4   Application of Models

• Two of the available models, the two-dimensional viscous incompressible flow model and the one-dimensional viscous compressible flow model, are not considered to be well suited to the modelling of fluid transmission lines. The two-dimensional incompressible flow model is accurate in predicting instantaneous velocity profiles, and the series impedance is readily obtained from it. However, it is necessary to introduce further assumptions concerning the nature of propagation to obtain approximate expressions for the characteristic impedance and propagation operator. As the compressibility effects do not complicate the solution, the two-dimensional viscous compressible model should be used in preference.

 
• The one-dimensional viscous compressible flow model is based upon the assumption that disturbances propagate as plane waves along the tube. Because it ignores the effects of velocity profile in the line, the actual losses occurring due to viscosity are grossly underestimated. For most practical applications this model and the one-dimensional inviscid compressible model could be considered to be equivalent.

 
• The 'exact' first-order model contains more detail than is necessary to cover the majority of engineering applications. Gerlach and Parker [45] have shown that the radial propagation modes are damped out within a couple of pipe radii of the end of the transmission line. Gerlach [61] and Urata [46] have demonstrated that the solution obtained using this model is identical to the solution to the two-dimensional viscous compressible model, assuming that plane wave attenuation effects and the radial pressure distribution are negligible.

 
• In their review of fluid transmission line models Goodson and Leonard [62] have shown that the choice between the two-dimensional viscous compressible model and the one-dimensional inviscid compressible model is determined by the relative size of the viscous losses occurring in the line compared with the resistive losses occurring at the load impedance. When considering the frequency response of a blocked line, they recommended that the viscous model should be used in all cases except where the driving frequency is well below the fundamental frequency of the line. The use of the two-dimensional thermal viscous compressible model is determined by the ratio of specific heats and the Prandtl number of the fluid. Further study reported in [55] and [10] carried out using alternating flow test rig confirmed that two-dimensional viscous compressible is most suitable for modelling of long transmission lines When the Prandtl number is small and the ratio of specific heats is greater than unity, as is the case for most gases, the thermal effects become significant and this model should be used. For liquids, the ratio of specific heats is generally close to one and the thermal losses are negligible.

 
• The one-dimensional linear resistance compressible model can be applied in many cases but it is difficult to give general considerations on its use, as the limits of application depend on the chosen form of resistance coefficient. This in turn limits the frequency response over which the model can accurately be applied.