## 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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