Nozzle flow

The nozzle flow is the flow of a fluid, i. H. a gas or a liquid , referred to by a nozzle . The fluid is accelerated while the pressure decreases. In a nozzle flow, potential is converted into kinetic energy .

The mass flow flowing through the nozzle is determined by the pre-pressure , the counter pressure and the narrowest cross-section of the nozzle: at constant pre-pressure, the mass flow increases with decreasing counter-pressure in the outflow space, until at a certain critical pressure ratio the speed in the narrowest cross-section is precisely the speed of sound of the fluid reached. If the back pressure is further reduced below the critical value, the mass flow remains constant. ${\displaystyle p_{i}}$ ${\displaystyle p_{a}}$ ${\displaystyle A}$

One-dimensional calculation model

The one-dimensional calculation model (with the coordinate along the nozzle axis in the direction of flow) is based on the conservation laws of mass, momentum and energy as well as the equations of state of the fluid. Local thermodynamic equilibrium is assumed at each point . ${\displaystyle x}$ ${\displaystyle x}$

It is also assumed that the flow velocity , pressure and temperature are uniform in each cross section perpendicular to the direction of flow: ${\displaystyle w}$ ${\displaystyle p}$ ${\displaystyle T}$ ${\displaystyle (y,z)}$

• ${\displaystyle w(y,z)}$ is constant
• ${\displaystyle p(y,z)}$ is constant
• ${\displaystyle T(y,z)}$ is constant

Flow velocity, pressure and temperature are therefore only dependent on the first spatial dimension . ${\displaystyle x}$

The model results in relationships between integral (global) input and output variables (mass flow, average speed, temperature).

The conservation of mass is ensured by the continuity equation , i. H. the mass flow through each cross section is the same: ${\displaystyle {\dot {m}}}$ ${\displaystyle A}$

${\displaystyle {\dot {m}}(x)=A(x)\cdot w(x)\cdot \rho (x)}$ is constant

Here referred to

• ${\displaystyle \rho (x)}$the density .

The conservation of energy is ensured by the Bernoulli equation :

${\displaystyle {\frac {w^{2}}{2}}+h=h_{0}}$ is constant

in which

• ${\displaystyle h}$the specific enthalpy of the fluid in the flow state with the velocity ${\displaystyle w}$
• ${\displaystyle h_{0}}$denotes the specific enthalpy of the fluid in the boiler state ( ). ${\displaystyle w=0}$

If the flow is adiabatic and friction losses can be neglected, then the entropy of the fluid remains constant during the acceleration through the nozzle in a first approximation ( isentropic nozzle flow):

${\displaystyle S}$ is constant

If the density and the specific enthalpy are given as a function of the specific entropy and the pressure (equations of state of the fluid), then: ${\displaystyle \rho (s,p)}$ ${\displaystyle h(s,p)}$ ${\displaystyle s}$ ${\displaystyle p}$

${\displaystyle \Rightarrow w(p)={\sqrt {2\cdot \left(h_{0}-h(s_{0},p)\right)}}}$

where the specific entropy is in the boiler state. ${\displaystyle s_{0}}$

The flow cross-section depending on the pressure follows from: ${\displaystyle A}$

${\displaystyle A(p)={\frac {\dot {m}}{w(p)\cdot \rho (p)}}}$

The pressure and thus all other quantities are functions of the coordinate . ${\displaystyle p=p(x)}$ ${\displaystyle x}$

The flow cross section has a minimum at the pressure at which the flow velocity is equal to the speed of sound . ${\displaystyle A(p)}$ ${\displaystyle w}$ ${\displaystyle c_{s}}$

The speed of sound is defined by:

${\displaystyle c_{s}={\sqrt {\left({\frac {\partial p}{\partial \rho }}\right)_{s}}}}$

In addition, the following generally applies:

${\displaystyle \left({\frac {\partial h}{\partial p}}\right)_{s}={\frac {1}{\rho (s,p)}}}$

This results in:

${\displaystyle \left({\frac {\partial A}{\partial p}}\right)_{s}={\dot {m}}\cdot {\frac {c_{s}^{2}-w^{2}}{c_{s}^{2}\cdot w^{3}\cdot \rho ^{2}}}}$

Nozzle flow of an ideal gas

In the adiabatic flow of an ideal gas , the following relationship between density and pressure applies :

${\displaystyle p(\rho )=p_{i}\cdot \left({\frac {\rho }{\rho _{i}}}\right)^{\kappa }}$

With

• ${\displaystyle \kappa ={\frac {c_{p}}{c_{V}}}}$: Adiabatic exponent (for example, for air. ) ${\displaystyle \kappa =1{,}4}$
• ${\displaystyle p_{i}}$: Pressure in the preprint area
• ${\displaystyle \rho _{i}}$: Density in the preprint area.

If the flow velocity in the antechamber is neglected ( ), the following relationship between mass flow and pressure ratio results with adiabatic flow: ${\displaystyle w_{i}\approx 0}$

${\displaystyle {\dot {m}}=A\cdot \mu \cdot \Psi (p_{a}/p_{i},\kappa )\cdot {\sqrt {2\cdot p_{i}\cdot \rho _{i}}}}$

With

• the discharge coefficient , which describes the reduction in the actual mass flow due to the constriction of the jet and friction. The flow rate depends only on the nozzle geometry: for well-rounded nozzles is , for sharp-edged orifices , the value can be reduced to. ${\displaystyle \mu }$ ${\displaystyle \mu \approx 1}$ ${\displaystyle \mu =0{,}59}$
• the outflow function ${\displaystyle \Psi (x,\kappa )={\begin{cases}\left({\frac {2}{\kappa +1}}\right)^{\frac {1}{\kappa -1}}\cdot {\sqrt {\frac {\kappa }{(\kappa +1)}}}&x\leq \left({\frac {2}{\kappa +1}}\right)^{\frac {\kappa }{\kappa -1}}\\{\sqrt {{\frac {\kappa }{\kappa -1}}\cdot x^{1/\kappa }\cdot \left(x^{1/\kappa }-x\right)}}&\left({\frac {2}{\kappa +1}}\right)^{\frac {\kappa }{\kappa -1}}<x<1\end{cases}}}$

where the pressure ratio must be used here instead . The value corresponds to the critical pressure ratio described above. ${\displaystyle x}$ ${\displaystyle p_{a}/p_{i}}$ ${\displaystyle \left({\frac {2}{\kappa +1}}\right)^{\frac {\kappa }{\kappa -1}}}$

Application example: gas burner nozzle

The mass flow of a fuel gas- air mixture through the nozzle of a gas burner results with very good accuracy from:

${\displaystyle {\dot {m}}=A\cdot \mu \cdot \Psi \left({\frac {p_{0}}{p_{0}+p_{D}}},\kappa \right)\cdot \left(p_{0}+p_{D}\right)\cdot {\sqrt {\frac {2\cdot M}{R\cdot T}}}}$

in which

• ${\displaystyle A}$: smallest cross section of the nozzle
• ${\displaystyle p_{0}}$: Pressure in the outflow space (ambient pressure)
• ${\displaystyle p_{D}}$: Overpressure in front of the nozzle
• ${\displaystyle M}$: mean molar mass of the fuel gas-air mixture
• ${\displaystyle R=8{,}3145\,\mathrm {J/(mol\,K)} }$: Gas constant
• ${\displaystyle T}$: absolute temperature of the fuel gas-air mixture in front of the nozzle.

The effective nozzle cross-section , which is the only parameter that depends on the design of the nozzle, can be determined from measured values. ${\displaystyle A\cdot \mu }$