Before considering the enlargement of the thermal model to take account of further spatial effects, there is one term which complicates the solution of the present set of Eqs. (1.26)—(1.28).

1.3.3.1 The Term dTJdz

The term dTJdz in Eq. (1.27) complicates the solution because of the second implicit independent variable z. We can evade the complication by averaging the coolant temperature and assuming a mean gradient. This is good if the flux distribution is sinusoidal. Thus the usual approximation is

dTJdz = (rCout — TCJ/L = 2(Го — TC. J/L (1.39)

where Tc is now a channel volume weighted mean.

However when temperatures are used to predict reactivity changes, whole reactor averages are required rather than component averages and these need to be weighted, according to perturbation theory, by the square of the flux ф (3). Thus

rav = J Тф2 dVH ф2 dV (1.40)

and for a sinusoidal flux this gives

dTJdz = 2.33 (Tav — TC. JL (1.41)

This is a more accurate value for feedback calculations and the average temperature here has a different meaning. The effect is a multiplicative constant, 1.16, which can easily be included in the model. However the fast reactor has a relatively flat flux and this weighting is less significant.

1.3.3.2 Spatial Representation

The present core thermal model is very coarse since it has only a single fuel and a single coolant temperature. The spatial representation can be improved by including more points in the fuel and the structure within the unit cell. This is done by adding further heat balance equations in each of which the temperature is the average of a particular volume.

wflcfl dTjdt = (1 — y)(l — а)ф — htl(Tn — Tt2)

mt2ct2 dTjdt = y( 1 — а)ф + htl(Tn — Tt2) — hi2(Tt2 — Тсй)

WdCcd dTjdt = hf2(Tf2 — 7cd) — hcd(Tcd — Tc) — R(T*$ — T*4) (1.42)

mccc dTjdt = hcd{Tcd — Tc) + hs(Ts — Tc) — 2.33mcccvc(TB — Tc. m)/L m3ca dTjdt = аф — hs(Ts — Tc) + R(T*£ — T*4)

Now we have two points in the fuel, one each in the cladding and coolant and one in the structural material.

Notice the radiative cooling term of the cladding. The problem is now nonlinear because of this term and will require more elaborate solution even in the steady state. However radiation terms are usually very small except in accident conditions; they may therefore be neglected in opera­tional or near operational conditions.

Further axial representation and a better model for dTJdz demands a set of such equations at a number of points of the coolant channel, linked through the coolant equation term dTJdz by a finite difference representa­tion. One such representation is the Fox-Goodwin equation (8a).

dTCnJdz + dTJdt = 2(TCn+i — TCn)/Az (1.43)

This defines gradients and temperatures at successive points in the coolant channel (Fig. 1.8).

Lz Lz Lz Lz

‘ • ■——— 1 ■ Fig. 1.8. Finite difference nomenclature

n-i n n+i n+2 n+3 along the reactor coolant channel.

Coolant flow—

With this model in steady state, neglecting the term mccc(dTJdt) and omitting the radiative terms,

Tcn+i = TCn + (Az/2mcccvcA’)[hca(Tcdn + Tajn+1) + hs(TSn + F9b+i)] — BTCn

(1.44)

where

A’ = 1 + (Az/2vc)[(hcJmccc) + (hjmccc)] and В = (2A’ — 2)1 A’ (1.45)

thus defining the coolant temperatures at a point n + 1, in terms of heat input to that channel at that point n + 1, and upstream n.

Thus a set of such equations can be obtained at each axial point and for every channel whose representation is required. All of these equations depend on a knowledge of the inlet temperature and the power distribution for their solution.