Question

A nuclear fuel element of thickness 2L is covered with a steel cladding of thickness b. Heat generated within the nuclear fuel at a rate q. is removed by a fluid at T , which adjoins one surface and is characterized by a convection coefficient h. The other surface is well insu- lated, and the fuel and steel have thermal conductivities of kƒ and ks, respectively.

(a) Obtain an equation for the temperature distribution T(x) in the nuclear fuel. Express your results in terms of q., kƒ, L, b, ks, h, and T .
(b) Sketch the temperature distribution T(x) for the entire system.

Answer 1

The temperature distribution in the nuclear fuel element is determined by solving the steady-state heat conduction equation with the given boundary conditions. The resulting distribution, T(x), is parabolic within the fuel element and linear within the steel cladding, ending with the fluid temperature T at the convective boundary.

Step-by-step explanation:

The student is looking for an equation for the temperature distribution T(x) in a nuclear fuel element with given parameters, and a sketch of that temperature distribution. To find this temperature distribution within the nuclear fuel, one must consider the steady-state heat conduction equation, Fourier's Law, and the boundary conditions provided, including convection at the fluid interface and insulation at the other end.

Since heat generation is uniform within the fuel, the temperature profile within the fuel element will be parabolic. Using Fourier's Law for heat conduction, modified to include heat generation:

q = -kfA (dT/dx) + qgen

Here, q is the heat transfer rate, A is the cross-sectional area of the fuel, kf is the thermal conductivity of the fuel, dT/dx is the temperature gradient within the fuel, and qgen is the rate of heat generation per unit volume.

By integrating this equation, considering the insulated boundary at x = 0 where dT/dx = 0, and the convective boundary at x = 2L + b, where the heat transfer is q = hA(T - Tφ), with Tφ being the temperature at the fluid interface, one can solve for T(x).

Part (b) of the question asks for a sketch of T(x), which will show a parabolic rise in temperature from the insulated end to a peak within the fuel element, then a linear decrease within the steel cladding, and finally leveling off to the fluid temperature T at the convective boundary.