden layers, each comprising 24 neurons. This configuration was chosen by evaluating and comparing the loss function outcomes throughout the training phase²⁷. Each hidden layer uses the ReLU activation function (${f}_{ReLU}$ ):

$${f}_{ReLU}\left(z\right)=max\left(0,z\right)$$

(13)

The output from the k-th neuron in the i-th hidden layer is given by:

$${h}_{i,k}={f}_{ReLU}\left({\sum }_{j=1}^{24}{W}_{i,k,j}.{h}_{i-1,j}+{b}_{i,k}\right)$$

(14)

Here, ${W}_{i,k,j}$ represents the weight connecting the j-th neuron in the preceding layer to the k-th neuron in the i-th hidden layer, ${h}_{i-1,j}$ is the output from the j-th neuron in the previous layer (or the input values for the first hidden layer), and ${b}_{i,k}$ denotes the bias term for the k-th neuron in the i-th hidden layer. This equation is applied to each of the 16 hidden layers, where the output from one layer is passed as the input to the subsequent layer. The final output layer consists of a single neuron, which predicts Nu $\left(\widehat{y}\right)$ according to the following formula:

$$\widehat{y}={W}_{out}\cdot {h}_{out}+{b}_{out}$$

(15)

where ${h}_{out}$ represents the output from the last hidden layer (the 16th layer), ${W}_{out}$ is the weight linking the final hidden layer to the output, and ${b}_{out}$ is the bias term associated with the output neuron.

In the present study, the linear activation function is used for the output layer, meaning no non-linearity is applied to the output. The linear function ensures that the output is a continuous value, which is appropriate for regression tasks that predict a scalar of average Nusselt number.

The ANN is fine-tuned using the Adam optimizer, which is an enhancement of the stochastic gradient descent method. It dynamically adapts the learning rate by considering both the first and second moments of the gradients. This approach is especially useful for training deep neural networks. The update rule for each weight W and bias b is as follows:

$$W=W-\eta .\frac{{m}_{t}}{\sqrt{{\upsilon }_{t}}+\epsilon }$$

(16)

where ${m}_{t}$ represents the first moment estimate (the average of the gradients), ${\upsilon }_{t}$ denotes the second moment estimate (the uncentered variance of the gradients), $\eta$ is the learning rate, and $\epsilon$ is a small constant added to avoid division by zero.

The loss function used for regression is typically the mean squared error (MSE), described as:

$$MSE=\frac{1}{n}{\sum }_{i=1}^{n}{\left(\widehat{{y}_{i}}-{y}_{i}\right)}^{2}$$

(17)

Here, $\widehat{{y}_{i}}$ represents the predicted Nu, ${y}_{i}$ is the actual Nu from the dataset, and $n$ denotes the total number of training samples.

The objective of the training procedure is to reduce MSE by modifying the weights and biases. This is achieved through backpropagation, which calculates the gradients of the loss function concerning each weight and bias in the network. These gradients are then utilized by the Adam optimizer to update the model parameters. During backpropagation, the error at the output layer is determined as follows:

$${\delta }_{out}=\widehat{y}-y$$

(18)

This error is propagated backward through the network, leading to adjustments in the weights and biases of each layer in order to minimize the total error. The gradients are calculated using the chain rule and subsequently employed to update the weights.

The selection of 16 hidden layers in the ANN model was the result of a systematic tuning process aimed at achieving the highest prediction accuracy while maintaining model generalization. Various architectures with different numbers of hidden layers (ranging from 4 to 24) and neuron counts were tested. Each configuration was evaluated based on performance metrics such as mean squared error (MSE), mean absolute percentage error (MAPE), and the coefficient of determination (R²) on both validation and test sets. The architecture with 16 hidden layers and 24 neurons per layer produced the lowest validation loss and highest R² (~ 1), with no signs of overfitting. Increasing the number of layers beyond this point either offered negligible performance gains or caused instability in training. Conversely, using fewer layers led to underfitting and reduced accuracy. Thus, the 16-layer structure was selected as the most efficient and stable model for accurately capturing the nonlinear relationship between the input parameters and the average Nusselt number.

To ensure the generalizability of the ANN model and prevent overfitting, several techniques were employed. Early stopping was implemented by monitoring the validation loss with a patience of 50 epochs; training was terminated when no improvement was observed to avoid overfitting. Additionally, L2 regularization (weight decay) was applied to the weights to penalize large values and promote a simpler model. The dataset was divided into training, validation, and test sets using a 70/15/15 split to evaluate performance on unseen data. Although Dropout was initially considered, it was ultimately not used, as the relatively small input dimension and high-quality experimental data allowed the model to generalize well with early stopping and regularization alone, as evidenced by the low-test errors and R² ≈ 1.

The ANN hyperparameters were selected through a structured trial-and-error process aimed at minimizing validation loss. The learning rate was set to 0.001 based on standard practice for the ADAM optimizer and refined to ensure stable convergence. A batch size of 32 was chosen to balance convergence speed and stability. The final architecture, consisting of 16 hidden layers with 24 neurons each, was determined by incrementally increasing the depth and width of the network while comparing the performance of each configuration. The selected structure provided the best compromise between training accuracy and validation performance without signs of overfitting.

Performance evaluation metrics

The effectiveness of an ANN model is generally evaluated through several metrics that offer important information about its predictive accuracy. In this study, the evaluation criteria for the ANN model include Mean Squared Error (MSE), Mean Absolute Percentage Error (MAPE), Mean Squared Logarithmic Error (MSLE), Log-Cosh Loss, Coefficient of determination (${R}^{2}$), and the Pearson correlation coefficient (r).

In this study, MSE is employed as the loss function in the ANN model to quantify the average squared difference between the observed and predicted values. By squaring the errors, MSE places greater emphasis on larger deviations, which makes it highly sensitive to outliers. This property makes MSE an ideal choice for regression tasks where large errors are undesirable. The formula for its calculation is provided in Eq. 17.

The MAPE quantifies the average percentage difference between the predicted and actual experimental values, providing a metric that is both scale-invariant and easy to interpret. Although it expresses errors as a percentage of the true values, MAPE can be sensitive to small actual values, resulting in excessively large percentage errors. The calculation of MAPE is given by the subsequent formula:

$$MAPE=\frac{1}{n}{\sum }_{i=1}^{n}\left|\frac{{y}_{i}-\widehat{{y}_{i}}}{{y}_{i}}\right|x100$$

(19)

MSLE measures the mean squared logarithmic difference between actual and predicted values, emphasizing relative errors and penalizing under-predictions more than over-predictions. When anticipating relative distinctions is of higher importance than absolute ones, this method proves beneficial and exhibits decreased sensitivity to significant errors in comparison to MSE. MSLE is calculated using the following equation:

$$MSLE=\frac{1}{n}{\sum }_{i=1}^{n}{\left(log\left({y}_{i}+1\right)-log\left(\widehat{{y}_{i}}+1\right)\right)}^{2}$$

(20)

Log-Cosh Loss is a smooth approximation of the absolute error, merging the advantages of both L1-norm and L2-norm losses. It is less sensitive to outliers and penalizes larger errors more gradually than MSE. For small errors, it approximates ${\left(\widehat{{y}_{i}}-{y}_{i}\right)}^{2}/2$, while for larger errors, it behaves similarly to $\left|\widehat{{y}_{i}}-{y}_{i}\right|$. This makes Log-Cosh Loss more robust to outliers compared to MSE. The Log-Cosh loss is computed using the following equation:

$$log\left(cosh\left(x\right)\right)-cosh\left(loss\right)={\sum }_{i=1}^{n}log\left(cosh\left(\widehat{{y}_{i}}-{y}_{i}\right)\right)$$

(21)

where

$$cosh\left(x\right)=\frac{{e}^{x}+{e}^{-x}}{2}$$

(22)

The R² value represents the percentage of variation in the dependent variable (y) that is explained by the independent variables, serving as an indicator of the regression model’s goodness of fit. An R² value of 1 indicates a perfect fit, while an R² of 0 suggests the model performs no better than using the mean. If R² falls below 0, it indicates that the model is performing worse than simply using the mean, which implies a poor fit. The R² value is determined using the following formula:

$${R}^{2}=1-\frac{{\sum }_{i=1}^{n}{\left({y}_{i}-\widehat{{y}_{i}}\right)}^{2}}{{\sum }_{i=1}^{n}{\left({y}_{i}-\overline{y}\right)}^{2}}$$

(23)

where $\overline{y}$ represents the average of actual values.

The correlation coefficient (r) is commonly employed to evaluate the performance of ANN models in regression problems. It measures both the strength and direction of the linear relationship between the ANN’s predicted outputs and the actual target values. The r value ranges between -1 and 1, where 1 indicates a perfect positive linear relationship, -1 represents a perfect negative linear relationship, and 0 denotes no linear association. The r value can be determined using the following formula:

$$r=\frac{{\sum }_{i=1}^{n}\left({y}_{i}-\overline{y}\right)\left(\widehat{{y}_{i}}-\overline{\widehat{y}}\right)}{\sqrt{{\sum }_{i=1}^{n}{\left({y}_{i}-\overline{y}\right)}^{2}{\sum }_{i=1}^{n}{\left(\widehat{{y}_{i}}-\overline{\widehat{y}}\right)}^{2}}}$$

(24)

where $\overline{\widehat{y}}$ stands for the average of the predicted values.

Results and discussion

This experimental investigation aimed to assess the average Nusselt number on a target plate exposed to dual inclined circular air jet impingement. The analysis considered the influence of several key parameters: jet inclination angles varying from 0° to 60°, jet-to-jet spacing ranging from 2 to 8 times the nozzle diameter, Reynolds numbers between 10,000 and 40,000, and nozzle-to-surface distances also spanning 2 to 8 times the nozzle diameter.

Convergence of training dataset

Figure 3 presents the convergence behavior of the training dataset over 5000 epochs, though only the first 20 epochs are shown for better visual clarity. The figure displays the training progress along with qualitative outcomes derived from multiple evaluation metrics. In Fig. 3a, the Mean Squared Error (MSE) reaches a final value of 6.265, suggesting that the model successfully learns the relationship between inputs and outputs, resulting in low prediction errors. Similarly, Fig. 3b shows the Mean Absolute Percentage Error (MAPE) at 0.796%, indicating high prediction accuracy relative to the actual values.

Fig. 3

A convergence of the ANN training data sets for different performance evaluation metrics (a) MSE, (b)MAPE, (c) MSLE, (d) log-cosh loss, (e) R² and (f) r.

Figure 3c shows that the Mean Squared Logarithmic Error (MSLE) reaches a notably low value of 9.82 × 10⁻⁵, emphasizing the model’s effectiveness in reducing differences between log-scaled predicted and actual values—a particularly beneficial trait when handling target variables with wide-ranging magnitudes. In a similar fashion, Fig. 3d displays a log-cosh loss of 1.4198, which supports the model’s strong generalization ability. Given that this loss function is less influenced by outliers than MSE, it suggests the model is capable of managing data anomalies effectively. The R-squared value in Fig. 3e is 0.998, implying that the model accounts for nearly the entire variability in the output, reflecting its high precision in identifying data trends. Lastly, Fig. 3f shows a Pearson correlation coefficient of 0.999 between actual and predicted values, signifying a very strong positive linear correlation and confirming the model’s accuracy and consistency in estimating heat transfer coefficients.

Average Nusselt number

Effect of jet inclination angle ($\theta$)

To validate the reliability of the present experimental results, a comparison was conducted with the data reported by Attalla et al.¹⁶, who investigated a similar configuration involving double impinging air jets. Specifically, the comparison was carried out at a nozzle-to-plate distance (H/D) of 2 and jet spacings (S/D) of 2 and 8, for Reynolds numbers of 10,000 and 40,000. As shown in Fig. 4a for Re = 10,000 and Fig. 4b for Re = 40,000, the present data show strong agreement with Attalla et al.'s results, with deviations remaining below 5%. This consistency reinforces the accuracy and rationality of the current experimental methodology and supports the validity of the results obtained.

Fig. 4

Influence of jet inclination angles on the average Nusselt number at H/D = 2 for Reynolds numbers of (a) 10,000 and (b) 40,000, including comparison with the experimental results of Attalla et al.¹⁶ for validation.

Figure 4 displays line graphs comparing the Nu obtained from experimental measurements and ANN predictions as a function of air jet inclination angle, for a fixed nozzle-to-plate distance of 2 and Re of 10,000 and 40,000. The findings reveal an initial increase in the Nu as the inclination angle grows, followed by a decline beyond a certain angle. The strong agreement between experimental and predicted data demonstrates the ANN model’s effectiveness in capturing the complex interplay of parameters, enabling accurate forecasting of average Nusselt number behavior.

Likewise, Fig. 5 presents the surface plots depicting how variations in jet inclination angle affect the Nu distribution on the impingement plate, based on both experimental measurements and ANN-predicted outcomes for nozzle-to-plate distances of 2 and 8. The data reveal that changes in the inclination of the twin air jets substantially impact heat transfer performance by modifying flow behavior, jet interactions, and the resulting thermal distribution across the surface.

Fig. 5

Influence of jets inclination angles on the surface plots of average Nusselt number at a nozzle-to-plate distance of (a) 2 (b) 8.

When the inclination angle of the jets is small, the jets impinge nearly perpendicularly onto the surface, producing intense localized heat transfer but covering a limited area due to restricted jet dispersion. As the inclination angle increases, the jets interact more significantly before reaching the target surface, which promotes higher turbulence levels and a wider heat transfer distribution—although this comes with a drop in the peak heat transfer rate. At very high angles, the jets tend to lose their directional focus, which weakens the impingement effect and reduces the overall thermal performance. The most effective heat transfer typically occurs at inclination angles between 10° and 20°, depending on factors like Re and jet spacing. These moderate angles strike a balance between concentrated impact and adequate interaction, improving heat transfer across a broader area without excessive energy losses. The close agreement between experimental results and ANN predictions across the graphs highlights the ANN model’s strong capability to accurately forecast average Nusselt number behavior, especially at lower inclination angles.

The observed peak in average Nusselt number at moderate inclination angles (θ = 10°–20°) can be explained by considering the underlying flow field behavior associated with inclined jet impingement. At low to moderate angles, the jets maintain a strong component of momentum normal to the impingement surface, which forms a well-defined stagnation zone and promotes efficient heat transfer. Additionally, the inclined configuration introduces secondary flow structures such as horseshoe vortices, corner vortices, and wall-directed entrainment that enhance mixing near the surface. These vortices, commonly observed in oblique jet studies, intensify convective transport without causing excessive lateral jet spread. As the angle increases beyond 20°, the jets begin to skim over the surface rather than impinge forcefully, and the beneficial vortex structures are weakened or become misaligned. This leads to reduced turbulence near the stagnation zones and thus a decline in heat transfer efficiency.

In contrast to findings reported in some single-jet impingement studies, where heat transfer often increases with larger inclination angles due to improved lateral flow dispersion and mixing, our results reveal a different trend for the double inclined jet configuration. Specifically, the average Nusselt number tends to decrease when the inclination angle exceeds 20°. This divergence can be attributed to the interaction between the two inclined jets. At moderate inclination angles (10°–20°), the jets impinge effectively on the surface while also interacting constructively, creating a high turbulence region that enhances heat transfer. However, as the inclination angle increases beyond this range, the jets begin to skim along the surface and exhibit greater mutual interference, which leads to deflection, reduced stagnation pressure, and less effective momentum transfer to the impingement surface. This results in lower localized heat transfer rates and a less uniform temperature field, as confirmed by the infrared thermography images. Thus, the observed decrease in heat transfer performance at higher angles is a result of the specific flow dynamics induced by the dual inclined jet arrangement, which differs fundamentally from single jet behavior.

Effect of jet spacing (S/D)

Figure 6 illustrates line plots that depict the changes in Nu on the impingement surface as a function of jet spacing distance, using both experimental results and ANN predictions. The analysis is performed for various jet inclination angles, with a fixed H/D of 6 and Re of 10,000 and 30,000. The findings indicate that Nu rises as the jet spacing distance increases, reaching its peak at a spacing ratio (S/D) of 4. Beyond this point, increasing the spacing further leads to a reduction in Nu. The high degree of agreement between experimental observations and ANN-predicted values confirms the model’s reliability in capturing the impact of jet spacing on heat transfer behavior.

Fig. 6

Impact of jet spacing distance on the line plots of the average Nusselt number at a Reynolds number of (a) 10,000 and (b) 30,000.

Similarly, Fig. 7 illustrates how jet spacing distance (S/D) influences the surface distribution of the average Nusselt number on the impingement plate, considering various H/D and Re. The outcomes demonstrate that jet spacing plays a crucial role in determining heat transfer performance for double-inclined air jets. With increased spacing, the interference between neighboring jets diminishes,

Fig. 7

Impact of jet spacing distance on the surface plots of the average Nusselt number at a Reynolds number of (a) 10,000 and (b) 30,000.

allowing each jet to retain its distinct flow pattern. This reduction in interaction promotes more efficient localized heat transfer across the plate.

However, excessively large jet spacing distance can lead to reduced jet coverage on the impingement surface, resulting in uneven heat transfer and potential hot spots. On the other hand, closely spaced jets create stronger interactions, including merging and coalescing of the jet flows, which can diminish their momentum and reduce heat transfer effectiveness. The optimal jet spacing distance lies within a range where the balance between individual jet integrity and sufficient coverage of the plate is achieved. This optimal spacing often depends on the nozzle diameter ($D$) and is typically expressed as a multiple of $D$, with an optimal range around 3D to 4D. This range ensures high heat transfer rates with minimal interference effects and uniform temperature distribution on the impingement plate.

The optimal jet-to-jet spacing (S/D = 3–4) also plays a crucial role in minimizing destructive interference and enhancing thermal performance. At very small spacings (S/D < 3), the close proximity of the nozzles leads to strong jet interaction and collision, disrupting the impingement cores and creating unstable flow fields that degrade heat transfer. On the other hand, at large spacings (S/D > 4), the jets become too isolated from one another, resulting in diminished vortex interaction and less uniform temperature distribution. The 3D–4D range appears to provide a favorable compromise, allowing for constructive interference between wall jets and vortex interactions, which reinforces turbulence and maintains effective coverage of the impingement surface. These findings are consistent with previously reported results for multi-jet arrays, where moderate spacing maximizes both local and average Nusselt numbers.

For the case of the 30° jet shown in Fig. 6(a), the average Nusselt number (Nu) declines continuously beyond S/D = 4 without forming a clear minimum. This can be attributed to the oblique nature of the jet, which causes a reduction in the normal momentum component impinging on the surface. As the spacing increases, the interaction between jets diminishes, and the jet core loses coherence before reaching the impingement surface effectively. Consequently, no distinct interference zone develops, resulting in a gradual performance drop instead of a sharp local minimum. A similar phenomenon is observed for the 10° inclination case in Fig. 6(b), where the shallow angle leads to the.

jet gliding along the surface rather than producing strong impingement and vortex structures. These flow characteristics suppress the formation of a well-defined interference region, causing Nu to decrease steadily with spacing.

Effect of Reynolds number

Figure 8 displays line plots comparing Nu on the impingement plate obtained from both experimental measurements and ANN predictions as a function of Re, across different jet inclination angles, with a fixed H/D of 4 and S/D of 2 and 4. The findings indicate a clear trend of increasing Nu with rising Re. Additionally, Fig. 9 presents surface plots that highlight the influence of Re on the heat transfer characteristics of the impingement plate at H/D of 2 and 8.

Fig. 8

Effect of Reynolds number on the line plots of the average Nusselt number at a jet spacing of (a) 2 (b) 4.

Fig. 9

Effect of Reynolds number on the surface plots of the average Nusselt number at a nozzle-to-plate distance of (a) 2 (b) 8.

These findings demonstrate that the Reynolds number of double-inclined air jets plays a crucial role in shaping the heat transfer behavior on the impingement plate by modifying key flow characteristics such as turbulence levels, impingement patterns, and boundary layer evolution. As the Re increases, stronger fluid mixing and steeper velocity gradients near the target surface are observed, which enhances convective heat transfer by increasing the heat transfer coefficient. The resulting turbulence promotes more vigorous disruption of the thermal boundary layer, allowing the jet to penetrate more deeply and efficiently exchange heat with the surface. Conversely, at lower Re, the flow remains with limited mixing, which diminishes the overall heat transfer performance.

Effect of nozzle-to-plate distance (H/D)

Figure 10 displays line plots comparing Nu on the impingement plate, obtained from both experimental measurements and ANN predictions, as a function of H/D. The analysis is conducted for different jet inclination angles at a fixed jet spacing of 4 and Reynolds numbers of 10,000 and 30,000. The findings indicate that as the nozzle-to-plate distance increases, the average Nusselt number tends to decline.

Fig. 10

Effect of nozzle-to-plate distance on the line plots of the average Nusselt number at a Reynolds number of (a) 10,000 and (b) 30,000.

Additionally, Fig. 11 illustrates how the H/D influences the average Nusselt number on the impingement plate, using both experimental results and ANN predictions. These findings are visualized through surface and line plots.

Fig. 11

Effect of nozzle-to-plate distance on the surface plots of the average Nusselt number at a jet spacing distance of (a) 4 (b) 8.

The results highlight the crucial role of nozzle-to-plate distance in determining the heat transfer behavior during impingement by double-inclined air jets on a flat surface. When this distance is small,

the jets remain more focused and impinge on a limited surface area, resulting in intensified convective heat transfer and elevated turbulence levels. However, as H/D increases, the jet spreads out over a larger area, leading to a decrease in heat transfer efficiency because the jet’s velocity decreases and the turbulence intensity weakens, reducing the convective heat transfer rate.

It is important to note that the observed maximum Nusselt number at H/D = 2 in this study differs from the commonly reported peak at H/D ≈ 6 in literature focused on single, vertically impinging jets. This discrepancy can be attributed to the use of double inclined jets in the present configuration. At low H/D ratios, the inclined jets intersect and interact more strongly before reaching the target surface, generating enhanced turbulence and mixing near the stagnation zone. This intensifies the convective.

heat transfer at the impact area. As H/D increases, jet cores diverge and the strength of interaction diminishes, reducing the effectiveness of impingement. Hence, for inclined jet systems, a lower H/D may be optimal, contrasting with vertically aligned jets where optimal spacing is typically larger.

Assessment of ANN model accuracy through experimental data comparison

Figures 4, 5, 6, 7, 8, 9, 10, 11 display a comparison between the experimental data and the predictions from the ANN model for Nu distribution, considering different jet inclination angles, jet spacing distances, Reynolds numbers, and nozzle-to-plate distances. This comparison is shown through both line plots and surface plots of Nu. In the 2D line plots, experimental results are illustrated by solid lines, while dashed lines indicate the ANN predictions. For the surface plots, the experimental data points are depicted as discrete markers overlaying the surfaces generated by the ANN. The close match between the experimental data and the ANN predictions in both plot types demonstrates the model’s capability to effectively capture the complex interdependencies of the influencing variables, providing accurate and reliable predictions of the average Nusselt number distribution.

Figure 12 presents a scatter plot comparing the ANN-predicted average Nusselt numbers with the corresponding experimental values across different jet inclination angles. In an ideal scenario, all points would align perfectly along the diagonal, indicating a perfect match between predictions and experimental results. While some small deviations from the diagonal are anticipated, the points are closely grouped near the line, demonstrating a strong correlation between the ANN predictions and the experimental data.

Fig. 12

Scatter plots of average Nusselt number for the experimental and ANN data at different air jets inclination angles.

Figure 13 presents the regression statistics used to quantitatively assess the agreement between the experimental data and the ANN predictions. Displayed as 3D bar charts across different evaluation metrics, the figure provides a comprehensive numerical overview of how well the ANN predictions align with the experimental outcomes. The 3D bar plots showcase various metrics, highlighting the ANN model’s capability in predicting Nu distribution across different jet inclination angles and jet spacing distances.

Fig. 13

Regression statistics of the present ANN model using different performance evaluation metrics (a) MSE, (b) MAPE, (c) MSLE, (d) log-cosh loss, (e) R² and (f) r.

The model demonstrated excellent performance on all evaluation metrics, reflected by its low MSE, MAPE, MSLE, and log-cosh loss values, along with high R² values and strong correlation coefficients (r). Specifically, the highest MSE values, illustrated in Fig. 13a, are 12.73, 15.62, 9.146, and 12.875 for S/D values of 2, 4, 6, and 8, respectively, highlighting the model’s strong accuracy and precision. In a similar fashion, the highest MAPE values, presented in Fig. 13b, are 1.144%, 1.0348%, 1.11%, and 1.226% for S/D values of 2, 4, 6, and 8, respectively, reflecting minimal percentage errors and enhanced prediction accuracy. Additionally, in Fig. 13c, the maximum MSLE values of 2.057 × 10⁻⁴ and 2.052 × 10⁻⁴ for S/D of 6 and 8, respectively, show that the model’s predictions are in close alignment with the actual values on a logarithmic scale, supporting reliable predictions across a broad range of values.

The highest log-cosh loss values, presented in Fig. 13d, are 0.1628, 0.1567, 0.1235, and 0.15 for S/D ratios of 2, 4, 6, and 8, respectively, highlighting the model’s reduced sensitivity to outliers.

Moreover, as illustrated in Fig. 13e, the R² values are above 0.994 for all cases, suggesting that the model explains more than 99.4% of the variability in the dependent variables. Finally, Fig. 13f presents the correlation coefficients (r), all of which are greater than 0.9985, confirming a strong linear association between the predicted and experimental results.

Conclusions

An experimental study was conducted to evaluate the thermal behavior of double-inclined air jets impinging on a flat surface, investigating the effects of jet inclination angle, jet spacing, Reynolds number, and nozzle-to-plate distance on heat transfer across the plate. Based on these parameters, an ANN-based model was also developed to predict the average Nusselt number on the impingement plate. The key findings from the heat transfer analysis and ANN predictions are summarized below:

• The optimal jet inclination angles typically fall within the range of 10° to 20°, depending on the Reynolds number and jet spacing. These angles achieve a balance between focused jet impingement and effective jet interaction, enhancing heat transfer across a larger surface area without substantial energy losses.

• The ideal jet spacing was found to lie between 3 and 4D, ensuring high heat transfer efficiency with minimal interference effects and uniform temperature distribution on the impingement plate.

• The ANN model was designed with 16 hidden layers, each containing 24 neurons. Its architecture was optimized using the ADAM algorithm, employing ReLU activation functions within the hidden layers and a linear activation function at the output layer.

• Performance evaluation demonstrated outstanding prediction performance for the ANN model, with coefficients of determination (R²) and correlation values (r) approaching 1. Error metrics were notably low with MSE = 6.265, MAPE = 0.796%, MSLE = 9.82e-05, and log-cosh loss = 1.42.

• This research highlights the ability of the ANN model to accurately predict the average Nusselt number. This method offers a practical and dependable approach for enhancing thermal performance in a range of engineering applications.

• While this study focused on a flat surface and a fixed circular nozzle diameter of 9.5 mm to ensure experimental consistency, future work could expand the scope by considering curved impingement surfaces (e.g., turbine blades) and alternative nozzle geometries. Such extensions would help broaden the applicability of the developed ANN model to more complex and industrially relevant scenarios.