Lect 12 Economic Diameter I

8/4/2019 Lect 12 Economic Diameter I

1/13

DepartmentofMechanical

Engineering

ME435 ThermalEnergySystemsDesign

Economic Pipe Diameter I

Cost Optimization of Piping Systems

Economic Diameter

Economic Velocity

Current ME 435 Knowledge Base Engineering Economics

Time value of money

Friction effects in fluid flow (Head losses) Straight pipes Minor losses

Optimization Methods Simple calculus methods of function minimization

2

With this knowledge, we would like to determine the

economic diameter of pipe in a fluid system.

The economic diameter optimizes the total cost ofthe pipe system (minimization).


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Costs of a Pipe System

Straight pipes

Valves and fittings

Insulation (if needed)

Installation cost

Pump(s)

Hangers and supports

Others?

Maintenance

Energy costs

Others?

3

Initial Costs Annual Costs

Since these costs do notoccur at the same time in

the life of the pipesystem, interest factors

must be used to equateinitial and annual costs

Cost Optimization Strategy Develop an objective function of the form

Find the diameter that minimizes this function

4

TC f D

0T

dC

dD

Total cost of thepipe system

Pipe diameter

Solve forD

Method developed by Darby and Melson in 1982. See Reference on Table 4.1.This method is presented in Section 4.2 of Design of Fluid Thermal Systems.


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Building the Cost Function

1. Amortized cost of installed pipe per unitlength

2. Amortized cost of fittings, valves, supports,insulation, and pumps per unit length

3. Annual cost of maintenance per unit length

4. Annual cost of moving the fluid through thesystem

5

. . . .T AC L 1 2 3 4Then

Note: Amortized = annualized

Building the Cost Function1. Amortized cost of installed pipe per unit

length




6

. . . .T AC L 1 2 3 4Then


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1. Amortized Cost of Installed Pipe

7

, ,P p pA

AC C i n aC P

Define Cp as the initial cost of the pipe and installation per unitlength. Then, the amortized cost,ACp, of this initial cost perunit length is, amortization rate

(capital recoveryfactor)

Notice in Table 4.1 (a) Data are based on 1980 $/ft(b) There is a new pipe classification

(the ANSI Designation). The

class designation establishestemperature and pressure limitsfor different types of pipe.

1

1 1

n

n

i ia

i


8

y = 1.2457x1.1587

y = 1.7834x1.3627

1

10

100

1000

1 10 100

Installed

pipe

costs

in

1980

$/ft

Nominal Diameter (in)

300#

400#

600#

900#

1500#

Power (300#)

Power (1500#)

Converting Table4.1 to an equation

1

n

pC C DFrom Figure 4.21500# n = 1.35900# n = 1.32600# n = 1.29400# n = 1.20300# n = 1.14

Cost of an arbitrary reference pipe($/ftn+1). The nominal 12 pipe isselected as this reference. Valuesvary between 22 and 53 $/ftn+1 1980 dollars!


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9

1

n

P p AC aC aC D

Then, the resulting equation for the amortized cost ofinstalled pipe per unit length is,

Units

1 1$ $

ftft ft

n n

P n AC a C D

Building the Cost Function1. Amortized cost of installed pipe per unit

length




10

. . . .T AC L 1 2 3 4Then


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2. Amortized Cost of Fittings, etc.

11

This cost represents the amortized cost of fittings, valves,supports, insulation, and pumps per unit length.

Rather than treat these individually, in this method they arelumped together in a multiplication factor, F. Using thisfactor, the initial cost of these items is determined by,

1

n

F pC FC FC D

Initial cost of fittings,valves, supports,insulation, and pumps

per unit length of pipe.

Multiplier that representsthe best estimate of the

additional cost of theseitems. This value is usuallyaround 6 or 7.

2. Amortized Cost of Fittings, etc.

12

To amortize this cost, multiply by the amortization rate (i.e.,capital recovery factor),

1

n

F F AC aC aFC D

1 1$ $

ftft ft

n n

F n AC aF C D

Units


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13

. . . .T AC L 1 2 3 4Then

3. Annual Cost of Maintenance

14

The annual cost of maintenance per unit length is estimatedto be a fraction, b, of the initial cost per unit length of thesystem,

1 1 1 1n n n M P F AC b C C b C D FC D bC D F

Estimated fraction of initial costs per unitlength spent on annual maintenance.Typical value is about 1% (0.01).

Units

1 1$ $

ft 1ft ft

n n

M n AC b C D F


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15

. . . .T AC L 1 2 3 4Then

4. Annual Cost of Moving the Fluid

16

Unlike the first three categories, this cost represents a totalannual cost to move the fluid (not per unit length).

If the fluid is being moved by a pump, this annual cost canbe determined by,

2OP

C tWAC

This is the same asCOP in Equation (4.9)

C2 = cost of energyt= annual operating time of the pump

= power transferred to the fluid = efficiency of the pump

W


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17

The power input to the fluid can be found by applying theenergy equation to a general pump/pipe system as shownbelow,

W

2 21 21 2

1 2

2 22 2

1 21 21 2

minor losses in all head loss due to frictionfittings in the in all straight pip

control volume

2 2

2 2 2 2

c c cm f

c c c k k k k k

k k k

Wg Pg P gV V z z l l

mg g g g g

Wg Pg P g V L V V V z z K f mg g g g g g D g

es inthe control volume


18

Now, for a few assumptions,

(1) The pipe diameter is constant throughout(2) The minor losses are truly minor and can be neglected

Incorporating these assumptions, the energy equationbecomes,

22

1 1

2

2

2

2 21

1 2

22 2

2

c c c

c

Wg L Vf

mg D g

Wg

P gPg

L Vf

V

mg D g

Vz

g

H

g

z

g

H

g


10/13


19

Solving for the power term,

2

2 12

c c

g L VW m H H f

g D g

Substitution of this expression into the annual cost equationgives,

2

2 22 1

2OP

c

C tW mC t L V AC H H g f

g D

22

22

2 1 2

2

ftlbm $s

sfts ft-lbf ft$ ft

s ft 2lbm-ft

lbf-s

OP

c

Vm C tL

AC H H g f D

g

Units


20

2

22 1 2

1 42

OP

c

mC t L m AC H H g f g D D

The velocity in can be expressed in terms of the mass flow rate,

2 2

1 4 4V m m mV

A A D D

Substitution into the annual cost equation gives,


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21

. . . .T AC L 1 2 3 4Then

Putting it all Together

22

The total annual cost of the pipe system can be determined by,

T P F M OP AC AC AC AC L AC

Substituting the previously derived expressions,

2

21 1 1 2 1 2

1 41

2

n n n

T

c

mC t L m AC aC D aFC D bC D F L H H g f

g D D

After some algebra, this can be written as,

3

2 21 2 1 2 2 5

81 n

T

c c

mC t C t g f LmAC a b F C D L H H

g D g


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The Economic Pipe Diameter

23

3

2 21 2 1 2 2 5

31 2

1 2 2 6

81

81 5

n

T

c c

T n

c

mC t C t g f LmAC a b F C D L H H g D g

d AC C t f Lmn a b F C D L

dD D g

Setting this derivative equal to zero and solving forD givesthe pipe diameter that minimizes the total annual cost of thesystem

1

3 52

2 2

1

40

1

n

opt

c

fm C tD

n a b F C g

This is known as the Economic Pipe Diameter

Observations

The pipe length is not in the economic diameter Viscosity does not appear directly, but it is part of the problem

because it determines the friction factor through the Reynoldsnumber

The head loss does not appear in the economic diameter The determination of the economic diameter requires an

iterative solution Very detailed and complicated desktop solution Easy for an equation solver to solve!

24

3

2 21 2 1 2 2 5

1

3 52

2 2

1

81

40

1

n

T

c c

n

opt

c

mC t C t g f LmAC a b F C D L H H

g D g

fm C tD

n a b F C g


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The Economic Velocity

25

The mass flow rate of the fluid in the system operating apipe diameter near the economic diameter is,

Solving for the velocity,

Significance: Table 6.4 Reasonable velocities for variousfluids calculated by using optimum economic diameterequations.

2

4

opt

opt opt

Dm AV V

2 2

4 4opt

opt opt

m VV

D D

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Lect 12 Economic Diameter I

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