The transfer of heat through any material is governed by the heat balance equation:

qc dT/dt = к V2T + H (1.18)

where H is the heat production term. This, like Eq. (1.3), is also time and space dependent.

Fig. 1.6. Schematic representation of the thermal distribution through a cross section of a cylindrical fuel pin.

Considering a cylindrical fuel pin in which heat is produced (Fig. 1.6), Eq. (1.18) can be used to derive the average fuel temperature in the follow­ing way. In the radially symmetrical pin

qc dT/dt = аф + k[(d2T/dr2) + r-‘(dT/dr)]

and by defining a volume-averaged fuel temperature and volume-averaged flux (or power) as

T = (2л/A) f Tr dr and ф = (2л/A) f фг dr (1.20)

Jo Jo

where

A = 2л r dr

J 0

Equation (1.18) becomes

Aoc(df/dt) = афА + 2лкЯ(дТ/дг)лШ (1.21)

By now assuming that

(дТ/дг) ]atiJ = (TBUl[~ Tc)/A

we have

me df/dt = а’ф — (2л Rk/A)(TBai{ — Tc)

and defining the heat-transfer coefficient h = 2лЯк/A, the average fuel temperature is given by

Now the heat-transfer coefficient h’ must include not only an allowance h for the film drop heat transfer between the surface temperature reurf and the coolant temperature Te as before, but also an allowance for conduction between the point of average fuel temperature T and the surface of the fuel. So

1 /К = (1 /h) + (/%лк) (1.25)

This adjustment is made so that Eq. (1.24) enables Г to be directly calculated whereas using Eq. (1.23) would have required an additional equation to obtain ГзигГ from f and Tc. Thus for the average fuel temperature, the equation becomes (Fig 1.7)

m[C{(dTf/dt) = аф— h^Tt — Tc) (1-26)

In a similar manner the average coolant temperature equation becomes

mcec(dTc/dt) = ht(Tt — Tc) + hB(Ts — Tc) — mcecvc(dTJdz) (1.27)

Fig. 1.7. Thermal distribution through a cross section of a cylindrical fuel pin and cladding.

where the last term accounts for the transfer of heat as the fluid moves down the channel with velocity v0. Average temperatures of any other material such as structural components in the core can be established in an exactly similar manner; i. e.,

mBc3(dTJdt) = Ьф — hs(Ts — T0) (1.28)

where in this case Ьф accounts for any direct у heating of the structure. Equation (1.27) accounts for heat deposited in the coolant from the fuel element h((Tt — Tc) and from the structural material hs{Ts — Tc).

1.3.1.1 Steady-State Temperatures

With all time derivative terms set to zero, by adding the steady-state temperatures of this system (fuel, coolant, and structure) we can obtain

Tc = Tc. m + [(a + b) J*Ф dz]lmcccvc	(1.30)
Tf = Tc + аф/hf	(1.31)
TS = TC + Ьф/hg	(1.32)

Thus one can obtain a complete distribution of temperatures as a function of z if the heat input distribution ф(г) and the coolant inlet temperature Tc. are known.

It is worth noting that the inlet temperature is effectively a base tempera­ture for the system. Any increase in coolant inlet temperature is reflected by an identical increase in all the other temperatures in the steady state—if the power is retained constant. In practice, we shall see in the next section that, in the absence of external controls, the power ф will also change as a function of temperature because of feedback reactivity changes. However this change of power is a secondary effect.