Eliminating the Diameter
Bennen and Myers (I) suggest that a plot or conelation of the friction factor./, a arunction of Refl' 5 would be useful in solving for line size if the flowrate and pressure drop are known. This dimension less quantity ( Ref 1S ) is drawn from two quantities already establishe d, namely, the Reynolds number and the Darcy friction factor. The velocity, v, appears in both Re and/ It ca n be eliminated by using the definition of the velocity:
V c:c -Q = - Q- - ~ - 4Q-
A n;D¼ ;r,D 2
The cliameler has now been eliminate d. Butf, and thus Ref IS, depends on the relative roughness. which presupposes knowledge of the pipe diameter. To get around t his , a new dimensionless quan1ity , the flow function , is introduced:
which can be calculated from known quantities. The absolute roughnes , E, is a function of the nalure of the pipe , which i known, and is independent of the line diameter.
A complementary dimensionless quantity that el imitates dependence on the flow is also needed. This has already been established in the form of the Karman number, Ka:
Many texts provide plots of the friction factor a function of the Karman number with the relative roughness as a variable. This may not be particularly helpful, however, since it presumes knowledge of the pipe diameter, which may not be the case. To work around this, another new dimension less quantity, the friction/roughness function, is defined:
Guidelines for sizing pipe (e.g., Peters and Timmerhaus (2)) include typica l velocity in the range of 3-10 ft/s for liquids and 50--150 ft/s for gases. A quick check of Reynolds numbers calculated using velocities within these range s and properties of commo n industrial liquids (e.g.. organ ic liquids with viscosities of approximately IeP and densities on the same order of magnitude as that of water) show that flow meeting the se guidelines are indeed turbulent; similar comments apply for industrial gases. Most design courses guide the student to design for turbulent flow in pipelines, since this is perhaps the most common situation in industry.
Thus, it is reasonable to assume full turbulence and, as a corollary. define a slightly modified version of <I> to use the fully turbulent Darcy friction factor, f-t. This modified quantity <I>, is the product of ft1/5 and the relative roughness E/D. This quantity will prove useful in one of the later variations of the classic flow problem presented later.
For the calculation of flow based on pipe size and pressure drop, values of fT as a function of line size or relative roughness are tabulated in the literature (3) and listed in Tables 1 and 2.
Now we have dimensionless quantity, O, that is independent of line size that can be used to correlate another dimensionless quantity, Ka. This latter dimensionless quantity if independent of flow.
Plotting log Ka as a function of log 0 and 0 7 yield a family of parallel curves. Multiple regression analysis shows that these curves can be represented by polynomials with log 0 as the in dependent variable and the form:
log Ka= a+ b( log 0 ) + c (log 0 )2 + d(log0 )3 + ... (9)
We'll call this relationship the correlation polynomial. A cubic polynomial is generally sufficient to correlate log Ka as a function of log 0 (for reasons that will be dis ussed later). Multiple regression analysis shows that a, b, c, d ... are well-correlated by polynomial. i n log <l>T. Thus:
y =A+ B( log <I>r) + C(log<l>r )
+D ( log <l)r )3 + E ( log <l>r )4 (10)
where y is a generic coefficient for Eq. 9. We'll call this relationship the gene rating polynomial to distinguish it from the correlating polynomial that relates log Ka 10 log
- A quartic (fourth-degree) polynomial is sufficiently accurate for most work. 171e coefficients A, B, C. D andE for Eq. IO are given in Table 3.
Knowing the physical definition of the system (line [equivalent length, pre sure drop, nature of the pipe), the flowrate and the fluid 's physical properties (density, viscosity) allows 0, log 0 , <J:>7 and log <1>7 to be determined. Log <J:>7 can be used to calc u late the coefficients a, b, c and d (via Eq. 10 ) and the tabulated values for A, B, C, D and E, which are used with Eq. 9 10 yield log Ka and therefore Ka. Finally, knowing Ka leads directly to a theoretical line size:
It is unlikely that the diameter calculated in this way will be equal 10 a commercial pipe size. In most cases the pressure drop specification is a maximum allow able pressure drop, so the next larger pipe size should be chosen. (Conversely if a minimum pressure drop were specified, the next smaller line size would be chosen.)