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Uploaded by station43.cebu on December 6, 2022

Heat Transfer

Basics and Practice

• © 2012
• Peter Böckh 0 ,
• Thomas Wetzel 1

Karlsruhe, Germany

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Karlsruhe Institute of Technology KIT, Karlsruhe, Germany

• A practical approach to calculating heat transfer problems
• With numerous practical examples and worked out solutions
• Theoretical background only where necessary
• Includes supplementary material: sn.pub/extras

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Table of contents (8 chapters)

Front matter, introduction and definitions.

• Peter von Böckh, Thomas Wetzel

Thermal conduction in static materials

Forced convection, free convection, condensation of pure vapors, boiling heat transfer, thermal radiation, heat exchangers, back matter.

• Heat transfer
• heat conduction
• heat exchanger
• thermal radiation
• transient heat conduction

About this book

The book provides an easy way to understand the fundamentals of heat transfer. The reader will acquire the ability to design and analyze heat exchangers. Without extensive derivation of the fundamentals, the latest correlations for heat transfer coefficients and their application are discussed. The following topics are presented - Steady state and transient heat conduction - Free and forced convection - Finned surfaces - Condensation and boiling - Radiation - Heat exchanger design - Problem-solving After introducing the basic terminology, the reader is made familiar with the different mechanisms of heat transfer. Their practical application is demonstrated in examples, which are available in the Internet as MathCad files for further use. Tables of material properties and formulas for their use in programs are included in the appendix. This book will serve as a valuable resource for both students and engineers in the industry.

The author’s experience indicates that students, after 40 lectures and exercises of 45 minutes based on this textbook, have proved capable of designing independently complex heat exchangers such as for cooling of rocket propulsion chambers, condensers and evaporators for heat pumps.

Authors and Affiliations

Peter Böckh

Thomas Wetzel

About the authors

Peter von Böckh: University Karlsruhe, Germany: MS in physics, PHD in thermal engineering under Prof. Dr. Ing. J. M. Chawla, theme critical two phase flow. 14 years with BBC/ABB, Switzerland: development of steam power plant heat exchangers and systems, management of retrofit projects in leading postion, since 1991 professorship for Thermal Energy Systems at the University of Applied Science in Basle: department mechanical engineering.

Thomas Wetzel: University of Hannover, Germany: MS in electrical engineering, PHD under Prof. Dr. A. Mühlbauer, theme: heat and mass transfer in molten semiconductor silicon. Wacker Silitronic AG, Munich and Behr GmbH & Co. KG, Stuttgart: department head in development of vehicle air conditioning systems, since 2009 Professorship for heat and mass transfer at the University Karlsruhe (TH) respectively Karlsruhe Institute of Technology (KIT), Institute for Thermal Process Engineering

Bibliographic Information

Book Title : Heat Transfer

Book Subtitle : Basics and Practice

Authors : Peter Böckh, Thomas Wetzel

DOI : https://doi.org/10.1007/978-3-642-19183-1

Publisher : Springer Berlin, Heidelberg

eBook Packages : Engineering , Engineering (R0)

Copyright Information : Springer-Verlag Berlin Heidelberg 2012

Softcover ISBN : 978-3-642-19182-4 Published: 08 October 2011

eBook ISBN : 978-3-642-19183-1 Published: 12 October 2011

Edition Number : 1

Number of Pages : XIII, 276

Number of Illustrations : 115 b/w illustrations

Additional Information : translation from German

Topics : Engineering Thermodynamics, Heat and Mass Transfer , Engineering Fluid Dynamics

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A Heat Transfer Textbook, 6th edition

Solutions manual

Solutions to more than 520 problems are on the following links.

Solutions for Chapter 1 (v1.01, 16 MB, February 2023)

Solutions for Chapter 2 (v1.0, 13 MB, August 2020)

Solutions for Chapter 3 (v1.0, 15 MB, August 2020)

Solutions for Chapters 4-10 (v1.07, 19 MB, 3 April 2024) Solutions for all problems in Chapters 4, 5, 6, 10, and most in Chapters 7, 8, 9.

Solutions for Chapter 11 (v1.07, 4 MB, 3 April 2024)

If additional solutions become available, they will be posted here.

The solutions that are handwritten were prepared decades ago, and some use property data that don’t precisely match today’s Appendix A. In most instances, the differences are small.

Between them, the authors have spent more than 100 years using various textbook solutions manuals. In our experience, none are without errors. If you happen to find one in our solutions manual, do let us know.

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1 Introduction

2 physics informed neural network, 3 results and discussion, 4 conclusions and future work, acknowledgment, conflict of interest, data availability statement, solving inverse heat transfer problems without surrogate models: a fast, data-sparse, physics informed neural network approach.

Contributed by the Computers and Information Division of ASME for publication in the J ournal of C omputing and I nformation S cience in E ngineering .

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Oommen, V., and Srinivasan, B. (March 10, 2022). "Solving Inverse Heat Transfer Problems Without Surrogate Models: A Fast, Data-Sparse, Physics Informed Neural Network Approach." ASME. J. Comput. Inf. Sci. Eng . August 2022; 22(4): 041012. https://doi.org/10.1115/1.4053800

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Physics informed neural networks have been recently gaining attention for effectively solving a wide variety of partial differential equations. Unlike the traditional machine learning techniques that require experimental or computational databases for training surrogate models, physics informed neural network avoids the excessive dependence on prior data by injecting the governing physical laws as regularizing constraints into the underlying neural network model. Although one can find several successful applications of physics informed neural network in the literature, a systematic study that compares the merits and demerits of this method with conventional machine learning methods is not well explored. In this study, we aim to investigate the effectiveness of this approach in solving inverse problems by comparing and contrasting its performance with conventional machine learning methods while solving four inverse test cases in heat transfer. We show that physics informed neural network is able to solve inverse heat transfer problems in a data-sparse manner by avoiding surrogate models altogether. This study is expected to contribute toward a more robust and effective solution for inverse heat transfer problems. We intend to sensitize researchers in inverse methods to this emerging approach and provide a preliminary analysis of its advantages and disadvantages.

In several engineering applications, the measured temperature and velocity data are readily available. However, the inherent states, such as the thermophysical properties, boundary conditions, or heat sources, which lead to these final readings remain concealed. Researchers have developed and adopted several inverse heat transfer approaches that take advantage of the available measurements and extract information about the unknown input conditions that cannot be directly measured by performing experiments. Examples include estimation of tumor parameter from breast thermogram, inferring the thermophysical properties of a gas undergoing combustion, predicting the surface temperature on re-entry vehicle. This type of problem is frequently encountered in nearly all fields of engineering sciences, and they do not have a unique solution. In other words, several combinations of input parameters can give rise to the same temperature distribution. Furthermore, a small variation in the input parameters can drastically increase fluctuations in the predicted output. This ill-posed nature makes solving inverse heat transfer problems challenging.

One of the methods for performing inverse parameter estimation task is to execute the forward and inverse models iteratively until convergence (see Fig. 1 ). The parameter to be estimated, λ , is randomly initialized and fed to the forward model. Now, for the given λ , the forward model predicts the required field T ^ as a function of the independent variable x ¯ ⁠ . The inverse model obtains a better estimate, λ new from the error ( T − T ^ ) 2 ⁠ . The inverse models can be broadly categorized as gradient-based methods and evolutionary methods. Levenberg Marquardt algorithm [ 1 ] and conjugate gradient methods [ 2 ] are gradient-based optimizers, whereas genetic algorithm [ 3 , 4 ], Markov Chain Monte Carlo method [ 5 , 6 ], repulsive particle swarm optimization algorithm [ 7 ], and golden section search method [ 8 ] belong to the category of evolutionary algorithms. Unlike the evolutionary approaches, the gradient-based techniques converge to a solution at a much faster pace. However, it is very likely for the gradient-based methods to get stuck at local optima, which is never the case with the evolutionary approaches. In other words, gradient-based inverse models are employed when there is a constraint on computational resources, whereas evolutionary methods are preferred when prediction accuracy is given a higher priority than the time taken for arriving at the optimal solution.

Conventional method for solving inverse parameter estimation problems

Very often in the case of forward models involving real-world engineering applications, an analytical expression for the temperature distribution does not exist even if the problem is closed (number of equations equals number of variables). This can be due to the geometric complexity of the domain considered or the nature of the governing physical laws. This makes it necessary to rely on finite element [ 1 , 2 , 5 ], finite difference [ 4 , 6 , 8 ], finite volume [ 7 ], or adomian decomposition [ 3 ] methods for solving the forward problem. These techniques have evolved through research in the past decades and have achieved robustness in its capabilities to solve forward problems. However, the computational costs associated with such solvers are often high. Further, while solving inverse problems, the forward model has to be executed numerous times, leading to prohibitively costly simulations.

This can be a significant setback in engineering applications, where the complete model has to be executed every time a new input is given. To address this issue, researchers have recently turned to employing artificial neural networks and machine learning algorithms, in general, for learning the forward problem. Fitting a surrogate artificial neural network model and using it as a substitute for the forward models [ 9 , 10 ] discussed earlier or learning the direct mapping from the measurement space to the parameter space can substantially reduce the computational costs involved in solving the inverse problem. Ardizzone et al. [ 11 ] use invertible neural networks to estimate the entire posterior over the parameter space. However, these strategies do not take advantage of the governing laws of physics, which are already available and, therefore, has to rely on vast chunks of experimental or simulation data to work successfully. Exhaustive experimentation or costly simulations need to be performed extensively for generating databases required for training these surrogate models.

The traditional methods for solving inverse heat transfer problems, although accurate and robust, excessively relies on computationally expensive forward models. In this study, as a remedy for this excessive dependency, we implement physics informed neural network (PINN) [ 12 ], which takes advantage of the governing physics laws, for solving inverse heat transfer problems. Here, a neural network model is trained in a data-efficient manner by incorporating the prior knowledge about the phenomenon available in the form of the governing partial differential equations as a regularizing constraint. In this manner, expensive forward model simulations that need to be performed numerous times for generating datasets can be completely avoided.

The capability of PINN and its variants in solving a wide variety of engineering problems has been investigated in the literature [ 13 – 21 ]. PINN has also been used for solving several real-world inverse problems in nano-optics and metamaterials [ 22 , 23 ], inferring material properties in functional materials, discovering missing physics in reactive transport, characterizing surface crack on wings of air-crafts [ 24 ] and a wide variety of seismic inversion problems [ 25 ]. However, a study that investigates the advantages and disadvantages of PINN and compares the performance of PINN with traditional methods for solving inverse problems is not well explored. In this study, we compare the time taken by conventional machine learning approach for solving an inverse problem with that of PINN. We perform experiments to investigate the effect of number of collocation points and the width of the neural network architecture on model performance. We also analyze the capability of PINN to solve inverse problems when the measurement data are noisy.

Physics informed neural networks is a neural network architecture that takes independent variables such as position and time as the input and predicts the dependent field variables as output, which is temperature in this case. The framework relies on the universal function approximation capabilities of neural networks [ 26 ]. The loss associated with the governing equations can be computed efficiently by employing automatic differentiation [ 27 ]. This loss term is added with the mean squared error computed between the true and predicted temperature distribution. The total loss, calculated at sparse collocation points within the domain, is minimized by PINNs without depending on meshes. The detailed mathematical formulation of PINNs is found in Ref. [ 12 ]. PINN successfully incorporates the prior knowledge about the underlying physics into a cost function. Just like any other NN model, PINN also minimizes the cost and will be able to model the underlying phenomenon correctly. In this study, the cost function is minimized by “L-BFGS” [ 28 ] algorithm, which is a gradient-based second-order optimization algorithm that belongs to the family of Quasi-Newton methods.

X ∈ R n s a m p l e s * d ⁠ , W j ∈ R n h j − 1 * n h j ⁠ , and b j ∈ R 1 * n h j ⁠ , where d is the dimension of the input vector X, which consists of both space and time inputs ( X = [ x , t ]) and n h j is the number of neurons at the j th hidden layer. W and b are the weights and biases, respectively.

Schematic of PINN with two hidden layers containing five neurons each. Position and time are fed as input, and temperature is given as output.

L-BFGS [ 28 ] method determines Δ W j , Δ b j , and Δ λ j to minimize the loss term L . The weights are updated during back-propagation [ 31 ]. When the loss term has been satisfactorily reduced (i.e., L ≤ ɛ ), W j , b j , and λ j would have adjusted themselves so that the output of the neural network, T ^ ⁠ , would become approximately equal to the actual temperature field T . At this stage, the value stored by λ is the parameter estimated by PINN. However, λ is estimated by a gradient-based optimizer, and therefore, PINN may get stuck at a local minimum. One way to overcome this issue is to set a sufficiently low loss value as the convergence criteria and stop the training procedure only when the model loss goes below this threshold. In this manner, PINN can accurately predict the parameter to be estimated.

Shin et al. in Ref. [ 32 ] provide the mathematical background behind the convergence and generalization of PINNs. The test cases provided in Ref. [ 32 ] restates the theoretical findings and consistency of the PINN methodology.

Estimation of surface heat transfer coefficient in steady-state rectangular fins.

Estimation of convection to conduction coefficient ratio ( h / k ) in annular fins

Estimation of heat generation number in a rectangular fin with temperature-dependent thermal conductivity and heat generation

Estimation of the volumetric rate of internal heat generation within a rectangular porous fin

Estimation of temperature-dependent thermal conductivity in transient heat conduction problem.

3.1 Test Case 1: Rectangular Fin.

3.1.1 the conventional method..

The forward model is simulated for 500 random different values of m using the “Heat Transfer in Solids” module in comsol v5.4. The mesh generated consists of 202 elements. Fifteen such temperature vectors obtained from these simulations are plotted in Fig. 3 . These temperature vectors are given as input to an artificial neural network model with one hidden layer containing five neurons. The surrogate neural network model is trained on the simulated dataset to perform function approximations task, enabling it to estimate the value of m for any given temperature. L-BFGS algorithm is used for minimizing the mean squared error loss between the true and predicted m .

( a ) The temperatures simulated in comsol for 500 different values of m . The plot shows 15 such temperature profiles and ( b ) Temperature predicted by PINN for P base = 172.4 W.

3.1.2 PINN-Based Method.

PINN with architecture [1,5,1] and hyperbolic tan activation function is used for predicting the value of m . The governing equation is satisfied at N f = 20 equi-spaced collocation points. Temperature readings (simulated) were taken from N u = 9 points along x . While computing the total loss, w PDE = 0.75 and w u = 0.25 are considered.

Table 1 compares the accuracy and computational time of PINN with that of the conventional method. It takes 35 s for creating the dataset and an additional 2.5 s for training the surrogate neural network model. Meanwhile, PINN completes the entire task in just 2.6 s.

Comparing the accuracy and time taken by PINN with conventional method

3.1.3 Experiment on the Number of Collocation Points Required by PINN and Error Analysis.

For this test case, we have seen that PINN is much faster than the conventional method. To investigate the robustness of PINN, we ran the model 1000 times from different initial weights. Simulated temperature data are added with Gaussian white noise ∼ N ( 0 , 1 ∘ C). We also try to understand the effect of the number of collocation points N f on the performance of PINN. The results are tabulated in Table 2 .

The mean and standard deviation of retrieved m for P base = 36 W

3.1.4 Observations.

Clearly, PINN is much faster than the conventional approach. But the standard deviation of the retrieved values of m is toward the higher end. This suggests that PINN may be sensitive to initial weights. Although it questions the robustness of PINNs, we can get around this issue by only considering the models whose total loss goes below a sufficiently low threshold.

3.1.5 Experiment Investigating the Minimum Number of Training Samples Required by Conventional Method.

Here, we investigate the minimum number of high fidelity training samples that is required for training the surrogate neural network model. We train the surrogate model on datasets comprising 10, 20, …, 500 training samples and plot the mean squared error associated with the predictions corresponding to each of these models.

3.1.6 Observations.

From Fig. 4 , we observe that the conventional model becomes as accurate as the best-performing PINN model only when there are more than 130 noiseless training samples. If the data are noisy, the required minimum number of training samples is expected to be higher. Generation of 130 samples still require 19 s, whereas PINN can solve the entire problem in just 2.6 s.

Change in the predicted mean squared error (MSE) with respect to number of noiseless training samples

3.2 Test Case 2: Annular Fins.

3.2.1 the conventional method..

The forward model is simulated for 200 different values of k using the “Heat Transfer in Solids” module in comsol v5.4. The mesh consists of 202 elements. The temperatures obtained through simulations are plotted in Fig. 5 and is used for training the surrogate neural network model with one hidden layer containing five neurons. The trained neural network predicts h / k for any given temperature vector. L-BFGS algorithm is used for minimizing the mean squared error loss between the true and predicted h / k .

( a ) The temperatures of cylindrical fin simulated in COMSOL for 500 different values of k . The plot shows 15 such temperature profiles. ( b ) PINN solution for test case 2: cylindrical fin.

3.2.2 PINN-Based Method.

PINN with architecture [1,5,1] is used. The governing equation is satisfied at N f = 20 collocation points. Temperature readings were randomly taken from N u = 9 points along r *. While computing the total loss, w PDE = 0.75 and w u = 0.25 are considered.

Table 3 compares the accuracy and computational time of PINN with that of the conventional method. It takes 20 s for creating the dataset and an additional 2.5 s for training the surrogate neural network model. However, PINN completes the entire task in 2.7 s.

Test case 2—comparing the accuracy and time taken by PINN with conventional method

3.2.3 Experiment on the Width of PINN Architecture and Error Analysis.

PINN is faster than the conventional method and solves the inverse problem with substantially less number computations. Again, to investigate the robustness of PINNs, we run the model from different initial weights. To make the synthetic data realistic, Gaussian white noise ∼ N ( 0 , 1 ∘ C ) is added to the temperatures simulated in comsol . We also investigate the effect of the number of neurons in the hidden layer of PINN. The results are compiled in Table 4 . In test case 1, we saw that the model performance is very sensitive to the weight initialization. To address this problem, here we only consider those models whose total loss after convergence is less than 5 × 10 −5 .

The mean and standard deviation of retrieved h / k

3.2.4 Observations.

From Table 3 , we observe that PINN is faster than the conventional method. From the experiments on the width of the neural network architecture, we are unable to find any significant variation in the model performance. But, for all the architectures considered in Table 4 , the mean of the estimated parameter is very close to the actual value and the standard deviation is small. This suggests that PINN is able to confidently make accurate predictions.

3.3 Test Case 3: Rectangular Fin With Heat Generation.

3.3.1 the conventional method..

The forward model is simulated for 200 different values of q 0 using the “Heat Transfer in Solids” module in comsol v5.4. All the temperatures simulated in this fashion is plotted and shown in Fig. 6 . The mesh consists of 202 elements. The temperatures obtained through simulations is used for training the surrogate neural network model with one hidden layer containing five neurons. The trained neural network is then capable of predicting the heat generation number G for any given temperature vector. L-BFGS algorithm is used for minimizing the mean squared error loss between the true and predicted G .

( a ) PINN solution for test case 3: Rectangular fin with heat generation, ( b ) the temperatures of fin with heat generation simulated in comsol for 500 different values of q 0 . The plot shows 15 such temperature profiles.

3.3.2 PINN-Based Method.

PINN with architecture [1,5,1] is solves Eq. (19) for estimating G . We consider a case with ɛ C = 0.4, N = 1, and ɛ G = 0.6. The governing equation is satisfied at N f = 20 collocation points ⁠ . Temperature readings were randomly taken from N u = 9 points along x *. While computing the total loss, w PDE = 0.17 and w u = 0.83 are considered.

Table 5 compares the accuracy and computational time of PINN with that of the conventional method. It takes 22 s for creating the dataset and an additional 2.9 s for training the surrogate neural network model, whereas PINN completes the entire task in 3.1 s.

Test case 3—comparing the accuracy and time taken by both methods

3.3.3 Investigating the Effect of Measurement Error.

Here, we investigate the performance of PINNs on noisy measurements. Hence, the temperature field simulated in comsol is added with Gaussian white noise, making the pseudo measurement data erroneous. PINN is run rigorously from 1000 random and different initial weights for the same problem to obtain a more conclusive inference about the robustness of this method. The results of this investigation is tabulated in Table 6 .

The mean and standard deviation of retrieved G

3.3.4 Observations.

From Table 5 , PINN solves the problem much faster than the conventional approach. From Table 6 , PINN performs well even on noisy measurement data. As expected, on increasing the standard deviation of the added Gaussian white noise, the accuracy of prediction decreases and standard deviation increases. Nevertheless, PINN does a fair job in retrieving the heat generation number G , and this is shown in Table 6 and Fig. 7 .

Temperature predicted by PINN plotted along with the noisy measurements (noise ∼ N ( 0 , 2 ∘ C ⁠ ))

3.4 Test Case 4: Porous Fin Subjected to Radiative Heat Transfer.

In this test case, we consider a challenging real-world problem. Porous fins are one of the most effective ways of increasing the rate of heat transfer. Often it performs better than the conventional solid fins. They are widely used in refrigeration, internal combustion engines, and air conditioning. In test case 4, PINN is employed to estimate the volumetric rate of internal heat generation in a porous fin subject to convective and radiative heat loss to the surroundings [ 8 ].

In this test case, PINN has to estimate Q 0 , the dimensionless volumetric internal heat generation rate (heat generation number). The constants of volumetric internal heat generation, c 1 = c 2 = c 3 = 0.40. The dimensionless parameter for convection and conduction N cc = 4.9023. The dimensionless parameter for radiation and conduction N rc = 7.5339. The porous parameter S h = 0.4199, index for variable heat transfer coefficient m = 1. The dimensionless ambient temperature θ 0 = 0.2727 and the variable emissivity coefficient α = 0.23 K −1 . A total of 0.1% Gaussian white noise error is added to synthetic data generated from the RK-4 method to make it realistic.

3.4.1 Conventional Method.

The forward model is simulated using the RK-4 method for 200 different random values of Q 0 . All the temperature vectors generated from the simulations are plotted in Fig. 8(a) . These temperature vectors are fed as input to a surrogate neural network model which performs the regression task to predicts ( Q 0 ) as its output. L-BFGS algorithm is used for minimizing the mean squared error loss between the true and predicted Q 0 .

( a ) The temperature profiles of porous fin obtained by solving Eq. (36) using RK-4 method for 200 different random values of Q 0 . The plot shows 15 such temperature profiles and ( b ) PINN solution for porous fin.

Temperature field predicted by PINN with respect to ( a ) time and ( b ) location in transient 1D heat conduction. Thermal conductivity is a quadratic function of temperature.

3.4.2 PINN-Based Method.

Architecture of the PINN used is [1,7,14,7,1] . The governing equation is satisfied at N f = 20 collocation points. Temperature readings (synthetic) were taken from N u = 9 points ⁠ . While computing the total loss, w PDE = 0.25 and w u = 0.75 are considered. The temperature field predicted by PINN is shown in Fig. 9(b) . Gaussian white noise error ∼ N ( 0 , 0.1 ∘ C ) has been added to make the pseudo measurement data realistic.

Table 7 compares the accuracy and computational time of PINN with that of the conventional method. It takes 20 s for creating the dataset using the RK-4 method and an additional 4 s for training the surrogate neural network model. However, PINN completes the entire task in 6 s.

Test case 4—comparing the accuracy and time taken by both methods

3.4.3 Error Analysis.

For the last test case as well, PINN is much faster than the conventional method. To investigate the robustness of PINNs, we ran the model 1000 times from different initial weights. The results are tabulated in Table 8 . The distribution of the retrieved value of Q 0 is presented in Fig. 10 .

The distribution of the retrieved values of Q 0

The mean and standard deviation of the retrieved Q 0

3.4.4 Observations.

From Table 7 , PINN is much faster compared to the conventional method. From Table 8 , the mean and MAP of the retrieved Q 0 is very close to the true value and the standard deviation of prediction is very low. This suggests that PINN is making accurate and confident predictions. Also note that the distribution associated with the retrieved values of Q 0 closely resembles a Gaussian distribution.

3.5 Test Case 5: Transient Heat Conduction.

We consider thermal conductivity, k ( θ ), as a quadratic function of temperature.

3.5.1 PINN-Based Method.

From the given initial and boundary temperature measurements, PINN estimates k 1 and k 3 . k 2 does not impact the solution. PINN with architecture [2,5,10,15,10,5,1] and hyperbolic tan activation function is used. Here, the location x and time t is fed as input to the PINN. The governing equation is satisfied at N f = 300 collocation points, across a uniform grid with 30 points in x and 10 in time t . Temperature readings (synthetic) were taken from N u = 30 points ⁠ . While computing the total loss, w PDE = 0.5 and w u = 0.5 are considered. PINN takes 6 s to solve the inverse problem (see Table 9 ). The prediction made by PINN can be seen in Fig. 9 .

Test case 5—PINN predictions in transient heat conduction problem

3.5.2 Observation.

PINN is able to accurately retrieve the parameters to be estimated in a transient problem governed by partial differential equations.

Conventional data-based surrogate models are data intensive and require large experimental or simulation databases. PINN uses known governing equations as regularizing constraints and hence are data efficient in terms of the database required to solve them.

Surrogate models require multiple forward solutions, whereas PINN is “one-shot.” In other words, PINN directly solves both the forward and inverse problems together without the need for additional simulations.

PINNs can handle complex geometries and do not require additional meshing. This adds robustness as the additional numerical error inherent in mesh quality does not affect PINN.

The field predicted by PINN is an analytical field and satisfies the governing equation in a least squared sense. This is unlike CAE solutions, which are, in essence, polynomial interpolations at nonmesh locations.

PINN can perform with very few collocation points and provide robust solutions even with noise. In all our test cases, we had a very low standard deviation despite very few measurement points.

Traditional machine learning-based methods solve the problem by minimizing only the L u term in Eq. (5) . PINN exploits the freely available knowledge about the governing physical laws by adding L PDE as a regularization term in Eq. (5) and will not overfit the training data. This enables PINN to generalize well across data coming from unseen distributions.

PINN relies on gradient-based optimizers for solving the inverse problem and is therefore prone to converging to a suboptimal solution. Methods and strategies that help prevent PINN from getting stuck at local minimum should be explored.

Theoretical proofs for the robustness of PINNs for inverse problems are not available. The results elsewhere in the literature and our results here are primarily empirical.

More complex applications, in three dimensions, with real experimental data would be particularly useful in proving the efficacy of this method.

The time comparisons presented in this work suggests that PINN is faster and hence computationally cheaper than the traditional methods. However, these results are empirical. Research focusing on calculating the computational complexity of PINN-based approach is required to rigorously prove the efficacy of this method.

We believe that these are important open problems and will be pursuing them in future publications. We, however, hope that our preliminary results discussed earlier generate sufficient interest in the engineering community to pursue PINNs for solving real-world inverse problems.

https://github.com/vivek-brown/PINN-for-inverse-HT

This work was supported by Robert Bosch Centre for Data Science and Artificial Intelligence, Indian Institute of Technology Madras, Chennai (Project No. CR1920ME615RBCX008832). We thank Drs. Vikas Dwivedi, Gaurav Kumar Yadav, Sreehari M, and Sushrut Ranade for the valuable suggestions and discussions.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The data and information that support the findings of this article are freely available. 1

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Solving the Neumann boundary value problem

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Alexandr Kanareykin; Solving the Neumann boundary value problem. AIP Conf. Proc. 24 April 2024; 3154 (1): 020038. https://doi.org/10.1063/5.0201213

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The paper is devoted to solving the Neumann boundary value problem for the Poisson equation in an elliptic body. In this case, heat transfer occurs under boundary conditions of the second kind. Based on the methods of differentiation and integration, a solution was obtained to the problem of the distribution of the temperature field of the body under study. The resulting solution has an analytical form containing hypergeometric and hyperbolic functions. The reliability of the obtained result is confirmed by the fact that the general solution of the problem coincides with the solution obtained in one of the author’s works for the case of a temperature field distribution in a body with an elliptical cross section of infinite length under boundary conditions of the third kind.

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Heat Transfer Basics: A Concise Approach to Problem Solving 1st Edition, Kindle Edition

Concise introduction to heat transfer, with a focus on worked example problems to aid in reader comprehension and student learning

Heat Transfer Basics covers the essential topics of heat transfer in a focused manner, starting with an introduction to heat transfer that explains its relationship to thermodynamics and fluid mechanics and continuing on to key topics such as free convection, boiling and condensation, radiation, heat exchangers, and more, for an accessible and reader-friendly yet comprehensive treatment of the subject.

Each chapter features multiple worked out example problems, including derivations of key governing equations and comparisons of worked solutions with computer modeled results, which helps students become familiar with the types of problems they will encounter in the field. Throughout the book, figures and diagrams liberally illustrate the concepts discussed, and practice problems allow students to test their understanding of the content. The text is accompanied by an online instructor’s manual.

Heat Transfer Basics includes information on:

• One-dimensional, steady-state conduction, covering the plane wall, the composite wall, solid and hollow cylinders and sphere, conduction with and without internal energy generation, and conduction with constant and temperature-dependent thermal conductivity
• Heat transfer from extended surfaces, fins of uniform and variable cross-sectional area, fin performance, and overall fin efficiency
• Transient conduction, covering general lumped capacitance solution method, one- and multi-dimensional transient conduction, and the finite-difference method for solving transient problems
• Free and forced convection, covering hydrodynamic and thermal considerations, the energy balance, and thermal analysis and convection correlations

More advanced than introductory textbooks yet not as overwhelming as textbooks targeted at specialists, Heat Transfer Basics is ideal for students in introductory and advanced heat transfer courses who do not intend to specialize in heat transfer, and is a helpful reference for advanced students and practicing engineers.

• ISBN-13 978-1119840268
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• Publication date November 1, 2023
• Language English
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Jamil Ghojel, Ph.D. is a retired academic with 25 years’ experience of teaching undergraduate and graduate mechanical and aerospace engineering courses in heat engines and heat transfer. He has held positions at the University of Damascus (Syria), the University of Michigan (USA), the University of Melbourne (Australia), and Monash University (Australia). He has conducted extensive research on heat engines and is the author of the Wiley-ASME Press book Fundamentals of Heat Engines: Reciprocating and Gas Turbine Internal Combustion Engines .

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• ASIN ‏ : ‎ B0CMFTBQSG
• Publisher ‏ : ‎ Wiley; 1st edition (November 1, 2023)
• Publication date ‏ : ‎ November 1, 2023
• Language ‏ : ‎ English
• File size ‏ : ‎ 82334 KB
• Simultaneous device usage ‏ : ‎ Up to 3 simultaneous devices, per publisher limits
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• Sticky notes ‏ : ‎ On Kindle Scribe
• Print length ‏ : ‎ 553 pages
• #483 in Chemical Engineering (Kindle Store)
• #1,862 in Mechanical Engineering (Kindle Store)
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Victor Mukhin

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Title : Active carbons as nanoporous materials for solving of environmental problems

However, up to now, the main carriers of catalytic additives have been mineral sorbents: silica gels, alumogels. This is obviously due to the fact that they consist of pure homogeneous components SiO2 and Al2O3, respectively. It is generally known that impurities, especially the ash elements, are catalytic poisons that reduce the effectiveness of the catalyst. Therefore, carbon sorbents with 5-15% by weight of ash elements in their composition are not used in the above mentioned technologies. However, in such an important field as a gas-mask technique, carbon sorbents (active carbons) are carriers of catalytic additives, providing effective protection of a person against any types of potent poisonous substances (PPS). In ESPE “JSC "Neorganika" there has been developed the technology of unique ashless spherical carbon carrier-catalysts by the method of liquid forming of furfural copolymers with subsequent gas-vapor activation, brand PAC. Active carbons PAC have 100% qualitative characteristics of the three main properties of carbon sorbents: strength - 100%, the proportion of sorbing pores in the pore space – 100%, purity - 100% (ash content is close to zero). A particularly outstanding feature of active PAC carbons is their uniquely high mechanical compressive strength of 740 ± 40 MPa, which is 3-7 times larger than that of  such materials as granite, quartzite, electric coal, and is comparable to the value for cast iron - 400-1000 MPa. This allows the PAC to operate under severe conditions in moving and fluidized beds.  Obviously, it is time to actively develop catalysts based on PAC sorbents for oil refining, petrochemicals, gas processing and various technologies of organic synthesis.

Victor M. Mukhin was born in 1946 in the town of Orsk, Russia. In 1970 he graduated the Technological Institute in Leningrad. Victor M. Mukhin was directed to work to the scientific-industrial organization "Neorganika" (Elektrostal, Moscow region) where he is working during 47 years, at present as the head of the laboratory of carbon sorbents.     Victor M. Mukhin defended a Ph. D. thesis and a doctoral thesis at the Mendeleev University of Chemical Technology of Russia (in 1979 and 1997 accordingly). Professor of Mendeleev University of Chemical Technology of Russia. Scientific interests: production, investigation and application of active carbons, technological and ecological carbon-adsorptive processes, environmental protection, production of ecologically clean food.   

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