In the previous section expressions for the axial distribution In the previous section expressions for the axial distribution
of coolant temperature have been derived. It has been of coolant temperature have been derived. It has been
shown that the axial distribution of coolant temperature shown that the axial distribution of coolant temperature
varies with the shape of the axial heat flux distribution.
varies with the shape of the axial heat flux distribution.
In particular, substituting Eqs. (VIII.3) and (VIII.4) into In particular, substituting Eqs. (VIII.3) and (VIII.4) into
(IX.12) gives the following expression for the temperature (IX.12) gives the following expression for the temperature
of the clad outer surface:
of the clad outer surface:
TTC0C0(z) = (q”(z) = (q”00.P.PHH.H) /(π.W.Cp) x [sin(πz /H) + sin(πH .H) /(π.W.Cp) x [sin(πz /H) + sin(πH /2H)] + q”0 /h.cos (πz / H) + T
/2H)] + q”0 /h.cos (πz / H) + Tlbilbi (X.1) (X.1)
77
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Figure X.2: Repartition of the temperature across the fuel rod.
Figure X.2: Repartition of the temperature across the fuel rod.
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Figure X.3: Represents the temperature of the cladding outer surface as Figure X.3: Represents the temperature of the cladding outer surface as function of axial distance.
function of axial distance.
79
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
It should be noted that the temperature of the clad outer It should be noted that the temperature of the clad outer
surface gets i/ H)ts maximum value
surface gets i/ H)ts maximum value TCo,max TCo,max at a certain at a certain location
location zCo,maxzCo,max. This location can be found from Eq. (X.1) . This location can be found from Eq. (X.1) using the following condition:
using the following condition:
dTdTC0C0(z) /dz = 0(z) /dz = 0 (X.2)(X.2)
It is convenient to represent the cladding outer temperature It is convenient to represent the cladding outer temperature
as:as:
TTC0C0 (z) = A + Bsin (πz /H) + C (z) = A + Bsin (πz /H) + CC0C0.cos(πz /H).cos(πz /H) (X.3)(X.3) Where,
Where,
A = Bsin(πz / 2H) + T A = Bsin(πz / 2H) + Tlbilbi B = (q”
B = (q”00.PH.H) / (π.W.Cp).PH.H) / (π.W.Cp) (X.4)(X.4) CCC0C0 = q” = q”00 / h / h
By combining the equations (X.2) & (X.3), we have:
By combining the equations (X.2) & (X.3), we have:
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Bcos(πz
Bcos(πzCOCO,Max / H) - C,Max / H) - CC0C0sin(πzsin(πzCo,Max Co,Max /H) = 0 (X.5)/H) = 0 (X.5) Which is equivalent to the following equation:
Which is equivalent to the following equation:
Tan(πz
Tan(πzCo,Max Co,Max / H)/ H) Co Co = B / C = B / C (X.6)(X.6) Thus:
Thus:
zzCo,Max Co,Max = (H/π)arctan(B /C= (H/π)arctan(B /CC0C0)) (X.7)(X.7)
It should be noted that a physically meaningful solution of the It should be noted that a physically meaningful solution of the
above equation should be positive and less than H.
above equation should be positive and less than H.
Noting that:
Noting that:
sin(πz
sin(πzCo,Max Co,Max /H) = ± tan(πz/H) = ± tan(πzC0C0,Max /H) / (1 + tan,Max /H) / (1 + tan22(πz(πzCoCo,Max ,Max /H))/H))1/21/2
= ± ((B/C= ± ((B/CC0C0) / (1 + (B /C) / (1 + (B /CC0C0))22))1/21/2 andand
cos (πz
cos (πzC0,Max C0,Max / H) = ± ( 1/ (1 + tan2(πz/ H) = ± ( 1/ (1 + tan2(πzC0,Max C0,Max / H)/ H) = ± (1 = ± (1 / (1 + (B /C / (1 + (B /CC0C0) )22) )1/21/2
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THERMAL-HYDRAULIC IN NUCLEAR REACTOR
The maximum temperature of the cladding outer surface The maximum temperature of the cladding outer surface
becomes (taking only + sign ):
becomes (taking only + sign ):
TTC0,Max C0,Max = A + (B= A + (B22 + C + CC0C022))1/21/2 (X.8)(X.8) Using constants
Using constants AA, , B B and and CCCoCo given by Eq. (X.4), the maximum given by Eq. (X.4), the maximum clad outer temperature is obtained as:
clad outer temperature is obtained as:
TTC0,Max C0,Max = (q”= (q”00.P.PHH.H) / π.W.Cp)sin(πH / 2H) + T.H) / π.W.Cp)sin(πH / 2H) + Tlbilbi + + ((q”((q”00.P.PHH.H /π.W.Cp).H /π.W.Cp)22 + (q”0 /h)2) + (q”0 /h)2)1/21/2
(X.9) (X.9) oror
(π.W.Cp(T
(π.W.Cp(TC0C0,Max - T,Max - Tlbilbi)) / q”)) / q”00.P.PHH.H = sin(πH /2H) + ((1 + .H = sin(πH /2H) + ((1 + (π.W.Cp) / P
(π.W.Cp) / PHH.H.h)2).H.h)2)1/21/2
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Since the clad maximum temperature is located on the inner Since the clad maximum temperature is located on the inner
surface, it is of interest to find it as well. The axial surface, it is of interest to find it as well. The axial
distribution of the clad inner temperature can be obtained distribution of the clad inner temperature can be obtained
from Eqs. (IX.11) and X.1) as:
from Eqs. (IX.11) and X.1) as:
• TTCiCi = ΔT = ΔTCC + T + TC0C0(z)(z)
= (q’/2λ= (q’/2λCC)lnr)lnrC0C0/r/rCi Ci + (q” + (q”00.P.PHH.H / π.W.Cp).[sin(πz /H) + .H / π.W.Cp).[sin(πz /H) + sin (πH 2/H] + (q"
sin (πH 2/H] + (q"00 /h).cos(πz /h) + T /h).cos(πz /h) + Tlbilbi
= (q”= (q”00.P.PHH.H / π.W.Cp).[sin(πz /H) + sin (πH /2H] + .H / π.W.Cp).[sin(πz /H) + sin (πH /2H] + q”0(rC0/λ
q”0(rC0/λCC)ln(rC0/r)ln(rC0/rCiCi) +1/h)cos(πz/H) + T) +1/h)cos(πz/H) + Tlbi lbi
(X.10)(X.10)
Equation (X.10) ca,n be expressed:
Equation (X.10) ca,n be expressed:
TTCiCi(z) = A + Bsin(πz / H) + C(z) = A + Bsin(πz / H) + CCiCicos(πz /H)cos(πz /H) (X.11)(X.11) 83
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Where A & B are given by equation (X.3) and:
Where A & B are given by equation (X.3) and:
CCCiCi = q” = q”00(r(rC0C0/λ/λCC)ln(r)ln(rC0C0/r/rCiCi +1/h) +1/h) (X.12)(X.12)
Using the same approach as in the case of the clad outer Using the same approach as in the case of the clad outer
temperature, the location of the maximum temperature on temperature, the location of the maximum temperature on
the clad inner surface is found as:
the clad inner surface is found as:
zzCi,Max Ci,Max = (H/π)arctyan(B/C= (H/π)arctyan(B/CCiCi)) (X.13)(X.13) and the maximum corresponding temperature is:
and the maximum corresponding temperature is:
TTCi,Max Ci,Max = (q”= (q”00.P.PHH.H) / (π.H) / (π.W.Cp)sin(πH/2H) + T.W.Cp)sin(πH/2H) + Tlbilbi + +
((q”((q”00.P.PHH.H)/(π.W.Cp).H)/(π.W.Cp)22 + [q” + [q”00(r(rC0C0/λ/λCC)ln(r)ln(rC0C0/r/rCiCi) + ) + 1/h]2)
1/h]2)1/2 1/2 (X.14)(X.14)
In a similar manner the fuel maximum temperature at the In a similar manner the fuel maximum temperature at the
center of the fuel pellet is given by:
center of the fuel pellet is given by:
TTFcFc(z) = A + Bsin (πz/H) + C(z) = A + Bsin (πz/H) + CFcFccos(πz/H)cos(πz/H) (X.15)(X.15)
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
where where
CCFCFC = q” [(r = q” [(rC00C00/λc)ln(r/λc)ln(rC0C0/r/rCiCi) + (r) + (rC0C0/λ/λGG)ln(r)ln(rG0G0/r/rGiGi) + ) + rrCoCo/2λ/2λFF + 1/h (X.16) + 1/h (X.16)
The maximum fuel temperature is located at:
The maximum fuel temperature is located at:
zzFc,Max Fc,Max = (H/π)arctan(B/C= (H/π)arctan(B/CFcFc)) (X.17)(X.17) and its value is:
and its value is:
TTFC,Max FC,Max = (q”= (q”00.P.PHH.H/π.W.Cp).sin (πH/2H) + T.H/π.W.Cp).sin (πH/2H) + Tlbilbi + +