**3. Theoretical study on pressure fluctuation of high-pressure common rail fuel injection system**

The fluctuation characteristics of fuel pressure in the pipeline of high-pressure common rail fuel injection system can be represented as one-dimensional partial differential equations of unstable compressible flow, as shown in Eq. (1).

$$\begin{cases} \frac{\partial p}{\partial t} + u \frac{\partial p}{\partial \mathbf{x}} + a^2 \rho \frac{\partial u}{\partial \mathbf{x}} = \mathbf{0} \\ \frac{\partial p}{\partial \mathbf{x}} + \rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial \mathbf{x}} + 2\kappa \rho u = \mathbf{0} \end{cases} \tag{1}$$

where *u* is the fuel flow velocity, *a* is the fuel pressure wave propagation velocity and *κ* is the fuel flow resistance coefficient.

*Pressure Fluctuation Characteristics of High-Pressure Common Rail Fuel Injection System DOI: http://dx.doi.org/10.5772/intechopen.102624*

The above-mentioned partial differential equations can be converted into ordinary differential equations as follows.

$$\begin{cases} \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)\_{\mathrm{R}} = u + a, & \frac{\mathrm{d}p}{\mathrm{d}t} + a\rho \frac{\mathrm{d}u}{\mathrm{d}t} + 2a\kappa\rho u = 0\\ \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)\_{\mathrm{L}} = u - a, & \frac{\mathrm{d}p}{\mathrm{d}t} - a\rho \frac{\mathrm{d}u}{\mathrm{d}t} - 2a\kappa\rho u = 0 \end{cases} \tag{2}$$

where (d*x*/d*t*)R and (d*x*/d*t*)L are the trace lines of pressure waves propagating to the right and left inside the pipeline. The above equations directly reflect the propagation relationship of the pressure wave in the pipeline of high-pressure common rail fuel injection system. When the fuel pressure wave propagates to a specific position in the pipeline, the pressure at this position rises, that is, d*p* is positive, and the pressure wave is a compression wave. Conversely, the pressure drops. d*p* is negative, and it is an expansion wave. Assuming that the right direction of *x* axis is the positive direction, the fuel pressure wave propagating along the forward direction is called the right–traveling wave, while the fuel pressure wave propagating along the negative direction is called the left–traveling wave. The right–traveling wave is represented by d*p*<sup>R</sup> and d*u*R, and it has the following relationship.

$$\mathbf{d}p\_{\mathrm{R}} = \mathfrak{a}\rho \mathbf{d}u\_{\mathrm{R}} \tag{3}$$

While the left–traveling wave is represented by d*p*<sup>L</sup> and d*u*L, which has the following relationship.

$$\mathbf{d}p\_{\mathbf{L}} = -a\rho \mathbf{d}u\_{\mathbf{L}} \tag{4}$$

The pressure wave in the pipeline of high-pressure common rail fuel injection system is divided into left-traveling wave and right-traveling wave according to the direction of propagation and divided into compression wave and expansion wave according to the change of fuel pressure caused by propagation. Therefore, pressure waves in the pipeline can be divided into the following four types.

#### a. Right-traveling compression wave

This kind of fuel pressure wave propagates along the positive direction of *x*-axis, where d*p*<sup>R</sup> is positive and d*u*<sup>R</sup> is also positive, that is, the fuel pressure and velocity along the x-forward propagating direction increase.

#### b. Right-traveling expansion wave

This kind of fuel pressure wave propagates along the positive direction of the *x* axis, but d*p*<sup>R</sup> is negative and d*u*<sup>R</sup> is also negative, that is, the fuel pressure and velocity along the x-forward propagating direction decrease.

#### c. Left-traveling compression wave

This kind of fuel pressure wave propagates backward along the *x*-axis, where d*p*<sup>L</sup> is positive and d*u*<sup>L</sup> is negative, that is, the fuel velocity decreases along the x-forward propagating direction. However, the fuel pressure and velocity of left-traveling compression wave along with the propagating direction increase.

#### d. Left-traveling expansion wave

This kind of fuel pressure wave propagates backward along the *x*-axis, where d*p*<sup>L</sup> is negative and d*u*<sup>L</sup> is positive, that is, the fuel velocity along the x-forward propagating direction increases.

Supposing the right-traveling wave arriving at *x* position of the fuel pipeline at moment *t* is d*p*<sup>R</sup> and d*u*R, and the left-traveling wave is d*p*<sup>L</sup> and d*u*L. According to the pressure wave synthesis theory, the total pressure variation d*p* and velocity variation d*u* are

$$\begin{cases} \mathbf{d}p = \mathbf{d}p\_{\mathbb{R}} + \mathbf{d}p\_{\mathbb{L}}\\ \mathbf{d}u = \mathbf{d}u\_{\mathbb{R}} + \mathbf{d}u\_{\mathbb{L}} \end{cases} \tag{5}$$

where d*p* and d*u* are called synthetic pressure waves at (*x*, *t*). In fact, after the fuel pressure wave is synthesized, each single-traveling wave continues to propagate along its determined direction. Therefore, the above equation can also be understood as the pressure fluctuation in a certain position in the pipeline can be decomposed into lefttraveling waves and right-traveling waves.

When the pressure wave propagates to the boundary surface, another returned pressure wave can be obtained based on the pressure wave and the boundary condition at the moment, which is the reflected wave. Assuming that the right end of the highpressure common rail fuel injection system is closed, the boundary condition is *u* = 0 and d*u* = 0. At this time, if there is a right-traveling wave d*p*<sup>R</sup> arriving on the left, there must be a left-traveling reflected pressure wave d*p*L, so that d*u* = d*u*<sup>R</sup> + d*u*<sup>L</sup> = 0. It can be seen that the fuel reflected pressure wave from the pipeline end boundary surface is the result of coupling the boundary condition with the propagating fuel pressure wave.

There are three boundary conditions for pressure wave propagation and reflection in the pipeline of high-pressure common rail fuel injection system. The boundary type at common rail and high pressure fuel pipeline is outlet isobaric end (outlet opening end). The boundary type is the closed-end when the nozzle needle valve is closed. The boundary type is the orifice flow ends when the needle opening nozzle and injector injection.

#### a. Outlet isobaric end

There are the following equations when the right-traveling wave d*p*<sup>R</sup> reaches the open end of the right end.

$$\begin{cases} \mathbf{d}p\_{\mathbb{R}} = a\rho \mathbf{d}u\_{\mathbb{R}} \\ \mathbf{d}p\_{\mathbb{L}} = -a\rho \mathbf{d}u\_{\mathbb{L}} \\ \mathbf{d}p = \mathbf{d}p\_{\mathbb{R}} + \mathbf{d}p\_{\mathbb{L}} = \mathbf{0} \\ \mathbf{d}u = \mathbf{d}u\_{\mathbb{R}} + \mathbf{d}u\_{\mathbb{L}} \end{cases} \tag{6}$$

Thus, d*p*<sup>L</sup> = �d*p*R, d*u*<sup>L</sup> = d*u*R, d*u* = 2d*u*R, d*p* = 0.

It can be seen that the signs of the incident pressure wave and reflected pressure wave are opposite when the pressure wave propagates to the common rail. However, the absolute value of the amplitude of the pressure wave is the same and the wave velocity is the same. The changing of the pressure at the pipeline end is zero and the changing of the velocity is twice that of the incident pressure wave. The reflection with the property of incident wave and reflected wave is opposite (an expansion wave, the other is compression wave) and the absolute value of the amplitude of the pressure wave is the same is called complete negative reflection. Therefore, the pressure wave reflection at common rail and high pressure fuel pipelines is a complete negative reflection.
