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5. Let $E^\prime$ be the rate at which heat is generated per unit volume in a solid sphere of radius $R$ (due to radioactive decay for example). The resulting increase in the temperature of the sphere creates a temperature gradient which causes heat to be conducted through the sphere to the environment that surrounds the sphere. If the rate of heat conduction is significant in the radial direction only and the rate of heat conduction in this direction obeys Fourier's law of heat conduction, \begin{equation} \frac{dQ}{dt} = -kA\frac{\partial T}{\partial r}, \end{equation} where $T(r, t)$ is the temperature at time $t$ at a point in the sphere a distance $r$ from the centre of the sphere, $k$ is the conductivity of the material of which the sphere is made and $A$ is the area of the sphere normal to the direction of heat conduction then, by considering the flow of heat into and out of a volume element in the sphere, derive the partial differential equation that must be solved to determine the temperature at any point in the sphere. Heat is generated at a rate of $E^\prime = 1.1 \times 10^6 \text{ W/m}^3$ in a radioactive ma- terial that has thermal conductivity $k_s = 26 \text{ W/m } \cdot \text{C}^\circ$. The radioactive material is in the form of a solid sphere with diameter 8.7 cm. The spheri- cal radioactive material is surrounded by a layer of ceramic that is 4.5 mm thick and has thermal conductivity $k_c = 1.0 \text{ W/m } \cdot \text{C}^\circ$. If the outer surface of the ceramic loses heat to the environment at a rate given by Newton's law of cooling where $T_\infty = 16^\circ \text{C}$ is the temperature of the environment and $h = 13 \text{ W/m}^2 \cdot \text{C}^\circ$ is the heat transfer coefficient, then determine (rounded to two decimal places) the temperature in the steady state at the center of the sphere.

          5. Let $E^\prime$ be the rate at which heat is generated per unit volume in a solid
sphere of radius $R$ (due to radioactive decay for example). The resulting
increase in the temperature of the sphere creates a temperature gradient
which causes heat to be conducted through the sphere to the environment
that surrounds the sphere. If the rate of heat conduction is significant in
the radial direction only and the rate of heat conduction in this direction
obeys Fourier's law of heat conduction,
\begin{equation}
\frac{dQ}{dt} = -kA\frac{\partial T}{\partial r},
\end{equation}
where $T(r, t)$ is the temperature at time $t$ at a point in the sphere a distance
$r$ from the centre of the sphere, $k$ is the conductivity of the material of
which the sphere is made and $A$ is the area of the sphere normal to the
direction of heat conduction then, by considering the flow of heat into
and out of a volume element in the sphere, derive the partial differential
equation that must be solved to determine the temperature at any point
in the sphere.
Heat is generated at a rate of $E^\prime = 1.1 \times 10^6 \text{ W/m}^3$ in a radioactive ma-
terial that has thermal conductivity $k_s = 26 \text{ W/m } \cdot \text{C}^\circ$. The radioactive
material is in the form of a solid sphere with diameter 8.7 cm. The spheri-
cal radioactive material is surrounded by a layer of ceramic that is 4.5 mm
thick and has thermal conductivity $k_c = 1.0 \text{ W/m } \cdot \text{C}^\circ$.
If the outer surface of the ceramic loses heat to the environment at a rate
given by Newton's law of cooling where $T_\infty = 16^\circ \text{C}$ is the temperature of
the environment and
$h = 13 \text{ W/m}^2 \cdot \text{C}^\circ$
is the heat transfer coefficient, then determine (rounded to two decimal
places) the temperature in the steady state at the center of the sphere.

5. Let E^' be the rate at which heat is generated per unit volume in a solid
sphere of radius R (due to radioactive decay for example). The resulting
increase in the temperature of the sphere creates a temperature gradient
which causes heat to be conducted through the sphere to the environment
that surrounds the sphere. If the rate of heat conduction is significant in
the radial direction only and the rate of heat conduction in this direction
obeys Fourier's law of heat conduction,

(dQ)/(dt) = -kA(∂ T)/(∂ r),

where T(r, t) is the temperature at time t at a point in the sphere a distance
r from the centre of the sphere, k is the conductivity of the material of
which the sphere is made and A is the area of the sphere normal to the
direction of heat conduction then, by considering the flow of heat into
and out of a volume element in the sphere, derive the partial differential
equation that must be solved to determine the temperature at any point
in the sphere.
Heat is generated at a rate of E^' = 1.1 × 10^6 W/m^3 in a radioactive ma-
terial that has thermal conductivity ks = 26 W/m ·C^∘. The radioactive
material is in the form of a solid sphere with diameter 8.7 cm. The spheri-
cal radioactive material is surrounded by a layer of ceramic that is 4.5 mm
thick and has thermal conductivity kc = 1.0 W/m ·C^∘.
If the outer surface of the ceramic loses heat to the environment at a rate
given by Newton's law of cooling where T∞ = 16^∘C is the temperature of
the environment and
h = 13 W/m^2 ·C^∘
is the heat transfer coefficient, then determine (rounded to two decimal
places) the temperature in the steady state at the center of the sphere.

Added by Stephanie C.

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