**6. Fuel centerline temperature calculations**

In order to calculate the fuel centerline temperature, steady-state one-dimensional heattransfer analysis was conducted. The MATLAB and NIST REFPROP software were used for programming and retrieving thermophysical properties of a light-water coolant, respectively. First, the heated length of the fuel channel was divided into small segments of one-millimeter lengths. Second, the temperature profile of the coolant was calculated. Third, sheath-outer and inner surface temperatures were calculated. Fourth, the heat transfer through the gap between the sheath and the fuel was determined and used to calculate the outer surface temperature of the fuel. Finally, the temperature of the fuel in the radial and axial directions was calculated. It should be noted that the radius of the fuel pellet was divided into 20 segments. The results will be presented for fuel-sheath gap widths of zero, 20 *μ*m and 36 *μ*m. Moreover, the fuel centerline temperature profiles have been calculated based on a no-gap condition in order to determine the effect of gap conductance on the fuel centerline temperature. Figure 12 illustrates the methodology based on which fuel centerline temperature was calculated. The following section provides more information about each step shown in Fig. 12.

As shown in Fig. 12, the convective heat transfer between the sheath and the coolant is the only heat transfer mode which has been taken directly into consideration. In radiative heat transfer, energy is transferred in the form of electromagnetic waves. Unlike convection and conduction heat transfer modes in which the rate of heat transfer is linearly proportional to temperature differences, a radiative heat transfer depends on the difference between absolute temperatures to the fourth power. The sheath temperature is high11 at SCWR conditions; therefore, it is necessary to take into account the radiative heat transfer.

In the case of the sheath and the coolant, the radiative heat transfer has been taken into consideration in the Nusselt number correlation, which has been used to calculate the HTC. In general, the Nusselt number correlations are empirical equations, which are developed

<sup>11</sup> It might be as high as 850°C.

Thermal Aspects of Conventional and Alternative Fuels

examined AHFPs envelope a wide range of power profiles.

in SuperCritical Water-Cooled Reactor (SCWR) Applications 143

It should be noted that there are many power profiles in a reactor core. In other words, the axial heat flux profile in each fuel channel differs from those of the other fuel channels. This variation in power profiles is due to the radial and axial power distribution, fuel burn-up, presence of reactivity control mechanisms, and refuelling scheme. Thus, a detailed design requires the maximum thermal power in the core, which can be determined based on neutronic analysis of the core which is beyond the scope of this chapter. However, the four

Fig. 13. Power ratios along heated length of fuel channel (based on Leung (2008)).

this problem through iterations, Newton's law of cooling should be used.

The calculation of the sheath temperature requires HTC values along the heated length of the fuel channel. In this study, the Mokry et al. correlation, shown as Eq. (17), has been used to determine HTC. The average Prandtl number in the Mokry correlation is calculated based on the average specific heat using Eq. (18). In Eq. (18) *µ* and *k* are the dynamic viscosity and thermal conductivity of the coolant at bulk temperature. The experimental data, based on which this correlation was developed, was obtained within conditions similar to those of proposed SCWR concepts. The experimental dataset was obtained for supercritical water flowing upward in a 4-m-long vertical bare tube. The data was collected at a pressure of approximately 24 MPa for several combinations of wall and bulk fluid temperatures. The temperatures were below, at, or above the pseudocritical temperature. The mass flux ranged from 200-1500 kg/m2s; coolant inlet temperature varied from 320 to 350°C, for heat flux up to 1250 kW/m2 (Mokry et al., 2009). The Mokry correlation requires iterations to be solved, because it contains two unknowns, which are HTC and sheath wall temperature. To solve

From a safety point of view, it is necessary to know the uncertainty of a correlation in calculating the HTC and sheath wall temperature. As shown in Fig. 14, the uncertainty associated in the prediction of the HTC using the Mokry et al. correlation is ±25%. In other

**6.2 Sheath temperature** 

based on experiments conducted in water using either bare tubes or tubes containing electrically heated elements simulating the fuel bundles. To develop a correlation, surface temperatures of the bare tube and/or simulating rods are measured along the heated length of the test section by the use of thermocouples or Resistance Temperature Detectors (RTDs). These measured surface temperatures already include the effect of the radiative heat transfer; therefore, the developed Nusselt number correlations represent both radiative and convection heat transfer modes. Consequently, the radiative heat transfer has been taken indirectly into consideration in the calculations.

Fig. 12. Fuel centerline temperature calculations.

#### **6.1 Bulk-fluid temperature profile**

The temperature profile of the coolant along the heated length of the fuel channel can be calculated based on the heat balance. Equation (16) was used to calculate the temperature profile of the coolant. The NIST REPFROP software Version 8.0 was used to determine the thermophysical properties at a bulk-fluid temperature corresponding to each one-millimeter interval.

$$h\_{i+1} = h\_i + \frac{p \cdot q\_{\chi}}{\dot{m}} \cdot \Delta x \tag{16}$$

In Eq. (16), *q*x is the axial heat flux value, which is variable along the heated length of the fuel channel if a non-uniform Axial Heat Flux Profile (AHFP) is used. In the present chapter, four AHFPs have been applied in order to calculate the fuel centerline temperature in fuel channels at the maximum channel thermal power. These AHFPs are cosine, upstreamskewed cosine, downstream-skewed cosine, and uniform. The aforementioned AHFPs were calculated based on power profiles listed in Leung (2008) while the downstream-skewed AHFP was determined as the mirror image of the upstream-skewed AHFP. A local heat flux can be calculated by multiplying the average heat flux by the corresponding power ratio from Fig. 13.

based on experiments conducted in water using either bare tubes or tubes containing electrically heated elements simulating the fuel bundles. To develop a correlation, surface temperatures of the bare tube and/or simulating rods are measured along the heated length of the test section by the use of thermocouples or Resistance Temperature Detectors (RTDs). These measured surface temperatures already include the effect of the radiative heat transfer; therefore, the developed Nusselt number correlations represent both radiative and convection heat transfer modes. Consequently, the radiative heat transfer has been taken

The temperature profile of the coolant along the heated length of the fuel channel can be calculated based on the heat balance. Equation (16) was used to calculate the temperature profile of the coolant. The NIST REPFROP software Version 8.0 was used to determine the thermophysical properties at a bulk-fluid temperature corresponding to each one-millimeter

> +1 *<sup>x</sup> i i p q h =h + <sup>Δ</sup><sup>x</sup> m*

In Eq. (16), *q*x is the axial heat flux value, which is variable along the heated length of the fuel channel if a non-uniform Axial Heat Flux Profile (AHFP) is used. In the present chapter, four AHFPs have been applied in order to calculate the fuel centerline temperature in fuel channels at the maximum channel thermal power. These AHFPs are cosine, upstreamskewed cosine, downstream-skewed cosine, and uniform. The aforementioned AHFPs were calculated based on power profiles listed in Leung (2008) while the downstream-skewed AHFP was determined as the mirror image of the upstream-skewed AHFP. A local heat flux can be calculated by multiplying the average heat flux by the corresponding power ratio

(16)

indirectly into consideration in the calculations.

Fig. 12. Fuel centerline temperature calculations.

**6.1 Bulk-fluid temperature profile** 

interval.

from Fig. 13.

It should be noted that there are many power profiles in a reactor core. In other words, the axial heat flux profile in each fuel channel differs from those of the other fuel channels. This variation in power profiles is due to the radial and axial power distribution, fuel burn-up, presence of reactivity control mechanisms, and refuelling scheme. Thus, a detailed design requires the maximum thermal power in the core, which can be determined based on neutronic analysis of the core which is beyond the scope of this chapter. However, the four examined AHFPs envelope a wide range of power profiles.

Fig. 13. Power ratios along heated length of fuel channel (based on Leung (2008)).

## **6.2 Sheath temperature**

The calculation of the sheath temperature requires HTC values along the heated length of the fuel channel. In this study, the Mokry et al. correlation, shown as Eq. (17), has been used to determine HTC. The average Prandtl number in the Mokry correlation is calculated based on the average specific heat using Eq. (18). In Eq. (18) *µ* and *k* are the dynamic viscosity and thermal conductivity of the coolant at bulk temperature. The experimental data, based on which this correlation was developed, was obtained within conditions similar to those of proposed SCWR concepts. The experimental dataset was obtained for supercritical water flowing upward in a 4-m-long vertical bare tube. The data was collected at a pressure of approximately 24 MPa for several combinations of wall and bulk fluid temperatures. The temperatures were below, at, or above the pseudocritical temperature. The mass flux ranged from 200-1500 kg/m2s; coolant inlet temperature varied from 320 to 350°C, for heat flux up to 1250 kW/m2 (Mokry et al., 2009). The Mokry correlation requires iterations to be solved, because it contains two unknowns, which are HTC and sheath wall temperature. To solve this problem through iterations, Newton's law of cooling should be used.

From a safety point of view, it is necessary to know the uncertainty of a correlation in calculating the HTC and sheath wall temperature. As shown in Fig. 14, the uncertainty associated in the prediction of the HTC using the Mokry et al. correlation is ±25%. In other

Thermal Aspects of Conventional and Alternative Fuels

Assumption to start the iteration: , *T T sheath wall o bulk* 50 C

iterations and required time to complete the execution of the code.

initial guess for the inner surface temperature of the sheath.

the sheath and the coolant.

**6.2.2 Inner-sheath temperature** 

in SuperCritical Water-Cooled Reactor (SCWR) Applications 145

0.684 0.904 b b

*p sheath,T bulk,T*

*sheath bulk*

<sup>b</sup> 0.0061 *<sup>w</sup>*

*p*

*μc H -H ,c = k T -T*

The developed MATLAB code uses an iterative technique to determine the sheath-wall temperature. Initially, the sheath-wall temperature is unknown. Therefore, an initial guess is needed for the sheath-wall temperature (i.e., 50°C above the bulk-fluid temperature). Then, the code calculates the HTC using Eq. (17), which requires the thermophysical properties of the light-water coolant at bulk-fluid and sheath-wall temperatures. Next, the code calculates a "new" sheath-wall temperature using the Newton's law of cooling shown as Eq. (19). In the next iteration, the code uses an average temperature between the two consecutive temperatures. The iterations continue until the difference between the two consecutive temperatures is less than 0.1 K. It should be noted that the initial guessed sheath-wall temperature could have any value, because regardless of the value the temperature converges. The only difference caused by different guessed sheath-wall temperatures is in the number of

As mentioned previously, the thermophysical properties of the coolant undergo significant changes as the temperature passes through the pseudocritical point. Since the operating pressure of the coolant is 25 MPa, the pseudocritical point is reached at 384.9°C. As shown in Fig. 16, the changes in the thermophysical properties of the coolant were captured by the Nusselt number correlation, Eq. (16). The Prandtl number in Eq. (16) is responsible for taking into account the thermophysical properties of the coolant. Figure 16 shows the thermophysical properties of the light-water coolant along the length of the fuel channel. The use of these thermophysical properties in the Nusselt number correlation indicates that the correlation takes into account the effect of the pseudocritical point on the HTC between

The inner surface temperature of the sheath can be calculated using Eq. (20). In Eq. (20), *k* is the thermal conductivity of the sheath, which is calculated based on the average temperature of the outer and inner wall surface temperatures. This inner-sheath temperature calculation is conducted through the use of an iteration, which requires an

> *sheath wall i sheath wall o* , , *o i*

*k / 2πL*

(20)

*T T Q = ln(r r )* 0.564

**Pr** (18)

*q=h T T sheath wall o bulk* , (19)

**Nu Re Pr** (17)

*b*

words, the HTC values calculated by the Mokry correlation are within ±25% deviation from the corresponding experimental values. However, the uncertainty associated with wall temperature is smaller and lies within ±15%. Figure 15 shows the uncertainty in the prediction of the wall temperature associated with the Mokry et al. correlation.

Fig. 14. Uncertainty in predicting HTC based on the Mokry et al. correlation (Mokry et al., 2011).

Fig. 15. Uncertainty in predicting wall temperature using the Mokry et al. correlation (Mokry et al., 2011).

#### **6.2.1 Outer-surface temperature of sheath**

The following sequence of equations can be used in order to calculate the outer surface temperature of the sheath along the heated length of the fuel channel.

words, the HTC values calculated by the Mokry correlation are within ±25% deviation from the corresponding experimental values. However, the uncertainty associated with wall temperature is smaller and lies within ±15%. Figure 15 shows the uncertainty in the

Fig. 14. Uncertainty in predicting HTC based on the Mokry et al. correlation (Mokry et al.,

s

+15%

s


G= 500 kg/m<sup>2</sup> s G=1000 kg/m<sup>2</sup>

G=1500 kg/m<sup>2</sup>

Twcalc

300

**6.2.1 Outer-surface temperature of sheath** 

400

500

600

700

Twexp 300 400 500 600 700

The following sequence of equations can be used in order to calculate the outer surface

Fig. 15. Uncertainty in predicting wall temperature using the Mokry et al. correlation

temperature of the sheath along the heated length of the fuel channel.

2011).

(Mokry et al., 2011).

prediction of the wall temperature associated with the Mokry et al. correlation.

Assumption to start the iteration: , *T T sheath wall o bulk* 50 C

$$\mathbf{Nu}\_{\rm b} = 0.0061 \,\mathrm{Re}\_{\rm b}^{0.904} \overline{\mathbf{Pr}}\_{\rm b}^{0.684} \left(\frac{\rho\_w}{\rho\_\flat}\right)^{0.564} \tag{17}$$

$$\overline{\mathbf{Pr}} = \frac{\mu \overline{c\_p}}{k}, \overline{c\_p} = \frac{H\_{\text{sheath},T} - H\_{\text{bulk},T}}{T\_{\text{sheath}} - T\_{\text{bulk}}} \tag{18}$$

$$q = h \left( T\_{\text{she}th, \text{wall }o} - T\_{\text{bulk}} \right) \tag{19}$$

The developed MATLAB code uses an iterative technique to determine the sheath-wall temperature. Initially, the sheath-wall temperature is unknown. Therefore, an initial guess is needed for the sheath-wall temperature (i.e., 50°C above the bulk-fluid temperature). Then, the code calculates the HTC using Eq. (17), which requires the thermophysical properties of the light-water coolant at bulk-fluid and sheath-wall temperatures. Next, the code calculates a "new" sheath-wall temperature using the Newton's law of cooling shown as Eq. (19). In the next iteration, the code uses an average temperature between the two consecutive temperatures. The iterations continue until the difference between the two consecutive temperatures is less than 0.1 K. It should be noted that the initial guessed sheath-wall temperature could have any value, because regardless of the value the temperature converges. The only difference caused by different guessed sheath-wall temperatures is in the number of iterations and required time to complete the execution of the code.

As mentioned previously, the thermophysical properties of the coolant undergo significant changes as the temperature passes through the pseudocritical point. Since the operating pressure of the coolant is 25 MPa, the pseudocritical point is reached at 384.9°C. As shown in Fig. 16, the changes in the thermophysical properties of the coolant were captured by the Nusselt number correlation, Eq. (16). The Prandtl number in Eq. (16) is responsible for taking into account the thermophysical properties of the coolant. Figure 16 shows the thermophysical properties of the light-water coolant along the length of the fuel channel. The use of these thermophysical properties in the Nusselt number correlation indicates that the correlation takes into account the effect of the pseudocritical point on the HTC between the sheath and the coolant.

#### **6.2.2 Inner-sheath temperature**

The inner surface temperature of the sheath can be calculated using Eq. (20). In Eq. (20), *k* is the thermal conductivity of the sheath, which is calculated based on the average temperature of the outer and inner wall surface temperatures. This inner-sheath temperature calculation is conducted through the use of an iteration, which requires an initial guess for the inner surface temperature of the sheath.

$$Q = \frac{T\_{\text{sheath,wall \'i \'i}} - T\_{\text{sheath,wall \'o \'o}}}{\frac{\ln(r\_o \ne r\_{\text{i}})}{2\pi Lk}} \tag{20}$$

Thermal Aspects of Conventional and Alternative Fuels

fuel-sheath gap, and *s* is an exponent dependent on gas type.

material. *A* and *n* are equal to 10 and 0.5.

*r*

*h =*

in degrees Kelvin.

**6.4 Fuel centerline temperature** 

in SuperCritical Water-Cooled Reactor (SCWR) Applications 147

The fuel-sheath gap is very small, in the range between 0 and 125 *µ*m (Lassmann and Hohlefeld, 1987). CANDU reactors use collapsible sheath, which leads to small fuel-sheath gaps approximately 20 *µ*m (Lewis et al., 2008). Moreover, Hu and Wilson (2010) have reported a fuel-sheath gap width of 36 *µ*m for a proposed PV SCWR. In the present study, the fuel centerline temperature has been calculated for both 20-*µ*m and 36-*µ*m gaps. In Eq. (22), *g* is the temperature jump distance, which is calculated using Eq. (23) (Lee et al., 1995).

273.15

Where, *g* is the temperature jump distance, *y*i is the mole fraction of the *i*th component of gas, *g*o,i is the temperature jump distance of the *i*th component of gas at standard temperature and pressure, *T*g is the gas temperature in the fuel-sheath gap, *P*g is the gas pressure in the

In reality, the fuel pellets become in contact with sheath creating contact points. These contact points are formed due to thermal expansion and volumetric swelling of fuel pellets. As a result, heat is transferred through these contact points. The conductive heat transfer rate at the contact points are calculated using Eq. (24) (Ainscough, 1982). In Eq. (24), *A* is a constant, *P*a is the apparent interfacial pressure, *H* is the Mayer hardness of the softer

2

*f sheath f e*

*kk RR*

() 2

*k k <sup>P</sup> h =A*

*<sup>n</sup> f sheath <sup>a</sup> <sup>c</sup>*

The last term in Eq. (21) is the radiative heat transfer coefficient through the gap, which is calculated using Eq. (25) (Ainscough, 1982). It should be noted that the contribution of this heat transfer mode is negligible under normal operating conditions. However, the radiative heat transfer is significant in accident scenarios. Nevertheless, the radiative heat transfer through the fuel-sheath gap has been taken into account in this study. In Eq. (25), *<sup>f</sup> ε* and *sheath ε* are surface emissivities of the fuel and the sheath respectively; and temperatures are

*<sup>i</sup> o,i <sup>g</sup> <sup>1</sup> <sup>y</sup> <sup>T</sup> <sup>=</sup> g g <sup>P</sup>*

*g i*

0.5

*s+*

2 2 1 2

*sh ath /*

*/*

 

(25)

*4 4*

*f,o sheath,i f sheath*

*f sheath f sheath f,o sheath,i*

*σε ε T -T*

*ε + ε -ε ε T -T*

Equation (26) can be used to calculate the fuel centerline temperature. The thermal conductivity in Eq. (26) is the average thermal conductivity, which varies as a function of temperature. In order to increase the accuracy of the analysis, the radius of the fuel pellet has been divided into 20 rings. Initially, the inner-surface temperature is not known, therefore, an iteration loop should be created to calculate the outer-surface temperature of the fuel and the

thermal conductivity of the fuel based on corresponding average temperatures.

*H*

(24)

0.101

(23)

Fig. 16. Thermophysical properties of light-water coolant as function of temperature.

#### **6.3 Gap conductance**

Heat transfer through the fuel-sheath gap is governed by three primary mechanisms (Lee et al., 1995). These mechanisms are 1) conduction through the gas, 2) conduction due to fuelsheath contacts, and 3) radiation. Furthermore, there are several models for the calculation of heat transfer rate through the fuel-sheath gap. These models include the offset gap conductance model, relocated gap conductance model, Ross and Stoute model, and modified Ross and Stoute model.

In the present study, the modified Ross and Stoute model has been used in order to determine the gap conductance effects on the fuel centerline temperature. In this model, the total heat transfer through the gap is calculated as the sum of the three aforementioned terms as represented in Eq. (21):

$$h\_{\text{total}} = h\_{\text{g}} + h\_{\text{c}} + h\_{r} \tag{21}$$

The heat transfer through the gas in the fuel-sheath gap is by conduction because the gap width is very small. This small gap width does not allow for the development of natural convection though the gap. The heat transfer rate through the gas is calculated using Eq. (22).

$$h\_{\mathcal{g}} = \frac{k\_{\mathcal{g}}}{1.5 \left(R\_1 + R\_2\right) + t\_{\mathcal{g}} + g} \tag{22}$$

Where, *h*g is the conductance through the gas in the gap, *k*g is the thermal conductivity of the gas, *R*1 and *R*2 are the surface roughnesses of the fuel and the sheath, and *t*g is the circumferentially average fuel-sheath gap width.

Fig. 16. Thermophysical properties of light-water coolant as function of temperature.

Heat transfer through the fuel-sheath gap is governed by three primary mechanisms (Lee et al., 1995). These mechanisms are 1) conduction through the gas, 2) conduction due to fuelsheath contacts, and 3) radiation. Furthermore, there are several models for the calculation of heat transfer rate through the fuel-sheath gap. These models include the offset gap conductance model, relocated gap conductance model, Ross and Stoute model, and

In the present study, the modified Ross and Stoute model has been used in order to determine the gap conductance effects on the fuel centerline temperature. In this model, the total heat transfer through the gap is calculated as the sum of the three aforementioned

The heat transfer through the gas in the fuel-sheath gap is by conduction because the gap width is very small. This small gap width does not allow for the development of natural convection though the gap. The heat transfer rate through the gas is calculated using Eq. (22).

> ( ) *g*

Where, *h*g is the conductance through the gas in the gap, *k*g is the thermal conductivity of the gas, *R*1 and *R*2 are the surface roughnesses of the fuel and the sheath, and *t*g is the

*k*

*12g*

*g*

circumferentially average fuel-sheath gap width.

*h =*

*total <sup>g</sup> c r h hhh* (21)

*1.5 R + R +t + <sup>g</sup>* (22)

**6.3 Gap conductance** 

modified Ross and Stoute model.

terms as represented in Eq. (21):

The fuel-sheath gap is very small, in the range between 0 and 125 *µ*m (Lassmann and Hohlefeld, 1987). CANDU reactors use collapsible sheath, which leads to small fuel-sheath gaps approximately 20 *µ*m (Lewis et al., 2008). Moreover, Hu and Wilson (2010) have reported a fuel-sheath gap width of 36 *µ*m for a proposed PV SCWR. In the present study, the fuel centerline temperature has been calculated for both 20-*µ*m and 36-*µ*m gaps. In Eq. (22), *g* is the temperature jump distance, which is calculated using Eq. (23) (Lee et al., 1995).

$$\frac{1}{g} = \sum\_{i} \left[ \frac{y\_i}{g\_{o,i}} \right] \left( \frac{T\_g}{273.15} \right)^{s+0.5} \left( \frac{0.101}{P\_g} \right) \tag{23}$$

Where, *g* is the temperature jump distance, *y*i is the mole fraction of the *i*th component of gas, *g*o,i is the temperature jump distance of the *i*th component of gas at standard temperature and pressure, *T*g is the gas temperature in the fuel-sheath gap, *P*g is the gas pressure in the fuel-sheath gap, and *s* is an exponent dependent on gas type.

In reality, the fuel pellets become in contact with sheath creating contact points. These contact points are formed due to thermal expansion and volumetric swelling of fuel pellets. As a result, heat is transferred through these contact points. The conductive heat transfer rate at the contact points are calculated using Eq. (24) (Ainscough, 1982). In Eq. (24), *A* is a constant, *P*a is the apparent interfacial pressure, *H* is the Mayer hardness of the softer material. *A* and *n* are equal to 10 and 0.5.

$$h\_c = A \frac{2k\_f \left(k\_{\text{sheath}}\right)}{\left(k\_f + k\_{\text{sheath}}\right) \left[\left(R\_f^2 + R\_{\text{sheath}}^2\right) / 2\right]^{1/2}} \left(\frac{P\_a}{H}\right)^n \tag{24}$$

The last term in Eq. (21) is the radiative heat transfer coefficient through the gap, which is calculated using Eq. (25) (Ainscough, 1982). It should be noted that the contribution of this heat transfer mode is negligible under normal operating conditions. However, the radiative heat transfer is significant in accident scenarios. Nevertheless, the radiative heat transfer through the fuel-sheath gap has been taken into account in this study. In Eq. (25), *<sup>f</sup> ε* and *sheath ε* are surface emissivities of the fuel and the sheath respectively; and temperatures are in degrees Kelvin.

$$\mathbf{M}\_r = \frac{\sigma \,\varepsilon\_f \cdot \varepsilon\_{\text{sheath}}}{\varepsilon\_f + \varepsilon\_{\text{sheath}} \cdot \varepsilon\_f \cdot \varepsilon\_{\text{sheath}}} \cdot \frac{\left(T\_{f,o}^4 \cdot T\_{\text{sheath},i}^4\right)}{\left(T\_{f,o} \cdot T\_{\text{sheath},i}\right)}\tag{25}$$

#### **6.4 Fuel centerline temperature**

Equation (26) can be used to calculate the fuel centerline temperature. The thermal conductivity in Eq. (26) is the average thermal conductivity, which varies as a function of temperature. In order to increase the accuracy of the analysis, the radius of the fuel pellet has been divided into 20 rings. Initially, the inner-surface temperature is not known, therefore, an iteration loop should be created to calculate the outer-surface temperature of the fuel and the thermal conductivity of the fuel based on corresponding average temperatures.

Thermal Aspects of Conventional and Alternative Fuels

channels as well.

material of the sheath.

in SuperCritical Water-Cooled Reactor (SCWR) Applications 149

approximately 35% less that of the SCW channels, the sheath and the fuel centerline temperatures will be definitely lower than those of the SCW channels. As a result, if a fuel and sheath meet their corresponding temperature limits under the operating conditions of the SCW channels with the maximum thermal power, they will be suitable for the SRH

For the SCW fuel channels, the fuel centreline temperature has been calculated at cosine, upstream-skewed cosine, downstream-skewed cosine, and uniform axial heat flux profiles. These heat flux profiles have been calculated based on the Variant-20 fuel bundle. Each of the 42 fuel elements of the Variant-20 fuel bundle has an outer diameter of 11.5 mm while the minimum required thickness of the sheath has been determined to be 0.48 mm. Therefore, the inner diameter of the sheath is 10.54 mm. Inconel-600 was chosen as the

The examined fuels were UO2, MOX, ThO2, UC, UN, UO2-SiC, UO2-C, and UO2-BeO. For each fuel, the fuel centerline temperature was analysed at the aforementioned AHFPs. Since the maximum fuel centerline temperature was reached at downstream-skewed cosine AHFP for all the examined fuels, only the results associated with this AHFP have been presented in this section. Figures 17 through 19 show the coolant, sheath, and fuel centerline temperature profiles as well as the heat transfer coefficient profile along the heated length of the fuel channel for UO2, UC, and UO2-BeO fuels. Each of these three fuels represents one fuel category (i.e., low, enhanced, high thermal-conductivity fuels). It should be noted that

In addition, Figure 20 shows the maximum fuel centerline temperatures of all the examined fuels. As shown in Figure 20, the maximum fuel centerline temperatures of all examined low thermal-conductivity fuels exceed the temperature limit of 1850°C. On the other hand, enhanced thermal-conductivity fuels and high thermal-conductivity fuels show fuel centerline temperatures below the established temperature limits of 1850°C and 1500°C, respectively.

the results presented in Figs. 17 through 19 are based on a 20-*µ*m fuel-sheath gap.

Fig. 17. Temperature and HTC profiles for UO2 at downstream-skewed cosine AHFP.

$$T\_{r,i+1} = \frac{Q\_{\text{gen}}\left(r\_i^2 - r\_{i+1}^2\right)}{4 \cdot k\_{\text{avg}}} + T\_{r,i} \tag{26}$$

#### **7. Results: Fuel centerline and sheath temperatures**

There are two temperature limits that a fuel and a fuel bundle must meet. First, the sheath temperature must not exceed the design limit of 850°C (Chow and Khartabil, 2008). Second, when UO2 fuel is used, the fuel centerline temperature must be below the industry accepted limit of 1850°C (Reisch, 2009) at all normal operating conditions.

Previously, it was mentioned that the industry accepted temperature limit for UO2 fuel is 1850°C; however, this temperature limit might be different for fuels other than UO2. There are several factors that may affect a fuel centerline temperature limit for a fuel. These factors include melting point, high-temperature stability, and phase change of the fuel. For instance, the accepted fuel centerline temperature limit of UO2 fuel is approximately 1000°C below its melting point. As a result, the same fuel centerline temperature limit has been established for the other low thermal-conductivity fuels and enhanced thermal-conductivity fuels. In regards to ThO2, the melting point is higher than that of UO2, but a high uncertainty is associated with its melting point. Therefore, as a conservative approach, the same temperature limit has been established for ThO2. Similarly, the corresponding limit for UC fuel would be 1500°C, because the melting point of UC is approximately 2505°C. UN fuel decomposes to uranium and gaseous nitrogen at temperatures above 1600°C. Therefore, the fuel centerline temperature limit for UN should be lower than that of UO2 under normal operating conditions. Ma (1983) recommends a temperature limit of 1500°C for UN.

A steady-state one-dimensional heat transfer analysis was conducted in order to calculate the fuel centerline temperature at SCW fuel channels. Based on the proposed core configuration SCW fuel channels are located at the center of the core. Consequently, the thermal power in some of these fuel channels might be by a factor higher than the average channel power of 8.5 MWth. Therefore, in the present study, a thermal power per channel of 9.8 MWth has been considered for the SCW fuel channels with the maximum thermal power. This thermal power is approximately 15% (i. e. 10% above the average power and 5% uncertainty) above the average thermal power per channel. The conditions based on which the calculations have been conducted are as follows: an average mass flow rate of 4.4 kg/s, a constant pressure of 25 MPa, a coolant inlet temperature of 350°C, a thermal power per channel of 9.8 MWth.

The presented analysis does not take into account the pressure drop of the coolant. The main reason for not taking the pressure drop into consideration is that the pressure drop is inversely proportional to the square of mass flux. In a CANDU fuel channel, the pressure drop is approximately 1.75 MPa (AECL, 2005). In addition, the mass flux in an SCWR fuel channel is approximately 5 times lower than that of a CANDU reactor. Therefore, the pressure drop of a SCWR fuel channel should be significantly lower than 1.75 MPa. As a result, the pressure drop has not been taken into consideration.

In addition, this study does not determine the sheath and the fuel centerline temperatures for the SRH fuel channels mainly due to the fact that the average thermal power in SRH channels is 5.5 MWth (see Table 1). Since the thermal power in SRH channels is

4 *i i+1 gen r,i+1 r,i avg*

**7. Results: Fuel centerline and sheath temperatures** 

limit of 1850°C (Reisch, 2009) at all normal operating conditions.

channel of 9.8 MWth.

*Q r -r*

There are two temperature limits that a fuel and a fuel bundle must meet. First, the sheath temperature must not exceed the design limit of 850°C (Chow and Khartabil, 2008). Second, when UO2 fuel is used, the fuel centerline temperature must be below the industry accepted

Previously, it was mentioned that the industry accepted temperature limit for UO2 fuel is 1850°C; however, this temperature limit might be different for fuels other than UO2. There are several factors that may affect a fuel centerline temperature limit for a fuel. These factors include melting point, high-temperature stability, and phase change of the fuel. For instance, the accepted fuel centerline temperature limit of UO2 fuel is approximately 1000°C below its melting point. As a result, the same fuel centerline temperature limit has been established for the other low thermal-conductivity fuels and enhanced thermal-conductivity fuels. In regards to ThO2, the melting point is higher than that of UO2, but a high uncertainty is associated with its melting point. Therefore, as a conservative approach, the same temperature limit has been established for ThO2. Similarly, the corresponding limit for UC fuel would be 1500°C, because the melting point of UC is approximately 2505°C. UN fuel decomposes to uranium and gaseous nitrogen at temperatures above 1600°C. Therefore, the fuel centerline temperature limit for UN should be lower than that of UO2 under normal

operating conditions. Ma (1983) recommends a temperature limit of 1500°C for UN.

A steady-state one-dimensional heat transfer analysis was conducted in order to calculate the fuel centerline temperature at SCW fuel channels. Based on the proposed core configuration SCW fuel channels are located at the center of the core. Consequently, the thermal power in some of these fuel channels might be by a factor higher than the average channel power of 8.5 MWth. Therefore, in the present study, a thermal power per channel of 9.8 MWth has been considered for the SCW fuel channels with the maximum thermal power. This thermal power is approximately 15% (i. e. 10% above the average power and 5% uncertainty) above the average thermal power per channel. The conditions based on which the calculations have been conducted are as follows: an average mass flow rate of 4.4 kg/s, a constant pressure of 25 MPa, a coolant inlet temperature of 350°C, a thermal power per

The presented analysis does not take into account the pressure drop of the coolant. The main reason for not taking the pressure drop into consideration is that the pressure drop is inversely proportional to the square of mass flux. In a CANDU fuel channel, the pressure drop is approximately 1.75 MPa (AECL, 2005). In addition, the mass flux in an SCWR fuel channel is approximately 5 times lower than that of a CANDU reactor. Therefore, the pressure drop of a SCWR fuel channel should be significantly lower than 1.75 MPa. As a

In addition, this study does not determine the sheath and the fuel centerline temperatures for the SRH fuel channels mainly due to the fact that the average thermal power in SRH channels is 5.5 MWth (see Table 1). Since the thermal power in SRH channels is

result, the pressure drop has not been taken into consideration.

2 2

*T = + T <sup>k</sup>* (26)

approximately 35% less that of the SCW channels, the sheath and the fuel centerline temperatures will be definitely lower than those of the SCW channels. As a result, if a fuel and sheath meet their corresponding temperature limits under the operating conditions of the SCW channels with the maximum thermal power, they will be suitable for the SRH channels as well.

For the SCW fuel channels, the fuel centreline temperature has been calculated at cosine, upstream-skewed cosine, downstream-skewed cosine, and uniform axial heat flux profiles. These heat flux profiles have been calculated based on the Variant-20 fuel bundle. Each of the 42 fuel elements of the Variant-20 fuel bundle has an outer diameter of 11.5 mm while the minimum required thickness of the sheath has been determined to be 0.48 mm. Therefore, the inner diameter of the sheath is 10.54 mm. Inconel-600 was chosen as the material of the sheath.

The examined fuels were UO2, MOX, ThO2, UC, UN, UO2-SiC, UO2-C, and UO2-BeO. For each fuel, the fuel centerline temperature was analysed at the aforementioned AHFPs. Since the maximum fuel centerline temperature was reached at downstream-skewed cosine AHFP for all the examined fuels, only the results associated with this AHFP have been presented in this section. Figures 17 through 19 show the coolant, sheath, and fuel centerline temperature profiles as well as the heat transfer coefficient profile along the heated length of the fuel channel for UO2, UC, and UO2-BeO fuels. Each of these three fuels represents one fuel category (i.e., low, enhanced, high thermal-conductivity fuels). It should be noted that the results presented in Figs. 17 through 19 are based on a 20-*µ*m fuel-sheath gap.

In addition, Figure 20 shows the maximum fuel centerline temperatures of all the examined fuels. As shown in Figure 20, the maximum fuel centerline temperatures of all examined low thermal-conductivity fuels exceed the temperature limit of 1850°C. On the other hand, enhanced thermal-conductivity fuels and high thermal-conductivity fuels show fuel centerline temperatures below the established temperature limits of 1850°C and 1500°C, respectively.

Fig. 17. Temperature and HTC profiles for UO2 at downstream-skewed cosine AHFP.

Thermal Aspects of Conventional and Alternative Fuels

sheath gap width.

in SuperCritical Water-Cooled Reactor (SCWR) Applications 151

Fig. 20. Maximum fuel centerline temperatures of examined fuels based on a 20–*μ*m fuel-

Fig. 21. HTC and sheath-wall temperature profiles as function of AHPF.

sheath and fuel centerline temperatures based on a downstream-skewed AHFP.

A comparison between the examined non-uniform AHFPs shows that in terms of the sheath and fuel centerline temperatures, upstream-skewed cosine AHFP is the most ideal heat flux profile. On the other hand, the downstream-skewed cosine AHFP results in the highest temperatures. Thus, for design purposes, it is a conservative approach to determine the

Fig. 18. Temperature and HTC profiles for UC at downstream-skewed cosine AHFP.

In regards to sheath temperature, the sheath temperature reached its maximum at downstream-skewed cosine AHFP. Figure 21 provides a comparison between the sheath temperature profiles for the four studied AHFPs. Figure 21 also shows the HTC profiles corresponding to each examined AHFPs. As shown in Fig. 21, unlike uniform AHFP, HTC reaches its maximum value in the beginning of the fuel channel for non-uniform AHFPs (i.e., downstream-skewed cosine, cosine, and upstream-skewed cosine AHFPs). This increase in HTC is due to the fact the sheath temperature reaches the pseudocritical temperature. In contrast, with uniform AHFP, the sheath temperature is above the pseudocritical temperature from the inlet of the fuel channel. Consequently, the peak in HTC at uniform AHFP occurs when the coolant reaches the pseudocritical temperature.

Fig. 19. Temperature and HTC profiles for UO2–BeO at downstream-skewed cosine AHFP.

150 Nuclear Reactors

Fig. 18. Temperature and HTC profiles for UC at downstream-skewed cosine AHFP.

In regards to sheath temperature, the sheath temperature reached its maximum at downstream-skewed cosine AHFP. Figure 21 provides a comparison between the sheath temperature profiles for the four studied AHFPs. Figure 21 also shows the HTC profiles corresponding to each examined AHFPs. As shown in Fig. 21, unlike uniform AHFP, HTC reaches its maximum value in the beginning of the fuel channel for non-uniform AHFPs (i.e., downstream-skewed cosine, cosine, and upstream-skewed cosine AHFPs). This increase in HTC is due to the fact the sheath temperature reaches the pseudocritical temperature. In contrast, with uniform AHFP, the sheath temperature is above the pseudocritical temperature from the inlet of the fuel channel. Consequently, the peak in HTC at uniform AHFP occurs when the coolant reaches the pseudocritical temperature.

Fig. 19. Temperature and HTC profiles for UO2–BeO at downstream-skewed cosine AHFP.

Fig. 20. Maximum fuel centerline temperatures of examined fuels based on a 20–*μ*m fuelsheath gap width.

Fig. 21. HTC and sheath-wall temperature profiles as function of AHPF.

A comparison between the examined non-uniform AHFPs shows that in terms of the sheath and fuel centerline temperatures, upstream-skewed cosine AHFP is the most ideal heat flux profile. On the other hand, the downstream-skewed cosine AHFP results in the highest temperatures. Thus, for design purposes, it is a conservative approach to determine the sheath and fuel centerline temperatures based on a downstream-skewed AHFP.

Thermal Aspects of Conventional and Alternative Fuels

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