91Ó°ÊÓ

Radioactive wastes \(\left(k_{\mathrm{rw}}=20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\) are stored in a spherical, stainless steel \(\left(k_{\mathrm{ss}}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\) container of inner and outer radii equal to \(r_{i}=0.5 \mathrm{~m}\) and \(r_{o}=0.6 \mathrm{~m}\). Heat is generated volumetrically within the wastes at a uniform rate of \(\dot{q}=10^{5} \mathrm{~W} / \mathrm{m}^{3}\), and the outer surface of the container is exposed to a water flow for which \(h=\) \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\infty}=25^{\circ} \mathrm{C}\). (a) Evaluate the steady-state outer surface temperature, \(T_{s, o}\) (b) Evaluate the steady-state inner surface temperature, \(T_{s, i^{*}}\) (c) Obtain an expression for the temperature distribution, \(T(r)\), in the radioactive wastes. Express your result in terms of \(r_{i}, T_{s, i}, k_{\mathrm{rw}}\), and \(\dot{q}\). Evaluate the temperature at \(r=0\). (d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation \(\left(\dot{q}=2 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\right)\) in a stainless steel container of equivalent inner radius \(\left(r_{i}=0.5 \mathrm{~m}\right)\). Safety considerations dictate that the maximum system temperature not exceed \(475^{\circ} \mathrm{C}\) and that the container wall thickness be no less than \(t=0.04 \mathrm{~m}\) and preferably at or close to the original design \((t=0.1 \mathrm{~m})\). Assess the effect of varying the outside convection coefficient to a maximum achievable value of \(h=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for \(h\) and \(t\), respectively.

Step by step solution

01

(a) Calculate the outer surface temperature, \(T_{s, o}\)

First, let's begin by calculating the heat generated in the radioactive waste, \(Q_{\text{rw}}\), using the volumetric heat generation rate, \(\dot{q}\), and the inner radius of the container, \(r_i\): \[ Q_{\text{rw}} = \dot{q} \cdot \frac{4}{3} \pi r_i^3 \] Now, let's calculate the heat transfer on the outer surface, \(Q_{\text{out}}\), using the convection coefficient, \(h\); the outer surface temperature, \(T_{s, o}\); and the ambient temperature, \(T_\infty\): \[ Q_{\text{out}} = h A_{s, o} (T_{s, o} - T_\infty) \] Here, \(A_{s, o}\) represents the outer surface area, which is equal to \(4\pi r_o^2\). In steady-state, the heat generated in the waste should equal the heat transferred from the outer surface, so we can set \(Q_{\text{rw}} = Q_{\text{out}}\) and solve for \(T_{s, o}\).

02

(b) Calculate the inner surface temperature, \(T_{s, i^{*}}\)

The heat conducted through the stainless steel container wall can be expressed as: \[ Q_{\text{rw}} = k_{\text{ss}} A_{s, i} \frac{T_{s, i} - T_{s, o}}{t}, \] Where \(T_{s, i}\) is the inner surface temperature with \(A_{s, i} = 4\pi r_i^2\), and \(t = r_o - r_i\) is the thickness of the container wall. We have already calculated \(Q_{\text{rw}}\) and \(T_{s, o}\), and can now solve for \(T_{s, i}\).

03

(c) Calculate the temperature distribution, \(T(r)\)

To obtain an expression for the temperature distribution within the radioactive wastes, first assume steady-state, non-radial heat conduction. The general equation for the temperature distribution in spherical coordinates can be represented as: \[ T(r) = \frac{1}{6} \left(\frac{\dot{q} r^2}{k_{\text{rw}}} - C_1 r + C_2 \right) \] We can use the initial condition that \(T(r_i) = T_{s,i}\) and solve for the constants \(C_1\) and \(C_2\). Then, calculate the temperature at \(r=0\).

04

(d) Assess the proposed extension and recommend suitable conditions

First, investigate the feasibility of the new design by figuring out if the maximum system temperature will not exceed \(475^\circ\text{C}\), and the container wall thickness will be no less than \(t=0.04\text{m}\). Analyze how changes in the convection coefficient and wall thickness affect the system and determine if the proposed extension of the design will be feasible. If the modified design is feasible, provide recommendations for suitable operating and design conditions, specifically the values for the outside convection coefficient, \(h\), and the container wall thickness, \(t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conduction

Conduction is the transfer of heat through a material without any movement of the material itself. This process involves the transfer of energy from high-temperature regions to low-temperature regions and relies on the collision of particles within the material. In the context of the radioactive waste storage, heat conduction occurs through the stainless steel container walls. The waste generates heat, and this heat is conducted through the container material to the outer environment. The formula for heat conduction through a solid material is given by Fourier’s Law:\[Q = -k A \frac{dT}{dx}\]where - \(Q\) is the heat transfer rate (in Watts), - \(k\) is the thermal conductivity of the material,- \(A\) is the area through which heat is conducted, - \(dT/dx\) is the temperature gradient.For our spherical container, we use spherical coordinates which require an adaptation of Fourier's Law to account for radial variation in temperature and the increasing area as the radius increases.

Convection

Convection is the heat transfer between a solid surface and a fluid (either liquid or gas) that is in motion. This process involves the bulk movement of the fluid, which carries energy away from the surface, enhancing the heat transfer. In the given exercise, convection is important at the outer surface of the spherical container, where it interacts with the surrounding water.The rate of heat transfer by convection can be expressed by Newton’s Law of Cooling:\[Q = h A (T_{s} - T_\infty)\]where- \(Q\) is the rate of heat transfer (in Watts),- \(h\) is the convection heat transfer coefficient,- \(A\) is the surface area,- \(T_s\) is the surface temperature,- \(T_\infty\) is the temperature of the fluid far away from the surface.In the exercise, the water flow with a convection coefficient \(h = 1000\, \mathrm{W/m^{2}\cdot K}\) provides a medium for heat dissipation from the container, significantly influencing the outer surface temperature \(T_{s,o}\). Adjusting the convection coefficient by increasing the water flow velocity offers a means to control the system’s thermal performance.

Thermal Conductivity

Thermal conductivity is a material property that indicates the ability of a material to conduct heat. It plays a crucial role in determining the rate of heat transfer across a material when a temperature difference exists. Substances with high thermal conductivity, such as metals, can transfer heat more readily compared to those with low thermal conductivity, like plastics or gases.For our radioactive waste storage problem, we deal with two different thermal conductivities:- \(k_{\text{rw}} = 20\, \mathrm{W/m \cdot K}\) for the radioactive wastes,- \(k_{\text{ss}} = 15\, \mathrm{W/m \cdot K}\) for the stainless steel container.These values indicate how effectively each material can conduct heat. The higher the thermal conductivity, the more efficient the heat transfer through that material. In the step-by-step solution, thermal conductivity is used to compute the temperatures at different locations (inner and outer surfaces) of the container by relating the heat flow to the conductive properties of the materials involved.

Spherical Coordinates

Spherical coordinates provide a system to measure positions in three-dimensional space using three numbers: radius \(r\), polar angle \(\theta\), and azimuthal angle \(\phi\). This system is particularly useful for problems involving radially symmetrical objects such as our spherical container.In the exercise, spherical coordinates offer a suitable framework for analyzing the radial heat conduction within the spherical radioactive waste container. The symmetry of the sphere simplifies the mathematical description of the temperature distribution, as it depends only on the radial distance \(r\) from the center.The temperature distribution equation for radial heat conduction in spherical coordinates is derived by expressing the heat transfer problem in terms of \(r\) alone, simplifying the heat equation and allowing us to integrate based on boundary conditions:\[T(r) = \frac{1}{6} \left(\frac{\dot{q} r^2}{k_{\text{rw}}} - C_1 r + C_2 \right)\]This approach enables us to solve for temperature variations throughout the waste material, providing vital information on temperature gradients inside the sphere.