INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017
DESIGN AND CFD ANALYSIS OF CONVERGENT AND DIVERGENT NOZZLE
1P.VINOD KUMAR,2B.KISHORE KUMAR 1 PG Scholar, Department ofMECH,NALANDA INSTITUTION OF ENGINEERING AND TECHNOLOGY KantepudiSattenapalli, GUNTUR,A.P, India, Pin: 522438 2 Assistant Professor, Department of MECH,NALANDA INSTITUTION OF ENGINEERING AND TECHNOLOGY,Kantepudi,Sattenapalli, GUNTUR,A.P, India, Pin: 522438 Abstract narrowed, increasing the speed of the jet to the speed Nozzle is a device designed to control the of sound, and then expanded again. Above the speed rate of flow, speed, direction, mass, shape, and/or the of sound (but not below it) this expansion caused a pressure of the Fluid that exhaust from them. further increase in the speed of the jet and led to a Convergent-divergent nozzle is the most commonly very efficient conversion of heat energy to motion. used nozzle since in using it the propellant can be The theory of air resistance was first proposed by Sir heated incombustion chamber. In this project we Isaac Newton in 1726. According to him, an designed a new Tri-nozzle to increase the velocity of aerodynamic force depends on the density and fluids flowing through it. It is designed based on velocity of the fluid, and the shape and the size of the basic convergent-Divergent nozzle to have same displacing object. Newton’s theory was soon throat area, length, convergent angle and divergent followed by other theoretical solution of fluid motion angle as single nozzle. But the design of Tri-nozzle is problems. All these were restricted to flow under optimized to have high expansion co-efficient than idealized conditions, i.e. air was assumed to posses single nozzle without altering the divergent angle. In constant density and to move in response to pressure the present paper, flow through theTri-nozzle and and inertia. Nowadays steam turbines are the convergent divergent nozzle study is carried out by preferredpower source of electric power stations and using SOLID WORKS PREMIUM 2014.The nozzle large ships, although they usually have a different geometry modeling and mesh generation has been design-to make best use of the fast steam jet, de done using SOLID WORKS CFD Software. Laval’s turbine had to run at an impractically high Computational results are in goodacceptance with the speed. But for rockets the de Laval nozzle was just experimental results taken from the literature. what was needed. 1. Introduction to nozzle A nozzle is a device designed to control the direction or characteristics of a fluid flow (especially Swedish engineer of French descent who, in trying to to increase velocity) as it exits (or enters) an enclosed develop a more efficient steam engine, designed a chamber. A nozzle is often a pipe or tube of varying turbine that was turned by jets of steam. The critical cross sectional area and it can be used to direct or component – the one in which heat energy of the hot modify the flow of a fluid (liquid or gas). Nozzles are high-pressure steam from the boiler was converted frequently used to control the rate of flow, speed, into kinetic energy – was the nozzle from which the direction, mass, shape, and/or the pressure of the jet blew onto the wheel. De Laval found that the most stream that emerges from them. A jet exhaust efficient conversion occurred when the nozzle first produces a net thrust from the energy obtained from
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017 combusting fuel which is added to the inducted air. gases is directly backwards, as any sideways This hot air is passed through a high speed nozzle, a component would not contribute to thrust. propelling nozzle which enormously increases its 2. Literature review kinetic energy. The goal of nozzle is to increase the CONVERGENT-DIVERGENT nozzle is kinetic energy of the flowing medium at the expense designed for attaining speeds that are greater than of its pressure and internal energy. Nozzles can be speed of sound. the design of this nozzle came from described as convergent (narrowing down from a the area-velocity relation (dA/dV)=-(A/V)(1-M^2) M wide diameter to a smaller diameter in the direction is the Mach number ( which means ratio of local of the flow) or divergent (expanding from a smaller speed of flow to the local speed of sound) A is area diameter to a larger one). A de Laval nozzle has a and V is velocity The following information can be convergent section followed by a divergent section derived from the area-velocity relation – and is often called a convergent-divergent nozzle 1. For incompressible flow limit, i.e. for M tends to ("con-di nozzle"). Convergent nozzles accelerate zero, AV = constant. This is the famous volume subsonic fluids. If the nozzle pressure ratio is high conservation equation or continuity equation for enough the flow will reach sonic velocity at the incompressible flow. narrowest point (i.e. the nozzle throat). In this 2. For M < 1, a decrease in area results in increase of situation, the nozzle is said to be choked. velocity and vice vera. Therefore, the velocity Increasing the nozzle pressure ratio further increases in a convergent duct and decreases in a will not increase the throat Mach number beyond Divergent duct. This result for compressible subsonic unity. Downstream (i.e. external to the nozzle) the flows is the same as that for incompressible flow. flow is free to expand to supersonic velocities. Note 3. For M > 1, an increase in area results in increases that the Mach 1 can be a very high speed for a hot of velocity and vice versa, i.e. the velocity increases gas; since the speed of sound varies as the square root in a divergent duct and decreases in a convergent of absolute temperature. Thus the speed reached at a duct. This is directly opposite to the behavior of nozzle throat can be far higher than the speed of subsonic flow in divergent and convergent ducts. sound at sea level. This fact is used extensively in 4. For M = 1, dA/A = 0, which implies that the rocketry where hypersonic flows are required, and location where the Mach number is unity, the area of where propellant mixtures are deliberately chosen to the passage is either minimum or maximum. We can further increase the sonic speed. Divergent nozzles easily show that the minimum in area is the only slow fluids, if the flow is subsonic, but accelerate physically realistic solution. sonic or supersonic fluids. Convergent-divergent One important point is that to attain supersonic nozzles can therefore accelerate fluids that have speeds we have to maintain favorable pressure ratios choked in the convergent section to supersonic across the nozzle. One example is attain just sonic speeds. This CD process is more efficient than speeds at the throat, pressure ratio to e maintained is allowing a convergent nozzle to expand (Pthroat / P inlet)=0.528. supersonically externally. The shape of the divergent section also ensures that the direction of the escaping
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017
Table1: speeds vs mach number 3.1 Conical Nozzles
Reg Subs Trans So Super Hyper High- ime onic onic nic sonic sonic hyper Ma <1.0 0.8- 1.0 1.0- 5.0- sonic ch 1.2 5.0 10.0 >10.0
From table.1 at transonic speeds, the flow field around the object includes both sub- and supersonic Fig3.1 conical nozzles parts. The transonic period begins when first zones of 1. Used in early rocket applications because of M>1 flow appear around the object. In case of an simplicity and ease of construction. airfoil (such as an aircraft's wing), this typically 2. Cone gets its name from the fact that the walls happens above the wing. Supersonic flow can diverge at a constant angle decelerate back to subsonic only in a normal shock; 3. A small angle produces greater thrust, because it this typically happens before the trailing edge. (Fig.a) maximizes the axial component of exit velocity and As the speed increases, the zone of M>1 flow produces a high specific impulse increases towards both leading and trailing edges. As 4. Penalty is longer and heavier nozzle that is more M=1 is reached and passed, the normal shock reaches complex to build the trailing edge and becomes a weak oblique shock: 5. At the other extreme, size and weight are the flow decelerates over the shock, but remains minimized by a large nozzle wall angle – Large supersonic. A normal shock is created ahead of the angles reduce performance at low altitude because object, and the only subsonic zone in the flow field is high ambient pressure causes overexpansion and flow a small area around the object's leading edge separation The governing continuity, momentum, and energy 6. Primary Metric of Characterization: Divergence equations for this quasi one-dimensional, steady, Loss isentropic flow can be expressed, respectively 3.2 BELL and Dual Bell 3. Types of nozzles Types of nozzles are several types. They could be based on either speed or shape. a. Based on speed The basic types of nozzles can be differentiated as • Spray nozzles • Ramjet nozzles b. Based on shape The basic types of nozzles can
be differentiated as Fig3.2 (1): BELL and Dual Bell • Conical This nozzle concept was studied at the Jet Propulsion • Bell Laboratory in 1949. In the late 1960s, Rocket dyne • Annular
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017 patented this nozzle concept, which has received 2. Match exit and atmospheric pressure as closely as attention in recent years in the U.S. and Europe. The desired. design of this nozzle concept with its typical inner 3. Permit afterburner operation without affecting base nozzle, the wall in section, and the outer nozzle main engine operation—requires variable throat extension can be seen. This nozzle concept offers an area nozzle. altitude adaptation achieved only by nozzle wall in 4. Allow for cooling of walls if necessary. section. In flow altitudes, controlled and symmetrical 5. Mix core and bypass streams of turbofan if flow separation occurs at this wall in section, which necessary. results in a lower effective area ratio. For higher 6. Allow for thrust reversing if desired. altitudes, the nozzle flow is attached to the wall until 7. Suppress jet noise, radar reflection, and infrared the exit plane, and the full geometrical area ratio is radiation (IR) if desired. used. Because of the higher area ratio, an improved 8. Two-dimensional and axisymmetric nozzles, thrust vacuum performance is achieved. However, vector control if desired. additional performance losses are induced in dual- 9. Do all of the above with minimal cost, weight, and bell nozzles. boat tail drag while meeting life and reliability goals. 10.4 Introduction to convergent and divergent nozzle A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube that is pinched in the middle, making a carefully balanced,
asymmetric hourglass shape. It is used to accelerate a Fig 3.2(2) performance losses are induced in dual- hot, pressurized gas passing through it to a higher bell nozzles. speed in the axial (thrust) direction, by converting the 3.3 Functions of Nozzle heat energy of the flow into kinetic energy. Because The purpose of the exhaust nozzle is to increase the of this, the nozzle is widely used in some types velocity of the exhaust gas before discharge from the of steam turbines and rocket engine nozzles. It also nozzle and to collect and straighten the gas flow. For sees use in supersonic jet engines. large values of thrust, the kinetic energy of the Similar flow properties have been applied to jet exhaust gas must be high, which implies a high streams within astrophysics. exhaust velocity. The pressure ratio across the nozzle controls the expansion process and the maximum uninstalled thrust for a given engine is obtained when the exit pressure (Pe) equals the ambient pressure (P0).The functions of the nozzle may be summarized by the following list: Fig4: convergent and divergent nozzle 1. Accelerate the flow to a high velocity with 5. Conditions for operation minimum total pressure loss.
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017
A de Laval nozzle will only choke at the rate is constant. The gas flowthrough a de Laval throat if the pressure and mass flow through the nozzle is isentropic (gas entropy is nearly constant). nozzle is sufficient to reach sonic speeds, otherwise In a subsonic flow the gas is compressible, no supersonic flow is achieved, and it will act as and sound will propagate through it. At the "throat", a Venturi tube; this requires the entry pressure to the where the cross-sectional area is at its minimum, the nozzle to be significantly above ambient at all times gas velocity locally becomes sonic (Mach number = (equivalently, the stagnation pressure of the jet must 1.0), a condition called choked flow. As the nozzle be above ambient). cross-sectional area increases, the gas begins to In addition, the pressure of the gas at the exit expand, and the gas flow increases to supersonic of the expansion portion of the exhaust of a nozzle velocities, where a sound wave will not propagate must not be too low. Because pressure cannot travel backwards through the gas as viewed in the frame of upstream through the supersonic flow, the exit reference of the nozzle (Mach number > 1.0). pressure can be significantly below the ambient 5.1 Fluid flow inside convergent and divergent pressure into which it exhausts, but if it is too far nozzle below ambient, then the flow will cease to A converging-diverging nozzle ('condi' be supersonic, or the flow will separate within the nozzle, or CD-nozzle) must have a smooth area law, expansion portion of the nozzle, forming an unstable with a smooth throat, dA/dx=0, for the flow to remain jet that may "flop" around within the nozzle, attached to the walls. The flow starts from rest and producing a lateral thrust and possibly damaging it. accelerates subsonically to a maximum speed at the In practice, ambient pressure must be no throat, where it may arrive at M<1 or at M=1, as for higher than roughly 2–3 times the pressure in the converging nozzles. Again, for the entry conditions supersonic gas at the exit for supersonic flow to leave we use 'c' (for chamber) or 't' (for total), we use 'e' for the nozzle. the exit conditions, and '*' for the throat conditions when it is choked (M*=1). If the flow is subsonic at the throat, it is subsonic all along the nozzle, and exit pressure pe naturally adapts to environmental pressure p0 because pressure-waves travel upstream faster (at the speed of sound) than the flow (subsonic), so that pe/p0=1. But now the minimum exit pressure for subsonic flow is no longer pe=pt(2/(γ+1))γ/(γ−1)
(pe/p0=0.53 for γ=1.4), since the choking does not Fig5: mach number condition take place at the exit but at the throat, i.e. it is the throat condition that remains valid, Its operation relies on the different properties of p*=pt(2/(γ+1))γ/(γ−1), e.g. p*/p0=0.53 for γ=1.4; now gases flowing at subsonic and supersonic speeds. the limit for subsonic flow is pe,min,sub>p* because Thespeed of a subsonic flow of gas will increase if of the pressure recovery in the diverging part. the pipe carrying it narrows because the mass flow
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017
However, if the flow is isentropic all along the discharge pressure. That is the normal nozzle, be it fully subsonic or supersonic from the situation for a nozzle working at low throat, the isentropic equations apply altitudes (assuming it is adapted at higher But if the flow gets sonic at the throat, altitudes); it also occurs at short times after several downstream conditions may appear. The ignition, when chamber pressure is not high control parameter is discharge pressure, p0. Let enough. consider a fix-geometry CD-nozzle, discharging a Adapted nozzle, where exit pressure equals given gas from a reservoir with constant conditions discharge pressure (evolution f). Notice that, (pt,Tt). When lowering the environmental pressure, as exit pressure pe only depends on chamber p0, from the no flow conditions, p0=pt, we may have conditions for a choked nozzle, a fix- the following flow regimes (a plot of geometry nozzle can only work adapted at a pressurevariation along the nozzle is sketched in Fig. certain altitude (such that p0(z)=pe). 2): Expansion waves appear at the exit, to expand the exhaust to the lower back pressure (evolution e); this • Subsonic throat, implying subsonic flow all along to the exit (evolution a in Fig 2). is the normal situation for nozzles working under
vacuum. This type of flow is named 'under-expanded' Sonic throat (no further increase in mass-flow-rate because exit pressure is not low-enough, and whatever low the discharge pressure let be). additional expansion takes place after exhaust.
Flow becomes supersonic after the throat, but, before 5.2 Choked flow exit, a normal shockwave causes a sudden transition Chokingisa compressible flow effect that obstructs to subsonic flow (evolution c). It may happen that the the flow, setting a limit to fluid velocity because flow detaches from the wall (see the corresponding theflow becomes supersonic and perturbations cannot sketch). move upstream; in gas flow, choking takes place o Flow becomes supersonic after the throat, when a subsonic flow reaches M=1, whereas in liquid with the normal shockwave just at the exit flow,chokingtakes place when an almost section (evolution d). incompressible flow reaches the vapour pressure (of o Flow becomes supersonic after the throat, the main liquid or of a solute), and bubbles appear, and remains supersonic until de exit, but with the flow suddenly jumping to M>1 there, three cases may be distinguished: Going on with gas flow and leaving liquid Oblique shock-waves appear at the exit, to flow aside, we may notice that M=1 can only occur in compress the exhaust to the higher back a nozzle neck, either in a smooth throat where dA=0, pressure (evolution e). The types of flow or in a singular throat with discontinuous area slope with shock-waves (c, d and e in Fig. 2) are (a kink in nozzle profile, or the end of a nozzle). named 'over-expanded' because the Naming with a '*' variables the stage where M=1 (i.e. supersonic flow in the diverging part of the the sonic section, which may be a real throat within nozzle has lowered pressure so much that a recompression is required to match the
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017 the nozzle or at some extrapolated imaginary throat expansion of steam in nozzle neither heat is supplied downstream of a subsonic nozzle. or rejected work. As steam passes through the nozzle 5.3 Area ratio it loses its pressure as well as heat. Nozzle area ratio ε (or nozzle expansion The work done is equal to the adiabatic heat ratio) is defined as nozzle exit area divided by throat drop which in turn is equal to Rankine area. area, ε≡Ae/A*, in converging-diverging nozzles, or 6.2 Velocity of Steam: divided by entry area in converging nozzles. Notice Steam Enters nozzle with high pressure and that ε sodefined is ε>1, but sometimes the inverse is very low velocity (velocity is generally neglected). also named 'area ratio' (this contraction area ratio is Leaves nozzle with high velocity & low pressure bounded between 0 and 1); however, although no All this is due to the reason that heat energy confusion is possible when quoting a value (if it is >1 at steam is converted into K.E as it passes through refers to Ae/A*, and if it is <1 refers to A*/Ae), one nozzle. must be explicit when saying 'increasing area ratio' The final or outlet velocity at steam can be found as (we keep toε≡Ae/A*>1). follows, To see the effect of area ratio on Mach Let number, (14) is plotted in Fig. 1 for ideal C-Velocity of steam at section considered (m/sec) monoatomic (γ=5/3), diatomic (γ=7/5=1.40), and h - enthalpy at steam at inlet low-gamma gases as those of hot rocket exhaust h - enthalpy at steam at outlet
(γ=1.20); gases like CO2 and H2O have intermediate h - heat drop during expansion at steam (h − h ) values (γ=1.3). Notice that, to get the same high (for 1 kg of steam) Mach number, e.g. M=3, the area ratio needed is Gain in K.E. = adiabatic heat drop * * A /A=0.33 for γ=1.67 and A /A=0.15 for γ=1.20, i.e. = h more than double exit area for the same throat area C=√(2 * 1000*h ) (that is why supersonic wind tunnels often use a = 44.72√h monoatomic working gas. In practice there is loss due to friction in the nozzle and its value from 10-15% at total heat drop. Due to this the total heat drop is minimized.
Let heat drop after reducing friction loss be kh
Velocity (C) = 44.72 √ kh Ratio A*/A (i.e. throat area divided by local area) vs. 6.3 Discharge Through The Nozzle And Condition Mach number M, for γ=1.20 (beige), γ=1.40 (green), For Its Maximum Value: and γ=1.67] (red). p - initial pressure at steam v - initial volume at 1 kg of steam at P (m ) 6. THEORETICAL BACK GROUND p -steam pressure at throat 6.1 Flows through Nozzles: v - volume at 1kg steam at P (m ) The steam flow through the nozzles may be A-area at cross section at nozzle at throat (m ) assumed as adiabatic flow. Since during the C- velocity at steam (m/s)
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INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 2 / SEP 2017
Steam passing through nozzle follows adiabatic process in which If m is the mass of steam discharged in kg/sec p = constant m = n= 1.135 for saturated steam by substituting value of c &v