Thermal analysis in an air conditioning duct : An experimental design approach

The main purpose of this work is to perform a thermal analysis in an air conditioning duct to verify the influence of the thermal properties of the insulating material on the minimum thermal insulation thickness necessary to avoid the condensation of water vapor present in the air. The mathematical formulation is based on Fourier’s law and the first law of thermodynamics. A response surface, a contour plot and a mathematical model for the analyzed response variable, were obtained from an experimental design. Results indicate that the reduction of thermal conductivity and increase of emissivity of the insulating material contribute to the reduction of the minimum thermal insulation thickness.


Resumen
Este trabajo tiene como objetivo realizar un análisis térmico en un conducto de climatización para comprobar la influencia de las propiedades térmicas del material aislante sobre el espesor mínimo de aislamiento térmico necesario para evitar el efecto de condensación del vapor de agua presente en el aire. La formulación matemática se basa en la ley de Fourier y la primera ley de la termodinámica. Del diseño experimental se obtuvo una superficie de respuesta, una curva de contorno y un modelo matemático para la variable de respuesta analizada. Los resultados indican que la reducción de la conductividad térmica y el aumento de la emisividad del material aislante contribuyen a la reducción del espesor mínimo de aislamiento térmico necesario para evitar el efecto de condensación.

Introduction
Indoor air quality is a global concern of the modern world. To provide air quality, it is important to have control of various parameters such as temperature, relative humidity, atmospheric pollutant concentrations and air changes per hour. Several studies indicate that air quality and thermal comfort significantly influence worker productivity, rates of respiratory disease, allergies, and asthma symptoms (Fisk & Rosenfeld, 1997;Huizenga et al., 2006;Lan et al., 2011) In certain situations, where it is impossible to maintain acceptable comfort levels with natural ventilation alone, artificial climate control becomes essential for people to perform their activities optimally, in comfortable conditions and with a sense of well-being.
In central air-conditioning systems, air is cooled in the evaporator and conducted to the environments to be air conditioned through a structural assembly known as the ductwork.
Such systems are commonly employed in the air conditioning of commercial buildings, malls, convention centers, large hotels, pharmaceutical industries, hospitals, etc.
Metal ducts used for air conditioning transport must be thermally insulated for two main reasons: reduce heat transfer from the outside environment to the interior of the duct and avoid condensation of water vapor in the air when in contact with the cold surface of the duct.
Research indicates that if the thickness of the thermal insulation is not sufficient to prevent the effect of condensation, fungal germination and proliferation will occur in the air Research, Society andDevelopment, v. 9, n. 12, e20291211140, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i12.11140 conditioning ducts, compromising the air quality in the air-conditioned environment (Morey & Williams, 1991;Pasanen et al., 1993). Other factors that may affect the growth of microorganisms in duct networks are: high temperature levels, relative humidity and dust present in the air that is blown into the air-conditioned environment (Li et al., 2010). Gomez et al. (2020) evaluated the influence of temperature and relative humidity of the outside air, and insufflation temperature (internal) on the minimum thickness of rock wool, coated with an aluminum film, required to avoid the effect of condensation in an air conditioning duct. The authors observed that an increase in the temperature and relative humidity of the external air and a reduction in the internal air temperature contribute to the increase in the minimum thickness of rock wool necessary to avoid condensation of water vapor present in the external air when in contact with the surface of the duct.
In this sense, the purpose of this work is to perform a thermal analysis in a rectangular duct for air conditioning transport, to quantify the influence of the thermal conductivity and the emissivity of the insulating material on the minimum thermal insulation thickness necessary to avoid the effect of condensation. It is a purely theoretical research with a predominantly quantitative analysis, which is based on works reported in the literature (Pereira et al., 2018;Gomez et al., 2020).

Methodology
For this analysis, consider Figure 1, which illustrates the temperature distribution in the cross section of a thermally insulated duct for air conditioning transport. It was considered a steady state, one-dimensional heat transfer from the external environment to the interior of the duct and no internal energy generation, making the temperature profiles along both the insulating material and in the duct to be linear.
Heat transfer occurs by convection and radiation on the external surface of the thermal insulation, by conduction on the insulating material and on the duct wall, and by convection on the internal surface of the duct, as illustrated in the equivalent thermal circuit ( Figure 2).
The minimum thermal insulation thickness required to prevent the effect of condensation (Linsu) is obtained considering that the temperature on the surface outside the thermal insulation, in contact with the external environment, is equal to the dewpoint temperature (TDP), which in turn is determined as a function of temperature (Tair_ext = 32ºC) and relative humidity in the external environment (ϕair_ext = 65%). Research, Society and Development, v. 9, n. 12, e20291211140, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i12.11140  (1) where q" is the heat flux, Tair_ext = 32ºC, Tair_int = 15ºC and TDP = 24.59ºC are external air, internal air and dewpoint temperatures, considering data for the city of João Pessoa. hc_ext and hr_ext are the convection and radiation heat transfer coefficients, respectively, on the external surface of the thermal insulation. hc_int is the convection heat transfer coefficient on the internal surface of the duct. kduct = 52 W/mK and kinsu are the thermal conductivity of the duct (galvanized steel) and insulating material, respectively. Lduct = 0.7 mm is the duct thickness Research, Society and Development, v. 9, n. 12, e20291211140, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i12.11140 6 and Linsu is the minimum thermal insulation thickness required to prevent the effect of condensation. Rearranging Equation 1, we have: (2) The convective heat transfer coefficient on the external surface of the thermal insulation is given by Eq. 3, as follows: ( 3) where L = 1.0 m is the characteristic length, which corresponds to the height of the duct sidewall and kf is the thermal conductivity of the outside air.
Modeling the sidewalls of the duct as vertical flat plates and considering laminar flow (10 4 ≤ RaL ≤ 10 9 ), we used the correlation recommended by Churchill and Chu (1975) to calculate the average Nusselt number, as follows: (4) where Pr is the Prandtl number and RaL is the Rayleigh number, the latter defined by Equation 5.
where g is the gravity acceleration, β is the volumetric expansion coefficient, ν is the kinematic viscosity and α is the thermal diffusivity of air. To calculate the values of wet air properties (kf, Pr, β, ν and α), a methodology proposed by Tsilingiris (2008) was used, valid for temperatures between 0 and 100ºC, and it was considered an altitude of 0 m (sea level).
The radiation heat transfer coefficient on the outer surface of the thermal insulation is calculated using an Equation 6, as follows: Research, Society and Development, v. 9, n. 12, e20291211140, 2020 (CC BY 4. where insu = 0.04 is the emissivity of the insulating material and is the Stefan-Boltzmann constant ( = 5.67×10 -8 W/(m 2 ×K 4 )).
The convection heat transfer coefficient on the inner surface of the duct (hc_int) was calculated as a function of the Nusselt number for internal and turbulent flow (Cengel, 2014), as follows: (7) where kf_int is the thermal conductivity of the air inside the duct, Dh is the hydraulic diameter of the duct, f is the friction factor, Re is the Reynolds number and Print is the Prandt number of the air inside the duct. The air velocity inside the duct used to calculate the Reynolds number was 6 m/s. In order to evaluate the influence of the thermal conductivity (kinsu) and emissivity (εinsu) of the insulating material on the minimum thermal insulation thickness necessary to avoid the effect of condensation (Linsu), an experimental design was developed.
The actual and coded levels of the experimental design input variables are presented in Table 1, where (-1) and (+1) represent the lowest and highest levels, respectively, and (0) represent the center point level. From the results obtained from the experimental design matrix, the STATISTICA ® 7 program was used to calculate the main and interaction effects of the input variables in the response variable. In addition, it was used to assist in the analysis of variance (ANOVA), as well as in obtaining the mathematical model, the response surface and the contour plot, as will be presented. Table 3 lists the significance levels (α) and the statistically significant coefficients of the main and interaction factors on the response variable under analysis. At a significance level (α) of 0.05, all factors are considered statistically significant, ie there is a probability of at least 95% accuracy in assuming that these factors influence the Linsu response variable.

Results and Discussions
Significance levels represent the probability of error in accepting that a given factor influences the studied response. As an example, for the effect of thermal conductivity of the insulating material (kinsu) on the Linsu response variable, this probability is 1.234%. Thus, as the probability of error is less than the default value of 5%, the result is considered statistically significant (Rodrigues & Iemma, 2009). Thus, the empirical mathematical model for the Linsu response variable, as a function of the coded values of the input variables, as presented in Table 1, is given by Eq. 8, as follows: From the mathematical model presented above, it is possible to quantify the main and interaction effects of the input variables in the response variable. Thus, it is possible to predict the minimum thermal insulation thickness necessary to avoid the effect of condensation on the duct, as a function of its thermal properties.
The higher the coefficient of a given factor, the greater its contribution to the response variable. Thus, in the proposed range of variation for the thermal properties of the insulating material, as shown in Table 1, the thermal conductivity (kinsu) has a slightly higher influence than the emissivity (εinsu) on the Linsu response variable.
To use the mathematical model, the coded values for the input variables must be used, according to Table 1, that is, their values may vary from -1 to +1. If it is desired to estimate, for example, the minimum thickness of rockwool blanket with aluminum foil (kinsu = 0.0372 W/mK and εinsu = 0.04) necessary to avoid the effect of condensation on the duct, use the Equation 8 with the coded levels, as follows: (9) Research, Society andDevelopment, v. 9, n. 12, e20291211140, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i12.11140 For the variable εinsu, the coded value -1 was extracted directly from Tab. 1, while for the kinsu variable, a linear interpolation between the lower and central levels was performed, obtaining the coded value of -0.57. Table 4 shows the main results of the analysis of variance (ANOVA) for the Linsu response variable. Such results are important in the evaluation of the coefficient of determination (R 2 ), the statistical significance and the adjustment of the mathematical model obtained from the experimental design.    From the analysis of Figures 3 and 4, it is observed that the lower the emissivity (εinsu) and the higher the thermal conductivity (kinsu) of the insulating material, the greater the thickness of thermal insulation required to avoid the effect of condensation on the air conditioning duct (Linsu). It is also observed that the influence of emissivity (εinsu) on the response variable increases as the thermal conductivity of the insulating material (kinsu) increases.

Final Considerations
In view of the results presented, it can be concluded that by analysis of variance (ANOVA), it was possible to determine a mathematical model with very satisfactory coefficient of determination (R 2 ), well-adjusted and statistically significant regression for the response variable under analysis. Also, the analysis developed by the experimental design methodology was important to prove that the thermal conductivity (kinsu) and emissivity (εinsu) of the insulating material influence the minimum thermal insulation thickness necessary to avoid the effect of condensation on air conditioning ducts. Furthermore, in the proposed ranges for the thermal properties of the insulating material, the thermal conductivity (kinsu) is the input variable that has the greatest influence on the response variable under analysis.
In view of the importance of this topic, the authors suggest that further research be carried out to assess the influence of other variables on the minimum thickness of thermal insulation necessary to avoid the effect of condensation on air conditioning ducts. Also, numerical computer simulations could be used to study, in more detail, the fluid dynamic behavior of the phenomenon under analysis.