Estimation of Head Loss

In piping system design, a central role is played by Equation 2.30. If we limit consideration to liquids modeled as incompressible, flowing through and com- pletely filling circular ducts, Equation 2.30 takes the form

(2.84) + ^- -

+

h, = 0

- I

P 2 m

where the term h,, the head is given by Equation 2.3 repeated here for ease of reference:

(2.31)

pipe friction resistances

With the assumption of one-dimensional flow, the velocities appearing in these expressions are frequently written in terms of the volumetric flow rate Q and cross-sectional area A : V = Q = 4Q

Equation 2.31 represents the head loss as the sum of two contributions called, respectively, the major and minor losses. The major loss summation term accounts for pipe friction. The minor loss summation term accounts for flow through various resistances to be discussed later.

Referring to the first summation on the right side of Equation 2.31, we see that the major losses are evaluated using the friction factor for each of the pipes making up the overall The friction factor is a function of two dimensionless groups: the relative roughness where denotes the roughness of the pipe's inner wall surface, and the Reynolds number

where denotes The graphical representation of the friction factor function developed from experimental data, called a Moody diagram, is pre- sented in Figure 2.5, together with a sampling of values for the roughness.

Three regimes are identified on Figure 2.5: laminar, critical, and turbulent.

The laminar flow friction factor is a straight line on the log-log plot and is given by = an analytical result that is valid to a Reynolds number

Fanning friction factor f used here and differs from the Darcy friction factor by a factor of 4: 4f =

roughness parameter is denoted by k, instead of the usual ^E to avoid overlap with other uses in this book for the symbol E .

4

4

8 E

C

c

100

2.6 MODELING AND DESIGN OF PIPING SYSTEMS

of about 2000. The Reynolds number in the critical zone, where the flow may be either laminar or turbulent, is about 2000-4000. In the transition from laminar to turbulent flow, the friction factor increases sharply and then de- creases gradually for smooth pipes as the Reynolds number increases. For rough pipes, the turbulent friction factor is determined by the relative rough- ness together with the Reynolds number. At sufficiently high Reynolds num- bers, however, the turbulent friction factor is determined by the relative rough- ness alone:

-

This is indicated on Figure 2.5 as rough zone.

To assist computer-aided design and analysis, the friction factor function (Re,, D) is available in several alternative mathematical forms. The brook equation is often cited for turbulent flow:

- - 1 - -4 log

+

1.256

Re, (2.85)

Since this expression is implicit in f , iteration is required to obtain the friction factor for a specified Re, and

The literature includes a number of alternative expressions giving the fric- tion factor explicitly. For example, an expression that represents the friction factor continuously for laminar through turbulent flow is the following

=

[

⁺

where

1

+

and

16

(2.86)

A plausible empirical approach for evaluating friction factors of noncircular ducts is to use the hydraulic diameter in place of D in the graphical and analytical approaches considered above. The hydraulic diameter is defined by

cross-sectional area wetted perimeter

4 (2.88)

Turning now to the second term on the right side of Equation 2.31, note that the minor losses are evaluated using a loss coefficient K for each of the

THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

resistances included within the overall system. Typically encountered resis- tances include pipe inlets and outlets, enlargements and contractions, pipe bends and tees, valves, and other fittings. Experimental loss coefficient data are abundant, but scattered among various sources, including handbooks [7, manufacturers’ data books and textbooks [3, 4, Table 2.1 gives a sampling of loss coefficients for use in Equation 2.3 Except as noted, the velocity downstream of the resistance is used to evaluate the associated loss. As different sources may give somewhat different values for the loss coefficient for the same application, the values of Table 2.1 should be con- sidered as only representative.

2.6.3 Piping System Design and Design Analysis

Single-path and multiple-path piping systems such as pictured in Figure 2.4 can be analyzed (or specified) by using Equation 2.84, together with Equation 2.31 and appropriate friction factor and loss coefficient data. Let us consider this for the case of single-path systems. Although such systems may consist of pipes having different lengths and diameters as well as various fittings, the discussion is considerably simplified if we think only of a single pipe of length L and diameter D without elevation change. Then, Equations 2.84 and 2.31 combine to read

2

(2.89) where we have written the velocity and Reynolds number in terms of the volumetric flow rate: V = Re, = and A p denotes the pressure drop from inlet to outlet: Ap -

If the pipe roughness and fluid properties are specified, Equation 2.89 in- volves four quantities: Ap, Q, L, and D, any one of which may be regarded as the unknown. Depending on the choice for the unknown quantity, the solution may be direct or iterative:

Direct Solutions

Ap unknown; L, Q, and D known 2. L unknown; Ap, Q, and D known Iterative Solutions

3. Q unknown; L, and D known 4. D unknown; Ap, L, and Q known

Cases 1 and 3 exhibit an analysis character since the geometry (L, D) is known and the corresponding pressure drop or flow rate is to be determined.

2.6 MODELING AND DESIGN OF PIPING SYSTEMS 103

Table 2.1 Loss coefficients for use with Equation 2.31"

Resistance K

Changes in Cross-Sectional Area Rounded pipe entrance

Contraction^b

e

0.1 AR 0.5

Valves and Fittings Gate valve, open

Globe valve, open Check valve (ball), open

elbow, standard 90" elbow, rounded

0.04-0.28

1

0.04-0.08

0.2 6-10 70

"Adapted from References 3, 4, and 7.

ratio, AR = (smaller area).

'When AR = 0, K = 1.0. Velocity used to evaluate the loss is the velocity upstream of the expansion.

104 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

Cases 2 and 4 exhibit a design character since Q and Ap are known and the appropriate geometry or is to be determined. Let us consider these four cases in order:

Cases I and 2. By inspection of Equation 2.89, it is evident that Ap, or L, can be obtained directly using the respective known quantities. In particular, since Q and are known in each of these cases, the friction factor required by Equation 2.89 can be obtained readily from the Moody diagram or an expression such as Equations 2.85 and 2.86.

Case 3. The unknown volumetric flow rate Q appears in Equation 2.89 both explicitly as the quadratic term and implicitly in the friction factor function via the Reynolds number. Thus an iterative solution is required. Since most practical pipe flow applications involve turbulent flow and the turbulent flow friction factor only weakly depends on the Reynolds number, the first iteration may be made using the fully rough- turbulent friction factor corresponding to the known value for the relative roughness. With this trial value for Equation 2.89 is readily solved for the volumetric flow rate. The Reynolds number is computed for this value of Q and a new value for f determined. Equation 2.89 is then solved for a second value for Q. The iterative procedure continues until convergence is attained. Convergence tends to occur quickly, however, and often with as few as two iterations.

Case 4. The unknown diameter appears in Equation 2.89 both explic- itly as and implicitly in the friction factor function via both the Reyn- olds number and the relative roughness. Thus an iterative solution is required: Iteration may begin by assuming a first-trial pipe diameter. With this, the Reynolds number and relative roughness are calculated and then used to determine the friction factor. The pressure drop Ap is evaluated next from Equation 2.89 and compared to the known value for Ap. If the calculated value for A p is smaller than the known value, another iteration would proceed with a smaller assumed diameter. If the calcu- lated value for Ap is greater than the known value, another iteration would proceed with a greater assumed diameter. Iteration would continue until satisfactory convergence is achieved between the calculated and known A p values.

Fluid mechanics texts typically provide solved examples illustrating these cases 41.

The cases we have just considered provide a reasonable basis for the anal- ysis of more complex piping configurations. For example, a system consisting of a series of single-path elements as in Figure can be analyzed by applying such pipe flow fundamentals in a systematic, sequential manner from the inlet of the overall system to its outlet. Multiple-path systems such as in Figure also can be analyzed with these fundamentals, but we must

2.6 MODELING AND DESIGN OF PIPING SYSTEMS 105

ognize additionally that (i) the total flow rate is the sum of the individual flow rates in the multiple paths: = Q ,

+ +

and (ii) the pressure drop for each of the multiple paths is the same: - = ( A p ) , ^{= =} Although the pipe flow governing relations have relatively simple mathe- matical forms, iterative solutions of them can be time consuming, especially for multiple-path systems. Accordingly, computer-aided approaches are de- sirable. The literature contains many specialized computer programs for this purpose, as for example the programs provided in Reference 12. One partic- ularly systematic computer-aided approach for complex flow networks is based on the highly effective Hardy-Cross method [ Several software development companies market generalized Hardy-Cross piping system pro- grams executable on microcomputers and in some versions on programmable hand-held calculators.

In document Bejan (Page 114-120)