11.7. Appendices

11.7.1. Appendix 11.1: Explicit Formulas for Stiffness Matrix and Load Vector

The stiffness matrix is defined in Section 11.2.4.2, Eq. (11.2-78).

(11.7-1)\[ \left\lbrack K \right\rbrack = \int_{\text{A}_{\text{c}}}{\left\lbrack B \right\rbrack^{T} \left\lbrack C \right\rbrack \left\lbrack B \right\rbrack \text{dA}}\]

where the form of [B] and [C] matrices depends on the type of fuel element considered. Introducing new variables

(11.7-2)\[ d = \frac{b}{r} - 1\]
(11.7-3)\[ e = 1 - \frac{a}{r}\]

the matrix [B] can be written in a compact form as

(11.7-4)\[\begin{split} \left\lbrack B \right\rbrack = \frac{1}{l}\begin{bmatrix} - 1 & 1 & 0 \\ d & e & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\end{split}\]

where \(a\), \(b\), and \(l\) are explained in Figure 11.2.2. As noted in Section 11.2.4.1, this form of matrix [B] is valid for both continuous and cracked elements.

Expanding the matrix triple product in Eq. (11.7-1) and using the symmetry of [C] yields

(11.7-5)\[\begin{split} \left\lbrack K \right\rbrack = \frac{1}{l^{2}}\int_{A_{c}}^{}\begin{bmatrix} \left( C_{11} - 2C_{12}d + C_{22}d^{2} \right) & \left( - C_{11} + C_{12}\left( d - e \right) + C_{22}\text{ed} \right) & \left( - C_{13} + C_{23}d \right)l \\ \left( - C_{11} + C_{12}\left( d - e \right) + C_{22}\text{ed} \right) & \left( C_{11} - 2C_{12}e + C_{22}e^{2} \right) & \left( - C_{13} + C_{23}e \right)l \\ \left( - C_{13} + C_{23}d \right)l & \left( C_{13} + C_{23}e \right)l & C_{33}l^{2} \\ \end{bmatrix} \text{dA}\end{split}\]

Inspection of this equation indicates that all elements of \(\left\lbrack K \right\rbrack\) can be expressed in terms of following six integrals

(11.7-6)\[ I \equiv \int_{\text{A}_{\text{c}}}{\text{dA} = \pi l \left( a + b \right)}\]
(11.7-7)\[ I_{\text{d}} \equiv \int_{\text{A}_{\text{c}}}{\left( d \right) \text{dA} = \pi l^{2}}\]
(11.7-8)\[ I_{\text{d}2} \equiv \int_{\text{A}_{\text{c}}}{\left( d^{2} \right) \text{dA} = \pi l \left\lbrack \frac{2b^{2}}{l} \ln\left( \frac{b}{a} \right) - 3b + a \right\rbrack}\]
(11.7-9)\[ I_{\text{e}} \equiv \int_{\text{A}_{\text{c}}}{\left( e \right) \text{dA} = \pi l^{2}}\]
(11.7-10)\[ I_{\text{e}2} \equiv \int_{\text{A}_{\text{c}}}{\left( e^{2} \right) \text{dA} = \pi l \left\lbrack b - 3a + \frac{2a^{2}}{l} \ln\left( \frac{b}{a} \right) \right\rbrack}\]
(11.7-11)\[ I_{\text{ed}} \equiv \int_{\text{A}_{\text{c}}}{\left( \text{ed} \right) \text{dA} = \pi l \left\lbrack a + b - \frac{2ab}{l} \ln\left( \frac{b}{a} \right) \right\rbrack}\]

The domain of these integrations is shown in Figure 11.2.2. In terms these newly defined variables, the stiffness matrix can be rewritten as

(11.7-12)\[\begin{split} \left\lbrack K \right\rbrack = \frac{1}{l^{2}}\begin{bmatrix} \left( C_{11}I - 2C_{12}I_{\text{d}} + C_{22}I_{\text{d}2} \right) & \left( - C_{11}I + C_{12}\left( I_{\text{d}} - I_{\text{e}} \right) + C_{22} I_{\text{ed}} \right) & \left( - C_{13}I + C_{23}I_{\text{d}} \right)l \\ \left( - C_{11}I + C_{12}\left( I_{\text{d}} - I_{\text{e}} \right) + C_{22}I_{\text{ed}} \right) & \left( C_{11}I - 2C_{12}I_{\text{e}} + C_{22} I_{\text{e}2} \right) & \left( C_{13}I + C_{23}I_{\text{e}} \right)l \\ \left( - C_{13}I + C_{23}I_{\text{d}} \right)l & \left( C_{13}I + C_{23}I_{\text{e}} \right)l & C_{33} I\ l^{2} \end{bmatrix}\end{split}\]

The load vector for a pressure load acting in the radial direction in given by Eq. (11.2-67)

(11.7-13)\[ t_{\text{i}} = \int_{\text{C}} \sigma_{\text{r}} N_{\text{i}} {\hat{i}}_{\text{r}} \cdot \hat{n}\text{dC}\]

(i=1,2). The shape functions \(N_{1}\) and \(N_{2}\) are defined in Eq. (11.2-63) and Eq. (11.2-64), respectively. For the case of an element bordering the central cavity, the curve \(C\) corresponds to the inner radius of the element, we have

(11.7-14)\[ \sigma_{\text{r}} = - p_{\text{cav}}\]
(11.7-15)\[ {\hat{i}}_{\text{r}} \cdot \hat{n} = {\hat{i}}_{\text{r}} \cdot \left( - {\hat{i}}_{\text{r}} \right) = - 1\]
(11.7-16)\[ N_{1} = 1\]
(11.7-17)\[ N_{2} = 0\]
(11.7-18)\[ \int_{\text{C}}{\text{dC} = 2 \pi r_{\text{cav}}}\]

Hence,

(11.7-19)\[ t_{1} = 2 \pi p_{\text{cav}} r_{\text{cav}}\]
(11.7-20)\[ t_{2} = 0\]

A similar derivation for an element experiencing a pressure \(p_{\text{out}}\) directed radially inward shows that

(11.7-21)\[ t_{1} = 0\]
(11.7-22)\[ t_{2} = 2 \pi p_{\text{cav}} r_{\text{out}}\]

where \(r_{\text{out}}\) is the outer radius of the element.

11.7.2. Appendix 11.2: List of Input Variables for Standalone FPIN2 Calculation

*** Location Numbers for Integer Data ***

(Number in parentheses following value name states maximum value)

*** COMMON/INPUTI/ ***

Location Number

Name

Value

Description

1

IFTYPE

=1

Uranium - 5% fissium fuel

=2

Uranium - 10% zirconium fuel

=3

Uranium - 15% plutonium - 10% zirconium fuel

=4

User supplied mixture of U-Pu-Zr fuel (input required at loc. numb. 5001+ & 5401+)

2

(Not currently used)

3

ICTYPE

=1

20% CW type 316 cladding

=2

D9 cladding

=3

HT9 cladding

4

(Not currently used)

5

IDTOPT

=0

Compute at equally spaced time values (use NDT and DTIME)

=1

Increments in time are user-supplied (use NDT and TTABLE)

6

IHTOPT

=0

Perform heat transfer calculation including coolant and wall

=1

Perform heat transfer calculation with input values of clad outer surface temperature. (Use NCLADT and COMMON/TEMPIN)

7-14

(8)

(Not currently used)

15

ICRACK

=0

No fuel cracking

=1

Radial fuel cracks included

16

IFPLAS

=0

Allow creep-plastic strains in fuel

=1

Suppress creep-plastic strains in fuel

17

ICPLAS

=0

Allow creep-plastic strains in clad

=1

Suppress creep-plastic strain in clad

18

IFSWEL

=0

Allow swelling-hotpressing strains in fuel

=1

Suppress swelling-hotpressing strains in fuel

19

ICSWEL

=0

Allow swelling strains in clad

=1

Suppress swelling strains in clad

20

IDZ

=0

Axial segments are of equal height DZ=Z/NDZ

=1

User-supplied axial segment heights

21

ILRGST

=0

Large strain analysis

=1

Small perturbation analysis

22

IFCSLP

=0

Fuel-clad locked when gap is closed

=1

Independent fuel-clad axial displacement

23

(Not currently used)

24

IOUTSW

=0

No detailed printing of results - summary of results only

=1

Normal detailed printout under IFREQA, NFREQA, and IFREQB control

25

IFREQA

Initial print frequency, number of time steps between normal detailed printout

26

NFREQA

Total number of time steps under IFREQA control

27

IFREQB

Final print frequency

28

IGRAPH

=0

Do not write graphics file

=1

Write datafile FT12F001 for processing by a graphics program

29-30

(2)

(Not currently used)

31

NDT

No. of time steps or no. of entries in TTABLE. If IDTOPT=0, there is no limitation on NDT. If IDTOPT=1, NDT is limited to 1980.

32

NDZ

(20)

No. of axial segments

33

NDRF

(20)

No. of radial elements in cladding (minimum value is 6)

34

NDRC

(10)

No. of radial elements in cladding (minimum value is 3)

35-40

(6)

(Not currently used)

41

NQ

(25)

No. of values in power vs. time table

42

NCOOLT

(25)

No. of values in coolant inlet temperature versus time table

43

NCOOLF

(25)

No. of values in coolant inlet flow versus time table

44

NCLADT

(50)

No. of values in clad outer surface temperature versus time table

45-100

(56)

(Not currently used)

101-500

IETYPI(I,J)

(20,20)

Fuel element type. Required only when ICRACK=1, default element type is 1 (Loc. Numb. = 100+I+20*(J-1)).

*** Location Numbers for Decimal Data ***

(Symbols in parentheses after array variables are maximum storage allocation and name of integer variable specifying dimension. Numbers within quotation marks are recommended values)

*** COMMON/CNTLIN/ ***

Location Number

Name

Value

Description

1

TZERO

Initial time (sec)

2

DTIME

=1

Computation time step (if IDTOPT=0) (sec)

3-10

(8)

(Not currently used)

11-2000

TTABLE

(1990 NDT)

Computation time steps (if IDTOPT=1) (sec) maximum of 1990 values

*** COMMON/GEOMIN/ ***

Location Number

Name

Value

Description

2001

ZFUEL

Length of fuel column (cm)

2002-2010

(9)

(Not currently used)

2011-2030

DZ

(20 NDZ)

Height of axial segments (cm) (required only when IDZ=1)

2031-2050

FUELIR

(20 NDZ)

Inner radius of fuel (cm)

2051-2070

FUELOR

(20 NDZ)

Outer radius of fuel (cm)

2071-2090

CLADIR

(20 NDZ)

Inner radius of clad (cm)

2091-2110

CLADOR

(20 NDZ)

Outer radius of clad (cm)

(Note: values #2111-2112 not required when IHTOPT=1)

2111

WALLIR

Inner radius of outer wall (cm)

2112

WALLOR

Outer radius of outer wall (cm)

2113

ZPLENM

Total length of plenum (cm)

2114

ZPLNA

Length of sodium in plenum (cm)

2115

PLGASR

Gas constant for plenum gasses (Bar-cc/gm-K)

2116

PLTREF

Temperature at which PINT specified (K). Used to correct PINT to value consistent with initial plenum temperature.

2117-2200

(84)

(Not currently used)

2201-2600

FECTI(I,J)

(20,20 NDZ)

Initial radial crack strain in fuel (required only when ICRACK=1)

(Loc. Numb.=2200+I+20*(J-1))

2601-3000

(400)

(Not currently used)

*** COMMON /DRIVIN/ ***

Location Number

Name

Value

Description

3001

PEXT

External pressure. Assumed constant during transient. (Bar)

3002

PINT

Initial value of internal pin press. (Bar)

3003-3004

(2)

(Not currently used)

(Note: values 3005-3006 not needed when IHTOPT=1)

3005

TSINK

Temperature of outer heat sink (K)

3006

HSINK

Heat transfer coefficient-wall to outer heat sink (Watts/cm2-K)

3007

QCONST

Energy generation constant. The energy generation rate for fuel radial segment I, axial segment J at time K is calculated as QCONST*QR(I)*QAX(J)*QT(K). The method of dimensioning and normalizing the four factors is arbitrary so long as the product dimension is in Watts/gm.

3008

BURNUP

Peak fuel burnup (atom %)

3009-3010

(2)

(Not currently used)

3011-3410

QR(I,J)

(20,20 NDZ)

Radial profile of energy generation rate. Values are required for NDRF radial elements in each axial segment. (See: QCONST for units) (Loc. Numb. = 3010+I+20*(J-1))

3411-3430

QAX

(20 NDZ)

Axial profile of energy generation rate (see: QCONST)

3431-3455

TQT

(25 NQ)

Time values in power vs. time table (sec)

3456-3480

QT

(25 NQ)

Power values in power vs. time table (see: QCONST)

(Note: values 3481-3580 not needed when IHTOPT=1)

3481-3505

TTIN

(25 NCOOLT)

Time values in coolant inlet temperature versus time table (sec)

3506-3530

TIN

(25 NCOOLT)

Coolant inlet temperatures (K)

3531-3555

TCFIN

(25 NCOOLF)

Time values in coolant inlet flow versus time table (sec)

3556-3580

CFIN

(25 NCOOLF)

Coolant inlet flow (gm/cm2-sec)

3581-4000

(420)

(Not currently used)

*** COMMON /MISCIN/ ***

Location Number

Name

Value

Description

4001

GBFRAC

Fraction of FISGAS on grain boundaries (default value is 0.1)

4002-4039

(38)

(Not currently used)

4040

GASCON

Gas constant for central cavity gases (Bars-cc/gm-K)

4041-4060

CLDFLU

(20 NDZ)

Axial profile of clad fluence used in subroutine CFAIL (1022 n/cm2)

4061-4080

CLDHRD

(20 NDZ)

Pre-transient hardness parameter used in clad flow stress calculation. (Default value is 0.223, the value appropriate to 20% CW unirradiated stainless steel.)

4081-4480

PORES(I,J)

(20,20 NDZ)

Dist. of total fuel porosity same radial grid as QR (do not include crack volume input)

(Loc. Numb. = 4080+I+20*(J-1))

4481-4880

FISGAS(I,J)

(20,20 NDZ)

Dist. of fission gas (gm/cc) in fuel closed porosity and in solution on same radial grid as QR (do not include fission gas in open porosity). (Loc. Numb. = 4480+I+20*(J-1))

4881-5000

(120)

(Not currently used)

5001-5400

FRACPU(I,J)

(20,20 NDZ)

Dist. of plutonium (wt. %) in fuel (required when IFTYPE=4) (Loc. Numb. = 5000+I+20*(J-1))

5401-5800

FRACZR(I,J)

(20,20 NDZ)

Dist. of zirconium (wt. %) in fuel (required when IFTYPE=4) (Loc. Numb. = 5400+I+20*(J-1))

5801-6000

(200)

(Not currently used).

*** COMMON TEMPIN/ ***

Location Number

Name

Value

Description

(Note: values 6001-7050 are only required when IHTOPT=1)

6001-6050

TVALUE

(50 NCLADT)

Time values for clad surface temperature table (include TZERO) (sec)

6051-7050

TCSURF(J,K)

(20 NDZ,50 NCLADT)

Clad outer surface temperature for axial segment J at time K (K) (Loc. Numb. = 6050+J+20*(K-1))

*** Location Numbers for Debug Data ***

(All debug options default to zero. Therefore, input is required only if a debug option is to be used. Debug input is identified with location numbers larger than 9000 and it follows same conventions as regular input. Integer debug data is read in with regular integer data and decimal debug data is read in with regular decimal data.)

*** COMMON /DEBUGI/ — Integer Debug Data ***

Location Number

Name

Value

Description

9001

IDBOUT

=0

No debug output

=1

Add debug output to regular IOUTSW=2 output

9002

IDBSTP

=0

Program stops when molten cavity freezes

=1

Ignore this program stop

9003

IDBFPL

=0

Use recommended fuel flow stress Eq. (11.3-16)

=1

Use simple power law fuel creep: \(\dot{\varepsilon} = C_{0}*\sigma_{\text{e}}^{C_{1}}\) (Decimal values 9001 and 9002 are required)

9004

IDBFDV

=0

Use recommended fuel swelling - hotpressing (fuel swelling option for metal fuel is the simple grain boundary swelling model (ANL-IFR-27) & (ANL/RAS 83-33)

=1

Use equilibrium swelling model (ANL-IFR-6 and -23)

=2

Use simple power law fuel swelling: \(\dot{\varepsilon} = C_{0}*\sigma_{\text{m}}^{C_{1}}\) (Decimal values 9003 & 9004 are required)

9005

IDBCPL

=0

Use recommended clad flow stress

=1

Use ideal plastic flow for clad: \(\sigma_{\text{y}} = C_{0} + C_{1}{\overline{\varepsilon}}^{p}\) (Decimal values 9005 & 9006 are required)

=2

Use high-temperature power-law creep

=3

Use simple power law clad creep: \(\dot{\varepsilon} = C_{0}*\sigma_{\text{e}}^{C_{1}}\) (Decimal values 9005 & 9006 are required)

9006

IGPRES

=0

Open fuel-clad gap pressure and plenum pressure remain at input values

=1

User supplied gap pressure. This option allows FPIN2 to calculate pressure transients in gas pressurized cladding tubes. Plenum pressure is set equal to \(p_{\text{gap}}\).

9007

IGAPCL

=0

Use fuel-clad opening/closure model

=1

Fuel-clad gap always closed

9008

ICPROP

=0

Use material property correlations

=1

Use temperature independent material properties

9009

ISKIPM

=0

Perform complete thermal/mechanical calculations

=1

Bypass mechanical calculation, heat trans. only

9010

IGCLOS

=0

Use gap closure routine at 100% fuel melting

=1

Do not close gap (if open) at 100% fuel melting

9011-9050

(40)

(Not currently used)

9051

NGPRES

(25)

No. of values in \(p_{\text{gap}}\) vs. time table

9052-9100

(49)

(Not currently used)

9101-9120

IDBOTA

(20 NDZ)

Axial debug print vector (0=no prt, 1=prt)

9121-9140

IDBOTF

(20)

Fuel radial debug print vector

9141-9150

IDBOTC

(10)

Clad radial debug print vector

*** COMMON /DEBUGD/ — Decimal Debug Data ***

Location Number

Name

Value

Description

9001

FPLC0

Fuel power law creep constant \(C_{0}\)

9002

FPLC1

Fuel power law creep constant \(C_{1}\)

9003

FDVC0

Fuel power law swelling constant \(C_{0}\)

9004

FDVC1

Fuel power law swelling constant \(C_{1}\)

9005

CIPLC0

Clad idealized flow stress constant \(C_{0}\)

9006

CIPLC1

Clad idealized flow stress constant \(C_{1}\)

9007

HTERR

Relative convergence criterion for heat transfer calculation (ND) “0.0005”

9008

EPSCAV

Rel. convergence criterion for cavity pressure (ND) “0.001”

9009

EPSFE

Rel. convergence criterion for finite element analysis (ND) “0.0005”

9010

EPTEST

Rel. convergence criterion for plastic-creep strains (ND) “0.0005”

9011

EVTEST

Rel. convergence criterion for swelling strains (ND) “0.0005”

9012-9020

(9)

(Not currently used)

9021-9045

TGPRES

(25 NG PRES)

Time values in \(p_{\text{gap}}\) table

9046-9070

GPRES

(25 NGPRES)

Fuel-cladding gap pressure in \(p_{\text{gap}}\) table