CORRESPONDENCE
Minimum Critical Velocity for One-Phase Flow of liquids SIR: The theory of Gutierrez and Lynn (1969) is a special case of a general relationship describing pressure drop in one-dimensional homogeneous equilibrium twophase flow. This general expression is (Wallis, 1969)
-
-dP dz
-
2CfGz -+ Dpm
-G-v ~-,dq --
(hfgA dz
G2 dA Apm dz
+ gpm cos 0)
aT
ah
- q = Cp(ap)* = o - T
($)x=o
(g)
(8)
For water a t low pressures the second term on the right-hand side of Equation 8 may be neglected and we may also invoke the Clausius-Clapeyron relationship.
4
(9) Using Equations 8 and 9 in Equation 7, we have
uf(min) = Vfhf, (C,T)-”’
where
(10)
VfP
The various terms in the numerator of Equation 1 express the effects of friction, heat addition, area change, and gravity on the pressure gradient. The denominator reflects compressibility and flashing effects. “Choking” occurs when the denominator is equal to zero-that is, when
For the particular case of the inception of flashing from single-phase liquid, we have x = 0 and pm = pi = l / u f . Moreover, the liquid can usually be regarded as incompressible and we may assume that au,/ap = 0. The critical velocity is then given by Equation 3 as
Moreover, from thermodynamics we have
ah h - iR - (ax),
(5)
which is in a useful form for calculation purposes. At low pressures, the variation of ufp with temperature exerts the dominant influence on the variation of ut. I t is good to see these predictions confirmed by the work of Gutierrez and Lynn. Nomenclature
A = area c, = friction factor c, = specific heat a t constant pressure D = duct hydraulic diameter g = acceleration due to gravity G = mass flux h = enthalpy P = pressure 9 = rate of heat transfer T = temperature u = velocity u = specific volume x = quality z = distance along duct P = density @ = defined in Equation 2 SUBSCRIPTS f = liquid g = vapor m = mean
and
literature Cited
Combining Equations 4, 5, and 6 we get the same result as Gutierrez and Lynn.
I n addition, we may use the following thermodynamic identity, which is a more exact version of the one used by Gutierrez and Lynn (Tribus, 1961). 486
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
Gutierrez, A., Lynn, S., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 8, 486-91 (1969). Tribus, M., “Thermostatics and Thermodynamics,” Exercise 1, p. 253, Van Nostrand, Princeton, N. J., 1961. Wallis, G. B., “One-Dimensional Two-Phase Flow,” Problem 2.25, p. 39, McGraw-Hill, New York, 1969.
G. B . Wallis Dartmouth College Hamuer, N . H . 03755
