On the eect of friction in steady ¯ow of dense gases in pipes D. Stojkovic a , V.D. Djordjevic b, * , P.S. Cvijanovic a a Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Yugoslavia b Faculty of Mechanical Engineering, University of Belgrade, 27. Marta 80, 11120 Belgrade, Yugoslavia Received 4 January 2000; accepted 30 September 2000 Abstract Theeectoffrictioninsteady,one-dimensional¯owofarealgasinpipesistreatedinthepaper,withanemphasisonthedense gas eects. In addition to the well-known fundamental derivative of gas-dynamics C, another derivative, de®ned as C 1 1 q oc=oq T =c; is shown to play a very important role in studying these non-isentropic ¯ows. In both of two characteristic cases of ¯ow, isothermal and adiabatic one, pressure and density variations along the pipe are not aected by the dense gas eect, their variations being qualitatively the same as in perfect gas theory. Variations of the temperature and of the Mach number are, however, severely in¯uenced by the dense gas eect. Perhaps the most important result from the practical point of view is that isothermal ¯ow in the C 1 < 0 region is choking-free, and that adiabatic ¯ow in the C < 0 is almost choking-free. Analytic con- siderations derived in the paper are supported by a number of numerical results for a van der Waals gas. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Dense gases; Eect of friction in pipes; Isothermal ¯ow; Adiabatic ¯ow 1. Introduction Although most ¯ow phenomena in gas-dynamics can be accurately enough characterized by using the model of a per- fect gas, there are several phenomena of great practical im- portance which require some re®ned gas models for their description. Such ¯ow phenomena usually take place in the vicinity of the saturated vapor line, in the single-phase regime of ¯ow, and are governed by one of the several equations of state existing in the literature, like van der Waals, Redlich± Kwong, Martin±Hou equations of state, and others. Among them gases for which the quantity: C 1 q c oc oq s ; 1 mayattainnegativevaluesconstituteaveryimportantclassof ¯uids which under certain conditions exhibit very unexpected and peculiar behavior in that the most ¯ow properties of perfect gases are inverted. Such a behavior is shown to be in- herent in both isentropic and non-isentropic ¯ows, and the most striking examples are the non-monotone variation of the Mach number with density in isentropic nozzle ¯ow, and the appearance of expansion shocks, the partial disintegration of both compression and expansion shocks, shock splitting, etc., in non-isentropic ¯ow. That is why the quantity C isgiventhe namefundamentalderivativeofgas-dynamicsseeThompson, 1971), while the gases with C < 0 behavior are given several names in the literature, like: Bethe±Zel'dovich±Thompson BZT) ¯uids, ¯uids with negative non-linearity, and dense gases. Given a concrete equation of state, C can readily be eval- uatedfromitsde®nition1).Foraperfectgas C ispositiveand equal c 1=2 const:, while for real gases it is in general a functionoftwoarbitrarilychosenstatevariables.Calculations performedinthe p,V) planeshowthat C < 0behaviorcanbe exhibited by gases which are characterized by relatively high values of the ratio c c v0 =R only, typical minimum value of this ratiobeingabout17foravanderWaalsgas.Sohighvaluesof this ratio are expected to be observed in gases of relatively complex molecular structure, and, indeed, calculations with most hydrocarbons and ¯uorocarbons performed by using several equations of state, s. Lambrakis and Thompson 1972) and Cramer 1989), clearly show the existence of a C < 0 re- gioninthe p; V planeforthesegases.Thisregionisatongue- shaped one and extends over a general neighborhood of the saturatedvaporlineatpressuresandtemperaturesoftheorder of their critical values. As far as the non-isentropic ¯ows of dense gases are con- cerned,themostoftheliteratureuntilnowhasbeenaddressed tothestudyofshocksseefore.g.,ThompsonandLambrakis, 1973; Cramer and Sen, 1986). However, the other non-isen- tropic ¯ows, like those in¯uenced by the friction and/or the heat exchanged with the environment, are not less important from the practical point of view. Several engineering problems which include the ¯ow of a real gas in the dense gas regime, International Journal of Heat and Fluid Flow 22 2001) 480±485 www.elsevier.com/locate/ijh * Corresponding author. Tel.: +381-11-33-70-371; fax: +381-11-33- 70-364. 0142-727X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII:S0142-727X00)00077-1