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An equation model for calculating the adiabatic temperature of the wet-bulb thermometer has been obtained empirical fit through a meteorological database, specificly a trough relative humidity and air temperature. A comparison of the results of calculations with the use of this equation and from meteorological database was made. The model deducted of the comparison is valid for a dry bulb temperature range of 3°C to 35°C and for relative humidity percentage in a range of 7% to 97%. Normalized errors are less than 5.5%. It means a maximum variation of 0.55°C from data. However, this variation from error represents only 3.6% of the data sample. The equation model was satisfactory.

The condensation and vaporization processes are important in different conventional and non-conventional sys- tems of energy, using fossil fuel or renewable energies, and in systems applied to refrigeration, evaporation, air conditioning, humidification, condensation, dehumidification and airing. For example, as in [_{db}), relative humidity (ϕ), local altitude (Z) is known and is shown by ASHRAE [_{wb}), according to the metho- dology, it is necessary a numerical implementation [

In the present work, an equation to determine wet bulb temperature for mentioned conditions, is presented. It was obtained by processing numerical data of a Meteorological Station located in the region Ciénega of Chapala in Michoacán México, with latitude 20˚0'52.24''N and length 102˚44.37'37.11''W. During the numerical analysis 91,519 data were processed, for each of the variables involved, and they correspond to a typical local year.

The considered variables for the analysis were dry bulb temperature and relative humidity, in ranges from 3˚C to 35˚C, and 7% to 97% respectively, with local altitude Z = 1526 m [

For each humidity range T_{wb} vs T_{db} were plotted and a linear adjusted equation was obtained, with general form: T_{wb} = AT_{db} + B.

Then, coefficients for A and B were correlated, for each humidity range considered (average of the range) as shown in

In Equation (1), relative humidity is considered dimensionless, with T_{db} and T_{wb} given in ˚C. Adjusted coeffi- cient values for each polynomial, are shown in

Equation (1) has in its general form a linear behavior for dry bulb temperature. But coefficients A_{i} and B_{i} de-

Percentage distribution for humidity ranges

Polynomial coefficients for A and B with coefficients of determination

. Adjustment equations from T_{wb} and T_{db} for each interval humidity

Φ | Equation | R^{2} |
---|---|---|

6 - 10 | T_{wb} = 0.4658T_{db} − 2.177 | 0.8672 |

11 - 15 | T_{wb} = 0.5506T_{db} − 3.5846 | 0.8711 |

16 - 20 | T_{wb} = 0.593T_{db} − 3.5889 | 0.9144 |

21 -25 | T_{wb} = 0.6456T_{db} − 3.8737 | 0.9643 |

26 - 30 | T_{wb} = 0.6845T_{db} − 3.8201 | 0.9828 |

31 - 35 | T_{wb} = 0.7277T_{db} − 3.9253 | 0.9868 |

36 - 40 | T_{wb} = 0.7555T_{db} − 3.6281 | 0.9915 |

41 - 45 | T_{wb} = 0.7876T_{db} − 3.4778 | 0.9935 |

46 - 50 | T_{wb} = 0.814T_{db} − 3.2415 | 0.9944 |

51 - 55 | T_{wb} = 0.8329T_{db} − 2.8555 | 0.9957 |

56 - 60 | T_{wb} = 0.8547T_{db} − 2.5688 | 0.9965 |

61 - 65 | T_{wb} = 0.8772T_{db} − 2.3067 | 0.9971 |

66 - 70 | T_{wb} = 0.8951T_{db} − 1.9578 | 0.9973 |

71 - 75 | T_{wb} = 0.9127T_{db} − 1.6419 | 0.9974 |

76 - 80 | T_{wb} = 0.9293T_{db} − 1.327 | 0.9979 |

81 - 85 | T_{wb} = 0.9464T_{db} − 1.0134 | 0.998 |

86 - 90 | T_{wb} = 0.9642T_{db} − 0.7506 | 0.998 |

91 - 95 | T_{wb} = 0.9772T_{db} − 0.4249 | 0.9967 |

96 - 100 | T_{wb} = 0.9852T_{db} − 0.1798 | 0.9999 |

. Fit coefficients for proposal equation

Coefficient | Value |
---|---|

A_{0} | 0.3652 |

A_{1} | 1.5181 |

A_{2} | 1.5164 |

A_{3} | 0.6334 |

B_{0} | −0.5194 |

B_{1} | −29.956 |

B_{2} | 84.459 |

B_{3} | −85.009 |

B_{4} | 31.063 |

pend on relative humidity, and they are 3th and 4th degree polynomials.

_{db}, ϕ, T_{wb} for data station and it is compared with the obtained values from Equation (1), for mentioned ranges, with acceptable fit.

In order to evaluate Equation (1), two errors are estimated: normalized error (E_{n}) and real error (E_{r}). These are obtained from base data and with calculated data, according to the proposed equation. Normalized error evalua- tion was obtained using the following equation:

_{n}) and is equivalent to maximum variation of 0.6˚C as shown in the main vertical axis (E_{r}). Absolute average error is 0.065% (En_av) about meteorological station, representing an average variation of a hundredth degree Celsius from real error (Er_av = 0.011˚C).

Also,

_{n} range (X_{j}) according to

Comparison of T_{db}, ϕ, T_{wb} correlation of meteorological data (a); and from proposed model according to Table 1(b)

Comparison between real error and normalized error evaluated with data from meteorological station

X_{j} vs f_{j} corresponding to data shown in Table 2

. Absolute frequency with its corresponding percentage to E_{n} data

X_{j} | Absolute frequency, f_{j} | Absolute percentage, % |
---|---|---|

3 to 2.5 | 0 | 0 |

2.5 to 2.0 | 5 | 0.0054 |

2.0 to 1.5 | 65 | 0.071 |

1.5 to 1.0 | 1735 | 1.89 |

1.0 to 0.5 | 29,115 | 31.81 |

0.5 to 0.0. | 16,757 | 18.3 |

0.0 to −0.5 | 21,663 | 23.67 |

−0.5 to −1.0 | 20,730 | 22.66 |

−1.0 to −1.5 | 908 | 0.99 |

−1.5 to −2.0 | 201 | 0.22 |

−2.0 to −2.5 | 143 | 0.16 |

−2.5 to −3.0 | 79 | 0.086 |

−3.0 to −6.0 | 110 | 0.12 |

DB = 91,519 | 99.98 |

difference of real error and normalized error are in an acceptable value (only 3.6% of total data are out of 1.0 and −1.0).

A direct equation was obtained in order to predict wet bulb temperature, from relative humidity and dry bulb temperature. Methodology used to obtain the model, was in an empiric way, grouping relative humidity in ranges of 5 units, and for each range wet and dry bulb temperatures were correlated, obtaining a general linear adjust. Also, coefficients A and B were evaluated. Quality of generated data for the model was settled through error analysis. Normalized errors are less than 5.5%. It means a maximum variation of 0.55˚C from data. The normalized error applies only to 3.6% of the data considered in the present work. The model has an acceptable fit. It is valid for a range of temperature and relative humidity of 3˚C to 100˚C and 7% to 97% respectively. The pressure was calculated from the local altitude and the model behavior was not explored at different pressures. The equation obtained represents an important tool that facilitates the analysis and engineering calculations of various important energy processes.