False Transients Applied to PDEs

The method will be illustrated using Equation 16.7, with constant p and /r, to find the velocity profile in a rectangular duct. Equation 16.7 is converted to an ODE by using second-order approximations for the spatial derivatives. The result is 

This set of ODEs can be solved by any convenient method. The computational template is shown in Eigure 16.3. Five points, centered around (x, y, t) on thex-y plane, are used to project ahead to one point, V (x, y, t + At), at the new time. 

Set Ax = Ay (this does not imply a square duct with W = H). Then 

Determine the axial velocity profile for laminar flow in a square duct. Also determine the pressure drop per unit length for water in a square duct with H = pm (each side is 2 pm) 

SOLUTION Code for Example 16.1 uses Euler s method for solution. Some results are as 

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For circular pipes, Rh = R- The reader is cautioned that some definitions of Rh omit the factor of 2 shown in Equation 3.22 so that the result must be multiplied by 2 for use in equations such as 3.18 and 3.19. The use of Rh is not recommended for laminar flow, but alternatives are available in the literature. Also, the method of false transients applied to PDEs in Chapter 16 can be used to calculate laminar velocity profiles in ducts with noncircular cross sections. For turbulent, low-pressure gas flows in rectangular ducts, the American Society of Heating, Refrigerating and Air Conditioning Engineers recommends use of an equivalent diameter defined as... 

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