The fission energy includes the fission fragment kinetic energy, the neutron kinetic energy, fission y-rays, /3- and y-rays from fission product decay. This energy distributes itself between fuel, moderator, reflector and shield, according to core size and composi­tion.

For dynamics calculations only the heat production in fuel and moderator is usually considered, and we can assume for HTRs that 92% of the energy is produced in the fuel and 8% in the moderator. Within the fuel and moderator blocks the time-dependent heat-conduction equation is used:

iI’=-^V2T + — (12.17)

dt Cp Cp

where T = temperature,

P = power per unit volume, c = specific heat, p = density,

к = thermal conductivity, t = time.

Integrating (12.17) over a given volume V and using Green’s theorem on the V2 term we obtain

3T=JL± f dT _P

dt cp V Js dn cp

where S is the surface corresponding to volume V and dT/dn the derivative in the direction perpendicular to this surface. Approximate solutions are obtained by sub­dividing the reactor components (fuel and moderator) into axial and radial zones in which the temperatures are supposed to be time-dependent but not space-dependent. The fuel is then subdivided in concentric radial zones. For each zone j eqn. (12.18) can be simplified to

The heat transfer equations for the coolant channels can be obtained considering the conservation of mass and of enthalpy together with the state equation of gases<2) applied to the axial zone Az of the coolant channel under consideration.

m, — nit-, = p, — p,

AAZi At

2 p ^ = cp(rihTc. i — conservation of enthalpy,

k = 1 Rk

p — = RTd state equation of gases

P

coolant specific heat,

number of different types of fuel elements giving power to the coolant, gas constant,

temperature of outer radial zone of type к fuel element, coolant temperature at axial level /,

heat resistance between outer radial zone of element к and gas, coolant mass flow at axial level і in the channel under consideration, coolant channel cross-section, height of axial zone, pressure.

In the second of these equations there is no accumulation term with time, i. e., no derivative A ■ Az ■ cp{d(pT)/dt}, because the helium coolant is supposed to behave as a perfect gas and therefore pT is constant, if we neglect pressure changes.

By substitution (and still neglecting pressure changes) we obtain the following heat transfer equation for the coolant:

The error introduced neglecting pressure changes is negligible even in case of rapid depressurizations, but it is in any case rather easy to treat these terms in computer codes.

Many coefficients of eqns. (12.19) and (12.20) are temperature-dependent. This difficulty is overtaken by iterative procedures correcting the coefficients at every iteration. The heat resistances Rk of eqn. (12.20) depend also on the mass flow.

These equations are also valid for pebble-bed reactors. In this case the coolant channel does not physically exist, but can be thought as an ideal vertical column in the pebble bed, containing a certain number of pebbles, with an area A free for the coolant passage. The fact of using heat resistances makes eqns. (12.19) and (12.20) independent of the fuel geometry (so that the same computer code can be used for different geometries), while this is not true for eqn. (12.17). The fuel geometry must obviously be considered in order to calculate the heat resistances.

In some cases the fuel elements are cooled inside and outside. In this case the distribution of the inner and outer mass flow must be determined iteratively imposing equal pressure drop along both ways of the channels. This same condition of equal pressure drop has to be applied when calculating the steady-state coolant flow partition between the different channels of the core. This calculation must take into account the
given gas outlet temperature profile in the core and the different pressure drops in the channels caused by acceleration losses, friction and discontinuity losses, gagging scheme. During transients caused either by reactivity excursions or mass flow variations, the split of the mass flow between different channels (and between the two semichannels of a tubular pin design) can vary. A mathematical solution of this problem can be found in ref. 2.

When only the zero-dimensional form of the neutron kinetics equation is considered, it is possible to make an approximation for the temperatures analogous to the one made for the fluxes in § 12.3, supposing that

T(r, t) = r(r)T(t)

where r(r) is the stationary temperature distribution.

This approximation can be sufficient in some cases but is not as good as the similar one for the fluxes. The axial temperature profile is usually distorted during the accident and the consideration of those distortions can result in corrections in the maximum temperature excursion ДТта* of the order of 10-15%. If a zero-dimensional heat — transfer calculation is performed the axial index і disappears from eqn. (12.19) (a certain number j of radial zones must be considered in any case).

Equation (12.20) is substituted by

І = cprh(T2 ~ T t) + c,,Ap~H — — T’ (12.21)

k = l Kk lit

with Tc = (Tc2- Tc,)/2,

Tci = coolant inlet temperature,

Tc2 = coolant outlet temperature,

H = core height.

In all these equations heat radiation has not been considered.

If a gap is present in the fuel it may be necessary to consider radiation. This implies the introduction of terms of the type

aiTf-TU)

to describe the heat radiation between region j and j + 1. In some cases it is sufficient to approximate this term with (T, — Tj+l)IRj choosing an appropriate resistance Rt, which will then be temperature dependent. In fuel element configurations where the mod­erator is physically separated from the fuel (e. g. pin and block design) the heat radiation becomes important especially when dealing with temperature excursions after a trip. In that case the exact treatment of heat radiation is necessary because after shut-down the coolant flow is strongly reduced.

The thermal power is usually considered to be proportional to the neutron flux. In reality some 8% of the power is due to fission product decay and is therefore released with various delays. As in the case of the delayed neutron precursors these fission products can be grouped in a certain number of groups (e. g. 2 in ref. 2). Also the repartition between fuel and moderator is different for the decay heat which, being due to absorption of /3- and y-rays, is more uniformly distributed (70% in fuel, 30% in moderator compared to 94% in fuel and 6% in moderator for the prompt heat).

A detailed treatment of decay heat, separate from prompt heat, is necessary in case of temperature calculations after shut-down, or accident during start-up (when no fission products are yet available).

In the heat-transfer equations appear two parameters: the mass flow m and the coolant inlet temperature Tc 1. The mass flow, together with the control-rod position, is one of the quantities through which the control system acts on the reactor core.

The gas-inlet temperature can be influenced by the feedback of the heat exchangers and secondary system. In most core dynamics studies this inlet temperature is supposed to be constant (or at most, a known function of the average plant power level). This is usually a good approximation which allows the separation of the core from the secondary system.

In eqns. (12.20) and (12.21) the left-hand term can receive contributions from different element types. In reactors with prismatic fuel usually each channel receives heat from only one type of fuel, or at most from the fuel pins and from the moderator. In the case of pebble-bed reactors various types of elements can give heat to the same channel.

In reactors with continuous or frequent reloading the different types of elements should represent the different burn-up stages. Both temperature distribution and temperature coefficient feedback are different in fresh and depleted elements. The grouping of all elements in one average class may lead to errors of the order of 20 to 30% in the maximum temperature increase following an accident.’3’ It is often sufficient to group the fuel elements in two classes, the fresh ones in one class and all the others in a second class. The treatment of different fuel element classes implies in the case of prismatic reactors the treatment of different channel types in each region.