Optimization and Improvement of Sodium Heated Once-through Steam Generator Transient Analysis Code Based on the JFNK Algorithm

Abstract

The sodium heated once-through steam generator (OTSG) is a vital barrier separating sodium and the water loop in the sodium-cooled fast reactor (SFR). In view of the timeliness requirement of OTSG operation performance evaluation, Fortran95 programming language is used to optimize and improve the solution algorithm of the home-made transient analysis code, named TCOSS, for SFR sodium heated OTSG, which is modified into the JFNK algorithm for solving large sparse nonlinear matrices. It includes a matrix preconditioning module, a Krylov subspace formation module, a GMRES algorithm module and an inexact Newton iteration module. The correctness and efficiency of the algorithm model were verified using benchmarks such as the B1-B transient, ETEC shutdown experiments and seven-tube prototype experiments. The calculation speed was increased by more than four times.

1. Introduction

The sodium-cooled fast reactor (SFR) generally uses a sodium heated once-through steam generator (OTSG), which has many advantages, such as no vapor/water separation device and compact structure; good static performance and stable steam pressure; good operation maneuverability and fast power in rise and fall; and superheated steam produced to improve thermal efficiency [1]. In the OTSG, the phase transformation process of water mainly includes superhot water, nuclear boiling, film boiling, single-phase steam and so on. Therefore, the heat transfer condition in the OTSG is more complex than the natural circulation mode of the traditional pressurized water reactor (PWR) steam generator (SG), and the sodium-water heat exchange mechanism is very different. At the same time, the OTSG is a significant barrier to separate sodium and the water loop in the SFR. The integrity of the OTSG heat exchange tube in the SFR is much more important than that of the PWR because of the intense chemical reaction between sodium and water [2]. So, it is very important to accurately simulate the sodium heated OTSG.

A series of universal OTSG thermal hydraulic programs have been developed internationally, including DESOPT, developed and designed by the Indira Gandhi Atomic Research Center in India [3]; ONCESG, developed by the Korea Atomic Energy Research Center [4] and PSM-W, developed by Westinghouse for its prototype OTSG [5]. In addition, Xi’an Jiaotong University developed the TCOSS program [6] to carry out two-phase flow OTSG thermal hydraulic and transient calculation in SFR, and conducted a large number of numerical simulation calculations, providing necessary data to support the design optimization of the OTSG structure [7,8].

With the completion of the comprehensive performance experiments carried out on the prototype OTSG test facility at Xi’an Jiaotong University [9], it is necessary to develop corresponding safety analysis codes to predict the accident process for the actual safe operation of the SFR. Different from the envelope design method in the OTSG design and analysis stage, the OSTG performance evaluation in the operation stage focuses more on the measured data feedback and real-time response. However, the existing TCOSS code Gear algorithm has a slow calculation speed and cannot meet the timeliness requirements of in-service performance evaluation. The original algorithm should be optimized and improved to increase the computational efficiency on the premise of meeting the computational accuracy and accuracy.

The JFNK (Jacobian-free Newton–Krylov) algorithm, as a specific class of imprecise Newton methods, was first proposed by Peter N. Brown and Youcef Saad in 1990 [10] and has developed into an efficient algorithm for solving large sparse nonlinear equations. It is nested using the inexact Newton iteration method [11] and Krylov subspace iteration method [12]. The emergence of this kind of algorithm provides an effective way to solve nonlinear equations.

This algorithm has two significant advantages. One is the use of the Krylov subspace iteration to solve the Newton equation inaccurately to reduce the amount of calculation and improve the calculation speed. Second, the iterative process of Krylov subspace method only needs to use the product of the Jacobian matrix and vector, and it is similar to the vector function difference, thus the Jacobian matrix has no use for computing and reserving, which can greatly save storage capacity and improve calculation efficiency.

2. Mathematical and Physical Model

2.1. SG Model

The sodium heated OTSG is divided into the evaporator and superheater modules. The two modules are composed of the sodium plenum, water-steam plenum, shell, tube bundle and so on, but the size is different. Its structure is shown in Figure 1 [9].

Liquid sodium gets into the down superheater plenum from the inlet connection and then runs through the shell side from top to bottom. Subsequently, the liquid enters the evaporator from top to bottom, and eventually flows out using the outlet pipe. The feedwater enters the heat exchange tubes from botton to top in the lower plenum of the evaporator, absorbing the sodium-side heat. Then it reaches the required temperature and generates superheated steam, which discharges using the evaporator upper plenum. The whole process of subcooled water from the inlet to the outlet of the evaporator includes single-phase water, nuclear boiling, film boiling and superheated steam. In the superheater, it’s the superheated steam.

Considering that the thermal parameters change significantly only in the flow direction, the TCOSS code [7] equates the OTSG to a single-tube model. During the modeling process, it is assumed that the axial thermal conductivity of the metal tube wall and fluid is ignored. The sodium coolant is treated as an incompressible fluid. The heat dissipation phenomenon of OTSG is ignored. The compressible model is adopted in the water region, and the compressible one-dimensional homogeneous flow model is adopted in the two-phase region. Therefore, corresponding fluid, heat transfer, pressure drop and heat transfer tube wall models are established. The equivalent OTSG heat transfer tube is as shown in Figure 2.

The staggered grid technology [13] is used to solve the thermal parameters, wherein, the main control volume stores pressure and temperature, and mass flowrate is stored in the momentum control volume. The control equations are integrated along the OTSG axial control volume one by one, and transformed into nonlinear total differential equations with variable coefficients for each control volume, namely:

$$x^{\prime }=F\left(t,x,x^{\prime }\right)$$

(1)

It can be simplified as:

$$F\left(x\right)=0$$

(2)

F is an n-dimensional vector quantity function, namely $F(x)={(F_1(x),F_2(x),\cdots ,F_n(x))}^T$, and $x={(x_1,x_2,\cdots ,x_n)}^T$ is an n-dimensional vector quantity. The Gear algorithm used by the TCOSS code adopts the Newton iteration method to solve nonlinear equations implicitly, which needs to consider the eigenvalues of the matrix. The calculation efficiency of the code will be reduced while the calculation accuracy is guaranteed, and it cannot meet the timeliness requirements of the OTSG in service period operation evaluation.

2.2. JFNK Algorithm Model

The JFNK algorithm is a kind of inner and outer iteration method. The outer iteration adopts the inexact Newton iteration while Krylov subspace iteration is adopted in the inner iteration.

The inexact Newton algorithm [14] can approximately solve F(x) using the iterative method, namely:

$$F\prime (x^{(k)})s^{(k)}=-F(x^{(k)})$$

(3)

The approximate solution is:

$$x^{(k+1)}=x^{(k)}+s^{(k)}$$

(4)

The convergence conditions need to be:

$$\Vert F(x^{(k)})+F\prime (x^{(k)})s^{(k)}\Vert \le {\eta }_k\Vert F(x^{(k)})\Vert ,{\eta }_k\in (0,1)$$

(5)

When the calculated residual $\Vert F(x^{(k)})\Vert$ and the forced factor ${\eta }_k$ reach the set convergence condition, the next iteration can be carried out. The convergence conditions are very flexible, which can obviously reduce the computational cost of the TCOSS code, and it is well combined with the Krylov subspace iteration method.

The Krylov subspace iteration algorithm is on account of the Krylov subspace. Consider the following linear equations:

Ax = b

(6)

Then, the m-dimensional Krylov line subspace is:

$$K_m(A,r^{(0)})=span\left\{r^{(0)},A{r}^{(0)},\cdots ,A^{m-1}{r}^{(0)}\right\}$$

(7)

The new approximate solution is:

$$x^{(m)}=x^{(0)}+{\displaystyle \sum _{i=0}^{m-1}{\beta }_i{A}^i{r}^{(0)}}$$

(8)

A is a $n\times n$ invertible square matrix, considering replacing A⁻¹ with the m − 1 polynomial q_m−1(A) of square matrix A, thus simplifying the calculation, namely:

$$x^{(m)}=x^{(0)}+q_{m-1}(A)r^{(0)}$$

(9)

For the $n\times n$ reversible square matrix A [15], there must be a polynomial q(t) whose degree is not greater than n, such that q(A) = 0. Therefore, the minimum polynomial degree of q(A) = 0 does not exceed n, thus the Krylov subspace iteration method can achieve convergence within n steps at most. In addition, the time of the minimum polynomial is far less than n in the sparse matrix.

The Arnoldi method [15] is often chosen to compute the orthonormal basis of Krylov subspace, so as to compute a new approximate solution of x^(m). This method needs to be modified to reduce rounding errors. The orthonormal basis of calculation is:

$$V_m=\left[{\nu }^{(1)},{\nu }^{(2)},\cdots ,{\nu }^{(m)}\right]$$

(10)

The Arnoldi process to form the $(m+1)\times m$ upper Hessenberg matrix ${\overline{H}}_m$. $H_m$ can be obtained by removing the last column of ${\overline{H}}_m$, and the eigenvalue of matrix A can be replaced by the eigenvalue of square matrix $H_m$, there is:

$$A{V}_m=V_m{H}_m+w^{(m)}{(e^{(m)})}^T=V_{m+1}\overline{H_m}$$

(11)

$$V_m^{T}A{V}_m=H_m$$

(12)

The generalized minimum residual method (GMRES) is the most suitable method for solving Jacobian asymmetric problems in the Krylov subspace methods [16], which can keep good stability while minimizing the iterative steps [17]. On account of the above modified Arnoldi method, the new approximate solution is:

$$x^{(m)}=x^{(0)}+V_m{y}^{(m)}$$

(13)

Definition:

$$\psi (y)={\Vert b-A{x}^{(m)}\Vert }_2={\Vert r^{(0)}-A{V}_m{y}^{(m)}\Vert }_2={\Vert \theta {\nu }^{(1)}-A{V}_m{y}^{(m)}\Vert }_2$$

(14)

Get:

$$\psi (y)={\Vert V_{m+1}(\theta e^{(1)}-\overline{H_m}{y}^{(m)})\Vert }_2={\Vert \theta e^{(1)}-\overline{H_m}{y}^{(m)}\Vert }_2$$

(15)

Since x^(m) is the vector Z in x⁽⁰⁾ + K_m(A,r⁽⁰⁾) that minimized the Euclid norm of b − Az, the solution of equations in GMRES algorithm can be converted to the least squares problem:

$$y^{(m)}=\underset{z}{\arg \min }{\Vert \theta e^{(1)}-\overline{H_m}z\Vert }_2$$

(16)

The product $F\prime (x)\nu$ of the Jacobian matrix and vector in the Krylov method can be approximately determined using the difference of vector function $F(x)$ [18], without constructing and storing the Jacobian matrix:

$$F\prime (x)\nu \approx \frac{F(x+\epsilon \nu )}{\epsilon },\epsilon \ne 0$$

(17)

where $\epsilon$ refers to the difference step length and its proper selection can improve the calculation accuracy and ensure the stability of the algorithm. $\epsilon$ is too large to be a good approximation of the product. If it is too small, it is easy to be covered by floating point error, and it will produce a large error beyond the precision range of the computer. Considering the ratio relation of x and v and the amount of calculation of the algorithm, this paper takes as follows [19]:

$$\epsilon =\frac{\sqrt{(1+\Vert x\Vert ){\epsilon }_{mach}}}{\Vert \nu \Vert }$$

(18)

The increase in iterations in the GMRES algorithm will result in a drastic increase in the amount of computation. The preconditioning of the matrix [19] can transform complex problems into equivalent problems which are easy to be solved. Therefore, the matrix eigenvalues are centrally distributed within a parameter range and the iterations are reduced, reducing the amount of computation in the algorithm. Using the right precondition, then:

$$\{\begin{array}{c}A{M}^{-1}y=b\\ x=M^{-1}y\end{array}$$

(19)

Considering that the residual of the equation will not change in the right preconditioning, it is beneficial to make the calculation convergence in the GMRES algorithm and accelerate the iteration speed. In this paper, the right preconditioning was selected to replace A in the GMRES algorithm with $A{M}^{-1}$ and V_m with $M^{-1}{V}_m$.

2.3. JFNK Algorithm Logic

The computational process of the JFNK algorithm is shown in Figure 3. Firstly, the linear equations need to be determined according to the input, and then the JFNK calculation module is called to solve the linear equations. The JFNK calculation module includes the matrix preconditioning module, the Krylov subspace formation module, the GMRES algorithm module and the Newton iteration module. JFNK is used to complete the solution of linear equations and then enter the next time step until the transient calculation is completed. In the JFNK algorithm module, it is necessary to constantly update x and recalculate the residual F(x) of the discrete conservation equation.

2.4. TCOSS Code Logic

TCOSS is a one-dimensional transient analysis code of a once-through steam generator for sodium cooled fast reactor. In the early version of TCOSS code, the Gear algorithm was adopted to solve the equation set in the OTSG thermal-hydraulic system [6], but the computation speed was slow and could not meet the timeliness requirements. Therefore, the original algorithm should be optimized and improved to increase the calculation efficiency.

As mentioned above, the JFNK algorithm uses only the “operation of multiplying the Jacobi matrix with vectors”, which can be computed by finite difference approximately. As a result, it is possible to eliminate the need to compute and store the Jacobi matrix, thus greatly saving memory and improving computational efficiency [20]. Therefore, the JFNK algorithm is considered to improve the computational speed, which can be verified below.

The TCOSS code algorithm was developed and optimized using FORTRAN 95 language. The simulation process is shown in Figure 4. Steady-state and transient numerical methods are the same, but the difference lies in the setting of boundary conditions. Firstly, the control volume is divided according to the input parameters, and the initial parameters of each control volume are obtained, including the flow area, temperature, pressure, etc. According to the calculation function set, the OTSG steady-state calculation, transient analysis calculation or restart transient calculation can be carried out.

Transient calculation is used to obtain the time response of thermal parameters in OTSG, establish differential equations and use the JFNK algorithm to solve various equations. Thermal-hydraulic parameters such as temperature and pressure of last time step are taken to initial approximation of the Newton iteration and backtracking method in the above algorithm to ensure convergence [14]. The transient simulation time step and the total simulation time are firstly set, and then the derivatives of enthalpy, pressure and mass flowrate are computed on the basis of the flow heat transfer and pressure drop relations according to the divided grid. According to the input boundary conditions, the JFNK algorithm is used to solve the ordinary various equations to obtain the parameter values of each step.

3. Benchmark Verification

After the algorithm optimization is accomplished, some verification should be carried out to ensure the rationality and accuracy of its calculation function, and to guarantee the precise of the computation while improving the speed. Verification refers to the overall performance verification of the TCOSS code, including steady-state transient calculation. The B1-B transient benchmark [21] and the ETEC shutdown experiment [5] were selected for verification

To fairly compare the performance improvements of the TCOSS code, a consistent test environment was utilized. The codes ran on Windows 10 Pro operating system with a CPU (Intel(R) Core(TM) i7-10700, manufacturer: Intel Corporation, Santa Clara, CA, USA) and 16 GB memory (3200 MHz).

3.1. B1-B Transient

Westinghouse developed the SG transient analysis program PSM-W, which was verified on its prototype OTSG, and gave the B1-B transient benchmark. The mass flow rate, pressure and temperature changes of the fluid on both sides were taken as input values for transient calculation.

The running time of B1-B transient was 1100 s, and the ratio of the calculation speed of the two algorithms was $v_{JFNK}/v_{Gear}=17.78$. It can be seen that the JFNK algorithm greatly improved the calculation speed of B1-B transient. The calculated outlet temperature was compared with the PSM-W program value as shown in Figure 5. The results with two codes were basically identical to those of the PSM-W program, which preliminarily verified the feasibility of using the JFNK algorithm in transient calculations.

In the early transient stage, the outlet steam temperature in the TCOSS code was higher than that calculated using the PSM-W program and tended to decrease first and then rise. The reasons for the change were preliminarily speculated: the decrease in sodium mass flowrate in the early stage of the transient caused the decrease in outlet steam temperature; the decrease in feedwater mass flowrate led to the rise of temperature; the rise of inlet sodium temperature caused the rise in temperature; the rise in feedwater temperature caused the rise in outlet steam temperature.

In the transient first 10 s, the rise of inlet sodium temperature was very small and can be ignored and the influence of feedwater temperature rise on outlet steam temperature can also be ignored. Therefore, the parameters that affect the change of steam temperature at the outlet were mainly sodium and feedwater mass flowrate, and the main parameter was sodium mass flowrate. That is, the decrease in outlet steam temperature resulting from the sodium mass flowrate decrease was greater than that from the feedwater mass flowrate decrease. Similarly, in the transient 10–20 s, it can be considered that the outlet steam temperature rise from the feedwater mass flowrate decrease was higher than that resulting from the sodium mass flowrate decrease.

3.2. ETEC Shutdown Experiment

The Energy Technology Engineering Center (ETEC) carried out the shutdown experiment of sodium heated OTSG. The sodium mass flowrate, inlet sodium temperature and feedwater mass flowrate, inlet feedwater temperature and inlet feedwater pressure were adjusted in the experiment and the above transient boundary was taken as input parameters. The transient calculation of ETEC shutdown experiment was carried out using TCOSS code using the Gear algorithm and the JFNK algorithm. The results were compared with the outlet sodium and steam temperature in the experiment to verify the correctness of the calculation of the optimized code.

The running time of the ETEC shutdown experiment was 14,500 s, and the ratio of the calculation speed of the numerical simulation of the two algorithms was $v_{JFNK}/v_{Gear}=15.53$, indicating that the JFNK algorithm significantly improved the ETEC transient calculation speed of the TCOSS code. The comparison between the outlet sodium temperature and steam temperature calculated using the two algorithms and the experimental values is shown in Figure 6. The temperature-change curves calculated using the two codes coincided well, and the change trend was basically consistent with the experiment.

Due to a certain time interval in the recording of experimental data, the real peak value was not recorded, so the peak value of the outlet sodium temperature using the program was higher than the experiment. Apart from the peak point, the calculated outlet sodium temperature was in accord with the experiment. For outlet steam temperature, the JFNK algorithm had small burr phenomena in some positions, which is because the physical property of water is pressure sensitive, and sometimes the outlet pressure does not iterate to stability. The control tolerance can be adjusted to increase the iterations so that the pressure calculation results converge.

On the whole, the calculated values of the TCOSS code using the JFNK algorithm were in accord with the experiment, and the integral change tendency was coincident, which verified the feasibility and accuracy of the code for OTSG transient experimental characteristics analysis.

4. Transient Experiment Verification

4.1. Experimental Introduction

To furtherly prove the OTSG analysis code based on the optimized algorithm, power-volume modeling was carried out for the evaporators in the China Demonstration Fast Reactor (CDFR), and transient experiments for the evaporator seven-tube prototype were carried out. The number of heat transfer tubes was seven and the structural parameters are as shown in

Table 1. Structural parameters of the seven-tube prototype.

Structural Parameters	Design Value
Tube length/m	17.7
Outer tube diameter/m	0.016
Inner tube diameter/m	0.011
Tube pitch/m	0.033
Tube number	7

.

The heat transfer tube is a straight tube. The resistance coefficient of the throttle at the entrance of each heat exchange tube, the mass flowrate at the water of the tube, the temperature and pressure at the inlet of the tube were adjusted to be consistent with the CDFR evaporator so that the process of water flow and heat transfer could be more truly simulated. The resistance coefficient of the restrictor was very large to ensure the flow distribution between pipes. The flow direction of the shell-side liquid sodium was opposite to that on water for sufficient convective heat exchange.

In the experiment, the outlet steam temperature of three heat transfer tubes in the middle of the seven-tube prototype and two symmetric tubes around it was measured, as shown in Figure 7. Transient experimental conditions included multi-parameter change transient of sodium temperature, water temperature, pressure and mass flowrate. The calculation accuracy and calculation speed of the OTSG analysis code after the optimized algorithm were verified by the experimental results.

4.2. Multi-Parameter Change Transient

Table 2. Transient experiment conditions 1~3.

Number	Sodium Mass Flowrate /m3∙h−1	Feedwater Mass Flowrate /kg∙s−1	Inlet Sodium Temperature /°C	Outlet Steam Pressure /MPa	Feedwater Temperature /°C	Note
1	22.64	0.586	457	14.4	210	Inlet sodium temperature decreased
2	22.64	0.586	467	14.4	210	Inlet sodium temperature increased
3	22.64	0.586	467	14.4	190	Feedwater temperature decreased

shows the experiment conditions given in the experimental outline, and each condition was a single thermal parameter transient change. Working conditions from one to three had continuity in time, so they were considered together.

Inlet temperature as well as mass flowrate of sodium side and inlet temperature, mass flowrate and outlet pressure of water side were taken as the transient boundary conditions of input, as shown in Figure 8. The comparison results are as shown in Figure 9. The running time is 3583 s, and the ratio of speed was $v_{JFNK}/v_{Gear}=4.75$. The JFNK algorithm significantly improved the calculation speed of the TCOSS code. The water inlet pressure calculated using the two algorithms coincided completely with the experimental change, and the difference was within 0.5 Mpa. The calculated outlet temperatures also agreed well.

The outlet steam temperature variation was basically consistent with the experiment and the deviation was within 10 °C. When the inlet sodium temperature decreased from 500 to 750 s, the inlet temperature and mass flowrate of water remained unchanged, so the outlet steam temperature decreased. In addition, it could be seen that there are small burrs in the calculated results of the outlet steam temperature with the JFNK algorithm. Since the water-side properties are pressure sensitive, the number of iterations can be increased by adjusting the control tolerance to make the water-side outlet pressure iterate to stability.

The variation trend of sodium temperature was basically the same as the experiment, and the deviation value was within 10 °C. This confirmed the TCOSS code’s accuracy after the optimization algorithm to simulate the sodium-water convective heat exchange. Because the heat dissipation in the experiment was not considered in the TCOSS code, the calculated heat exchange was larger than the actual heat transfer. Therefore, the calculated sodium temperature at the highest point was larger and that at the lowest point was smaller. In general, this transient condition preliminarily verified that the JFNK algorithm can significantly improve the transient calculation speed of the TCOSS program while ensuring the accuracy of the code.

The comparison of CPU time between the Gear algorithm and the JFNK algorithm is shown in

Table 3. Comparison of CPU time between the Gear algorithm and the JFNK algorithm.

Case	CPU Time/s		Speed Ratio
	Gear Algorithm	JFNK Algorithm
B1-B transient benchmark	19,558	1100	17.78
ETEC shutdown experiment	225,185	14,500	15.53
Transient experiment of evaporator seven-tube prototype	17,019	3583	4.75

. In general, the JFNK algorithm can significantly improve the transient calculation speed of the TCOSS code, and its calculation speed is about five times that of the Gear algorithm in the seven-tube prototype. In addition, the calculation of outlet parameter changes under different transient conditions using the optimized code and is basically coincident with the experiment, which confirms the correctness of the code for the CDFR evaporator transient calculation, and its calculation accuracy is not lower than that of the Gear algorithm. This shows that the program can not only improve the speed but also ensure the accuracy of calculation and can be used for studying the OTSG thermal hydraulic characteristics.

5. Conclusions

In view of the timeliness requirement of OTSG evaluation, the numerical calculation algorithm of the TCOSS transient code in the SFR OTSG was optimized and improved using the JFNK algorithm utilizing Fortran95 programming language to improve the calculation speed of the code. Some verification analysis was carried out on the OTSG calculation code after the optimization algorithm, including classical benchmarks, such as the B1-B transient benchmark and the ETEC shutdown experiment, and the CDFR evaporator seven-tube prototype experiment verification. The below conclusions can be drawn:

Whether the Gear algorithm or the JFNK algorithm is adopted, the TCOSS code can obtain good calculation results, agreeing well with the benchmarks and the CDFR prototype experiments, which verified the TCOSS accuracy.

The optimized TCOSS code with the advanced JFNK algorithm can improve the calculation speed up to 15 times for B1-B and ETEC benchmarks, and almost five times for CDFR prototype experiments.