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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
(*p _{1}, ρ_{1}*, and

*T*) 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

_{1}*(p*and

_{2}, ρ_{2}, T_{2}, v_{2},*A*) at the exit of the nozzle. Also indicated is

_{2}*p*, the pressure outside the tank.

_{2}'

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 p_{2}/p_{1} must be equal to the "critical pressure
ratio" as defined by

Eq. 1

$\left(\frac{{p}_{2}}{{p}_{1}}\right)}_{c}={\left(\frac{2}{k+1}\right)}^{k/(k-1)$where

(p_{2}/p_{1})_{c} = critical pressure ratio

k = specific heat ratio

If flow through the throat is subsonic, the ratio *(p _{2}/p_{1})_{c}* will be larger than (

*p'*.

_{2}/p_{1})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, p_{1} > p'_{2}). If the pressure
drop is small [(*p _{2}/p_{1}*) >(

*p*], 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 (

_{2}/p_{1})_{c}*p*). In this case the weight flow rate can be determined from the equation

_{2}= p'_{2}Eq. 2

$G={A}_{2}\sqrt{\frac{2gk}{k-1}{p}_{1}{\gamma}_{1}[{\left(\frac{{p}_{2}}{{p}_{1}}\right)}^{2/k}-{\left(\frac{{p}_{2}}{{p}_{1}}\right)}^{(k+1)/k}]}$where:

*G* = weight flow rate, lbs/ft^{2}, (N/m^{2})

*A _{2}* = throat area, ft

^{2}, (m

^{2})

*g*= acceleration of gravity, ft/sec

^{2}(m/sec

^{2})

*k*= specific heat ratio

*p*= pressure inside the tank, lbs/ft

_{1}^{2}, (N/m

^{2})

*p*= pressure inside the tank, lbs/ft

_{2}^{2}, (N/m

^{2})

*γ*= specific weight of fluid inside the tank, lbs/ft

_{1}^{3}, (N/m

^{3})

R = Gas constant, ft/°R, (m/K)

If the pressure drop increases (either by increasing

*p*or decreasing

_{1}*p'*, or both), flow through the nozzle will remain subsonic until the point is reached where the ratio

_{2}*p'*/

_{2}*p*is equal to the critical pressure ratio (

_{1}*p*/

_{2}*p*)

_{1}_{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 (

*p*=

_{2}*p'*). In this case, the weight flow rate can be determined from the equation

_{2}Eq. 3

$G=\frac{{A}_{2}{p}_{1}}{\sqrt{{T}_{1}}}\sqrt{\frac{gk}{R}{\left(\frac{2}{k+1}\right)}^{(k+1)/(k-1)}}$where T_{1} 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}/p_{1} is equal to the critical
pressure ratio (p'_{2}/p_{1})_{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 (p_{2} > p'_{2}). However, the weight flow rate will

not increase. Thus, no matter how much p_{1} is increased or p'_{2} is decreased, if the ratio p'_{2}/p_{1} is less
than the critical pressure ratio (p_{2}/p_{1})_{c} the weight flow rate will be the same as that where the ratio
p'_{2}/p_{1} 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 p_{1} is the pressure that makes the ratio p'_{2}/p_{1} equal to
the critical pressure ratio (p_{2}/p_{1})_{c}.

Schaum's Outline of Fluid Mechanics and Hydraulics

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