Abstract— This article aims to develop a rigorous

mathematical model for analyzing the transient response of the

high pressure steam pipeline network in a refinery. A

sequentially iterative fully implicit method has been propounded

to deal with the unsteady nonlinear equations in a pipeline. The

proposed method is then combined with the modified

Hardy-Cross method to study the transient response in a looped

pipeline network. A complex high pressure steam network

problem is used for demonstrating the applicability of the

proposed solution method in analyzing the transient response in

a pipeline network. This analysis is critical for optimizing

operation and control of the steam distribution systems.

I. INTRODUCTION

The subject of unsteady flow has been studied for a long time and numerous mathematical models have been set up to deal with different kinds of flowing scenarios. Researchers continue to develop numerical methods that focus on the unsteady nature of these systems. Even though many related design problems could be solved using steady-state modeling, the scale and complexity of modern fluid distribution systems are seldom operated under steady flow conditions. Therefore, transient analysis plays an important role when drawing guidelines for pipeline design standards. Their design and simulation should always be on the premise that the flow is unsteady.

Previous researches attempted different numerical methods to solve the compressible unsteady one-dimensional flow. Wylie and Streeter used the method of characteristics (MOC)[1]. An advantage of the MOC is that it can handle discontinuities in the simulation. However, Fincham and Goilwater (1979)[2] pointed out that the MOC has an obvious disadvantage in calculations concerning large pipeline networks, because the time-steps are restricted by the stability criterion such that the time step needs to be small enough to satisfy the Courant condition. There are also several explicit finite-difference methods such as the first-order Lax-Wendroff method (Bender 1979)[3], and the second-order Lax-Wendroff method (Poloni et al. 1987)[4]. Guy (1967)[5] proposed a partially implicit algorithm based on the Crank-Nicolson method. Chaczykowski (2001)[6] applied a fully implicit method. The major advantages of using an implicit method is that they are unconditionally stable and hence impose no restrictions on the maximum allowable time step. These methods do, however, require the solution of a set of nonlinear equations simultaneously (usually by Newton-Raphson linearization) at each time step.

*Research supported by the Ministry of Science and Technology, Taiwan.

Authors are with the Department of Chemical Engineering, National

Taiwan University, Taipei ROC (corresponding author: C.L. Chen, phone:

+886-2-33663039; fax:+886-2-23623040; e-mail: CCL@ntu.edu.tw).

High-pressure steam distribution networks are systems with hundreds or thousands of meters of pipe. In an oil refinery, a variety of process units generate steam and numerous process units consume it. The objective of this study is to simulate non-isothermal, one-dimensional compressible high-pressure steam flows through a steam distribution system. The method of solution proposed in this study is the iterative fully implicit finite-difference method, which is very suitable for gas system simulations because it gives necessary opportunity for large time step size. The algorithm used to solve the non-linear finite-difference equations is based on the Newton-Raphson Method.

II. DEVELOPMENT OF THE MATHEMATICAL MODEL

A. Governing Equations

To describe the fluid motion in a steam pipeline, the following assumptions are made: 1) one-dimensional single-phase flow, where the longitudinal viscous and conductive effects are neglected; 2) unsteady, compressible flow that include the effect of wall friction and heat transfer. This initial assumption is examined in details elsewhere, but briefly it may be stated that for high Reynolds number flows, as in gas transmission lines, the one-dimensional assumption has been shown to be very good for steady and slowly varying flows.

The governing equations for the one-dimensional, compressible, non-isothermal flow is expressed as follows[7]:

For continuity equation,

0)(

v

xt

(1)

For linear momentum,

singA

w

x

P

x

vv

t

v

(2)

For conservation of energy,

A

wvQ

x

Pv

t

P

x

hv

t

h

(3)

where

Dvvf

w

8

Churchill (1977)[8] discussed the calculation of friction factor (f) for steam in details and the applicable ranges of Reynolds number and pipe roughness for different correlations. In this research, the friction factor equation is used as follows:

Transient Response Analysis of High Pressure Steam

Distribution Networks in A Refinery

Chiao-Ying Chang, Shih-Han Wang, Yu-Cheng Huang, Cheng-Liang Chen*

6th International Symposium onAdvanced Control of Industrial Processes (AdCONIP)May 28-31, 2017. Taipei, Taiwan

978-1-5090-4396-5/17/$31.00 ©2017 IEEE 418

12

1

5.1

12

)(

1

Re

88

CBf (4)

where

169.0

ε27.0

Re

7ln457.2

DB

16

Re

37530

C

Q is the heat flow out the pipe per unit length of pipe per unit time as follows:

D

TTUAQ a )(4 (5)

Equations (1) to (3) may be written with pressure, temperature and velocity as the dependent variables by using the equation of state for a real gas and the thermodynamic identity described by Zemansky (1968)[9].

RT

Pz

(6)

dP

T

TdTCdh

P

p

1 (7)

Where ρ is the density of steam, which is a function of temperature and pressure. Vukalovich et al. (1968)[10] provide the correlation as

5 41

0 1

1 14.260321148

ji irrj

j icr r

iaTP

p

(8)

The coefficients aji are given in Table І and the subscripts r refers to the reduced properties. The resulting set of hyperbolic governing equations are

x

va

x

Pv

t

Ps

2

A

wvQ

T

z

z

T

TC

a

PP

s

1

2

(9)

sin1

gA

w

x

P

x

vv

t

v

(10)

x

v

T

z

z

T

C

a

x

Tv

t

T

PP

s

1

2

A

wvQ

P

z

z

P

PC

a

TP

s

1

2

(11)

The parameter as which represents the isentropic wave speed is

2

11PPT

s

T

z

z

T

Tc

P

P

z

z

P

zRTa

(12)

TABLE I. COEFFICIENTS USED IN STEAM DENSITY EQUATION

aji i=1 i=2 i=3 i=4

j=0 -0.8853863 -24.246870 -1.9973940 -0.7527841

j=1 -4.4293200 -156.31350 -4.7518400 -1.9478900

j=2 -9.9615830 -399.15670 -63.776670 -2.3244200

j=3 -13.689410 -503.60670 -156.21240 -11.629330

j=4 -7.6042130 -313.50630 -152.97240 -12.017370

j=5 -1.7971360 -77.098100 -53.969370 -3.8431900

B. Numerical Resolution

Several different methods of solution for the three governing equations have been developed, and the choice is partly dependent upon the individual. In this study, a modified fully implicit method has been developed and used to solve for the transient problems. The fully implicit method consists of transforming Equations (9)-(11) from partial differential equations to algebraic equations by using finite-difference approximations for the partial derivatives. Fig. 1 shows a mesh used in the calculation. The pipe has N nodes and n time levels. (Abbaspour et al., 2008)[11].

The partial derivatives with respect to time and length are approximated by

t

YYYY

t

Y n

i

n

i

n

i

n

i

2

1

11

1 (13)

x

YY

x

Y n

i

n

i

11

1 (14)

The individual terms at an interface between the nodes are approximated by

2

11

1

n

i

n

i YYY (15)

The parameter Y represents P, T, v in Eqs. (13)-(15). Substituting Eqs. (13)-(15) into Eqs. (9)-(11) results in three sets of equations for each node and 3N-3 equations are derived for a pipe without considering the end node. The number of unknown values at time level n+1, which consists of pressure, temperature and velocity at each node, is 3N. Since three equations will come from the boundary conditions, there are 3N equations and 3N unknowns. Thus, the Newton-Raphson method can be applied to solve these nonlinear equations simultaneously. However, for a complex gas network, the matrix becomes quite large and costs much time when executing the simulation. This study proposed an iterative calculation method to reckon these nonlinear equations sequentially rather than simultaneously, where the detailed calculation procedures will be described in the next section.

419

C. Overall Network Calculation Procedures

In real steam distribution systems, beside single pipelines, looped networks also play an important role. Since single pipelines are a special case of the looped networks, a method which can be used to simulate looped networks is proposed in this study. The main ideal is to combine the Hardy-Cross method which was proposed by Wang et al. (2015)[12] with the iterative implicit method in order to analyze a looped network. The overall analysis procedures for the compressible, non-isothermal steam system are described as follows, where the procedures are also shown in Fig. 2. In Fig.2, the yellow block represents the iterative implicit method and the pink block refers to the modified Hardy-Cross method which is used for the compressible fluid system.

1. Given all the parameters of the pipe and the steam.

2. Mesh length and time into grids to get Δt and Δx.

3. Calculate the initial steady-state condition.

4. Do the calculation at time t=t0 + Δt.

5. Make initial guesses of the pressure and the temperature at each node, and use the modified Hardy-Cross method to calculate the steam flow distribution.

6. Execute the calculation for pipe j=j+1.

7. Set up the boundary conditions of pressure and temperature at the upstream (node 1) as well as the fluid velocity at the downstream (node N) of the pipe.

8. Make initial guesses of the upstream fluid velocity.

9. Execute the calculation at node i=i0 + 1of pipe j.

10. Make initial guesses of the pressure and the temperature at this node. Use the initial guesses to determine the friction factor f, the density ρ, and the compressibility factor z at this node.

11. Substituting Eqs.(13)-(15) into Eqs.(9)-(11) and solve these three nonlinear equations simultaneously by the Newton-Raphson method.

12. Repeat steps 10 to 12 until the pressure, the temperature and the fluid velocity differences between iterations are smaller than the defined convergence criterion.

13. Repeat steps 9 to 13 until finishing the calculation of all nodes in pipe j.

14. Repeat steps 8 to 14 until the difference between the resulting downstream fluid velocity and the boundary condition are smaller than the defined convergence criterion.

15. Repeat steps 6 to 15 until finishing the calculation of all pipes.

16. Repeat steps 5 to 16 until the pressure and the temperature differences at each node between iterations are smaller than the defined convergence criterion.

17. Repeat steps 4 to 17 until reaching a new steady-state.

The above method can be used to analyze the steam distribution systems including singe pipelines, branch systems and looped network systems. Several examples will be discussed using the proposed method in the next section.

n

n+1

1 2 3 i-1 i+1i N

Δ t

N-1

Δ x

Figure 1. Mesh of the calculation.

Set up boundary conditions (P)1,(T)1,(v)N

t ≥ tmax?

Output the solutions

No

Yes

Input data and parameters

Mesh length(Δx) and time(Δt) into grids

Get initial steady-state conditions

t=t+Δt (n=n+1)

For section i=i+1

Guess pressure (Pg)i, temperature (Tg)i at node i

Solve three governing equations by Newton-Raphson method

│(P)i-(Pg)i│ and │(T)i-(Tg)i│and │(v)i-(vg)i│≤ tol ?

Yes

No

No Yes

Update (Pg)i

(Tg)iand (vg)

i

Guess the flow velocity at the upstream (vg)1

i ≥ I?

│(vg)N -(v)N│≤ tol?

Yes

No

Update

(vg)1=(vg)

1-0.5*│(vg)N -(v)N│

Use modified Hardy Cross method to get the flow rate distribution

j ≥ jmax?

Is ΔP, ΔT at each node

small enough between iterations?

No

NoYes

Yes

Update P at each node

For pipe j=j+1

Iterative Implicit Method

Hardy-Cross Method

Figure 2. Overall simulation procedures

III. SINGLE PIPELINE ANALYSIS

The studied example, as illustrated in Fig. 3, is characterized by a simple straight isothermal pipe segment of

5,000 m in length and 0.6096 m internal diameter, holding a

high pressure steam at a pressure of 126 (atm) and a

temperature of 811.15 (K). The friction factor f is assumed to

be 0.013, the compressibility factor z at specific temperature

and pressure is assumed to be 0.91. The demand changes are

shown in Fig. 4. At the initial steady-state, the outlet flow is

maintained at 25.0 kg/s (90 ton/hr). At time t = 600 (s), the

steam outflow steps up from 25.0 (kg/s) to 33.3 (kg/s), while

the inlet pressure is maintained at 126 (atm).

A. Comparison with Original Implicit Method

The iterative implicit method is compared with the original implicit method in this section. Fig. 5 shows the variation of the mass flow rate at the supply end with respect to time and compares the results from the iterative method to the original implicit method. The simulation results show that the present method and the original implicit method give similar results. Furthermore, the simulation time of the proposed iterative implicit method and the original implicit method is 0.0156 second and 0.3961 second respectively. For a complex looped network, the present method is more suitable because of the less simulation time it costs.

420

Figure 3. Pipe information

Figure 4. Boundary comdition at the demand end

Figure 5. Comparison of present work with original implicit method

B. Example for Non-isothermal Single Pipeline

The present method is applied to the non-isothermal pipeline. The scenario is the same as the case discussed earlier and the overall heat transfer coefficient U is 0.137 W/m2 K. To validate the simulation results, the steady-state result for temperature is compared with the temperature equation developed by Coulter and Bardon (1979)[13]. The temperature equation is defined as

baLbTT INOUT )exp( (16)

Where

pmC

DUa

L

PP

aTb INOUTJ

a

The Joule-Thompson coefficient ηJ is set to be 7.84x10-6 (K/Pa). Based on this assumption, Fig. 6 illustrates the comparison between the temperature result for steady-state condition with the Coulter and Bardon equation. As shown in Fig. 6, the result of temperature matches quite well with the Coulter and Bardon equation.

Figure 6. Temperature comparison for present work with Coulter-Bardon

equation

IV. NETWORK SYSTEMS ANALYSIS

In this section, a real looped high pressure steam network in a large-scale oil refinery is discussed using the method presented in the previous section. The schematic diagram of this network is shown in Fig. 7.

This network has 45 pipe sections, 26 nodes, 8 steam sources (those red rectangles in figure 7), 11 steam sinks (those blue rectangles in figure 7) and one loop. The ambient temperature of the location where this network is situated is 298.15 (K), and the absolute roughness of all pipes is assumed to be 0.0000457 (m). The overall heat transfer coefficient is calculated according to the inner diameter of pipeline sections.

The original steady-state of this looped network is shown in Fig.7. The situation is that boiler #3, one of the primary steam sources, is shut down for maintenance or malfunction and the amount of steam deficiency is compensated by increasing steam supplies from boiler #1 and/or boiler #2. Different operating strategies are discussed in the following section.

A. Different Operating Strategy: The steam deficiency is

compensated by boiler #1 and boiler #2

Figures 8-10 show the new steady-state results when the steam deficiency is compensated 100%, 50% and 0% by boiler #1 respectively. Fig.s 11-13 are the transient response results for unit #1, unit #6 and unit #11. The percentages on the figures represent the percentages of deficiency which boiler #1 compensates.

From Fig. 11, the temperature at unit #1 drops rapidly in the beginning and reaches to a new steady-state in about 50 seconds. As the percentages of deficiency which boiler #1 compensates increased, the new steady-state temperature will be higher. The reason is that the temperature at boiler #1 is relatively high compares with boiler #3, therefore the steady-state temperature will be higher than the original operating scenario. Besides, the flow rate at pipeline segment #29 will decrease as the percentages of deficiency which boiler #1 compensates decreased. This causes the heat loss at pipeline segment #29 to be larger, thus the final steady-state temperature for 100%-compensation will be higher in comparison with 50%-compensation.

Fig. 12 shows that unit #6 will reach a new steady-state in about 10 seconds. The new steady-state temperature will be higher if the percentages of deficiency which boiler #1 compensates are larger than 40%; otherwise the temperature will be lower. The reason for the temperature at unit #6 to be higher is the same as unit #1. And the reason why the temperature will become lower than the original steady-state

421

when the percentages of deficiency which boiler #1 compensates are less than 40% is because the flow direction at pipeline segment #32 will change, therefore the steam which flows to unit #6 contains both steam from pipeline segments #31 and #32. The steam temperature at pipeline segment #32 is lower than the steam from boiler #1 which causes the temperature at unit #6 to be lower than the original steady-state.

Fig. 13 shows the temperature variation at unit #11, and it shows that as the percentages of deficiency which boiler #1 compensates decreased, the time to the new steady-state will increase and the new steady-state temperature will decrease. The reason for the temperature to be lower is because when boiler #3 is under maintenance, the main steam source for unit #11 will be unit #8. The steam temperature at unit #8 is lower than unit #3, therefore the new steady-state temperature will be lower.

V. CONCLUSION

This study proposed an iterative fully implicit method for

the calculation of unsteady and non-isothermal steam flow,

and combined the iterative method with the modified Hardy

Cross method to deal with the looped network system.

Compared with traditional fully implicit method, the present

method neglects the complicated calculation when solving

numerous equations simultaneously and hence shortens the

computation time. The current study shows that the present

method is suitable for large scale steam distribution in an oil

refinery, and can be used to analyze the transient situations.

Unit#2T=649.1 P=41.9 M=78.0

Unit#3T=656.6 P=42.1 M=7.0

Unit#4T=653.1 P=41.5 M=36.0

Unit#1T=661.9 P=41.5 M=76.0

Unit#6T=651.0 P=41.6 M=94.0

Unit#7T=643.1 P=41.7 M=38.0

Unit#10T=651.0 P=41.9 M=17.0

Boiler#1T=683.6 P=42.2 M=101.0

Unit#5T=634.2 P=41.8 M=71.0

Boiler#2T=653.1 P=41.7 M=70.0

Unit#8T=653.5 P=42.0 M=91.0

Unit#9T=648.9 P=41.5 M=20.0

Boiler#3T=663.0 P=41.9 M=108.0

Unit#11T=651.0 P=41.5 M=79.0

Unit#12T=646.2 P=41.5 M=7.0

Unit#14T=638.7 P=41.5 M=88.0

Unit#13T=648.8 P=41.5 M=32.0

Unit#16T=641.1 P=41.7 M=40.0

Unit#15T=635.7 P=41.7 M=5.0

1

2 3 5

4

6 7 8 9

10

12

11

13

14 15 16 17 18 19

20

21

22

23

25

24

26 27 28 29 30

45

44 43 42 41 40 39 38 37 36

31 32 33 34

35

76.0

40.0

52.6 48.4 119.4 25.4 95.4

57.4

17.412.475.643.650.6

59.0

20.0

74.0

91.0

109.61.6

72.4

12.65.6

Figure 7. Original steady-state conditions

Figure 8. New steady-state conditions when boiler #1 compensates 100% of the amount of steam deficit

Figure 9. New steady-state conditions when boiler #1 compensates 50% of

the amount of steam deficit

Figure 10. New steady-state conditions when boiler #1 compensates 0% of the amount of steam deficit

422

Figure 11. Temperature variations at unit #1 where boile r#1 compensates

different percentages of the amount of steam deficit

Figure 12. Temperature variations at unit #6 where boiler #1 compensates

different percentages of the amount of steam deficit

Figure 13. Temperature variations at unit #11 where boiler #1 compensates

different percentages of the amount of steam deficit

NOMENCLATURE

A = cross section area of the pipe (m2)

D = pipe diameter (m) P = fluid pressure (N/m2)

U = overall heat transfer coefficient (W/m2 K)

R = specific gas constant (J/kg K)

T = temperature (K)

Ta = ambient temperature (K)

CP = specific heat capacity (J/kg K)

Q = heat flow out the pipe (J/m s)

as = isentropic wave speed (m/s)

v = fluid velocity (m/s)

f = friction factor of the pipe

h = specific enthalpy (W/m2 K) t = time (s)

w =frictional force per unit length of pipe (N/m)

x = distance along the pipe (m)

z = compressibility factor

ρ = density of the fluid (kg/m3)

ε = pipe roughness (m)

ηJ = Joule-Thompson coefficient (K/Pa)

ACKNOWLEDGMENT

The authors would like to thank the financial support from

the Ministry of Science and Technology (MOST) of ROC

under Grant NSC102-2221-E-002-216-MY3, and the support

from Formosa Petrochemical Company.

REFERENCES

[1].Wylie, E. B.; Streeter, V. L., Fluid transients. McGraw-Hill International

Book Co.: New York, 1978; Vol. 1.

[2].Fincham, A. E.; Goldwater, M. H., Simulation models for gas

transmission networks. Transactions of the Institute of Measurement and

Control 1979, 1, (1), 3-13.

[3].Bender, E., Simulation of dynamic gas flows in networks including

control loops. Computers & Chemical Engineering 1979, 3, (1–4),

611-613.

[4].Poloni, M.; Winterbone, D. E.; Nichols, J. R. In Calculation of pressure

and temperature discontinuity in a pipe by the method of characteristics

and the two-step differential Lax-Wendroff methods, American Society of

Mechanical Engineers, Fluids Engineering Division (Publication) FED,

1987; 1987; pp 1-7.

[5].Guy, J. In Computation of unsteady gas flow in pipe networks, Proceeding

of symposium,,, Efficient methods for practising chemical engineers”,

Symposium series, 1967; 1967; pp 139-145.

[6].Chaczykowski, M.; Osiadacz, A., Simulation of non-isothermal transient

gas flow in a pipeline. Archives of thermodynamics 2001, 22, (1-2), 51-70.

[7].Thorley, A. R. D.; Tiley, C. H., Unsteady and transient flow of

compressible fluids in pipelines-a review of theoretical and some

experimental studies. International Journal of Heat and Fluid Flow 1987,

8, (1), 3-15.

[8].Churchill, S. W., Friction-factor equation spans all fluid-flow regimes.

Chemical engineering 1977, 84, (24), 91-92.

[9].Zemansky, M. W., Heat and Thermodynamics: An Intermediate Textbook

for Students of Physics, Chemistry, and Engineering. 5 ed.; McGraw-Hill:

1968.

[10].Vukalovich, M. P., Aleksandrov, A. A. , Tracgtengerts, M. S., Equations

of state for superheated steam for industrial computations using electronic

computers. Teploenergetika 1968, 9, 86-90.

[11].Abbaspour, M.; Chapman, K. S., Nonisothermal transient flow in natural

gas pipeline. Journal of Applied Mechanics, Transactions ASME 2008, 75,

(3), 0310181-0310188.

[12].Wang, S.-H.; Wang, W.-J.; Chang, C.-Y.; Chen, C.-L., Analysis of a

Looped High Pressure Steam Pipeline Network in a Large-Scale Refinery.

Industrial & Engineering Chemistry Research 2015, 54, (37), 9222-9229.

[13].Coulter, D. M.; Bardon, M. F., Revised equation improves flowing gas

temperature prediction. Oil & Gas Journal 1979, 77, (9), 107-108.

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