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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.

Steam generators (SGs) of the vertical recirculating type (UTSG) are the dominant types that are used in the pressurized water reactors (PWRs). SGs are heat exchangers containing thousands of tubes to transfer heat from the primary coolant which represents the hot side to the secondary coolant which represents the cold side to produce steam, that produce electricity by powering turbine generators. A large number of nuclear power plants (NPPs) have from 2 to 6 SGs; however some have up to 12 SGs [

Water/Steam thermodynamic properties are required in equation derivation of SG models. These properties are available in the form of plots or tables. Extracting data directly from these plots or tables is not suitable for dynamic modeling. Therefore, most of the researchers expressed them in linear mathematical forms, which can be effective for simulation purposes. In UTSG mathematical models, the relations of the specific volumes/densities of the vapor and liquid, and their specific enthalpy of the vapor and liquid and saturated temperature that depend on the saturation pressure (P_{sat}) are required. Typically, such relationships can be found for pressure intervals in thermodynamic properties tables [

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 [

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 [

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 [

The most popular design for steam generators is UTSG [

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.

2) Primary flow (W_{pi}) is constant.

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}).

The current mathematical model of UTSG is based on the geometrical parameters and the numerical constants for the four regions listed in

Symbol | Description | Value and Units |
---|---|---|

A_{d} | Area of the coolant in the accumulation tank | 3 m^{2} |

A_{dw} | Area of coolant in recirculation region | 10.3 m^{2} |

C_{m} | Specific heat capacity of metal tubes | 460 J/(kg˚C) |

C_{s} | Conversion factor between Wso and Psat | 80 kg/(s MPa) |

K | thermal conductivity of metal tube | 55.0012 Jsm˚C |

L | height of U-tube | 10.83 m |

L_{d} | Height of accumulation tank | 10.83 m |

L_{sb} | height of liquid | 1.057 m |

M_{3} | Mass of half of total metal tube upward | 25,600 kg |

M_{4} | Mass of half of total metal tube downward | 25,600 kg |

N | total number of U-tubes | 3388 |

R_{avg} | (R_{in} + R_{out})/2 | 10.357 cm |

R_{in} | inner radius of U-Tube | 9.75 cm |

R_{out} | outer radius of U-Tube | 11 cm |

T_{fi} | Temperature of inlet feed water | 223˚C |

T_{p} | Temperature of primary water | 312˚C |

U_{ms} | heat transfer coefficient between the metal tubes and the secondary water | J/(s m^{2}˚C) |

V_{dr} | Volume of drum | 124.5 m^{3} |

V_{r} | Internal volume of Liquid/Vapor separator | 13.3 m^{3} |

V_{s} | volume of the secondary coolant in the effective heat exchange region | 94.4 m^{3} |

W_{p} | Flow rate of primary water | 4950 kg/s |

ρ_{d} | metal tube density | 8490 |

ρ_{m} | Density of metal tube | 8500 kg/m^{3} |

Symbol | Description | Value and Units |
---|---|---|

T_{1} | Temperature of primary coolant flowing upward | 576 K |

T_{2} | Temperature of primary coolant flowing downward | 569 K |

T_{3} | Temperature of metal tube of coolant flowing upward | 571 K |

T_{4} | Temperature of metal tube of coolant flowing downward | 565 K |

T_{5} | Temperature of recirculated water | 535 K |

T_{6} | Temperature of water in the mixing region | 535 K |

L_{s} | Level of water of mixing region of secondary coolant | 3.2 m |

x_{b} | Quality of coolant at the end of the boiling region | 0.233 |

P_{sat} | Saturation Pressure of secondary water in boiling region | 5.9 MPa |

Symbol | Description | Units |
---|---|---|

A_{ms} | Surface area in contact between secondary water and metal tubes | m^{2} |

A_{pm} | Surface area in contact between and metal tubes and primary coolant | m^{2} |

C_{p} | specific heat of subcooled water | J/(kg˚C) |

h_{f} | saturated water specific enthalpy | kJ/kg |

h_{fg} | H_{f} − h_{g} | kJ/kg |

h_{g} | saturated vapor specific enthalpy | kJ/kg |

h_{s} | average enthalpy of secondary water | kJ/kg |

L_{b} | height of boiling column | m |

M_{d} | Mass of water in annular tank | kg |

M_{dw} | mass of drum water | kg |

M_{p} | Mass of half of the primary water | kg |

M_{s} | mass of secondary coolant | kg |

Q ˙ m s 1 | upward metal tube to secondary coolant heat transfer rate | J/s |

Q ˙ m s 2 | downward metal tube to secondary coolant heat transfer rate | J/s |

T_{s} | Average bulk mean temperature in the secondary lump | ˚C |

T_{sat} | saturation temperature | ˚C |

U_{ms} | Coefficient of heat transfer between the secondary coolant and metal tubes | J/(s m^{2}˚C) |

U_{pm} | Coefficient of heat transfer between the primary coolant and metal tubes | J/(s m^{2}˚C) |

W_{1} | mass flow rate of fluid from the downcomer to the riser | Kg/s |

W_{2} | Mass flow rate of outlet secondary water | Kg/s |

W_{3} | mass flow rate of secondary coolant leaving separator | kg/s |

W_{fi} | Flow rate of feed water | kg/s |

W_{so} | Flow rate of saturated steam | kg/s |

ρ_{f} | saturated liquid density | kg/m^{3} |

ρ_{g} | saturated vapor density | kg/m^{3} |

υ_{f} | saturated liquid specific volume | m^{3}/kg |

υ_{fg} | υ_{f} − v_{g} | m^{3}/kg |

v_{g} | saturated vapor specific volume | m^{3}/kg |

Nodalization scheme of the UTSG model is shown in

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 (

d d t ( M p ∗ C p ∗ T 1 ) = W p i ∗ C p ( T p − T 1 ) − U p m ∗ A p m ∗ ( T 1 − T 3 ) (1)

M p ∗ C p ∗ d T 1 d t = W p i ∗ C p ∗ T p − W p i ∗ C p ∗ T 1 − U p m ∗ A p m ∗ T 1 + U p m ∗ A p m ∗ T 3 (2)

U p m = 1 1 h i + R i n k ∗ log ( R a v g R i n ) (3)

so

d T 1 d t = ( − W p i M p − U p m ∗ A p m M p ∗ C p ) T 1 + ( U p m ∗ A p m M p ∗ C p ) T 3 + ( W p i M p ) T p i (4)

In the same way, we have

d T 2 d t = ( − W p i M p − U p m ∗ A p m M p ∗ C p ) T 2 + ( U p m ∗ A p m M p ∗ C p ) T 4 + ( W p i M p ) T 1 (5)

Also, U-Tube metal region equations derivation is done by applying the energy conservation equation on both METAL_UP and METAL_DOWN, which represent the inverted U-metal tube up flow and downflow respectively as shown in

d d t M 3 ∗ C m ∗ T 3 = U p m ∗ A p m ∗ ( T 1 − T 3 ) − U m s ∗ A m s ∗ ( T 3 − T s ) (6)

U m s = 1 1 h d + R o u t k ∗ log ( R o u t R a v g ) (7)

d T 3 d t = ( − U p m ∗ A p m − U m s ∗ A m s M 3 ∗ C m ) T 3 + ( U p m ∗ A p m M 3 ∗ C m ) T 1 + ( U m s ∗ A m s M 3 ∗ C m ) T s (8)

In the same way, we have

d T 4 d t = ( − U p m ∗ A p m − U m s ∗ A m s M 4 ∗ C m ) T 4 + ( U p m ∗ A p m M 4 ∗ C m ) T 2 + ( U m s ∗ A m s M 4 ∗ C m ) T s (9)

Secondary coolant region equations derivation is done applying energy and mass conservation equations on the three secondary coolant regions: Effective heat transfer region (EHT), upper tank region (ACCU_T), and Inlet annular tank region (IA_T) as shown in

EHT region of secondary cooling water is the area of heat accumulation in the metal tubes. Because of the heat build-up, the secondary cooling water is pushed up, and the energy conservation equation can be written as follows:

d d t ( h s ∗ M s ) = Q ˙ m s 1 + Q ˙ m s 2 + W 1 ∗ C p s ∗ T 5 − W 2 ∗ h e (10)

V s d d t ( ρ s ∗ h s ) = U m s ∗ A m s ∗ ( T 3 − T s ) + U m s ∗ A m s ∗ ( T 4 − T s ) + W 1 ∗ C p s ∗ T 5 − W 2 ∗ h e (11)

Now the following equations are deduced for the average thermodynamic properties of the secondary coolant in the effective heat exchange mass.

L = L s b + L b (12)

T s = T 5 + T s a t 2 ∗ L s b L + T s a t ∗ L b L (13)

ρ s = ρ d + ρ f 2 ∗ L s b L + L b ( υ f + x e 2 ∗ υ f g ) ∗ L (14)

h s = C p s ∗ T 5 + T s a t 2 ∗ L s b L + ( h f + x e 2 ∗ h f g ) ∗ L b L (15)

h e = h f + x e ∗ h f g (16)

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 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:

d ( V r ∗ ρ r ) d t = W 2 − W 3 (17)

d ( L d w ∗ A d w ∗ ρ f ) d t = W f i + ( 1 − x e ) ∗ W 3 − W 1 (18)

d ( M d w ∗ C p s ∗ T 6 ) d t = W f i ∗ M d w ∗ C p s ∗ T f i + ( 1 − x e ) ∗ W 3 ∗ C p s ∗ T s a t − W 1 ∗ C p s ∗ T 6 (19)

d ( ( V d r − A d w ∗ L d w ) ∗ ρ g ) d t = x e ∗ W 3 − W s o (20)

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:

d ( M d ∗ C p s ∗ T 5 ) d t = W 1 ∗ C p s ∗ T 6 − W 1 ∗ C p s ∗ T 5 (21)

We can rewrite Equation (21) as:

d T 5 d t = W 1 M d ∗ T 6 − W 1 M d ∗ T 5 (22)

From Equations (18) and (19):

d T 6 d t = ( W 3 ( x e − 1 ) ∗ W 1 − W f i L d w ∗ A d w ∗ ρ f ) T 6 + W f i L d w ∗ A d w ∗ ρ f ∗ T f i + ( 1 − x e ) ∗ W 3 L d w ∗ A d w ∗ ρ f ∗ T s a t (23)

From Equation (20):

W 3 = 1 x e ∗ { ( ( V d r − A d w ∗ L d w ) ∗ d ρ g d P + ρ g ρ f ∗ A d w ∗ L d w ∗ d ρ f d P ) / ( 1 + ( 1 − x e ) x e ∗ ρ g ρ f ) ∗ d P s a t d t − ρ v ρ L ∗ ( W f i − W 1 ) G 2 + W s o G 2 A g } (24)

From Equations (17) and (24):

W 2 = { 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 } ∗ d P s a t d t − { V r ∗ v f g ( v f + x e ∗ v f g ) 2 } ∗ d x e d t − 1 x e ∗ ρ g ρ f ∗ W f i − W 1 ( 1 + 1 − x e x e ∗ ρ g ρ f ) + 1 x e ∗ W s o ( 1 + 1 − x e x e ∗ ρ g ρ f ) (25)

From Equation (18):

d L d w d t = ( − 1 ρ f ∗ d ρ f d P ∗ d P s a t d t ) ∗ L d w + 1 ρ f ∗ A d w ∗ ( W f i + ( 1 − x e ) ∗ W 3 − W 1 ) (26)

From Equations (10), (12), (24), (25), and (26), the state equations of both P_{sat} and x_{e} can be written as:

d x e d t = ( V s ∗ G 5 + ( 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 ) ) ∗ U m s ∗ A m s G 1 ∗ ( T 3 + T 4 ) + ( W 1 ∗ C p s + V s ∗ ρ s ∗ C p s ∗ L s b 2 ∗ L ∗ W 1 M d − U m s ∗ A m s ∗ L s b L ) ∗ T 6 + ( ( W 1 ∗ C p s + V s ∗ ρ s ∗ C p s ∗ L s b 2 ∗ L ∗ W 1 M d − U m s ∗ A m s ∗ L s b L ) ∗ G 7 G 1 ) ∗ T 5 + G 8 (27)

and

d P s a t d t = ( V s ∗ ( L b L ∗ v f g 2 ( v f + x e 2 ∗ v f g ) 2 ) ) ∗ U m s ∗ A m s G 1 ∗ ( T 3 + T 4 ) − ( ( V s ∗ L b 2 L ∗ υ f g / ( υ f + x e 2 ∗ υ f g ) 2 + G 2 ) ∗ V s ∗ ρ s ∗ ( C p s ∗ L s b ) ∗ W 1 2 ∗ L ∗ M d ∗ G 1 ) ∗ T 6 + G 3 ∗ T 5 + G 4 (28)

where

G 1 = V s ∗ { L s b 2 ∗ L ∗ d ρ f d P − L b L ∗ ( d v f d P + x e 2 ∗ d v f g d P ) ( v f + x e 2 ∗ v f g ) 2 } + { 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 } ∗ V s ∗ ( ρ s ∗ ( L b 2 ∗ L ∗ h f g ) − h s ∗ ( L b L ∗ v f g 2 ( v f + x e 2 * v f g ) 2 ) )

− h e ∗ ( V r ∗ v f g ( v f + x e ∗ v f g ) 2 ) + { V s ∗ ( L b L ∗ v f g 2 ( v f + x e 2 ∗ v f g ) 2 ) + ( V r ∗ v f g ( v f + x e ∗ v f g ) 2 ) } ∗ { V s ∗ ( ρ s ∗ G 4 + h s ∗ G 5 ) + h e ∗ ( 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 ) } (29)

G 2 = ( h e x e ∗ ρ g ρ f ∗ W f i − W 1 1 + 1 − x e x e ∗ ρ g ρ f − 2 ∗ U m s ∗ A m s ∗ ( L s b 2 ∗ L + L b L ) ) ∗ T s a t (30)

G 3 = ( V s ∗ ( L b L ∗ v f g 2 ( v f + x e 2 ∗ v f g ) 2 ) + V r ∗ v f g ( v f + x e ∗ v f g ) 2 ) G 1 ∗ ( W 1 ∗ C p s + V s ∗ ρ s ∗ ( C p s ∗ L s b 2 ∗ L ) ∗ W 1 M d − U m s ∗ A m s ∗ L s b L ) (31)

G 4 = C p s ∗ L s b 2 ∗ L ∗ d T s a t d P + L b L ∗ ( d h f d P + x e 2 ∗ d h f g d P ) (32)

G 5 = L s b 2 ∗ L ∗ d ρ f d P − L b L ∗ ( d v f d P + x e 2 ∗ d v f g d P ) ( v f + x e 2 ∗ v f g ) 2 (33)

G 6 = ( ( V d r − A d w ∗ L d w ) ∗ d ρ g d P + ρ g ρ f ∗ A d w ∗ L d w ∗ d ρ f d P ) / ( 1 + 1 − x e x e ∗ ρ g ρ f ) (34)

G 7 = V s ∗ G 5 + 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 (35)

G 8 = − ( V s ∗ ( ρ s ∗ ( C p s ∗ L s b 2 ∗ L ∗ d T s a t d P + L b L ∗ ( d h f d P + x e 2 ∗ d h f g d P ) ) + h s ∗ G 5 ) + h e ∗ ( 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 ) ) ∗ ( W 1 + 1 x e ∗ ρ g ρ f ∗ W f i − W 1 1 + 1 − x e x e ∗ ρ g ρ f ) − G 2 ∗ G 7 G 1

+ ( V s ∗ ( ρ s ∗ G 4 + h s ∗ G 5 ) + h e ∗ ( 1 x e ∗ G 6 − V r ∗ ( d v f d P + x e ∗ d v f g d P ) ( v f + x e ∗ v f g ) 2 ) ) − h e ∗ G 7 G 1 ∗ x e ∗ ( 1 + 1 − x e x e ∗ ρ g ρ f ) ∗ W s o (36)

From Equations (15) to (36), we need to calculate the terms: ρ V , ρ L , h V , h L and T s a t , which are functions of P_{sat}. Also we need their derivatives: d ρ V d P s a t , d ρ L d P s a t , d h V d P s a t , d h L d P s a t and d T s a t d P s a t .

We used MATLAB Curve Fitting Toolbox to provide an accurate functions or equations of steam/water properties taken from [

The formulae which are fitted are:

1) Vapor Saturated Density in terms of P_{sat}, ρ_{V}(P_{sat}) as shown in

2) Liquid Saturated Density in terms of P_{sat},ρ_{L}(P_{sat}) as shown in

3) Vapor Saturated Enthalpy in terms ofP_{sat},h_{V}(P_{sat}) as shown in

Goodness of fit | Coefficients (with 95% confidence bounds) |
---|---|

SSE: 1.81e−07 R-square: 1 Adjusted R-square: 1 RMSE: 6.206e−05 | a = 14.41 (14.4, 14.42) b = −0.9291 (−0.9293, −0.9289) c = −0.005371 (−0.005414, −0.005329) |

ρ V = 14.41 ∗ ( P s a t ) − 0.9291 − 0.005371 (37) |

Goodness of fit | Coefficients (with 95% confidence bounds) |
---|---|

SSE: 1.621e−10 R-square: 0.9997 Adjusted R-square: 0.9997 RMSE: 1.898e−06 | p_{1} = −5.342e−16 (−6.484e−16, −4.2e−16) p_{2} = 1.383e−12 (1.148e−12, 1.617e−12) p_{3} = −1.276e−09 (−1.437e−09, −1.116e−09) p_{4} = 8.258e−07 (7.85e−07, 8.665e−07) p_{5} = 0.001053 (0.00105, 0.001056) |

ρ L = − 5.342 e − 16 ∗ ( P s a t ) 4 + 1.383 e − 12 ∗ ( P s a t ) 3 − 1.276 e − 9 ∗ ( P s a t ) 2 − 8.258 e − 7 ∗ P s a t + 0.001053 (38) |

4) Liquid Saturated Enthalpy in terms of P_{sat},h_{V}(P_{sat}) as shown in

5) Saturated Temperature in terms of P_{sat},,T_{sat}(P_{sat},) as shown in

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

Goodness of fit | Coefficients (with 95% confidence bounds) |
---|---|

SSE: 9.775 R-square: 0.9997 Adjusted R-square: 0.9996 RMSE: 0.4943 | p_{1} = 1.141e−22 (8.235e−23, 1.458e−22) p_{2} = −5.663e−19 (−7.121e−19, −4.205e−19) p_{3} = 1.199e−15 (9.165e−16, 1.481e−15) p_{4} = −1.413e−12 (−1.712e−12, −1.114e−12) p_{5} = 1.016e−09 (8.273e−10, 1.205e−09) p_{6} = −4.604e−07 (−5.327e−07, −3.881e−07) p_{7} = 0.0001316 (0.0001152, 0.0001481) p_{8} = −0.02346 (−0.02553, −0.02139) p_{9} = 2.531 (2.407, 2.655) p_{10} = 2665 (2663, 2668) |

h V = 1.141 e − 22 ∗ ( P s a t ) 9 − 5.663 e − 19 * ( P s a t ) 8 + 1.199 e − 15 ∗ ( P s a t ) 7 − 1.413 e − 12 ∗ ( P s a t ) 6 + 1.016 e − 9 ∗ ( P s a t ) 5 − 4.604 e − 7 ∗ ( P s a t ) 4 + 0.0001316 ∗ ( P s a t ) 3 − 0.02346 ∗ ( P s a t ) 2 + 2.531 ∗ P s a t + 2665 (39) |

Goodness of fit | Coefficients (with 95% confidence bounds) |
---|---|

SSE: 515.8 R-square: 0.9998 Adjusted R-square: 0.9998 RMSE: 3.313 | a = 146.5 (135.2, 157.7) b = 0.3114 (0.3027, 0.32) c = 143.5 (120.3, 166.7) |

h L = 146.5 ∗ ( P s a t ) 0.3114 + 143.5 (40) |

Goodness of fit | Coefficients (with 95% confidence bounds) |
---|---|

SSE: 11.73 R-square: 0.9999 Adjusted R-square: 0.9999 RMSE: 0.4997 | a = 84.17 (81.84, 86.51) b = 0.2052 (0.2025, 0.208) c = −37.1 (−40.27, −33.93) |

T s a t = 84.17 ∗ ( P s a t ) 0.2052 − 37.1 (41) |

one, which indicates better fit to the data, and finally, the RMSE, which expresses the standard deviation.

The above Equations ((4), (5), (8), (9), (22), (23), (26)-(28)) define the following main UTSG variables:

· T_{1}: temperature of the upward direction primary coolant in inverted U-Tube.

· T_{2}: temperature of the downward direction primary coolant in inverted U-Tube.

· T_{3}: temperature of the metal tube containing upward direction primary water.

· T_{4}: temperature of the metal tube containing downward direction primary water.

· 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:

d X d t = A ∗ X + B (42)

and state vector is:

X = [ T 1 T 2 T 3 T 4 T 5 T 6 L d w x P s a t ] (43)

and state matrix is:

A = [ A 1 , 1 0 A 1 , 3 0 0 0 0 0 0 A 2 , 1 A 2 , 2 0 A 2 , 4 0 0 0 0 0 A 3 , 1 0 A 3 , 3 0 0 0 0 0 0 0 A 4 , 2 0 A 4 , 4 0 0 0 0 0 0 0 0 0 A 5 , 5 0 0 0 0 0 0 0 0 A 6 , 5 A 6 , 6 0 0 0 0 0 0 0 0 0 A 7 , 7 0 0 0 0 A 8 , 3 A 8 , 4 A 8 , 5 A 8 , 6 0 0 0 0 0 A 9 , 3 A 9 , 4 A 9 , 5 A 9 , 6 0 0 0 ] (44)

B = [ B 1 0 B 3 B 4 B 5 0 B 7 B 8 B 9 ] (45)

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.

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 _{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 _{i} = 500 seconds tell final time t_{f} = 1500 seconds. _{1}, T_{2}, T_{3}, T_{4}, P_{sat}, and x. respectively.

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

the pressurized water reactor. The main advantage of using such highly precision expressions for deducing the thermodynamic relationships is the reduction in the simulation time as these functions provide a very fast calculation and control the safety usage of the PWR. This is because the properties of the vapor are called millions of times within the framework of the model while running and used the interpolation technique with the steam tables. The evidence of the high precision is shown through the statistical quantities computed in the fitting process with R-square being less than 1 in all cases, and can be easily used on their own or through their derivatives or integrals. These computed functions increase the precision for the calculation of the thermodynamic properties of steam/water. Moreover, this model was tested with both static persistence and 5% transient changes and gave typical and reasonable results that reflect the dynamic characteristics of the steam generator in real time. It shows a good adaptability in a wide range of industrial control.

The authors declare no conflicts of interest regarding the publication of this paper.

Alramady, A.M., Al-Sharif, S. and Nafee, S.S. (2021) Modeling of UTSG in the Pressurized Water Reactor Using Accurate Formulae of Thermodynamic Properties. Journal of Applied Mathematics and Physics, 9, 947-967. https://doi.org/10.4236/jamp.2021.95065