Cooling design of an aero-engine fuel centrifugal pump at shut-off

Abstract

Because of the problem of a high fuel temperature rise at shut-off, a conventional centrifugal pump could not be used as an aero-engine fuel pump for the afterburner fuel supply. In this study, a centrifugal pump with a cooling bypass was designed to address this problem. Experiments and numerical simulations were conducted to assess the design. The results showed that the circulation cooling achieved by a small amount of cooling fuel from a cooling bypass could effectively solve the problem. When the pump with a cooling bypass operated at shut-off, a large vapour core appeared at the centre of the impeller due to cavitation. This large vapour core reduced both the pump power consumption and the heat generated by friction, thereby greatly reducing the fuel temperature rise in the pump. Further research works showed that the radius of the vapour core was a function of the flow rate and the pump rotational speed. Moreover, the cooling fuel flow rate was regulated by the size of the nozzle hole. It is clear that a centrifugal pump with a cooling bypass overcomes the problem of the high fuel temperature rise in the pump at shut-off.

Introduction

Compared with gear pumps or piston pumps, which are often used as aero-engine fuel pumps for main engine or afterburner fuel supplies, centrifugal pumps offer the attractive advantages of large flow rates, simple structures, light weight and low cost. ¹ However, because of the problem of the high fuel temperature rise at shut-off, a conventional centrifugal pump cannot be used as an aero-engine fuel pump for the afterburner fuel supply (called an afterburner fuel pump). The shut-off condition is characterised by the following features:

1. The pump outlet valve is closed.

2. The pump is full of fuel.

3. The pump flow rate is zero (no fuel delivery).

4. The pump impeller is still in rotation at a high rotational speed.

A conventional centrifugal pump operates at shut-off only at the moment of start-up to minimise the starting load on the driver. However, an afterburner fuel pump in an aero-engine has to operate at shut-off for a long time. However, the pump is usually driven directly by the main engine, without the clutch, for the purposes of weight reduction and mechanical simplicity. Hence, the pump shaft rotates all the time with the main engine, regardless of whether or not fuel is supplied for the afterburner. However, the afterburner is usually turned on (fuel supply) for a short time only in the case of an emergency to augment extra engine thrust. In comparison, the afterburner is turned off (interrupting fuel supply) for a long time under normal flight conditions to save fuel. If a conventional centrifugal pump operates at shut-off for a long time, friction both between components and between the impeller and the fuel will cause the fuel trapped in the impeller chamber to generate a considerable increase in temperature, which is very dangerous for an aero-engine. ² To ensure the operating safety of an aero-engine, its design requirements typically stipulate that the pump inlet fuel temperature remains below 60°C and that the fuel temperature rise in the pump remains below 40°C. ³

For the purpose of energy saving, the conventional fuel centrifugal pump usually runs near the rated operating condition. During the all course of the flight, the fuel temperature rise in the centrifugal pump for the aircraft fuel system is below 14°C. ⁴ With the decrease in the flow rate (far from the rated operating point), the fuel temperature rise increases greatly. ⁵ Therefore, the minimum continuous operating flow is limited. ⁶ For the conventional fuel centrifugal pump, the operating time at shut-off is so short that the temperature rise in the pump is small and can be neglected, so the problem of the temperature rise in a conventional centrifugal pump at shut-off has rarely been studied. To date, studies on the shut-off condition for a centrifugal pump have mainly focused on the shut-off head^7–10 and the shut-off power. ¹¹

However, in the case of the afterburner fuel centrifugal pump, the problem of the high fuel temperature rise at shut-off cannot be ignored. Dowty Fuel Systems Limited ¹² designed a centrifugal pump with an inlet throttle valve at the impeller eye to overcome the problem. The flow rate was regulated by the opening degree of the inlet throttle valve, the function of which was to generate a core of fuel vapour at the impeller eye. However, the control on the opening degree of the inlet throttle valve was very complex. Therefore, both the pump weight and the structural complexity greatly increased.

In this study, a centrifugal pump with a cooling bypass was designed to solve the problem of the high fuel temperature rise at shut-off. Experiments and numerical simulations were conducted to assess the design.

Structure design

Original pump

A single-stage, single-suction centrifugal pump was adopted herein as the aero-engine fuel centrifugal pump. This pump was similar to the high-speed partial-emission pump designed by Barske ¹³ during the Second World War for the rocket fuel supply. Figure 1 shows the structure of the pump (the specific speed $n_{\mathrm{s}}=21.5$ ) before design optimisation (referred to hereinafter as an original pump). Figure 1(a) denotes that the original pump is composed of the inlet, impeller chamber, impeller, volute, rotation shaft, diffuser and outlet. An inlet valve was located between the inlet and the impeller chamber, while an outlet valve was placed at the diffuser exit. Figure 1(b) shows the three-dimensional structure of the impeller, volute and diffuser. The impeller in the impeller chamber consisted of five open straight blades, which were slightly curved forward. Table 1 lists the design parameters of the impeller blades. An annular volute with the same cross-sectional area was used to collect the fuel from the impeller outlet. Meanwhile, a conical diffuser with a 10° cone angle was used to reduce the flow speed and recover the static pressure head through the increasing cross-sectional area. The impeller and the rotation shaft were the rotating components. The others were the static components.

Figure 1.

Structure of original pump: (a) whole structure and (b) structure of impeller, volute and diffuser.

Table 1.

Design parameters of impeller blades.

Geometry	Symbol	Relative dimensions
Inlet radius	R 1	0.38
Outlet radius	R 2	1
Inlet height	b 1	0.47
Outlet height	b 2	0.17
Blade thickness	h	0.056

Cooling bypass

To address the problem of the high fuel temperature rise in the pump at shut-off, a tentative idea for circulation cooling by a small amount of fuel from a cooling bypass was considered. A structural improvement to the original pump was then made to facilitate this. Figure 2(a) shows the structure of the cooling bypass. The cooling bypass was composed of two branch pipes, a fuel filter, two nozzles (equal diameter) and a conical return pipe. The branch pipes and the nozzles connected the pump inlet and the impeller chamber. The fuel filter was located between branch pipe-1 and branch pipe-2 to prevent the nozzles from the clogging caused by the fuel impurity. The return pipe was located on the side wall of the diffuser. Figure 2(b) displays the position of the nozzle holes at the bottom plate of the impeller chamber. Figure 2(c) presents the original pump with the cooling bypass (hereinafter referred to as an improved pump). Table 2 lists the design parameters of the cooling bypass. The structural design of the cooling bypass is clearly very simple, showing that it does not add significant extra weight to the pump.

Figure 2.

Structure of improved pump: (a) cooling bypass, (b) position of nozzle holes and (c) pump with cooling bypass.

Table 2.

Cooling bypass design parameters.

Geometry	Symbol (diameter)	Relative dimensions
Nozzle	D 1	1
Branch pipe-1	D 2	4
Branch pipe-2	D 3	2
Fuel filter	D 4	8
Return pipe (inlet)	D 5	5.5
Return pipe (outlet)	D 6	2.5

Figure 3(a) shows that, in the improved pump at shut-off, a small amount of fuel (called cooling fuel) from the pump inlet enters the branch pipe, flows through the shaft sleeve, sprays from the nozzles into the impeller chamber and returns through the return pipe to the fuel tank. The heat generated by friction in the pump was carried away by the cooling fuel. The cooling fuel flowing through the shaft sleeve also played an important role in lubricating and cooling the high-speed rotating shaft.

Figure 3.

Fuel flow direction: (a) improved pump at shut-off and (b) original pump at a low flow rate.

Although the cooling fuel also entered the impeller chamber from the cooling bypass, the operating state of the improved pump at shut-off (the inlet and outlet valves being closed) was very different from that of the original pump at a low flow rate (the inlet and outlet valves being open). The flow rate of a centrifugal pump was usually adjusted by the outlet valve, which allowed a small amount of fuel (also called the cooling fuel) to enter the impeller chamber of the original pump from the pump inlet and return from the outlet of the diffuser to the fuel tank (Figure 3(b)). In other words, when the afterburner was turned off, a small amount of cooling fuel was allowed to directly enter the impeller chamber of the original pump from the pump inlet to carry away the heat generated by friction in the pump. This seemed to be a sound solution to the problem of the high fuel temperature rise in the pump at shut-off. However, the study results showed that this was not a feasible solution. A detailed analysis would be provided later.

Experimental assessment

Experimental equipment

An aero-engine fuel centrifugal pump test rig (Figures 4 and 5) was set up to experimentally measure the fuel temperature rise in the pumps. The test rig was a closed circulation system using China No. 3 jet fuel (RP-3). A booster pump was used to pressurise the fuel and deliver it from the fuel tank to the test pump. A heat exchanger was connected in series to the pipeline system to decrease the fuel temperature rise in the pipeline system. The pump rotational speed, achieved using a two-stage speed-increasing gear box driven by an electric motor, was measured by a tachometer JFXM/AHMVO with the uncertainty of ±15 r/min. The flow rate, regulated by the flow control valve JFXAM-100C, was measured by a flow meter JFXGY-15 with the uncertainty of ±0.05 m³/h. The pump shaft power was measured by a power meter JFXRH-3P51 with the uncertainty of ±10 W. The fuel inlet and outlet temperatures were measured by temperature sensors JFX/P-20 with the uncertainty of ±0.1°C. The fuel inlet and outlet pressures were measured by pressure gauges Y-100 with the uncertainty of ±0.01 MPa.

Figure 4.

Schematic diagram of aero-engine fuel centrifugal pump test rig.

Figure 5.

Real picture of aero-engine fuel centrifugal pump test rig: (a) test pump (without measuring instruments) and (b) test bed.

Experimental results

Three groups of comparative tests were conducted using the test rig at different non-dimensional flow rates ( $\tilde{Q}=0.01,0.03\mathrm{and}0.05$ ; $\tilde{Q}=Q_{\mathrm{v}}/Q_{\mathrm{opt}}$ ) and the pump rated rotational speed to compare the cooling effects of the original pump and the improved pump. Figure 6 shows that the fuel temperature rise in both the original and improved pumps increases with the decrease in the fuel flow rate. The fuel temperature rise largely increased from 45.3°C to 85.6°C in the original pump, but only slightly increased from 17.9°C to 23.6°C in the improved pump.

Figure 6.

Fuel temperature rise versus non-dimensional flow rate (by experiment).

The fuel temperature rise in the original pump at low flow rates obviously exceeded the maximum permissible temperature rise (40°C). Although the fuel temperature rise in the original pump might fall below 40°C as the flow rate increased by expanding the outlet valve opening, this delivered more heat to the fuel tank (as a heat sink). The fuel temperature rise in the improved pump at shut-off was below the maximum permissible temperature rise and, at any given flow rate, was much lower than that in the original pump. In addition, the improved pump delivered less heat to the fuel tank than the original pump at the same flow rate. It is clear that the circulation cooling by a small amount of cooling fuel from the cooling bypass solves the problem of the high fuel temperature rise in the pump at shut-off.

Numerical simulations

Mesh and independence verification

The meshes of the computational domains (Figure 7) of both the original and improved pumps were generated using ICEM CFD 14.0 software. Given the structural complexity of the pumps, unstructured multi-block tetrahedral meshes with prism grids near the walls were used. The computational domain of the original pump (Figure 7(a)) was divided into three parts: the inlet, impeller and volute domains. The impeller domain was a dynamic rotation domain with an anticlockwise direction. The inlet and volute domains were static domains. Dynamic–static interfaces (Figure 7(b)) were established between the dynamic and static domains. The computational domain of the improved pump (Figure 7(c)) included the inlet, impeller and volute domains. It also included a branch domain and a return domain.

Figure 7.

Mesh of computational domains: (a) original pump, (b) interfaces between domains and (c) improved pump.

A non-dimensional height ‘Y plus’ was examined to limit the model error of the turbulence model near the wall boundary layer and to ensure that the Y plus of the first layer mesh near the wall was between 5 and 20. Four mesh schemes from sparse to dense (Table 3) were employed to test the mesh independence by solving the pump outlet pressure under the design operation condition.

Table 3.

Mesh elements in different computational domains.

	Number	Inlet	Impeller	Volute	Bypass	Return	Total
Original pump	1	78,457	84,764	63,541	–	–	226,762
	2	297,773	318,453	217,306	–	–	833,532
	3	577,167	699,379	477,232			1,753,778
	4	898,785	998,953	791,589	–	–	2,689,327
Improved pump	1	78,374	84,654	63,841	43,117	4921	274,907
	2	297,441	318,331	217,564	165,476	13,435	1,012,247
	3	577,013	699,135	477,445	391,463	21,307	2,166,363
	4	898,466	998,841	792,133	653,371	28,132	3,370,943

The comparison of the pump outlet pressure between the simulation and experimental results (Figure 8) indicated that the simulation error caused by the mesh gradually decreased with an increase in the number of mesh elements. The No. 4 mesh was adopted herein considering the limitation of computing ability.

Figure 8.

Pump outlet pressure versus mesh element number.

Simulation process

The fuel temperature rise in the pump was related to the flow field. Considering the cavitation effects,^14,15 the real flow pattern in the pump was a vapour–liquid two-phase flow, which could be described by the fluid mechanics governing equations (i.e. continuity, momentum and energy equations). A shear stress transport (SST) turbulence model ¹⁶ was introduced to close the above equations dealt by the Reynolds average. Moreover, a cavitation model based on the Rayleigh–Plesset equation^17,18 was utilised to describe the cavitation process. The transient numerical simulations were conducted using ANSYS CFX 14.0 software. The simulation parameters and boundary conditions were as follows:

1. At the inlet: Total pressure, liquid volume fraction ( $a_{\mathrm{l}}=1$ ), vapour volume fraction ( $a_{\mathrm{g}}=0$ ) and inlet temperature ( $T_1$ , obtained from measurement).

2. At the outlet: Mass flow rate, liquid volume fraction ( $a_{\mathrm{l}}=1$ ) and vapour volume fraction ( $a_{\mathrm{g}}=0$ ).

3. At the interfaces: The frame change was set to be transient rotor–stator, and the mesh connection was set to be a general grid interface (GGI).

4. Solver control: The time step ( $\delta t=T_{\mathrm{n}}/360$ , where $T_{\mathrm{n}}$ is the impeller rotation period) was set to 6.1728e−6 s. The total time was set to 0.008888 s (corresponding to four rotation cycles). The maximum number of iterations per time step was set to 20. The residual target is set to 1e−6.

5. Cavitation setup: The Zwart–Gerber–Belamri cavitation model ¹⁹ was adopted. The bubble radius ( $R_{\mathrm{b}}$ ) was set to 1e−6 m. Meanwhile, the fuel saturation vapour pressure ( $P_{\mathrm{sat}}$ ) was set to 3680 Pa. ²⁰

Simulation results

The numerical simulations were conducted under the same operation conditions as the experiments. Figure 9 shows that the fuel temperature rise curve from the simulation results is similar to that from the experimental results (Figure 6). However, the simulation values were lower than the experimental values at the same flow rates. It was because that the heat generated by the friction between the impeller shaft and the bushing had not been considered in the simulations, but the heat was transferred to the fuel.

Figure 9.

Fuel temperature rise versus non-dimensional flow rate (by simulation).

Results and discussion

Analysis of the cooling principle

Explaining the big difference in the fuel temperature rises of the original and improved pumps at the same flow rates (Figure 6), which might be related to the flow fields in the pumps, was not easy. Flow visualisation ²¹ was known to be a good method of analysing the flow fields, but applying visualisation instruments to a high-pressure, high-speed, small-volume pump was very difficult. Therefore, numerical simulation analyses were adopted herein. Taking the operation condition $\tilde{Q}=0.01$ as an example, Figures 10–12 show the temperature contours, static pressure distribution contours and vapour volume fraction contours of the pump chamber in both the original and improved pumps, respectively.

Figure 10.

Temperature contour of pump chamber cross section: (a) original pump and (b) improved pump.

Figure 11.

Static pressure distribution contour of pump chamber cross section: (a) original pump and (b) improved pump.

Figure 12.

Vapour volume fraction contour of pump chamber: (a) original pump and (b) improved pump.

Figure 10(a) shows that the fuel temperature in the original pump chamber is approximately 88°C, while Figure 10(b) displays that the fuel temperature in the original pump chamber is approximately 41°C. Figure 11(a) shows a small low-pressure region at the inlet of the impeller in the original pump, while Figure 11(b) presents a large low-pressure region at the centre of the impeller in the improved pump. Figure 12(a) shows no vapour-phase region in the pump chamber of the original pump, whereas Figure 12(b) illustrates a large vapour-phase region (referred to hereinafter as a vapour core ²² ) at the centre of the impeller in the improved pump. It is clear that there are great differences in the flow fields of the original and improved pumps at the same flow rate.

Due to the existence of the large vapour core at the centre of the impeller (i.e. the impeller chamber was nearly empty), the pump shaft power ( $N_{\mathrm{sh}}$ ) was greatly reduced. At $\tilde{Q}=0.01$ , $N_{\mathrm{sh}}$ of the original pump was 24.1 kW, whereas that of the improved pump was 7.8 kW. $N_{\mathrm{sh}}$ is divided into two parts, that is, effective power ( $N_{\mathrm{eff}}$ ) and loss power ( $N_{\mathrm{loss}}$ )

$$N_{\mathrm{sh}}=N_{\mathrm{eff}}+N_{\mathrm{loss}}$$

(1)

$N_{\mathrm{eff}}$ , gained by the fuel to increase the energy head, is calculated as follows

$$N_{\mathrm{eff}}=Q_{\mathrm{v}}(P_{\mathrm{T},\mathrm{out}}-P_{\mathrm{T},\mathrm{in}})$$

(2)

where $P_{\mathrm{T},\mathrm{out}}$ is the pump outlet total pressure ( $P_{\mathrm{T},\mathrm{out}}=P_{\mathrm{out}}+(\rho |{\mathbf{V}}_{\mathrm{out}}{|}^2/2)$ , where $P_{\mathrm{out}}$ is the pump outlet static pressure and ${\mathbf{V}}_{\mathrm{out}}$ is the pump outlet absolute velocity), and $P_{\mathrm{T},\mathrm{in}}$ is the pump inlet total pressure ( $P_{\mathrm{T},\mathrm{in}}=P_{\mathrm{in}}+(\rho |{\mathbf{V}}_{\mathrm{in}}{|}^2/2)$ , where $P_{\mathrm{in}}$ is the pump inlet static pressure and ${\mathbf{V}}_{\mathrm{in}}$ is the pump inlet absolute velocity).

For calculation simplicity, all $N_{\mathrm{loss}}$ are assumed to increase the fuel temperature rise ( $\Delta T$ ), which is calculated by

$$\Delta T=\frac{q}{C_{\mathrm{P}}{Q}_{\mathrm{V}}\rho }$$

(3)

where q is the heat rate and $C_{\mathrm{P}}$ is the fuel specific heat (C_P = 2.223 kJ/(kg °C) ²³ ). Therefore, $\Delta T$ is determined as follows

$$\Delta T=\frac{N_{\mathrm{sh}}-Q_{\mathrm{v}}\left[(P_{\mathrm{out}}-P_{\mathrm{in}})\right]}{C_{\mathrm{p}}{Q}_{\mathrm{V}}\rho }$$

(4)

The calculation results showed that, at $\tilde{Q}=0.01$ , $\Delta T$ of the original pump was approximately 103.3°C, whereas that of the improved pump was approximately 33.7°C. Compared with the experiment results (Figure 6), $\Delta T$ was higher than the experimental values because only a part of $N_{\mathrm{loss}}$ (e.g. loss power caused by friction) was converted to q absorbed by the fuel.

Analysis of vapour core

The fuel at the impeller inlet was pushed towards the volute because of the centrifugal force, thereby forming a low-pressure region (Figure 11). When the pressure in the low-pressure region decreased to $p_{\mathrm{sat}}$ , cavitation was produced (Figure 12). For the original pump at a low flow rate, the cooling fuel from the pump inlet filled the low-pressure region. Hence, no vapour-phase region existed in the pump chamber (Figure 12(a)). For the improved pump at shut-off, the cooling fuel from the cooling bypass could not fill the low-pressure region, causing the vapour core to appear (Figure 12(b)). Because of the impeller rotation, the pressure was higher on the blade pressure side than on the blade suction side. Therefore, the vapour core boundary was not round but zigzag. The distance from the spin axis to the intersection point between the vapour core boundary and the angle bisector of two blades was defined as the vapour core radius ( $R_{\mathrm{d}}$ , Figure 13).

Figure 13.

Definition of vapour core radius.

The vapour core was formed due to cavitation. With respect to cavitation, we naturally thought of cavitation erosion. ²⁴ Cavitation erosion occurred when vapour bubbles, which were formed at a low pressure, subsequently imploded at a high pressure. If the implosions occurred near the surface of the impeller, they would cause damage to the impeller. ²⁵ For the conventional centrifugal pump, the cavitation erosion usually occurred at the leading edge of the impeller blades. ²⁶ However, for the improved pump at shut-off, this was not of concern. Figure 14 shows the change in the vapour core with the impeller rotation, where $\theta$ is the rotation angle of the impeller blades and $T_{\mathrm{b}}$ is the rotation period of the impeller blades. The change period of the vapour core was obviously equal to $T_{\mathrm{b}}$ . Furthermore, the vapour core existed all the time. Although the vapour core shape varied with the impeller rotation, the vapour core interior was unchanged. An impeller wrapped by a vapour core in this manner would not suffer from cavitation erosion.

Figure 14.

Change of vapour core with impeller rotation: (a) $\theta =0\circ$ , $t=0T_{\mathrm{b}}$ , (b) $\theta =18\circ$ , $t=(1/4)T_{\mathrm{b}}$ , (c) $\theta =36\circ$ , $t=(1/2)T_{\mathrm{b}}$ and (d) $\theta =54\circ$ , $t=(3/4)T_{\mathrm{b}}$ .

Centrifugal pumps generally have three kinds of blades (i.e. backward-curved, radial and forward-curved). Figure 15 shows the velocity triangles at the impeller inlet and outlet for a backward-curved blade, a straight radial blade and a straight radial blade with a vapour core. In Figure 15, $R_1$ , ${\mathbf{v}}_1$ , ${\mathbf{u}}_1$ , ${\mathbf{w}}_1$ and ${\mathbf{v}}_{1\mathrm{u}}$ are the inlet radius, inlet absolute velocity, inlet circular velocity, inlet relative velocity and projection component of ${\mathbf{v}}_1$ in ${\mathbf{u}}_1$ direction, respectively. $R_2$ , ${\mathbf{v}}_2$ , ${\mathbf{u}}_2$ , ${\mathbf{w}}_2$ and ${\mathbf{v}}_{2\mathrm{u}}$ are the outlet radius, outlet absolute velocity, outlet circular velocity, outlet relative velocity and projection component of ${\mathbf{v}}_2$ in ${\mathbf{u}}_2$ direction, respectively. ${\mathbf{v}}_{\mathrm{d}}$ , ${\mathbf{u}}_{\mathrm{d}}$ and ${\mathbf{w}}_{\mathrm{d}}$ are the interface absolute velocity, interface circular velocity and interface relative velocity, respectively.

Figure 15.

Velocity triangles at impeller inlet and outlet: (a) backward-curved blade, (b) straight radial blade and (c) straight radial blade with vapour core.

Assuming an infinite number of blades, ideal fluid behaviour and neglecting all types of losses, the theoretical head equation ²⁷ for a centrifugal pump with backward-curved blades is calculated by

$$H_{\mathrm{T}\infty }=\frac{1}{g}({\mathbf{u}}_2{\mathbf{v}}_{2\mathrm{u}}-{\mathbf{u}}_1{\mathbf{v}}_{1\mathrm{u}})$$

(5)

where $H_{\mathrm{T}\infty }$ is the theoretical head and g is the acceleration of gravity. The impeller blades herein were assumed to be approximately straight radial blades (Figure 15(b)). Hence, ${\mathbf{v}}_{1\mathrm{u}}$ was equal to ${\mathbf{u}}_1$ ( $|{\mathbf{u}}_1|=2\pi n{R}_1$ ) and ${\mathbf{v}}_{2\mathrm{u}}$ was equal to ${\mathbf{u}}_2$ ( $|{\mathbf{u}}_2|=2\pi n{R}_2$ ). Because of the existence of the vapour core (Figure 15(c)), ${\mathbf{u}}_1$ should be equal to ${\mathbf{u}}_{\mathrm{d}}$ ( $|{\mathbf{u}}_{\mathrm{d}}|=2\pi n{R}_{\mathrm{d}}$ ).

The operating point of the pump was related to the pipeline system characteristics. The pipeline system energy head ( $H_{\mathrm{c}}$ ) is calculated by

$$H_{\mathrm{c}}=K{Q_{\mathrm{V}}}^2$$

(6)

where K is the load coefficient. Accordingly, $R_{\mathrm{d}}$ is determined as follows

$$R_{\mathrm{d}}=\sqrt{{R_2}^2-\frac{\mathrm{Kg}{Q_{\mathrm{v}}}^2}{4\psi {\pi }^2{n}^2}}$$

(7)

where $\psi$ is the head coefficient (ψ = 0.67−0.75). ²⁸ Equation (7) shows that $R_{\mathrm{d}}$ is a function of $Q_{\mathrm{V}}$ and n. The smaller the $Q_{\mathrm{V}}$ , the larger the $R_{\mathrm{d}}$ . When $Q_{\mathrm{V}}$ and n were constants, $R_{\mathrm{d}}$ remained unchanged. Apparently, the larger the $R_{\mathrm{d}}$ , the better the cooling effect. However, if $R_{\mathrm{d}}\ge R_2$ , the cooling fuel would not acquire enough pressure to return to the fuel tank.

Control on the cooling fuel flow rate

The pump rotational speed at shut-off was considered nearly constant. Hence, according to equation (7), $R_{\mathrm{d}}$ only depended on $Q_{\mathrm{V}}$ . In other words, $R_{\mathrm{d}}$ was changed by controlling $Q_{\mathrm{V}}$ . The design flow rate of the cooling fuel from the cooling bypass is approximately 1% of $Q_{\mathrm{opt}}$ . For the purposes of simplicity, there are no flow control devices (e.g. the flow control valves) in the bypass cooling fuel circuit (Figure 3(a)). The nozzle acted as a throttle device. The nozzle flow rate is calculated by the formula ²⁹ as follows

$$Q_{\mathrm{v}}=C\pi {R_{\mathrm{n}}}^2\sqrt{\frac{2}{\rho }(P_{\mathrm{i}}-P_{\mathrm{o}})}$$

(8)

where C is the discharge coefficient, $R_{\mathrm{n}}$ is the nozzle hole radius and $P_{\mathrm{i}}$ and $P_{\mathrm{o}}$ are the nozzle inlet and outlet pressures, respectively. C is a function of the Reynolds number ( $R_{\mathrm{e}}$ ) and the length to diameter ratio ( $L/D$ ). $R_{\mathrm{e}}$ is defined by

$$R_{\mathrm{e}}=\frac{Q_{\mathrm{v}}D}{A\nu }$$

(9)

where A is the nozzle hole cross-sectional area ( $A=\pi D^2/4$ ) and $\nu$ is the kinematic viscosity.

According to the literature ³⁰

$$C={[1.23+\frac{58L}{R_{e}D}]}^{-1}$$

(10)

for $L/D$ from 2 to 5 and $R_{\mathrm{e}}$ from 100 to 1.5 × 10⁵, claiming the accuracy within 1.5%. In this article, $L/D=3.5$ and $R_{\mathrm{e}}=7.4\times {10}^4$ . Therefore, $C=0.81$ .

As equation (8) shows, $Q_{\mathrm{V}}$ is related to $R_{\mathrm{n}}$ , $P_{\mathrm{i}}$ and $P_{\mathrm{o}}$ . $P_{\mathrm{i}}$ was considered a constant ( $P_{\mathrm{i}}=0.6\mathrm{MPa}$ ), while $P_{\mathrm{o}}$ was considered $P_{\mathrm{sat}}$ . Therefore, $Q_{\mathrm{V}}$ only depended on $R_{\mathrm{n}}$ (i.e. $Q_{\mathrm{V}}$ was regulated only by $R_{\mathrm{n}}$ ). In the experiments (Figure 4), for the original pump at the low flow rates, $Q_{\mathrm{V}}$ was regulated by the flow control valve. For the improved pump at shut-off, to eliminate the influence of the load of the pipeline system on $R_{\mathrm{n}}$ , $Q_{\mathrm{V}}$ was regulated not by the flow control valve but by $R_{\mathrm{n}}$ . Therefore, three different nozzle holes ( $R_{\mathrm{n}}$ , $1.7R_{\mathrm{n}}$ and $2.2R_{\mathrm{n}}$ ) were adopted to meet the three different flow rates.

The nozzle exit position must be considered. Figure 16 illustrates the position and the state of the nozzle exit at the three different flow rates. A nozzle exit located outside the vapour core might lead to $P_{\mathrm{o}}\ge P_{\mathrm{i}}$ . If $P_{\mathrm{o}}\ge P_{\mathrm{i}}$ , the cooling fuel would not enter the impeller chamber. For normal circulation of the cooling fuel, the nozzle exit should be in the vapour core.

Figure 16.

Vapour volume fraction contour of pump chamber in axial plane: (a) $\tilde{Q}=0.01$ , (b) $\tilde{Q}=0.03$ and (c) $\tilde{Q}=0.05$ .

Conclusion

In this study, a centrifugal pump with a cooling bypass was designed to act as an aero-engine fuel centrifugal pump for the afterburner fuel supply. The study focused on how to solve the problem of the high fuel temperature rise in the pump at shut-off. The main conclusions drawn from the research results were as follows:

1. When the pump with a cooling bypass operates at shut-off, a small amount of cooling fuel from the cooling bypass entered the impeller chamber and carried away the heat generated by friction in the pump. The circulation cooling by a small amount of cooling fuel from the cooling bypass solved the problem of the high fuel temperature rise in the pump at shut-off.

2. At shut-off, a large vapour core appeared at the centre of the impeller due to cavitation. This reduced both the pump power consumption and the heat generated by friction, thereby greatly reducing the fuel temperature rise in the pump.

3. A formula for the vapour core radius was derived. The vapour core radius was a function of the flow rate and the pump rotational speed. The vapour core stably existed in the pump at shut-off.

4. The cooling fuel flow rate was regulated by the size of the nozzle hole. For normal circulation of the cooling fuel, the nozzle exit should be located in the vapour core.

The problem of the high fuel temperature rise in the centrifugal pump at shut-off is solved, making it possible for a centrifugal pump to be used as the afterburner fuel pump in aero-engines. Considering the many advantages of the centrifugal pump, a centrifugal pump used as the main engine fuel pump or both as the main engine and the afterburner fuel pump are worthy of future research.