Dynamic characteristics of the recessed chamber within a gas–liquid coaxial injector

Abstract

In this article, a linear dynamic characteristic model of the recessed chamber in a gas–liquid shear coaxial injector is developed. The gaseous injector flow before the recessed chamber is considered to be steady, and we just investigate the response of pressure drop across the recessed chamber to the mass flow rate fluctuation of the liquid injector. The transfer function of the recessed chamber is obtained, considering the distribute characteristics of liquid droplets within the recessed chamber. The results show that the amplitude of pressure drop pulsation would decrease as the liquid flow pulsation frequency increases, when the variation of liquid velocity is not taken into account. While considering the variation of liquid velocity, the result is opposite. The amplitude of pressure drop oscillation on the boundary of the recessed chamber increases with the increase in liquid velocity oscillation frequency.

Introduction

As the demand for large thrust liquid rocket engines increases, some highly effective propellant such as liquid hydrogen, liquid oxygen, and kerosene as well as some new techniques such as staged combustion cycle are widely used. The application of these new techniques facilitates the use and research of coaxial injector technique. For a gas–liquid coaxial swirl injector, the liquid propellant is injected through a center swirl injector, and the gaseous propellant is injected with high velocity through an annular gap around the center injector. The hollow cone liquid sheet ejected out of center injector impinges on the surrounding gas stream, providing better atomization and higher performance. In industrial applications, the center injector always retracts with respect to the exit surface of the outer injector, forming a recessed chamber. ¹ The recessed chamber of coaxial injector can enhance the mixing of propellants, subsequently affect the stability of flame and result in favorable and stable combustion.^2–5

As shown in Figure 1, the oxidant and fuel of recessed coaxial injector flow in each tube. They meet in the mixed recessed chamber, break up, mix, and eject out of the nozzle. The recessed length of oxidant nozzle outlet’s section to fuel nozzle outlet’s is the principal section which influences the interaction of propellant in the entrance section of combustion chamber, and the interaction of gas–liquid phase in the recessed chamber plays an important role in completeness and stability of combustion in the combustion chamber.

Figure 1.

Schematic diagram of a shear coaxial injector.

There have been many studies on the spray and combustion characteristics of the coaxial swirl injector with the recess configuration;^6–10 however, the dynamic characteristics of this type of injector have been less studied.

Chen et al. ¹¹ analyzed the propagation law of fluid oscillation in the gas–liquid jet; however, the calculating work is large. Actually, frequency-domain analysis is easier than time-domain analysis while solving dynamic characteristics of the injector, which means calculating the frequency response in real frequency domain is easier than solving differential equation. When a sinusoidal input of a certain frequency is put in the injector, the amplitude ratio of the output and input and the phase shift are called frequency characteristics or frequency response. The characteristic is the manifestation of dynamic characteristics of the injector in real frequency domain, and hydromechanics equations can be transformed into frequency characteristic equations via Laplace transformation. Fu and Yang ¹² developed a linear dynamic characteristics model of a gas–liquid coaxial swirl injector. They assumed the gas–liquid two-phase flow in the recessed chamber is homogeneous flow and obtained the transfer function of the recessed chamber. This article analyzes the interaction and the dynamic process of gas and liquid in the recessed chamber of gas–liquid coaxial injector linearly, and the distribution of dispersed phase within the recessed chamber is taken into account, that is, the liquid droplets within the recessed chamber are considered to be inhomogeneous.

Theoretical framework

In the recessed chamber, because the momentum of the gas is much larger than the momentum of the liquid, the liquid film within the liquid nozzle exit will break up and turn into droplets on account of aerodynamic force. The droplets will produce the medium motion-resistance force, which will cause the appearance of pressure difference in the gas in the entrance and exit. Droplet distribution within the recessed chamber is nonuniform, that is the parameters cannot be considered to be lumped in the recessed chamber when ${\lambda }_l<L_{\mathrm{rec}}$ . Because the frequency correlates with the wavelength, it is necessary to estimate the range of disturbance frequency while considering the nonuniform distribution of droplets along the recessed chamber. The liquid enters the mixing zone at the speed of $v_l$ , which is slower than sound velocity apparently, so the disturbance wavelength of mass flow rate of liquid phase can be estimated through the formula below

$${\lambda }_l=\frac{v_l}{f_l}$$

(1)

where ${\lambda }_l$ is the wavelength of oscillation wavelength and $f_l$ is the frequency of oscillation wavelength.

Let $L_{\mathrm{rec}}$ be the length of the recessed chamber, and if ${\lambda }_l<L_{\mathrm{rec}}$ (it means $f_l>v_l/L_{\mathrm{rec}}$ ), the distribution of droplets cannot be ignored. Suppose that the ejecting velocity of liquid at the liquid nozzle is 30 m/s, and the length of the recessed chamber is 10 mm, the range of disturbance frequency is $f_l>3000\mathrm{Hz}$ while the mass distribution characteristics of droplets should be taken into account. Thus, it shows that the distributed characteristics of flow parameters cannot be ignored in medium-high frequency range. For the detailed derivation of the transfer function, we can refer to the literature. ⁵

Supposing the velocity of liquid in the recessed chamber is constant

The pressure drop of the recessed chamber whose length is dx can be calculated as

$$\mathrm{dp}=F({\rho }_l,{\rho }_g,v_g-v_l,\dots )\mathrm{dx}$$

(2)

where ${\rho }_l$ is the density of liquid and ${\rho }_g$ is the density of gas, and $v_g-v_l$ is the velocity difference in liquid and gas in the recessed chamber.

The total pressure loss in the recessed chamber can be obtained by integration of dp from $x_1$ to $x_2$

$$p_1-p_2=\underset{x_1}{\overset{x_2}{\int }}F({\rho }_l,{\rho }_g,v_g-v_l,\dots )\mathrm{dx}$$

(3)

Linearize equation (3) and express it through relative amount of disturbance. The velocity of the gas phase is relatively higher than that of the liquid phase. According to $\lambda =v/f$ , the wavelength for disturbance of gas phase is greater than that of liquid phase. For example, if v_g = 80 m/s, f = 1000 Hz, then the disturbance wavelength equals to 80 mm, which is much greater than the common recessed chamber length. Therefore, it is reasonable to consider that ${\rho }_g$ and $v_g$ are only relevant to t (it means that the gas in the recessed chamber is viewed as lumped parameter), and we can obtain

$$\delta {\overline{p}}_1-\delta {\overline{p}}_2=k_1\frac{{\rho }_l}{p_d}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}\delta {\rho }_{l}dx}+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g-k_4\frac{v_l}{p_d}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}\delta {\overline{v}}_{l}dx}$$

(4)

where the quantity with overbar “−” denotes the relative quantity, prefix “ $\delta$ ” denotes the perturbation quantity, $p_d=p_1-p_2$ , $k_1=\partial F/\partial {\rho }_l$ , $k_2=(\partial F/\partial \underset{g}{\overset{.}{m}})(x_1-x_2)$ , $k_3=(\partial F/\partial \underset{g}{\overset{.}{v}})(x_2-x_1)$ , and $k_4=\partial F/\partial v_l$ .

Also, we can obtain equation (5) through ${\rho }_l={\stackrel{.}{m}}_l/(A{v}_l)$

$$\delta {\overline{\rho }}_l=\delta {\overline{\stackrel{·}{m}}}_l-\delta {\overline{v}}_l$$

(5)

In consideration of equation (5), equation (4) can be expressed as

$$\delta {\overline{p}}_1-\delta {\overline{p}}_2=k_1\frac{{\rho }_l}{p_d}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}\delta {\overline{\stackrel{.}{m}}}_{l}dx}+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g-\left[k_1\frac{{\rho }_l}{p_d}+k_4\frac{v_l}{p_d}\right]{\displaystyle \underset{x_1}{\overset{x_2}{\int }}\delta {\overline{v}}_{l}dx}$$

(6)

The liquid infinitesimal in section x at moment t is ejected out from the section of nozzle (x = 0) at the moment of $(t-\tau )$ . Let the velocity of liquid infinitesimal be $v_{l(t-\tau )}$ . If the change in translation speed of liquid infinitesimal from 0 to x is ignored, apparently, delay time $\tau$ is equal to the time for the liquid infinitesimal moving from section 0 to section x

$$\tau =\frac{x}{v_{l(t-\tau )}}$$

(7)

In the axial direction of the recessed chamber, the velocity of liquid entering the recessed chamber can be hypothesized as $v_0$ which is a constant. In this situation, on account of equation (7), equation (6) can be expressed as

$$\delta {\overline{p}}_1=\delta {\overline{p}}_2+k_1\frac{{\rho }_l}{p_d}\underset{0}{\overset{x}{\int }}\delta {\overline{\stackrel{·}{}}m}_{l(t-\tau )}\mathrm{dx}+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g$$

(8)

When the initial conditions are zero, conduct Laplace transform with equation (8) and obtain

$$\delta {\overline{p}}_1=\delta {\overline{p}}_2+k_1\frac{{\rho }_l}{p_d}{\displaystyle \underset{0}{\overset{x}{\int }}\delta {\overline{\stackrel{.}{m}}}_l\exp \left(\frac{-sx}{v_0}\right)dx}+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g$$

(9)

Calculate the integral term in equation (9), and we can obtain the relation of relative disturbance quantity of the parameters on the boundary of the recessed chamber

$$\delta {\overline{p}}_1=\delta {\overline{p}}_2+W_m\left(s\right)\delta {\overline{\stackrel{·}{m}}}_l+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g$$

(10)

In the equation above, $W_m(s)=k_1({\rho }_l/p_d)(v_0/s)(1-\exp (-\mathrm{sx}/v_0))$ is the transfer function of pressure drop oscillation in the recessed chamber which is caused by liquid flow oscillation.

Velocity of liquid in the recessed chamber changes

In actual situation, the velocity of liquid entering the recessed chamber changes. At the same time, the delay time $\tau$ will also change accordingly. Expressing spouting velocity as the sum of the mean value and perturbation velocity, we can obtain equation (11) as follows

$$v_{l(t-\tau )}=v_{0(t-\tau )}+\delta v_{l(t-\tau )}$$

(11)

In this situation, for velocity fluctuating $\delta v_l$ which is small enough, equation (6) can be expressed as the following equation

$$\delta {\overline{p}}_1=\delta {\overline{p}}_2+k_1\frac{{\rho }_l}{p_d}{\displaystyle \underset{0}{\overset{x}{\int }}\left[\delta {\overline{\stackrel{.}{m}}}_{l(t-x/v_0)}+x/v_0\frac{d\delta {\overline{v}}_l}{dt}\right]dx}+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g-\left[k_1\frac{{\rho }_l}{p_d}+k_4\frac{v_l}{p_d}\right]{\displaystyle \underset{0}{\overset{x}{\int }}\delta {\overline{v}}_{l(t-x/v_0)}dx}$$

(12)

When the initial condition is zero, we conduct Laplace transform with equation (12) and obtain

$$\delta {\overline{p}}_1=\delta {\overline{p}}_2+k_1\frac{{\rho }_l}{p_d}\left[{\displaystyle \underset{0}{\overset{x}{\int }}\delta {\overline{\stackrel{.}{m}}}_l\exp \left(-\frac{sx}{v_0}\right)dx}+{\displaystyle \underset{0}{\overset{x}{\int }}\frac{sx}{v_0}\delta {\overline{v}}_l\exp \left(-\frac{sx}{v_0}\right)dx}\right]+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g-\left[k_1\frac{{\rho }_l}{p_d}+k_4\frac{v_l}{p_d}\right]{\displaystyle \underset{0}{\overset{x}{\int }}\delta {\overline{v}}_l\exp \left(-\frac{sx}{v_0}\right)dx}$$

(13)

The equation above can be simplified as equation (14)

$$\delta {\overline{p}}_1-\delta {\overline{p}}_2=W_m\left(s\right)\delta {\overline{\stackrel{·}{m}}}_l+W_v\left(s\right)\delta {\overline{v}}_l+k_2\frac{{\rho }_g}{p_d}\delta {\overline{\rho }}_g+k_3\frac{v_g}{p_d}\delta {\overline{v}}_g$$

(14)

In the equation above

$$W_v\left(s\right)=\frac{v_0}{s}\left[k_1\frac{{\rho }_l}{p_d}\left(1-e^{-s\frac{x}{v_0}}\left(1+\frac{sx}{v_0}\right)\right)-\left(k_1\frac{{\rho }_l}{p_d}+k_4\frac{v_l}{p_d}\right)\left(1-e^{-s\frac{x}{v_0}}\right)\right]$$

(15)

$W_v(s)$ is the transfer function considering the oscillation of delay time, which is caused by the oscillation of liquid spouting velocity while entering the recessed chamber.

Equation (14) manifests that considering the oscillation of delay time, although there does not exist flow disturbance at the section of nozzle exit, there is pressure drop oscillation on the boundary of the recessed chamber. This is because, if liquid spouting velocity while entering the recessed chamber is variable, at the certain time t, the total mass of liquid at section x is not only decided by the liquid mass entering at time $(t-\tau )$ , whose average velocity is $v_0$ , but also decided by the particle entering before time $(t-\tau )$ , whose velocity is relatively slow. Therefore, considering the oscillation of liquid spouting velocity, although the flow into the recessed chamber, spouted by the nozzle, is a constant, there exists the oscillation of flow and density at section x.

Results and discussion

Figure 2(a) shows the amplitude–frequency curve of $W_m(s)$ , Figure 2(b) expresses the phase frequency response curve of $W_m(s)$ , and Figure 2(c) expresses the hodograph of $W_m(s)$ . According to this figure, when the frequency increases, the module value of $W_m(s)$ decreases and reach to zero periodically at the points where $\omega x/v_0=2k\pi$ . Also, in the situation of these oscillation frequencies satisfying $\omega x/v_0=2k\pi$ , the pressure drop oscillation on the boundary of the recessed chamber obtains the minimum.

Figure 2.

Dynamic characteristics of the recessed chamber when the velocity change in liquid is not considered ( ${\rho }_l=1000\mathrm{kg}/{\mathrm{m}}^3$ , $p_d=0.1\mathrm{MPa}$ , $v_0=10\mathrm{m}/\mathrm{s}$ , and $x=10\mathrm{mm}$ ): (a) amplitude–frequency curve, (b) phase–frequency response curve, and (c) hodograph.

According to Figure 2(c), when angular frequency of disturbance increases, the module value of $W_m(s)$ decreases and reaches zero periodically at the points where $\omega x/v_0=2k\pi$ . However, only the case $k=1$ conforms to the actual situation. Therefore, only in the situation of these oscillation frequencies when $\omega x/v_0=2\pi$ , the pressure drop oscillation on the boundary of the recessed chamber obtains the minimum. In the situation of these oscillations, the amplitude of pressure drop decreases because when $v_0$ is constant, the amplitude of liquid flow oscillation along the direction of the length of the recessed chamber has simple harmonic characteristics. At the point where $\omega x/v_0=2\pi$ in the streamwise direction of the recessed chamber, there are integer numbers of disturbance waves. Therefore, the increase in pressure drop at the positive half-wave of oscillation of $\delta {\stackrel{·}{m}}_l$ and the decrease in pressure drop at the negative half-wave of oscillation of $\delta {\stackrel{·}{m}}_l$ counteract with each other. In high-frequency oscillation region, all components of pressure drop oscillation, which have diploid times of period, equal zero. Therefore, the integral of oscillation of pressure drop will be decided only by the components which cannot exist in integral number of period in the recessed chamber. With the increase in oscillation frequency, the wavelength of liquid flow oscillation will decrease, and the oscillation amplitude of pressure drop will decrease accordingly.

Figure 3 shows the dynamic characteristics of the recessed chamber calculated using equation (15), which consider the variation of liquid velocity along the recessed chamber. It shows the dynamic respond of pressure drop to the pulsation of liquid velocity. Figure 3(a) shows the amplitude–frequency diagram of $W_v(s)$ , Figure 3(b) shows the phase–frequency diagram of $W_v(s)$ , and Figure 3(c) shows the hodograph of $W_v(s)$ . According to this figure, when disturbance frequency increases, the amplitude of pressure drop disturbances on the boundary of the recessed chamber increases, and the phase difference between pressure drop and liquid spouting velocity also increases. The hodograph starts from positive real axis. When $\omega$ increases, the curve rotates clockwise in spiral line. The distance between the endpoint and the origin is module value of frequency characteristic. Comparing the amplitude of pressure drop considering velocity change and the one irrespective of velocity change, when considering velocity change, the amplitude of pressure drop on the boundary of the recessed chamber is less than the one when taking no account of velocity change.

Figure 3.

Dynamic characteristics of the recessed chamber considering liquid velocity change ( ${\rho }_l=1000\mathrm{kg}/{\mathrm{m}}^3$ , $p_d=0.1\mathrm{MPa}$ , $v_0=10\mathrm{m}/\mathrm{s}$ , and $x=10\mathrm{mm}$ ): (a) amplitude–frequency characteristic, (b) phase–frequency characteristic, and (c) hodograph.

Conclusion

The interaction of gas phase and liquid phase in the recessed chamber of gas–liquid coaxial injector is investigated with the frequency method. The dynamic response of pressure drop pulsation in the recessed chamber to the liquid flow pulsation is calculated, in the condition of considering and not considering the variation of liquid velocity. When disturbance frequency is in medium-high frequency range, the distribution characteristic of the drop in the recessed chamber must be taken into consideration. The calculation results show that when not considering the variation of liquid velocity across the recessed chamber, the amplitude of pressure drop pulsation on the boundary of the recessed chamber decreases in general, with the increase in liquid velocity oscillation frequency. At points where $\omega x/v_0=2k\pi =1,2,\dots$ , the amplitude of pressure drop oscillation becomes zero periodically. Also, the phase difference between the pressure drop oscillation and liquid velocity oscillation increases linearly with the increase in disturbance frequency. While considering liquid velocity change in the recessed chamber, the amplitude of pressure drop oscillation on the boundary of the recessed chamber increases with the increase in liquid velocity oscillation frequency. Also, the phase difference between them also increases with the increase in disturbance frequency.