1. Chapter 21 Theoretical Rates in Closed-Channel Flow

2. Many a time did I stand such a pipe and exert myself to invent how to force these pipes so reveal the secret of their hidden action. Clemens Herschel (1898)

3. 21.1 General Remarks In closed-channel flow ( as in pipes, ducts, etc.), the system is usually full of fluid, and consequently, the fluid is completely bounded. For one-dimensional steady flow, continuity can be expressed as Under the same conditions, a general energy accounting for a reversible thermodynamic process (i.e., when mechanical friction and fluid turbulence are negligible) yields, in the absence of mechanical work and elevation change. (21.1) (21.2)

4. Geometric considerations indicate the usefulness of the definitions It is necessary also to consider the very important differences that exist between the highly compressible gases and the relatively in compressible liquids [1]-[4]. They must be considered separately when evaluating densities, velocities, and flow rates. (21.3) (21.4) (21.5)

5. It is conventional in a study of flow rates to examine theoretical relations first. In the interest of simplicity, we also idealize the fluids so that a liquid is taken to exhibit constant density, whereas it is assumed that the equation of state of a gas is given by (21.6)

6. 21.2 CONSTANT-DENSITY FLUIDS Equation(21.3), when integrated between two arbitrary positions for the constant-density case, yields Which is a form of the Bernoulli equation. Combining equation(21.2),(21.5), and (21.7) results in Thus the theoretical rate of flow of a constant-density fluid in a closed channel is (21.7) (21.8) (21.9)

7. Where mi deal is in lbm/s, A 2 is in 2, g is in lbm-ft/lbf-s 2, ρis in lbm/f t 2,and p is in lbf/m 2. The flow rate is to be directly proportional to the square root of the pressure drop p1-p2.however,the pressure drop across a constant area section will be very small indeed, even in the presence of frictional losses. To obtain a measurable pressure drop, the flow is usually obstructed in a manner similar to the way in which open- channel flow is obstructed.

8. The obstruction and the required static pressure taps make up the closed-channel fluid meter. The venturi, the nozzle, and the square-edged orifice plate (and their associated pressure taps) are the most common closed-channel fluid meters, although porous plugs or simple restrictions in the walls of a flow tube can suffice to establish a suitable pressure drop (Figure 21.1)

9. 21.3 Compressible Fluids When the thermodynamic process between two arbitrary positions in a system is isentropic (i.e, when there is no heat transfer, no mechanical friction, no fluid turbulence, and no unrestrained expansion), the ideal gas of equation(21.6) also can be characterized by Where r is the static pressure ratio p2/p1 and r is the ratio of specific heats cp/cv. The general energy equation(21.3) under these conditions can be integrated to yield (21.10) (21.11)

10. Which with equation(21.2)can be expressed as thus the theoretical rate of flow a compressible fluid in a closed channel, according to equations(21.1), (21.10), and (21.12), is (21.12) (21.13)

11. For the same units as in equation (21.9).note that in equations(21.10)through (21.17)the velocities and densities are those based on an isentropic process between the total pressure of equation(21.23)and the thermodynamic state of interest. equation(21.13)also can be given in the useful form(3),(4) (21.14)

13. FIGURE 21.1 Types of fluid meter for closed-channel flow. (a) Herschel-type venturi tube. (b) Long-radius flow nozzle. (c) HEI flow nozzle. (d) Square-edged orifice. (e) Porous plug flow meter. (f) Restrictive-type flow meter. (source from ASME(5))

14. With the same units as given under equation(21.9), except that R is in lbf-ft/lbm- o R and T c is in o R. Alternatively, if the general energy equation(21.3) is integrated between the actual throat static pressure and the isentropic total pressure of equation(21.23), we have the general relation (21.15) which can be expressed as (21.16)

15. Thus the theoretical rate of flow of a compressible fluid in a closed channel is, according to equation(21.1),(21.10), and (21.16)[3],[4], In terms of the generalized compressible flow function Г, which has been defined [4],[6] as (21.17) (21.18)

16. Equation(21.17) also can be given in the simplified form Note that in equations(21.18) and (21.19) the actual total pressure at meter inlet is used. The constant in equation(21.19) is simply which takes values at standard gravity conditions of (21.19) (21.20) (21.21) (21.22)

17. For brief tablulations of the t function see table 21.1. For more complete tabulations see(4),(6). Note that p in equation(21.1) is the isentropic total pressure in the fluid meter, defined in general as (7),(8) (21.23) In the ideal case Cc is usually set equal to unity.

18. 21.4 CRITICAL FLOW RELATIONS The flow rate of a compressible fluid was seen [equation(21.13)] to be dependent in general on the ratio of the downstream static P 2 to the upstream static pressure P 1. The variation in flow rate with changes in the static pressure ratio is important in studying the critical flow of gases through nozzles. First the square of the isentropic flow rate [equation(21.13)] is differentiated with respect to r to obtain (21.24)

19. Theoretical rates in closed-channel flow The critical static-pressure ratio (the one that yields the maximum isentropic flow rate for given fluid conditions at inlet and for a given geometry) is obtained by setting equation(21.24) equal to zero. The result is (21.25) When the asterisk signifies the condition of maximum flow rate. Note that if the geometry is such that β->0,then p1- >p0,and equation(21.25) leads to the familiar critical point function of thermodynamics (21.26)

20. Thus theory reveals and experiment agrees that the flow rate of a convergent nozzle (where C C =1) attains constancy and is maximized at the critical pressure ratio, equation(21.25). at this critical pressure ratio, the fluid velocity equals the local velocity of sound, and the flow no longer responds to changes in the downstream pressure[8]. Although in the case of a flow nozzle the throat static pressure is called for in equations(21.13)-(21.23),it is customary (and usually preferred, see, for example, [9])to measure the lower pressure in the larger diameter discharge pipe. This is usually called the back pressure P b.If the flow is subsonic, p 2 can be taken as the back pressure.

21. On the other hand, if the nozzle is choked (i.e, if for a given inlet pressure the flow is maximum and also independent of the back pressure), the throat static pressure must be greater than the back pressure. In fact, whenever the measured static pressure ratio Pb/P1 is less than or equal to r of equation(21.13)-(21.23),is r of equation(21.25).on the other hand, if Pb/P1. Venturis also are operated as critical flow meter with certain advantages noted in the literature[10]. To verity that critical flow conditions exist in the venturi, it is only necessary to show that throat conditions are independent of the overall pressure ratio across the venturi.

22. Contrary to the behavior of the convergent and convergent- divergent passages of nozzles and venturis, the square-edged orifice meter does not exhibit a maximum flow rate. For example, Perry[11]and Cunningham [12]both indicate that the flow rate (for constant upstream conditions) continues to increase at all pressure ratios between the critical ratio of equation(21.25)and zero. This range is thus defined as the “supercritical” range of pressure ratios. The study of critical flowmeters for compressible flow measurements is a complex and rapidly changing subject for which a rapidly growing literature is developing[7], [10]- [14].