Thermal energy balance steady state conduction

Write the steady-state thermal energy balance for an incompressible fluid which includes transport by convection and conduction. Neglect viscous dissipation. 

A two-dimensional bar has the geometry shown in Fig. P5-114 with. specified temperature T,iOn the upper surface and Tg on the lower surfaces, and insulation on the sides. Tiie thermal conductivity of tlie upper part of the bar is while that of the lower part is kg. For a grid defined by A.r = Ay = /, write the simplest form of the matrix equation, AT = C, used to find the steady-state temperature field in the cross section of the bar. Identify on the figure the grid nodes where you write tlie energy balance. 

Finally, we often assume that the diffusivity, thermal conductivity and partial molar enthalpies are independent of temperature and composition to produce the following coupled mass and energy balances for the steady-state problem... 

In the preceding sections we assumed that the flow was in a steady state, where the amount of a quantity flowing into any volume element is balanced by an equal amount of the quantity flowing out in the same time interval. For diffusion this means that the concentration in any volume element is independent of time, dN/dt = 0. For thermal conductivity it means that energy does not accumulate in any volume element, or that dT/dt = 0. 

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