# Inverse Estimation of Time-Dependent Heat Flux in Stagnation Region of Annular Jet on a Cylinder Using Levenberg–Marquardt Method

Document Type : Research Article

Authors

1 Department of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, I.R. IRAN

2 Department of Electrical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, I.R. IRAN

Abstract

All solving methods available in the literature are formulated for direct solution of stagnation point flow and its heat transfer impinging on the surfaces with known boundary conditions. In this study for the first time, an anumerical code based on Levenberg–Marquardt method is presented for solving the inverse heat transfer problem of an annular jet on a cylinder and estimating the time-dependent heat flux using temperature distribution at a specific point. Also, the effect of noisy data on the final results is studied. For this purpose, the numerical solution of the dimensionless temperature and the convective heat transfer in a radial incompressible flow on a cylinder rod is carried out as a direct problem.In the direct problem, the free stream is steady with an initial flow strain rate of k. Using similarity variables and appropriate transformations, momentum and energy equations are converted into Semi-similar equations. The new equation systems are then discretized using an implicit finite difference method and solved by applying the Tri-Diagonal Matrix Algorithm (TDMA). The heat flux is then estimated by applying the Levenberg–Marquardt parameter estimation approach. This technique is an iterative approach based on minimizing the least-square summation of the error values, the error being the difference between the estimated and measured temperatures. Results of the inverse analysis indicate that the Levenberg–Marquardt algorithm is an efficient and acceptably stable technique for estimating heat flux in axisymmetric stagnation flow. This method also exhibits considerable stability for noisy input data. The maximum value of the sensitivity coefficient is related to the estimation of exponential heat flux and its value is 0.1619 also the minimum value of the sensitivity coefficient is 5.62´10-6 which is related to the triangular heat flux. The results show that the parameter estimation error in calculating the triangular and trapezoidal heat flux is greater than the exponential and sinus–cosines heat flux because the maximum value of RMS error is obtained for these two cases, which are 0.481 and 0.489, respectively the reason for the increase in the errors in estimating these functions is the existence of points where the first derivative of the function does not exist. The problem is particularly important in pressure-lubricated bearings.

Keywords

Main Subjects

#### References

[1] Huang C. H., Wang P., A Three-Dimensional Inverse HeatConduction Problem in Estimating Surface Heat Flux by ConjugateGradient Method, Int. J. Heat Mass Trans.,42(18): 3387-3403 (1999).
[2] Shiguemori E.H., Harter F.P., Campos Velho H.F., Dasilva J.D.S., Estimation of Boundary Condition in Conduction Heat Transfer by Neural Networks, Tendências em Matemática Aplicada e Computacional.3: 189-195(2002).
[3] Volle F., Maillet D., Gradeck M., Kouachi A., Lebouché M., Practical Application of Inverse Heat Conduction for Wall Condition Estimation on a Rotating Cylinder, Int. J. Heat Mass Trans., 52(2): 210-221 (2009).
[4] Golbahar HaghighiM.R., EghtesadM., MalekzadehP., NecsulescuD.S., Three-Dimensional Inverse Transient Heat Transfer Analysis of Thick Functionally Graded Plates, Ene. Convers. Manage, 50(3): 450-457 (2009).
[5] Su J., Neto A., Two Dimensional Inverse Heat Conduction Problem of Source StrengthEstimation in Cylindrical Rods, Applied Mathematical Modeling., 25(10): 861- 872 (2001).
[6] Hsu P.T., Estimating the Boundary Condition in a 3D Inverse Hyperbolic Heat Conduction Problem, Applied Mathematics and Computation, 177(2): 453- 464 (2006).
[7] Shi J., Wang J., Inverse Problem of Estimating Space and Time Dependent Hot Surface Heat Flux in Transient Transpiration Cooling Process, Int. J. Thermal Sc., 48(7): 1398-1404 (2009).
[8] Ling X., Atluri S.N., Stability Analysis for Inverse Heat Conduction Problems, Computer Modeling in Engineering & Sciences., 13(3):219-228 (2006).
[9] Jiang B.H., Nguyen T.H., Prud’homme M., Control of the Boundary Heat Flux During the Heating Process of a Solid Material, Int. Commun. Heat Mass Transfer., 32(6): 728-738 (2005).
[10] Chen S.G., Weng C.I., Lin J., Inverse Estimation of Transient Temperature Distribution in the End Quenching Test, J. Mater. Process. Technol., 86(3): 257-263 (1999).
[11] Plotkowski A., Krane M.M., The Use of Inverse Heat Conduction Models for Estimation of Transient Surface Heat Flux in Electroslag Remelting, J. Heat Transfer., 137(3):031301 (2015).
[13] Khaniki H.B., Karimian S.M.H., Determining the Heat Flux Absorbed by Satellite Surfaces with Temperature Data, Journal of Mechanical Science and Technology., 28: 2393-2398 (2014).
[14] Beck J., Black well B., Clair C. St., "Inverse Heat Conduction", John Wiley & Sons, Inc., New York, (1985).
[16] Mohammadiun M., Rahimi A.B., Khazaee I., Estimation of the Time-Dependent Heat Flux Using Temperature Distribution at a Point by Conjugate Gradient Method, Int. J. Thermal Sc., 50(11): 2443-2450 (2011).
[17] Tai B.L., Stephenson D.A., Shih A.J., An Inverse Heat Transfer Method for Determining Work Piece Temperature in Minimum Quantity Lubrication Deep Hole Drilling, J. Manuf. Sci. Eng., 134(2): 021006 (2012).
[19] Mohammadiun H., Molavi H., Talesh Bahrami H.R., Mohammadiun M., Real-Time Evaluation of Severe Heat Load Over Moving Interface of Decomposing Composites, J. Heat Transfer., 134(11): 111202 (2012).
[20] Mohammadiun M., Molavi H., Talesh Bahrami H.R., Mohammadiun H., Application of Sequential Function Specification Method in Heat Flux Monitoring of Receding Solid Surfaces, Heat Transfer Eng., 35(10): 933–941(2014).
[21]WuT. S., LeeH. L.,Chang W.J.,Yang Y.C., An Inverse Hyperbolic Heat Conduction Problem in Estimating Pulse Heat Flux with a Dual-Phase-Lag Model, Int. Commun. Heat Mass Transfer.,60: 1-8 (2015).
[22]  Cuadrado D. G., Marconnet A., Paniagua G., Non-Linear Non-Iterative Transient Inverse Conjugate Heat Transfer Method Applied to Microelectronics, Int. J. Heat Mass Trans.,152: 119503 (2020).
[25] Brociek R., Słota D., Król M., Matula G., Kwas´ny W., Comparison of Mathematical Models with Fractional Derivative for the Heat Conduction Inverse Problem based on the Measurements of Temperature in Porous Aluminum, Int. J. Heat Mass Trans., 143: 118440 (2019).
[26] Perakis N., Strauß J., Haidn O.J., Heat Flux Evaluation in a Multi-Element CH4/O2Rocket Combustor Using an Inverse Heat Transfer Method, Int. J. Heat Mass Trans., 142: 118425 (2019).
[27] Sochinskii A., Colombet D., Medrano Muñoz M., Ayela F., Luchier N., Flow and Heat Transfer around a Diamond-Shaped Cylinder at Moderate Reynolds Number, Int. J. Heat Mass Trans., 142: 118435 (2019).
[28]Hiemenz K., Die Grenzchicht an Einem in den Gleichformingen Flussigkeitsstrom Eingetauchten Geraden Kreiszylinder, Dingler,s Polytechn. Journal, 326: 391-393 (1911).
[29]Homann F. Z., Der Einfluss Grosser Zahighkeit bei der Strmung um Den Zylinder und um Die Kugel, Zeitschrift für Angewandte Mathematik und Mechanik, 16: 153-164 (1936).
[30] Howarth L., The Boundary Layer in Three Dimensional Flow. Part II. The Flow Near a Stagnation Point, Philos. Mag., 42(7): 1433-1440 (1951).
[31] Davey A., Boundary Layer Flow at a Saddle Point of Attachment, J. Fluid Mech., 10(4): 593-610 (1961).
[32] Wang C., Axisymmetric Stagnation Flow on a Cylinder, Q. Appl. Math., 32(2): 207-213 (1974).
[33] Gorla R.S.R., Nonsimilar Axisymmetric Stagnation Flow on a Moving Cylinder, Int. J. Eng. Sci., 16(6): 397-400 (1978).
[35] Gorla R.S.R., Heat Transfer in Axisymmetric Stagnation Flow on a Cylinder, Appl. Sci. Res., 32(5): 541-553 (1976).
[36] Gorla R.S.R., Unsteady Viscous Flow in the vicinity of an Axisymmetric Stagnation-Point on a Cylinder, Int. J. Eng. Sci., 17(1): 87-93 (1979).
[37] Cunning G.M., Davis A.M.J., Weidman P.D., Radial Stagnation Flow on a Rotating Cylinder with  Uniform Transpiration, J. Eng. Math., 33(2): 113-128 (1998).
[38] Takhar H.S., Chamkha A.J., Nath G., Unsteady Axisymmetric Stagnation-Point Flow of a Viscous Fluid on a Cylinder, Int. J. Eng. Sci., 37(15): 1943-1957 (1999).
[42] Abbasi A.S., Rahimi A.B., Non-Axisymmetric Three- Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate, J. Fluids Eng., 131(7): 074501.1– 074501.5 (2009).
[43] Abbasi A.S., Rahimi A.B., Three-Dimensional Stagnation- Point Flow and Heat Transfer on a Flat Plate with Transpiration, J. Thermophys. Heat Transfer, 23(3): 513–521 (2009).
[44] Abbasi A.S., Rahimi A.B., Niazmand H., Exact Solution of Three-Dimensional Unsteady Stagnation Flow on a Heated Plate, J. Thermophys. Heat Transfer, 25(1): 55–58 (2011).
[45] Abbasi A.S., Rahimi A.B., Investigation of Two-Dimensional Stagnation-Point Flow and Heat Transfer Impinging on a Flat Plate, J. Heat Transfer, 134(6): 064501-1-o64501-5 (2012).
[46] Mohammadiun H., Rahimi A.B., Stagnation-Point Flow and Heat Transfer of a Viscous, Compressible Fluid on a Cylinder, J. Thermophys. Heat Transfer, 26(3): 494-502 (2012).
[47] Mohammadiun H., Rahimi A.B., Kianifar A., Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous Compressible Fluid on a Cylinder with Constant Heat Flux, Sci. Iran., Trans. B, 20(1): 185–194 (2013).
[48] Rahimi A.B., Mohammadiun H., Mohammadiun M., Axisymmetric Stagnation Flow and Heat Transfer of a Compressible Fluid Impinging on a Cylinder Moving Axially, J. Heat Transfer, 138(2): 022201-1-022201-9 (2016).
[49] Rahimi A.B., Mohammadiun H., Mohammadiun M., Self-Similar Solution of Radial Stagnation Point Flow and Heat Transfer of a Viscous, Compressible Fluid Impinging on a Rotating Cylinder, Iran. J. Sci. Technol. Trans. Mech. Eng., 43(1): S141-S153 (2019).
[50]Mohammadiun H., Amerian V., Mohammadiun M., Rahimi A.B., Similarity Solution of Axisymmetric Stagnation-Point Flow and Heat Transfer of a Nanofluid on a Stationary Cylinder with Constant Wall Temperature, Iran. J. Sci. Technol. Trans. Mech. Eng.,41(1): 91-91 (2017).
[51] Mohammadiun H., Amerian V., Mohammadiun M., Khazaee I., Darabi M., Zahedi M., AxisymmetricStagnation-Point Flow and Heat Transfer of Nano-Fluid Impingingon a Cylinder with Constant Wall Heat flux, Thermal Science, 23(5):3153-3164  (2019).
[52] Zahmatkesh R., Mohammadiun H., Mohammadiun M., Dibaei Bonab M.H., Investigation of Entropy Generation in Nanofluid’s Axisymmetric Stagnation Flow over a Cylinder with Constant Wall Temperature and Uniform Surface Suction-Blowing, Alexandria Eng. J., 54(4): 1483-1498 (2019).
[54] Bilal M., Sharma S., Aneja M., Cattaneo-Christov Heat Flux Model of Eyring Powell Fluid Along with Convective Boundary Conditions, Iran. J. Chem. Chem. Eng. (IJCCE), 40 (3):971-979 (2021).
[55] Nematollahzadeh A., Jangara H., Exact Analytical and Numerical Solutions for Convective Heat Transfer in a Semi-Spherical Extended Surface with Regular Singular Points, Iran. J. Chem. Chem. Eng. (IJCCE), 40 (3):980-989 (2021).
[56] Madelatparvar M., Hosseini salami M., Abbasi F., Numerical Study on Parameters Affecting the Structure of Scaffolds Prepared by Freeze-Drying Method, Iran. J. Chem. Chem. Eng. (IJCCE),39(2): 271-286 (2020).
[57] Hussain Z., Zaman M., Nadeem M., Ullah A., CFD Modeling of the Feed Distribution System of a Gas-Solid Reactor, Iran. J. Chem. Chem. Eng. (IJCCE), 38(1): 233-242 (2019).
[58] Ozicik M.N., Orlande H.R.B., "Inverse Heat Transfer Fundamentals and Application", Taylor & Francis, New York, (2000).
[60] Beck J.V., Arnold K.J., "Parameter Estimation in Engineering and Science", John Wiley & Sons, Inc., New York, (1977).
[61] Hong L., Wang C.Y., Annular Axisymmetric Stagnation Flow on a Moving Cylinder, Int. J. Eng. Sci., 47(1): 141–152 (2009).
[62] Zahmatkesh R., Mohammadiun H., Mohammadiun M., Dibaei Bonab M.H., Sadi M., Theoretical Investigation of Entropy Generation in Nanofluid’s Axisymmetric Stagnation Flow over a Cylinder with Constant Wall Heat Flux and Uniform Surface Suction-Blowing, Iran. J. Chem. Chem. Eng. (IJCCE), 40(6): 1893-1908 (2021).
[63] Cuadrado D. G., Marconnet A., Paniagua G., Non-linear Non-Iterative Transient Inverse Conjugate Heat Transfer Method Applied to Microelectronics, Int. J. Heat Mass Trans., 152: 119503 (2020).
[64] Ashraf M., Ali K., Numerical Simulation of Micropolar Flow in a Channel under Osciatory Pressure Gradient, Iran. J. Chem. Chem. Eng. (IJCCE), 39(2): 261-270 (2020).
[65] Safaei H., Sohrabi M., Falamaki C., Royaee S.J., A New Mathematical Model for the Prediction of Internal Recirculation in Impinging Streams Reactors, Iran. J. Chem. Chem. Eng. (IJCCE),39 (2): 249-259 (2020).
[66] Habibi M. R., Amini M., Arefmanesh A., Ghasemikafrudi E., Effects of Viscosity Variations on Buoyancy-Driven Flow from a Horizontal Circular Cylinder Immersed in Al2O3-Water Nanofluid, Iran. J. Chem. Chem. Eng. (IJCCE), 38(1): 213-232 (2019).