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.

(A11.1‑1)

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

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

(A11.1‑2)

\[d = \frac{b}{r} - 1\]

(A11.1‑3)

\[e = 1 - \frac{a}{r}\]

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

(A11.1‑4)

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

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. A11.1-1 and using the symmetry of [C] yields

(A11.1‑5)

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

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

(A11.1‑6)

\[I \equiv \int_{\text{A}_{\text{c}}}{\text{dA} = \pi l \left( a + b \right)}\]

(A11.1‑7)

\[I_{\text{d}} \equiv \int_{\text{A}_{\text{c}}}{\left( d \right) \text{dA} = \pi l^{2}}\]

(A11.1‑8)

\[I_{\text{d}2} \equiv \int_{\text{A}_{\text{c}}}{\left( d^{2} \right) \text{dA} = \pi l \left\lbrack \frac{2b^{2}}{l} \ln\left( \frac{b}{a} \right) - 3b + a \right\rbrack}\]

(A11.1‑9)

\[I_{\text{e}} \equiv \int_{\text{A}_{\text{c}}}{\left( e \right) \text{dA} = \pi l^{2}}\]

(A11.1‑10)

\[I_{\text{e}2} \equiv \int_{\text{A}_{\text{c}}}{\left( e^{2} \right) \text{dA} = \pi l \left\lbrack b - 3a + \frac{2a^{2}}{l} \ln\left( \frac{b}{a} \right) \right\rbrack}\]

(A11.1‑11)

\[I_{\text{ed}} \equiv \int_{\text{A}_{\text{c}}}{\left( \text{ed} \right) \text{dA} = \pi l \left\lbrack a + b - \frac{2ab}{l} \ln\left( \frac{b}{a} \right) \right\rbrack}\]

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

(A11.1‑12)

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

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

(A11.1‑13)

\[t_{\text{i}} = \int_{\text{C}} \sigma_{\text{r}} N_{\text{i}} {\hat{i}}_{\text{r}} \cdot \hat{n}\text{dC}\]

(i=1,2). The shape functions \(N_{1}\) and \(N_{2}\) are defined in Eqs. 11.2-63 and 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

(A11.1‑14)

\[\sigma_{\text{r}} = - p_{\text{cav}}\]

(A11.1‑15)

\[{\hat{i}}_{\text{r}} \cdot \hat{n} = {\hat{i}}_{\text{r}} \cdot \left( - {\hat{i}}_{\text{r}} \right) = - 1\]

(A11.1‑16)

\[N_{1} = 1\]

(A11.1‑17)

\[N_{2} = 0\]

(A11.1‑18)

\[\int_{\text{C}}{\text{dC} = 2 \pi r_{\text{cav}}}\]

Hence,

(A11.1‑19)

\[t_{1} = 2 \pi p_{\text{cav}} r_{\text{cav}}\]

(A11.1‑20)

\[t_{2} = 0\]

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

(A11.1‑21)

\[t_{1} = 0\]

(A11.1‑22)

\[t_{2} = 2 \pi p_{\text{cav}} r_{\text{out}}\]

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			(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			(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		No. of axial segments
33	NDRF		No. of radial elements in cladding (minimum value is 6)
34	NDRC		No. of radial elements in cladding (minimum value is 3)
35-40			(Not currently used)
41	NQ		No. of values in power vs. time table
42	NCOOLT		No. of values in coolant inlet temperature versus time table
43	NCOOLF		No. of values in coolant inlet flow versus time table
44	NCLADT		No. of values in clad outer surface temperature versus time table
45-100			(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			(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			(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			(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			(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			(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			(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			(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			(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			(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			(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			(Not currently used)
9051	NGPRES		No. of values in \(p_{\text{gap}}\) vs. time table
9052-9100			(Not currently used)
9101-9120	IDBOTA	(20 NDZ)	Axial debug print vector (0=no prt, 1=prt)
9121-9140	IDBOTF		Fuel radial debug print vector
9141-9150	IDBOTC		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			(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