International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013 doi: 10.14355/ijnese.2013.0304.02
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Study on Influence of Rewetting on Conduction Heat Transfer for AHWR Fuel Bundle Re-flooding Phenomena Mithilesh Kumar1, D. Mukhopadhay1, A. K. Ghosh2, Ravi Kumar3 Reactor Safety Division1, Bhabha Atomic Research Centre, Mumbai, India Raja Ramanna Fellow2, Bhabha Atomic Research Centre, Mumbai, India Indian Institute of Technology, Roorkee3, India mithiles@barc.gov.in
1
Abstract Numerical study on re-flooding behavior of AHWR fuel bundle has been carried out to assess influence of circumferential and axial rewetting on the conduction heat transfer. As the AHWR fuel bundle quenching under accident condition is designed primarily with radial jets at several axial locations, however, bottom re-flooding still remain as an another option. A 3-D (r, θ, z) transient conduction fuel pin model has been developed to carry out the study with a Finite Difference Method (FDM) technique with Alternating Direction Implicit (ADI) scheme. Both the analyses are carried out with same fluid temperature and heat transfer coefficients as boundary conditions. It has been found from the analyses that for radial jet, the circumferential conduction is significant and overall the fuel temperature falls in the quench plane with the initiation of quenching event. Axial conduction is found to be very poor for bottom reflooding case and the fuel temperature only falls when the quench front reaches the location. Sensitivity studies with respect to the direction of solving by ADI are carried out to assess directional influence on the solution. The study shows that direction (r- θ-z) of solution is insensitive to the solution for axial and circumferential conduction solutions. Keywords Re-flooding; Rewetting; Heat Conduction; Numerical Method; Sensitivity Study
Introduction Rewetting of hot surface is a process in which a liquid wets a hot surface by displacing its own vapour that otherwise prevents the contact between the solid and liquid phases. This has generated immense interest in studying rewetting through both theoretical simulation by Yamanouchi, Coney M.W.E. and experimental studied was carried by Yamanouchi, Duffey R.B. Falling film rewetting for several vertical geometries such as plates (Coney M.W.E., Tien C.L.),
rods [Blair J.M.] and tubes [Satapathy A.K.] have been modeled by a number of researchers. In general, in all models, a moving rewetting front that divides the solid into two distinct regions is considered. Most of the models also consider a constant rewetting velocity that reduces the problem into a quasi-static one. Initial efforts were made to formulate one-dimensional conduction models [Yamanouchi] that are reasonably successful in correlating rewetting phenomena at low Peclet number. Tien and Yao presented the asymptotic solutions of a two-dimensional conduction model which clearly demonstrates the different physical pictures for the cases of high and low coolant flow rates. A variety of techniques have been used for solving two-dimensional conduction models for falling film rewetting. Some of the important studies are elaborated. Because of mathematical difficulty, most two-dimensional analyses are either approximate or numerical ones. Duffey and Porthouse first considered for solving the rewetting problem by separation of variables. They retained only the first term in the series solution. However, Coney M.W.E reported that using a small number of terms in a series yields inaccurate results due to slow convergence of the series. An approximate solution to the same model for a cylindrical rod was presented by Blair J.M. Tien and Yao first applied the Wiener–Hopf technique to a twodimensional rewetting problem of a rectangular slab, while an exact solution to the same problem was presented by Castiglia et al., employing the method of separation of variables. Numerical solutions of conduction controlled rewetting were provided by Satapathy et al., Thompson, and Raj and Date by using the finite difference technique. Heat Balance Integral Method (HBIM) is one of many semi-analytical methods used to solve conduction problems (Eckert and Goodman). This is analogous to classical integral
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International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
technique used for fluid flow and convective heat transfer analysis. A numerical study has been made to investigate the effect of internal heating and precursory cooling during quenching of an infinite tube was studied by Satapathy A. K (2000).
location. However, the developed computational model is equally applicable for 2nd and 3rd circle pins with a different boundary conditions.
An experimental assessment of rewetting of a Advanced Heavy Water Reactor (AHWR) specific nuclear fuel bundle with radial jets has been carried out by Patil N. D (2013). The experimentation involved cooling of a 54-rod bundle by in-bundle injection and demonstrated that quenching occurs for all fuel pins along the radial direction. However, the study did not bring out the radial jet influence on circumferential rewetting of a fuel pin. The numerical study presented in this paper aimed to bring out a comparison in rewetting pattern for a radial jet rewetting versus axial rewetting for AHWR fuel bundle. The proposed reactor is a vertical pressure tube type, heavy water moderated boiling light water coolant reactor. Fuel bundle is housed in the pressure tube (PT) which in turn is housed in calandria tube (CT). The 3.5 m height fuel bundle has 54 fuel pins arranged in three different concentric rings as shown in Fig.1. The inner, middle and outer rings having 12, 18 and 24 fuel pins and centre of the fuel pin have a water rod which is used to inject emergency coolant in radial direction at different elevations of the fuel bundle under a pipe break scenario. The water rod has eight holes of 1.5 mm dia. at one elevation along the circumference of water rod. The radial injection is designed for 13 different axial locations along the 3.5 m length of water rod. More details of AHWR fuel bundle and it’s injection arrangement have been furnished by Sinha et al. Bottom re-flooding is also an option with the designer where water may be injected from bottom most holes instead on 13 axial elevations. Under this study, a 3-D (r, θ, z) transient conduction fuel pin model has been developed based on Finite Difference Method (FDM) technique with Alternating Direction Implicit (ADI) scheme. The number of nodes in r, θ & z direction has been considered to be 50, 20 and 50 respectively. A 1.6 ms time step has been used as predicted from stability criteria. A total time of 4 hrs is required for individual rewetting study. Single pin from first circle has been considered for which bottom and radial reflooding was studied. The 1st circle pin is selected as it experiences a strong radial jet and experience a large circumferential temperature gradient. For bottom rewetting case, all the pins will experience the same velocity front irrespective of its 86
FIG. 1 VIEW OF AHWR FUEL BUNDLE
Both the analyses are carried out with same fluid temperature and heat transfer coefficients as boundary conditions. The single fuel pin considers for circumferential and axial rewetting. The three dimensional partial differential equation for unsteady state conduction equation for cylindrical rod is as follows, ∂ 2T 1 ∂T 1 ∂ 2T ∂ 2T Q ′′′ 1 ∂T (1) + + + + = K α ∂t ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2 Alternating Direction Implicit (ADI) numerical method has been adopted. With an ADI method, the heat diffusion equation is first solved implicitly in the rdirection while leaving the other two directions explicit. The heat diffusion equation is then solved implicitly in a similar way in the θ and z direction. This scheme reduces a three-dimensional problem to a series of one-dimensional implicit problem. Fig 2 shows the three different steps in ADI. Many methods [Peaceman D.W, Brian, Douglas J., Chang, M] are also available in ADI for solving the heat diffusion equation, as Peaceman-Rachford pureADI method, Brain method, Douglas method, and based on superposition principle by Jules Thibault (1994). All above schemes having the same problem of conditionally stable criteria as right side of equations have a negative coefficient. This paper discussed conduction effect on axial, circumferential direction as well as the study on sensitivity on the direction of solution(r- θ- z) in steady state as well as in transient which is not found in open literature. This study is required to assess the time frame to reach the steady state condition.
International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
n +1/3 n +1/3 Ti n+1,+1/3 j , k − 2Ti , j , k + Ti −1, j , k
Ti ,nj +1, k
∆r 2 − 2Ti ,nj , k + Ti ,nj −1, k
ri2 ∆θ 2 n +1/3 n Q ′′′ 1 Ti , j , k − Ti , j , k + = k ∆t / 3 αf
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n +1/3 n +1/3 1 Ti +1, j.k − Ti −1, j ,k + 2∆r ri Ti ,nj , k +1 − 2Ti ,nj , k + Ti ,nj , k −1
+
+
∆z 2
(1.1)
Or ∆r n +1/3 T = F0 ∆t 1 − + 2r i −1, j , k i ∆r n +1/3 2 Fθ ∆t F0 ∆t 1 + T + 1− − 2 Fz ∆t Ti n, j , k + 2 2r i +1, j , k ri i Fθ ∆t n Q ′′′∆t n n n Ti , j +1, k + Ti , j −1, k + Fz ∆t Ti , j , k +1 + Ti , j , k −1 + 2 ri 3ρ f C p
(1 + 2F ∆t ) T
n +1/3 i, j ,k
0
(
FIG. 2 ADI STEPS TO SOLVE THE CONDUCTION EQUATION
The tri-diagonal matrix has been used to solve the equation corresponding to given direction. Normally in ADI scheme, first solution is obtained in r-direction in an implicit manner and leaves both directions explicit. But here all the possible six combinations, likes r- θ- z, r- z- θ, θ- r- z, θ- z- r, z- r- θ and z - θ- r have been used to solve the heat diffusion equation and compared results. 3-D Model Development Finite Difference Formulation FDM is used to formulate the heat diffusion equation in cylindrical Co-ordinate (Eq. no. 1) and further semi implicit used to formulate in r-ө-z direction. The differential equation has been divided into the three category (1) centre of fuel pin (2) between centre of fuel pin and surface and (3) surface of fuel pin. The
length increment in r, ө, z directions are ∆r , ∆θ ∆z respectively. The semi implicit scheme has been used to formulate the differential equation in r, ө and z directions and is represented by equation numbers 1.1, 1.2, and 1.3 respectfully. As per ADI norms, first 1/3 time increment has been taken to solve the temperature at different grid points in r-direction next 1/3 time increment for solving the temperature in ө direction and last time step, for solving the temperature in z-direction. For stability criteria, the coefficient of Ti ,nj+.k1/3 , Ti ,nj+.k2/3 , Ti ,nj+.k1 in right side of the equation must be positive. Finite difference formulation between centre and surface of fuel rod r-direction
)
(
)
Condition for stability 2 Fθ ∆t − 2 Fz ∆t ≥ 0 1 − 2 ri Ө-direction n +1/3 n +1/3 Ti n+1,+1/3 j , k − 2Ti , j , k + Ti −1, j , k
Ti n, j++2/3 1, k
∆r 2 − 2Ti n, j+, k2/3 + Ti n, j+−2/3 1, k
+
+ ri2 ∆θ 2 n + 2/3 n +1/ 3 Q ′′′ 1 Ti , j , k − Ti , j , k + = ∆t / 3 k αf
n +1/3 n +1/3 1 Ti +1, j , k − Ti −1, j ,k + 2∆r ri n +1/3 n +1/3 Ti n, j+,1/3 k +1 − 2Ti , j , k + Ti , j , k −1
∆z 2
(1.2)
Or 2 Fθ ∆t n + 2/3 Fθ ∆t n + 2/3 Ti , j +1, k + Ti n, j+−2/3 1 + Ti , = j ,k 1, k + 2 2 r r i i ∆r n +1/3 T 1 − 2 F0 ∆t − 2 Fz ∆t Ti n, j+,1/3 1+ + k + F0 ∆t 2r i +1, j , k i Q ′′′∆t ∆r n +1/3 n +1/3 F0 ∆t 1 − T + Fz ∆t Ti n, j+,1/3 k +1 + Ti , j , k −1 + 2r i −1, j , k 3ρ f C p i Condition for stability
(
(
)
)
(
)
(1 − 2F ∆t − 2F ∆t ) ≥ 0 0
z
z- direction n + 2/3 n + 2/3 Ti n+1,+ 2/3 j , k − 2Ti , j , k + Ti −1, j , k
+
Ti n, j++2/3 1, k
+
∆r 2 − 2Ti n, j+, k2/3 + Ti n, j+−2/3 1, k
ri2 ∆θ 2 n +1 n + 2/3 1 Ti , j , k − Ti , j , k Q ′′′ + = k ∆t / 3 αf
n + 2/3 n + 2/3 1 Ti +1, j , k − Ti −1, j , k 2∆r ri Ti n, j+,1k +1 − 2Ti n, j+,1k + Ti n, j+,1k −1 + (1.3) ∆z 2
or Z-direction
(
)
(
)
Ti ,nj+,1k 1 + 2 Fz ∆t = Fz ∆t Ti ,nj+,1k +1 + Ti ,nj+,1k −1 + ∆r n + 2/3 ∆r n + 2/3 F0 ∆t 1 + Ti +1, j , k + F0 ∆t 1 − T 2r 2r i −1, j ,k i i
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International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
2 Fθ ∆t n + 2/3 Fθ ∆t n + 2/3 n + 2/3 1 − 2 F0 ∆t − Ti , j , k + 2 Ti , j +1, k + Ti , j −1, k 2 r r i i Q′′′∆t + 3ρ f C p
(
)
Condition for stability
2 Fθ ∆t 1 − 2 F0 ∆t − ≥0 ri2 Centre and Surface of the Fuel Pin The centre line temperature for fuel pin has been calculated by using the concept of energy balance and finite difference formulation is used to calculate the centre line temperature. It is shown by Equation 1.4. The centre line grid point represented by i and next to centre line temperature is represented by i+1 and is shown in Fig. 3. In similar way, surface temperature of fuel pin has been formulated in Equation 1.5. n n ∆r ∆r Ti , j , k − Ti +1 Q ′′′π − 2π k s ∆r 2 2 1 + n n 2 T T − ∆r i, j ,k i, j ,k = π ρ f cp t 2 ∆ 2
(1.4)
or
(
)
= Ti ,nj+,1k 4 F0Ti n+1, j +1, k +1 + Ti n, j , k 1 − 4 Fo +
Q ′′′∆t
ρ f Cp
Condition for stability
(1 − 4F ) ≥ 0 o
FIG. 3 DISCRITISATION FOR CENTRE LINE TEMPERATURE
At surface of fuel pin Implicit Scheme
(
2π R f K f Tin−1+1/3 − Tin +1/3
) + 2π R h
f o
∆r
(T
inf
− Ti
n +1/3
)
(
)
n +1/3 2 − Tin (1.5) ∆r ∆r Ti Q′′ ρ f C pπ R 2f − R f − +π R 2f − R f − = ∆t 2 2 3 Or 8 Fo R f ∆t 8 Fo R f ho ∆r ∆t n +1/3 1.0 + T + i K f 4 R f − ∆r 4 R f − ∆r 8 Fo R f ∆t n +1 Q ′′′α∆t 8 Fo R f ho ∆r ∆t Ti −1 + Tinf + Ti n = + 3K f K f 4 R f − ∆r 4 R f − ∆r 2
(
(
88
)
)
(
)
(
)
Boundary Condition and Solver Once the formulation of differential equations is over then with help of boundary, it is required to calculate the temperature at different nodes of fuel pin. There are two cases considered for validation of numerical code; the first of which is constant coolant temperature and heat transfer coefficient while second case is coolant temperature varying along the length of fuel pin when it flows along the length of pin. The governing equation which calculates the coolant average temperature between the corresponding nodes of fuel pin is represented by Eq.1.7. Energy balance at surface of fuel pin ho A(Ts − Tinf = ) mC p (Tout − Tin )
(1.6)
Tin + Tout 2 h A ho ATs h A Tout ( o = + 1) + (1 − o )Tin 2mC p 2mC p mC p Tinf =
(1.7) (1.8)
Equation no 1.8 is used for calculating the outlet coolant temperature for a given volume. After applying the boundary condition, it is required to solve the discretised partial differential in each direction by ADI method. At each incremental time step, the differential equations form a tri-diagonal matrix is solved by Thomas Algorithm (Eq 1.9). In first time increment (i.e. 1/3 s), it solves temperature in rdirection at different grids points and takes old values of temperature in ө, z direction. Similarly, for higher time step (2/3 s), this algorithm solves in ө direction and remaining directions take previous value. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ : : ∗
∗ ∗
T1,nj+,1/3 k n +1/3 T2, j , k T3,n +j ,1/3 k n +1/3 T4, j , k = . . . Tnn++1,1/3j , k
b1 b 2 b3 b4 . . . bn +1
(1.9)
Different heat transfer correlations have been used to determine the heat transfer coefficient at the surface of fuel pin. Thom correlation (Eq. 2.0) is used for nucleate boiling and Modified Bromley correlation (Eq. 2.1) is used for film boiling. p 6.2 e = ho 0.79
4
3 ∆Tsat
k g h fg ρ g g (∆ρ Lg ) ho = 0.62 µ g L∆Tsat
(2.0) 0.25
(2.1)
International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
gσ c L = 2π g (∆ρ Lg )
Steady State Computation for Uniform and Non-uniform Heat Generation Fig 4 shows the 3D temperature behavior in the fuel pin in different cases, the first of which is uniform heat generation (1.16E+8 W/m3) with a constant heat transfer coefficient and coolant temperature and second case is with sinusoidal heat generation with variation of coolant temperature along the length of the fuel pin.
direction. Here sensitivity studies against the direction of solution like r- θ- z, θ-r- z and z- r- θ have been carried out. In θ-r- z, ADI first solves the temperature in θ direction implicitly and then goes to r and z direction and similar ways for left last cases. It has been found that there is insensitivity of temperature prediction by ADI method by changing the directions of solution. 780
760
Temperature (K)
Where
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740
r_theta_Z Theta_r_Z Z_r_theta
720
700
680
660 0
20
40
60
80
100
Time (s) Steady state temperature of surface of fuel pin
FIG. 5 STEADY STATE SURFACE TEMPERATURE DISTRIBUTION BY SENSITIVITY STUDIES IN ADI
380
Temperature in fuel pin has been predicted uniform across the fuel pin and at any cross section minimum and maximum temperatures are at the surface and centre of fuel pin respectively. For sinusoidal heat generation, as shown in Fig. 4, the fuel surface temperature starts increasing along the length of the fuel pin but in half of the fuel pin temperature first increases very fast duo to increasing the heat flux, later, it is not decreased even decreases of flux due to integration of heat energy in the fuel pin which results in maximum fuel pin surface temperature in-between second half of the fuel pin and its end. Sensitivity Study on Direction of Solution Sensitivity steady has been carried for calculation of temperature and fuel surface temperature is shown in Fig. 5. As practice in open literature, ADI first solves in r- direction implicitly for calculation of temperature then goes to θ-direction and finally it solves in z
360
Temperature (oC)
FIG. 4: TEMPERATURE DISTRIBUTION ON SOLID FUEL PIN IN (A) UNIFORM (B) NON UNIFORM (SINUSOIDAL) HEAT GENERATION
Tmax = 380 K at 3.09 m
340 320
25 Nodes 50 Nodes 75 Nodes
300 280 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Length (m) FIG. 6 STEADY STATE SURFACE TEMPERATURE DISTRIBUTION ALONG THE LENGTH BY SENSITIVITY STUDIES IN NUMBER OF NODES
Further sensitivity studies have been carried out with variation of numbers nodes along the length of furl pin. It has been found that ADI method is insensitivity and shown in Fig. 6. Benchmarking of 3-D Model A benchmarking exercise has been carried out for the 3-D conduction model with the available 1-D analytical solution for a steady state case. As analytical solution of 3-D is not available, the benchmarking is
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carried out with 1-D solution. The analytical solution for estimating the coolant temperature and fuel pin surface temperature along the length of fuel pin is represented by equation no. 2.2 and 2.3 [Glasstone S et al] and used to calculate the fluid temperature and surface temperature for 1-D analytical solution. The heat flux along the fuel pin has been considered as a cosine profile. The Fourier conduction equation and its analytical solution for calculating the temperatures along the radius of fuel pin solution are given in Equation 2.4 and 2.5 respectively. Table 1 provides the input parameters for this exercise. Tinf = Ts =
Qmax A L
π mc p
(1 − cos(
πx l
370 360 340 330 320
Coolant temperature
310 300 290
Q ′′′
= 0 + Kf
250 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time (s)
(2.4)
FIG. 7 BENCHMARKING OF COOLANT AND SURFACE TEMPERATURE PREDICTION
Fig. 8 shows the comparison of temperature distribution in r-direction with 3-D numerical model and analytical solution for fuel rod. Result shows a good agreement between the two solutions.
At outer surface
950
TABLE 1 INPUT DATA FOR BENCHMARK EXERCISE:
Value 0.0041 m 3.5 m 2.8 W/m K 10730 kg/m3 0.28 KJ/kg K 300 oC 500 W/m2K πx 8 l
Tmax = 938oC
850
Analytical Numerical
800
750
0.000
0.001
Fig 7 shows the coolant temperature and fuel surface temperature variation along the length of the fuel pin. The calculated coolant temperature and surface temperature using equation 1.7 and 1.8, acts as a
0.003
0.004
FIG. 8 STEADY STATE TEMPERATURE DISTRIBUTION ALONG THE CENTRE OF FUEL PIN 200
) (W/m3)
Two step benchmarking exercise has been done. In the first step, boundary conditions used for calculating the fuel radial temperature distribution for 3-D model and analytical solution are compared and in the subsequent step the radial temperature distributions are compared.
0.002
Radius (m)
1st Ring Numerical 2nd Ring Numerical 3rd Ring Numerical 1st Ring Experimental 2nd Ring Experimental 3rd Ring Experimental
150
Temperature (oC)
3.2X 10 sin(
900
Temperature (oC)
Heat generation in body is heat transfer to coolant due to convection = Q ′′′π R 2f l 2π R f lh(Ts − Tinf ) Q ′′′R 2f (2.5) R2 T ( R= ) (1 − 2 ) + Ts Rf 4K f
90
350
280
(2.3)
Apply boundary condition dT =0 dr r =0
Axial Heat generation rate
Analytical Analytical Numerical Numerical
380
260
1-D Fourier conduction equation
Parameter Radius of fuel pin Length of fuel pin Fuel thermal conductivity Fuel density Fuel specific heat Inlet Coolant temperature Heat transfer coefficient at surface
Surface temperature
390
270
πx
1 d dT r r dr dr
400
(2.2)
)) + Tin
Q V πx sin( ) + max 1 − cos + Tin l l ho A π mc p
QmaxV
boundary conditions for the 3-D code. A good agreement is found between the numerical code result with analytical solution obtained using equation 2.2 and 2.3
Temperature (oC)
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100
50
0 50
100
150
200
Time (s)
FIG. 9 BENCHMARKING RADIAL FUEL SURFACE TEMPERATURE PREDICTION OF DIFFERENT RINGS
International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
The exercise shows that the multi pin model is able to capture the rewetting pattern well. However, in this study, circumferential rewetting pattern could not be produced as measurements across the circumference of a simulated fuel pin are not done/reported. Study on Influence of Rewetting on Conduction Heat Transfer Assessment Bottom Re-flooding The schematic for bottom re-flooding is shown in Fig. 10. The schematic shows the rewetting front travels from bottom to top.
starting from bottom (0.0 m) to top (3.5 m). 800 700
0.076 m 0.648 m 1.292 m 2.052 m 2.656 m 3.5 m
600
Temperature (oC)
Further, this single pin model has been extended to multiple pins model considering that pins are arranged in three concentric rings with radial jet simulation. A validation exercise has been carried out with experimental results reported by Patil N. D. The experimental case with initial temperature of 168°C for all pins with an injection flow rate of 73 lpm has been considered. The validation exercise results are shown in Figure 9 at an elevation at 2.4 m.
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500 400 300 200 100 0 10
20
30
40
Time (s)
FIG. 11 FUEL TEMPERATURE TRANSIENT AT DIFFERENT AXIAL LOCATION
The result shows for the chosen boundary conditions axial conduction is negligible. Cooling by rewetting does not influence the upstream node temperatures. The change in upstream node temperature only happens with the arrival of the rewetting front. 800 r_theta_z r_theta_z r_theta_z r_z_theta r_z_theta r_z_theta theta_r_z theta_r_z theta_r_z theta_z_r theta_z_r theta_z_r z_r_theta z_r_theta z_r_theta z_theta_r z_theta_r z_theta_r
700 3.5 m
Temperature (oC)
600 500 2.052 m
400 300
0.076 m
200 100 0 5
10
15
20
25
30
35
40
45
FIG. 12 TRANSIENT SURFACE TEMPERATURE AT DIFFERENT AXIAL LOCATIONS - A SENSITIVITY STUDIES FIG. 10 SCHEMATIC FOR BOTTOM REFLOODING
For this study, boundary conditions like re-flooding velocity is assumed to be 0.35 m/s (Jagdish,2010), hence over a period of 10 s the rewetting front will reach the top of the heated fuel pin. As the initial fuel temperature (750°C, Jagdish, 2013) is considered to be high (as in a practical situation) as compared to Leiden frost temperature (220°C), a linear variation of heat transfer coefficient from 500 W/m2 K (only steam) to 30000 W/m2 K (only water) and variation of fluid temperature from 300oC (only steam) to 30oC (only water) are assumed over 20 s period at the node where rewetting front has reached. Fig. 11 shows the transient cooling temperature for six axial locations
As the axial conduction phenomenon is predicted by numerically technique, influence of ADI solution technique on the above mentioned findings is investigated. The ADI solutions are obtained with all possible directions combinations like r- θ- z, r- z- θ, θr- z, θ- z- r, z- r- θ and z - θ- r. Fig. 12 shows that the axial conduction behavior is insensitive to direction of solution direction. Hence there is no influence of numerical scheme on the axial conduction behavior. Radial Re-flooding The schematic for radial re-flooding is shown in Fig. 13. The schematic shows the radial direction water jet travel from centre to the periphery, attempting to rewet the front and rear surface of fuel pin over the
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minute of the beginning of rewetting of font section. The influence of one sided rewetting has caused the maximum temperature of fuel pin to shift from centre to un-rewetted portion of fuel pin, as shown in Fig. 15.
issuing jet plane.
FIG. 13 SCHEMATIC FOR RADIAL REFLOODING 800 700
r-thita-z 0.004
Temp
4
500
3
400 300 200 100
500 450 400 350 300 250 200 150 100 50
0.002
2
Radius (m)
Temperature (oC)
600
1
0
-0.002
0 0
20
40
60
80
-0.004
Time (s)
92
-0.002
0
0.002
0.004
Radius (m)
FIG. 15 INFLUENCE OF CIRCUMFERENTIAL REWETTING ON FUEL CENTRE LINE TEMPERATURE
As the circumferential conduction phenomenon is predicted by numerically technique, and influence of ADI solution technique on the above mentioned findings is investigated. 800 700 600 r_theta_Z r_theta_Z r_theta_Z r_theta_Z theta_r_z theta_r_z theta_r_z theta_r_z z_r_theta z_r_theta z_r_theta z_r_theta
400 300 200 100
47.25 47.00 46.75
o
500
TemperatureC) (
For this study, boundary conditions like high rewetting heat transfer coefficient of 30000 W/m2 K (only water) is applied over half of the circumference (180°) of fuel pin where the water jet is likely to impinge. At the same time, a low heat transfer coefficient of 500 W/m2 K (only steam) is applied on the other half of the circumference (180°) where the jet is unlikely to reach. A linear variation of heat transfer coefficient from 500 W/m2 K (only steam) to 30000 W/m2 K (only water) for the water impinged surface is considered as described earlier. Variation of fluid temperature from 300oC (only steam) to 30oC (only water) is assumed over 20 s period at the node where rewetting front has reached. Fig. 14 shows the transient cooling temperature for six circumferential locations covering the half of circumference which experiences the water jet. The result shows that for the chosen boundary conditions, influence of circumferential conduction is significant. Temperature of the un-wetted portion is dropped by 250°C with rewetting of the other half of the fuel pin over a
-0.004
Temperature (oC)
FIG. 14 TRANSIENT SURFACE TEMPERATURE ALONG THE CIRCUMFERENCE OF FUEL PIN
46.50 46.25 46.00 45.75 45.50 31.79
31.80
31.81
31.82
Time (s)
0 0
20
40
60
80
Time (s) FIG. 16 TRANSIENT SURFACE TEMPERATURE AT DIFFERENT CIRCUMFERENTIAL LOCATIONS- A SENSITIVITY STUDIES
International Journal of Nuclear Energy Science and Engineering Volume 3 Issue 4, December 2013
The ADI solutions are obtained with all possible directions combinations like r- θ- z, r- z- θ, θ- r- z, θ- zr, z- r- θ and z - θ- r as done for axial conduction case. Fig. 16 shows that the axial conduction behavior is insensitive to direction of solution direction. Hence there is no influence of numerical scheme on the axial conduction behavior.
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Q ′′′ = Local heat generation in fuel (W/m3)
r = Fuel radius changing in r-direction R f = outer radius of fuel pin (m) T = temperature (K) Ts = Fuel surface temperature (K) Tout =coolant outlet temperature (K) Tin= coolant inlet temperature (K)
Conclusion
Tinf = Average coolant temperature
Influence of Rewetting on Conduction Heat Transfer for AHWR Fuel bundle Re-flooding Phenomena has been studied with the help of 3-D conduction numerical model. A circumferential conduction is found to be significant for radial jet rewetting cases as against a very poor axial conduction during axial rewetting situation. The conservative assessment shows that radial jet is able to bring down the unrewetted portion temperature by 250°C where the temperature at different axial location falls only with arrival of rewetting front. A radial direction jet is found to be more effective to cool the hot surface as against the bottom axial reflooding. A sensitivity study carried out with respect to ADI technique shows that the solution is insensitive to the direction of solution; thus eliminating the uncertainty associated with direction of solution method of ADI.
∆Tsat = Temperature difference between fuel surface and saturation temperature of coolant (oC) V = Volume (m3)
ρ g = Density of steam (kg/m3) ρ = density fuel (Kg/m3) σ = surface tension (N/m)
θ = angle (radians) α = thermal diffusivity (m2/s) ∆ρ Lg = density difference between liquid and steam (kg/m3) ∆r = distance increment in fuel (m) ∆z = distance increment in z direction(m)
∆θ = angle increment in circumference direction (radians) t= time (s) ∆t =time increment(s)
Nomenclature A = area (m ) C p = specific heat capacity (J/kgK) 2
k f ∆t Fo = ρ C p ( ∆r )2 k f ∆t Fθ = ρ C p ( ∆θ )2 k f ∆t Fz = ρ C p ( ∆z )2 g = local acceleration due to gravity (m/s2)
gc = gravitation constant (m/s2) ho = heat transfer coefficient at outer surface (W/m2K) h fg = Latent heat of vaporization (W/m)
Subscripts f = fuel i =mesh point in x-direction j = mesh point in θ-direction k = mesh point in z-direction Superscript n= time step ACKNOWLEDGMENT:
Authors acknowledged Dr. R. K. Singh, Mr. H.G. Lele and Dr. P. K. Vijayan for their valuable suggestion during the development of code. REFERENCES
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