Eliminate Iteration from Flow Problems

Friction in Pipe Flow - The Classic Approach

The Bernoulli equation, also known as the mechanical

energy balance, is the basis for understanding flow in pipes:

v²         g          AP

A-+-Az+-+/111 + W -0

2g,    g, p

The first term, accounting for kinetic energy changes. is usually small compared to the other four terms, and thus can be assumed to be negligible. Note that for line sizing puqmses, the envelope for a mechanical energy balance typically excludes rotating equipment (e.g., a pump or a turbine) and focuses on the terminal pressure conditions, elevation changes, and friction losses: the rotating equipment is then sized to meet the conditions derived from this mechanical energy balance.

The second tc1m in the energy balance refers to elevation changes; those are typically defined as pait of the conceptual layout of the proposed line. The third term, refer^ring to the difference in terminal pressures, is also typically defined as part of the piping layout. This approach focuses on the fourth term, lw.r, the work lost due to friction:

In this equat1on,Jis the Darcy friction factor: 

The Darcy friction factor equals four times the Fanning

friction factor,f_f'am,;,.8. To use the Fanning friction factor, substitute 4/;0,,,,1,.8 for/wherever the latter appears.

The friction factor is a function of both the fluid

Reynolds number, Re: Dvp/�1, and the relative roughness t/D. The relationship between these quantities and the friction factor is expressed graphically in a Moody plot, or mathematically in various empirical relations (e.g., the Colebrook or Churchill equations). To solve the types of problems considered here , it is necessary 10 introduce dimensionless   quantities that do not depend on the line size, D, or the fluid velocity, v.

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 ^ft^1/5 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

1. 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.)

Example 1

A refinery needs to move 60,000 bbl/d (1 ,750 gal/ min) of a product with an AP I specific gravity of 30 (approximately 54.7 l b/ft3) and a viscosity of 1.8 cP. The pressure at the new 1ie -in point (i.e., the source) for this new line is 90 psig. Due to the design of existing equipment, the discharge pressure downstream cannot exceed 15 psig. The proposed line routing (approximately 12,000 equivalent ft, accounting for fittings) is essentially flat. Refinery specifications call for carbon steel pipe to be used. What size pipe is needed?

Solution. The data for the problem are given in Table 4.

The calculations are detailed in the box on the next page. 

1. Calculate 0 using Eq. 6, and then log 0 . 0 = 9.562,
2. log 0 = 0.9805.
   1.  Calculate <t> us in g Eq. 8, and then lo g <I>. <t> = 1 2,511. lo g <'J) = 4 .0 9 73.
   2.  Use the generating polynomial, Eq. 10, 10 get coefficients for the Ka-O relationship (for simplicity, use the quadratic form). These coefficients are a=3.5707, b=1.0363, and c= -0.00058.
   3.  Use these coefficients in Eq. 9 and value of O calculated in Step 1 to generate Ka: log Ka = 3.5707 + (1.0363)(0.9805) + (-0.0058)(0.9805)² = 4.5862.

3. So, KA = 38,59.89054

   5. Solve for D using Eq. 1. D= 0.882 ft = 10.589 in.

   Since minimum pressure drop was specified, the next-smaller commercially available line size would be chosen to ensure that this minimum drop criterion is met. Thus, a 10-in line (actual ID = 10.02 in. for Sch. 40 pipe) should be selected.
4. 
   

Estimating Flow

This approach can also be used to calculate the expected flow through a pipe of a specified size with a known pressure drop. The terms used to calc ula1e Ko (and 1hus log Ka) are known. Again fully turbulent flow is assumed, since that is common in industry. Then EID can be calculated based on the known line size and the nature of the pipe, and fr can be obtained from the tabulated value, as mentioned previously. That allows <J)7 to be calculated. As well as the coefficients a, b, c and d for the correlating polynomial relating log Ka to log O.

This new approach uses the fully turbulent friction factor for a given line size as a contributing parameter to the generating equation (Eq. 10). The results of the generating equation are ultimately employed 10 get a value of 0 and thus the now. One could then reevaluate the friction factor based on this flow and the known line size and use the generating equation again, but it's up to the user to evaluate whether this would have practical value.

In the traditional approach (specifying physical properties and terminal conditions), one may calculate flow directly by postulating complete turbulence in addition to the known constraints (pressure drop, line length, physical properties). In that case, it is necessary to verify that the calculated flow does indeed yield a Reynolds number that qualifies as fully turbulent. Again, further refinement by iteration would be at the user's discretion.

It was suggested previously that a cubic polynomial should suffice to relate log Ka to log 0, based on the observation that a cubic polynomial is the highest order equation that can relatively easily be solved analytically. (It is possible to solve a quartic polynomial analytically, but the solution is considerably more involved than solving a cubic polynomial. There is no general analytic solution for higher-order polynomials - and it's unlikely that there would be much, if any, benefit to using one.) Methods of solving cubic and quadratic equations are described in detail at hnp:1/mathworld.wolfram.com (4).

Of the three roots arising from the solution of a cubic equation, only one i or interest. Values of log ? within 1he scope of this correlation fall within the approximate range of - 3.5 < log O <7.5.

With log 0 and thus 0 known, a flowrate can be found by solving Eq. 6:

Example 2

Chilled water at 45°F flows from a constant-level reservoir through a 2-in. Sch. 40 steel pipe, the end or which is open to the atmosphere. The pipe has an equivalent length of 175 ft, and the outlet is 35 ft below the liquid level in the reservoir. Neglecting any kinetic energy contributions, determine the flow.

Solution. The data for this problem appear in Table 5. In the interest of space, the calculations are not detailed here.

1. Using Eq. 7, calculate Ka= 16,80 1.2. Then, log Ka = 4.2253.

1. Since the nature of 1he pi pe and 1he line size are known, EID, as well as the fully turbulent friction factor for that relative roughness, are also known. Therefore, <I> is obtained from it definitio n, Eq. 8: <I>= lf1'5( E/D) J- t=

2 ,536.98 1. So log (I)= 3.404317.

1. Use <I> and the gene rating polynomial {Eq. I 0) 10 calculate the coefficient   a, band c of 1he co1Tela1ing polynomial (Eq. 9) . For the purpose of illustration, use a quadratic generating polynomial. Thus: a= 2.9216 , b = 0.970,

and c = 0.002223.

1. Solve the quadratic correlating polynomial for 0:

where f:I = b = 0.970 , y = a - log Ka = - 1 . 3034,  a = c =0.002223. Thus, x = log 0 = 1 .3 40 and -437.766. The first root reflects the use of the positive sig n of the radical; the second root, the negative sign. Only the first root has any physical significance, so with log 0 = I .340 , 0 = 2 1 .878.

Use Eq. 12 10 derive the flowrate: Q = 0.259 f tl/s =1 16 .4 gal/min

The Special Case of Smooth Pipe

By definition, smooth pipe has zero roughness, which would render the definition of 0 use less. However, Gilmont's work (5) ca n be modified to co1Telat e Ka as a function of 0 *, where 0 * = Rej11S , to obtain a  relationship where one variable is independent of the line size and the other independent of the flowrate. Like the 0-Ka correlation, this 0 * -Ka correlation c n be represented well by a cubic polynomial. Therefore, the techniques discussed above for rough pipe can be applied to smooth pipe using the data at the bottom of Table 3 and the relationship in Figure I .