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Numerical simulation of injection rate of each nozzle hole of multi-hole diesel injector
Applied Thermal Engineering 108 (2016) 793–797
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical simulation of injection rate of each nozzle hole of multi-hole diesel injector Xiwen Wu a,b, Jun Deng b, Huifeng Cui d, Fuying Xue a, Liying Zhou a,c, Fuqiang Luo a,⇑ a
School of Automobile and Traffic Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China Zhenjiang Watercraft College of PLA, Zhenjiang 212013, China c School of Mechanical Engineering, Guiyang University, Guiyang 550005, China d Jiangling Motors Co., Ltd., Nanchang 330001, China b
h i g h l i g h t s A three-dimensional gas-liquid two-phase model of cavitation flow was developed. Taking the effect of injection conditions on bubble number density into account. The model can be used to simulate the injection rate of each nozzle hole accurately.
a r t i c l e
i n f o
Article history: Received 6 April 2016 Revised 20 June 2016 Accepted 19 July 2016 Available online 20 July 2016 Keywords: Diesel engine Multi-hole injector Each nozzle hole The injection rate Numerical simulation
a b s t r a c t The relative differences in injection rates within nozzle holes of multi-hole diesel injectors significantly influences the combustion and emission characteristics of diesel engines. To study systematically the injection rate of each nozzle hole of a multi-hole diesel injector, a three-dimensional gas-liquid twophase model of cavitation flow was developed, taking the influence of injection conditions on bubble number density into consideration. To verify validity of the model, the injection rate of each nozzle hole of the injector was experimentally measured on a fuel injection system based on the transient measurement of spray momentum flux. The compared results of the measured and simulated injection rates of each nozzle hole shows that the developed model has relatively high precision and can be used to simulate the injection rate of each nozzle hole accurately. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In order to ensure effective combustion and lower emission for diesel engines, a study on optimization of the combustion spray process based on injection rates and spray characteristics is crucial [1,2]. In fact, spray development and fuel air interactions are all directly affected by injection rates, which further influences the combustion process [3]. Therefore, further understanding of injection rates is of utmost importance to design improvement and performance optimization of diesel engines. With regards to several techniques used to measure injection rates, such as Charge measuring [14] and Laser Doppler Velocimeter [15] methods, the Bosch tube [4–9] and Zeuch measuring [10– 13] methods are the most common. All the methods mentioned give accurate testing results of injection rates for multi-hole diesel ⇑ Corresponding author. E-mail address: [email protected] (F. Luo). http://dx.doi.org/10.1016/j.applthermaleng.2016.07.136 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
injectors. Research work and literatures in relations to the possible differences in the injection rate of each nozzle of the multi-hole diesel injector are limited. For the multi-hole diesel injector, discrepancies [16] in the injection rates within each nozzle hole occurs due to workmanship errors and differences in hydraulic conditions, which affects the uniform distributions of fuel (in time and space) in the combustion chamber, thereby leading to thermal load inconsistency with deterioration of the combustion and emission process [16–19]. A deformational testing method [16,17] was proposed by Marcˇicˇ for testing the injection rates in each hole of the multi-hole diesel injector, Payri et al. also develop a hole to hole mass flow measuring bench [20], but few reports covering relevant measuring units and methods which have been experimentally validated, possess adequate response characteristics and potential to be applied widely. With the rapid development of computer technology and computational fluid dynamic (CFD), the multi-dimensional numerical simulation has already become an effective means of relevant
794
X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
theoretical research in relation to internal combustion engines. To study the injection rate of each nozzle hole of a multi-hole diesel injector systematically, a three-dimensional gas-liquid two-phase model of cavitation flow was developed, the injection rate of each nozzle hole of the injector was experimentally measured (on an experimental rig based on the transient measurements of spray momentum flux) and used to validate the developed model. In Section 2 of this manuscript, the mathematical model for fuel flow characteristics within the injector nozzle holes of the diesel engine is presented. A three-dimensional gas-liquid two-phase model of cavitation flow is developed in the next section, which predicts the output. The prediction accuracy of the model is validated using the measurements of the transient spray momentum flux of each nozzle hole obtained from experimental data. The investigated outcomes are then summarized in the conclusion section. 2. Mathematical model The two-fluid model approach was used for computations of flow characteristics of the diesel fuel within the nozzle holes of the diesel engine. From the gas-liquid two-phase approach [21], the continuity and momentum equations are as follows: 2 X @ ak qk Ckl þ r ak qk v k ¼ @t l¼1;l–k
ð1Þ
@ ak qk v k þ r ak qk v k v k ¼ ak rp þ r ak ðsk þ T tk Þ þ ak qk g @t 2 2 X X Mkl þ v k Ckl þ l¼1;l–k
where the value for the parameter k is either 1, for gas phase only, or 2 for only liquid phase. The summation of the volume fractions at P phase k(ak) is 2k¼1 ak ¼ 1. qk and vk respectively represent the density and velocity at phase k. Ckl is the interfacial mass transfer between the phases k and l, T tk is the Reynolds stress at phase k and M kl represents the interfacial momentum transfer between phases k and l. From the two-fluid model approach, the gas phase pressure (p1) and the liquid phase pressure (p2) are equal and can be represented by P, that is:
ð3Þ
For phase k, the shear stress (sk) is:
sk ¼ lk ðrv k þ rv
T kÞ
2 r vk 3
ð7Þ
6
1:0 10
where pi and po are the injection and ambient (back) pressures respectively. The linearized Rayleigh-Plesset equation was used for the _ as stated determination of the rate of change of bubble radius R, in Eq. (8).
sffiffiffiffiffiffiffiffiffiffiffiffi _R ¼ signðDpÞ 2jDpj 3q2
ð8Þ
Dp is the effective pressure difference with regards to bubble number growth and collapse due to pressure fluctuations. It is expressed as:
2 Dp ¼ psat p C E q2 k2 3
ð9Þ
where C E is the Egler coefficient. The expression for the interfacial momentum of the gas-fluid two-phase flow take the form:
M12 ¼ F D12 þ F TD 12 ¼ M 21
ð10Þ
1 F D12 ¼ F D21 ¼ C D q2 A000 i jv r jv r 8
ð11Þ
TD F TD 12 ¼ F 21 ¼ C TD q2 k2 ra1
ð12Þ
1 3
2
2 000 000 3 3 A000 i ¼ pDb N ¼ ð36pÞ N a1
ð13Þ
where F D12 is the drag force between gas and fluid phases, the turbulent dispersion force is F TD 12 , CD is the drag coefficient, the turbulent dispersion coefficient is CTD and bubble diameter is Db. The standard k-e model is used to calculate the turbulence of the core region. The transport equations for the turbulence kinetic energy k and the turbulence kinetic energy diffusivity e are respectively expressed as:
@ ak qk kk þ r ak qk v k kk ¼ r ak @t
ð5Þ
lk þ
ltk rkk þ ak Pk rk
2 X
ð6Þ
where N 000 is the bubble number density, R is the mean bubble radius in cavitation region and R_ is the rate of change of bubble radius. For the bubble number density N 000 (taking the influence of injection condition into consideration [24]), Eq. (7) was used.
K kl
Ckl
ð14Þ
l¼1;l–k
þ ek
lk þ
2 X
2 X ltk rek þ Dkl re l¼1;l–k
Ckl þ ak C 1 Pk
l¼1;l–k
where ltk is the viscosity of turbulence. For the interfacial mass transfer between gas phase and liquid phase, the string cavitation model was used for computations, that is:
2 X l¼1;l–k
@ ak qk ek þ r ak qk v k ek ¼ r ak @t
The Reynolds stress at phase k (T tk ) is:
þ ak PB;k ak qk ek þ
ð4Þ
lk being the viscosity at phase k.
C12 ¼ q1 N000 4pR2 R_ ¼ C21
3
pi po
þ kk
2 2 T tk ¼ qk v 0k v 0k ¼ ltk ðrv k þ rv Tk Þ r v k I qk kk I 3 3
1
ð2Þ
l¼1;l–k
p ¼ p1 ¼ p2
N000 ¼ 1:0 1012
þ ak C 3 maxðPB;k ; 0Þ
ek kk
ek kk
ak C 2 qk
e2k kk
ak C 4 qk ek r v k ð15Þ
where kk is the turbulence kinetic energy at phase k, ek is the diffusivity of the turbulence kinetic energy at phase k, PB,k is the generation component of the turbulence kinetic energy caused by buoyancy, the Prandtl number for the turbulence kinetic energy is r,k, Kkl is the component of transmission between k phase and l phase, re is the Prandtl number for the e equation, C1, C2, C3, C4 are constants and Dkl is the interface exchange component of the e.equation
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X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
3. Numerical simulations of injection rate of each nozzle hole The measuring bench (depicted in Fig. 1) used for measuring the transient spray momentum flux of each nozzle hole [22], a P-type multi-hole microsac injector with the opening pressure of 20 MPa were used in the experimental set up, the basic injector parameters obtained from the manufaturer’s drawing are as follows: the number of holes is 5, the hole diameter is 0.2 mm, the hole length is 0.8 mm, the inclination angle is shown in Fig. 2, the inlet rounding of each hole is 0.15 lm and the k-factor of the each hole is 0. The measured results of the spray force for each nozzle hole and the upstream pressure of the microsac injector are displayed in Figs. 3 and 5 at constant operating conditions(the fuel injection quantity per cycle is 50 mm3/cyc, while the pump speed is 1000 r/min) respectively. According to reference [22], spray force, spray momentum and injection rate have the following relationships:
ð17Þ
hence, the computed results of the injection rate for each nozzle hole are shown in Fig. 4. 2.2
Hole1
Hole2
Hole4
Hole3 Hole5
2.0 1.8 1.6
Spray force (N)
The turbulence for the near-wall regions were determined by using standard wall equations.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 _VðtÞ ¼ Fðt þ t Þ Ageo qf ðtÞ
1.4 1.2 1.0 0.8 0.6 0.4 0.2
_ MðtÞ ¼ Fðt þ t0 Þ
ð16Þ
0.0 5
10
15
20
25
Cam rotation angle ( oCaA) Fig. 3. Test result of spray force for each nozzle hole.
1.6
Hole1
Hole2 Hole3 Hole5
Hole4
10
15
20
1-oil mist dispersal chamber, 2-magnetic stand, 3-angle adjustment knob, 4-distance adjusting screw, 5-force sensor and target, 6-clamp-on pressure sensor,
Injection rate (mm 3/ oCaA)
1.4 1.2 1.0 0.8 0.6 0.4 0.2
7,8-charge amplifier, 9-injector, 10-high pressure fuel pipe, 11-injection pump᧷ 0.0 5
12-fuel adjusting mechanism, 13-data acquisition system, 14-monitoring computer,
25
Cam rotation angle ( oCaA)
15-pump test-bed, 16-controller of pump test-bed. Fig. 4. Computed result of injection rate for each nozzle hole. Fig. 1. Spray momentum flux test rig.
Injection pressure MPa
50
Injection pressure
40
30
20
10
0 0
8
16
24
32
Cam rotation angle [ CaA ] Fig. 2. Schematic diagram of the experimental injector.
Fig. 5. The profile curve of injection pressure.
40
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X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
For the injector shown in Fig. 2, the three-dimensional structured hexahedral mesh grids of the nozzle is obtained by using multiple block coupling method. During the numerical simulation, the profile of injection pressure changes from set pressure boundary conditions (injection pressure for inlet and standard atmospheric pressure for outlet) for each nozzle hole are shown in Fig. 5. Compared results of fuel injection quantity per cycle of each nozzle hole between simulated values (qsimulation obtained using the integral method) and experimentally measured ones (qmeasure) are shown in Fig. 6. Relative errors (4hole) of the nozzle holes for the fuel injection quantity per cycle are computed based on the values of qsimulation and qmeasure using Eq. (18) and then integrated into Fig 7. The relative error, experimental and simulated results
Experimental result
1.6
0.8
0.4
5
10
15
20
Experimental result
0.8
0.4
0.0
25
5
10
Injection rate [ mm3/ CaA ]
0.4
10
15
20
25
Simulation result
1.2
0.8
0.4
0.0
25
Experimental result
5
10
15
Cam rotation angle CaA
Cam rotation angle CaA
(d) hole 4
(c) hole 3 1.6
Injection rate [ mm3/ CaA ]
Injection rate [ mm3/ CaA ]
1.6
Simulation result
0.8
5
20
(b) hole 2
1.2
0.0
15
Cam rotation angle CaA
(a) hole 1 Experimental result
Simulation result
1.2
Cam rotation angle CaA
1.6
ð18Þ
For model validation, simulated and experimentally measured results of the injection rate of each nozzle hole were compared in Figs. 6 and 7. The results show a high degree of accuracy between them, meanwhile, for the fuel injection quantity per cycle of each nozzle hole, a relative error of less than 5% between both results were obtained From the comparison shown in Figs. 6 and 7, the relatively high precision of the simulated results are due to the three-dimensional
Simulation result
1.2
0.0
qsimulation qmeasure 100% qmeasure
Dhole ¼
Injection rate [ mm3/ CaA ]
Injection rate [ mm3/ CaA ]
1.6
for fuel injection quantity per cycle of each nozzle hole is graphically displayed in Fig. 7.
Experimental result
Simulation result
1.2
0.8
0.4
0.0
5
10
15
20
25
Cam rotation angle CaA
(e) hole 5 Fig. 6. Comparisons between experimentally measured and simulated injection rate of each nozzle hole.
20
25
X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
Simulation result 11
4
Relative error
10 2
9 8
0
Relative error [%]
Cycle fuel injection quantity [ mm3]
Appendix A. Supplementary materila
Experimental result
12
797
7 6
1
2
3
4
5
-2
Nozzle hole serial number Fig. 7. Comparisons between experimentally measured and simulated cycle fuel injection quantity of each nozzle hole.
gas-liquid two-phase model of cavitation flow used, which took the influence of injection conditions on bubble number density into consideration. Thereby ascertaining the fact that, the transient spray velocity of each nozzle hole of a multi-hole diesel injector, can be accurately simulated.
4. Conclusions In this paper, a three-dimensional gas-liquid two-phase model of cavitation flow was developed, taking the influence of injection conditions on bubble number density into consideration. Experimentally measured results of injection rate based on the transient measurements of spray momentum flux for each nozzle hole were obtained and used to verify and predict the accuracy of the developed model. Compared results shows that, the developed model has a relatively high accuracy and hence can be used to simulate the injection rate of each nozzle hole accurately.
Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 51476072), Scientific Research Innovation Foundation for Graduate Students of Jiangsu Province (CXLX13_654 and KYLX_1034), a Project Supported by the Priority Academic Program Development of Jiangsu High Education Institutions, and the Science and Technology Foundation of Guizhou Province (20147186).
Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.applthermaleng. 2016.07.136. References [1] L. Postrioti, F. Mariani, M. Battistoni, Experimental and numerical momentum flux evaluation of high pressure Diesel spray, Fuel 98 (2012) 149–163. [2] J. Dernotte, C. Hespel, F. Foucher, S. Houillé, C. Mounaïm-Rousselle, Influence of physical fuel properties on the injection rate in a Diesel injector, Fuel 96 (2012) 153–160. [3] O. Armas, C. Mata, S. Martínez-Martínez, Effect of diesel injection parameters on instantaneous fuel delivery using a solenoid-operated injector with different fuels, Rev. Fac. Ing. Univ. Antioquia 64 (2012) 9–21. [4] K. Jangsik, B. Choongsik, Effect of nozzle hole geometry on non-evaporating diesel spray characteristics at high-pressure injection, Atomization Sprays 22 (1) (2012) 1–21. [5] J. Lee, K. Min, Effects of needle response on spray characteristics in high pressure injector driven by piezo actuator for common-rail injection system, J. Mech. Sci. Technol. 19 (2005) 1194–1205. [6] A.E. Catania, A. Ferrari, M. Manno, E. Spessa, Experimental investigation of dynamics effects on multiple-injection common rail system performance, ASME Trans. J. Eng. Gas Turb. Power 130 (3) (2008) 032806. [7] Y. Guanjun, L. Liguang, D. Jun, et al., Research on the effect of the parameters of common-rail system on the injection rate, Lect. Notes Electr. Eng. 189 (2013) 411–420. [8] R. Radu, M. Gheorghe, C. Petru, et al., Comparative analysis of two injection systems fueled with biodiesel, SAE Paper 2012-01-1657, 2012. [9] M. Costa, L., A. Montanaro, et al., Multiple injection in a mixed mode GDI boosted engine, SAE Paper 2010-01-1496, 2010. [10] C. Arcoumanis, M.S. Baniasad, Analysis of consecutive fuel injection rate signals obtained by the Zeuch and Bosch methods, SAE Paper 930921, 1993. [11] A. Takamura, T. Ohta, S. Fukushima, et al., A study on precise measurement of diesel fuel injection rate, SAE Paper 920630, 1992. [12] S. Ishikawa, Y. Ohmori, S. Fukushima, et al., Measurement of rate of multipleinjection in CDI diesel engines, SAE Paper 2000-01-1257, 2000. [13] T. Ikeda, Y. Ohmori, A. Takamura, Y. Sato, Measurement of the rate of multiple fuel injection with diesel fuel and DME, SAE Paper 2001-01-0527, 2001. [14] M. Marcˇicˇ, A new method for measuring fuel-injection rate, Flow Meas. Instrum. 10 (1999) 159–165. [15] G. Brenn, D. Trimis, M. Weclas, et al., Methods and tools for advanced fuel spray production and investigation, Atomization Sprays 1 (1997) 43–75. [16] M. Marcˇicˇ, Sensor for injection rate measurements, Sensors 6 (2006) 1367– 1382. [17] M. Marcˇicˇ, Measuring method for diesel multihole injection nozzles, Sens. Actuat. A 107 (2) (2003) 152–158. [18] F. Payri, R. Payri, F.J. Salvador, et al., Comparison between different hole to hole measurement techniques in a diesel injection nozzle, SAE Paper 2005-012094, 2005. [19] M. Marcˇicˇ, Deformational injection rate measuring method, Rev. Sci. Instrum. 9 (2002) 3373–3377. [20] F. Payri, R. Payri, F.J. Salvador, et al., Comparison between different hole to hole measurement techniques in a diesel injection nozzle, SAE Paper 2005-012094, 2005. [21] X. Wang, W.H. Su, Numerical investigation on relationship between injection pressure fluctuations and unsteady cavitation processes inside high-pressure diesel nozzle holes, Fuel 89 (2010) 2252–2259. [22] Fuqiang Luo, Huifeng Cui, Shaofeng Dong, Transient measuring method for injection rate of each nozzle hole based on spray momentum flux, Fuel 125 (2014) 20–29.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical simulation of injection rate of each nozzle hole of multi-hole diesel injector Xiwen Wu a,b, Jun Deng b, Huifeng Cui d, Fuying Xue a, Liying Zhou a,c, Fuqiang Luo a,⇑ a
School of Automobile and Traffic Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China Zhenjiang Watercraft College of PLA, Zhenjiang 212013, China c School of Mechanical Engineering, Guiyang University, Guiyang 550005, China d Jiangling Motors Co., Ltd., Nanchang 330001, China b
h i g h l i g h t s A three-dimensional gas-liquid two-phase model of cavitation flow was developed. Taking the effect of injection conditions on bubble number density into account. The model can be used to simulate the injection rate of each nozzle hole accurately.
a r t i c l e
i n f o
Article history: Received 6 April 2016 Revised 20 June 2016 Accepted 19 July 2016 Available online 20 July 2016 Keywords: Diesel engine Multi-hole injector Each nozzle hole The injection rate Numerical simulation
a b s t r a c t The relative differences in injection rates within nozzle holes of multi-hole diesel injectors significantly influences the combustion and emission characteristics of diesel engines. To study systematically the injection rate of each nozzle hole of a multi-hole diesel injector, a three-dimensional gas-liquid twophase model of cavitation flow was developed, taking the influence of injection conditions on bubble number density into consideration. To verify validity of the model, the injection rate of each nozzle hole of the injector was experimentally measured on a fuel injection system based on the transient measurement of spray momentum flux. The compared results of the measured and simulated injection rates of each nozzle hole shows that the developed model has relatively high precision and can be used to simulate the injection rate of each nozzle hole accurately. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In order to ensure effective combustion and lower emission for diesel engines, a study on optimization of the combustion spray process based on injection rates and spray characteristics is crucial [1,2]. In fact, spray development and fuel air interactions are all directly affected by injection rates, which further influences the combustion process [3]. Therefore, further understanding of injection rates is of utmost importance to design improvement and performance optimization of diesel engines. With regards to several techniques used to measure injection rates, such as Charge measuring [14] and Laser Doppler Velocimeter [15] methods, the Bosch tube [4–9] and Zeuch measuring [10– 13] methods are the most common. All the methods mentioned give accurate testing results of injection rates for multi-hole diesel ⇑ Corresponding author. E-mail address: [email protected] (F. Luo). http://dx.doi.org/10.1016/j.applthermaleng.2016.07.136 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
injectors. Research work and literatures in relations to the possible differences in the injection rate of each nozzle of the multi-hole diesel injector are limited. For the multi-hole diesel injector, discrepancies [16] in the injection rates within each nozzle hole occurs due to workmanship errors and differences in hydraulic conditions, which affects the uniform distributions of fuel (in time and space) in the combustion chamber, thereby leading to thermal load inconsistency with deterioration of the combustion and emission process [16–19]. A deformational testing method [16,17] was proposed by Marcˇicˇ for testing the injection rates in each hole of the multi-hole diesel injector, Payri et al. also develop a hole to hole mass flow measuring bench [20], but few reports covering relevant measuring units and methods which have been experimentally validated, possess adequate response characteristics and potential to be applied widely. With the rapid development of computer technology and computational fluid dynamic (CFD), the multi-dimensional numerical simulation has already become an effective means of relevant
794
X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
theoretical research in relation to internal combustion engines. To study the injection rate of each nozzle hole of a multi-hole diesel injector systematically, a three-dimensional gas-liquid two-phase model of cavitation flow was developed, the injection rate of each nozzle hole of the injector was experimentally measured (on an experimental rig based on the transient measurements of spray momentum flux) and used to validate the developed model. In Section 2 of this manuscript, the mathematical model for fuel flow characteristics within the injector nozzle holes of the diesel engine is presented. A three-dimensional gas-liquid two-phase model of cavitation flow is developed in the next section, which predicts the output. The prediction accuracy of the model is validated using the measurements of the transient spray momentum flux of each nozzle hole obtained from experimental data. The investigated outcomes are then summarized in the conclusion section. 2. Mathematical model The two-fluid model approach was used for computations of flow characteristics of the diesel fuel within the nozzle holes of the diesel engine. From the gas-liquid two-phase approach [21], the continuity and momentum equations are as follows: 2 X @ ak qk Ckl þ r ak qk v k ¼ @t l¼1;l–k
ð1Þ
@ ak qk v k þ r ak qk v k v k ¼ ak rp þ r ak ðsk þ T tk Þ þ ak qk g @t 2 2 X X Mkl þ v k Ckl þ l¼1;l–k
where the value for the parameter k is either 1, for gas phase only, or 2 for only liquid phase. The summation of the volume fractions at P phase k(ak) is 2k¼1 ak ¼ 1. qk and vk respectively represent the density and velocity at phase k. Ckl is the interfacial mass transfer between the phases k and l, T tk is the Reynolds stress at phase k and M kl represents the interfacial momentum transfer between phases k and l. From the two-fluid model approach, the gas phase pressure (p1) and the liquid phase pressure (p2) are equal and can be represented by P, that is:
ð3Þ
For phase k, the shear stress (sk) is:
sk ¼ lk ðrv k þ rv
T kÞ
2 r vk 3
ð7Þ
6
1:0 10
where pi and po are the injection and ambient (back) pressures respectively. The linearized Rayleigh-Plesset equation was used for the _ as stated determination of the rate of change of bubble radius R, in Eq. (8).
sffiffiffiffiffiffiffiffiffiffiffiffi _R ¼ signðDpÞ 2jDpj 3q2
ð8Þ
Dp is the effective pressure difference with regards to bubble number growth and collapse due to pressure fluctuations. It is expressed as:
2 Dp ¼ psat p C E q2 k2 3
ð9Þ
where C E is the Egler coefficient. The expression for the interfacial momentum of the gas-fluid two-phase flow take the form:
M12 ¼ F D12 þ F TD 12 ¼ M 21
ð10Þ
1 F D12 ¼ F D21 ¼ C D q2 A000 i jv r jv r 8
ð11Þ
TD F TD 12 ¼ F 21 ¼ C TD q2 k2 ra1
ð12Þ
1 3
2
2 000 000 3 3 A000 i ¼ pDb N ¼ ð36pÞ N a1
ð13Þ
where F D12 is the drag force between gas and fluid phases, the turbulent dispersion force is F TD 12 , CD is the drag coefficient, the turbulent dispersion coefficient is CTD and bubble diameter is Db. The standard k-e model is used to calculate the turbulence of the core region. The transport equations for the turbulence kinetic energy k and the turbulence kinetic energy diffusivity e are respectively expressed as:
@ ak qk kk þ r ak qk v k kk ¼ r ak @t
ð5Þ
lk þ
ltk rkk þ ak Pk rk
2 X
ð6Þ
where N 000 is the bubble number density, R is the mean bubble radius in cavitation region and R_ is the rate of change of bubble radius. For the bubble number density N 000 (taking the influence of injection condition into consideration [24]), Eq. (7) was used.
K kl
Ckl
ð14Þ
l¼1;l–k
þ ek
lk þ
2 X
2 X ltk rek þ Dkl re l¼1;l–k
Ckl þ ak C 1 Pk
l¼1;l–k
where ltk is the viscosity of turbulence. For the interfacial mass transfer between gas phase and liquid phase, the string cavitation model was used for computations, that is:
2 X l¼1;l–k
@ ak qk ek þ r ak qk v k ek ¼ r ak @t
The Reynolds stress at phase k (T tk ) is:
þ ak PB;k ak qk ek þ
ð4Þ
lk being the viscosity at phase k.
C12 ¼ q1 N000 4pR2 R_ ¼ C21
3
pi po
þ kk
2 2 T tk ¼ qk v 0k v 0k ¼ ltk ðrv k þ rv Tk Þ r v k I qk kk I 3 3
1
ð2Þ
l¼1;l–k
p ¼ p1 ¼ p2
N000 ¼ 1:0 1012
þ ak C 3 maxðPB;k ; 0Þ
ek kk
ek kk
ak C 2 qk
e2k kk
ak C 4 qk ek r v k ð15Þ
where kk is the turbulence kinetic energy at phase k, ek is the diffusivity of the turbulence kinetic energy at phase k, PB,k is the generation component of the turbulence kinetic energy caused by buoyancy, the Prandtl number for the turbulence kinetic energy is r,k, Kkl is the component of transmission between k phase and l phase, re is the Prandtl number for the e equation, C1, C2, C3, C4 are constants and Dkl is the interface exchange component of the e.equation
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X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
3. Numerical simulations of injection rate of each nozzle hole The measuring bench (depicted in Fig. 1) used for measuring the transient spray momentum flux of each nozzle hole [22], a P-type multi-hole microsac injector with the opening pressure of 20 MPa were used in the experimental set up, the basic injector parameters obtained from the manufaturer’s drawing are as follows: the number of holes is 5, the hole diameter is 0.2 mm, the hole length is 0.8 mm, the inclination angle is shown in Fig. 2, the inlet rounding of each hole is 0.15 lm and the k-factor of the each hole is 0. The measured results of the spray force for each nozzle hole and the upstream pressure of the microsac injector are displayed in Figs. 3 and 5 at constant operating conditions(the fuel injection quantity per cycle is 50 mm3/cyc, while the pump speed is 1000 r/min) respectively. According to reference [22], spray force, spray momentum and injection rate have the following relationships:
ð17Þ
hence, the computed results of the injection rate for each nozzle hole are shown in Fig. 4. 2.2
Hole1
Hole2
Hole4
Hole3 Hole5
2.0 1.8 1.6
Spray force (N)
The turbulence for the near-wall regions were determined by using standard wall equations.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 _VðtÞ ¼ Fðt þ t Þ Ageo qf ðtÞ
1.4 1.2 1.0 0.8 0.6 0.4 0.2
_ MðtÞ ¼ Fðt þ t0 Þ
ð16Þ
0.0 5
10
15
20
25
Cam rotation angle ( oCaA) Fig. 3. Test result of spray force for each nozzle hole.
1.6
Hole1
Hole2 Hole3 Hole5
Hole4
10
15
20
1-oil mist dispersal chamber, 2-magnetic stand, 3-angle adjustment knob, 4-distance adjusting screw, 5-force sensor and target, 6-clamp-on pressure sensor,
Injection rate (mm 3/ oCaA)
1.4 1.2 1.0 0.8 0.6 0.4 0.2
7,8-charge amplifier, 9-injector, 10-high pressure fuel pipe, 11-injection pump᧷ 0.0 5
12-fuel adjusting mechanism, 13-data acquisition system, 14-monitoring computer,
25
Cam rotation angle ( oCaA)
15-pump test-bed, 16-controller of pump test-bed. Fig. 4. Computed result of injection rate for each nozzle hole. Fig. 1. Spray momentum flux test rig.
Injection pressure MPa
50
Injection pressure
40
30
20
10
0 0
8
16
24
32
Cam rotation angle [ CaA ] Fig. 2. Schematic diagram of the experimental injector.
Fig. 5. The profile curve of injection pressure.
40
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X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
For the injector shown in Fig. 2, the three-dimensional structured hexahedral mesh grids of the nozzle is obtained by using multiple block coupling method. During the numerical simulation, the profile of injection pressure changes from set pressure boundary conditions (injection pressure for inlet and standard atmospheric pressure for outlet) for each nozzle hole are shown in Fig. 5. Compared results of fuel injection quantity per cycle of each nozzle hole between simulated values (qsimulation obtained using the integral method) and experimentally measured ones (qmeasure) are shown in Fig. 6. Relative errors (4hole) of the nozzle holes for the fuel injection quantity per cycle are computed based on the values of qsimulation and qmeasure using Eq. (18) and then integrated into Fig 7. The relative error, experimental and simulated results
Experimental result
1.6
0.8
0.4
5
10
15
20
Experimental result
0.8
0.4
0.0
25
5
10
Injection rate [ mm3/ CaA ]
0.4
10
15
20
25
Simulation result
1.2
0.8
0.4
0.0
25
Experimental result
5
10
15
Cam rotation angle CaA
Cam rotation angle CaA
(d) hole 4
(c) hole 3 1.6
Injection rate [ mm3/ CaA ]
Injection rate [ mm3/ CaA ]
1.6
Simulation result
0.8
5
20
(b) hole 2
1.2
0.0
15
Cam rotation angle CaA
(a) hole 1 Experimental result
Simulation result
1.2
Cam rotation angle CaA
1.6
ð18Þ
For model validation, simulated and experimentally measured results of the injection rate of each nozzle hole were compared in Figs. 6 and 7. The results show a high degree of accuracy between them, meanwhile, for the fuel injection quantity per cycle of each nozzle hole, a relative error of less than 5% between both results were obtained From the comparison shown in Figs. 6 and 7, the relatively high precision of the simulated results are due to the three-dimensional
Simulation result
1.2
0.0
qsimulation qmeasure 100% qmeasure
Dhole ¼
Injection rate [ mm3/ CaA ]
Injection rate [ mm3/ CaA ]
1.6
for fuel injection quantity per cycle of each nozzle hole is graphically displayed in Fig. 7.
Experimental result
Simulation result
1.2
0.8
0.4
0.0
5
10
15
20
25
Cam rotation angle CaA
(e) hole 5 Fig. 6. Comparisons between experimentally measured and simulated injection rate of each nozzle hole.
20
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X. Wu et al. / Applied Thermal Engineering 108 (2016) 793–797
Simulation result 11
4
Relative error
10 2
9 8
0
Relative error [%]
Cycle fuel injection quantity [ mm3]
Appendix A. Supplementary materila
Experimental result
12
797
7 6
1
2
3
4
5
-2
Nozzle hole serial number Fig. 7. Comparisons between experimentally measured and simulated cycle fuel injection quantity of each nozzle hole.
gas-liquid two-phase model of cavitation flow used, which took the influence of injection conditions on bubble number density into consideration. Thereby ascertaining the fact that, the transient spray velocity of each nozzle hole of a multi-hole diesel injector, can be accurately simulated.
4. Conclusions In this paper, a three-dimensional gas-liquid two-phase model of cavitation flow was developed, taking the influence of injection conditions on bubble number density into consideration. Experimentally measured results of injection rate based on the transient measurements of spray momentum flux for each nozzle hole were obtained and used to verify and predict the accuracy of the developed model. Compared results shows that, the developed model has a relatively high accuracy and hence can be used to simulate the injection rate of each nozzle hole accurately.
Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 51476072), Scientific Research Innovation Foundation for Graduate Students of Jiangsu Province (CXLX13_654 and KYLX_1034), a Project Supported by the Priority Academic Program Development of Jiangsu High Education Institutions, and the Science and Technology Foundation of Guizhou Province (20147186).
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