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COLEBROOK-WHITE FORMULA FOR PIPE FLOWS Grant Keady

∗

ABSTRACT: Flow resistance laws, as used for example in water-supply pipe networks, are formulae relating the volume flow rate q along a pipe to the pressure-head difference t between its ends, q = ψ(t). ψ is monotonic. The simple Hazen-Williams power “law” is often used: in appropriate circumstances the more complicated Colebrook-White law (CW) may better represent aspects of the experimental data. Result 1, the first and easiest-to-state result in the paper, is that φCW , the inverse to ψCW , can be expressed in terms of the Lambert W -function (Corless et al. 1993). One use of this and of related results, in connection with convex optimization problems describing equilibrium flows in pipe networks, is summarised in Result 2.

INTRODUCTION There is a substantial, and growing, body of knowledge concerning general network flow problems. See, e.g. Rockafellar 1984. In network flow problems, there is a potential function p defined over the nodes and a flow q defined over the arcs. Furthermore the flow on any arc a is a given increasing function of the ‘tension’ t on the arc, the ‘tension’ being defined as the difference in potential between the arc’s initial and terminal node. In the case of equilibrium flow in pipe networks, the potential is the head. Two of the commonly used empirical rules giving the flow as a function of head-difference are Colebrook-White and Hazen-Williams. ‘Power-law nonlinearity’ flow-laws are where the flow qa on an arc a ∗

Mathematics Department, University of Western Australia, Nedlands, 6907, Australia.

[email protected]

1

joining node i to node k satisfies, for some positive constant ka , qa = ka σ(p(i) − p(k), sa ) , σ(t, s) = t|t|s−1 , s > 0.

(1)

Hazen-Williams’ means a power-law flow-law with s = 27/50. (The ‘rough-pipe limit’ of Colebrook-White is equation (3) with c3 = 0: this is a power-law flow-law with s = 1/2.) The first part of this paper, in the next subsection, is concerned with elementary mathematical facts about the Colebrook-White rule. These are of use, for example, in connection with the convex optimization formulations of the network-flow problem. In the final section of the paper we give reasons why the convex optimization approach to the network problem is important.

A single pipe Consider the case of a single pipe. Let D be the diameter, L the length, and K the roughness of the pipe. Let g denote the acceleration due to gravity and ν the kinematic viscosity of water. Define π c1 = ln(10)

s

gD5 , 2L

10K c2 = , 37D

s

c3 = ν

L . 2gD3

(2)

Note c2 < 1. Let q denote the flow down the pipe, and t head difference between the ends of the pipe. The Colebrook-White rule for the turbulent flow down a pipe states that an approximation to the flow rate q along the pipe is, √ c3 q = ψCW (t) = −c1 t ln(c2 + √ ), t

(3)

when t is sufficiently large positive that the argument of the ln is less than one, i.e. t ≥ t0 =

c23 . (1 − c2 )2

(4)

(ln denotes the natural logarithm.) The simplest way of extending this so that it is defined for all positive values of t is to define ψCW (t) = 0 for 0 < t ≤ t0 . (Physically this is zero flow with a nonzero head difference, but one shouldn’t be applying a turbulent flow drag law with really tiny flow rates. The simplest extension was chosen to remove the even-more-unphysical flows in the wrong direction suggested by the original formula (3) at tiny t, which would have spoiled the convex optimisation formulation treated later in the paper.) 2

In our general considerations of network flows, we often treat more general flow laws q = ψ(t). As in the Colebrook-White case, we consider only ψ which are nondecreasing functions of t and which have ψ(0) = 0. The extension to negative t is always done so that ψ is an odd function of t. However, for the remainder of this section we need only consider ψCW . It is sometimes useful to express t in terms of the flow q, t = φ(q). This is done as follows. RESULT 1. ψCW is nondecreasing on (0, ∞), increasing on (t0 , ∞), ψCW → ∞ as t → ∞. For t > t0 the inverse function φCW to function ψCW defined by equation (3) is

φCW (q) =

2

c3

,

c2 q

q exp( c c ) c1 c3 1 3 ) q W( c1 c3

for q > 0.

(5)

− c2

Here W denotes the Lambert W -function, defined by W (x) exp(W (x)) = x (and we need only x > 0, W (x) > 0). φCW is increasing on (0, ∞), φCW (q) → ∞ as q → ∞. This was discovered by running the single line of Maple: psi:= -c1*sqrt(t)*log(c2+c3/sqrt(t)); solve(q=psi,t); selecting the physically-relevant solution, and then checking by hand-calculation. φCW has, of course, been approximated numerically by engineers for some time. There are many other uses of the Lambert W -function, and it seems possible that exchanges concerning the numerical methods used in engineering to calculate φCW , and the numerical methods used in these other uses of W could be mutually beneficial. (By way of caution, we comment that different applications have different needs. In the pipe-network application, speed is more likely to be important than great accuracy. Result 1 is not presented in any expectation of immediate gains in efficiency of numerical computation of φCW over what engineers currently use in well-tested codes. Result 1 does, however, enable engineers without access to a pre-written Colebrook-White code to use the standard publicly-available Lambert W function routines and thereby quickly produce their own Colebrook-White codes.) See Corless et al. 1993, and Barry et al. 1995, for references, including methods for efficient numeric evaluation of W . We are not sure of the final uses of Result 1, and we confine our own suggested uses to just one - following from the fact that φCW can be integrated in closed form - at the end of the paper.

3

NETWORKS Pipe networks of various kinds arise in practice (e.g. Brebbia and Ferrante 1983, Chau 1995). There are a variety of practical problems. For the rest of this paper, consider a network where the details of all pipes – Ka , La and Da –are given as are all consumptions at the nodes. A reference head at one node is also given, and the problem is to determine the heads at all remaining nodes. Some computer codes (e.g.

Brebbia and Ferrante 1983) continue to use Newton-

iteration methods with full Newton steps for solving for these equilibrium flows. For these methods, finding a good starting guess is often a serious difficulty. It is, we feel, better to exploit the fact that the problem is a convex optimization problem, a consequence of which is that globally convergent algorithms based on partial Newton steps are available. The remaining notes in this paper indicate that this can be done even when the analytically unpleasant Colebrook-White law is used.

Convex optimization formulations For reasons of space, this subsection is not self-contained. For readers who do not regularly work in the area, the cited references will be essential. The equilibrium flow problem, for monotonic φ and ψ, can be formulated as a convex optimization problem: see Collins et al. 1978, Calvert and Keady 1993 and 1995, Rockafellar 1984. There is a useful convex duality theory. The references above will suffice for the general problem. The integrals of φ and of ψ are needed to form the dual pair of optimization problems. The objective function in a primal optimization problem involves the indefinite integral of ψ: that for the dual optimization problem involves the indefinite integral of φ. As the integral of a power-law is very easy, several people have implemented numeric codes for convex optimization formulations with Hazen-Williams flow laws. It happens, though we have not seen it noted elsewhere, that for the Colebrook-White law, the integrals can also be evaluated in closed form. RESULT 2. For t > t0 , ΨCW (t) ≡

Z t

ˆ CW (t) − Ψ ˆ CW (t0 ), ψCW (tˆ)dtˆ = Ψ

0

4

(6)

where ˆ CW (t) √ 2c3 Ψ 2 √ c3 2c2 √ c3 = − 33 ln(c2 t + c3 ) − t t ln(c2 + √ ) + 23 t − t. c1 3 3c2 3c2 3c2 t

(7)

For q > 0, ΦCW (q) ≡

Z q

φCW (ˆ q )dˆ q = φCW (q)q − ΨCW (φCW (q)).

(8)

0

The result in equation (7) can be checked by running the Maple: int(psi,t); Equation (8) is just the formula for integration by parts: it expresses ΦCW in terms of the Lambert W function. For details of Result 2, equations (6) and (7), being used in some simple trial networks, see Keady 1995. One use of Results 1 and 2 may be to guide in the organisation of efficient numerics ˆ CW . There are a variety of other problems - sensitivity of solutions for the calculation of Φ to changes in La , Ka , Da as one example. Our hope is that our analytical Results on the Colebrook-White law will find some use with engineers designing pipe-networks.

APPENDIX. REFERENCES D.A. Barry, S.J. Barry and P.J. Culligan-Hensley, 1995. “ Algorithm 743: WAPR: A Fortran routine for calculating real values of the W -function, ACM Trans. on Math. Software, 21, 172-181. C.A. Brebbia and A.J. Ferrante, 1983. Computational hydraulics (Butterworth, London). B. Calvert and G. Keady, 1993. “Braess’s paradox and power-law nonlinearities in networks”, Jnl of the Australian Math. Soc., 35B, 1-22. B. Calvert and G. Keady, 1995. “Braess’s paradox and power-law nonlinearities in networks, Part II”. In Proceedings of the First World Congress of Nonlinear Analysts, WC528, Aug. 92, editor W. de Gruyter, Vol III, pp 2223-2230. K.W. Chau 1995. “Integrated CAD package for storm-water drainage networks”. J. Water Res. Planning and Management 121 (4), 336-339. M. Collins, R. Cooper, J. Helgason, J. Kennington and L. LeBlanc, 1978. “Solving the pipe network analysis problem using optimization techniques”, Mgmt. Sci. 24, 747-760. 5

R.M. Corless, G.H. Gonnet, D.E.G. Hare and D.J. Jeffrey, 1993. “Lambert’s W -function in Maple”, Maple Technical Newsletter 9, 12-22. G. Keady, 1995. “The Colebrook-White formula for pipe networks”. University of Western Australia, Mathematics Department Electronic Report. http://maths.uwa.edu.au/~keady/papers.html R.T. Rockafellar, 1984. Network flows and monotropic optimization (Wiley, New York).

APPENDIX. NOTATION The following symbols are used in this article. A subscript CW is used on φ, ψ, Φ and Ψ to denote the corresponding function for the particular Colebrook-White law. The subscript a below particularises a quantity to a specific arc – pipe – labelled a: where it is not important to specify the arc the subscript may be omitted. c1 , c2 , c3

constants in ψCW ;

W

Lambert W -function;

Da

diameter of pipe a;

ν

viscosity of water

g

acceleration due to gravity;

φ

inverse to ψ;

Ka

pipe roughness for a;

Φ

integral of φ;

La

length of pipe a;

ψ

flow function;

p

pressure head at nodes;

Ψ

integral of ψ;

qa

flow along pipe a;

ka

conductivity factor;

ta

pressure head difference along a;

sa

flow-law power-law exponent;

t0

CW cutoff pressure - see (4) ;

σ

(separable) flow function.

6

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∗

ABSTRACT: Flow resistance laws, as used for example in water-supply pipe networks, are formulae relating the volume flow rate q along a pipe to the pressure-head difference t between its ends, q = ψ(t). ψ is monotonic. The simple Hazen-Williams power “law” is often used: in appropriate circumstances the more complicated Colebrook-White law (CW) may better represent aspects of the experimental data. Result 1, the first and easiest-to-state result in the paper, is that φCW , the inverse to ψCW , can be expressed in terms of the Lambert W -function (Corless et al. 1993). One use of this and of related results, in connection with convex optimization problems describing equilibrium flows in pipe networks, is summarised in Result 2.

INTRODUCTION There is a substantial, and growing, body of knowledge concerning general network flow problems. See, e.g. Rockafellar 1984. In network flow problems, there is a potential function p defined over the nodes and a flow q defined over the arcs. Furthermore the flow on any arc a is a given increasing function of the ‘tension’ t on the arc, the ‘tension’ being defined as the difference in potential between the arc’s initial and terminal node. In the case of equilibrium flow in pipe networks, the potential is the head. Two of the commonly used empirical rules giving the flow as a function of head-difference are Colebrook-White and Hazen-Williams. ‘Power-law nonlinearity’ flow-laws are where the flow qa on an arc a ∗

Mathematics Department, University of Western Australia, Nedlands, 6907, Australia.

[email protected]

1

joining node i to node k satisfies, for some positive constant ka , qa = ka σ(p(i) − p(k), sa ) , σ(t, s) = t|t|s−1 , s > 0.

(1)

Hazen-Williams’ means a power-law flow-law with s = 27/50. (The ‘rough-pipe limit’ of Colebrook-White is equation (3) with c3 = 0: this is a power-law flow-law with s = 1/2.) The first part of this paper, in the next subsection, is concerned with elementary mathematical facts about the Colebrook-White rule. These are of use, for example, in connection with the convex optimization formulations of the network-flow problem. In the final section of the paper we give reasons why the convex optimization approach to the network problem is important.

A single pipe Consider the case of a single pipe. Let D be the diameter, L the length, and K the roughness of the pipe. Let g denote the acceleration due to gravity and ν the kinematic viscosity of water. Define π c1 = ln(10)

s

gD5 , 2L

10K c2 = , 37D

s

c3 = ν

L . 2gD3

(2)

Note c2 < 1. Let q denote the flow down the pipe, and t head difference between the ends of the pipe. The Colebrook-White rule for the turbulent flow down a pipe states that an approximation to the flow rate q along the pipe is, √ c3 q = ψCW (t) = −c1 t ln(c2 + √ ), t

(3)

when t is sufficiently large positive that the argument of the ln is less than one, i.e. t ≥ t0 =

c23 . (1 − c2 )2

(4)

(ln denotes the natural logarithm.) The simplest way of extending this so that it is defined for all positive values of t is to define ψCW (t) = 0 for 0 < t ≤ t0 . (Physically this is zero flow with a nonzero head difference, but one shouldn’t be applying a turbulent flow drag law with really tiny flow rates. The simplest extension was chosen to remove the even-more-unphysical flows in the wrong direction suggested by the original formula (3) at tiny t, which would have spoiled the convex optimisation formulation treated later in the paper.) 2

In our general considerations of network flows, we often treat more general flow laws q = ψ(t). As in the Colebrook-White case, we consider only ψ which are nondecreasing functions of t and which have ψ(0) = 0. The extension to negative t is always done so that ψ is an odd function of t. However, for the remainder of this section we need only consider ψCW . It is sometimes useful to express t in terms of the flow q, t = φ(q). This is done as follows. RESULT 1. ψCW is nondecreasing on (0, ∞), increasing on (t0 , ∞), ψCW → ∞ as t → ∞. For t > t0 the inverse function φCW to function ψCW defined by equation (3) is

φCW (q) =

2

c3

,

c2 q

q exp( c c ) c1 c3 1 3 ) q W( c1 c3

for q > 0.

(5)

− c2

Here W denotes the Lambert W -function, defined by W (x) exp(W (x)) = x (and we need only x > 0, W (x) > 0). φCW is increasing on (0, ∞), φCW (q) → ∞ as q → ∞. This was discovered by running the single line of Maple: psi:= -c1*sqrt(t)*log(c2+c3/sqrt(t)); solve(q=psi,t); selecting the physically-relevant solution, and then checking by hand-calculation. φCW has, of course, been approximated numerically by engineers for some time. There are many other uses of the Lambert W -function, and it seems possible that exchanges concerning the numerical methods used in engineering to calculate φCW , and the numerical methods used in these other uses of W could be mutually beneficial. (By way of caution, we comment that different applications have different needs. In the pipe-network application, speed is more likely to be important than great accuracy. Result 1 is not presented in any expectation of immediate gains in efficiency of numerical computation of φCW over what engineers currently use in well-tested codes. Result 1 does, however, enable engineers without access to a pre-written Colebrook-White code to use the standard publicly-available Lambert W function routines and thereby quickly produce their own Colebrook-White codes.) See Corless et al. 1993, and Barry et al. 1995, for references, including methods for efficient numeric evaluation of W . We are not sure of the final uses of Result 1, and we confine our own suggested uses to just one - following from the fact that φCW can be integrated in closed form - at the end of the paper.

3

NETWORKS Pipe networks of various kinds arise in practice (e.g. Brebbia and Ferrante 1983, Chau 1995). There are a variety of practical problems. For the rest of this paper, consider a network where the details of all pipes – Ka , La and Da –are given as are all consumptions at the nodes. A reference head at one node is also given, and the problem is to determine the heads at all remaining nodes. Some computer codes (e.g.

Brebbia and Ferrante 1983) continue to use Newton-

iteration methods with full Newton steps for solving for these equilibrium flows. For these methods, finding a good starting guess is often a serious difficulty. It is, we feel, better to exploit the fact that the problem is a convex optimization problem, a consequence of which is that globally convergent algorithms based on partial Newton steps are available. The remaining notes in this paper indicate that this can be done even when the analytically unpleasant Colebrook-White law is used.

Convex optimization formulations For reasons of space, this subsection is not self-contained. For readers who do not regularly work in the area, the cited references will be essential. The equilibrium flow problem, for monotonic φ and ψ, can be formulated as a convex optimization problem: see Collins et al. 1978, Calvert and Keady 1993 and 1995, Rockafellar 1984. There is a useful convex duality theory. The references above will suffice for the general problem. The integrals of φ and of ψ are needed to form the dual pair of optimization problems. The objective function in a primal optimization problem involves the indefinite integral of ψ: that for the dual optimization problem involves the indefinite integral of φ. As the integral of a power-law is very easy, several people have implemented numeric codes for convex optimization formulations with Hazen-Williams flow laws. It happens, though we have not seen it noted elsewhere, that for the Colebrook-White law, the integrals can also be evaluated in closed form. RESULT 2. For t > t0 , ΨCW (t) ≡

Z t

ˆ CW (t) − Ψ ˆ CW (t0 ), ψCW (tˆ)dtˆ = Ψ

0

4

(6)

where ˆ CW (t) √ 2c3 Ψ 2 √ c3 2c2 √ c3 = − 33 ln(c2 t + c3 ) − t t ln(c2 + √ ) + 23 t − t. c1 3 3c2 3c2 3c2 t

(7)

For q > 0, ΦCW (q) ≡

Z q

φCW (ˆ q )dˆ q = φCW (q)q − ΨCW (φCW (q)).

(8)

0

The result in equation (7) can be checked by running the Maple: int(psi,t); Equation (8) is just the formula for integration by parts: it expresses ΦCW in terms of the Lambert W function. For details of Result 2, equations (6) and (7), being used in some simple trial networks, see Keady 1995. One use of Results 1 and 2 may be to guide in the organisation of efficient numerics ˆ CW . There are a variety of other problems - sensitivity of solutions for the calculation of Φ to changes in La , Ka , Da as one example. Our hope is that our analytical Results on the Colebrook-White law will find some use with engineers designing pipe-networks.

APPENDIX. REFERENCES D.A. Barry, S.J. Barry and P.J. Culligan-Hensley, 1995. “ Algorithm 743: WAPR: A Fortran routine for calculating real values of the W -function, ACM Trans. on Math. Software, 21, 172-181. C.A. Brebbia and A.J. Ferrante, 1983. Computational hydraulics (Butterworth, London). B. Calvert and G. Keady, 1993. “Braess’s paradox and power-law nonlinearities in networks”, Jnl of the Australian Math. Soc., 35B, 1-22. B. Calvert and G. Keady, 1995. “Braess’s paradox and power-law nonlinearities in networks, Part II”. In Proceedings of the First World Congress of Nonlinear Analysts, WC528, Aug. 92, editor W. de Gruyter, Vol III, pp 2223-2230. K.W. Chau 1995. “Integrated CAD package for storm-water drainage networks”. J. Water Res. Planning and Management 121 (4), 336-339. M. Collins, R. Cooper, J. Helgason, J. Kennington and L. LeBlanc, 1978. “Solving the pipe network analysis problem using optimization techniques”, Mgmt. Sci. 24, 747-760. 5

R.M. Corless, G.H. Gonnet, D.E.G. Hare and D.J. Jeffrey, 1993. “Lambert’s W -function in Maple”, Maple Technical Newsletter 9, 12-22. G. Keady, 1995. “The Colebrook-White formula for pipe networks”. University of Western Australia, Mathematics Department Electronic Report. http://maths.uwa.edu.au/~keady/papers.html R.T. Rockafellar, 1984. Network flows and monotropic optimization (Wiley, New York).

APPENDIX. NOTATION The following symbols are used in this article. A subscript CW is used on φ, ψ, Φ and Ψ to denote the corresponding function for the particular Colebrook-White law. The subscript a below particularises a quantity to a specific arc – pipe – labelled a: where it is not important to specify the arc the subscript may be omitted. c1 , c2 , c3

constants in ψCW ;

W

Lambert W -function;

Da

diameter of pipe a;

ν

viscosity of water

g

acceleration due to gravity;

φ

inverse to ψ;

Ka

pipe roughness for a;

Φ

integral of φ;

La

length of pipe a;

ψ

flow function;

p

pressure head at nodes;

Ψ

integral of ψ;

qa

flow along pipe a;

ka

conductivity factor;

ta

pressure head difference along a;

sa

flow-law power-law exponent;

t0

CW cutoff pressure - see (4) ;

σ

(separable) flow function.

6

We are a sharing community. So please help us by uploading **1** new document or like us to download:

OR LIKE TO DOWNLOAD IMMEDIATELY