Modern nuclear power reactors contain cylindrical fuel Modern nuclear power reactors contain cylindrical fuel elements that are composed of ceramic fuel pellets located elements that are composed of ceramic fuel pellets located in metallic tubes (so-called cladding). A cross-section over in metallic tubes (so-called cladding). A cross-section over a square lattice of fuel rods is shown in FIGURE 4-10. For a square lattice of fuel rods is shown in FIGURE 4-10. For thermal analyses it is convenient to subdivide the fuel rod thermal analyses it is convenient to subdivide the fuel rod assembly into sub-channels. The division can be performed assembly into sub-channels. The division can be performed in several ways; however, most obvious choices are so- in several ways; however, most obvious choices are so- called coolant centered sub-channels and rod-centered sub- called coolant centered sub-channels and rod-centered sub- channels. Both types of sub-channels are equivalent in channels. Both types of sub-channels are equivalent in terms of major parameters such as the flow cross-section terms of major parameters such as the flow cross-section area, the hydraulic diameter, the wetted perimeter and the area, the hydraulic diameter, the wetted perimeter and the heated perimeter. In continuation, the thermal analysis will heated perimeter. In continuation, the thermal analysis will
be performed for a single sub-channel.
be performed for a single sub-channel.
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THERMAL-HYDRAULIC IN NUCLEAR REACTOR
The stationary (time independent) heat conduction equation The stationary (time independent) heat conduction equation
for an infinite cylindrical fuel pin, in which the axial heat for an infinite cylindrical fuel pin, in which the axial heat
conduction can be ignored is as follows:
conduction can be ignored is as follows:
q” = (- 1/r).d/dr[λ
q” = (- 1/r).d/dr[λFFr.dTr.dTFF /dr] /dr] (IX.1)(IX.1) where
where F T F T is the fuel temperature, [K], is the fuel temperature, [K], F F l is the thermal l is the thermal conductivity of the fuel material, [W m-1 K-1],
conductivity of the fuel material, [W m-1 K-1], q q ¢ is the ¢ is the density of heat sources, [W m-3] and
density of heat sources, [W m-3] and r r is the radial is the radial
distance. Here the angular dependence of the temperature distance. Here the angular dependence of the temperature
is omitted due to the assumed axial symmetry of the is omitted due to the assumed axial symmetry of the
temperature distribution.
temperature distribution.
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Assuming that q”’ is constant a cross-section equation (IX.1) Assuming that q”’ is constant a cross-section equation (IX.1)
could be integrated to obtain:
could be integrated to obtain:
λλFFrdTrdTFF / dr = - r2 /2q”’ / dr = - r2 /2q”’ (IX.2)(IX.2)
If the fuel conductivity was constant, Eq. (IX.2) could be If the fuel conductivity was constant, Eq. (IX.2) could be
integrated and the temperature distribution would be integrated and the temperature distribution would be
obtained. However in typical fuel materials the fuel thermal obtained. However in typical fuel materials the fuel thermal
conductivity strongly depends on the temperature and this conductivity strongly depends on the temperature and this
is the reason why the temperature distribution can not be is the reason why the temperature distribution can not be
found from Eq. (IX.2) in a general analytical form. Instead, found from Eq. (IX.2) in a general analytical form. Instead,
Eq. (IX.2) is transformed and integrated as follows:
Eq. (IX.2) is transformed and integrated as follows:
λλFFrdTrdTFF = - r2 /2q”’dr ƒ = - r2 /2q”’dr ƒ→→ TFcTFcTF0TF0λλFFdTdTFF = - q” /2ƒ = - q” /2ƒ00TF0TF0rdr = rdr = (- r(- r22F0F0 /4)q”’ (IX.3) /4)q”’ (IX.3)
where the integration on the left-hand-side is carried out from where the integration on the left-hand-side is carried out from
the temperature at the centerline,
the temperature at the centerline, TFcTFc, to the temperature , to the temperature on the fuel pellet surface
on the fuel pellet surface TFoTFo==TF TF ((rForFo). Defining the ). Defining the average fuel conductivity as:
average fuel conductivity as: 7171
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
λλFF = 1 / (T = 1 / (TFcFc – T – Tfofo)ƒ)ƒTFoTFoTFcTFcλλFFdTdTFF (IX.4)(IX.4)
The temperature drop across the fuel pellet can be found as The temperature drop across the fuel pellet can be found as
follows:
follows:
ΔTΔTFF = T = TFcFc - T - TFoFo = q”’r = q”’r22FoFo /4λ /4λFF (IX.5)(IX.5)
In the thermal analysis of reactor cores, the power is often In the thermal analysis of reactor cores, the power is often
expressed in terms of the linear power density, that is, the expressed in terms of the linear power density, that is, the
power generated per unit length of the fuel element:
power generated per unit length of the fuel element:
Q’ = πr
Q’ = πr22FoFoq”’q”’ (IX.6)(IX.6) By combining equations ( IX.5) &( IX.-6), we have:
By combining equations ( IX.5) &( IX.-6), we have:
ΔTΔTFF = q’ /4π.λ = q’ /4π.λFF (IX.7)(IX.7)
Equation (IX.7) reveals that the fuel temperature drop is a Equation (IX.7) reveals that the fuel temperature drop is a
function of the linear power density and the averaged fuel function of the linear power density and the averaged fuel
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THERMAL-HYDRAULIC IN NUCLEAR REACTOR
In a similar manner the temperature drop across the gas gap In a similar manner the temperature drop across the gas gap
can be obtained. In particular, Eq. (IX.1) can be used to can be obtained. In particular, Eq. (IX.1) can be used to
describe the temperature distribution in the gas gap, describe the temperature distribution in the gas gap,
however, unlike for the fuel pellet, the heat source term is however, unlike for the fuel pellet, the heat source term is
equal to zero and the gas thermal conductivity can be equal to zero and the gas thermal conductivity can be
assumed constant, thus:
assumed constant, thus:
-1/2 (d/dr)λ
-1/2 (d/dr)λGGr(dTr(dTGG/dr) = 0 T/dr) = 0 T→→ GG(r) = C(r) = C11 /λ /λGGlnr +Clnr +C22
(IX.8) (IX.8)
The integration constant C, can be found from condition of The integration constant C, can be found from condition of
the heat flux continuity at r = r the heat flux continuity at r = rFoFo::
-λ-λGG(dT(dTGG/dr) = -C1 / r/dr) = -C1 / rFoFo = q’ / 2πr = q’ / 2πrFoFo C C→→ 11 = -q’/2π = -q’/2π (IX.9) (IX.9)
And the temperature drop in the gap can be expressed as And the temperature drop in the gap can be expressed as
follows:
follows:
ΔTΔTGG = T = TGG(r(rGiGi) –TG(r) –TG(rGoGo) = q’/2πλ) = q’/2πλGGlnrGo/rlnrGo/rGiGi (IX.10)(IX.10)7373
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Equation (IX.10) is applicable to the clad material as well, since the Equation (IX.10) is applicable to the clad material as well, since the assumptions on the heat generation and the thermal conductance assumptions on the heat generation and the thermal conductance
are valid in this case as well. Substituting the proper dimensions are valid in this case as well. Substituting the proper dimensions
and property data yields, and property data yields,
ΔTΔTcc = T = Tcc(r(rcici) - Tc (r) - Tc (rcoco) = q’ /2πλ) = q’ /2πλccln rln rcoco/r/rcici (IX-11)(IX-11) where
where rCo rCo is the outer clad radius and _is the outer clad radius and _C C is the clad thermal is the clad thermal conductivity.
conductivity.
Heat transfer from the clad surface to the coolant is described by the Heat transfer from the clad surface to the coolant is described by the
following following Equation:
Equation:
q” = h(T
q” = h(Tcoco – T – Tlblb)) (IX.12)(IX.12) where
where h h is the convective heat-transfer coefficient. Taking is the convective heat-transfer coefficient. Taking intoaccount that
intoaccount that
( ) Co q( ) Co q¢¢ = q¢¢ = q¢ 2p ¢ 2p r r , the temperature drop in the coolant boundary , the temperature drop in the coolant boundary layer is found as:
layer is found as:
THERMAL-HYDRAULIC IN NUCLEAR REACTOR
the total temperature drop from the center of the fuel pellet the total temperature drop from the center of the fuel pellet
to coolant is expressed as follows:
to coolant is expressed as follows:
ΔT = ΔT
ΔT = ΔTFF + ΔT + ΔTGG + ΔT + ΔTcc + ΔT + ΔTll
= q’/2π [1/2λ= q’/2π [1/2λFF + 1/λ + 1/λGG lnr lnrGoGo/r/rGiGi + 1/λ + 1/λCClnrlnrCoCo/r/rCCi + i + 1/r1/rCoCoh] (IX.14)h] (IX.14)
The total temperature drop in a fuel rod cross-section is The total temperature drop in a fuel rod cross-section is
represented in following figure IX.2.
represented in following figure IX.2.
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THERMAL-HYDRAULIC IN NUCLEAR REACTOR
Figure IX.1: Cross section of a square fuel lattice.
Figure IX.1: Cross section of a square fuel lattice.
THERMAL-HYDRAULIC IN NUCLEAR REACTOR