9.143 As background for describing the temperature variations along the path of the reactor coolant, it is useful first to consider a classical idealized channel with uniform fuel enrichment and no axial reflector. A generalized representation of such a coolant channel, with its associated effective heat-removal area, is shown in Fig. 9.19, where the different variables are identified. The x coordinate represents the direction of cool­ant flow in a vertical channel. The rate dq(x) at which heat is added to the stream in a differential length dx of any coolant passage is then given by

dq(x) = wcp dt, (9.43)

where w is the mass-flow rate of the coolant in the channel associated with one fuel rod, cp is the specific heat, and dt is the differential temperature increase in the coolant across the length dx. For simplicity it will be pos­tulated that all heat flow within the solid fuel is normal to the coolant stream, i. e., there is no heat conduction along the fuel rod parallel to the coolant channel.[14] Then dq(x) is the rate of heat generation in a volume Ac dx, where Ac is the cross-sectional area of the fuel rod (in the y, z plane). Since the volumetric heat source is, in general, a point function Q(x, y, z), it is possible to write

j Q(x, y, z) dy dz j dx,

the integration being carried over the fuel rod cross section.

9.144. If a local average heat source per unit volume Q(x) is defined

by

it follows that

dq(x) = ACQ(x) dx.

Fig. 9.19. Generalized representation of fuel rod and coolant flow channel. (Note that the rod and channel are actually vertical.)

Hence, from equations (9.43) and (9.44), the temperature distribution in the direction of coolant flow must satisfy the relation

dt_ = ACQ dx wcp

it being understood that Q is really Q{x). If the quantity AJwcp is inde­pendent of jc, the distribution of the coolant mixed-mean (or bulk) tem­perature tm along the channel will be given by

(9.45)

where the fluid entrance temperature, te at x = 0, is chosen as the datum. This expression gives the increase in the coolant temperature due to the heat added as it flows through the channel.

9.145. By equation (9.9), the local (solid) surface temperature ts is re­lated to the coolant bulk temperature tm by

(9.46)

where h is the heat-transfer coefficient for the given conditions and dAh is a small element of heat-transfer surface, i. e., the fuel-rod surface (not its cross section). Although h will vary to some extent along the length of the channel, it will be assumed to be constant and independent of x, in order to simplify the treatment without affecting the general conclusions. If p is the circumference of the rod, which may be clad or unclad, then

dAh = p dx, and equation (9.46) may be written as

dq = hp(ts — tm) dx.

From equations (9.44) and (9.47), it is seen that

QAc

hP ‘

For a clad fuel rod, the heat-generating volume is based on the pellet radius, a, and the heat-transfer area is based on the outer clad surface. It is then best to write

QV hAh’

where Ah and V are the heat-transfer area and heat-generating volume, respectively, associated with the coolant channel.