Name: Wai Ming Hui Kelvin

Derivations and ExplanationsObjectiveTo obtain temperature profile from engine surface to detector surface.Assumptionsa) No chemical reaction so thermal energy generation is zerob) Convective heat transfer and thermal conductivity is assumed to be constantc) Steady-state heat transfer is achieved after some time has passed to reach thermal equilibrium. This assumption will be applied after partial differential equation (PDE) is obtained.d) Engine surface is assumed to be blackbody, so emissivity equals to one.e) Only x-direction (the direction from engine surface to detector) is considered.Outline of Steps

Derivation from general heat equation

Accumulation of thermal energy (A)=input (I)-output (Ou)+thermal energy generation (GEN)A=I Ou----(1)Let ----(2)Let be the heat flux from engine surface----(3)Let ----(4)----(5)Substitute (2), (3) and (4) into (1)

Applying appropriate assumptionsSteady-state heat transfer is assumed so since there is no accumulation of thermal energy. Since steady state heat transfer also means that heat transfer is no longer time-dependent, the PDE can be expressed as an ordinary differential equation (ODE).

Convert to MATLAB format Due to the 3rd term T3, this 2nd order ODE is non-linear so MATLAB will be used to solve the ODE and obtain temperature profile. Since engine temperature at engine surface is known. This problem should be categorized as an initial value problem (IVP). Hence, ODE45 command in MATLAB will be used. But the equation needs to be converted into a form that ODE45 can use.Let y1=T, so So Therefore,

With initial condition at T(0)=753 KAnd the constants ath= 200 W/m2.K (convective heat transfer coefficient)k= 0.02816 W/m.K (thermal conductivity of air)=1 (emissivity; since it is blackbody so it will be 1)=5.67X10-8 W/m2.K4 (Stefan-Boltzmann constant)