Dev. Chem. Eng. Mineral Process., I I(1/2), pp. 121-126, 2003. Two-Phase Flow, Momentum and Energy Balances Revisited R.S. Raghunathan, P.L. Spedding” and R.K. Cooper School of Aeronautical Engineering, The Queen’s‘University of Belfast, Stranmillis Road, Berfast BT9 5AG, Nort~ernIreland, UK The apparent differences between the momentum balance and energy balance derivation for the case of two-phase flow are resolved by examination of the liquid holdup within the system. Introduction There has been a long running discussion over the apparent differences between the pressure, momentum and energy balances for two-phase flow. The problem does not arise with single-phase flow where consistency has been demonstrated between the various balances [ 1,2]. Pressure Drop in Two-phase Flow The total pressure drop in two-phase flow can be written as, (dP/dlh = (dP/dlf, + (dpldl), + (dP/dl), (1) where (dP/dl)* and (dP/dl),, are due to acceleration and hydrostatic head respectively and are commonly termed the reversible components, while the frictional term (dP/dl)F is the irreversible component of the pressure loss being mainly dissipated as heat within the system. Methods of calculating the acceleration term have been given by a number of workers [3-111 as: * Authorfor correspondence. R.S. Raghunathan, P.L. Spedding and R.K. Cooper while experimental measurement of this quantity have been reported by several workers [12-151. Generally in the case of adiabatic, steady state flow in a uniform duct at low gas turbulent Reynolds numbers the acceleration component of pressure drop is small and there is little difference between the results obtained by the various experimental methods. There exists a divergence of views about the absolute value of the hydrostatic head term with some workers indicating coincidence of the balances while others showed a significant difference depending on whether the derivation was based on the energy or momentum balance. For the energy balance: while the momentum balance yields: Equations (3) and (4) yield vastly different results since in general the liquid hoIdup, RL,is much greater than the corresponding (1 p) input condition [16]. In fact, it is the difference between these two parameters that lies at the centre of the enigma regarding the energy and momentum balances with two phase flow. The frictional pressure loss components (dP/dl)FErefers to the total mechanical energy dissipated into thermal energy, while (dP/dl)FMrefers to the wall shear stress. These two frictional terms are identical irrespective of the basis of their derivations. - Two-phase Momentum Balance Using the deviation of Harrison [17, 181 yields: -42w d +G,-[xV, dl +(I-x)VL]+gpTPsinO which agrees with Equation (2), and other derivations reported [5-11, 19-22]. 122 Two-Phase Flow,Momentum and Energy Balances Revisited Two-phase Energy Balance By adding the pressure, potential, kinetic and internal energies Harrison [17, 181 derived: dq dl dw - d [H,x+ H, (I - x)]+ d [-+xV: dl dl dl 2 (1-x)Vt ]+gsin, where the enthalpy is given by: dH =dq + dF+ dP/ p (7) and dF is the energy degraded into heat for an incompressible fluid or into both heat and pressure energy in the case of a compressible fluid. Substitution of Equation (7), taking dw = 0 and the pressure drop being equal in both phases as given by: The terms in Equation (8) correspond to the frictional energy loss, the acceleration and hydrostatic head. This latter will now be examined in detail. Hydrostatic Head and Potential Energy Many workers have used the two-phase density of Equation (4) in the potential energy term with the energy balance [6, 8-1 1, 21, 23, 241. However, others have used the homogeneous density of Equation (3) [25 - 281. As has been pointed out already the effects of this variation in the choice of density are not trivial. Within the energy balance the potential energy or head loss term represents the energy necessary to bring the inlet flow of the phases to the elevation at the outlet of the duct as shown in Figure 1. However also shown on this figure, there is an excess of liquid holdup in the body of the duct which acts like a ladder allowing the outlet flow of the liquid to take place. If it were not present then liquid outflow would not occur. This excess of liquid regurgitates continuously within the body of the duct adding both to the pressure energy and the degraded energy, i.e. the internal energy of the whole system. Hence if the homogeneous density of Equation (3) is used to calculate the potential energy, the effect of the liquid holdup in the conduit must also be considered by increasing the pressure and internal energies of the system accordingly in order to account for the energy needed to support the liquid regurgitating within the duct. A simpler, but approximate, way is to use the twophase density of Equation (4) in the potential energy calculation. 123 R.S. Raghunathan, P.L. Spedding and R.K. Cooper GAS FLOW LIQUID FLOW LIQUID HELD IN Figure 1. Showing diagrammatically the diflerence between delivered liquid and gas and in situ liquid and gas holdup. Conclusions The momentum balance and energy balance derivations for two-phase gas-liquid flow are shown to give identical results. The potential energy calculations in the energy balance reflect the energy expanded to raise the flowing fluids to the outlet elevation. However in order to achieve this outflow, particularly of the liquid phase, an amount of liquid must be maintained or held-up continuously regurgitating within the duct. The energy necessary for the support of this internal movement of liquid will be drawn from both the pressure energy and internal energy of the system. The pragmatic way of calculating the resultant energy output is to use the two-phase density in the potential energy calculation of the energy balance. 124 Two-Phase Flow, Momentum and Energy Balances Revisited Nomenclature D F g G 1 P 9 Q R X V W W P e Diameter (m) Energy degraded (J kgl) Gravity (m ss2) Mass flow rate (kg m-* s-I) Length (m) Pressure (kg m-1s-2) Heat added (J kg-') Volume flow (m3 s") Holdup Mass input ratio (WG/WT) Velocity (m s-l) Net work done by fluid (J kg-') Mass flow (kg s-') Volume input ratio (QG/QT) Angle to horiztonal (degrees) p T Density (kg m") Shear stress (kg rn-' s - ~ ) Subscripts Acceleration E Energy F Friction G Gas H Hydrostatic head L Liquid M Momentum T Total T P Two phase W Wall A References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. I I. 12. 13. 14. 15. 16. 17. 18. Bennett, C.O., and Myers, J.E. 1974. Momentum, Heat and Mass Transfer, 2"d Edition, McGrawHill, New York, p.40-70. Coulson, J.M., and Richardson, J.F. 1977. Chemical Engineering Volume 1, 3" Edition, Pergamon Press, UK, p.24-28. Johnson, H.A. 1955. Heat transfer and pressure drop for viscous-turbulent flow of oil-air mixtures in 1257-1264. horizontal pipe. Trans. Am. Inst. Mech. Eng. Hoogendoom, C.J., and Buitelaar, A.A. 1961. The effect of gas density and gradual vaporisation on gas-liquid flow in horizontal pipes. Chem. Eng. Sci. &5 208-221. Dukler, A.E., and Wicks, M. 1963. Gas-liquid flow in conduits. In Modem Chemical Engineering Volume 1, Physical Operations. Editor: A. Acrivos. Reinhold Publishing, New York, p.349-435. Scott, D.S. 1964. Properties of co-current gas liquid flow. Advances in Chemical Engineering 4 199277. Wallis, G.B. 1969. One Dimensional Two-Phase Flow. McGraw-Hill, New York. Butterworth, D. 1969. Two phase gas-liquid pressure drop. British Chem. Eng. (9) 89-92. Butterworth, D., and Hewitt, G.F. 1977. Two-Phase Flow and Heat Transfer. Oxford Press, UK. Hewitt, G.F. 1982. Liquid-gas systems. In Handbook of Multiphase Flow. Editor: G. Hetstroni, McGraw-Hill, New York. Klausner, J.F, Chao, B.T., and Soo, S.L. 1990. An improved method of simultaneous determination of frictional pressure drop and vapour volume fraction in vertical flow boiling. Experimental Thermal Fluid Sci. 3 404-415. Rose, S.C., and Griffith, P. 1965. Flow properties of bubbly mixtures. ASME paper 65-HJ-58. Andeen, G.B., and Griffith, P. 1965. Momentum flux in two-phase flow. Trans ASME J. Heat Transfer 90 2 1 1-220. Henry, R.E., and Franske, H.K. 1968. In discussion of reference (121. Trans. AMSE J. Heat Transfer 220-22. Baumeister, K.J., Graham, R.W., and Henry, R.E. 1973. Momentum flux in two-phase twocomponent low quality flow. AIChE Symposium Series @ (131) 46-54. Spedding, P.L., and Chen, J.J.J. 1984. Holdup in two-phase flow. Int. J. Multiphase Flow J.Q 307339. Harrison, B.E., and Dean, R.B. 1975. Transmission of geothermal two-phase flow. Inst. Eng. Aust. Thermofluids Conference, December, pp. 1 15-1 19. Harrison, B.E. 1975. Methods for the Analysis of Geothermal Two-Phase Flow. MEng Thesis, University of Auckland, New Zealand. 125 R.S. Raghunathan, P.L. Spedding and R.K. Cooper 19. Mamaev, V. 1965. Some problems in the hydro-dynamics ofjoint transport of gas and liquid. Int. Chem. Eng. 5 3 18-322. 20. Hagedorn, A.R., and Brown, K.E. 1965. Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conducts. J. Petroleum Technol. April 475-484. 21. Brown, F.C., and Kranich, W.L. 1968. A model for the prediction of velocity and void fraction profiles in two-phase flow. AIChE J. & 750-758. 22. David, M.D. 1990. Wall friction for two-phase bubbly flow in rough and smooth tubes. Int. J. Multiphase Flow 16220-229. 23. Vohr, J. 1968. The energy equation for two-phase flow. AIChE J. 4 280-281. 24. Lamb, D.E., and White, J.L. 1968. Use of momentum and energy equations in two-phase flow. AlChE J. 8 261-263. 25. Lombardi, C., and Pedrocchi, E. 1972. A pressure drop correlation in two-phase flow. Energie Nucleare 19 91-99. 26. Lombardi, C., and Ceresa, I. 1978. A generalized pressure drop correlation in two-phase flow. Energie Nucleare 25181-198. 27. Bonfanti, F., Ceresa, I,. and Lombardi, C. 1979. Two-phase pressure drops in the low flow rate region. Energie Nucleare 3 481492. 28. Lombardi, C., and Carsana, C.G. 1992. A dimensionless pressure drop correlation for two-phase mixtures flowing uphill in vertical ducts covering wide parameter ranges. Heat and Technology 125- 141. 126

1/--страниц