Modeling of the boiler economizer

The boiler economizer is a tube heat exchanger located in the final part of the convective duct. In the economizer, water flowing into the boiler is preheated by flue gases. The paper presents the boiler economizer mathematical model with distributed parameters, which can be used to simulate its operation. The developed mathematical model makes it possible to determine temperatures of the tube and working medium of the boiler economizer. In addition, the non-linear mathematical model of the entire boiler allows to analyze the influence of ash fouling of individual boiler heating surfaces on the economizer operation. The proposed model can also be used for monitoring heat and flow parameters of the economizer in on-line mode.


Introduction
The subject of this paper is the boiler economizer mathematical model which can be applied in the economizer design and operation calculations. Literature in the field of hydraulic and thermal boiler calculations is huge. Both analytical methods and modeling CFD are used for mathematical modeling of processes in boilers [1,2]. Computational modeling is an excellent way to optimize boiler design and performance. However, analytical methods are more suitable to monitor the operation of the boiler on-line. The reason for this is the computation time of CFD simulations. A lot of attention is paid in the literature to modeling of steam superheaters. The complexity of heat transfer processes in the steam superheaters causes some difficulties in mathematical modeling of superheaters. The steam superheater cannot be calculated using the method based on the logarithmic mean temperature difference between the fluids (the LMTD method) or the ε-NTU method (effectiveness -the number of transfer units). It is caused by a large dependence of the water steam specific heat on pressure and temperature [3][4][5]. In works [6,7] a standard method for calculating steam boilers was presented. In this method, superheaters are calculated as common heat exchangers assuming constant physical properties of the liquid. Jan Taler et al. [8] proposed a transient mathematical model for the combustion chamber for the optimization of the plant start-up time. In the work [9] also discuss the optimal boiler start-up by using mathematical models of the critical pressure components of a steam boiler. Corrosion of economizer tubes is the subject of many works [10,11]. The available literature contains a little information about the economizer modeling. In the work [12] models of economizer with smooth ducts and economizer ducts embossed with turbulence inducing ribs were presented. Stevanovic et al. [13] proposed the numerical model which allows to demonstrate how the high-pressure economizer can be used to raise the primary control reserve in coal-fired thermal power plants.

Mathematical model of the boiler economizer
Having passed through steam superheaters, flue gases are directed onto the boiler economizer, located in the final part of the convective duct. The economizer is a tube heat exchanger where water flowing into the boiler is preheated by flue gases. Thereby, the flue gas temperature is reduced (flue gas waste heat recovery). The OP-210M boiler has a two-stage economizer made of tubes arranged in a staggered configuration. The economizer tubes are made of Russian steel 20, whose heat conductivity coefficient is approximated using the relation -7 2.5 35.762267 6.01214 10 where w k is in W/(m•K) and T in o C.
Temperature-dependent changes in the heat conductivity coefficient for steel 20 are shown in Fig. 1. The economizer first and second stage are made of tubes connected with fins along the entire length (membrane heating surface). Due to the temperature field symmetry, the economizer tubes can be treated as longitudinally finned ones (Fig. 2).  Basic data concerning the economizer first and second stage are listed in Table 1. The economizer mathematical modeling is similar to modeling the steam superheater. However, it has to be taken into account that the working medium flowing in tubes is water. It should also be emphasized that due to high specific heat of this medium, the heat flux that has to be supplied to water from flue gases is very high. The presence of fins on the superheater tubes will be taken into consideration by introducing an equivalent (weighted) heat transfer coefficient on the flue gas side.
The weighted heat transfer coefficient on the flue gas side related to the outer surface of a smooth tube (with no longitudinal fins) is determined from the condition of equality between the heat flux absorbed by the outer surface of the smooth tube (assuming that the heat transfer coefficient is equal to the weighted one) and the total heat flux absorbed by the fins and the smooth tube surface in between the fins ( Fig. 2 and Fig. 3) The symbols in Eq. (2) are as follows: zr hweighted heat transfer coefficient, flue gas side, (W/m 2 K), g h -heat transfer coefficient, flue gas side, (W/m 2 K), After transformations, Eq. (2) gives the following relation for the weighted heat transfer coefficient on the flue gas side The efficiency of a straight fin with constant thickness is expressed as [14][15][16][17]   where parameter m is described using the following equation 2 g fin fin h m k   (5) and the fin height is found from the relation (Fig. 2 Using the weighted heat transfer coefficient zr h , the economizer is calculated as if it was made of smooth tubes.
For the staggered tube arrangement, the Nusselt number is found from the following relation [18] where the friction factor for smooth tubes is given by the Filonienko equation [20]   2 1.82 log Re 1.64 The heat transfer coefficient at the tube inner surface for transition regime from laminar to turbulent and for turbulent flow can also be determined using the correlation proposed by Taler [21]       where: in d -inner diameter of the tube, L -tube length.
The mean Nusselt number for laminar flow Num with respect to tube length L can be expressed for uniform wall heat as [21]   where the symbol  designates the gamma function [22].
Eq. (18)  The economizer was modeled in the same way as the superheaters, using the developed distributed parameter mathematical model proposed in [23,24]. The general assumptions made in the development of a numerical model of the boiler economizer are as follows: the water and gas flow is one dimensional, the physical properties of fluids are functions of temperature, axial heat conduction in the tube wall and fluid is negligible, the temperature and flue gas velocity are constant over the channel cross-section before the economizer, and heat transfer coefficients on the inner and outer tube surfaces are uniform. By using the partial differential equations describing the space and time changes of steam Ts, tube wall Tw, ash layer Ta and flue gas temperatures Tg can be obtained  the steam temperature at the outlet of the control volume (Fig. 4)   where x  is the control volume length.
The resulting system of nonlinear algebraic equations (19), (21)  A mathematical model of the device was developed. The flow arrangement and division of the economizer second stage into finite volumes is shown in Fig. 6. Considering the flow arrangement, the economizer second stage can be classified as a parallel-cross-flow heat exchanger. The economizer tubes are in a staggered configuration. The tube arrangement is shown in Fig. 5.

Conclusions
The proposed mathematical model is suitable for modeling multi-pass boiler economizers and various flow systems. Due to the short calculation time, the method can be used for monitoring heat and flow parameters of the economizer in on-line mode. The developed mathematical model makes it also possible to determine temperatures of the tube outer and inner surfaces and external fouling for all heating surfaces of the boiler. The developed mathematical model allows to determine parameters of working mediums and the temperature of the wall, which is essential in design calculations. The knowledge of the metal temperature under the different boiler loads enables a correct selection of the steel grade. Determining the tube walls temperature also makes it possible to avoid overheating of the tubes, e.g. at the boiler low loads.