Question

Consider a uranium nuclear fuel element $(k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\rho=19,070 \mathrm{~kg} / \mathrm{m}^{3}\(, and \)\left.c_{p}=116 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( of radius \)10 \mathrm{~cm}$ that experiences a volumetric heat generation at a rate of $4 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}$ because of the nuclear fission reaction. The nuclear fuel element initially at a temperature of \(500^{\circ} \mathrm{C}\) is enclosed inside a cladding made of stainless steel material $\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8055 \mathrm{~kg} / \mathrm{m}^{3}\right.\(, and \)\left.c_{p}=480 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( of thickness \)4 \mathrm{~cm}$. The fuel element is cooled by passing pressurized heavy water over the cladding surface. The pressurized water has a bulk temperature of \(50^{\circ} \mathrm{C}\), and the convective heat transfer coefficient is $1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming one-dimensional transient heat conduction in Cartesian coordinates, determine the temperature in the fuel rod and in the cladding after 10,20 , and \(30 \mathrm{~min}\). Use the implicit finite difference formulation with a uniform mesh size of \(2 \mathrm{~cm}\) and a time step of $1 \mathrm{~min}$.

Short answer

Question: Based on the provided step-by-step solution, describe the key process involved in determining the temperature in a uranium nuclear fuel rod and its surrounding stainless steel cladding at different time intervals using one-dimensional transient heat conduction in Cartesian coordinates. Answer: The key process involves setting up the heat conduction equation for the fuel element and cladding, applying the initial and boundary conditions, discretizing the heat conduction equation using the implicit finite difference scheme, solving the finite difference equations, and then obtaining the temperature distribution in the fuel rod and cladding at different time intervals.

Step-by-step solution

Set up the heat conduction equation for the fuel element and cladding

The governing equation for transient heat conduction in a solid is given by: $$\rho c_p \frac{\partial T}{\partial t} = k \nabla^2 T + q$$ where \(T\) is temperature, \(\rho\) is the density, \(c_p\) is the specific heat capacity, \(k\) is thermal conductivity, and \(q\) is the volumetric heat generation rate. Since we assume one-dimensional transient heat conduction in Cartesian coordinates, the governing equation reduces to: $$\rho c_p \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2} + q$$

Apply the initial and boundary conditions

The initial temperature of the fuel element and cladding is given as \(T_{initial}=500^{\circ}C\). Boundary conditions: 1. At the interface between the fuel element and cladding (x = 10 cm), heat transfer is continuous: $$k_{fuel}\frac{\partial T_{fuel}}{\partial x} = k_{cladding}\frac{\partial T_{cladding}}{\partial x}$$ 2. At the outer surface of the cladding (x = 14 cm), convective cooling with heavy water takes place: $$h(T_{surface} - T_{water}) = k_{cladding}\frac{\partial T_{cladding}}{\partial x}$$ where \(h\) is the convective heat transfer coefficient and \(T_{water}\) is the bulk temperature of pressurized water.

Discretize the heat conduction equation using the implicit finite difference scheme

We discretize the spatial domain using a mesh size of 2 cm and the time domain with a time step of 1 minute. The discretized form of the heat conduction equation using the implicit finite difference scheme is: $$\rho c_p \frac{T_i^{t+1} - T_i^t}{\Delta t} = k \frac{T_{i+1}^{t+1} - 2T_i^{t+1} + T_{i-1}^{t+1}}{\Delta x^2} + q$$ where \(T_i^t\) and \(T_i^{t+1}\) represent the temperature at spatial index \(i\) at the current time step \(t\) and the next time step \(t+1\), respectively.

Solve the finite difference equations

To solve the finite difference equations, we can use an iterative method such as the Gauss-Seidel or Jacobi method until the solution converges to the desired accuracy. Alternatively, we can also use a matrix method to solve the resulting system of linear equations.

Obtain the temperature distribution in the fuel rod and cladding at different time intervals

After solving the finite difference equations, we can obtain the temperature distribution in the fuel rod and the cladding for different time intervals, (10 minutes, 20 minutes, and 30 minutes). Using the given mesh size and time step, we can extract the temperatures and plot them as a function of space and time. By following these steps, we can determine the temperature in the uranium nuclear fuel rod and its surrounding stainless steel cladding at the specified times using the implicit finite difference method for one-dimensional transient heat conduction in Cartesian coordinates.