Plenum Gas Effect on Fuel Temperature

Abstract

All key phenomena in a fuel element are dominated by the temperature distribution. Fuel thermal expansion, fission gas-induced swelling, and release are directly related to the temperature distribution of the fuel. The fuel-cladding heat transfer coefficient has two components (a) heat transfer through the plenum and (b) heat transfer in case of contact. The gap width, in turn, is affected by thermal expansion, cracking and healing of the fuel, fuel densification, and fuel swelling. As the thermal and mechanical properties of the fuel are interdependent, inaccuracy in fuel-cladding temperature difference directly affects the reactor operating margins. A quantitative, as well as qualitative assessment of the fission heat transport across the fuel and embodiment of that knowledge in computer code, allows for a more realistic prediction of fuel performance. This knowledge helps in reducing the operating margins and leads to an improved operating economy of the reactor.

1. Introduction

A good understanding of the factors governing the temperature distribution within a nuclear fuel element is important to predict the fuel temperature in all operating conditions of a nuclear reactor. The temperature distribution influences fuel performance in terms of solid-state reactions, e.g., grain-growth, densification, etc., and the temperature gradient results in fuel deformation or crack in low temperature zones. Oxide fuel is particularly disadvantageous due to its low density and low thermal conductivity [1]. Hence, a large temperature difference between the center and the surface of the rod is required for efficient heat extraction to make electric power generation economical. These constraints are at odds with each other. We intend to operate the reactor at the largest possible power density consistent with maintaining the fuel and coolant temperature below limits set by safety considerations. In accident conditions, we need to have enough margin so that the fuel does not lose integrity due to high temperature and poor heat transfer arrangement. Hence, the length of time and the fuel element that can be utilized in the reactor core is determined by the ability of the fuel element to withstand radiation damage and thermal and mechanical stresses experienced in the reactor environment and not so much on the depletion of fissile material. This is true for reactors utilizing enriched uranium as well as those using natural uranium as fissile fuel material.

Uranium metal is superior to oxide as far as density and conductivity is concerned, but the phase change at a low temperature of 600$°\mathrm{C}$ followed by a large volume change means that the fuel clad will be under severe stress. This has led to the investigation of other refractory compounds of uranium, such as uranium carbide, uranium nitride. For any type of fuel being used in the reactor, the fuel performance computer codes are needed to assure the continued safe operation of the reactor. With increasing demands of nuclear fuel efficiency, new fuel designs are being studied and the reliability of these new designs is in the interest of fuel manufacturers [1].

In this chapter, we will look into existing fuel analysis computer codes to develop an appreciation of the fuel characteristics. In the last section of this chapter, we will discuss a new code Fuel Characteristics Calculator (FCCAL) [2] and its suitability in the analysis of oxide fuel.

2. Classification of fuel performance codes

There are several available computer codes to analyze the thermal and mechanical behavior of fuel for different types of reactors viz., LWR, CANDU, VVER, etc. some of these codes are available in the public domain while some are proprietary and not available publicly.

The fuel rod behavior is determined by thermal, mechanical, and physical processes such as densification, swelling, fission gas generation, fission gas release, and irradiation damage. The fuel performance analysis code covers these aspects through thermal and mechanical components of fuel performance. The codes may be 1D, 2D, or 3D. However, experience shows that one-dimensional codes are most widely used for fuel analysis. The codes can be further classified as steady-state and/or transient codes. Examples of steady-state codes are FRAPCON, TRANSURANUS, COMETHE, etc. these codes calculate the radial temperature profile and fission gas release to the fuel plenum. Mechanical properties like creep deformation and irradiation growth can also be calculated using these codes. The transient codes like GRASS-SST [3] can calculate these parameters and additionally calculate cladding plastic stress-strain behavior, the effect of annealing, the behavior of oxide and hydrides during temperature ramps, phase changes, and large cladding deformation such as ballooning. The transient codes neglect long-term phenomena like creep deformation. Let us first discuss two computer codes for an understating of how we go about fuel characteristics quantification.

2.1 GAPCON-THERMAL

GAPCON-THERMAL-II (GT-II) [4] is an updated version of the older GAPCON-THERMAL-I (GT-I) [5] code that is widely used for calculating light water reactor fuel thermal performance. GT-I has been modified to improve upon the uncertainty in the calculation of power history and burn up. GT-II is an American National Standards Institute (ANSI) compliant Fortran-77 code. We can calculate the thermal behavior, fuel plenum conductance, temperature and pressure, and fuel stored energy using this code. There are models for power history, fission gas generation and release, fuel relocation, and densification in the code. For the power history simulation, the code uses constant power for each finite time step. At any time step other than the first for each axial node, the current fission gas release, relocation, and densification values are compared with the values used in the previous step. Relocation and densification displacement will not decrease if lesser values are subsequently calculated. The fission gas release algorithm depends upon all previous fission gas release values. The fission gas generation does not require that the simulation starts from zero. Transmutation of U-238 to plutonium and subsequent fission gas release due to fission of plutonium is also available in the code models. The fuel diameter is a function of power (Kw/ft), as fabricated cold-gap thickness (inches) and burnup (MWD/MTM). The model is based on linear regression analysis of experimental data. The fuel densification correlation calculates the reduction of fuel radius as a function of burnup and differential fuel density.

GT-II calculates the gap conductance, temperature, pressure, and stored thermal energy based on the power history of the fuel. The plenum gap conductance for each equal-length, user-designated axial region is determined by an iterative scheme. Radial temperature is calculated using finite difference. The solution procedure consists of iterative convergence for each axial region, followed by iterative convergence on the fuel gas release for each time-power step. Empirical, theoretical, and physical models are used for fuel gas release calculation.

2.2 Fuel design analysis (FUDA)

The computer code FUDA [6] is used for the design analysis of fuel for licensing application of CANDU type reactors in India. The code is used for fuel performance evaluation as well as to optimize the fuel design and fabrication parameters of Indian reactor fuel. The code is valid for the burnup of 50,000 MWD/Te of oxide fuel. Natural uranium and thorium oxide fuel can be analyzed using this code. There are models for computation of fuel temperature, thermal expansion, and clad stress parameters in the code.

The code uses the finite difference method for temperature and computation of thermal expansion. The clad stress, local stress, and ridge analysis is carried out by finite element technique. Fuel expansion is calculated by the two-zone model in which the stress in uranium oxide is ignored. Uranium oxide deformation is assumed to occur above a certain temperature as plastic, and below this temperature, the fuel element is assumed to behave as elastic solid with radial cracking. The extent of plasticity is governed by fuel temperature, stress due to cladding strength, and the coolant pressure in a time-dependent manner. Global clad stress and strain due to fuel thermal expansion, swelling and densification are calculated by models and correlation used in Notley [7]. The creep and stress relaxation in the time zone at constant power operation is calculated using semi-empirical formula considering a thermal and thermal creep including the effect of irradiation. Fuel sheath interfacial pressure is then calculated based on gas pressure and strains. Using global diametral changes, local deformation of the fuel element and sheath is calculated considering hourglass phenomena in the fuel element. The finite element method using asymmetrical 8-node isoparametric elements is used for calculating deformation, stress, and strain in the element and the clad.

Fuel gap conductance is calculated by the Ross and Stoute model [8] taking care of the physical gap existing in the fuel plenum. Plenum gap conductance consists of (a) conduction through solid-solid contact points (b) conduction through solid-gas contact points and, (c) radiation exchange between the element and clad. For plenum gap conductance, the URGAP model of K. Lassmann [9] has been used. The fission gas release is calculated using two methods (a) temperature-dependent release model and (b) physical model based on diffusion and grain growth mechanism.

To estimate the local flux perturbation, the Bessel function is used. The heat transfer from the clad surface to the coolant is calculated using Dittus-Boelter [10] equation. Using the fuel element-clad heat transfer coefficient, new temperature distribution across the fuel and clad is calculated. The corresponding internal gas pressure is calculated using the new temperature distribution and when successive internal gas pressure is within ±5% then the pressure and temperature results are assumed to converge and the iteration is stopped. For improving accuracy, the pellet is divided into 100 rings radially and the fuel temperature and pressure are calculated for each ring. The code is validated against the results of benchmarked codes ELESIM and ELESTRES.

3. Fuel characteristics calculator (FCCAL)

The main thrust of FCCAL [2] is to analyze the plenum gas conductivity with fission gas accumulation and its analytical evaluation. With irradiation of the fuel inside the reactor core, fission noble gases Xenon and Krypton accumulate in the plenum gap which changes the gap conductivity from the initial fuel behavior that is for the Helium-filled during manufacturing. The analytical model is a better approximation over the use of correlations to estimate the effect of noble gases in the plenum gap. The change in conductivity is observable in the fuel in CANDU reactors where on-power refueling is practiced and the old fuel bundles move to higher power generating regions of the core. The fresh fuel along with the old bundles leads to a higher fission rate and hence the release of trapped noble gases towards the plenum. This results in a higher temperature of the old bundles even without an appreciable change in the power. We will discuss these phenomena in the sections below.

3.1 Fission gas release

Fission gases are considered to be released from the fuel when they reach any space that is connected to the free volume within the fuel pin. The released gases accumulate in the fuel-cladding gap, the central void, and porosity within the fuel which communicates directly with the fuel-pin gas space [1]. Cracks or interlinked gas bubbles or pores are an important type of open porosity. The fission gas that has been released from the fuel is responsible for the change in plenum gap conductivity and is assumed to have the following properties. (a) Once the gas is released, the probability of its re-entering the solid from the free volume is negligible (b) the gas pressure in open porosity is equal to that in the free volume of the pin. Because of the insolubility of Xenon and Krypton in solids, there is no effect of plenum fission gas pressure on the rate of gas escape from the fuel (c) while the fission gas contained by the fuel tends to cause swelling, fission gas that has been released promotes shrinkage in the fuel by pressurizing the solid pellets leading to collapse of the internal porosity and bubbles. FCCAL carries out an explicit calculation for changes in gap gas conductivity due to a binary mixture of Helium and Xenon. As the heat generated in the pellet is transferred across the fuel to the coolant, the heat transfer across discontinuities is calculated in the following steps. (i) Heat transfer from the meat of the pellet to the pellet surface. It is estimated by the heat transfer coefficient of the natural uranium oxide pellet (${\mathrm{h}}_{\mathrm{P}})$. (ii) Heat transfer across the plenum gap and the Zircalloy clad $\left({\mathrm{h}}_{\mathrm{Gg}}+{\mathrm{h}}_{\mathrm{Gs}}+{\mathrm{h}}_{\mathrm{s}}\right)$. ${\mathrm{h}}_{\mathrm{Gg}}$ is the heat transfer coefficient due to plenum gas,${\mathrm{h}}_{\mathrm{Gs}}$ is the heat transfer coefficient of the solid-solid contact points between the pellet and the sheath, and${\mathrm{h}}_{\mathrm{s}}$ is the heat transfer of the Zircalloy sheath. (iii) Heat transfer from the clad outer surface to the coolant is estimated by the heat transfer coefficient of the coolant film near the fuel surface $\left({\mathrm{h}}_{\mathrm{cf}}\right)$. Total heat transfer coefficient $\left({\mathrm{h}}_{\mathrm{T}}\right)$ is the sum of terms in (i), (ii), and (iii) i.e.

$${\mathrm{h}}_{\mathrm{T}}={\mathrm{h}}_{\mathrm{P}}+{\mathrm{h}}_{\mathrm{Gg}}+{\mathrm{h}}_{\mathrm{Gs}}+{\mathrm{h}}_{\mathrm{s}}+{\mathrm{h}}_{\mathrm{cf}}$$E1

3.1.1 Fuel pellet conductivity and temperature calculation

PHWR fuel is made of ceramic containing ${\mathit{UO}}_2$ with 0.7% U-235. It has poor heat conductivity properties as compared to carbide or metallic uranium. The heat transfer coefficient is dependent upon temperature as well as fission product accumulating inside the pellet. Moreover, as the fuel undergoes irradiation, cracks develop which changes the conductivity. A widely used correlation for the calculation of temperature-dependent pellet conductivity is as follows [11].

$$0<\mathrm{T}\le 1650°\mathrm{C}$$

$${\mathrm{h}}_{\mathrm{P}}=\mathrm{\eta }\frac{{\mathrm{B}}_1}{{\mathrm{B}}_2+\mathrm{T}}+{\mathrm{B}}_3{\mathrm{e}}^{\left({\mathrm{B}}_4\mathrm{T}\right)}$$E2

$$1650\le \mathrm{T}\le 2940°\mathrm{C}$$

$${\mathrm{h}}_{\mathrm{P}}=\mathrm{\eta }\left[{\mathrm{B}}_5+{\mathrm{B}}_3{\mathrm{e}}^{{\mathrm{B}}_4\mathrm{T}}\right]$$E3

$$\mathrm{\eta }=\left[\frac{1-\mathrm{\beta }\left(1-\frac{\mathrm{\rho }}{{\mathrm{\rho }}_{\mathrm{TD}}}\right)}{1-\mathrm{\beta }\left(1-0.95\right)}\right]$$E4

Where η is the porosity factor and $\mathrm{\beta }=2.58-0.58\times {10}^{-3}\times \mathrm{T}$. The constants for a different fuel types are shown in Table 1.

Fuel	${\mathrm{B}}_1$$\mathrm{w}/\mathrm{cm}$	${\mathrm{B}}_2°\mathrm{C}$	${\mathrm{B}}_3\mathrm{w}/\mathrm{cm}°\mathrm{C}$	${\mathrm{B}}_4°{\mathrm{C}}^{-1}$	${\mathrm{B}}_5\mathrm{w}/\mathrm{cm}°\mathrm{C}$
${\mathrm{UO}}_2$	40.4	464	$1.216\times {10}^{-4}$	$1.867\times {10}^{-3}$	$0.0191$
$\left(\mathrm{U},\mathrm{Pu}\right){\mathrm{O}}_2$	33.0	375	$1.540\times {10}^{-4}$	$1.710\times {10}^{-3}$	0.0171

Table 1.

Correlation constants for different fuel types.

This correlation is based on the data pooled from ten sources and an analytical expression is generated based on this data. The integral of ${\mathrm{UO}}_2$ thermal conductivity between $0°C$ and the melting point $2850°C$ is analytically determined in MATPRO. Assuming that the electronic contribution ${\mathrm{B}}_3{\mathrm{e}}^{\left({\mathrm{B}}_4\mathrm{T}\right)}$ has the value of $2\times {10}^{-3}\mathrm{w}/\mathrm{cmK}$at 1$500°\mathrm{C}$, a least-squares value of $97\mathrm{w}/\mathrm{cm}$ is obtained for the integral of ${\mathrm{h}}_{\mathrm{p}}$ from $0°\mathrm{C}$ to the melting point. Data points were fit to an equation including a temperature-dependent, modified Loeb porosity correction.

3.1.2 Equivalent conductivity and temperature drop across plenum gap

Heat transfer coefficient ${\mathrm{h}}_{\mathrm{Gg}}$ due to fission gas accumulating in the gap between the pellet and the sheath is a function of fission gas diffusing from the pellet towards the plenum gap. For fresh fuel, the conductivity is a function of helium thermal conductivity but the fission gas changes the gap conductivity. Change in the composition of the plenum gas is a function of fission gas accumulating in the plenum and it is estimated using the industry standard for estimation of the fraction of Xe and Kr [12] diffusing to the plenum as shown in Table 2.

Lower temperature limit	Higher temperature limit	Fraction
<1400$°\mathrm{C}$	—	0.05
1400$°\mathrm{C}$	1500$°\mathrm{C}$	0.10
1500$°\mathrm{C}$	1600$°\mathrm{C}$	0.10
1600$°\mathrm{C}$	1700$°\mathrm{C}$	0.10
1700$°\mathrm{C}$	1800$°\mathrm{C}$	0.10
1800$°\mathrm{C}$	2000$°\mathrm{C}$	0.10
>2000$°\mathrm{C}$	—	0.98

Table 2.

Temperature-dependent fraction of fission gas Xe and Kr in the plenum gap.

The burnup-dependent yield of the fission product noble gases is input through a data file in FCCAL. To estimate the cumulative effect of fission gas in the plenum gap n-component gas mixture model is applied [13, 14, 15] as shown in Eq. (5).

$${\mathrm{\lambda }}_{\mathrm{mix}}=\sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{{\mathrm{\lambda }}_{\mathrm{t}}}{1+{\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{\phi }}_{\mathrm{ij}}\frac{{\mathrm{X}}_{\mathrm{i}}}{{\mathrm{X}}_{\mathrm{j}}}}$$E5

Where, ${\mathrm{\lambda }}_{\mathrm{mix}}$ and ${\mathrm{\lambda }}_{\mathrm{t}}$ are the thermal conductivities of the mixture gas and the individual component gases respectively, ${\mathrm{X}}_{\mathrm{i}}$ and ${\mathrm{X}}_{\mathrm{j}}$ are the mole fractions of the component gases and ${\mathrm{\phi }}_{\mathrm{ij}}$ is constant. For the binary gas mixture consisting of He-Kr or He-Xe Eq. (1) may be written as:

$${\mathrm{\lambda }}_{\mathrm{mix}}=\frac{{\mathrm{\lambda }}_1}{1+{\mathrm{\phi }}_{12}\frac{{\mathrm{X}}_2}{{\mathrm{X}}_1}}+\frac{{\mathrm{\lambda }}_2}{1+{\mathrm{\phi }}_{21}\frac{{\mathrm{X}}_1}{{\mathrm{X}}_2}}$$E6

Where, subscript 1 is for heavier gas of the binary mixture. Values of ${\mathrm{\phi }}_{\mathrm{ij}}$ [16] is shown in Table 3.

${\mathrm{\phi }}_{\mathrm{He}-\mathrm{Xe}}$ = 3.4284	${\mathrm{\phi }}_{\mathrm{Xe}-\mathrm{He}}$= 0.3849
${\mathrm{\phi }}_{\mathrm{He}-\mathrm{Kr}}$ = 2.7863	${\mathrm{\phi }}_{\mathrm{Kr}-\mathrm{He}}$ = 0.4909

Table 3.

Mixture dependent constants for n (=2) component gas mixture equivalent conductivity formula.

The values of ${\mathrm{\phi }}_{\mathrm{ij}}$ for the component, mixture gases are independent of composition and temperature as shown by Gambhir and Saxena [17, 18] and are given by the following expression.

$$\frac{{\mathrm{\phi }}_{\mathrm{ij}}}{{\mathrm{\phi }}_{\mathrm{ji}}}=\frac{{\mathrm{\lambda }}_{\mathrm{i}}}{{\mathrm{\lambda }}_{\mathrm{j}}}\frac{59{\mathrm{M}}^2+88\mathrm{M}+150}{150{\mathrm{M}}^2+88\mathrm{M}+59}$$E7

$$\mathrm{M}=\frac{{\mathrm{M}}_2}{{\mathrm{M}}_1}$$E8

These formulas are important because we can estimate the ${\mathrm{\lambda }}_{\mathrm{mix}}$ of multi-component gas mixtures if the thermal conductivity values of the corresponding binary and pure components are known. Moreover, these formulas help us to obtain ${\mathrm{\lambda }}_{\mathrm{mix}}$ value at high temperature from knowledge of pure $\mathrm{\lambda }$ values at that temperature. Thus ${\mathrm{\phi }}_{\mathrm{ij}}$values determined at some lower temperature can be used to calculate ${\mathrm{\lambda }}_{\mathrm{mix}}$ at some higher temperature.

3.1.3 Zircalloy sheath conductivity and temperature drop across the coolant

Zircalloy sheath heat transfer coefficient ${\mathrm{h}}_{\mathrm{s}}$is calculated based on Eq. (9) [12].

$${\mathrm{h}}_{\mathrm{s}}=\left[7.51+2.09\times {10}^{-2}-1.45\times {10}^{-5}{\mathrm{T}}^2+7.67\times {10}^{-9}{\mathrm{T}}^3\right]$$E9

Where, ${\mathrm{h}}_{\mathrm{s}}$ is in W/mK and T is in K. Coolant heat transfer coefficient ${\mathrm{h}}_{\mathrm{cf}}$ is calculated by the Dittus-Boelter correlation [10] as shown in Eq. (10).

$$\mathrm{Nu}={\left(\frac{\mathrm{hDe}}{\mathrm{k}}\right)}_{\mathrm{b}}=\mathrm{C}{\left(\frac{\mathrm{DeG}}{\mathrm{\mu }}\right)}_{\mathrm{b}}^{0.8}{\left(\frac{{\mathrm{c}}_{\mathrm{p}}\mathrm{\mu }}{\mathrm{k}}\right)}_{\mathrm{b}}^{\mathrm{n}}$$E10

Where,

h = heat transfer coefficient, Btu/(hr${\text{ft}}^3°\mathrm{F}$);

De = equivalent diameter, ft.;

k = thermal conductivity of fluid, Btu/hr. ft. $°F$;

${\mathrm{c}}_{\mathrm{p}}$ = specific heat of fluid, Btu/lb.;

G = mass velocity, lb./hr. ${\text{ft}}^2$;

$\mathrm{\mu }$ = fluid viscosity, lb./hr. ft.;

$\frac{\mathrm{De}\mathrm{G}}{\mathrm{\mu }}>\mathrm{10,000}$ and $\frac{\mathrm{L}}{\mathrm{De}}>60$;

B = bulk conditions.

All other terms have their standard notational convention.

3.1.4 FCCAL code methodology

FCCAL is written in Fortran computer programming language. It is used to calculate fuel centerline temperature, fuel surface temperature, and average coolant temperature in steady-state as well as transient conditions as a function of coolant flow rate (Kg/s), bundle power (Kw), Fuel burnup (MWD/TeU), average channel power (Kw) and actual coolant temperature. For the start of the iteration, a guess value of fuel centerline temperature is assumed taking into account the fact that the fuel centerline temperature cannot be less than the coolant temperature for a reactor operating at steady power. The heat generated in the fuel is taken away by the coolant and transferred to the steam drum and further goes on to generate electricity. Equivalent thermal conductivity across the pellet, plenum gap, clad, and heat transfer to the coolant is calculated. The results are compared with the actual instrument measurements and the guess temperature is accordingly re-evaluated. The guess temperature is accepted to be correct when the error in the code computed values and the instrumented value is within ±2$°\mathrm{C}$. The instrumented temperature is measured from platinum resistance temperature detectors. An error of this magnitude is acceptable as the instrument measurement error is ±2$°\mathrm{C}$.

4. Results

The thermal conductivity of the binary mixture of He-Xe and He-Kr is calculated using the code and the temperature profile is shown in Figure 1. The fuel assembly that is analyzed has the same power generation rate and is computed for the same fuel irradiation history.

Figure 1.

The temperature profile of fuel assembly for He-Xe binary and He-Kr binary.

As the yield of Kr is small as compared to Xe yield hence its effect on the total temperature is small and He dominates the equivalent heat transfer characteristics. For fuel-producing nearly equal power but different irradiation history, we observe that the assembly has a different temperature (Figure 2). This is attributable to poor heat conduction properties of the He-Xe binary and relative dominance of Xe in the equivalent conductivity. The fuel parameters computed using FCCAL are compared with MATPRO-10 typical parameter values and agree with MATPRO predictions.

Figure 2.

Assembly producing similar power but different irradiation history and hence different amount of xenon gas buildup in the plenum gap.

5. Conclusion

A model for calculation of fuel temperature profile using binary gas mixture is presented in this chapter along with a discussion of two benchmarked codes for fuel characteristics evaluation. Computing the effect of fission gas products on the overall fuel temperature is presented with the rod irradiation phenomena. From the analysis, it is clear that the code FCCAL can be used for the calculation of fuel centerline temperature, fuel surface temperature, and average coolant temperature of the Pressurized Heavy Water Reactor (PHWR). A better approximation can be obtained by incorporating the fuel cracking and deformation. The effect of fuel heat loss due to irradiation although negligible in steady-state assumes significance in severe transients.

Acknowledgments

The author acknowledges that he is an employee of the atomic power plant and is involved in the PHWR operational aspects.