Journal of Thermal Analysis and Calorimetry

, Volume 137, Issue 6, pp 2029–2043 | Cite as

Modeling of the frost deposition by natural convection on horizontal ultra-low-temperature surfaces

  • Alireza ZendehboudiEmail author
  • S. H. Hosseini


The present work develops a reliable predictive model for precise estimation of the frost layer thickness by free convection on horizontal ultra-low-temperature surfaces. Wall temperature, relative humidity, time, and air temperature are considered as the input variables, and six well-known heuristic models are developed to estimate the desired output. The comparative results demonstrate that the least square support vector machine incorporating the genetic algorithm (GA-LSSVM) outperforms the other approaches. The coefficient of determination of 0.9998 and 0.9976, average absolute relative deviation of 0.8536% and 9.4002%, root mean squared error of 0.0115 and 0.0486, and relative root mean square error of 1.4479 and 5.8989 are the results of training and testing stages of the suggested model, respectively. A new test condition is studied to verify applicability of the proposed approach in computing the values that have not been evaluated in the experiments. It is observed that a decrease in the wall temperature causes a decrease in the frost layer thickness on horizontal surfaces under ultra-low-temperature conditions. The non-dominated sorted genetic algorithm II is also employed and combined with LSSVM model to study a sensitivity analysis. According to the Pareto optimal solutions, the time, wall temperature, air temperature, and relative humidity are, respectively, the most influential parameters on the frost layer thickness.


Frost layer thickness Ultra-low-temperature surface Genetic algorithm Variable sequencing Modeling Leverage approach 



Average absolute relative deviation


Adaptive neuro-fuzzy inference system


Artificial neural network


Genetic algorithms


Group method of data handling


Least square support vector machine


Multilayer perceptron


Non-dominated sorted genetic algorithm II


Particle swarm optimization


Root mean squared error


Relative root mean square error


Standardized residual


Support vector machine

List of symbols






Temperature (°C)


Time (min)


Average of actual output


Actual output


Diagonal element of the hat matrix


Hat matrix


Warming leverage

K(χ, χi)

Kernel function


Lagrangian function


Total number of data

Oi, j

Output of the ith node in ANFIS


Predicted output


Coefficient of determination

Greek symbols


Relative humidity (%)


Frost layer thickness (mm)


Input variable




Membership function


Firing strength


Normalized firing strength


Feature map


Lagrange multiplier


Regularization parameter







Supplementary material

10973_2019_8087_MOESM1_ESM.xlsx (25 kb)
Supplementary material 1 (XLSX 24 kb)


  1. 1.
    Zendehboudi A, Li X. Robust predictive models for estimating frost deposition on horizontal and parallel surfaces. Int J Refrig. 2017;80:225–37.CrossRefGoogle Scholar
  2. 2.
    Ameen FR, Coney JER, Sheppard CGW. Experimental study of warm-air defrosting of heat-pump evaporators. Int J Refrig. 1993;16:13–8.CrossRefGoogle Scholar
  3. 3.
    Amer M, Wang C-C. Review of defrosting methods. Renew Sustain Energy Rev. 2017;73:53–74.CrossRefGoogle Scholar
  4. 4.
    Wang F, Liang C, Zhang X. Research of anti-frosting technology in refrigeration and air conditioning fields: a review. Renew Sustain Energy Rev. 2018;81:707–22.CrossRefGoogle Scholar
  5. 5.
    Schneider HW. Equation of the growth rate of frost forming on cooled surfaces. Int J Heat Mass Transf. 1978;21:1019–24.CrossRefGoogle Scholar
  6. 6.
    Lee K-S, Kim W-S, Lee T-H. A one-dimensional model for frost formation on a cold flat surface. Int J Heat Mass Transf. 1997;40:4359–65.CrossRefGoogle Scholar
  7. 7.
    Sengupta S, Sherif SA, Wong KV. Empirical heat transfer and frost thickness correlations during frost deposition on a cylinder in cross-flow in the transient regime. Int J Energy Res. 1998;22:615–24.CrossRefGoogle Scholar
  8. 8.
    Yang D-K, Lee K-S. Dimensionless correlations of frost properties on a cold plate. Int J Refrig. 2004;27:89–96.CrossRefGoogle Scholar
  9. 9.
    Hermes CJ. An analytical solution to the problem of frost growth and densification on flat surfaces. Int J Heat Mass Transf. 2012;55:7346–51.CrossRefGoogle Scholar
  10. 10.
    Liu Z, Dong Y, Li Y. An experimental study of frost formation on cryogenic surfaces under natural convection conditions. Int J Heat Mass Transf. 2016;97:569–77.CrossRefGoogle Scholar
  11. 11.
    Li L, Liu Z, Li Y, Dong Y. Frost deposition on a horizontal cryogenic surface in free convection. Int J Heat Mass Transf. 2017;113:166–75.CrossRefGoogle Scholar
  12. 12.
    Ahmadi MH, Ahmadi MA, Nazari MA, Mahian O, Ghasempour R. A proposed model to predict thermal conductivity ratio of Al2O3/EG nanofluid by applying least squares support vector machine (LSSVM) and genetic algorithm as a connectionist approach. J Therm Anal Calorim. 2018. Scholar
  13. 13.
    Baghban A, Habibzadeh S, Ashtiani FZ. Toward a modeling study of thermal conductivity of nanofluids using LSSVM strategy. J Therm Anal Calorim. 2018;5:10. Scholar
  14. 14.
    Esfe MH, Ahangar MRH, Toghraie D, Hajmohammad MH, Rostamian H, Tourang H. Designing artificial neural network on thermal conductivity of Al2O3–water–EG (60–40%) nanofluid using experimental data. J Therm Anal Calorim. 2016;126:837–43.CrossRefGoogle Scholar
  15. 15.
    Varamesh A, Hemmati-Sarapardeh A, Dabir B, Mohammadi AH. Development of robust generalized models for estimating the normal boiling points of pure chemical compounds. J Mol Liq. 2017;242:59–69.CrossRefGoogle Scholar
  16. 16.
    Baghban A, Ahmadi MA, Shahrakia BH. Prediction carbon dioxide solubility in presence of various ionic liquids using computational intelligence approaches. J Supercrit Fluids. 2015;98:50–64.CrossRefGoogle Scholar
  17. 17.
    Tian Y, Fu MY, Wu F. Steel plates fault diagnosis on the basis of support vector machines. Neurocomputing. 2015;151:296–303.CrossRefGoogle Scholar
  18. 18.
    Kim K, Jung K, Park S, Kim HJ. Support vector machine-based text detection in digital video. Pattern Recognit. 2001;34:527–9.CrossRefGoogle Scholar
  19. 19.
    Cao Z, Han H, Gu B, Ren N. A novel prediction model of frost growth on cold surface based on support vector machine. Appl Therm Eng. 2009;29:2320–6.CrossRefGoogle Scholar
  20. 20.
    Tahavvor AR, Yaghoubi M. Prediction of frost deposition on a horizontal circular cylinder under natural convection using artificial neural networks. Int J Refrig. 2011;34:560–6.CrossRefGoogle Scholar
  21. 21.
    Zendehboudi A, Wang B, Li X. Application of smart models for prediction of the frost layer thickness on vertical cryogenic surfaces under natural convection. Appl Therm Eng. 2017;115:1128–36.CrossRefGoogle Scholar
  22. 22.
    Esfe MH, Naderi A, Akbari M, Afrand M, Karimipour A. Evaluation of thermal conductivity of COOH-functionalized MWCNTs/water via temperature and solid volume fraction by using experimental data and ANN methods. J Therm Anal Calorim. 2015;121:1273–8.CrossRefGoogle Scholar
  23. 23.
    Ahmadi MA, Soleimani R, Bahadori A. A computational intelligence scheme for prediction equilibrium water dew point of natural gas in TEG dehydration systems. Fuel. 2014;137:145–54.CrossRefGoogle Scholar
  24. 24.
    Tatar A, Naseri S, Bahadori M, Hezave AZ, Kashiwao T, Bahadori A, et al. Prediction of carbon dioxide solubility in ionic liquids using MLP and radial basis function (RBF) neural networks. J Taiwan Inst Chem Eng. 2016;60:151–64.CrossRefGoogle Scholar
  25. 25.
    Ivakhnenko AG. Polynomial theory of complex systems. IEEE Trans Syst Man Cybernet. 1971;SMC-1:364–78.CrossRefGoogle Scholar
  26. 26.
    Pourkiaei SM, Ahmadi MH, Hasheminejad SM. Modeling and experimental verification of a 25 W fabricated PEM fuel cell by parametric and GMDH-type neural network. Mech Ind. 2016;17(1):105.CrossRefGoogle Scholar
  27. 27.
    Ahmadi MH, Ahmadi M-A, Mehrpooya M, Rosen MA. Using GMDH neural networks to model the power and torque of a stirling engine. Sustainability. 2015;7:2243–55.CrossRefGoogle Scholar
  28. 28.
    Jang JSR. ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybernet. 1993;23:665–85.CrossRefGoogle Scholar
  29. 29.
    Baghban A, Adelizadeh M. On the determination of cetane number of hydrocarbons and oxygenates using Adaptive Neuro Fuzzy Inference System optimized with evolutionary algorithms. Fuel. 2018;230:344–54.CrossRefGoogle Scholar
  30. 30.
    Baghban A, Pourfayaz F, Ahmadi MH, Kasaeian A, Pourkiaei SM, Lorenzini G. Connectionist intelligent model estimates of convective heat transfer coefficient of nanofluids in circular cross-sectional channels. J Therm Anal Calorim. 2018;132:1213–39.CrossRefGoogle Scholar
  31. 31.
    Ahmadi MH, Tatar A, Nazari MA, Ghasempour R, Chamkha AJ, Yan W-M. Applicability of connectionist methods to predict thermal resistance of pulsating heat pipes with ethanol by using neural networks. Int J Heat Mass Transf. 2018;126:1079–86.CrossRefGoogle Scholar
  32. 32.
    Suykens J, Vandewalle J. Least squares support vector machine classifiers. Neural Process Lett. 1999;9:293–300.CrossRefGoogle Scholar
  33. 33.
    Baghban A, Jalali A, Mohammadi AH, Habibzadeh S. Efficient modeling of drug solubility in supercritical carbon dioxide. J Supercrit Fluids. 2018;133:466–78.CrossRefGoogle Scholar
  34. 34.
    Ahmadi MH, Nazari MA, Ghasempour R, Madah H, Shafii MB, Ahmadi MA. Thermal conductivity ratio prediction of Al2O3/water nanofluid by applying connectionist methods. Colloids Surf A. 2018;541:154–64.CrossRefGoogle Scholar
  35. 35.
    Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput. 2002;6:182–97.CrossRefGoogle Scholar
  36. 36.
    Ahmadi MH, Mohammadi AH, Dehghani S, Barranco-Jiménez MA. Multi-objective thermodynamic-based optimization of output power of Solar Dish-Stirling engine by implementing an evolutionary algorithm. Energy Convers Manag. 2013;75:438–45.CrossRefGoogle Scholar
  37. 37.
    Ahmadi MH, Ahmadi MA, Mohammadi AH, Feidt M, Pourkiaei SM. Multi-objective optimization of an irreversible Stirling cryogenic refrigerator cycle. Energy Convers Manag. 2014;82:351–60.CrossRefGoogle Scholar
  38. 38.
    Toghyani S, Kasaeian A, Ahmadi MH. Multi-objective optimization of Stirling engine using non-ideal adiabatic method. Energy Convers Manag. 2014;80:54–62.CrossRefGoogle Scholar
  39. 39.
    Ahmadi MH, Ahmadi MA, Sadatsakkak SA. Thermodynamic analysis and performance optimization of irreversible Carnot refrigerator by using multi-objective evolutionary algorithms (MOEAs). Renew Sustain Energy Rev. 2015;51:1055–70.CrossRefGoogle Scholar
  40. 40.
    Ahmadi MH, Ahmadi MA. Multi objective optimization of performance of three-heat-source irreversible refrigerators based algorithm NSGAII. Renew Sustain Energy Rev. 2016;60:784–94.CrossRefGoogle Scholar
  41. 41.
    Ahmadi MH, Ahmadi MA, Pourfayaz F. Thermodynamic analysis and evolutionary algorithm based on multi-objective optimization performance of actual power generating thermal cycles. Appl Therm Eng. 2016;99:996–1005.CrossRefGoogle Scholar
  42. 42.
    Tatar A, Shokrollahi A, Mesbah M, Rashid S, Arabloo M, Bahadori A. Implementing radial basis function networks for modeling CO2-reservoir oil minimum miscibility pressure. J Nat Gas Sci Eng. 2013;15:82–92.CrossRefGoogle Scholar
  43. 43.
    Rousseeuw PJ, Leroy AM. Robust regression and outlier detection. New York: Wiley; 2005.Google Scholar
  44. 44.
    Eslamimanesh A, Gharagheizi F, Mohammadi AH, Richon D. Assessment test of sulfur content of gases. Fuel Process Technol. 2013;110:133–40.CrossRefGoogle Scholar
  45. 45.
    Bagheri-Chokami Y, Farahani N, Mirkhani SA, Ilani-Kashkouli P, Gharagheizi F, Mohammadi AH. A chemical structure-based model for estimating speed of sound in liquids. J Therm Anal Calorim. 2014;116:529–38.CrossRefGoogle Scholar
  46. 46.
    Gramatica P. Principles of QSAR models validation: internal and external. QSAR Comb Sci. 2007;26:694–701.CrossRefGoogle Scholar
  47. 47.
    Shateri M, Ghorbani S, Hemmati-Sarapardeh A, Mohammadi AH. Application of Wilcoxon generalized radial basis function network for prediction of natural gas compressibility factor. J Taiwan Inst Chem Eng. 2015;50:131–41.CrossRefGoogle Scholar
  48. 48.
    Hornik K, Stinchombe M, White H. Multi-layer feed forward networks are universal approximations. Neural Netw. 1989;2:359–66.CrossRefGoogle Scholar
  49. 49.
    Li M-F, Tang X-P, Wu W, Liu H-B. General models for estimating daily global solar radiation for different solar radiation zones in mainland China. Energy Convers Manag. 2013;70:139–48.CrossRefGoogle Scholar
  50. 50.
    Shokrollahi A, Tatar A, Safari H. On accurate determination of PVT properties in crude oil systems: committee machine intelligent system modeling approach. J Taiwan Inst Chem Eng. 2015;55:17–26.CrossRefGoogle Scholar
  51. 51.
    Zendehboudi A, Tatar A. Utilization of the RBF network to model the nucleate pool boiling heat transfer properties of refrigerant-oil mixtures with nanoparticles. J Mol Liq. 2017;247:304–12.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Building ScienceTsinghua UniversityBeijingChina
  2. 2.Department of Chemical EngineeringIlam UniversityIlamIran

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