Lecture_5.pdf

NUCE 402: Introduction to Nuclear System and Operation Chapter 3.1 (Heat Removal from Nuclear Reactors – Convection) Dr. Ahmed Alkaabi Previous Lectures  Conduction  Fourier’s law q  kT  From energy balance...

NUCE 402: Introduction to Nuclear System and Operation Chapter 3.1 (Heat Removal from Nuclear Reactors – Convection) Dr. Ahmed Alkaabi Previous Lectures  Conduction  Fourier’s law q  kT  From energy balance q Steady state heat  2T  0 k conduction equation  Solutions d 2T q q 2  0 T  x  Tm  Plate-type dx 2 kf 2k f d 2T xa 0 T  Ts  (Ts  Tc ) dx 2 b Tm  Tc a b q  Tm  Tc  q(  ) a  b 2k f A k c A  Cylindrical 2k f A k c A 1 Tm  Ts  q( ) 4k f H T T Tm ( z )  Tc ( z ) q m c q( z )  ln( 1  b / a ) R f  Rc generalization 2 (a  b) H ( R f  Rc ) Ts  Tc  q kc 2H 2 Convection  Newton’s law of cooling  Transfer of heat from a heated solid to a moving fluid q  h(Tc  Tb ) Heat flux Heat transfer coefficient Btu/hr-ft2 Btu/hr-ft2-oF  Heat transfer coefficients  Function of liquid, flow types, coolant temp., etc.  Typical values in power reactor  Light and heavy water: 5,000~8,000 Btu/hr-ft2-oF 25~45 kW/m2-K  Gases: 10~100 Btu/hr-ft2-oF 0.055~0.55 kW/m2-K  Liquid sodium: 4,000~50,000 Btu/hr-ft2-oF 20~300 kW/m2-K Total rate of heat flow across A 1 q  qA  hA(Tc  Tb ) Introduce the thermal resistance, Rh Rh  hA (Tc  Tb ) q Rh 3 Thermal Resistance  Total thermal resistance  For plate-type fuel elements Tm  Tc Tm  Tb q q a b a b 1    Tb 2k f A k c A 2k f A kc A hA a b 1 then R    2k f A kc A hA  For clad cylindrical fuel elements Tm  Tb Tm  Tc Tm  Tb q q q R 1 ln( 1  b / a ) 1 ln( 1  b / a) 1    4k f H 2kc H 4k f H 2kc H hA 1 ln( 1  b / a) 1 then R    where A  2 (a  b) H 4k f H 2kc H hA  For space dependent heat source T ( z )  Tb ( z ) Tm ( z )  Tb ( z ) q( z )  m q( z )  R 2 (a  b) H ( R f  Rc  Rh ) Tc ( z )  Tb ( z ) q( z )  q( z)  h(Tc ( z)  Tb ( z)) 2 (a  b) HRh 4 Temperature along the Coolant Channel  Coolant channel  A coolant volume associated with a single fuel rod  Heat generated within the fuel is Ac transferred  Coolant temperature will rise  Consider a coolant slab of t = dz Volume of coolant: Ac dz Ac : c  x area of fuel channel Mass of coolant: Ac dz Amount of heat needed to increase dTb: Ac dz c p dTb 1 Rate of heat addition to increase dTb = dq: Ac dz c p dTb dt Therefore, Ac c p dTb dq  c p dTb  qAf dz  => Coolant flow rate z where, q  qmax  cos( ) for fuel rod at the center H 5  Temperature along the Coolant Channel  For the central fuel rod  Bulk coolant temperature qA f dq  c p dTb  qAf dz dTb  dz z c p where, q  qmax  cos( ) H Tb  Tb  Tb 0 Tb q’’’ z  A f qmax z  cos( )dz  H / 2 c H p Vf Therefore, the maximum coolant temp.  A f H  qmax z  Tb  Tb 0  1  sin( )  V f 2qmax c p   H  Tb ,max  Tb 0  Cc c p Tb0  Cladding temperature Ac 1 dq dq  qAf dz  qdAc  qCc dz  h(Tc  Tb )Cc dz Tc  Tb  Substituting Tb and the integration hCc dz  V f  qmax z  qmax  A f z qA f Tc  Tb 0  1  sin( )   cos( )  c p  H  hCc H hCc  A f qmax z  V f qmax z z Tc  Tb  cos( )   V f cos( ) cos( )  Rh qmax hCc H hA H H 6 Temperature along the Coolant Channel z z rewrite  V f cos( Tc  Tb  Rh qmax  A f H cos( )  Rh qmax ) H H  For fuel rod central temp. =q Tm  Tb z  A f H cos( )  qmax Substituting Tb and the integration R H Total thermal resistance  V f  qmax z  z Tm  Tb 0  1  sin( )   q  V R cos( ) c p  max f H  H  Factors affecting maximum fuel temp.  Maximum heat generation at mid-point  Largest heat transfer q’’’  Slow decreasing away from the mid-point  Rising bulk temperature along z  Tm must increase to provide q” => Maximum Tm occurs above the mid-point 7 Temperature along the Coolant Channel  Locations of maximum temp. H zc ,max  cot 1 (c p Rh )  H z m ,max  cot 1 (c p R )   Values of maximum temp. 1  1  2   V f Rh  Tc ,max  Tb 0  qmax    c p Rh    1  1  2   V f R  Tm ,max  Tb 0  qmax    c p R    Limitations: - qualitative simple power distribution constant k and h does not account for boiling - not applicable to liquid metal coolants  If thermal resistance is small, the maximum temp. can be lowered  increase conductivity of fuel/cladding  increase heat transfer coefficient of coolant 8 Heat Transfer Coefficients  Heat removal by coolant  Conduction  When every portion of fluid moves parallel to the channel => Laminar flow  No net motion of fluid in radial direction => conduction dominate  Convection  When there are significant velocity fluctuation in radial direction => Turbulent flow  Heat carried away from the wall to the bulk of liquid => convection dominate  In NPP  Coolant is pumped  Forced convection => turbulent flow  More or less uniform bulk temperature  Rapid drop of coolant temperature in the vicinity of fuel 9  Heat Transfer Coefficients  Reynolds number  Useful in characterizing the flow of a fluid  Dimensionless parameter De Re  Fluid viscosity  lb/hr-ft, or poise Equivalent diameter (g/cm-sec) c  x area of coolant channel De  4  wetted perimeter of coolant channel s 2  a 2 s 2  a 2  Condition for turbulent (typical) De  4 2 2a a  Laminar: Re < 2000  Mixture: 2000 < Re < 10000 High Re => more turbulent => large h  Turbulent: Re > 10000 => high rate of heat transfer to coolant  Values of heat transfer coefficients  Determined by experiments  Provided as experimental correlations  Need other dimensionless parameters hDe cp Nusselt number Nu  Prandtl number Pr  k k 10  Heat Transfer Coefficients  For flow through long straight channel under turbulent conditions k Nu  C Re Pr m n h( )C Re m Pr n De  For long, straight, circular tube Limitations: - valid for only for the reference k h  0.023( ) Re 0.8 Pr 0.4 temperatures De - not applicable when conduction become non-negligible Dittus-Boelter Eq. - significant error if channel deviate significantly from circular shape Liq. Na: 46.4 Btu/hr-ft-oF  For liquid metals Water: 0.381 Btu/hr-ft-oF  Large conductivity He: 0.115 Btu/hr-ft-oF  Less steep temp. gradient in coolant Correlation for Liq. Metal through a hexagonal lattice of rods Nu  0.66  3.126(s / d )  1.184(s / d ) 2  0.0155(Pe)0.86 Valid for s/d > 1.35 Ratio of lattice pitch Pe  Re Pr to rod diameter Peclet number 11 Summary  Newton’s law of cooling q  h(Tc  Tb )  Total thermal resistance Tb a b 1 R   2k f A kc A hA Tm  Tb q 1 ln( 1  b / a) 1 R R   4k f H 2kc H hA  Temp. along the channel z  V f   V f cos( Tc  Tb  Rh qmax ) qmax z  H Tb  Tb 0  1  sin( ) c p  H  z  V f cos( ) Tm  Tb  Rqmax H  Heat transfer coefficients  Dimensionless numbers  High Re value for turbulent flow k h  0.023( ) Re 0.8 Pr 0.4 D  hDe cp De Re  e Nu  Pr   k k Dittus-Boelter Eq. 12