FLOW OF GASES AND STEAM THROUGH DIFFUSERS

What are diffusers and applications of diffuser theory

A diffuser is a channel with a continuously changing flow cross-section. Fluid flow in the diffuser is a process that primarily involves an increase in pressure and a decrease in kinetic energy. According to Hugoniot theorem, a different shape of diffuser is suitable for supersonic inlet velocities than for subsonic inlet velocities. In the case of supersonic inlet velocity, the flow must first slow down to the speed of sound in the tapering part of the diffuser, see Figure 374.

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left-diffuser for subsonic speeds; right-diffuser for supersonic speeds. A [m²] diffuser flow area; V [m·s^-1] gas velocity; M [Mach] Mach number; A* [m²] critical cross-section of supersonic diffuser in which gas reaches speed of sound (critical state). The index _i denotes the state at the inlet of the diffuser, the index _e denotes the state at the outlet of the diffuser.

Energy parameters of diffusers

The energy parameters of diffusers, such as the values ​​of state quantities, mass flow, critical velocity and efficiency, can be determined from the energy balance of the gas in the h-s chart of the diffuser, from which most of the quantities can be directly read. Many calculation procedures can be taken from the nozzle calculations given in the article Flow of gases and steam through nozzles. The energy balance of a diffuser during liquid flow can be obtained using the Bernoulli equation.

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h-s chart of gas compression in diffuser

The compression in the diffuser is affected by energy dissipation or losses. An h-s chart can be used to identify the actual gas states as it flows through the diffuser and the losses, where the comparative (ideal) process is an isentropic compression with the same pressure and velocity at the outlet as the actual compression, see Figure 1274, pp. 5.4. The pressure loss L_p is then defined as the loss between the total outlet and inlet pressure of the diffuser. To overcome the loss L_p and achieve the same pressure as in lossless compression, the kinetic energy at the diffuser inlet must be increased by just the value of L_h.

Similarity in diffuser efficiencies

Similar diffusers under similar operating conditions will also have similar efficiencies and because it is the similarity coefficient. This similarity can be used in the design of a new diffuser to predict its parameters based on an estimate of its efficiency. The accuracy of such a design depends on the degree of similarity of the diffusers being compared.

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Energy balance of diffuser at liquid flow

In the case of liquids or insignificant changes in gas density, the energy balance of the diffuser is based on Bernoulli equation. In the diffuser, the liquid does not do any external work, so the total energy of the liquid in front of the diffuser must be equal to the total energy of the liquid at the outlet of the diffuser plus losses, see Formula 415.

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g [m·s^-2] gravitational acceleration; H_{i, e} [J·kg^-1] head of fluid at inlet or outlet; z [m] height of diffuser axis from reference plane; ρ [kg·m^-3] density.

Hydraulic efficiency of diffuser

In these cases, the diffuser efficiency, referred to as the hydraulic efficiency, can be defined as the ratio between the total energy of the fluid at the outlet and the inlet of the diffuser (Formula 411).

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Diffuser shapes

In practice, only two diffuser shapes are used. The simplest shape is the conical diffuser with a constant diffuser angle. The other diffusers, also known as cornut diffusers, have a variable angle diffuser depending on the pressure gradient requirement of the diffuser.

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Pressure gradient in diffuser

The properties of diffusers depend on the pressure gradient distribution in the diffuser, which can be determined for the case of lossless flow and ideal gas using Equation 432. In the case of actual processes, the pressure gradient can be calculated using thermodynamic data of real gases, see Problem 441, pp. 5.7.

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κ [1] ratio of heat capacities. The derivation of this equation is given in Appendix 432. The equation is derived under the simplifying assumption that the flow velocity has only an axial direction throughout the cross section and for an ideal gas.

Calculated radius of diffuser with constant pressure gradient - so-called cornut shape [Frass, 1989, p. 156]. r [mm]; x [mm].

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Two basic shapes of cornut diffusers

(a) diffuser with constant pressure gradient, see its calculation in Problem 441; (b) diffuser with linear decrease in pressure gradient.

Flow separation from walls of cornut diffusers

Cornut diffusers have a sharp widening at the outlet (see Problem 441), so they can be expected to be more sensitive to boundary layer separation from the wall than cone diffusers. Measurements show that this is the case for long diffusers, but the opposite is true for short diffusers (cone diffusers with α>18°) [Dejč, 1967, p. 392].

Diffusers with constant pressure gradient

Diffusers with the constant pressure gradient (Figure 430a) also have a more uniform velocity profile than cone diffusers and are therefore also used upstream of coolers or heat exchangers with the requirement for uniform distribution of mass flux over the flow area of the exchanger [Goroščenko, 1952, p. 67], [Frass, 1989, p. 155].

Diffusers with linear pressure gradient

In the diffuser designed to linearly decrease the pressure gradient (Figure 430b), the pressure gradient decreases gradually as the boundary layer energy decreases (approximately linearly), and is therefore the shape with the lowest probability of flow separation [Dejč, 1967, s. 388].

Practical solutions for cornut diffusers

Continuous shape changes of diffusers with variable diffuser angles are difficult to manufacture and are therefore replaced by a combination of two or more conical diffusers with different diffuser angles, see Figure 831, pp. 5.8, [Dejč, 1967, s. 393].

Increasing kinetic energy in the boundary layer using centrifugal force and suction

The boundary layer can also be stabilised by the tangential velocity component and the centrifugal force will cause a higher pressure at the diffuser walls - in the case of a potential vortex, a nearly stationary core is formed in the axis of the diffuser, which in the case of liquids is filled with saturated vapour. Typical examples are water turbine draft tubes, in which a small tangential component of the flow at the turbine outlet. The flow at the outlet of the diffuser can also be stabilised by suction gas through openings in the diffuser walls, etc (see [Japikse and Baines, 1995]).

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Pressure loss in diffusers

The flow separation will also affect the magnitude of the pressure loss L_p of the diffuser (see Equation 1274, pp. 5.4 for definition). Pressure loss is also a function of diffuser length and diffuser angle.

Effect of cone diffuser angle on pressure loss

Figure 631 shows the dependence of the pressure loss L_p of the diffuser with the change of the diffuser angle α. This pressure loss is compared with the pressure loss at the flow through two connected channels without a diffuser (α=90°). In this way, it is possible to evaluate up to which diffuser angle it makes sense to use diffusers and when not to use it. According to Figure 631, the pressure drop of the cone diffuser from a certain angle can be greater than that of the flow through two connected channels without a diffuser. This is due to the fact that the internal friction loss decreases with the diffuser angle α, but the vorticity loss at boundary layer separation increases with the angle α. Thus, in a flow through two connected channels without a diffuser, only the vortices at separation are generated [Maštovský, 1964, p. 88], which cause an increase in entropy by the same mechanism as in the throttling of the flow through the orifice.

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A scale chart is given in [Dejč, 1967, p. 382].

Solutions for short diffuser shapes with regard to pressure loss

If it is necessary to shorten the diffuser, it is more cost effective to use the combination shown in Figure 427 than to increase the diffuser angle. This solution can be likened to a smooth cornut diffuser on Figure 430a, pp. 5.7.

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Non-design states of subsonic diffusers

Figure 554 shows the two non-design conditions of the subsonic diffuser, denoted by a, b (index n indicates the design condition). These non-design conditions are induced by a change in the inlet velocity V_i for the same inlet stagnation pressure, where V_ia<V_in<V_ib=a. The velocity V_ib is sonic respectively critical. For each case, the backpressure also changes, if it were still the same (p_e=p_en), there would be no flow equilibrium. If we want to maintain backpressure, then we need to use inlet flow control-such a typical application is a diffuser valve. At less than the critical pressure p* a shock wave is generated behind the narrowest cross-section and, in addition, when the backpressure drops below p_ec, the diffuser becomes a Laval nozzle, see Hugoniot theorem.

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Effect of inlet velocity change on function of subsonic diffuser

N-area function of Laval nozzle.

Non-design states of supersonic diffusers

Figure 654 shows two non-design conditions of the supersonic diffuser, denoted by a, b (index n denotes the design condition), with V_ia<V_in<V_ib>a. For each case, the backpressure is also varied so that the subsonic section of the diffuser does not produce a shock wave.

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Effect of inlet velocity change on supersonic diffuser function

In the case-a, the convergent section of the diffuser is not able to accommodate such a large amount of gas, so a normal shock wave will be generated before the diffuser, which will increase the pressure to supercritical and reduce the velocity to subsonic – convergent part act as a nozzle and the divergent section of the diffuser will function as a Laval nozzle in the non-design condition.

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Supersonic profile cascades for compressors

Supersonic profile cascades are rarely used due to their low efficiency and poor controlled operation. Their use is justified, for example, in single-stage compressors with very high compression ratios, see Figure 770.

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Example of supersonic turbocompressor arrangement

1-radial compressor impeller; 2-diffuser blades with supersonic profile.

Ejectors and injectors

Ejectors and injectors are jet machines that are used as vacuum pumps or pumps. The function of ejectors or injectors is based on transferring part of the kinetic energy of the driving fluid to the fluid being drivenin the mixing zone. This happens approximately at the neck of the diffuser, see Figure 112, where the driven fluid is drawn into the jet of the driving fluid, the whole process being accompanied by relatively high losses manifested by an increase in the internal thermal energy of the working fluid. In the diffuser section of the machine, kinetic energy is transformed into pressure energy. The difference between an ejector and an injector is that the pressure at the outlet of the ejector is lower than the pressure of the driving fluid at the inlet. In contrast, the pressure at the outlet of the injector is higher than the pressure of the driving fluid.

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General chart of ejector or injector

A-driving fluid; B-driven fluid; 1-inlet zone; 2-diffuser neck (mixing zone); 3-outlet diffuser.

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Example of using injector as feed pump

Injectors are used as feed water pumps for steam boilers of steam locomotives. The water is pumped to higher pressure using the steam injector, which has an inlet pressure lower than the outlet pressure of the diffuser p_e. This is possible because of the very high kinetic energy the steam can gain in the nozzle during expansion, see Problem 410. The steam transfers this kinetic energy to the water in the mixing chamber (neck of diffuser) and condenses at the same time. A necessary condition for the operation of such a pump is that the vapour still condenses in the neck of the diffuser, or that only liquid without vapour bubbles flows through the diffuser, otherwise the required pressure cannot be achieved. The driving vapour will completely condense in the diffuser neck if an adequate amount of cold water is added. This means that the pump performance decreases with the temperature of the intake water.

η_A-2 [1] expansion efficiency in nozzle and momentum transfer in mixing chamber (derivation in Appendix 410, §4).

Ram-powered engines

Ram-powered engines use the supersonic diffuser in the engine mouth to compress air during supersonic flight. The compressed air is then burned in a combustion chamber with fuel and the hot exhaust gases expand in the nozzle and create thrust. Unlike turbocompressor jet engines, they do not contain a turbocompressor and turbine section. When moving at supersonic speed, the values of the achieved pressures change significantly, hence we distinguish between the design of the ramjet jet engine suitable for lower supersonic speeds and the scramjet jet engine more suitable for very high supersonic speeds.

X-43A: Scramjet-powered aircraft

The experimental X-43A Scramjet-powered unmanned aircraft reached Mach 6,83 during a 10-minute flight. It reached its operating speed using a booster rocket at an altitude of 30 000 m. The entire system was launched from a B-52B bomber at a lower altitude. The X-43A aircraft uses the effect of a diagonally cut Laval nozzle, i.e. the creation of expansion waves that replace the opposite wall of the nozzle - the aircraft is lighter.

References