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Flow through Convergent Nozzle Equations and Calculator

Isentropic Flow through Convergent Nozzle Equations and Calculator

Isentropic flow of compressible fluid from a large tank through a convergent nozzle, as shown in Fig. 1 indicates that the pressure, mass density, and temperature (p1, ρ1, and T1) at a point within the tank. Since the tank is "large," velocity here is assumed to be near zero. Also indicated on Fig. 1 are the same parameters as well as the velocity of flow and area of the nozzle (p2, ρ2, T2, v2, and A2) at the exit of the nozzle. Also indicated is p2', the pressure outside the tank.

Convergent nozzle
Figure 1, Convergent nozzle.

Preview Isentropic Flow through Convergent Nozzle Calculator

In a convergent nozzle, flow through the nozzle's throat will be either sonic or subsonic. If flow is sonic, the Mach number is equal to unity, and the ratio p2/p1 must be equal to the "critical pressure ratio" as defined by

Eq. 1

( p 2 p 1 ) c = ( 2 k + 1 ) k / ( k 1 )

where

(p2/p1)c = critical pressure ratio
k = specific heat ratio

If flow through the throat is subsonic, the ratio (p2/p1)c will be larger than (p'2/p1).

Obviously, in order to have appreciable flow from the tank through the nozzle out of the tank, pressure inside the tank must be greater than pressure outside the tank (that is, p1 > p'2). If the pressure drop is small [(p2/p1) >(p2/p1)c], flow through the nozzle will be subsonic and the pressure at the exit of the nozzle will be the same as the pressure outside the tank (p2 = p'2). In this case the weight flow rate can be determined from the equation

Eq. 2

G = A 2 2 g k k 1 p 1 γ 1 [ ( p 2 p 1 ) 2 / k ( p 2 p 1 ) ( k + 1 ) / k ]

where:

G = weight flow rate, lbs/ft2, (N/m2)
A2 = throat area, ft2, (m2)
g = acceleration of gravity, ft/sec2 (m/sec2)
k = specific heat ratio
p1 = pressure inside the tank, lbs/ft2, (N/m2)
p2 = pressure inside the tank, lbs/ft2, (N/m2)
γ1 = specific weight of fluid inside the tank, lbs/ft3, (N/m3)
R = Gas constant, ft/°R, (m/K)

If the pressure drop increases (either by increasing p1 or decreasing p'2, or both), flow through the nozzle will remain subsonic until the point is reached where the ratio p'2/p1 is equal to the critical pressure ratio (p2/p1)c, At this point, flow through the nozzle will be sonic and the pressure at the exit of the nozzle will be the same as the pressure outside the tank (p2 = p'2). In this case, the weight flow rate can be determined from the equation

Eq. 3

G = A 2 p 1 T 1 g k R ( 2 k + 1 ) ( k + 1 ) / ( k 1 )

where T1 is the absolute temperature of the fluid inside the tank, R is the gas constant, and other terms are as defined above for equation 2.

If the pressure drop increases further [beyond the point where the ratio p'2/p1 is equal to the critical pressure ratio (p'2/p1)c , flow through the nozzle will remain sonic and the pressure at the exit of the nozzle will be greater than the pressure outside the tank (p2 > p'2). However, the weight flow rate will
not increase. Thus, no matter how much p1 is increased or p'2 is decreased, if the ratio p'2/p1 is less than the critical pressure ratio (p2/p1)c the weight flow rate will be the same as that where the ratio p'2/p1 is equal to the critical pressure ratio. In this case the weight flow rate can be determined from
equation 3 provided the value substituted for p1 is the pressure that makes the ratio p'2/p1 equal to the critical pressure ratio (p2/p1)c.

Schaum's Outline of Fluid Mechanics and Hydraulics

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