The one-dimensional temperature transport equation for advection and diffusion is given as: aT T r +U...

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The one-dimensional temperature transport equation for advection and diffusion is given as: aT T r +U t ax Ox = (4) (i) Based on the above equation, formulate the Finite Difference Approximation (FDA) with explicit formulations, using forward difference in time; upwind difference (backward) in space for advection; central difference in space for the diffusion term. [3 Marks] (ii) Apply the Von Neumann stability analysis to FDA written in part (i) if u = 0.5 ms . Given that Ax=0.2m, At = 0.1s and the thermal conductivity a=0.25, specify the stability criterion for the FDA written and elaborate whether this FDA is stable (i.e. conditionally or unconditionally) or unstable. [12 Marks] (iii) Followed by the conditions in part (ii), consider the temperature change for a one-dimensional hypothetical rod of 1 m where heat is applied at both ends. The boundary conditions are given by: T(x=0,t)= T(x=1,1)=100C T(x,t=0)=20C for 0 The one-dimensional temperature transport equation for advection and diffusion is given as: aT T r +U t ax Ox = (4) (i) Based on the above equation, formulate the Finite Difference Approximation (FDA) with explicit formulations, using forward difference in time; upwind difference (backward) in space for advection; central difference in space for the diffusion term. [3 Marks] (ii) Apply the Von Neumann stability analysis to FDA written in part (i) if u = 0.5 ms . Given that Ax=0.2m, At = 0.1s and the thermal conductivity a=0.25, specify the stability criterion for the FDA written and elaborate whether this FDA is stable (i.e. conditionally or unconditionally) or unstable. [12 Marks] (iii) Followed by the conditions in part (ii), consider the temperature change for a one-dimensional hypothetical rod of 1 m where heat is applied at both ends. The boundary conditions are given by: T(x=0,t)= T(x=1,1)=100C T(x,t=0)=20C for 0