9.146.

In a bare cylindrical or rectangular parallelepiped reactor core, the neutron flux has a cosine distribution in the axial direction (Table 3.2). The volumetric heat-source distribution in this (*) direction will thus be represented by a cosine function, provided the fuel enrichment is uniform along the length of each rod. Previously, the origin of the coordinates was chosen at the center of the core, but here, however, it is convenient to take the origin at the point where the coolant enters the channel (Fig. 9.20). The heat-source distribution along a particular channel, i. e., in the

axial (or flow) direction, can then be expressed in sinusoidal form; thus,

Q = бо sin—,

where Q is the volumetric heat-release rate at the point x, and Q0 is the maximum value at the center of the channel; L is the total channel length, assuming no reflector. Upon inserting this expression for Q into equation (9.45) and performing the integration, the result is

(9.50)

which gives the rise in temperature of the coolant as it flows through the channel.

9.147. The temperature of the coolant increases continuously along the channel, as shown in the bottom curve of Fig. 9.21; the bulk coolant temperature rise, represented by Дfc, is thus seen to be

2Q0V

TTWCp

It is of interest to note that this result can be derived in another manner. Since Q0 is the maximum value of the heat-release rate per unit volume, the average is 2<20/’П’ (cf. §9.16), and the total heat-release rate for the given channel is then 2Q0Vhr. In the steady state this must, of course, be equal to the rate at which heat is removed by the coolant, i. e., wcpAtc, so that equation (9.51) follows immediately.

9.148. The temperature difference between the solid and the fluid at any point is obtained from equations (9.48) and (9.49) as

Q0V. ttx

4-г — = /ЙГ“Т

which is seen to be analogous to equation (9.49) for the source distribution. The value of ts — tm, consequently, passes through a maximum when x = ViL, i. e., in the middle of the channel, as shown in the second curve of Fig. 9.21. This result is, of course, to be expected since at any point the fluid-solid temperature difference will be proportional to the radial heat — flow rate at that point, and this is a maximum in the center of the fuel channel.

9.149.

The value of ts — te, i. e., the fuel surface temperature at any point with respect to the value at the channel entrance, is obtained by adding equations (9.50) and (9.52), so that

It is apparent that ts — te must pass through a maximum at some point beyond the middle of the channel, as shown in the top curve of Fig. 9.21. This maximum represents the highest surface temperature of the fuel ele­ment. The point at which the maximum is attained is found by differen­tiating equation (9.53) with respect to x and setting the result equal to zero; thus,

L ttw’cA

*ma* = (9’54)

From this expression it is seen that the position of the maximum value of ts approaches the end of the channel, i. e., *max —» L, as w and cp decrease, and h and Ah increase.

9.150. The maximum surface temperature in the given channel can now be obtained by insertion of Jtmax into equation (9.53). Assuming the coolant — flow rate to be the same in all channels, the maximum fuel-element surface temperature occurs in the channel where the neutron flux is a maximum.

In this case, Q0 is the volumetric heat-release rate at the center of the reactor, and the corresponding acceptable maximum value of ts will affect the maximum reactor power.

9.51. It should be noted that the surface temperature ts is less than the temperature in the middle of the fuel element. The difference, represented by the temperature drop through the fuel region and the cladding, can be calculated by the methods described earlier (§9.47 et seq.). Because Q varies along the length of the fuel element, the point at which the surface temperature is a maximum will generally not be the same as that at which the interior temperature of the fuel has its maximum value.

9.152. For purposes of computation, it is convenient to represent ттх/ L by the symbol a, i. e.,

(9.55)

It can then be found from equation (9.54) that

where amax = TTJCmax/L. Utilizing this expression for tan amax, equation (9.53) can be reduced to

(ts)max — te = -%7(1 — sec amax). (9.57)

TTWCp

With these two equations, the value of the maximum surface temperature and its location can be readily evaluated.

9.153. This treatment is strictly applicable to a bare reactor. For a reflected reactor, the flux (or power) distribution is flatter than in an equivalent bare core, and the ratio of maximum to average specific power (or volumetric heat source) is lower, as shown in §9.17. One method of treating the problem of a reactor with a thick reflector is to assume a cosine source distribution, as in Fig. 9.21, but to suppose that the value of Q goes to zero at some distance, e. g., about one or two reflector diffusion lengths, beyond the boundary of the core. The integration of sin ttx/L, referred to in §9.146, then does not start from zero, but from x equal to roughly two diffusion lengths in the reflector. Of course, as a result of fuel burnup, the enrichment will no longer be axially uniform. The high-flux region will “flatten” and the axial variations described will change accordingly.