Modeling of UTSG in the Pressurized Water Reactor Using Accurate Formulae of Thermodynamic Properties

In nuclear power plants, steam generators are important parts. This paper introduces a U-tube steam generator (UTSG) model implemented completely using MATLAB environment. The UTSG consists of four regions: upward and downward primary regions, upward and downward metal tube regions, and secondary regions, which contain heat transfer region, steam separation region, and subcooled water region. Governing equations are derived by applying energy and mass conservation equations in all regions. Accurate functions that describe the relationships between thermodynamic properties of the saturated steam are introduced instead of interpolation method that is widely used. Steady state and one transient case are presented as well.

Recently, many researchers reported the modeling of the UTSG using different platforms. For example, the derivation of the UTSG model was done using MATLAB as a part of a complete power plant simulator [4]. They used steam tables from Ref. [5] to get water/steam properties using interpolation functions programmed in C++ and FORTRAN languages. Reference [6] used an external software package to calculate the physical parameters of water and vapor. Then, a dynamic model for steady-state and transient responses was programmed on a personal computer using MATLAB/SIMULINK. While in Ref. [7], a dynamic model of AP1000 steam generator introduced using the following assumption: linear approximate profiles for volume weighted specific volume and specific enthalpy are used in secondary side tube bundle region. Other researchers reported the using of MATLAB to model Steam Generators as in [8] [9] [10] [11] [12] to do the same for calculating water/steam properties.
In the present work, a mathematical model for the UTSG is introduced using a home-made computer model written fully using MATLAB programming environment. It is well known that if there are many measurement data, MATRIXx environment provides a quick solution to the least square's problems, so matrix calculations can be performed easily using commercial software packages such as MATHCAD and MATLAB [1]. MATLAB is widely used in modeling of steam generators because of its interactive programmable matrix calculator with graphics capabilities [13].
The Steam Generator is divided into four regions according to Ali's second intermediate model (model C) for the UTSG, which is utilized in this work using a MATLAB Program [14]. The regions are: upward and downward primary regions which represent primary water pumped from the reactor cooling pump going up then down inside thousands of inverted U tubes, and during this movement heat transferred through upward and downward metal tube regions.
Finally, the heat is transferred to the secondary regions where the cold feed water uses this heat to generate steam; then steam is separated during its way to turbine. Governing equations are derived by applying energy, mass, and momentum conservation equations on all regions. The most popular design for steam generators is UTSG [1]. Figure 1 represents a schematic diagram of an inverted U-shaped vertical steam generator, in which all parts of the steam generator are contained within a cylindrical vessel and the pre-cooling water coming from the reactor core enters the inlet at a pressure of about 15 MPa facing upward at the inverted U. This flowing water transfers the heat to the secondary water in the steam generator through the pipe walls without mixing. The primary cooling water enters the steam generator at approximately 583 -600 K and exits at 530 -560 K. Also, the steam mass ratio varies from 17% -33%. The higher this percentage, the higher the efficiency of the steam generator and the better for the turbine because this wet steam cannot be sent to the high -pressure turbine, which increases the need for high techniques to remove the liquid from the steam as much as possible [15].

Assumptions
The first step in the UTSG modeling is to make appropriate assumptions. These assumptions are introduced to smoothing calculations and minimizing the complexity. The basic assumptions on which our model is constructed are: 1) One dimensional flow for primary and secondary cooling water.
3) Steam exit flow (W so ) is proportion to saturated pressure (P sat ). 4) Thermal conductivity of the inverted metal tube is constant. 5) Heat transfer coefficients are assumed to be constant during transients. 6) Heat transfer between the tube bundle area and the downcomer is neglected. 7) Flow of exit Steam (W so ) = flow of feedwater (W fi ).

UTSG Parameters and Design State Values
The current mathematical model of UTSG is based on the geometrical parameters and the numerical constants for the four regions listed in Table 1 and Table 2, respectively [4]. In addition, Table 3 shows symbols and units of the variables used in the thermodynamic equations in the current model.

State Equations for UTSG Model
Nodalization scheme of the UTSG model is shown in Figure 2. The model governing the equations is derived by applying energy, mass, and momentum conservation equations.
Primary coolant equations derivation is done by applying the energy conservation equation to both PRIMARY_UP and PRIMARY_DOWN, which represent the moving of primary coolant up and down inside the inverted U-tube respectively ( Figure 2).
In the same way, we have In the same way, we have Now the following equations are deduced for the average thermodynamic properties of the secondary coolant in the effective heat exchange mass.
The heat transfer in the ACCU_T zone is not carried out immediately when the secondary coolant meets the metal tubes. It is in this region specifically that the secondary cooling water is in the state of saturation, as a mixture of water Journal of Applied Mathematics and Physics and steam. And dryers are used to separate the steam from the water to allow the steam to flow into the turbine while the rest of the water is returned to the inlet ring tank to be mixed with the feed water. Now by applying the continuity equations only to the mixture of water vapor and mass of vapor respectively, and by applying both the continuity equations and the energy balance to the mass of water only: Subcooled water in IA_T region of UTSG is a mixture of both recycled water from the previous zone and the main feed water. By applying the energy balance: ( ) We can rewrite Equation (21) as: From Equations (18) and (19): From Equation (20): From Equations (17) and (24)         We used MATLAB Curve Fitting Toolbox to provide an accurate functions or equations of steam/water properties taken from [16]. Also, the derivatives of 1) Vapor Saturated Density in terms of P sat , ρ V (P sat ) as shown in Figure 3 and Table 4.
2) Liquid Saturated Density in terms of P sat ,ρ L (P sat ) as shown in Figure 4 and Table 5.
3) Vapor Saturated Enthalpy in terms ofP sat ,h V (P sat ) as shown in Figure 5 and Table 6.
4) Liquid Saturated Enthalpy in terms of P sat ,h V (P sat ) as shown in Figure 6 and Table 7.
5) Saturated Temperature in terms of P sat ,,T sat (P sat ,) as shown in Figure 7 and Table 8.
It is clear that the results of the statistical data in Tables 4-8 show good compatibility according to four important statistical indicators; the first one is the (SSE), which represents the sum of squares due to the error and is considered a measure of the total deviation, the second one is R-Square, which is a measure of the variance of the data, the third is the Adjusted R-Square, which is closer to    Figure 6. Fitting of saturated liquid enthalpy in terms of P sat . Journal of Applied Mathematics and Physics  Figure 7. Fitting of saturated temperature in terms of P sat .
one, which indicates better fit to the data, and finally, the RMSE, which expresses the standard deviation. • T 4 : temperature of the metal tube containing downward direction primary water.

State Variables of UTSG
• T 5 : recirculated water temperature before mixing. • T 6 : annular tank temperature. • L dw : water level of recirculated water plus feedwater (mixing region). • x: steam quality at the exit.
• P sat : secondary coolant saturation pressure.
Now we can arrange these equations in matrix form to be suitable for MATLAB as the following: and state vector is: and state matrix is:

of Applied Mathematics and Physics
To make the model works independently, the model must be maintained in a closed-loop situation. This is done by adding the following two control conditions: 1) The flow rate of the exit steam is equal to the flow rate of thefeedwater (W so = W fi ).
2) The flow rate of the exit steam is proportional to the saturated pressure of the steam (W so /P sat = constant). Now, by solving Equation (42), and taking into consideration the two control conditions, the UTSG model is presented. It was totally implemented in MATLAB environment. In the next section, some results of running the model in both steady state and transient state will be presented.

Results
Simulations are carried out by running the MATLAB model for 3 hours (1800 sec). Simulations are programmed on a personal computer (Intel Core™ i5-11600T Processor (12M Cache, up to 4.10 GHz)). The first step is using the initial design data from Table 2 to simulate the steady-state condition. Figures   8-11 show the steady state simulation of the thermodynamic properties T 1 , T 2 , T 3 , T 4 , P sat , and x, respectively.
Simulation of the transient performance of the UTSG is also done. As shown in Figure 12, the transient case increases the steam flow by 5% starting from initial time t i = 500 seconds tell final time t f = 1500 seconds. Figure 13 through Figure 16 show the transient state simulation of the response of thermodynamic properties T 1 , T 2 , T 3 , T 4 , P sat , and x. respectively.

Conclusion
An accurate mathematical modeling of the U-Tube steam generator, which is introduced in this work, offers a good start to assist a complete simulation for