Session 15 FLOWS THROUGH NOZZLE Outline • Definition • Types of nozzle • Flow analysis • Ideal Gas Relationship • Mach Number and Speed of Sound • Isentropic flow of an ideal gas • De Laval nozzle • Nozzle efficiency • Calculation examples Definition Nozzle is a duct by flowing through which the velocity of a fluid increases at the expense of pressure drop. A duct which decreases the velocity of a fluid and causes a corresponding increase in pressure is called a diffuser Types of Nozzles There are three types of nozzles a. Convergent nozzle b. Divergent nozzle c. Convergent-divergent nozzle Flow Analysis • Incompressible flow • Compressible Flow The variation of fluid density for compressible flows requires attention to density and other fluid property relationships. The fluid equation of state, often unimportant for incompressible flows, is vital in the analysis of compressible flows. Also, temperature variations for compressible flows are usually significant and thus the energy equation is important. Curious phenomena can occur with compressible flows. Flow Analysis • For simplicity, the gas is assumed to be an ideal gas. • The gas flow is isentropic. • The gas flow is constant • The gas flow is along a straight line from gas inlet to exhaust gas exit • The gas flow behavior is compressible Ideal Gas Relationship The equation of state for an ideal gas is: For an ideal gas, internal energy, û is considered to be a function of temperature only Thus, Ideal Gas Relationship The fluid property enthalpy, ĥ is defined as Since for an ideal gas, enthalpy is a function of temperature only, the ideal gas specific heat at constant pressure, Cp can be expressed as And, Ideal Gas Relationship We see that changes in internal energy and enthalpy are related to changes in temperature by values of Cp and Cv. We turn our attention now to developing useful relationships for determining Cp and Cv Thus, Since, so, Ideal Gas Relationship For compressible flows, changes in the thermodynamic property entropy, s, are important. For any pure substance including ideal gases, the “first T ds equation” is where T is absolute temperature, s is entropy, is internal energy, p is absolute pressure, and is density If Cp and Cv are assumed to be constant for a given gas, can be integrated to get Ideal Gas Relationship The second law of thermodynamics requires that the adiabatic and frictionless flow of any fluid results in ds = 0. Constant entropy flow is called isentropic flow. For the isentropic flow of an ideal gas with constant and Thus, which is a useful relationship between temperature, density, and pressure for the isentropic flow of an ideal gas. We can conclude that Mach Number and Speed of Sound The Mach number is defined as the ratio of the value of the local flow velocity, V, to the local speed of sound, c. In other words, This discussion about pressure wave patterns suggests the following categories of fluid flow: 1. Incompressible flow: Ma ≤ 0.3. Unrestricted, nearly symmetrical and instantaneous pressure communication. 2. Compressible subsonic flow: 0.3 < Ma < 1. Unrestricted but noticeably asymmetrical pressure communication. 3. Compressible supersonic flow: Ma > 1. Formation of Mach wave; pressure communication restricted to zone of action. Mach Number and Speed of Sound When the continuity equation is applied to the flow through this control volume, the result is Mach Number and Speed of Sound We can also write, Neglecting the product dρ.dV, both being very small, ρdV = cdρ The momentum equation for the control volume gives, (p-(p+dρ))A = ρAc((c-dV)-c) so., we obtain Mach Number and Speed of Sound We learned earlier that, And thus, Thus for an ideal gas Isentropic flow of an ideal gas When fluid flows steadily through a conduit that has a flow cross-section area that varies with axial distance, the conservation of mass 1 continuity 2 equation And If we form the logarithm of both sides of the continuity equation the result is Isentropic flow of an ideal gas Differentiating results Since the flow being considered is isentropic, the speed of sound is related to variations of pressure with density, repeated here for convenience as combined with the definition of Mach number So Isentropic flow of an ideal gas To better understand why subsonic and supersonic duct flows are so different, this equation can be used Converging-diverging duct flow In the preceding section, we discussed the variation of density and velocity of the steady isentropic flow of an ideal gas through a variable area duct. We proceed now to develop equations that help us determine how other important flow properties vary in these flows. Converging-diverging duct flow We demonstrated that for the isentropic flow of an ideal gas The streamwise equation of motion for steady and frictionless flow can be expressed for an ideal gas as since the potential energy term, γ dz can be considered as being negligibly small in comparison with the other terms involved. Converging-diverging duct flow Consider the steady, one-dimensional, isentropic flow of an ideal gas with constant Cp and Cv through the converging-diverging nozzle of By using the ideal gas equation of state we obtain It is of interest to note that combining leads to results in Converging-diverging duct flow The definition of Mach number and the speed of sound relationship for ideal gases can be combined to yield We can also develop an equation for pressure variation. Since p/ρ =RT then we obtain Converging-diverging duct flow Combining leads to For density variation we consolidate to get Converging-diverging duct flow We have already used the stagnation state for which Ma = 0 as a reference condition. It will prove helpful to us to use the state associated with Ma = 1 and the same entropy level as the flowing fluid as another reference condition we shall call the critical state, denoted We can get a relationship for the critical temperature ratio, T*T0, by substituting Ma = 1. Thus, De Laval nozzle Its operation relies on the different properties of gases flowing at subsonic and supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is isentropic (gas entropy is nearly constant). At subsonic flow the gas is compressible; sound, a small pressure wave, will propagate through it. At the "throat", where the cross sectional area is a minimum, the gas velocity locally becomes sonic (Mach number = 1.0), a condition called choked flow. As the nozzle cross sectional area increases the gas continues to expand and the gas flow increases to supersonic velocities where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0). De Laval nozzle De Laval nozzle A de Laval nozzle will only choke at the throat if the pressure and mass flow through the nozzle is sufficient to reach sonic speeds, otherwise no supersonic flow is achieved. In addition, the pressure of the gas at the exit of the expansion portion of the exhaust of a nozzle must not be too low. Because pressure cannot travel upstream through the supersonic flow, the exit pressure can be significantly below ambient pressure it exhausts into, but if it is too far below ambient, then the flow will cease to be supersonic, or the flow will separate within the expansion portion of the nozzle, forming an unstable jet that may 'flop' around within the nozzle, possibly damaging it. In practice ambient pressure must be no higher than roughly 2-3 times the pressure in the supersonic gas at the exit for supersonic flow to leave the nozzle. Nozzle efficiency Due to friction between the fluid and walls of the nozzle, the expansion process is irreversible, although still approximately adiabatic. In nozzle design, the usual practice is to base all calculation on isentropic flow and then to make an allowance for friction using a coeficient or efficiency. Nozzle efficiency The nozzle efficiency, ηn, is defined as the ratio of the actual enthalpy drop to the isentropic enthalpy drop Nozzle efficiency If the actual velocity at exit from the nozzle is V1 and the velocity at exit when the flow is isentropic is V1s, then : When the inlet is very small, then Calculation examples 1. Steam is expanded in a set of nozzle from 10 bar. 300 0C to 2 bar. Are the nozzle convergent or convergent- divergent? Neglecting the initial velocity, find the minimum area of the nozzle to flow 1 kg/s of steam, Assume isentropic expansion. Calculation examples No.1No.1 Calculation examples 2. Steam expands in a set of nozzle from 10 bar, 300 0C to 1 bar. The convergent parts of the nozzle are sharp and frictionless. In the divergent parts, the friction loss may be taken as 0.15 of the isentropic enthalpy drop. If the steam floe rate is 1 kg/s and the initial velocity of steam is negligible, find the minimum area of the nozzle. If the exit diameter of nozzle is 25 mm, find the number of nozzles. Calculation examples No.2No.2 Calculation examples 3. Air at 7.8 bar and 180 0C expands through a convergent divergent nozzle into a space at 1.03 bar. The flow rate of air is 3.6 kg/s. Assuming isentropic flow throughout and neglecting the inlet velocity, calculate the throat and exit areas of the nozzle. Calculation examples No.3No.3 Calculation examples 4. Combustion gases expand in a propulsion nozzle from 3.8 bar and 450 0C to a back pressure of 1 bar at the rate of 16 kg/s.