(Solved): Please use Jupyter Notebook (Python) Heat is conducted along a metal rod positioned between two fxed ...

Heat is conducted along a metal rod positioned between two fxed temperature walls. Aside from conduction, heat is transferred between the rod and the surrounding air by convection. Based on a heat balance, the distribution of temperature along the rod is described by the following second-order differential equation. $$\frac{d^{2}T}{d{x}^2}+h^{?}\left(T_a?T\right)$$ where $$T=$$ temperature $$\left({}^{?}\mathrm{C}\right),h^{?}=$$ a bulk heat transfer coefficient reflecting the relative importance of corvection to conduction $$\left(0.01{\mathrm{m}}^{?3}\right),\mathrm{x}=$$ distance along the rod $$(10\mathrm{m})$$, and $$T_{\text{a }}=$$ temperature of the surrounding fluid $$\left(20^{?}\mathrm{C}\right)$$, a) Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. if $$T(0)=40^{?}\mathrm{C}$$ and $$\mathrm{T}(10)=200^{?}\mathrm{C}$$, , $$\mathrm{b}$$ bain anslutical solution (t.e. solve differential equation (1)). Write out all the detailed procodure and steps you have taken. b) Above heat transfer equation can be transformed into a sot of linear algobraic equations by conceptualizing the rod as consisting of a series of nodes. For example, the rod in Figure is divided into six equi-spaced nodes. Since the rod has a length of 10 , the spacing between nodes is $$Ax=2$$. Finite-difference approximations prowide a means to transtom derivatives into algebraic form. For example, the second derivative at each node can be approximated as $$\frac{d^{2}T}{d{x}^2}=\frac{T_{i+1}?2T_i+T_{i=1}}{?x^2}$$ where Tidesignates the temperature at node i. This approximation can be substituted into Eq. (1) to give $$\frac{T_{i+1}?2T_i+T_{i=1}}{?x^2}+h^{?}\left(T_e?T_1\right)=0$$ Develop linear equations applied to each of the nodes (for $$i=1,2,3,4)$$. Write out the linear equations you developed. c) Calculate node temperatures $$(\mathrm{T}1$$ - $$\mathrm{T}4)$$ with Python code. d) Plot the results of analytical solution (a) and temperatures obtained in (c) into one graph. Use black solid line for the analytical solution, and for the results of (c), use blue open circle. e) $$h^{?}=h/k$$, and $$h$$ is convective heat transfer coefficient and $$k$$ is the thermal conductivity of solid material. We can analyze the temperature distribution in the solid material by changing the thernal conductivity. Ler's use the $$k$$ values $$(50,25,5$$, and $$1\mathrm{W}/\mathrm{m}K\mathrm{K})$$ while using foxed $$h$$ value of $$1\mathrm{W}/{\mathrm{m}}^2\mathrm{K}$$, and plet temeerature disiribution. f) Explain the results (le, impact of different $$k$$ values on temperature distribution). (Hint: think how much temperature will be varied in the length direction when a solid bar contacts the hot surface on its one end While the other surfaces expose to the air at room temperature, and how much themal conductivity of the solid material will affect the temperalure distribution. You may consider the high k-value materials as steel bars, and the smaller k-value materials as wood bars to catch the effect of themal conductivity]