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20 points temperature . eCnsidet the rectangular slab R edge on the left , right, and {(I,u)lo < I < 10, 0 < y < 50} with given by the function top edge...

Question

20 points temperature . eCnsidet the rectangular slab R edge on the left , right, and {(I,u)lo < I < 10, 0 < y < 50} with given by the function top edges equal to 0? and the temperature at the bottom appropriate initial f(r) = 50. Write conditions. partial differential equation, along with all solve the equation. to find the steady state temperature of the slab_ Do not

20 points temperature . eCnsidet the rectangular slab R edge on the left , right, and {(I,u)lo < I < 10, 0 < y < 50} with given by the function top edges equal to 0? and the temperature at the bottom appropriate initial f(r) = 50. Write conditions. partial differential equation, along with all solve the equation. to find the steady state temperature of the slab_ Do not



Answers

A point $x_{0}$ at the centre of a large slab of material of thermal coductivity $k$, specific heat $C$ and density $\rho$ has an infinitely high temperature $T$ at a time $t_{0}$. If the heat diffuses through the medium at a rate given by $$ \frac{\partial T}{\partial t}=\frac{k}{\rho C} \frac{\partial^{2} T}{\partial x^{2}}=d \frac{\partial^{2} T}{\partial x^{2}} $$ show that the heat flow along the $x$ -aixs is given by $$ f(\alpha, t)=\frac{r}{\sqrt{\pi}} \mathrm{e}^{-(r \alpha)^{2}} $$ where $$ \alpha=\left(x-x_{0}\right) \quad \text { and } \quad r=\frac{1}{2 \sqrt{\mathrm{d} t}} $$ by inserting this solution in the differential equation. The solution is a Guassian function; its behaviour with $x$ and $t$ in this problem is shown in Fig. 10.12. At $\left(x_{0}, t_{0}\right)$ the function is the Dirac delta function. The Guassian curves decay in height and widen with time as the heat spreads through the medium, the total heat, i.e. the area under the Gaussian curve, remaining constant.

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