EXPANSION OF THE RANGE OF WET AIR І-D DIAGRAM FOR ENVIRONMENTAL SAFE HEAT PRODUCTION

For environment protection, condensing boilers may be used instead of emission of high temperature combustion products. For condensing combustion, high moisture content in exhaust gases is typical. I-d diagram can be used for condensing economizer development, but it is primary built for heating, ventilation and air conditioning. In the work, new approach to build I-d diagram is proposed that allow widening of the range of parameters without precision loose. It allows obtaining higher precision for calculation of condensing economizers.


Introduction
The most environment-safe technology of fuel combustion is condensing boiler. It saves from 3% (by the datasheets of ÖkoFEN, 2015) of hard fuel to 10...12% of gas fuel. There is a less volume of combustion products. Some pollutants are absorbed by the condensate. They can be utilised more easily than from the smoke. One of the problems developing the condensing heat exchangers is precise calculation of the thermodynamic processes. The most easier tool for wet air calculations is I-d diagram also known as Ramzin diagram or Mollier diagram. The main purpose of it is heating, ventilation and air conditioning (HVAC) systems. Often, laboratory researches of heat-mass transfer in the condensing heat exchangers are performed on very wet and hot air. Authors tried using the diagram for researches for new condensing polymer economizer [1]. Precision of the calculations was not enough because the basic parameters of enthalpy calculation are heat of vaporization and isobaric specific heat at temperature t = 0 °C. In is enough for HVAC systems because of low temperature rangeup to 40 °C. In the temperature range of condensing heat exchangersup to 100 °C and higher,the physical properties are different from the values above. To avoid difficult integration, the averaged values in the range of experimental studies are used. But an universal precision tool that can be used in the wide temperature range from HVAC to combustion systems may be preferred.

Basic equations for I-d diagram building
Relative humidity can be found from the following equation using the pressure of saturated vapour psat [Pa]: The work [2] uses very rough quadratic approximation of the pressure of saturated vapour. Moisture content d [kg/kg dry air] is dependent on partial pressure of the vapour pvap [Pa] and the pressure of the process p [Pa] (usually equal to the barometric pressure): In the equation (2) 0.623 is the ratio between the specific gas constant for the dry air and the vapour Rd.a. / Rvap. The precise value, used in this work, is Rd.a. / Rvap = 18.016/28.96 = 0.6221.
The standards of I-d diagram building are described in the work [2]. The enthalpy [kJ/kg dry air] can be calculated by the following simplified formula: In the equation (3) the numerical values are the physical properties at t = 0 °C: specific heat of the dry air cd.a. = 1.005 kJ/(kg K), of the moisture (vapour) cvap. = 1.8 kJ/(kg K) and the heat of vaporization of water r = 2500 kJ/kg. All of the values are dependent on temperature and will be represented below.
These equations are valid in temperature range of HVAC systems. If we need more precise calculations for wide temperature range, the equations will be more complex. If we pass any process from the starting point O (t = 0°C, d = 0) to some point C (with current enthalpy I [kg/kg dry air], temperature t [°C] and moisture content d [kg/kg dry air]), there are some non-answered questions, how to integrate the heat, required for the process. The most important is: what heat of vaporization [kJ/kg] we need? It is dependent on the temperature. However, the vapour is superheated except the curve φ = 1. Thus, what temperature does correspond to the vaporization?

Principles for wide-range I-d diagram building
To answer the question above, it is possible to use the property of isobaric processes that no work will is performed. All heat q [kJ/kg] necessary for a process is equal to the difference between ending and starting enthalpy [kJ/kg] and independent on the process curve. Therefore, we can introduce a basic process (possible or fictitious) from a point with known enthalpy (the best choice is a point with I = 0) to the calculation point C. The process may be easy calculated with minimum assumptions and simplifications ( fig. 1). We will start the basic process from the point O and perform it along three lines: OAisothermal humidification at (t = 0 °C) up to φ = 1 and the moisture content dA, kg/kg dry air. As we have very low moisture content (up to 0.001 kg / kg dry air), it is possible to use heat of vaporization at zero temperature r0 = r(t = 0 °C) without significant error. The necessary heat [kJ/kg dry air] is qOA = r0 dA. (4) AB -the process of humidification and heating along the curve φ = 1. The heat of the process consists of vaporization heat, heating of the dry air and the already vaporized water. The necessary heat [kJ/kg dry air] is where d is the differential operator (straight and bold in contrast with moisture contentitalic, not bold); dsat is the moisture content of the air at φ = 1. It is called by the different index because it can be found easy.
BCheating with constant moisture content d [kg/kg dry air]. The heat of the process consists of heating of the dry air and the vapour. The necessary heat [kJ/kg dry air] is The enthalpy [kJ/kg dry air] of the air at the calculation point C is equal to the sun of the equations (4-6) Replacing the moisture content by the temperature in the first integral of the equation (7) can simplify the expression but cause additional derivative ddsat / dt. The derivative can be found only using table data for saturated vapour pressure. This is possible only numerically with additional error and can be recommended only for rough calculations.
Physical properties of dry air and water vapour can be found by author's interpolations (preserving all table digits) of the data [3]. (11)

Method of wide-range I-d diagram building
From the equation (2)  (13) The equations (2) and (11) dives the opportunity to find moisture content on the line φ = 1 at known temperature t [°C] and pressure p [Pa].
If we create functions in computer algebra system (or using some programming language) by the equations (8-13), we can use it to calculate integrands in the expression (7). The integrals may be numerically computed using any quadrature formula. Desired precision can be achieved using some known adaptive quadrature algorithm. Therefore, the equation (7) can give enthalpy I [kJ/kg dry air] by known temperature t [°C], moisture content d [kg/kg dry air] and pressure [Pa]. The function that computes it, is enough to build isotherms (t = const) in coordinates (I, d).
To operate with relative humidity, partial pressure of the vapour is necessary. It can be found from the equation (2): Using the equations (14), solving the equation (11) and substituting the results to the function created by the equation (7)

Example of wide-range I-d diagram building
Let us build an I-d diagram for standard barometric pressure p = 101325 Pa (Fig. 2). The calculations and plotting are performed in SciLab 5.5.2 [5][6][7][8][9][10]. Let us compare it with I-d diagram in the work [2]. The maximum moisture content is 30 g/kg dry air or 0.03 kg/kg dry air. The maximum value of temperature that allows easy take enthalpy at the maximum moisture content is 48 °C. The corresponding enthalpy by the equations (7-13) is 124.79 kJ/kg dry air. By [2] it is 125.8 kJ/kg dry air. The deviation by enthalpy is 0.8%. At the same moisture content, but the temperature 32 °C the enthalpy is, corresponding, 107.77 kJ/kg dry air and 108.9 kJ/kg dry air. The deviation is 1.05%. It is enough for engineering calculations. At moisture content 15 g/kg dry air or 0.015 g/kg dry air and temperature 24 °C the enthalpy has no deviation: 62 kJ/kg dry air. At the same moisture content and temperature 60 °C the corresponding values of enthalpy are 99.30 kJ/kg dry air and 99.5 kJ/kg dry air. The deviation is comparable with line thickness on the diagram [2].
At moisture content 4 g/kg dry air or 0.004 g/kg dry air and temperature 4 °C the enthalpy has no deviation: 14.06 kJ/kg dry air. At the same moisture content and temperature 60 °C the corresponding values of enthalpy are 70.86 kJ/kg dry air and 70.8 kJ/kg dry air. The deviation is comparable with line thickness on the diagram [2]. Therefore, the enthalpy deviation is near to zero, but at the right part, close to d = 0.030 kg/kg dry air, it has tendency of increasing. At high moisture content, typical for condensing economizers, the deviation may reach 3% -energy saving of hard fuel condensing boilers. Let us test a point, typical for the economizers: t = 60°C, d = 0.1 kg/kg dry air. The equation [2] I = 1.005 t+(2500 + 1.8 t) d = 1.005 · 60 + (2500 + 1.8 · 60) · 0.1 = 321.1 kJ/kg dry air.
By the equations (7-13) the value is I = 313.33 kJ/kg dry air. The deviation is 7.8 kJ/kg dry air or 2.5%. It is comparable with energy saving on hard fuel condensing boiler (near to 3%). We can make a mistake calculating such devices up to twice. At moisture content d = 0.2 kg/kg dry air the deviation reaches 3.3%.
If we use standard I-d diagram, we need to include the deviation into the total uncertainty of the experimental results. This can be avoided using proposed method.
Therefore, the standard I-d diagrams are acceptable for rough calculations. For more precise calculations, the proposed method is recommended.