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. 

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

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 /can be obtained from the equations, because e/D is also unknown. Thus the following procedure is recommended for this problem  

Determine/from the Moody diagram or Churchill equation using the above values of NRe and e/D (if NRe 2000, use f = 16/NRe). 

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

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The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

Equation (7-25) is implicit for Dec, because the friction factor (/) depends upon Dec through the Reynolds number and the relative roughness of the pipe. It can be solved by iteration in a straightforward manner, however, by the procedure used for the unknown diameter problem in Chapter 6. That is, first assume a value for/ (say, 0.005), calculate Z>ec from Eq. (7-25), and use this diameter to compute the Reynolds number and relative roughness then use these values to find / (from the Moody diagram or Churchill equation). If this value is not the same as the originally assumed value, used it in place of the assumed value and repeat the process until the values of / agree. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

We will illustrate the procedure for solving the three types of pipe flow problems for high-speed gas flows unknown driving force, unknown flow rate, and unknown diameter. 

The procedure for an unknown diameter involves a trial-and-error procedure similar to the one for the unknown flow rate. 

As before, everything in this equation is known except for ARe>pl, which can be determined by iteration (or by using the solve spreadsheet or calculator command). When this is found, the unknown diameter is given by... 

Example 11-1 Unknown Velocity and Unknown Diameter of a Sphere Settling in a Power Law Fluid. Table 11-1 summarizes the procedure, and Table 11-2 shows the results of a spreadsheet calculation for an application of this method to the three examples given by Chhabra (1995). Examples 1 and 2 are unknown velocity problems, and Example 3 is an unknown diameter problem. The line labeled Equation refers to Eq. (11-32) for the unknown velocity cases, and Eq. (11-35) for the unknown diameter case. The Stokes value is from Eq. (11-9), which only applies for - Re,pi < 1 (e.g., Example 1 only). It is seen that the solutions for Examples 1 and 2 are virtually identical to Chhabra s values and the one for Example 3 is within 5% of Chhabra s. The values labeled Data were obtained by iteration using the data from Fig. 4 of Tripathi et al. (1994). These values are only approximate, because they were obtained by interpolating from the (very compressed) log scale of the plot. 

Table 11-1 Procedure for Determining Unknown Velocity or Unknown Diameter for Particles Settling in a Power Law Fluid...

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

A liquid with a specific gravity of 2.6 and a viscosity of 2.0 cP flows through a smooth pipe of unknown diameter, resulting in a pressure drop of 0.183 Ib /in. for 1.73 mi. What is the pipe diameter in inches if the mass rate of flow is 7000 Ib/h ... 

A liquid (/t = 0.001 pascal seconds, p = 4700 kg/m ) flows through a smooth pipe of unknown diameter at a mass flow rate of 0.82 kg/sec. The pressure drop for the 2.78-km-long pipe is 1262 pascals. What is the pipe s diameter ... 

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