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.
Given. De exclaim Abi is a temp rager function on line integral C minus gate de into nds Use a laid off he close Andi t X comma by Isaac will toe 16 years to the power buying by 50 and C is the rectangle with excess it will do minus 20 X is equal to 20 bicycle to minus phi. By going to fight, we need to compute the clothes. So that is compute Stephen Why is equal to minus Phi X is equal Duke de on my last 20 years doesn't equal to tease listen equals 20 So we guarantee point Let us find the yes the yes is it Will do De He's 14 square they physical door standing therefore at and physical toe zero comma minus one My coughs even minus cape There 30 is Andy s has given Byron minus 20 to 20 six game bye bye into either should improbable minus one by then 80. So this is it Will do Won't be the issue of the bubble minus one Brighton No, we will see on the gulf. Seemed to relax Physical 20 by people to t minus by was listening flotillas and equal to fire you and musical one comma zero again the yes will be Puerto even though my car c to minus care then that d and yes, it will be given by minus by 50 Diva says it will do. Zero No, Even seen from the border were C three years. Why is ableto by X is equal to minus T minus 20 is listening Porto dealers and equal to 20 and Mexico 20 come up when So my girl c three minus in the D in two nds is given by 20 to minus 20 minus six game by by additional bottle one But then Deitz doesn't want to will previously bubble one, but then into minus 48 g now along the left we have seen for X is equal to minus 20. Why is equal to minus T minus phi Listen, ecology has any going to fight and is equal to minus one comma zero Oh, we're full minus game there, Daddy. And yes, is given by by minus by Tzeitel Dainty because DDS do you so this is equal to zero. So let us find the total go minus k their daddy the Yes, and yes is a weirdo. What did you just were? One, But then bless zero minus all, won t it? He used to be 11 but then blast zero. So this is a Quito. Or did Goldman you do? Minus one by then. Minus you Super one. But then so this is approximately equal to minus 9.616 So here he closes this much, it is clear that no hits bring along the left and right because they're zero. So many stem, dangerous, constant. Certain creature is Go on, student along. Let Andi right, or we can say the Lord's own two sites. So therefore integral is zero.