Vishal Patel NETS 2015

Temperature Profile in Fuel and Tie-Tubes for Nuclear Thermal Propulsion Systems

•  Improved hot channel analysis of NTP core •  Take into account spatial geometry without

homogenization •  Historically simple methods performed

–  2D semi-analytical heat equation solve on single webbing wedge

–  Neglect tie-tubes –  Assume a heat flux out of tie-tubes

• CFD performed –  Not for LEU-NTP design

Problem Description

Geometry

•  Fuel  Element  •  Many  coolant  channels  

•  Moderator  Element  •  Downwards  H2  flow  path  •  Upwards  H2  flow  path  •  Insulator  and  structural  materials  

•  Clad  •  No  simple  symmetry  

Flow Model

•  Continuity  

•  Momentum  

•  Energy  

•  Friction  factor  

•  BC  

•  Closure  

Discretization and Solution Method

Flow Channel

Fuel Slice i-1 Fuel Slice i

•  First order forward difference for derivatives •  Fixed point iteration scheme to solve equations •  1st order accuracy expected

Heat Equation

•  Zero  heat  flux  boundaries  on  symmetry  planes  •  Constant  heat  source  in  fuel  only  •  Temperature  dependent  material  properties  for  fuel  if  non-­‐linear  

equation  solved  

)

Nu  

Simple Method

Sparrow,  E.M.  "Temperature  Distribution  and  Heat_Transfer  Results  for  an  internally  cooled,  Heat-­‐Generating  Solid,"  J.  Heat  Transfer,  vol.  82,  no.  4,  pp.  389-­‐392,  (Nov  1960).    

Linear heat equation solved on:

Same  flow  solution  and  coupling  scheme        No  tie-­‐tube  modeled  

• MATLAB PDE Toolbox •  2D heat equation solved using Galerkin FEM

–  Weighting functions and basis functions first order Lagrange

–  1st order triangle elements –  Midpoint rule to integrate

• Mesh generated based on geometry inputs • Adaptive mesh refinement

Finite Elements

Symmetries (Fuel to Moderator Ratios)

3  to  1   2  to  1   1  to  2   1  to  3  

•  Zero  flux  boundaries  used  on  black  triangle  boundaries  

Solution Method

Flow  Equations  

Heat  transfer  

coefficient  solution  

h  Heat  

Equation  

Guessed  q’’  

Initialize  

Convergence  Criteria:  Heat  transfer  coefficient  &  Wall  Temperatures  meet  a  specified  tolerance:    

where  I  is  the  iteration  number  

Convergence  Check  

•  Iterated  operator  splitting  scheme  implemented  

•  First order accuracy in flow solution implementation expected

Verification (1/4)

Verification (1/4)

1.7 1.8 1.9 2 2.1 2.2

−4.4

−4.2

−4

−3.8

−3.6

−3.4

log(Number of Nodes)

log

L 2 Erro

r

y = − 1*x − 1.8

Actual linear1st Order2nd Order

• Comparing using linear and non-linear heat equation –  Quantify (visually) error in appoximation

•  Increased mesh size should converge (in a mathematical sense)

Verification (2/4)

Verification (2/4)

0 2 4 6 8 10x 104

3370

3380

3390

3400

3410

3420

3430

3440

3450M

axim

um F

uel T

empe

ratu

re (K

)

Non−Linear PropertiesConstant Properties

Mesh  Size  (Number  of  Elements)  

• Nodalization scheme for flow channel should not affect the solution much

•  Solution defined here as maximum fuel temperature

Verification (3/4)

Verification (3/4)

30 40 50 60 70 80 902750

2752

2754

2756

2758

2760

2762

2764

2766

2768

Number of Axial Bins

Max

imum

Fue

l Tem

pera

ture

(K)

•  Relative error used as convergence criteria should not affect solution

Verification (4/4)

Verification (4/4)

10−10 10−8 10−6 10−4 10−2 1003340

3350

3360

3370

3380

3390

3400

3410

3420

3430

3440

Relative Error

Max

Fue

l Tem

p Fo

und

(K)

•  FEM/flow coupling compared to simple method using 2 to 1 geometry

• Average FEM/flow solution –  Simple temperature average for every coolant

channel node •  Simple method compares well on average

Comparing Solutions (1/2)

Simple vs FEM (Coolant Temperature)

−0.5 0 0.50

500

1000

1500

2000

2500

Fuel Length (m)

Hyd

roge

n Te

mpe

ratu

re (K

)

Simple SolutionNon−Linear PDELinear PDEMax PDE Coolant ChannelMin PDE Coolant Channel

Linear  PDE  Non-­‐Linear  PDE  

Max  PDE  Coolant  Channel  

Min  PDE  Coolant  Channel  

Simple  Solution  

• Maximum fuel temperature predicted compared

•  Simple solution under-predicts (bad)

Comparing Solutions (2/2)

Simple vs FEM (Fuel Temperature)

−0.5 0 0.5500

1000

1500

2000

2500

3000

Fuel Length (m)

Max

imum

Fue

l Tem

pera

ture

(K)

Simple SolutionNon−Linear PDELinear PDE

Simple  Solution  

Linear  PDE  

Non-­‐Linear  PDE  

Temperature Distribution

0.01 0.015 0.02 0.025 0.031700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

Fuel

Tem

pera

ture

Alo

ng L

ine

Sege

men

t (K)

Y axis (m)

• CFD to verify flow solutions •  Spatially dependent power distribution • Move to INL’s MOOSE framework •  Include cladding •  Find solutions to flatten temperature profile •  Use approximate symmetries to handle

different fuel to moderator configurations

Future Works

• Created new tool to analyze temperature distribution in NTRs with tie-tubes

• Discussed implementation details •  Showed previous 2D semi-analytical

approach to temperature distribution is not applicable

•  Presented sample result for temperature distribution and ideas to flatten it

Conclusions