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Research Papers: Flows in Complex Systems

Transmission Line Modeling of Laminar Liquid Wave Propagation in Tapered Tubes

[+] Author and Article Information
Travis Wiens

Department of Mechanical Engineering,
University of Saskatchewan,
Saskatoon, SK S7N 5A9, Canada
e-mail: t.wiens@usask.ca

Jeremy ven der Buhs

Bourgault Industries, Ltd.,
St. Brieux, SK S0K 3V0, Canada

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 17, 2018; final manuscript received March 18, 2019; published online April 15, 2019. Assoc. Editor: Luigi P.M. Colombo.

J. Fluids Eng 141(10), 101103 (Apr 15, 2019) (9 pages) Paper No: FE-18-1844; doi: 10.1115/1.4043235 History: Received December 17, 2018; Revised March 18, 2019

This paper presents an improved method of time-domain modeling of pressure wave propagation through liquid media in rigid tapered pipes. The method is based on the transmission line model (TLM), which uses linear transfer functions and delays to calculate the pressures and/or flows at the pipe inlet and outlet. This method is computationally efficient and allows for variable rate simulation. The proposed form of the model differs from previous TLM models in the literature, allowing it to accurately model both low and high frequency characteristics.

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References

Montgolfier, J. M. , 1803, “ Note Sur le Bólier Hydraulique et Sur la Manière D'en Calculer Les Effets [Note on the Hydraulic Ram, and How to Calculate Its Effects],” J. Mines, 13(73), pp. 42–51.
Johnston, N. , Pan, M. , Kudzma, S. , and Wang, P. , 2014, “ Use of Pipeline Wave Propagation Model for Measuring Unsteady Flow Rate,” ASME J. Fluids Eng., 136(3), p. 031203. [CrossRef]
Viersma, T. J. , 1980, Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines (Studies in Mechanical Engineering I), Elsevier Science, Amsterdam, The Netherlands.
Sutterby, J. , 1964, “ Laminar Newtonian and Non-Newtonian Converging Flow in Conical Sections,” Ph.D. thesis, University of Wisconsin, Madison, WI.
Trikha, A. K. , 1975, “ An Efficient Method for Simulating Frequency-Dependent Friction in Transient Liquid Flow,” ASME J. Fluids Eng., 97(1), pp. 97–105. [CrossRef]
Johnston, D. N. , 2006, “ Efficient Methods for Numerical Modeling of Laminar Friction in Fluid Lines,” ASME J. Dyn. Syst., Meas., Control, 128(4), pp. 829–834. [CrossRef]
Krus, P. , Weddfelt, K. , and Palmberg, J.-O. , 1994, “ Fast Pipeline Models for Simulation of Hydraulic Systems,” ASME J. Dyn. Syst. Meas. Control, 116(1), pp. 132–136. [CrossRef]
Johnston, N. , 2012, “ The Transmission Line Method for Modelling Laminar Flow of Liquid in Pipelines,” J. Syst. Control Eng., 226(5), pp. 586–597.
Johnston, N. , Pan, M. , and Kudzma, S. , 2014, “ An Enhanced Transmission Line Method for Modelling Laminar Flow of Liquid in Pipelines,” J. Syst. Control Eng., 228(4), pp. 193–206.
ven der Buhs, J. , and Wiens, T. , 2018, “ Modelling Dynamic Response of Hydraulic Fluid Within Tapered Transmission Lines,” ASME J. Dyn. Syst., Meas. Control, 140(12), p. 121008. [CrossRef]
Wiens, T. , and ven der Buhs, J. , 2018, “ An Improved Transmission Line Model for Dynamic Laminar Flow Through Tapered Tubes,” ASME Paper No. FPMC2018-8881.
Ducasse, E. , 2002, “ An Alternative to the Traveling-Wave Approach for Use in Two-Port Descriptions of Acoustic Bores,” J. Acoust. Soc. Am., 112(6), pp. 3031–41. [CrossRef] [PubMed]

Figures

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Fig. 1

Tapered tube layout and nomenclature

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Fig. 2

Layout of the standard TLM architecture. Equations (19) and (20) can be used to allow for selection of whether pressures and flows are inputs or outputs.

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Fig. 3

Sample pressure responses to a unit step in inlet flow with an open outlet, for taper ratios of 1 (top) and 0.5 (bottom). Plots are calculated using the MOC using no friction, distributed friction similar to Ref. [7], and frequency-dependent friction [6].

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Fig. 4

Layout of the new TLM architecture, showing addition of H transfer functions as well as different input and output E, F, and G transfer functions

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Fig. 5

H impulse responses for low damping (a) and higher damping (b) for varying taper ratios, calculated numerically using the MOC

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Fig. 6

Time-domain error maps while varying taper ratio and dissipation number, showing the error in inlet flow (a), outlet flow (b), inlet pressure (c), and outlet pressure (d)

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Fig. 7

A representative error map for data calculated on the lookup table grid points, as well as that interpolated at the midpoint between grid points, showing the very small effect of interpolation errors

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Fig. 8

Sample inlet pressure response for a nontapered tube with β = 1 × 10−3, with a unit step in inlet flow and open outlet. Responses calculated using the methods in Johnston's previous [8,9] work are also included. This shows the full response (a) and a detail of one cycle (b).

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Fig. 9

Sample inlet pressure response for a nontapered tube with β = 1 × 10−2, with a unit step in inlet flow and open outlet. This shows the full response (a) and a detail of one cycle (b).

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Fig. 10

Sample inlet pressure response for a nontapered tube with β = 1 × 10−1, with a unit step in inlet flow and open outlet. This shows the full response (a) and a detail of one cycle (b).

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Fig. 11

Sample inlet pressure response for a diverging tube with β = 1 × 10−3 and λ1=2, with a unit step in inlet flow and an open outlet

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Fig. 12

Sample inlet pressure response for a converging tube with β = 1 × 10−3 and λ1=0.5, with a unit step in inlet flow and an open outlet. Note that this is the same tube as in Fig. 11, but with the ends exchanged.

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Fig. 13

Sample inlet pressure response for a diverging tube with β = 1 × 10−1 and λ1=2, with a unit step in inlet flow and an open outlet

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Fig. 14

Sample inlet pressure response for a converging tube with β = 1 × 10−1 and λ1=0.5, with a unit step in inlet flow and an open outlet. Note that this is the same tube as in Fig. 13, but with the ends exchanged.

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Fig. 15

Time-domain errors for nontapered tubes (λ1 = 1) while varying dissipation number. Errors are shown for inlet flow (a), outlet flow (b), inlet pressure (c), and outlet pressure (d).

Tables

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