Research on a widely applicable function expression of the dry-wet bulb coefficient through deviation control theory

. Relying on the experimental data, by using the theory of error control and function approximation to establish new wet and dry bulb coefficient A(cid:3553)(v, t (cid:2911) ) . Through searching method and Iterative approximation, it finds an approximation function whose figure is closer to "experimental data" than previous one did. Hence, it increases the measurement accuracy of relative humidity by the wet and dry bulb method under the condition of the relative humidity in the range of 10℃≤t≤70℃, air flow rate 0.1m/s≤v≤4m/s, and relative humidity 30%≤U≤90%.


Foreword
In modern industry, agricultural production, meteorology, environmental protection, national defense, scientific research, aerospace and other fields, the control requirements of relative humidity have become more common and important technical support. Moreover, the accurate standard of relative humidity control is getting stricter, especially in food, medicine, tobacco, textile and some other industries. It puts forward higher precision requirements of relative humidity control in high temperature environment to ensure the product quality and improve the product performance.
Wet and dry bulb method is a classical, commonly used and simple method. This method used to calculate the relative humidity indirectly by measuring the temperature and speed. Measurement of temperature and speed has been mature technology, hence, this method has more stable and reliable performance, good maintainability, and low cost compared with other measuring methods [1]. It is very important to improve the accuracy of relative humidity measurement by wet and dry bulb method in high temperature environment.

Theory of measuring relative humidity by dry wet bulb method
Relative humidity refers to the percentage of the actual partial pressure of water vapor in wet air and the saturated partial pressure of water vapor at the same temperature. The wet and dry bulb method measures the dry bulb temperature, the wet bulb temperature, the smaller the deviation degree (D.D), and the flow speed of the wet air to find out the relative humidity through formula (1) or figure table. The measurement device is simple, it consists of two thermometers with the same specification. One is called dry bulb thermometer, which is used to measure the ambient temperature. Its indication is noticed as t , the other is wrapped by wet gauze, a connected with the water container, which is called wet bulb thermometer. Its indication is noticed as t . At the same time, the velocity of wet air flow is measured by anemometer, which is used for the relative humidity calculation.
The measuring principle: the water of the wet gauze covered on the warm bubble absorbs the heat in the wet air and then evaporates continuously, which changes the water from liquid state to gas state, reducing the temperature of the wet ball. In this way, the evaporation rate of water in the wet bulb and the moisture content in the surrounding wet air form a certain functional relationship -the classical formula of relative humidity measured by the dry wet bulb method: August apzohn dry wet bulb equation [2][3]. U= Among all symbols in the equation: U equals relative humidity (%),t stands for dry bulb temperature (℃),t stands for wet bulb temperature (℃), E w is the partial pressure of saturated water vapor at the wet bulb temperature (Pa), E s is the partial pressure of saturated water vapor at the dry bulb temperature (Pa), A is the coefficient of dry wet bulb (s / m ·℃), P a is the total pressure of the measured gas (Pa).
When it comes to the spatial measurement of open low altitude areas, P a usually equals atmospheric pressure 101325Pa, when the altitude of measurement is higher than 100m above sea level, the atmospheric pressure will change by more than 1%, in which P a can be chosen according to the annual average statistical value of local atmospheric pressure. If this data is not available or the data source is not reliable, formula (2) is able to help to calculate P a when the altitude is less than 5000m [4]. P a = 101325 ×(1-(162.1025×H)/(6357×1000+H)) 5.256 (2) In the equation, P a -local average atmospheric pressure, Pa, H-local altitude, m.
Wet and dry bulb coefficient A used to be calculated through equation below [2]: In the equation, v means Air velocity surround the wet bulb, unit measurement: m/s. Equation (3) is used when t ,<40℃ [5]. Precisely, the wet and dry bulb coefficient A is not a single value function of air speed, it is a multi-parameter composite function. The main factors affecting its figure are: 1. Air speed, 2. Size and shape of wet bulb thermometer, response speed, pollution condition, structure of upper water jacket, 3. Errors caused by radiation and heat transfer.
When t >40℃, the calculation error of A value will increase rapidly with the increase of temperature. When t reaches 70℃, the relative difference between the calculation result of equation (3) and the experimental data ranges from 20% to 34% under different wind speed conditions [6]. Therefore, some papers have questioned this measurement method and accuracy [7][8].
The experimental data shows that: under high temperature state, the wet and dry bulb coefficient A value is significantly related to the air flow velocity v and the dry bulb temperature t [6].

3
Deviation analysis between experimental data and original fitting function A (v, ) [6] 3.1 Experimental data and original fitting function A(v, )  (1), which is indirectly measured Experimental data table [6]. This method can reflect the parameters that affect the wet and dry bulb coefficient A macroscopically. Chart 1 shows the curve of a value data change under different v and t a states.
The A (v, t ) relation proved by reference is as follows [6].
In the equation: v is the velocity of air around the wet bulb, m/s, t a is the dry bulb temperature (℃).

Deviation analysis
Initially, the concept of degree of deviation (D.D) is introduced, which is defined as the proportion of the absolute value of the absolute difference between the actual data and the target data in the target data. Its formula (5) as follows. D.D= ×100% (5) In the equation, A is the target data, X is the actual data.
In this paper, the formula of deviation degree (D.D) is changed into formula (6), the relevant formula of average deviation degree (AD) and standard deviation degree (SD) are introduced.
In the equation： A , is the data of wet and dry bulb coefficient A of corresponding v and t in Table 1, A (v, t ) i, j is the data calculated according to formula (4) of A (v,t ) under the condition of corresponding v and t with A, i= 1,2,3,… k, j=1,2,3,… n, k=13,n=7, It uses excel to calculate and analyze the data of each state point of a value in formula A (v,t ) (4) and table 1 [6]: the maximum deviation (D.D) of the data of discovery formula (4) and table 1 is 2345.21%,and the minimum is 3.71%, the comparative analysis of 91 specific data shows that: 86 data with deviation (D.D) greater than 15%, the average deviation (AD) of deviation degree is 6.438, and the standard deviation (SD) of deviation degree is 7.575.
From the data analysis above, it is shown that the original fitting function A (v,t ), formula (4) has a very large deviation, and the structure of formula (4) has an issue of imbalance, which needs to be further corrected.

4
Improvement method and establishment of new calculation model 4

.1 Linear interpolation method to obtain the wet and dry bulb coefficient A under the given conditions of v and
This method is relatively simple and practical. Table 1 [6] can be used as the basic data table, then the wet and dry bulb coefficient a under the given V and t , conditions is able to calculated through linear interpolation method with equation (9).

Fig.2.Position coordinate diagram of linear interpolation method
According to the definition of linear interpolation method, the data of A-value of wet and dry bulb coefficient of non-node in Table 1 can be calculated by linear interpolation method (9) [6], and up to three times, it can calculate the corresponding value of A-value of wet and dry bulb coefficient A(v,t ) in the definition domain (10 ℃ ≤ t , ≤ 70 ℃, 0.02m/s ≤ v ≤ 4m / s).
The advantage of this method is that it is very convenient to use Excel for interpolation calculation under given conditions. However, if it is used for data flow calculation processing, or programming calculation, is not convenient, and the calculation takes long steps.

Design of new calculation method
Under the condition of ensuring multiple constraint functions, especially the theory of error minimization method is used as the constraint function, the system model is set as a rational function for approximation [9], which is an ideal solution to find the fitting approximation function A (v, t ).
In the equation: Y is the minimum error function, A v , t is the original function, A v , t is the new fitting approximation function.
The method above is able to be evolved to control the minimum value of deviation degree (D.D). The smaller the deviation degree (D.D) is, the higher the precision of fitting approximation function is getting, and vice versa.
The design of this method aims to establish objective function model and constraint condition function, and to obtain a new fitting approximation function A v , t by iterative approximation of search method [10].

Found of new calculation model
Under the condition of standard atmospheric pressure, in order to analyze formula (1) of measuring U by wet and dry bulb method, when the deviation degree (D.D) is less than 2%, the possible calculation deviation for U will be less than 0.5%, which is enough to meet the accurate requirements of measuring U by wet and dry bulb method. (Notice: the allowable range of indication error of the dry and wet meter used in the work is ± 5% R.H) [2]. Therefore, to find a new fitting approximation functionA v , t , to ensure that the deviation (D.D) 'value of all the experimental data in Table 1 is less than 2%, this problem can be solved [6]. In the equation: A , is the corresponding experimental data in Table 1,A (v, t ) , is the calculated value of the new fitting approximation function A (v, t ) under the condition of v [6], t corresponding to a in Table 1 [6], in which: i = 3,4,5,…k, j=1,2,3,… n, k=13,n=7.
According to the two-dimensional function approximation theory [11], the numerical distribution law in Table 1 conforms to the condition that the separated variables (in which the variable v is the main variable), so its structure can be greatly simplified. This calculation model is based on the image characteristic curve made by equation (4) In the equation, i=3,4,5,…k,j=1,2,3,…n, k=13,n=7, Through programming operation and data analysis of array by Excel, the coefficients of fitting approximation functionA (v, t )) that can meet the constraint function conditions are obtained. The specific coefficients are as follows: a=0.65275, b=0.0675, c=-1.9655×10 -5 , d=1.006×10 -6 , e=0.0945, f=7.67, Take all values into formula 20. The range of 10℃≤t ≤70℃，0.02m/s≤v≤0.05m/s is the check area defined for the existed function A (v, t ) in order to verify the applicability of the new function A (v, t ) formula (21) and the deviation degree (D.D)' data compared with chart 1 in the non defined area are shown in chart 2 (%) [6]: When v=0.05m/s，t =70℃, the deviation degree is relatively large, reaching 4.307%, and the deviation degree of the other 13 points are between 0.080% and 1.614%. The data show that the new function A (v, t ) formula (21) fits the chart 1. Take the corresponding conditions V and t in Table 1 into formula (21) [6], compared with the data in Table 1 [6], it calculates the deviation degree (D.D)' data (%) is shown in table 3, it's shown that: The maximum deviation degree (D.D)' is 1.777%, among them, there are 16 points in the deviation degree (D.D)' from 1% to 1.777%, and 61 points in the deviation degree (D.D)' from 0.0% to 1.0%, the average deviation degree (X ) in the definition domain is 0.678%, the average deviation degree (AD) is 0.342%, and the standard deviation degree (SD) is 0.434%.
All the data above meet the constraint function formula (16), formula (17), formula (18), formula (19). The data show that the new function A (v, t ) fit the experimental data table 1 well [6].  (2) is a classical calculation formula of dry wet bulb coefficient A. In order to further verify the consistency of the new fitting approximation function A (v, t ) equation (21) and equation (2) In the equation, i=1,2,3,…k,j=1,2,3,…n, k=13,n=5,formula (2)A(v) , has no relationship with t . Table 4 shows the deviation degree (D.D)'' data (%) obtained by comparing the new function A (v, t )formula (21) with formula (3), and the data analysis in Table 4 shows that: In the range of 10℃≤t a ≤30℃， 0.02m/s≤v≤4.0m/s, the maximum deviation (D.D)'' is 1.906%, hense, the new function A (v, t ) fit this region well.
When 30℃＜t , it is found that the deviation degree (D.D)'' will increase with the increase of t and V. When t =70℃, the maximum deviation degree (D.D)'' will exceed 33%. It is further proved that formula (3) is only suitable for the environment within t < 40℃.

Conclusions
The analysis above shows that the wet and dry bulb coefficient A is not only related to the wind speed v, but also to the ambient temperature t . Especially in the high temperature area, the wet and dry bulb coefficient A will decrease with the temperature increase, which will expand the error between the traditional calculation and the actual situation. In the middle and high altitude, low pressure working areas, the influence is even greater, up to 20% R.H error. Hence, considering the influence of wind speed and ambient temperature on the wet and dry bulb coefficient A, the wet and dry bulb method is getting more practical. It has positive impact on the environmental control of industrial and agricultural production in the new era, and achieve high-precision environmental state control through relatively low-cost moisture measuring device [12]. The new fitting approximation function A (v, t )formula (21) in this paper is not a unique relation expression. It is only a kind of expression function satisfying the constraints.
The new fitting approximation function A (v, t ) formula (21), in the range of 0.1m/s≤v≤4m/s,10℃≤t a ≤70℃, has enough high accuracy. Without considering other interference effects, such as temperature measurement, pressure measurement and speed measurement, the accuracy error of relative humidity measurement by dry wet ball method is less than 0.5%.