Elsevier

Energy and Buildings

Volume 225, 15 October 2020, 110282
Energy and Buildings

General correlation for frost thermal conductivity on parallel surface channels

https://doi.org/10.1016/j.enbuild.2020.110282Get rights and content

Highlights

  • A general model for frost thermal conductivity on parallel surface channels is proposed.

  • The comprehensiveness of the suggested model against other models is appraised.

  • It is the only correlation consists of dimensionless parameters.

  • A comprehensive sensitivity analysis of the related input variables is conducted.

Abstract

Growth of frost layer on cold fins of tube-fin heat exchangers leads to an increase in the pressure drop and a decrease in the frost thermal conductivity and thereby the heat transfer rate. There is a lack of a general model in the literature for estimating the frost thermal conductivity on parallel plate channels, including almost all parameters affecting this factor. In this study, for the first time, the general explicit semi-empirical correlations consist of dimensionless parameters are developed, which apply to parallel surface channels. The dimensionless input parameters include the wall temperature, air temperature, air velocity, frost porosity, relative humidity, specific heat of moist air, latent heat of sublimation, and operating time. The comparative results indicate that the best correlation predicts data points with an coefficient of determination, average absolute relative error, and relative root mean square error equal 0.9921, 2.755%, and 3.713%, respectively. Other available published correlations present higher deviations using the same dataset. Furthermore, to provide a good insight into this study, a sensitivity analysis is carried out employing the validated model. It is shown that the effective thermal conductivity of the frost layer is not only a function of frost density but also depends on a group of dimensionless parameters. It is observed that the thermal conductivity of the frost layer increases with the increase in the Reynolds number, Fourier number, air humidity, and it decreases with the increase in the dimensionless temperature, modified Jakob number, and porosity.

Introduction

The air source heat pumps have attracted much attention in recent decades due to their economic and efficiency benefits to heating requirements in commercial and residential buildings. Despite all advantages associated with air source heat pumps, the operation of an air source heat pump in heating mode at low ambient/outdoor temperatures in winter is associated with a relatively low heating coefficient in terms of performance and heating capacity, which creates some limitations when exploiting air source heat pumps in cold climate regions [1]. Frost accumulates on the surface of outside coil of the air source heat pump when the temperature of the surface in contact with air is less than the freezing point of water and the air dew point. A frosting process on the airside surface of an outdoor heat exchanger in an air source heat pump unit is divided into four stages, namely (1) occurring droplets on the fin surface, (2) occurring ice crystals on the fin surface, (3) forming low-density frost layer, and (4) forming high-density frost layer [2]. Accordingly, the frost layer causes a reduction in the heat transfer rate as a result of acting as a thermal insulator between the coil surface and the air [3]. On the other hand, the air channels become smaller, and defrosting operations will consume considerable energy to melt the frost layer and evaporate the melted ice [4]. In this context, determining the thermal conductivity of frost is considered to be one of the critical parameters in understanding the frost structure and rate of formation to improve the thermal performance of heat exchangers.

Even though progressions in the experimental technique in the assessment of frosting characteristics have been made, these procedures are mainly expensive and time-consuming, requiring accurate and advanced devices. Besides, as reported by Gall and Grillot [5], it is also experimentally challenging to measure frosting characteristics due to the unstable and fragile nature of the deposited frost. Accordingly, the accessibility of an accurate model can complement the experimental data in providing a more comprehensive understanding of the impact of different parameters on frost growth [6].

There are several empirical equations reported in the literature to predict the frost thermal conductivity (FTC) on the flat plates at different conditions. Yonko and Sepsy [7] developed a model that was also modified by other authors. In this model, the conductivity is assumed to be a function of frost density and can be utilized for frost density (ρf) lower than 575 kg/m3. Brian et al. [8] examined the relationship between the frost thermal conductivity to the frost density and wall temperature. The authors developed an empirical model for computing the frost thermal conductivity for ρf < 250 kg/m3. Sanders [9] figured out that the frost density is an influential parameter on the frost thermal conductivity, and suggested a correlation for calculating the frost thermal conductivity for ρf < 500 kg/m3. Ostin and Andersson [10] modified the Yonko and Sepsy [7] model for high frost densities ranging from 50 to 680 kg/m3. Lee et al. [11] considered frost density as the main parameter affecting the frost thermal conductivity and proposed an empirical model. For evaluation of frost thermal conductivity, Sturm et al. [12] reported two correlations as a function of frost density, which are limited to the range of the measured data used in the fitting process. Kim et al. [13] developed a correlation for thermal conductivity of frost based on their experimental data, which takes into account the effects of frost density, time, air relative humidity, and airflow velocity. Negrelli and Hermes [14] proposed a new correlation for predicting the frost thermal conductivity over flat surfaces based on the frost structure. Negrelli et al. [15] suggested an expression for frost thermal conductivity over parallel surface channels by considering the effects of wall temperature and frost porosity.

Reviewing the relevant literature shows that although there are several empirical equations in determining FTC on plates in the open literature, defining this factor is still quite challenging due to complexity of porosity, ignoring some affecting parameters in the models, and also the surface configuration that the frost has been formed on it [16]. In particular, the well-known FTC empirical equations have only been correlated to the frost density [7], [9], [10], [11], [12], frost density and wall temperature [8], wall temperature and frost porosity [14], [15], operating time, frost density, relative humidity, and air velocity [13]. It can be found that some critical parameters, such as air temperature, specific heat of moist air, and latent heat of sublimation have not been included in these equations or all the mentioned parameters are not culumated in a correlation, and also almost all these equations are not general as a function of dimensionless parameters, so that these equations are hardly applied to practice. Furthermore, among the mentioned well-known correlations for estimation of FTC, there are only two models, namely those of Ostin and Andersson [10] and Negrelli et al. [15], have been developed for the parallel surface channels. More recently, Zendehboudi et al. [17] estimated the FTC on the parallel surface channels using the intelligent model of GA-LSSVM with very high precision, yet this model is not able to provide an explicit form between the parameters. Despite the appropriate performance of data-driven approaches, such as ANNs, ANFIS, and SVM, these intelligent techniques are basically black-box model and are not able to provide the explicit equations. However, genetic programming (GP) approach is an alternative technique to eliminate the limitations of the mentioned approaches and generate several mathematical equations. Genetic programming as an evolutionary algorithm was firstly proposed by Koza [18]. In the recent years, GP has been widely used to find accurate and non-linear correlations in different engineering applications [19], [20], [21].

In the light of above discussion, the main target of this study is, for the first time, to develop a general and comprehensive model using GP method for estimating the FTC on the parallel surface channels comprising almost all parameters considered as dimensionless groups. A comparison of the same data samples with the most accurate published correlation for parallel surface channels is also presented, and the results of data analysis are elaborated. In the second part of this paper, a sensitivity analysis is carried out by utilizing the suggested predictive model to evaluate the influences of variation of each input variables on the frost thermal conductivity.

Section snippets

Genetic programming

GP is a repetitive algorithm that approaches the answer step by step. GP algorithm combines equations and generates new equations. In the following, implementation of the GP is described.

Results and discussion

First, by employing the available data set, several semi-empirical equations are developed, and then the precision of the proposed models is evaluated. As mentioned above, in this study, unlike the previous works, the maximum number of factors affecting the FTC has been considered in the models as dimensionless parameters. Therefore, as a starting point, a general equation comprising the maximum number of constants, which is capable of presenting the non-linear behavior of the involved

Concluding remarks

In the present study, a new systematic method based on genetic programming (GP) is developed to predict the frost thermal conductivity on the parallel plate channels. To fulfill this objective, the main dimensionless factors affecting the frost thermal conductivity including the dimensionless temperature, air relative humidity, Reynolds number, modified Jakob number, Fourier number, and porosity are used as the required inputs for the model. To demonstrate the proficiency of GP, its outcomes

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (23)

Cited by (11)

  • Frosting mechanism and behaviors on surfaces with simple geometries: A state-of-the-art literature review

    2022, Applied Thermal Engineering
    Citation Excerpt :

    The model can be used to predict average frost thickness on different configurations, such as vertical, horizontal, and parallel plates. Apart from frost thickness and density, black-box frosting models for predicting frost thermal conductivity were also developed based on intelligent methods [62,138]. As seen in Fig. 10 (e), a set of fins is composed of several parallel fins, which are thin plates with a fixed edge temperature.

  • Machine learning based models to predict frost characteristics on cryogenic surfaces under forced convection conditions

    2021, International Communications in Heat and Mass Transfer
    Citation Excerpt :

    Therefore, more investigation on the frost characteristics such as frost density and frost thickness at cryogenic condition and developing the reliable models in estimation of the mentioned factors can be helpful for the associated designers. Several experimental and analytical studies have been carried out to measure the frost characteristics [4,10–15]. Most of earlier works have focused on ordinary-low temperature under natural convection conditions [5,16,17].

  • Experimental and numerical investigation of frost formation on an array of square fins under natural convection condition

    2021, International Communications in Heat and Mass Transfer
    Citation Excerpt :

    The strategy is based on a new thermodynamic model that includes a dimensionless artificial neural network for predicting the frost mass on these types of heat pumps. Hosseini et al. [17] developed a relationship for frost conductivity as a function of a group of dimensionless parameters. They suggested that the dimensionless values to forecast frost conductivity variation are more effective than other models in other studies.

View all citing articles on Scopus
View full text