# Experimental Investigation on the Flow Resistance of a Staged Fuel Injector with Thermal Protection

Authors: Jingyu Zhang; Xinyi Wang; Yu Wu; Xiaomin He
Journal: International Journal of Aerospace Engineering
 (2021)

---

## Abstract

The multistaged rotating swirling usually adopts the staged combustion and multiple point fuel supply, which has the advantages of high efficiency and low emissions. However, the high inlet temperature rise will cause autooxidation and fuel coking. Multiple fuel circuit mutual cooling technology by using different flight conditions is an effective measure for the thermal protection of the nozzle. But the complicated fuel circuit configuration inevitably increase the flow resistance in the fuel jet. This paper presents an experimental investigation of flow resistance of a triple-stage rotating swirling nozzle, concerning the effects of various factors on total pressure loss and friction factor in nozzle fuel circuit. The factors include fuel velocityVf,i, inlet temperature Tf,i, and inlet Reynolds number Ref,i. The results show that the complicated fuel-cooled structure made flow resistance of the pilot fuel circuit much bigger than the main one. Meanwhile, the empirical correlations between the friction factor and the inlet Reynolds number have been fitted, which can be the reference for engineering design.


---

## Body

## 1. Introduction

As more performance requirements of aeroengine has been raised, the thrust-to-weight (T/W) ratio of advanced aeroturbine engine tends to be higher. To further improve the T/W ratio of engine, the possible way is to improve the pressure ratio of compressor and the temperature rise of combustor. The increasing of pressure ratio of compressor will increase the inlet temperature of combustor, which could make the fuel autoxidation and deposit formation inside the fuel nozzle. The fuel autoxidation and deposit formation could reduce the life of fuel nozzle [1], significantly. For example, the combustor inlet temperature has to be increased to 1000 K in the advanced high T/W ratio (16~20) engine [2], while the JP-8 fuel was found to start autoxidize and format deposit from 423 K [3]. The increase of wall temperature of T700 engine fuel nozzle from 450 K to 480 K will reduce sharply the nozzle life from 1000 hours to 20 hours [4]. In order to avoid the depositions of coking, decreasing of fuel circuit surface and increasing of the flow resistance, which made strong influence on the flow characteristics, thermal protection is indispensable to ensure the operation in the fuel nozzles.High efficiency and low emission are two major requirements in the combustor development for advanced aero-engines. Staged combustion is one of the technologies to achieve high efficiency and low emission [5]. In the staged combustion, two stage flames are usually formed inside the combustor: pilot flame and main flame. The pilot flame introduces a small quantity of fuel to stabilize the whole combustion zone; the main flame is stabilized by the pilot flame and carries the most of the combustion and heat-release of the combustor. Staged combustor can be classified as radially staged combustion and axially staged combustion, in which the combustor structure and combustion organization are different. GE Aviation has developed the Twin Annular Premixing Swirler (TAPS) combustor, which is a representative of axially staged combustion (Figure 1). Multiple Point Fuel Injection is adopted to achieve low emission combustion. TAPS produces two coannular swirling flows for the pilot flame and main flame, respectively. Figure 2 presents the fuel nozzle structure of TAPS. A pressure atomizer injector is installed at the center of the swirling cup for the pilot flame stage. The fuel injector for the main flame stage consists of a cyclone and a cavity with fuel injected by discrete transverse jets. The pilot swirling jet (s) interacts with the main swirling jet (s) separated suitably by a step height to provide thermal environment required to meet combustor’s critical design requirements [6].Figure 1
Combustion organization in TAPS [7, 8].Figure 2
Fuel nozzle structure in TAPS.Nowadays, there are many typical thermal protection methods, for example, increasing the velocity of fuel in low-speed zone [9], adopting oil cooling structure [7], designing the air gaps [10, 11], and using fuel addictive [12]. But every method has their own disadvantages, only one kind thermal protection measure cannot satisfy the requirement. Though the TAPS injector could get better combustion performance than the traditional fuel nozzles, it is difficult to implement the traditional thermal protection method due to the complicated structure. The TAPS uses “cold” fuel passage to cool the “hot” fuel passage [7, 13, 14]. By using the heat sink of fuel, this method can provide good thermal protection performance, especially in the low flow condition [15]. However, the fuel cooling structure has multicircuit and will bring many problems, such as fuel velocity increased, flow path complicated, and flow direction changed, which will increase the flow resistance [16]. Now, there are not many published literatures focused on the flow resistance characteristics. Therefore, the experimental investigation on flow resistance of typical fuel nozzle mutual cooling structure in a triple-stage rotating swirling combustor is developed, and the flow resistance characteristic of complicated fuel cooling structure is analyzed in this paper to provide technical reference and support for advanced fuel supply system design and fuel jet thermal protect structure design in the low emission combustor.


## 2. Physical Model

### 2.1. Test Model

As shown in Figure3, the test injector contains two-stage axis swirling jet in the pilot and one-stage radial swirling jet in the main mixer. After flowing along the fuel passageways, both main fuel and pilot fuel are divided into two paths and flow into the catch ring. Twenty plain orifice injectors extend radially along the main swirling jet. The pilot circuit is surrounded along the annular main fuel circuit, than come together in front of the nozzle. The pilot path adopts a centrifugal nozzle located at the annular center (Figure 4).Figure 3
Test injector model.Figure 4
Fuel paths model.The test sections of both main and pilot fuel path are processed separately. The test piece of main fuel path includes a fuel inlet pipe, a cylinder core, and shell. The fuel paths are formed by interference fit of the shell and the core to ensure well sealing (Figure5). The fuel enters the path from the inlet pipe (① in Figure 4). The cylinder core is slotted as the fuel catch ring path (③ and ④ in Figure 4). The holes around the cylinder torus above the housing are corresponded the 20 plain injectors in Figure 4.Figure 5
Test piece of main fuel path.
(a)
Injector shell(b)
Cylinder coreThe design of pilot test section is similar with the main one, the fuel paths are formed by slotting method (Figure6 ⑤, ⑥). ② is the fuel inlet, and the outlet of centrifugal nozzle is above the housing. The numbers of fuel path in Figures 5 and 6 correspond to the numbers in Figure 4.Figure 6
Test piece of main fuel path.
(a)
Injector shell(b)
Cylinder core


### 2.2. Test Case

The China RP-3 fuel is adopted in the present study. The density and viscosity of RP-3 fuel are affected by temperature slightly. Formulas of fuel density and viscosity coefficient with temperature have been fitted in this paper:(1)ρ=942.85715−0.37557T−6.38744×10−4T2,μ=0.0246−2.0210×10−4T+6.3619×10−7T2−8.9865×10−10T3+4.7747×10−13T4.The outlet pressure of experimental system is ambient pressure, the operation conditions is listed in Table1, and the range of inlet velocity has covered all the typical flight conditions.Table 1
Test case.
Fuel circuitInlet temperature (K)Inlet velocity (m/s)Main fuel circuit303-3481.0~3.9Pilot fuel circuit303-3730.5~3.6


## 2.1. Test Model

As shown in Figure3, the test injector contains two-stage axis swirling jet in the pilot and one-stage radial swirling jet in the main mixer. After flowing along the fuel passageways, both main fuel and pilot fuel are divided into two paths and flow into the catch ring. Twenty plain orifice injectors extend radially along the main swirling jet. The pilot circuit is surrounded along the annular main fuel circuit, than come together in front of the nozzle. The pilot path adopts a centrifugal nozzle located at the annular center (Figure 4).Figure 3
Test injector model.Figure 4
Fuel paths model.The test sections of both main and pilot fuel path are processed separately. The test piece of main fuel path includes a fuel inlet pipe, a cylinder core, and shell. The fuel paths are formed by interference fit of the shell and the core to ensure well sealing (Figure5). The fuel enters the path from the inlet pipe (① in Figure 4). The cylinder core is slotted as the fuel catch ring path (③ and ④ in Figure 4). The holes around the cylinder torus above the housing are corresponded the 20 plain injectors in Figure 4.Figure 5
Test piece of main fuel path.
(a)
Injector shell(b)
Cylinder coreThe design of pilot test section is similar with the main one, the fuel paths are formed by slotting method (Figure6 ⑤, ⑥). ② is the fuel inlet, and the outlet of centrifugal nozzle is above the housing. The numbers of fuel path in Figures 5 and 6 correspond to the numbers in Figure 4.Figure 6
Test piece of main fuel path.
(a)
Injector shell(b)
Cylinder core


## 2.2. Test Case

The China RP-3 fuel is adopted in the present study. The density and viscosity of RP-3 fuel are affected by temperature slightly. Formulas of fuel density and viscosity coefficient with temperature have been fitted in this paper:(1)ρ=942.85715−0.37557T−6.38744×10−4T2,μ=0.0246−2.0210×10−4T+6.3619×10−7T2−8.9865×10−10T3+4.7747×10−13T4.The outlet pressure of experimental system is ambient pressure, the operation conditions is listed in Table1, and the range of inlet velocity has covered all the typical flight conditions.Table 1
Test case.
Fuel circuitInlet temperature (K)Inlet velocity (m/s)Main fuel circuit303-3481.0~3.9Pilot fuel circuit303-3730.5~3.6


## 3. Experimental Method

### 3.1. Experimental System

As shown in Figures7 and 8, the experimental system includes the fuFel supply system, heating system, test section, and measurement system. The gear pump can supply maximum 14.7 MPa fuel pressure, fuel passes the filter (10 μm) and the regulator, and the valve is used to adjust the flow rate. In the heating system, the power can be controlled by the differences between thermometers 1 and 2. For the pilot fuel circuit where the flow rate is smaller, the SCR (Silicon controlled rectifier) heating system is adopted, of which the voltage and current range are 0~30 V and 0~300 A, respectively. For the main fuel circuit where the flow rate is higher, the oil-bath heater whose working temperature can be adjusted from 20°C to 300°C is adopted. The heating section is insulated by aluminosilicate fiber cladding whose thermal conductivity is lower than 0.17 W/(m·K). After being heated, the fuel enters in the test section, the electronic scale can record the increment of the mass flow in the unit time to get the fuel velocity where the collection frequency is almost 2 Hz.Figure 7
Test injector model.Figure 8
Experimental system.


### 3.2. Measurement System

#### 3.2.1. Measure Point Position in Experimental System

As shown in Figure7, the precise pressure gauge is located upstream of the test section, the pressure sensors and pressure difference sensors are arranged along the test section, and the accuracy and the working range of pressure measuring equipment is shown in Table 2. K-type thermocouples is adopted for getting the temperature of test system; the accuracy of the electronic scale is 0.1 g.Table 2
Accuracy and range of pressure measuring equipment.
Range (MPa)AccuracyPressure sensor0~30.3%0~60.3%Pressure difference sensor0~0.250.3%0~0.50.3%Precise pressure gauge0~60.4%


#### 3.2.2. Measure Point Position in Test Section

In the test section, the main fuel circuit has 8 measure points along the path, which is located at the nozzle inlet and both sides of the annular fuel path, respectively. The pilot circuit has 9 measure points, which is located at the path inlet, both sides of the ring, and the fuel convergence section (Figure9). The pressure in points 2, 3, and 4 is the average value of both symmetric side (P2=P2a+P2b/2). Pressure sensors are located at every point of main and pilot fuel circuit, and pressure difference sensors are located between different points.Figure 9
Schematic diagram of the measure point.
(a)
Main fuel path(b)
Pilot fuel path


#### 3.2.3. Flux Calibration

There is a functional relationship between the pressure differences of fuel supplying and mass flux, which can be used to calibrate the fuel mass flow rate during the experiment. The experiment has 4 calibrations in main fuel circuit and 7 times in the pilot. The results of calibrations has been shown in Figure10. Here is the empirical correlations of fuel mass flux mf and fuel pressure differences ΔPf, which is fitted by least square method application:(2)Mainfuelpath:mf=0.0308×ΔPf0.554Pilotfuelpath:mf=0.00546×ΔPf0.486where mf (kg/s) is the fuel mass flow rate and ΔPf is the fuel pressure differences between the injector inlet and outlet.Figure 10
Flux calibration.
(a)
Main fuel circuit(b)
Pilot fuel circuitThe comparisons of experimental parameters and fitting formulas are shown in Figure11; the average errors of fitting formulas are lower than 1%, which means the method of controlling mass flow rate and velocity by pressure is reliable.Figure 11
Relative error of empirical correlation.
(a)
Main fuel circuit(b)
Pilot fuel circuit


## 3.1. Experimental System

As shown in Figures7 and 8, the experimental system includes the fuFel supply system, heating system, test section, and measurement system. The gear pump can supply maximum 14.7 MPa fuel pressure, fuel passes the filter (10 μm) and the regulator, and the valve is used to adjust the flow rate. In the heating system, the power can be controlled by the differences between thermometers 1 and 2. For the pilot fuel circuit where the flow rate is smaller, the SCR (Silicon controlled rectifier) heating system is adopted, of which the voltage and current range are 0~30 V and 0~300 A, respectively. For the main fuel circuit where the flow rate is higher, the oil-bath heater whose working temperature can be adjusted from 20°C to 300°C is adopted. The heating section is insulated by aluminosilicate fiber cladding whose thermal conductivity is lower than 0.17 W/(m·K). After being heated, the fuel enters in the test section, the electronic scale can record the increment of the mass flow in the unit time to get the fuel velocity where the collection frequency is almost 2 Hz.Figure 7
Test injector model.Figure 8
Experimental system.


## 3.2. Measurement System

### 3.2.1. Measure Point Position in Experimental System

As shown in Figure7, the precise pressure gauge is located upstream of the test section, the pressure sensors and pressure difference sensors are arranged along the test section, and the accuracy and the working range of pressure measuring equipment is shown in Table 2. K-type thermocouples is adopted for getting the temperature of test system; the accuracy of the electronic scale is 0.1 g.Table 2
Accuracy and range of pressure measuring equipment.
Range (MPa)AccuracyPressure sensor0~30.3%0~60.3%Pressure difference sensor0~0.250.3%0~0.50.3%Precise pressure gauge0~60.4%


### 3.2.2. Measure Point Position in Test Section

In the test section, the main fuel circuit has 8 measure points along the path, which is located at the nozzle inlet and both sides of the annular fuel path, respectively. The pilot circuit has 9 measure points, which is located at the path inlet, both sides of the ring, and the fuel convergence section (Figure9). The pressure in points 2, 3, and 4 is the average value of both symmetric side (P2=P2a+P2b/2). Pressure sensors are located at every point of main and pilot fuel circuit, and pressure difference sensors are located between different points.Figure 9
Schematic diagram of the measure point.
(a)
Main fuel path(b)
Pilot fuel path


### 3.2.3. Flux Calibration

There is a functional relationship between the pressure differences of fuel supplying and mass flux, which can be used to calibrate the fuel mass flow rate during the experiment. The experiment has 4 calibrations in main fuel circuit and 7 times in the pilot. The results of calibrations has been shown in Figure10. Here is the empirical correlations of fuel mass flux mf and fuel pressure differences ΔPf, which is fitted by least square method application:(2)Mainfuelpath:mf=0.0308×ΔPf0.554Pilotfuelpath:mf=0.00546×ΔPf0.486where mf (kg/s) is the fuel mass flow rate and ΔPf is the fuel pressure differences between the injector inlet and outlet.Figure 10
Flux calibration.
(a)
Main fuel circuit(b)
Pilot fuel circuitThe comparisons of experimental parameters and fitting formulas are shown in Figure11; the average errors of fitting formulas are lower than 1%, which means the method of controlling mass flow rate and velocity by pressure is reliable.Figure 11
Relative error of empirical correlation.
(a)
Main fuel circuit(b)
Pilot fuel circuit


## 3.2.1. Measure Point Position in Experimental System

As shown in Figure7, the precise pressure gauge is located upstream of the test section, the pressure sensors and pressure difference sensors are arranged along the test section, and the accuracy and the working range of pressure measuring equipment is shown in Table 2. K-type thermocouples is adopted for getting the temperature of test system; the accuracy of the electronic scale is 0.1 g.Table 2
Accuracy and range of pressure measuring equipment.
Range (MPa)AccuracyPressure sensor0~30.3%0~60.3%Pressure difference sensor0~0.250.3%0~0.50.3%Precise pressure gauge0~60.4%


## 3.2.2. Measure Point Position in Test Section

In the test section, the main fuel circuit has 8 measure points along the path, which is located at the nozzle inlet and both sides of the annular fuel path, respectively. The pilot circuit has 9 measure points, which is located at the path inlet, both sides of the ring, and the fuel convergence section (Figure9). The pressure in points 2, 3, and 4 is the average value of both symmetric side (P2=P2a+P2b/2). Pressure sensors are located at every point of main and pilot fuel circuit, and pressure difference sensors are located between different points.Figure 9
Schematic diagram of the measure point.
(a)
Main fuel path(b)
Pilot fuel path


## 3.2.3. Flux Calibration

There is a functional relationship between the pressure differences of fuel supplying and mass flux, which can be used to calibrate the fuel mass flow rate during the experiment. The experiment has 4 calibrations in main fuel circuit and 7 times in the pilot. The results of calibrations has been shown in Figure10. Here is the empirical correlations of fuel mass flux mf and fuel pressure differences ΔPf, which is fitted by least square method application:(2)Mainfuelpath:mf=0.0308×ΔPf0.554Pilotfuelpath:mf=0.00546×ΔPf0.486where mf (kg/s) is the fuel mass flow rate and ΔPf is the fuel pressure differences between the injector inlet and outlet.Figure 10
Flux calibration.
(a)
Main fuel circuit(b)
Pilot fuel circuitThe comparisons of experimental parameters and fitting formulas are shown in Figure11; the average errors of fitting formulas are lower than 1%, which means the method of controlling mass flow rate and velocity by pressure is reliable.Figure 11
Relative error of empirical correlation.
(a)
Main fuel circuit(b)
Pilot fuel circuit


## 4. Data Analysis

Fuel inlet velocity is calculated by calibrated mass flux and the fuel density:(3)Vf,i=mfρf.i×Ai,where Vfi is fuel inlet velocity, ρf.i is fuel density, and Ai is fuel path area.Total pressure loss is defined as the total pressure difference of the first and the last measure point:(4)ΔP∗=Pi∗−Po∗,whereΔP∗is total pressure loss, P∗i is inlet total pressure, and P∗o is outlet environment pressure.Total pressure loss coefficient is defined as(5)λ=Pi∗−Po∗Pi∗,where λ is total pressure loss coefficient, Pi∗ is inlet total pressure, and P∗o is outlet environment pressure.Friction factor is defined as(6)ξ=Pi∗−Po∗1/2ρf.iVf.i2,where ξ is friction factor, Pi∗ is inlet total pressure and Pso is outlet environment pressure, ρf.i is fuel density, and Vfi is fuel inlet velocity.Table3 gives the uncertainty of measured and derived quantities in the experiment.Table 3
Uncertainty of quantities.
ParametersCalculated formulaError transfer formulaUncertaintyFuel velocityVf,i=mf/ρf,i×Ai=mf×RTf,i/Pi×AidVf,i/Vf,i=dmf/mf2+dTf,i/Tf,i2+−dPi/Pi2+−dAi/Ai21.29%Reynolds numberRei=ρf,iVf,idi/μ=mfdi/AiμdRei/Rei=dmf/mf2+ddi/di2+−dAi/Ai21.414%Total pressure loss coefficientλ=ΔP∗/Pi∗dλ/λ=dPi/Pi2+−dPi/Pi20.424%Friction factorζ=ΔP∗/1/2ρf,iVf,i2=2×ΔP∗×Ai/mf×Vf,idζ/ζ=dPi/Pi2+dAi/Ai2+−dmf/mf2+−dVf,i/Vf,i21.33%


## 5. Results and Discussion

### 5.1. Results of Main Path

#### 5.1.1. Pressure Distribution

Figure12 presents the pressure distribution of main fuel path with different fuel inlet temperature Tf.i and inlet velocity Vf.i. As shown in the figure, the pressure gradually decreases along the main fuel path for each inlet temperature and inlet velocity. With the same temperature, the absolute values of pressure in the path significant increase with the inlet velocity. In fact, the maximum pressure loss present between measure points 1 and 2, where the radius of paths have changed.Figure 12
Pressure distribution of main path.
(a)
Tf.i=306K(b)
Tf.i=353K


#### 5.1.2. Total Pressure Loss

The flow resistance losses include resistance loss along the path and local resistance loss. According to the boundary layer theory in turbulent pipe flow [17], with the increasing of velocity, the viscous sublayer become thinner which leads to the increases of frictional resistance and the resistance along flow direction. Meanwhile, the structure of flow direction changing, variable cross section, and diffluence in the main path, which can result in the increase of local resistance losses, will lead the total pressure loss grow along the main fuel path. According to the empirical estimation, the resistance loss along the path accounts for almost 30% of the total pressure loss. The most important factor of pressure loss in the main path is the complicated structure of flow path. The variation of total pressure loss with fuel inlet velocity in different temperature was shown in Figure 13. It presents linear relation between the total pressure loss and inlet velocity square of main path with the same inlet temperature, which satisfy the general law even in the fuel jet circuit with complicated structure. While the effect of inlet temperature on the pressure loss is shown negligible for the same inlet velocity in the figure.Figure 13
Total pressure loss variation with inlet velocity square in main fuel circuit.


#### 5.1.3. Total Pressure Loss Coefficient and Friction Factor

From Figure14 one can see the variation of total pressure loss coefficient with inlet Reynolds number in main fuel circuit. In general, the total pressure loss coefficient decreases slightly with the increase of Reynolds number at corresponding fuel temperature, and it stayed almost constant at 0.0055 for Reynolds number which is higher than 10000.Figure 14
Variation of total pressure loss coefficient of main fuel circuit.According to the classic Nicholas fluid experiments in rough pipe [18], for straight pipe whose relative roughness is d/Δ=0.001, the friction factor tends to be independent of the Reynolds number when the inlet ReynoldsnumberRe>1×106 which means the flow is in the completely turbulent regime. As mentioned in some researches [19], the flow mixing will be strengthened and the flow will be more turbulent as the fluid flowing through the boundary with variable shape. Compared with relatively straight pipe, the flow in the complex structure will enter the completely turbulent regime at a lower Reynolds number. For example, the pressure losses in closed-coupled fitting have been investigated in the paper [20]; it shows that the local friction factor is independent with Reynolds number for Re>1.5×105 in the test pipe. For the test nozzle with complicated structure of main path, the change curve of friction factor is shown in Figure 15; the friction factor of main fuel path decreases with the developed Reynolds number which is lower than 12000 (Rei≤12000). Then, the friction factor tends to be constant at 6.1 as Reynolds number higher than 12000 (Rei>12000), and the main fuel flow inside this structure is in completely turbulent regime. In addition, both total pressure loss coefficient and friction factor of main fuel path decrease slightly with the increase of inlet fuel temperature.Figure 15
Variation of friction factor of main fuel path.


### 5.2. Results of Pilot Path

#### 5.2.1. Pressure Distribution

Figure16 shows the pressure distribution of pilot fuel path in different fuel inlet temperature Tf.i and inlet velocity Vf.i, which is similar to the main fuel path. As a result, pressure along pilot path progressively decreases at the same condition. Furthermore, absolute value of pressure significantly elevate with the increase of inlet velocity. The highest pressure loss appears at the divided flow region and convergent flow region.Figure 16
Pressure distribution of pilot path.
(a)
Tf.i=306K(b)
Tf.i=363K


#### 5.2.2. Total Pressure Loss

The variation of total pressure in main and pilot path are similar (Figure17). For certain temperature, the total pressure loss grows with the inlet velocity. Compared with main path, the structure of pilot fuel path is more complicated, and there are more flow direction changing in pilot circuit of which the local pressure loss is bigger; at the same inlet velocity, their total pressure loss is almost 5 to 6 times larger than the main path. According to the empirical estimation, the local pressure loss accounts for almost 80% of the total pressure loss in pilot fuel circuit.Figure 17
Total pressure loss variation with inlet velocity square in pilot fuel circuit.From Figure17, the total pressure loss and inlet velocity square present linear relation. Compared with the main fuel path, the influence of temperature on total pressure loss is enhanced at the same inlet velocity. Higher temperature will change fuel properties and reduce the fuel viscosity, so that the slope of total pressure loss with the square of inlet flow rate is slightly reduced. However, the overall tendency of pressure loss in pilot path still satisfies the general law under the experimental conditions.


#### 5.2.3. Total Pressure Loss Coefficient and Friction Factor

In Figure18, the variation of total pressure loss coefficient at different temperature is ploted versus the inlet Reynolds number. Total pressure loss coefficient strongly decreases with the Reynolds number which is lower than 10000 (Rei<10000). For Reynolds number Rei>10000, the total pressure loss coefficient tends to be a constant at 0.025.Figure 18
Variation of total pressure loss coefficient of pilot fuel circuit.Figure19 shows that the friction factor varies with the inlet Reynolds number; for inlet Reynolds number Rei<10000, the friction factor strongly decreases as Reynolds number increases. Then, after the Reynolds number increasing higher than 10000, the friction factor reduces slightly and stays a constant value, which means the flow is in completely turbulent regime. Both total pressure loss coefficient and friction factors of pilot path are larger than those in the main path, because the more complicated circuit structure of pilot greatly increases the local resistance loss. For different fuel inlet temperature, both total pressure loss coefficient and friction factors in pilot path are close to unchanged under the same Reynolds number.Figure 19
Variation of friction factor of main fuel path.


### 5.3. Empirical Correlation

In the present investigation, in regard to investigated injector structure, the empirical correlations of total pressure loss fraction versus Reynolds number for the main and pilot fuel path friction factor for different Reynolds number have been developed by using least square method. The functional relation between main path friction factor and inlet Reynolds number is derived from polynomial, that is,(7)ξ=17.13−0.002×Ref.i+2.3×10−7×Ref.i2−9.8×10−12×Ref.i3+2×10−16×Ref.i4+1.7×10−21×Ref.i5,where ξ is friction factor and Ref.i is inlet Reynolds number.The comparison of measured data with the correlation is given in Figure20(a), the value of average data errors is lower than 5%, and Figure 21(a) described clearly the relative error of correlation.Figure 20
Comparison of correlation and measured data.
(a)
Main fuel circuit(b)
Pilot fuel circuitFigure 21
Relative error of empirical correlation.
(a)
Main fuel circuit(b)
Pilot fuel circuitHere is the empirical correlation between friction factor and Reynolds number of pilot fuel path, which is in exponential distribution.(8)ξ=283.123×Rei‐0.239where ξ is friction factor and Rei is inlet Reynolds number.As result of correlation shown in Figure20(b) and Figure 21(b), the value of average data errors for pilot fuel path is lower than 5%.


## 5.1. Results of Main Path

### 5.1.1. Pressure Distribution

Figure12 presents the pressure distribution of main fuel path with different fuel inlet temperature Tf.i and inlet velocity Vf.i. As shown in the figure, the pressure gradually decreases along the main fuel path for each inlet temperature and inlet velocity. With the same temperature, the absolute values of pressure in the path significant increase with the inlet velocity. In fact, the maximum pressure loss present between measure points 1 and 2, where the radius of paths have changed.Figure 12
Pressure distribution of main path.
(a)
Tf.i=306K(b)
Tf.i=353K


### 5.1.2. Total Pressure Loss

The flow resistance losses include resistance loss along the path and local resistance loss. According to the boundary layer theory in turbulent pipe flow [17], with the increasing of velocity, the viscous sublayer become thinner which leads to the increases of frictional resistance and the resistance along flow direction. Meanwhile, the structure of flow direction changing, variable cross section, and diffluence in the main path, which can result in the increase of local resistance losses, will lead the total pressure loss grow along the main fuel path. According to the empirical estimation, the resistance loss along the path accounts for almost 30% of the total pressure loss. The most important factor of pressure loss in the main path is the complicated structure of flow path. The variation of total pressure loss with fuel inlet velocity in different temperature was shown in Figure 13. It presents linear relation between the total pressure loss and inlet velocity square of main path with the same inlet temperature, which satisfy the general law even in the fuel jet circuit with complicated structure. While the effect of inlet temperature on the pressure loss is shown negligible for the same inlet velocity in the figure.Figure 13
Total pressure loss variation with inlet velocity square in main fuel circuit.


### 5.1.3. Total Pressure Loss Coefficient and Friction Factor

From Figure14 one can see the variation of total pressure loss coefficient with inlet Reynolds number in main fuel circuit. In general, the total pressure loss coefficient decreases slightly with the increase of Reynolds number at corresponding fuel temperature, and it stayed almost constant at 0.0055 for Reynolds number which is higher than 10000.Figure 14
Variation of total pressure loss coefficient of main fuel circuit.According to the classic Nicholas fluid experiments in rough pipe [18], for straight pipe whose relative roughness is d/Δ=0.001, the friction factor tends to be independent of the Reynolds number when the inlet ReynoldsnumberRe>1×106 which means the flow is in the completely turbulent regime. As mentioned in some researches [19], the flow mixing will be strengthened and the flow will be more turbulent as the fluid flowing through the boundary with variable shape. Compared with relatively straight pipe, the flow in the complex structure will enter the completely turbulent regime at a lower Reynolds number. For example, the pressure losses in closed-coupled fitting have been investigated in the paper [20]; it shows that the local friction factor is independent with Reynolds number for Re>1.5×105 in the test pipe. For the test nozzle with complicated structure of main path, the change curve of friction factor is shown in Figure 15; the friction factor of main fuel path decreases with the developed Reynolds number which is lower than 12000 (Rei≤12000). Then, the friction factor tends to be constant at 6.1 as Reynolds number higher than 12000 (Rei>12000), and the main fuel flow inside this structure is in completely turbulent regime. In addition, both total pressure loss coefficient and friction factor of main fuel path decrease slightly with the increase of inlet fuel temperature.Figure 15
Variation of friction factor of main fuel path.


## 5.1.1. Pressure Distribution

Figure12 presents the pressure distribution of main fuel path with different fuel inlet temperature Tf.i and inlet velocity Vf.i. As shown in the figure, the pressure gradually decreases along the main fuel path for each inlet temperature and inlet velocity. With the same temperature, the absolute values of pressure in the path significant increase with the inlet velocity. In fact, the maximum pressure loss present between measure points 1 and 2, where the radius of paths have changed.Figure 12
Pressure distribution of main path.
(a)
Tf.i=306K(b)
Tf.i=353K


## 5.1.2. Total Pressure Loss

The flow resistance losses include resistance loss along the path and local resistance loss. According to the boundary layer theory in turbulent pipe flow [17], with the increasing of velocity, the viscous sublayer become thinner which leads to the increases of frictional resistance and the resistance along flow direction. Meanwhile, the structure of flow direction changing, variable cross section, and diffluence in the main path, which can result in the increase of local resistance losses, will lead the total pressure loss grow along the main fuel path. According to the empirical estimation, the resistance loss along the path accounts for almost 30% of the total pressure loss. The most important factor of pressure loss in the main path is the complicated structure of flow path. The variation of total pressure loss with fuel inlet velocity in different temperature was shown in Figure 13. It presents linear relation between the total pressure loss and inlet velocity square of main path with the same inlet temperature, which satisfy the general law even in the fuel jet circuit with complicated structure. While the effect of inlet temperature on the pressure loss is shown negligible for the same inlet velocity in the figure.Figure 13
Total pressure loss variation with inlet velocity square in main fuel circuit.


## 5.1.3. Total Pressure Loss Coefficient and Friction Factor

From Figure14 one can see the variation of total pressure loss coefficient with inlet Reynolds number in main fuel circuit. In general, the total pressure loss coefficient decreases slightly with the increase of Reynolds number at corresponding fuel temperature, and it stayed almost constant at 0.0055 for Reynolds number which is higher than 10000.Figure 14
Variation of total pressure loss coefficient of main fuel circuit.According to the classic Nicholas fluid experiments in rough pipe [18], for straight pipe whose relative roughness is d/Δ=0.001, the friction factor tends to be independent of the Reynolds number when the inlet ReynoldsnumberRe>1×106 which means the flow is in the completely turbulent regime. As mentioned in some researches [19], the flow mixing will be strengthened and the flow will be more turbulent as the fluid flowing through the boundary with variable shape. Compared with relatively straight pipe, the flow in the complex structure will enter the completely turbulent regime at a lower Reynolds number. For example, the pressure losses in closed-coupled fitting have been investigated in the paper [20]; it shows that the local friction factor is independent with Reynolds number for Re>1.5×105 in the test pipe. For the test nozzle with complicated structure of main path, the change curve of friction factor is shown in Figure 15; the friction factor of main fuel path decreases with the developed Reynolds number which is lower than 12000 (Rei≤12000). Then, the friction factor tends to be constant at 6.1 as Reynolds number higher than 12000 (Rei>12000), and the main fuel flow inside this structure is in completely turbulent regime. In addition, both total pressure loss coefficient and friction factor of main fuel path decrease slightly with the increase of inlet fuel temperature.Figure 15
Variation of friction factor of main fuel path.


## 5.2. Results of Pilot Path

### 5.2.1. Pressure Distribution

Figure16 shows the pressure distribution of pilot fuel path in different fuel inlet temperature Tf.i and inlet velocity Vf.i, which is similar to the main fuel path. As a result, pressure along pilot path progressively decreases at the same condition. Furthermore, absolute value of pressure significantly elevate with the increase of inlet velocity. The highest pressure loss appears at the divided flow region and convergent flow region.Figure 16
Pressure distribution of pilot path.
(a)
Tf.i=306K(b)
Tf.i=363K


### 5.2.2. Total Pressure Loss

The variation of total pressure in main and pilot path are similar (Figure17). For certain temperature, the total pressure loss grows with the inlet velocity. Compared with main path, the structure of pilot fuel path is more complicated, and there are more flow direction changing in pilot circuit of which the local pressure loss is bigger; at the same inlet velocity, their total pressure loss is almost 5 to 6 times larger than the main path. According to the empirical estimation, the local pressure loss accounts for almost 80% of the total pressure loss in pilot fuel circuit.Figure 17
Total pressure loss variation with inlet velocity square in pilot fuel circuit.From Figure17, the total pressure loss and inlet velocity square present linear relation. Compared with the main fuel path, the influence of temperature on total pressure loss is enhanced at the same inlet velocity. Higher temperature will change fuel properties and reduce the fuel viscosity, so that the slope of total pressure loss with the square of inlet flow rate is slightly reduced. However, the overall tendency of pressure loss in pilot path still satisfies the general law under the experimental conditions.


### 5.2.3. Total Pressure Loss Coefficient and Friction Factor

In Figure18, the variation of total pressure loss coefficient at different temperature is ploted versus the inlet Reynolds number. Total pressure loss coefficient strongly decreases with the Reynolds number which is lower than 10000 (Rei<10000). For Reynolds number Rei>10000, the total pressure loss coefficient tends to be a constant at 0.025.Figure 18
Variation of total pressure loss coefficient of pilot fuel circuit.Figure19 shows that the friction factor varies with the inlet Reynolds number; for inlet Reynolds number Rei<10000, the friction factor strongly decreases as Reynolds number increases. Then, after the Reynolds number increasing higher than 10000, the friction factor reduces slightly and stays a constant value, which means the flow is in completely turbulent regime. Both total pressure loss coefficient and friction factors of pilot path are larger than those in the main path, because the more complicated circuit structure of pilot greatly increases the local resistance loss. For different fuel inlet temperature, both total pressure loss coefficient and friction factors in pilot path are close to unchanged under the same Reynolds number.Figure 19
Variation of friction factor of main fuel path.


## 5.2.1. Pressure Distribution

Figure16 shows the pressure distribution of pilot fuel path in different fuel inlet temperature Tf.i and inlet velocity Vf.i, which is similar to the main fuel path. As a result, pressure along pilot path progressively decreases at the same condition. Furthermore, absolute value of pressure significantly elevate with the increase of inlet velocity. The highest pressure loss appears at the divided flow region and convergent flow region.Figure 16
Pressure distribution of pilot path.
(a)
Tf.i=306K(b)
Tf.i=363K


## 5.2.2. Total Pressure Loss

The variation of total pressure in main and pilot path are similar (Figure17). For certain temperature, the total pressure loss grows with the inlet velocity. Compared with main path, the structure of pilot fuel path is more complicated, and there are more flow direction changing in pilot circuit of which the local pressure loss is bigger; at the same inlet velocity, their total pressure loss is almost 5 to 6 times larger than the main path. According to the empirical estimation, the local pressure loss accounts for almost 80% of the total pressure loss in pilot fuel circuit.Figure 17
Total pressure loss variation with inlet velocity square in pilot fuel circuit.From Figure17, the total pressure loss and inlet velocity square present linear relation. Compared with the main fuel path, the influence of temperature on total pressure loss is enhanced at the same inlet velocity. Higher temperature will change fuel properties and reduce the fuel viscosity, so that the slope of total pressure loss with the square of inlet flow rate is slightly reduced. However, the overall tendency of pressure loss in pilot path still satisfies the general law under the experimental conditions.


## 5.2.3. Total Pressure Loss Coefficient and Friction Factor

In Figure18, the variation of total pressure loss coefficient at different temperature is ploted versus the inlet Reynolds number. Total pressure loss coefficient strongly decreases with the Reynolds number which is lower than 10000 (Rei<10000). For Reynolds number Rei>10000, the total pressure loss coefficient tends to be a constant at 0.025.Figure 18
Variation of total pressure loss coefficient of pilot fuel circuit.Figure19 shows that the friction factor varies with the inlet Reynolds number; for inlet Reynolds number Rei<10000, the friction factor strongly decreases as Reynolds number increases. Then, after the Reynolds number increasing higher than 10000, the friction factor reduces slightly and stays a constant value, which means the flow is in completely turbulent regime. Both total pressure loss coefficient and friction factors of pilot path are larger than those in the main path, because the more complicated circuit structure of pilot greatly increases the local resistance loss. For different fuel inlet temperature, both total pressure loss coefficient and friction factors in pilot path are close to unchanged under the same Reynolds number.Figure 19
Variation of friction factor of main fuel path.


## 5.3. Empirical Correlation

In the present investigation, in regard to investigated injector structure, the empirical correlations of total pressure loss fraction versus Reynolds number for the main and pilot fuel path friction factor for different Reynolds number have been developed by using least square method. The functional relation between main path friction factor and inlet Reynolds number is derived from polynomial, that is,(7)ξ=17.13−0.002×Ref.i+2.3×10−7×Ref.i2−9.8×10−12×Ref.i3+2×10−16×Ref.i4+1.7×10−21×Ref.i5,where ξ is friction factor and Ref.i is inlet Reynolds number.The comparison of measured data with the correlation is given in Figure20(a), the value of average data errors is lower than 5%, and Figure 21(a) described clearly the relative error of correlation.Figure 20
Comparison of correlation and measured data.
(a)
Main fuel circuit(b)
Pilot fuel circuitFigure 21
Relative error of empirical correlation.
(a)
Main fuel circuit(b)
Pilot fuel circuitHere is the empirical correlation between friction factor and Reynolds number of pilot fuel path, which is in exponential distribution.(8)ξ=283.123×Rei‐0.239where ξ is friction factor and Rei is inlet Reynolds number.As result of correlation shown in Figure20(b) and Figure 21(b), the value of average data errors for pilot fuel path is lower than 5%.


## 6. Conclusions

In this paper, based on a typical triple-stage rotating swirling, the experiment on the resistance characteristics for main and pilot path flow has been implemented. This work studied the effect of different inlet velocity and temperature for investigated fuel circuit system. In addition, the empirical correlations between friction factor and inlet Reynolds number are obtained and good agreement was observed. Major findings are as follows:(1)
The local resistance loss dominates in both main fuel path and pilot fuel path. Due to complex configuration of pilot fuel circuit, the pilot oil-cooling structure design will strongly increase the local resistance loss. Their flow resistance loss is more than 4 times higher than that in the main fuel path at the same Reynolds number. Hence, the major point during the nozzle structure designing is decreasing the local resistance loss, and the total pressure loss of whole oil circuit will be decreased(2)
For main fuel circuit, the highest pressure loss occurred at the divided flow region; for pilot fuel circuit, the highest pressure loss appeared at the divided flow region and the convergent flow region(3)
Under the investigated conditions of present studies, the inlet temperature has slight influence on total pressure loss at different velocity. Both in main and pilot fuel circuit, the total pressure loss presents exponential growth with the inlet velocity and linear growth with the square of inlet velocity(4)
Both total pressure loss coefficient and friction factor reduce as the inlet Reynolds number rises. For main fuel path, the total pressure loss coefficient stayed almost constant at 0.0055 for Reynolds number which is higher than 10000, and the friction factor tends to be constant at 6.1 as Reynolds number is higher than 12000. For pilot fuel path, the total pressure loss stayed around 0.025, and the friction factor is almost 29 as Reynolds number is higher than 10000



---

*Source: 8881460-2021-03-31.xml*