# Preliminary Development of Thermal Power Calculation Code H-Power for a Supercritical Water Reactor

Authors: Fan Zhang; Dao-gang Lu; Dan-ting Sui; Bo Yuan; Chao Guo
Journal: Science and Technology of Nuclear Installations
 (2014)

---

## Abstract

SCWR (Supercritical Water Reactor) is one of the promising Generation IV nuclear systems, which has higher thermal power efficiency than current pressurized water reactor. It is necessary to perform the thermal equilibrium and thermal power calculation for the conceptual design and further monitoring and calibration of the SCWR. One visual software named H-Power was developed to calculate thermal power and its uncertainty of SCWR, in which the advanced IAPWS-IF97 industrial formulation was used to calculate the thermodynamic properties of water and steam. The ISO-5167-4: 2003 standard was incorporated in the code as the basis of orifice plate to compute the flow rate. New heat balance model and uncertainty estimate have also been included in the code. In order to validate H-Power, an assessment was carried out by using data published by US and Qinshan Phase II. The results showed that H-Power was able to estimate the thermal power of SCWR.


---

## Body

## 1. Introduction

SCWR (Supercritical Water Reactor) is one of the promising Generation IV nuclear systems, which has higher thermal power efficiency than current pressurized water reactor. Compared to the currently running water coolant reactors, SCWR has the higher thermal efficiency and safety, which makes it a more promising advanced nuclear energy system. The concept of SCWR was originally put forward by Westinghouse and GE (General Electric) in the 1950s and preliminarily studied by the United States and the former Soviet Union from 1950s to 1960s. In the 1990s, Dobashi et al. [1] proposed the concept of SCWR again and made a further development in this area.For reactor operating more safely, stably, and economically, the accurate calibration for real thermal power has important significance in reactor design and analysis. It is necessary to develop a special thermal power calculation code for SCWR, while there is no available program published in the world. This paper preliminarily developed visual software named H-Power to calculate SCWR thermal power and its uncertainty based on heat balance method, which has been applied to current PWRs [2].According to the various conceptual SCWR designs [3–5], H-Power divided SCWR into two groups: one is the single-loop type and the other is the double-loop type. The validation of H-Power consists of two parts: for the single-loop SCWR, the data published by Jacopo [6] was used, while for the double-loop SCWR, the values of Qinshan Phase II [7] were used; though it is not the SCWR, we just use it for preliminary validation since the exact detail design data was rarely published.IAPWS is an international nonprofit association concerned with the properties of water and steam, particularly thermodynamic properties. In 1997, IAPWS adopted the “IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam” for industrial use [8], called IAPWS-IF97 for short, which replaced the previous industrial formulation IFC-67 [9] published in 1967.We have done lots of researches and work in comparing IAPWS-IF97 with IFC-67 in several aspects. The result showed that IAPWS-IF97 improves significantly in boundary consistency and accuracy and calculation speed [10], which makes it more extensive use in power industry nowadays.H-Power adopted IAPWS-IF97 to calculate thermodynamic properties (including specific volume, specific enthalpy, and viscosity) of water. The code covers all range of IAPWS-IF97(
273.15
K
≤
T
≤
1073.15
K, p
≤
100
MPa, 1
073.15
K
≤
T
≤
2273.15
K, p
≤
50
MPa), which has been divided into five regions.


## 2. Theoretical Modeling

The modules of H-Power are given in Figure1, where READDT is the reading data subroutine, NUSOLV is numerical calculation subroutine, THPROP is the subroutine which calculates the thermodynamic properties of water and steam, FLOWRT is the subroutine for flow rate calculation, ERCALC subroutine is for the uncertainty estimation, and OUTPUT is the output data subroutine.Figure 1
H-Power modules diagram.H-Power provides a visual interface for users and has high module independence, which makes it a practical code in SCWR analysis.


### 2.1. Thermodynamic Properties Subroutine

#### 2.1.1. Subroutine Flow Chart

The range of IAPWS-IF97 is divided into five subareas; each of them has a basic equation. To make the program more concise and readable, this subroutine used the compiling method of modularization; that is, separate districts of thermodynamics equation were written into separate subroutine.We can directly obtain the thermophysical properties from pressure and temperature values using the basic equations except region 3. Subregion 3 does not have the formula which can calculate other various thermal parameters directly through the temperature and pressure; it can only rely on iteration calculation. In this subroutine, the first pressure calculation uses temperature and a small specific volume which was chosen as initial value. If the difference between the calculated value of pressure and user input values is positive, the value of specific volume increases incremental quantity ratio by half and continues the iteration until the pressure difference is less than condition of convergence. The iteration is over and other properties can be calculated.Through this kind of iteration and close proximity to the input value by increasing incremental quantity in halving bisection method, not only the disadvantages of the increased specific volume increment are too small to calculate quickly was avoided, but also the accuracy of calculation can be ensured.The subroutine of water and steam thermodynamic properties has been totally developed according to the flow chart shown in Figure2.Figure 2
Flow chart of thermodynamic properties subroutine based on IAPWS-IF97.


#### 2.1.2. Subroutine Calculation Verification

In order to do the verification, we list the calculation results together with those given by IAPWS-IF97 in Table1. We can see that the present results are in good agreement with those given by IAPWS-IF97, so that the thermodynamic properties calculated by H-Power are valid.Table 1
Comparison of calculation results between H-Power routine and IAPWS-IF97.
Region
Parameters
Given by
v/(m3 kg−1)
h/(kJ kg−1)
s/(kJ kg−1 K−1)
u/(kJ kg−1)
Subregion 1
T
=
300 K
IF97
0.00100215168
115.331273
0.392294792
112.324818
P
=
3 MPa
H-Power
0.0010021516
115.33127302
0.3922947924
112.32481798
T
=
300 K
IF97
9.71180894
e
-
4
184.142828
0.368563852
106.448356
P
=
80 MPa
H-Power
9.7118089
e
-
4
184.14282773
0.3685638523
106.44835621
Subregion 2
T
=
300 K
IF97
39.4913866
2549.91145
8.52238967
2411.69160
P
=
0.0035 MPa
H-Power
39.491386637
2549.9114508
8.5223896673
2411.6915976
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825
Subregion 3
T
=
663.15 K
P
=
24 MPa
IF97
0.005613
2500.75
5.0320
2366.0
H-Power
0.0056134399
2500.7519610
5.0320333072
2366.0291590
T
=
723.15 K
P
=
50 MPa
IF97
0.002487
2284.44
4.5892
2160.1
H-Power
0.0024874366
2284.4399286
4.58922040321
2160.0680674
Subregion 5
T
=
1500 K
IF97
1.3845509
5219.76855
9.65408875
4527.49310
P
=
0.5 MPa
H-Power
1.3845508987
5219.7685512
9.6540887533
4527.4931018
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825


### 2.2. Flow Rate Calculation Subroutine

The main feedwater flow rate of the steam generator can be measured by orifice plate, which is widely used in measurement of fluid flow. This part is based on the ISO 5167 standard in [11].As the schematic diagram shown in Figure3, upstream fluid pressure P
1 is obtained by (1) taking pressure loss Δ
ω caused by the existence of orifice plate into account:
(1)
P
1
=
P
me
+
P
a
+
Δ
ω
,
(2)
Δ
ω
=
Δ
P
×
(
1
-
β
1.9
)
.Figure 3
Pressure measuring schematic by orifice plate.H-Power adopted the following equation for flow rate measurement:(3)
q
m
=
C
1
-
β
4
ε
·
π
d
2
4
2
Δ
P
ρ
u
,
where q
m is mass flow rate, β is ratio of d (diameter of orifice at operating conditions) and D (diameter of internal pipe at operating conditions), ε is expansibility factor (for compressible fluid, ε
=
1), Δ
P is differential pressure, ρ
u is the density of upstream, and C is discharge coefficient determined by Reader-Harris/Gallagher equation as follows:
(4)
C
=
0.5961
+
0.0261
β
2
-
0.216
β
8
+
0.000521
(
1
0
6
β
R
e
D
)
0.7
+
(
0.0188
+
0.0063
A
)
β
3.5
(
1
0
6
R
e
D
)
0.3
+
(
0.043
+
0.080
e
-
10
L
1
-
0.123
e
-
7
L
1
)
(
1
-
0.11
A
)
β
4
1
-
β
4
-
0.031
(
M
2
′
-
0.8
M
2
′
1.1
)
β
1.3
+
⋯
.
If D
<
7
1.12 mm, (4) should add the following term:
(5)
0.011
(
0.75
-
β
)
(
2.8
-
D
25.4
)
.As we can see from the above equation, discharge coefficientC depends on the Reynolds number R
e
D, which is dependent on q
m in turn, so the calculation of mass flow rate q
m has to be iterated.Diameters in the formula for calculating should be corrected due to the difference of temperature between working condition and measurement. If there is drain hole (diameter isd
k) on the plate, the d should be corrected by the following equation:
(6)
d
=
d
m
{
1
+
0.55
(
d
k
d
m
)
2
}
,
where the d
m have already been corrected by temperature as follows:
(7)
d
m
=
d
0
+
d
0
λ
(
t
-
t
0
)
,
where d
0 is the measurement value under the standard temperature t
0
,
λ is the coefficient of linear expansion, and t is the operating temperature.R
e
D is the Reynolds number calculated by D (diameter of internal pipe at operating conditions), dimensionless parameters presenting the ratio of the inertia force and viscous force of upstream, calculated by the following equation:
(8)
R
e
D
=
4
q
m
μ
π
D
,
where μ is viscosity of the fluid at the working conditions, which is obtained by the thermodynamic properties subroutine.L
1
(
=

l
1
/
D
) is the ratio of the distance from the upstream tapping to the upstream face of the orifice plate and the pipe diameter, and the value selection of its conditions is as follows:
(9)
L
1
=
{
0
corner
tappings
,
1
D
and
D
2
tappings
,
25.4
D
flange
tappings
,
A
=
(
19000
β
R
e
D
)
0.8
,
M
′
2
=
2
L
2
′
1
-
β
.L
2
′
(
=

l
2
′
/
D
) is the ratio of the distance from the downstream tapping to the downstream face of the orifice plate and the pipe diameter, and the value selection of its conditions is as follows:
(10)
L
2
′
=
{
0
corner
tappings
,
0.47
D
and
D
2
tappings
,
25.4
D
flange
tappings
.The uncertainty of the feedwater flow measured and calculated by orifice plate and relevant equations will be given in Section3.2. The computer verification of this subroutine will be integrated in Section 4.2.


### 2.3. Single-Loop SCWR Program

In single-loop type SCWR, the coolant is supercritical water which will be operated above the critical point of water(
T
c
=
647.096
K
,
p
c
=
22.064
MPa
). The coolant remains single-phase when operated above the critical point, which is a stupendous advantage for it will eliminate coolant boiling throughout the system. The no need for recirculation results in the elimination of pressurizer, jet pumps, steam generators, and steam separators. This direct cycle type is greatly simplified as shown in Figure 4.Figure 4
Sketch of SCWR with one loop.The thermal power in this type can be calculated with thermal balance using measured values, including inlet and outlet temperatures, working pressure, flow rate of coolant, and the practical speed of pump. The thermal power(
W
) was obtained by the following equation:
(11)
W
=
Q
×
Ω
Ω
0
×
(
H
h
-
H
c
)
1000
(
MW
)
,
where Q (kg/s) is the flow rate of the coolant in the loop, Ω (tr/min) is the practical speed of pump, Ω
0 (tr/min) is the ideal speed of pump, and H
h
-
H
c (kj/kg) is the difference between the enthalpy at the hot leg and the cold leg. The enthalpies are calculated using THPROP subroutine.


### 2.4. Double-Loop SCWR Program

Some conceptual designs of double-loop SCWR have been put forward, like CANDU-SCWR with a steam generator proposed by Dr. William Fatoux [12] and a pressurized water reactor cooled with supercritical water in the primary loop proposed by Vogt et al. [13]. Double-loop SCWR drives a turbine by the steams from the steam generator in the second loop indirectly as shown in Figure 5.Figure 5
Sketch of SCWR with two loops.Heat balance method in this type of SCWR is based on the steam generator approximation enthalpy balance. H-Power calculate the thermal power of reactor by the theory of heat balance, which obtains thermal power (of generator) in second loop from primary loop through the measurement values of temperature, pressure, and flow rate in the second loop, counting the internal heat loss in at the same time. The advantages of the working conditions of second loop for easy measurement makes it an important method of obtaining thermal power of the double-loop SCWR.Figure6 shows the heat in and out the system (expressed by arrows), which can be described by the following heat balance equation:
(12)
W
R
=
∑
i
=
1
N
W
S
G
i
-
W
Δ
Pr
,
where the W
R is the core thermal power (
MW
), W
S
G
i is the thermal power obtained from the primary loop in the second loop in ith steam generator, and W
Δ
Pr is the thermal power coolant system obtained from other heat sources, (
MW
), which can be estimated from the design of reactor.Figure 6
Heat balance principle diagram of double-loop SCWR.The main physical process occurring in the steam generator is as follows: unsaturated water with the mass flow rateQ
e and specific enthalpy H
e flows into the steam generator through the U tube, exchanges heat with the coolant in primary loop, and mostly becomes wet steam with the mass flow Q
v and specific enthalpy H
v; after the increase of endothermic enthalpy, the others are the saturated blowdown water with mass flow rate Q
p and specific enthalpy H
p. The thermal power in ith steam generator can be calculated with following equation:
(13)
W
SG
=
(
H
v
-
H
e
)
Q
e
-
(
H
v
-
H
p
)
Q
p
,
where H
v is the wet steam enthalpy in steam generator export, H
p is the blowdown enthalpy, Q
p is the flow rate of blowdown in second loop, H
e is the feedwater enthalpy in second loop, and Q
e is the flow rate of feedwater in second loop.The feed water satisfies the law of conservation of mass as follows:(14)
Q
e
=
Q
v
+
Q
p
.Simplify (13) and (14):
(15)
W
SG
=
(
H
v
-
H
e
)
Q
e
-
(
H
v
-
H
p
)
Q
p
,
where feedwater enthalpy H
e and blowdown enthalpy Q
p are calculated by the thermodynamic properties subroutine, flow rate of feedwater Q
e can be obtained from flow rate calculation subroutine, and the flow rate of blowdown Q
p is a measured value.The wet steam always consists of saturated steam and saturated water, which is determined by vapour fractionx, so the enthalpy of wet steam is characterized by the following equation:
(16)
H
v
=
x
H
v
s
+
(
1
-
x
)
H
l
s
,
where the H
v
s is the enthalpy of saturated steam and H
l
s is the enthalpy of saturated water; both are calculated by the thermodynamic properties subroutine.


## 2.1. Thermodynamic Properties Subroutine

### 2.1.1. Subroutine Flow Chart

The range of IAPWS-IF97 is divided into five subareas; each of them has a basic equation. To make the program more concise and readable, this subroutine used the compiling method of modularization; that is, separate districts of thermodynamics equation were written into separate subroutine.We can directly obtain the thermophysical properties from pressure and temperature values using the basic equations except region 3. Subregion 3 does not have the formula which can calculate other various thermal parameters directly through the temperature and pressure; it can only rely on iteration calculation. In this subroutine, the first pressure calculation uses temperature and a small specific volume which was chosen as initial value. If the difference between the calculated value of pressure and user input values is positive, the value of specific volume increases incremental quantity ratio by half and continues the iteration until the pressure difference is less than condition of convergence. The iteration is over and other properties can be calculated.Through this kind of iteration and close proximity to the input value by increasing incremental quantity in halving bisection method, not only the disadvantages of the increased specific volume increment are too small to calculate quickly was avoided, but also the accuracy of calculation can be ensured.The subroutine of water and steam thermodynamic properties has been totally developed according to the flow chart shown in Figure2.Figure 2
Flow chart of thermodynamic properties subroutine based on IAPWS-IF97.


### 2.1.2. Subroutine Calculation Verification

In order to do the verification, we list the calculation results together with those given by IAPWS-IF97 in Table1. We can see that the present results are in good agreement with those given by IAPWS-IF97, so that the thermodynamic properties calculated by H-Power are valid.Table 1
Comparison of calculation results between H-Power routine and IAPWS-IF97.
Region
Parameters
Given by
v/(m3 kg−1)
h/(kJ kg−1)
s/(kJ kg−1 K−1)
u/(kJ kg−1)
Subregion 1
T
=
300 K
IF97
0.00100215168
115.331273
0.392294792
112.324818
P
=
3 MPa
H-Power
0.0010021516
115.33127302
0.3922947924
112.32481798
T
=
300 K
IF97
9.71180894
e
-
4
184.142828
0.368563852
106.448356
P
=
80 MPa
H-Power
9.7118089
e
-
4
184.14282773
0.3685638523
106.44835621
Subregion 2
T
=
300 K
IF97
39.4913866
2549.91145
8.52238967
2411.69160
P
=
0.0035 MPa
H-Power
39.491386637
2549.9114508
8.5223896673
2411.6915976
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825
Subregion 3
T
=
663.15 K
P
=
24 MPa
IF97
0.005613
2500.75
5.0320
2366.0
H-Power
0.0056134399
2500.7519610
5.0320333072
2366.0291590
T
=
723.15 K
P
=
50 MPa
IF97
0.002487
2284.44
4.5892
2160.1
H-Power
0.0024874366
2284.4399286
4.58922040321
2160.0680674
Subregion 5
T
=
1500 K
IF97
1.3845509
5219.76855
9.65408875
4527.49310
P
=
0.5 MPa
H-Power
1.3845508987
5219.7685512
9.6540887533
4527.4931018
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825


## 2.1.1. Subroutine Flow Chart

The range of IAPWS-IF97 is divided into five subareas; each of them has a basic equation. To make the program more concise and readable, this subroutine used the compiling method of modularization; that is, separate districts of thermodynamics equation were written into separate subroutine.We can directly obtain the thermophysical properties from pressure and temperature values using the basic equations except region 3. Subregion 3 does not have the formula which can calculate other various thermal parameters directly through the temperature and pressure; it can only rely on iteration calculation. In this subroutine, the first pressure calculation uses temperature and a small specific volume which was chosen as initial value. If the difference between the calculated value of pressure and user input values is positive, the value of specific volume increases incremental quantity ratio by half and continues the iteration until the pressure difference is less than condition of convergence. The iteration is over and other properties can be calculated.Through this kind of iteration and close proximity to the input value by increasing incremental quantity in halving bisection method, not only the disadvantages of the increased specific volume increment are too small to calculate quickly was avoided, but also the accuracy of calculation can be ensured.The subroutine of water and steam thermodynamic properties has been totally developed according to the flow chart shown in Figure2.Figure 2
Flow chart of thermodynamic properties subroutine based on IAPWS-IF97.


## 2.1.2. Subroutine Calculation Verification

In order to do the verification, we list the calculation results together with those given by IAPWS-IF97 in Table1. We can see that the present results are in good agreement with those given by IAPWS-IF97, so that the thermodynamic properties calculated by H-Power are valid.Table 1
Comparison of calculation results between H-Power routine and IAPWS-IF97.
Region
Parameters
Given by
v/(m3 kg−1)
h/(kJ kg−1)
s/(kJ kg−1 K−1)
u/(kJ kg−1)
Subregion 1
T
=
300 K
IF97
0.00100215168
115.331273
0.392294792
112.324818
P
=
3 MPa
H-Power
0.0010021516
115.33127302
0.3922947924
112.32481798
T
=
300 K
IF97
9.71180894
e
-
4
184.142828
0.368563852
106.448356
P
=
80 MPa
H-Power
9.7118089
e
-
4
184.14282773
0.3685638523
106.44835621
Subregion 2
T
=
300 K
IF97
39.4913866
2549.91145
8.52238967
2411.69160
P
=
0.0035 MPa
H-Power
39.491386637
2549.9114508
8.5223896673
2411.6915976
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825
Subregion 3
T
=
663.15 K
P
=
24 MPa
IF97
0.005613
2500.75
5.0320
2366.0
H-Power
0.0056134399
2500.7519610
5.0320333072
2366.0291590
T
=
723.15 K
P
=
50 MPa
IF97
0.002487
2284.44
4.5892
2160.1
H-Power
0.0024874366
2284.4399286
4.58922040321
2160.0680674
Subregion 5
T
=
1500 K
IF97
1.3845509
5219.76855
9.65408875
4527.49310
P
=
0.5 MPa
H-Power
1.3845508987
5219.7685512
9.6540887533
4527.4931018
T
=
2000 K
IF97
0.0311385219
6571.22604
8.53640523
5637.07038
P
=
30 MPa
H-Power
0.0311385218
6571.2260386
8.5364052311
5637.0703825


## 2.2. Flow Rate Calculation Subroutine

The main feedwater flow rate of the steam generator can be measured by orifice plate, which is widely used in measurement of fluid flow. This part is based on the ISO 5167 standard in [11].As the schematic diagram shown in Figure3, upstream fluid pressure P
1 is obtained by (1) taking pressure loss Δ
ω caused by the existence of orifice plate into account:
(1)
P
1
=
P
me
+
P
a
+
Δ
ω
,
(2)
Δ
ω
=
Δ
P
×
(
1
-
β
1.9
)
.Figure 3
Pressure measuring schematic by orifice plate.H-Power adopted the following equation for flow rate measurement:(3)
q
m
=
C
1
-
β
4
ε
·
π
d
2
4
2
Δ
P
ρ
u
,
where q
m is mass flow rate, β is ratio of d (diameter of orifice at operating conditions) and D (diameter of internal pipe at operating conditions), ε is expansibility factor (for compressible fluid, ε
=
1), Δ
P is differential pressure, ρ
u is the density of upstream, and C is discharge coefficient determined by Reader-Harris/Gallagher equation as follows:
(4)
C
=
0.5961
+
0.0261
β
2
-
0.216
β
8
+
0.000521
(
1
0
6
β
R
e
D
)
0.7
+
(
0.0188
+
0.0063
A
)
β
3.5
(
1
0
6
R
e
D
)
0.3
+
(
0.043
+
0.080
e
-
10
L
1
-
0.123
e
-
7
L
1
)
(
1
-
0.11
A
)
β
4
1
-
β
4
-
0.031
(
M
2
′
-
0.8
M
2
′
1.1
)
β
1.3
+
⋯
.
If D
<
7
1.12 mm, (4) should add the following term:
(5)
0.011
(
0.75
-
β
)
(
2.8
-
D
25.4
)
.As we can see from the above equation, discharge coefficientC depends on the Reynolds number R
e
D, which is dependent on q
m in turn, so the calculation of mass flow rate q
m has to be iterated.Diameters in the formula for calculating should be corrected due to the difference of temperature between working condition and measurement. If there is drain hole (diameter isd
k) on the plate, the d should be corrected by the following equation:
(6)
d
=
d
m
{
1
+
0.55
(
d
k
d
m
)
2
}
,
where the d
m have already been corrected by temperature as follows:
(7)
d
m
=
d
0
+
d
0
λ
(
t
-
t
0
)
,
where d
0 is the measurement value under the standard temperature t
0
,
λ is the coefficient of linear expansion, and t is the operating temperature.R
e
D is the Reynolds number calculated by D (diameter of internal pipe at operating conditions), dimensionless parameters presenting the ratio of the inertia force and viscous force of upstream, calculated by the following equation:
(8)
R
e
D
=
4
q
m
μ
π
D
,
where μ is viscosity of the fluid at the working conditions, which is obtained by the thermodynamic properties subroutine.L
1
(
=

l
1
/
D
) is the ratio of the distance from the upstream tapping to the upstream face of the orifice plate and the pipe diameter, and the value selection of its conditions is as follows:
(9)
L
1
=
{
0
corner
tappings
,
1
D
and
D
2
tappings
,
25.4
D
flange
tappings
,
A
=
(
19000
β
R
e
D
)
0.8
,
M
′
2
=
2
L
2
′
1
-
β
.L
2
′
(
=

l
2
′
/
D
) is the ratio of the distance from the downstream tapping to the downstream face of the orifice plate and the pipe diameter, and the value selection of its conditions is as follows:
(10)
L
2
′
=
{
0
corner
tappings
,
0.47
D
and
D
2
tappings
,
25.4
D
flange
tappings
.The uncertainty of the feedwater flow measured and calculated by orifice plate and relevant equations will be given in Section3.2. The computer verification of this subroutine will be integrated in Section 4.2.


## 2.3. Single-Loop SCWR Program

In single-loop type SCWR, the coolant is supercritical water which will be operated above the critical point of water(
T
c
=
647.096
K
,
p
c
=
22.064
MPa
). The coolant remains single-phase when operated above the critical point, which is a stupendous advantage for it will eliminate coolant boiling throughout the system. The no need for recirculation results in the elimination of pressurizer, jet pumps, steam generators, and steam separators. This direct cycle type is greatly simplified as shown in Figure 4.Figure 4
Sketch of SCWR with one loop.The thermal power in this type can be calculated with thermal balance using measured values, including inlet and outlet temperatures, working pressure, flow rate of coolant, and the practical speed of pump. The thermal power(
W
) was obtained by the following equation:
(11)
W
=
Q
×
Ω
Ω
0
×
(
H
h
-
H
c
)
1000
(
MW
)
,
where Q (kg/s) is the flow rate of the coolant in the loop, Ω (tr/min) is the practical speed of pump, Ω
0 (tr/min) is the ideal speed of pump, and H
h
-
H
c (kj/kg) is the difference between the enthalpy at the hot leg and the cold leg. The enthalpies are calculated using THPROP subroutine.


## 2.4. Double-Loop SCWR Program

Some conceptual designs of double-loop SCWR have been put forward, like CANDU-SCWR with a steam generator proposed by Dr. William Fatoux [12] and a pressurized water reactor cooled with supercritical water in the primary loop proposed by Vogt et al. [13]. Double-loop SCWR drives a turbine by the steams from the steam generator in the second loop indirectly as shown in Figure 5.Figure 5
Sketch of SCWR with two loops.Heat balance method in this type of SCWR is based on the steam generator approximation enthalpy balance. H-Power calculate the thermal power of reactor by the theory of heat balance, which obtains thermal power (of generator) in second loop from primary loop through the measurement values of temperature, pressure, and flow rate in the second loop, counting the internal heat loss in at the same time. The advantages of the working conditions of second loop for easy measurement makes it an important method of obtaining thermal power of the double-loop SCWR.Figure6 shows the heat in and out the system (expressed by arrows), which can be described by the following heat balance equation:
(12)
W
R
=
∑
i
=
1
N
W
S
G
i
-
W
Δ
Pr
,
where the W
R is the core thermal power (
MW
), W
S
G
i is the thermal power obtained from the primary loop in the second loop in ith steam generator, and W
Δ
Pr is the thermal power coolant system obtained from other heat sources, (
MW
), which can be estimated from the design of reactor.Figure 6
Heat balance principle diagram of double-loop SCWR.The main physical process occurring in the steam generator is as follows: unsaturated water with the mass flow rateQ
e and specific enthalpy H
e flows into the steam generator through the U tube, exchanges heat with the coolant in primary loop, and mostly becomes wet steam with the mass flow Q
v and specific enthalpy H
v; after the increase of endothermic enthalpy, the others are the saturated blowdown water with mass flow rate Q
p and specific enthalpy H
p. The thermal power in ith steam generator can be calculated with following equation:
(13)
W
SG
=
(
H
v
-
H
e
)
Q
e
-
(
H
v
-
H
p
)
Q
p
,
where H
v is the wet steam enthalpy in steam generator export, H
p is the blowdown enthalpy, Q
p is the flow rate of blowdown in second loop, H
e is the feedwater enthalpy in second loop, and Q
e is the flow rate of feedwater in second loop.The feed water satisfies the law of conservation of mass as follows:(14)
Q
e
=
Q
v
+
Q
p
.Simplify (13) and (14):
(15)
W
SG
=
(
H
v
-
H
e
)
Q
e
-
(
H
v
-
H
p
)
Q
p
,
where feedwater enthalpy H
e and blowdown enthalpy Q
p are calculated by the thermodynamic properties subroutine, flow rate of feedwater Q
e can be obtained from flow rate calculation subroutine, and the flow rate of blowdown Q
p is a measured value.The wet steam always consists of saturated steam and saturated water, which is determined by vapour fractionx, so the enthalpy of wet steam is characterized by the following equation:
(16)
H
v
=
x
H
v
s
+
(
1
-
x
)
H
l
s
,
where the H
v
s is the enthalpy of saturated steam and H
l
s is the enthalpy of saturated water; both are calculated by the thermodynamic properties subroutine.


## 3. Uncertainty Analysis of Double-Loop SCWR

The relative uncertainty of thermal power given in (17) is obtained by the uncertainty transmission formula:
(17)
Δ
W
W
=
[
∑
i
=
1
n
[
W
SG
i
W
Δ
W
SG
i
W
SG
i
]
2
+
[
W
Δ
Pr
W
Δ
(
W
Δ
Pr
)
W
Δ
Pr
]
2
]
1
/
2
,
where the Δ
W
SG
i
/
W
SG
i is the uncertainty of thermal power in steam generation in each second loop and Δ
(
W
Δ
Pr
)
/
W
Δ
Pr is the uncertainty of thermal power of coolant system in primary loop:
(18)
Δ
W
SG
i
Δ
W
≤
[
Δ
W
SG
Δ
W
]
M
,
where [
Δ
W
SG
/
Δ
W
]
M is the max uncertainty of each steam generation.Assuming that(19)
W
SG
i
W
=
1
i
.We can obtain the following equation:(20)
Δ
W
W
=
{
1
i
[
Δ
W
SG
W
SG
]
M
2
+
[
W
Δ
Pr
W
×
Δ
W
Δ
Pr
W
Δ
Pr
]
2
}
1
/
2
.


### 3.1. The Calculation ofΔ
W
S
G
/
W
S
G

The following equation can be obtained by error propagation formula:(21)
Δ
W
SG
W
SG
=
[
[
H
v
(
Q
e
-
Q
p
)
W
SG
Δ
H
v
H
v
]
2
+
[
H
e
Q
e
W
SG
Δ
H
e
H
e
]
2
+
[
H
p
Q
p
W
SG
Δ
H
p
H
p
]
2
+
[
Q
e
(
H
v
-
H
e
)
W
SG
Δ
Q
e
Q
e
]
2
+
[
Q
p
(
H
v
-
H
p
)
W
SG
Δ
Q
p
Q
p
]
2
]
1
/
2
,
Δ
H
v
H
v
=
[
[
x
H
v
(
H
v
s
-
H
e
s
)
Δ
x
x
]
2
+
[
H
v
s
H
v
x
Δ
H
v
s
H
v
s
]
2
+
[
H
e
s
H
v
(
1
-
x
)
Δ
H
e
s
H
e
s
]
2
]
1
/
2
.Assume that the uncertainty of saturated water in the wet steam is “1” as in (22) and inaccuracy of measurement should be in this range:
(22)
Δ
(
1
-
x
)
1
-
x
=
x
1
-
x
Δ
x
x
=
1
.
Equation (22) can be deduced by the above formula:
(23)
Δ
x
x
=
1
-
x
x
.Δ
H
v
s
/
H
v
s is the uncertainty of saturated steam enthalpy, mainly caused by the uncertainty of pressure measurement and water and steam thermodynamic properties as follows:
(24)
Δ
H
v
s
H
v
s
=
[
(
Δ
H
v
s
H
v
s
)
P
v
2
+
(
Δ
H
v
s
H
v
s
)
f
2
]
1
/
2
and the uncertainty caused by pressure measurement can be calculated by following equation:
(25)
(
Δ
H
v
s
H
v
s
)
P
v
=
P
v
H
v
s
|
∂
H
v
s
∂
P
v
|
Δ
P
v
P
v
,
where Δ
P
v is uncertainty caused by pressure measurement, root mean square of the uncertainty of pressure transmitter, data acquisition system, and calibration, determined by the design of pressure measurement:
(26)
∂
H
v
s
∂
P
v
=
τ
R
T
s
{
∑
i
=
1
43
n
i
J
i
(
τ
-
0.5
)
J
i
-
1
I
i
(
0.1
P
v
p
*
)
I
i
-
1
(
1
p
*
)
}
,
where I
i
,
J
i
,
n
i is given in IAPWS-IF97, and π
=
0.1
P
v
/
p
*
τ
=
T
*
/
t
e, p
*
=
1
MPa, T
*
=
5
40 K, and R
=
0.461526 kJ kg−1K−1.(
Δ
H
v
s
/
H
v
s
)
f
2 is the uncertainty caused by the calculation of water and steam thermodynamic properties, where Δ
H
v
s can be selected in the corresponding section of IAPWS-IF97.


### 3.2. The Calculation ofΔ
H
e
s
/
H
e
s

The uncertainty of saturated water enthalpyΔ
H
e
s
/
H
e
s and feedwater enthalpy Δ
H
e
/
H
e can be estimated by the method the same as Δ
H
v
s
/
H
v
smentioned above.The relative uncertainty of mass flow rateδ
q
m
/
q
m can be described by the following equation:
(27)
δ
q
m
q
m
=
(
(
δ
ρ
1
ρ
1
)
2
(
δ
C
C
)
2
+
(
δ
ε
ε
)
2
+
(
2
β
4
1
-
β
4
)
2
(
δ
D
D
)
2
+
(
2
1
-
β
4
)
2
(
δ
d
d
)
2
+
1
4
(
δ
Δ
p
Δ
p
)
2
+
1
4
(
δ
ρ
1
ρ
1
)
2
)
1
/
2
.The uncertainty of discharge coefficient is given by(28)
C
=
{
(
0.7
-
β
)
%
0.1
≤
β
≤
0.2
,
0.5
%
0.2
≤
β
≤
0.6
,
(
1.667
β
-
0.5
)
%
0.6
≤
β
≤
0.75
.IfD
<
71.12 mm, (10) should add the following term:
(29)
0.9
(
0.75
-
β
)
(
2.8
-
D
25.4
)
%
.Ifβ
>
0.5 and R
e
D
<
10000, the above values should add the following relative uncertainty:
(30)
+
0.5
%
.The uncertainty of expansibility factorε is 3.5
(
Δ
p
/
κ
p
1
)
%, for compressible fluid, ε
=
0.The uncertainty of internal diameter of the pipe line can be estimated by the following equation:(31)
σ
D
=
0.05
%
×
d
i
×
2
3
=
0.05
%
×
d
i
×
1.15
,
where 0.05
%
×
d
i is the error of the measuring instrument; “2” is an extension coefficient of the confidence interval of 95%; 3 is the constant of rectangular distribution.The volume elasticity coefficient of water is quite large determining that its compressibility is very small, so the influence of pressure on the density can be ignored. The uncertainty of feed water densityσ
ρ
/
ρ is determined by the uncertainty of temperature measurement and the calculation of water properties as follows:
(32)
(
Δ
ρ
ρ
)
=
[
(
Δ
ρ
ρ
)
t
e
2
+
(
Δ
ρ
ρ
)
f
2
]
1
/
2
and the uncertainty caused by temperature measurement can be calculated by following equation:
(33)
(
Δ
ρ
ρ
)
t
e
=
t
e
ρ
|
∂
ρ
∂
t
e
|
Δ
t
e
t
e
.The calculation procedures and method of the parameters are the same as theΔ
H
v
/
H
v mentioned above.The relative uncertainty ofΔ
(
W
Δ
Pr
)
/
W
Δ
Pr is estimated by experience values.


## 3.1. The Calculation ofΔ
W
S
G
/
W
S
G

The following equation can be obtained by error propagation formula:(21)
Δ
W
SG
W
SG
=
[
[
H
v
(
Q
e
-
Q
p
)
W
SG
Δ
H
v
H
v
]
2
+
[
H
e
Q
e
W
SG
Δ
H
e
H
e
]
2
+
[
H
p
Q
p
W
SG
Δ
H
p
H
p
]
2
+
[
Q
e
(
H
v
-
H
e
)
W
SG
Δ
Q
e
Q
e
]
2
+
[
Q
p
(
H
v
-
H
p
)
W
SG
Δ
Q
p
Q
p
]
2
]
1
/
2
,
Δ
H
v
H
v
=
[
[
x
H
v
(
H
v
s
-
H
e
s
)
Δ
x
x
]
2
+
[
H
v
s
H
v
x
Δ
H
v
s
H
v
s
]
2
+
[
H
e
s
H
v
(
1
-
x
)
Δ
H
e
s
H
e
s
]
2
]
1
/
2
.Assume that the uncertainty of saturated water in the wet steam is “1” as in (22) and inaccuracy of measurement should be in this range:
(22)
Δ
(
1
-
x
)
1
-
x
=
x
1
-
x
Δ
x
x
=
1
.
Equation (22) can be deduced by the above formula:
(23)
Δ
x
x
=
1
-
x
x
.Δ
H
v
s
/
H
v
s is the uncertainty of saturated steam enthalpy, mainly caused by the uncertainty of pressure measurement and water and steam thermodynamic properties as follows:
(24)
Δ
H
v
s
H
v
s
=
[
(
Δ
H
v
s
H
v
s
)
P
v
2
+
(
Δ
H
v
s
H
v
s
)
f
2
]
1
/
2
and the uncertainty caused by pressure measurement can be calculated by following equation:
(25)
(
Δ
H
v
s
H
v
s
)
P
v
=
P
v
H
v
s
|
∂
H
v
s
∂
P
v
|
Δ
P
v
P
v
,
where Δ
P
v is uncertainty caused by pressure measurement, root mean square of the uncertainty of pressure transmitter, data acquisition system, and calibration, determined by the design of pressure measurement:
(26)
∂
H
v
s
∂
P
v
=
τ
R
T
s
{
∑
i
=
1
43
n
i
J
i
(
τ
-
0.5
)
J
i
-
1
I
i
(
0.1
P
v
p
*
)
I
i
-
1
(
1
p
*
)
}
,
where I
i
,
J
i
,
n
i is given in IAPWS-IF97, and π
=
0.1
P
v
/
p
*
τ
=
T
*
/
t
e, p
*
=
1
MPa, T
*
=
5
40 K, and R
=
0.461526 kJ kg−1K−1.(
Δ
H
v
s
/
H
v
s
)
f
2 is the uncertainty caused by the calculation of water and steam thermodynamic properties, where Δ
H
v
s can be selected in the corresponding section of IAPWS-IF97.


## 3.2. The Calculation ofΔ
H
e
s
/
H
e
s

The uncertainty of saturated water enthalpyΔ
H
e
s
/
H
e
s and feedwater enthalpy Δ
H
e
/
H
e can be estimated by the method the same as Δ
H
v
s
/
H
v
smentioned above.The relative uncertainty of mass flow rateδ
q
m
/
q
m can be described by the following equation:
(27)
δ
q
m
q
m
=
(
(
δ
ρ
1
ρ
1
)
2
(
δ
C
C
)
2
+
(
δ
ε
ε
)
2
+
(
2
β
4
1
-
β
4
)
2
(
δ
D
D
)
2
+
(
2
1
-
β
4
)
2
(
δ
d
d
)
2
+
1
4
(
δ
Δ
p
Δ
p
)
2
+
1
4
(
δ
ρ
1
ρ
1
)
2
)
1
/
2
.The uncertainty of discharge coefficient is given by(28)
C
=
{
(
0.7
-
β
)
%
0.1
≤
β
≤
0.2
,
0.5
%
0.2
≤
β
≤
0.6
,
(
1.667
β
-
0.5
)
%
0.6
≤
β
≤
0.75
.IfD
<
71.12 mm, (10) should add the following term:
(29)
0.9
(
0.75
-
β
)
(
2.8
-
D
25.4
)
%
.Ifβ
>
0.5 and R
e
D
<
10000, the above values should add the following relative uncertainty:
(30)
+
0.5
%
.The uncertainty of expansibility factorε is 3.5
(
Δ
p
/
κ
p
1
)
%, for compressible fluid, ε
=
0.The uncertainty of internal diameter of the pipe line can be estimated by the following equation:(31)
σ
D
=
0.05
%
×
d
i
×
2
3
=
0.05
%
×
d
i
×
1.15
,
where 0.05
%
×
d
i is the error of the measuring instrument; “2” is an extension coefficient of the confidence interval of 95%; 3 is the constant of rectangular distribution.The volume elasticity coefficient of water is quite large determining that its compressibility is very small, so the influence of pressure on the density can be ignored. The uncertainty of feed water densityσ
ρ
/
ρ is determined by the uncertainty of temperature measurement and the calculation of water properties as follows:
(32)
(
Δ
ρ
ρ
)
=
[
(
Δ
ρ
ρ
)
t
e
2
+
(
Δ
ρ
ρ
)
f
2
]
1
/
2
and the uncertainty caused by temperature measurement can be calculated by following equation:
(33)
(
Δ
ρ
ρ
)
t
e
=
t
e
ρ
|
∂
ρ
∂
t
e
|
Δ
t
e
t
e
.The calculation procedures and method of the parameters are the same as theΔ
H
v
/
H
v mentioned above.The relative uncertainty ofΔ
(
W
Δ
Pr
)
/
W
Δ
Pr is estimated by experience values.


## 4. Validation of H-Power Code

### 4.1. Single-Loop Type Program

We have done the verification of single-loop program in H-Power using the main parameters design by Jacopo Buongiorno as in Table2 and assuming the ideal and practical pump speed according to the level of pump speed at present.Table 2
The main parameters reference design of the US SCWR.
Parameter
Value
Thermal power
3575 MWth
Operating pressure
25 MPa
Reactor inlet temperature
280°C
Reactor outlet temperature
510~550°C
Reactor mass flow rate
1843 kg/sThe calculation result of thermal power 3564.3396 MW given in Table3 shows a high consistency with the value 3575MW designed by US in Table 3.Table 3
The input and output values calculated by H-Power.
Input parameter
Value
Operating pressure
25 MPa
Reactor inlet temperature
280°C
Reactor outlet temperature
520°C
Reactor mass flow rate
1843 kg/s
Ideal pump speed
1900 tr/min
Practical pump speed
1850 tr/min
Output parameter
Value
Cold leg enthalpy
1230.2407 kj/kg
Hot leg enthalpy
3216.4985 kj/kg
Thermal power
3564.3396 MW


### 4.2. Double-Loop SCWR Program

Due to the lack of design data of SCWR, the validation of this part has been done by using the values of Qinshan Phase II [13] in Table 4 though it is not SCWR. Table 5 shows the comparison between the results of H-Power and the values of Qinshan Phase II.Table 4
Input parameters of H-Power using the values of Qinshan Phase II.
Parameters
Values
Orifice diameter in reference (m)
0.2417
Feedwater pipe diameter in reference (m)
0.36374
Steam pipe diameter in reference (m)
0.739
Orifice linear expansion coefficient (m/m°C)
0.0000166
Feedwater pipe linear expansion coefficient (m/m°C)
0.0000121
Steam pipe linear expansion coefficient (m/m°C)
0.0000121
Orifice reference temperature (°C)
20
Feedwater pipe reference temperature (°C)
20
Steam pipe reference temperature (°C)
20
Feedwater pressure difference (KPa)
182.1147
Feedwater temperature (°C)
228.920334
Feedwater pressure (MPa)
7.11855
Steam pressure (MPa)
6.941479
Blowdown flow rate (t/h)
20.757261Table 5
Comparison between the results of the values of Qinshan Phase II and H-Power.
Parameters
Qinshan Phase II
H-Power
Feedwater density (kg/m3)
833.12894
832.69758
Feedwater enthalpy (kj/kg)
986.18622
986.07354
Saturated water enthalpy (kj/kg)
1264.3947
1265.5310
Saturated steam enthalpy (kj/kg)
2774.1882
2773.0492
Steam enthalpy (kj/kg)
2772.6784
2771.5417
Blowdown enthalpy (kj/kg)
1264.3947
1265.531
Discharge coefficient
0.603068
0.606047
Feedwater flow rate (t/h)
1948.4789
1957.6866
Generator thermal power (MW)
958.2318
970.94089
Feedwater flow rate uncertainty
0.007236
0.006088Although the main calculation result trend of SCWR is same as CNP650 of Qinshan Phase II, there is many detail differences in thermodynamic properties of water and steam and uncertainties calculation. The inconsistencies in Table5 also come from the different input parameters in measuring instrument.It is a valid way to verify the H-Power while the main calculation process of both PWR and SCWR is the same and there is no particular data of SCWR fitting for the code. The result in Table5 shows that H-Power runs well and has a valid trend in thermal power and its relevance computation. So it can be used in SCWR design and calculation.


## 4.1. Single-Loop Type Program

We have done the verification of single-loop program in H-Power using the main parameters design by Jacopo Buongiorno as in Table2 and assuming the ideal and practical pump speed according to the level of pump speed at present.Table 2
The main parameters reference design of the US SCWR.
Parameter
Value
Thermal power
3575 MWth
Operating pressure
25 MPa
Reactor inlet temperature
280°C
Reactor outlet temperature
510~550°C
Reactor mass flow rate
1843 kg/sThe calculation result of thermal power 3564.3396 MW given in Table3 shows a high consistency with the value 3575MW designed by US in Table 3.Table 3
The input and output values calculated by H-Power.
Input parameter
Value
Operating pressure
25 MPa
Reactor inlet temperature
280°C
Reactor outlet temperature
520°C
Reactor mass flow rate
1843 kg/s
Ideal pump speed
1900 tr/min
Practical pump speed
1850 tr/min
Output parameter
Value
Cold leg enthalpy
1230.2407 kj/kg
Hot leg enthalpy
3216.4985 kj/kg
Thermal power
3564.3396 MW


## 4.2. Double-Loop SCWR Program

Due to the lack of design data of SCWR, the validation of this part has been done by using the values of Qinshan Phase II [13] in Table 4 though it is not SCWR. Table 5 shows the comparison between the results of H-Power and the values of Qinshan Phase II.Table 4
Input parameters of H-Power using the values of Qinshan Phase II.
Parameters
Values
Orifice diameter in reference (m)
0.2417
Feedwater pipe diameter in reference (m)
0.36374
Steam pipe diameter in reference (m)
0.739
Orifice linear expansion coefficient (m/m°C)
0.0000166
Feedwater pipe linear expansion coefficient (m/m°C)
0.0000121
Steam pipe linear expansion coefficient (m/m°C)
0.0000121
Orifice reference temperature (°C)
20
Feedwater pipe reference temperature (°C)
20
Steam pipe reference temperature (°C)
20
Feedwater pressure difference (KPa)
182.1147
Feedwater temperature (°C)
228.920334
Feedwater pressure (MPa)
7.11855
Steam pressure (MPa)
6.941479
Blowdown flow rate (t/h)
20.757261Table 5
Comparison between the results of the values of Qinshan Phase II and H-Power.
Parameters
Qinshan Phase II
H-Power
Feedwater density (kg/m3)
833.12894
832.69758
Feedwater enthalpy (kj/kg)
986.18622
986.07354
Saturated water enthalpy (kj/kg)
1264.3947
1265.5310
Saturated steam enthalpy (kj/kg)
2774.1882
2773.0492
Steam enthalpy (kj/kg)
2772.6784
2771.5417
Blowdown enthalpy (kj/kg)
1264.3947
1265.531
Discharge coefficient
0.603068
0.606047
Feedwater flow rate (t/h)
1948.4789
1957.6866
Generator thermal power (MW)
958.2318
970.94089
Feedwater flow rate uncertainty
0.007236
0.006088Although the main calculation result trend of SCWR is same as CNP650 of Qinshan Phase II, there is many detail differences in thermodynamic properties of water and steam and uncertainties calculation. The inconsistencies in Table5 also come from the different input parameters in measuring instrument.It is a valid way to verify the H-Power while the main calculation process of both PWR and SCWR is the same and there is no particular data of SCWR fitting for the code. The result in Table5 shows that H-Power runs well and has a valid trend in thermal power and its relevance computation. So it can be used in SCWR design and calculation.


## 5. Conclusions

The thermal power calculation code H-Power for SCWR was developed based on heat balance method, which can be easily applied in designs and operation. In this paper thermodynamic properties calculation subroutine based on IAPWS-IF97 has been verified, and the uncertainty analysis has been proposed. The validation of H-Power has been carried out in two parts: single-loop SCWR by the data given by Jacopo Buongiorno and double-loop SCWR by the values of Qinshan Phase II. Although Qinshan Phase II is a PWR, the result can show that H-Power is valid because the main calculation process of both PWR and SCWR is the same. In the future, the validation will be carried out using the values of SCWR when published.



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*Source: 279092-2014-03-26.xml*