Techniques for Convection Problems

For the centered-difference technique, the finite-difference analogs are written about the half point (z + 1/2, n -I-1/2). The reason for this is that this centered differencing develops second-order correct analogs for both the variable and its first derivatives. To get the first-order time partial, we develop Taylor series expansions for the functions u ti, Xn+112) and Xn+1/2) about the half point ( 1+1/2, a +i/2) then we have 

This is a second-order correct analog for dufdt. What is meant by second-order correct is that the linearization procedure truncates after the second-order terms in the Taylor series. By averaging the n + 1/2 point, we can obtain an expression only in terms of the grid points. 

Similarly, we can obtain a second-order expression for the distance derivative as 

One of two methods is usually used to obtain a finite-difference analog for the variable itself. The first just averages the four neighbors of t+i/2,n+i/2 to obtain 

We are interested in computing the dynamic response of the axial temperature profile of the fluid flowing in the inner tube for a step change in the inlet temperature T z = 0). The steam shall maintain a constant wall temperature of T in the exchanger. The describing differential equation is obtained by writing an energy balance around the tube and yields 

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