M 4QpdVq ir

5. Compare the value off from step 4 with the assumed value in step 1. If they do not agree, use the value of f from step 4 in step 2 and repeat steps 2-5 until they agree. Note that an iteration is required to determinefL in Eq. (6-65), but this procedure normally converges rapidly unless unusual conditions are encountered.

C. Unknown Diameter

In this problem, it is desired to determine the size of the pipe (D) that will transport a given fluid (Newtonian or non-Newtonian) at a given flow rate (Q) over a given distance (L) with a given driving force (DF). Because the unknown (D) appears in each of the dimensionless variables, it is appropriate to regroup these variables in a more convenient form for this problem.

1. Newtonian Fluid

The problem statement is

We can eliminate the unknown (D) from two of the three basic groups (NRe, e/D, and f) as follows:

Thus, the three basic groups for this problem are fNRe, NR, and NRe, with NRe being the dimensionless "unknown" (because it is now the only group containing the unknown D). D is unknown, so no initial estimate for f can be obtained from the equations, because e/D is also unknown. Thus the following procedure is recommended for this problem:

1. Calculate fNRe from known quantities using Eq. (6-78).

3. Calculate NRe:

4. Calculate D from NRe:

5. Calculate e/D.

6. Determine ffrom the Moody diagram or Churchill equation using the above values of NRe and e/D (if NRe < 2000, usef = 16/NRe).

7. Compare the value of ffrom step 6 with the assumed value in step 2. If they do not agree, use the result of step 6 for f in step 3 in place of 0.005 and repeat steps 3-7 until they agree.

2. Power Law Fluid

The problem statement is

The procedure is analogous to that for the Newtonian fluid. In this case, the combined group /NR^3^ (which we shall call K, for convenience) is independent of D:

' n2DF \32LQQ2

The following procedure can be used to find D:

3. Calculate NRe pl from

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