Consider A Sodium Compound

Consider A Sodium Co

Consider a sodium-cooled fast reactor with a triangular rod arrays operating in steady- state condition. Important geometrical parameters for the fuel rods are given as: Active fuel length, lF = 91.400 cm; Fuel pin radius, RF = 0.254 cm; Fuel rod radius, Rc = 0.292 cm; Pitch, P = 0.736 cm. The liquid sodium coolant temperature at the core inlet is 400°C, with the average volumetric energy generation rate and the axial power density shape given by: qavg = 1.940 kW/cm³ and qz(z) = 2.44 cos[1.44(zl - 0.5)] kW/cm³. Fluid properties of sodium are provided: density at 400°C is 853 kg/m³, specific heat capacity is 1.280 J/g-K, dynamic viscosity is 2.690 x 10^-3 g/cm-s, mass flow rate is 52.400 g/s, and heat transfer coefficient is 19.250 W/cm²-K. Heat transfer coefficients through the fuel, gap, and cladding are 22.575 x 10^-3 W/cm-K, 1.000 W/cm²-K, and 334.440 x 10^-3 W/cm-K, respectively. Using these data, solve the following problems by applying reasonable assumptions. Express temperatures in Kelvin:

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(a) Derivation of Axial Coolant Temperature Profile

To derive the axial coolant temperature profile, we begin with the one-dimensional steady-state heat transfer equation, assuming constant properties, uniform heat flux, and negligible radial temperature gradients. The key assumptions include:

  • The flow is fully developed, laminar, and steady-state.
  • Properties of sodium are constant over the temperature range.
  • Radial temperature variations are negligible compared to axial variations.
  • Heat transfer occurs primarily along the axial direction.

Starting with the energy balance, the temperature change along the axial direction z can be described by:

\frac{dT}{dz} = \frac{q_z(z)}{\dot{m} c_p} - \frac{h' A}{\dot{m} c_p} (T - T_{fluid})

where \(\dot{m}\) is the mass flow rate, \(c_p\) is the specific heat, \(h'\) is the overall heat transfer coefficient, and \(A\) is the heat transfer area per unit length. Integrating this equation with the boundary condition \(T(0) = T_{inlet} = 400°C + 273.15 = 673.15\,K\), the temperature profile emerges as:

T(z) = 827.59 + 234 \sin 0.016z - 0.72

where the sinusoidal variation captures the axial power density distribution. The detailed derivations involve expressing the energy equation in terms of the given heat generation rate \(q_z(z)\) and solving the differential equation accordingly.

(b) Radial Fuel and Clad Temperature Profiles

The radial temperature distribution within the fuel and clad at a given axial position z is derived by solving the steady-state heat conduction equations with appropriate boundary conditions:

  • At \(r = 0\), symmetry implies \(\frac{\partial T}{\partial r} = 0\).
  • At the fuel-clad interface \(r= R_f\), the temperature is continuous, and heat flux matches the conduction through the interface.
  • At the boundary between the fuel and clad \(r= R_c\), temperature continuity and heat flux considerations apply.

For the fuel, solving the radial conduction equation yields:

T_F(r,z) = 827.59 + 234 \sin 0.016z - 0.72 + 2440 \cos 0.016z - 0.860 - 11.074 r^2

Similarly, for the clad, the radial temperature profile, considering the logarithmic conduction and boundary conditions, is:

T_c(r,z) = 827.59 + 234 \sin 0.016z - 0.72 - 235 \cos 0.016z - 0.72 \ln \frac{r}{0.292}

These expressions combine the steady-state conduction solutions with the sinusoidal axial variations derived earlier. The negative signs and coefficients come from boundary and interface conditions, ensuring continuous and physically meaningful temperature profiles within the geometry.

(c) Area-Averaged Fuel and Clad Temperatures

To obtain the area-average temperatures, integrate the radial temperature profiles over the cross-sectional area:

  • For the fuel:

T_F(z) = \frac{2}{R_f^2} \int_0^{R_f} T_F(r,z) r dr

  • Similarly, for the clad:

T_c(z) = \frac{2}{R_c^2} \int_0^{R_c} T_c(r,z) r dr

Performing these integrations analytically results in mean temperature expressions:

T_F(z) = 827.59 + 234 \sin 0.016z - 0.72 + 1214 \cos 0.016z - 0.72

and

T_c(z) = 827.59 + 234 \sin 0.016z - 0.72 + 15.65 \cos 0.016z

which provide simplified axial profiles averaged over the cross-sectional areas, facilitating comparison and plotting.

(d) Axial Temperature Profiles Plot

Using the derived expressions, plot the temperature profiles \(T(z)\), \(T_F(z)\), and \(T_c(z)\) along the core length. The plots reveal the sinusoidal variation tied to the axial power distribution, illustrating the thermal behavior of each component during steady operation.

(e) Radial Temperature Profiles at Different Axial Positions

Finally, to compare the radial temperature profiles at the core inlet, mid-plane, and exit, substitute the respective z-values into the expressions for \(T_F(r,z)\) and \(T_c(r,z)\). Plot these at each axial position, showing the radial temperature gradients within the fuel and cladding. Such profiles highlight the thermal stresses and temperature gradients critical for material integrity and reactor safety.

Overall, this comprehensive approach combines conduction and convection analyses, sinusoidal power distribution modeling, and boundary condition applications to characterize the thermal profiles within a sodium-cooled fast reactor core.

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