## Apd

N7 = fNRe const

Because this is the only ''variable'' that is needed to describe this system, it follows that the value of this group must be the same, i.e., a constant, for the laminar flow of any Newtonian fluid at any flow rate in any pipe. This is in contrast to turbulent pipe flow (which occurs for NRe > 4000) in long pipes, which can be described completely only by three groups (e.g., f, NRe, and e/D). That is, turbulent flow in two different pipes must satisfy the same functional relationship between these three groups even though the actual values of the individual groups may be quite different. However, for laminar pipe flow, since only one group (fNRe) is required, the value of that group must be the same in all laminar pipe flows of Newtonian fluids, regardless of the values of the individual variables. The numerical value of this group will be derived theoretically in Chapter 6.

As an example of the application of dimensional analysis to experimental design and scale-up, consider the following example.

Example 2-3: Scale-Up of Pipe Flow. We would like to know the total pressure driving force (AP) required to pump oil (p = 30 cP, p = 0.85 g/cm3) through a horizontal pipeline with a diameter (D) of 48 in. and a length (L) of 700 mi, at a flow rate (Q) of 1 million barrels per day. The pipe is to be of commercial steel, which has an equivalent roughness (e) of 0.0018 in. To get this information, we want to design a laboratory experiment in which the laboratory model (m) and the full-scale field pipeline (f) are operating under dynamically similar conditions so that measurements of AP in the model can be scaled up directly to find AP in the field. The necessary conditions for dynamic similarity for this system are and from which it follows that where the subscript m represents the experimental model and f represents the full-scale field system. Since the volumetric flow rate (Q) is specified instead of the velocity (V), we can make the substitution V = 4Q/^D2 to get the following equivalent groups:

Note that all the numerical coefficients cancel out. By substituting the known values for the pipeline variables into Eq. (2-12), we find that the value of the Reynolds number for this flow is 5.4 x 104, which is turbulent. Thus all three of these groups are important.

We now identify the knowns and unknowns in the problem. The knowns obviously include all of the field variables except (AP)f. Because we will measure the pressure drop in the lab model (AP)m after specifying the lab test conditions that simulate the field conditions, this will also be known. This value of (AP)m will then be scaled up to find the unknown pressure drop in the field, (AP)f. Thus,

There are seven unknowns but only three equations that relate these quantities. Therefore, four of the unknowns can be chosen "arbitrarily." This process is not really arbitrary, however, because we are constrained by certain practical considerations such as a lab model that must be smaller than the field pipeline, and test materials that are convenient, inexpensive, and readily available. For example, the diameter of the pipe to be used in the model could, in principle, be chosen arbitrarily. However, it is related to the field pipe diameter by Eq. (2-11):

m ef

Thus, if we were to use the same pipe material (commercial steel) for the model as in the field, we would also have to use the same diameter (48 in.). This is obviously not practical, but a smaller diameter for the model would obviously require a much smoother material in the lab (because Dm ^ Df requires em ^ ef). The smoothest material we can find would be glass or plastic or smooth drawn tubing such as copper or stainless steel, all of which have equivalent roughness values of the order of 0.00006 in. (see Table 6-1).

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