A multi-fluid blast furnace model was simply introduced and was used to simulate several innovative ironmaking operations. The simulation results show that injecting hydrogen bearing materials, especially injecting na...A multi-fluid blast furnace model was simply introduced and was used to simulate several innovative ironmaking operations. The simulation results show that injecting hydrogen bearing materials, especially injecting natural gas and plastics, the hydrogen reduction is enhanced, and the furnace performance is improved simultaneously. Total heat input shows obvious decrease due to the decrease of heat consumption in direct reduction, solution loss and silicon transfer reactions. If carbon composite agglomerates are charged into the furnace, the temperature of thermal reserve zone will obviously decrease, and the reduction of iron-bearing burden materials will be retarded. However, the efficiency of blast furnace is improved just due to the decrease in heat requirements for solution loss, sinter reduction, and silicon transfer reactions, and less heat loss through top gas and furnace wall. Finally, the model is used to investigate the performance of blast furnace under the condition of top gas recycling together with plastics injection, cold oxygen blasting and carbon composite agglomerate charging. The lower furnace temperature, extremely accelerated reduction rate, drastically decreased CO2 emission and remarkably enhanced heat efficiency were obtained by using the innovative operations, and the blast furnace operation with superhigh efficiency can be realized.展开更多
The Euler-Euler model is less effective in capturing the free surface of flow film in the spiral separator,and thus a Eulerian multi-fluid volume of fluid(VOF)model was first proposed to describe the particulate flow ...The Euler-Euler model is less effective in capturing the free surface of flow film in the spiral separator,and thus a Eulerian multi-fluid volume of fluid(VOF)model was first proposed to describe the particulate flow in spiral separators.In order to improve the applicability of the model in the high solid concentration system,the Bagnold effect was incorporated into the modelling framework.The capability of the proposed model in terms of predicting the flow film shape in a LD9 spiral separator was evaluated via comparison with measured flow film thicknesses reported in literature.Results showed that sharp air–water and air-pulp interfaces can be obtained using the proposed model,and the shapes of the predicted flow films before and after particle addition were reasonably consistent with the observations reported in literature.Furthermore,the experimental and numerical simulation of the separation of quartz and hematite were performed in a laboratory-scale spiral separator.When the Bagnold lift force model was considered,predictions of the grade of iron and solid concentration by mass for different trough lengths were more consistent with experimental data.In the initial development stage,the quartz particles at the bottom of the flow layer were more possible to be lifted due to the Bagnold force.Thus,a better predicted vertical stratification between quartz and hematite particles was obtained,which provided favorable conditions for subsequent radial segregation.展开更多
A large number of submarine landslides with different scales have been identified in the canyon area of the submarine pipeline route of Liwan 3-1 gas field. There is still much chance that submarine slope failures wou...A large number of submarine landslides with different scales have been identified in the canyon area of the submarine pipeline route of Liwan 3-1 gas field. There is still much chance that submarine slope failures would happen, and the following mass movement would present great risk to the submarine pipeline. In view of this, a numerical prediction method based on Eulerian-Eulerian two-phase flow model is introduced to simulate the mass movement of potential submarine landslides. The sliding soil and ambient water are respectively simulated by Herschel-Bulkley rheology model and Newtonian fluid model. The turbulence is simulated using the k-e model. Compared with both the experiment data and Bing result, the two-phase flow model shows a good accuracy, and its result is more close to the actual situation; the dynamic coupling between soil and ambient water can be effectively simulated and the phenomena of hydroplaning and head detachment can be obtained. Finally, the soil movement of a potential submarine landslide is simulated as an example, according to the seismic profile in the canyon area. The result shows that the hydroplaning occurs during the movement process. The runout distance calculated by the two-phase flow model is 877 m, which is 27.1% larger than the Bing result. However, the peak front velocity of soil is relative small, with a maximum value of 8.32 m/s. The Bing program with a simple and rapid process can be used for a preliminary evaluation, while the two-phase flow model is more appropriate for an accurate assessment.展开更多
Based on the two-phase model (Eulerian-Eulerian model), the three dimensional fluid flow in water and that liquid steel systems stirred by one or two multiple gas jets are simulated. In the Eulerian-Eulerian two-phase...Based on the two-phase model (Eulerian-Eulerian model), the three dimensional fluid flow in water and that liquid steel systems stirred by one or two multiple gas jets are simulated. In the Eulerian-Eulerian two-phase model, the gas and the liquid phase are considered to be two different continuous fluids interacting with each other through the finite inter-phase areas. The exchange between the phases is represented by source terms in conversation equations. Turbulence is assumed to be a property of the liquid phase. A new turbulence modification - model is introduced to consider the bubbles movement contribution to and . The dispersion of phases due to turbulence is represented by introducing a diffusion term in mass conservation equation. The mathematical simulation agrees well with the experiment results. The study results indicate that the distance of two nozzles has big effect on fluid flow behavior in the vessel. Using two gas injection nozzles at the half radii of one diameter of the bottom generates a much better mixing than with one nozzle under the condition of the same total gas flow rate.展开更多
The reducibility of iron-bearing burdens was emphasized for improving the operation efficiency of blast furnace. The blast furnace operation of charging the burdens with high reducibility has been numerically evaluate...The reducibility of iron-bearing burdens was emphasized for improving the operation efficiency of blast furnace. The blast furnace operation of charging the burdens with high reducibility has been numerically evaluated using a multi-fluid blast furnace model. The effects of reaction rate constants and diffusion coefficients were investigated separately or simultaneously for clarifying the variations of furnace state. According to the model simulation results, in the upper zone, the indirect reduction of the burdens proceeds at a faster rate and the shaft efficiency is enhanced with the improvement under the conditions of interface reaction and intra-particle diffusion. In the lower zone, direct reduction in molten slag is restrained. As a consequence, CO utilization of top gas is enhanced and the ratio of direct reduction is decreased. It is possible to achieve higher energy efficiency of the blast furnace, and this is represented by the improvement in productivity and the decrease in consumption of reducing agent. The use of high-reducibility burdens contributes to a better performance of blast furnace. More efforts are necessary to develop and apply highreducibility sinter and carbon composite agglomerates for practical application at a blast furnace.展开更多
Experimental residence time distribution (RTD) measurement and computational fluid dynamics (CFD) simulation are the best methods to study the hydrodynamics of process flow systems. However, CFD approach leads to bett...Experimental residence time distribution (RTD) measurement and computational fluid dynamics (CFD) simulation are the best methods to study the hydrodynamics of process flow systems. However, CFD approach leads to better understanding of the flow structure and extent of mixing in stirred tanks. In the present study, CFD models were used to simulate the flow in an industrial gold leaching tank. The objective of the investigation was to characterize the flowfield generated within the tank after process intensification. The flow was simulated using an Eulerian-Eulerian multi-fluid model where the RANS standard kmixture model and a multiple reference frame approach were used to model turbulence and impeller rotation respectively. The simulated flowfield was found to be in agreement with the flow pattern of pitched blade axial-flow impellers that was used for mixing. The leaching tank exhibited good “off-bottom suspension” which reveals minimum deposition of gold ore particles on the bottom of the leaching tanks. Simulation results were consistent with experimental results obtained from a radioactive tracer investigation. CFD approach gave a better description of the flow structure and extent of mixing in a leaching tank. Hence it could be a preferred approach for flow system analysis where the cost of experimentation is high.展开更多
Removal of condensates from wet steam flow in the last stages of steam turbines significantly promotes stage efficiency and prevents erosion of rotors. In this paper, homogeneous spontaneous condensation in transonic ...Removal of condensates from wet steam flow in the last stages of steam turbines significantly promotes stage efficiency and prevents erosion of rotors. In this paper, homogeneous spontaneous condensation in transonic steam flow in the 2-D rotor-tip section of a stage turbine is investigated. Calculated results agree with experimental data reasonably well. On the basis of the above work, a 2-D numerical simulation of wet steam flow in adjacent root sections of a complex steam turbine stage was carded out. Computational results were analyzed and provide insights into effective removal of humidity.展开更多
This work applies the Eulerian-Eulerian model in conjunction with the kinetic theory of granular flow and inter-particle collisions to study the characteristics of pressure recovery for a dilute gas-particle suspensio...This work applies the Eulerian-Eulerian model in conjunction with the kinetic theory of granular flow and inter-particle collisions to study the characteristics of pressure recovery for a dilute gas-particle suspension flowing through a pipe with a sudden expansion at a relatively higher solid loading.To do so,the numerical procedure was validated against experimental results for relatively small dilute phase flows in a pipe with a sudden expansion;a satisfactory agreement was obtained.Initially,the effect of the two significant numerical parameters namely,the speculairty coefficient and the coefficient of restitution for particle-particle collisions were investigated and then,for fixed combinations of these parameters the effect of particle-phase volume fraction,particle density,particle size,and inlet slip ratio were studied.It was concluded that pressure recovery diminishes as the speculairty coefficient increases and the particle-particle coefficient of restitution decreases.Pressure recovery was found to increase with increasing solid volume fraction for coarse particles whereas,for fine particles,a critical volume fraction is obtained above which pressure recovery reduces.An increase in slip ratio is found to diminish pressure recovery.Similarly,at relatively higher solid volume fractions,a critical diameter was found,above and below which pressure recovery decreases.展开更多
The validity of single-fluid model in inertial confinement fusion simulations is studied by comparing the results of the multi- and single-fluid models. The multi-fluid model includes the effects of collision and inte...The validity of single-fluid model in inertial confinement fusion simulations is studied by comparing the results of the multi- and single-fluid models. The multi-fluid model includes the effects of collision and interpenetration between fluid species. By simulating the collision of fluid species, steady-state shock propagation into the thin DT gas and expansion of hohlraum Au wall heated by lasers, the results show that the validity of single-fluid model is strongly dependent on the ratio of the characteristic length of the simulated system to the particle mean free path. When the characteristic length L is one order larger than the mean free path A, the single-fluid model's results are found to be in good agreement with the multi-fluid model's simulations, and the modeling of single-fluid remains valid. If the value of L/A is lower than 10, the interpenetration between fluid species is significant, and the single-fluid simulations show some unphysical results; while the multi-fluid model can describe well the interpenetration and mix phenomena, and give more reasonable results.展开更多
In this study,a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed.Two new power-law liquid–solid dra...In this study,a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed.Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop.One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore,and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations.Predictions are compared with experimental data measured by Lali et al.,where the computed porosities from model B are closer to the measured data than other models.Furthermore,the predicted pressure drop rises as liquid velocity increases,while it decreases with the increase of particle size.Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient,and particle concentration,granular temperature,granular pressure,and granular viscosity go down accordingly.展开更多
The two-phase micro-bubble flow over an axisymmetric body is investigated using the OpenFOAM framework.The numerical model consists of an Eulerian-Eulerian two-fluid model with closure relationships for the interfacia...The two-phase micro-bubble flow over an axisymmetric body is investigated using the OpenFOAM framework.The numerical model consists of an Eulerian-Eulerian two-fluid model with closure relationships for the interfacial momentum transfer to capture the multiphase flow,a standard A:-£*model for the continuous phase and one turbulence model inside the OpenFOAM for the dispersed phase.The bubble sizes are calculated based on the solution of the transport equation of the interfacial area density.The simulations in this work are carried out with different air injection rates and different flow velocities.The effects of bubble size on drag reduction are analyzed.The numerical results are compared against some available experiments and other numerical simulations.The numerical results indicate that the airflow rate and air volume fraction within the boundary layer near the body play important roles in micro-bubble drag reduction.The frictional drag reduction effect by micro bubbles is larger for lower water speed,and the presence of the micro bubbles can increase the pressure resistance of the body.Drag reduction rates are generally higher when the bubble diameter is smaller.展开更多
Bubbles with different sizes have different dynamic and kinetic behavior in a two-phase bubbly flow. A common two-fluid model based on the uniform bubble size assumption is not suitable for a bubbly flow with non-unif...Bubbles with different sizes have different dynamic and kinetic behavior in a two-phase bubbly flow. A common two-fluid model based on the uniform bubble size assumption is not suitable for a bubbly flow with non-uniform bubble sizes. To deal with non-uniform bubbly flows, a multi-fluid model is established, with which bubbles are divided into several groups according to their sizes and a set of basic equations is derived for each group of bubbles with almost the same size. Through analyzing the bubble-bubble and bubble-pipe wall interactions, two new constitutive laws for the wall-force and pressure difference between the liquid phase and interface are developed to close the averaged basic equations. The respective phase distributions for each group of bubbles measured by a specially designed three-dimensional photographic method are used to check the model. Comparison between model-predicted values and experimental data shows that the model can describe laminar bubbly flow with non-uniform bubble sizes.展开更多
文摘A multi-fluid blast furnace model was simply introduced and was used to simulate several innovative ironmaking operations. The simulation results show that injecting hydrogen bearing materials, especially injecting natural gas and plastics, the hydrogen reduction is enhanced, and the furnace performance is improved simultaneously. Total heat input shows obvious decrease due to the decrease of heat consumption in direct reduction, solution loss and silicon transfer reactions. If carbon composite agglomerates are charged into the furnace, the temperature of thermal reserve zone will obviously decrease, and the reduction of iron-bearing burden materials will be retarded. However, the efficiency of blast furnace is improved just due to the decrease in heat requirements for solution loss, sinter reduction, and silicon transfer reactions, and less heat loss through top gas and furnace wall. Finally, the model is used to investigate the performance of blast furnace under the condition of top gas recycling together with plastics injection, cold oxygen blasting and carbon composite agglomerate charging. The lower furnace temperature, extremely accelerated reduction rate, drastically decreased CO2 emission and remarkably enhanced heat efficiency were obtained by using the innovative operations, and the blast furnace operation with superhigh efficiency can be realized.
基金the National Natural Science Foundation of China(Nos.51974065 and 52274257)the Open Foundation of State Key Laboratory of Mineral Processing(No.BGRIMMKJSKL-2020-13)the Fundamental Research Funds for the Central Universities(Nos.N2201008 and N2201004).
文摘The Euler-Euler model is less effective in capturing the free surface of flow film in the spiral separator,and thus a Eulerian multi-fluid volume of fluid(VOF)model was first proposed to describe the particulate flow in spiral separators.In order to improve the applicability of the model in the high solid concentration system,the Bagnold effect was incorporated into the modelling framework.The capability of the proposed model in terms of predicting the flow film shape in a LD9 spiral separator was evaluated via comparison with measured flow film thicknesses reported in literature.Results showed that sharp air–water and air-pulp interfaces can be obtained using the proposed model,and the shapes of the predicted flow films before and after particle addition were reasonably consistent with the observations reported in literature.Furthermore,the experimental and numerical simulation of the separation of quartz and hematite were performed in a laboratory-scale spiral separator.When the Bagnold lift force model was considered,predictions of the grade of iron and solid concentration by mass for different trough lengths were more consistent with experimental data.In the initial development stage,the quartz particles at the bottom of the flow layer were more possible to be lifted due to the Bagnold force.Thus,a better predicted vertical stratification between quartz and hematite particles was obtained,which provided favorable conditions for subsequent radial segregation.
基金The National Natural Science Foundation of China under contract No.41206058the National Science and Technology Major Project of China under contract No.2011ZX05056-001-02the Scientific Research Fund of the First Institute of Oceanography,State Oceanic Administration under contract No.GY0213G25
文摘A large number of submarine landslides with different scales have been identified in the canyon area of the submarine pipeline route of Liwan 3-1 gas field. There is still much chance that submarine slope failures would happen, and the following mass movement would present great risk to the submarine pipeline. In view of this, a numerical prediction method based on Eulerian-Eulerian two-phase flow model is introduced to simulate the mass movement of potential submarine landslides. The sliding soil and ambient water are respectively simulated by Herschel-Bulkley rheology model and Newtonian fluid model. The turbulence is simulated using the k-e model. Compared with both the experiment data and Bing result, the two-phase flow model shows a good accuracy, and its result is more close to the actual situation; the dynamic coupling between soil and ambient water can be effectively simulated and the phenomena of hydroplaning and head detachment can be obtained. Finally, the soil movement of a potential submarine landslide is simulated as an example, according to the seismic profile in the canyon area. The result shows that the hydroplaning occurs during the movement process. The runout distance calculated by the two-phase flow model is 877 m, which is 27.1% larger than the Bing result. However, the peak front velocity of soil is relative small, with a maximum value of 8.32 m/s. The Bing program with a simple and rapid process can be used for a preliminary evaluation, while the two-phase flow model is more appropriate for an accurate assessment.
文摘Based on the two-phase model (Eulerian-Eulerian model), the three dimensional fluid flow in water and that liquid steel systems stirred by one or two multiple gas jets are simulated. In the Eulerian-Eulerian two-phase model, the gas and the liquid phase are considered to be two different continuous fluids interacting with each other through the finite inter-phase areas. The exchange between the phases is represented by source terms in conversation equations. Turbulence is assumed to be a property of the liquid phase. A new turbulence modification - model is introduced to consider the bubbles movement contribution to and . The dispersion of phases due to turbulence is represented by introducing a diffusion term in mass conservation equation. The mathematical simulation agrees well with the experiment results. The study results indicate that the distance of two nozzles has big effect on fluid flow behavior in the vessel. Using two gas injection nozzles at the half radii of one diameter of the bottom generates a much better mixing than with one nozzle under the condition of the same total gas flow rate.
文摘The reducibility of iron-bearing burdens was emphasized for improving the operation efficiency of blast furnace. The blast furnace operation of charging the burdens with high reducibility has been numerically evaluated using a multi-fluid blast furnace model. The effects of reaction rate constants and diffusion coefficients were investigated separately or simultaneously for clarifying the variations of furnace state. According to the model simulation results, in the upper zone, the indirect reduction of the burdens proceeds at a faster rate and the shaft efficiency is enhanced with the improvement under the conditions of interface reaction and intra-particle diffusion. In the lower zone, direct reduction in molten slag is restrained. As a consequence, CO utilization of top gas is enhanced and the ratio of direct reduction is decreased. It is possible to achieve higher energy efficiency of the blast furnace, and this is represented by the improvement in productivity and the decrease in consumption of reducing agent. The use of high-reducibility burdens contributes to a better performance of blast furnace. More efforts are necessary to develop and apply highreducibility sinter and carbon composite agglomerates for practical application at a blast furnace.
文摘Experimental residence time distribution (RTD) measurement and computational fluid dynamics (CFD) simulation are the best methods to study the hydrodynamics of process flow systems. However, CFD approach leads to better understanding of the flow structure and extent of mixing in stirred tanks. In the present study, CFD models were used to simulate the flow in an industrial gold leaching tank. The objective of the investigation was to characterize the flowfield generated within the tank after process intensification. The flow was simulated using an Eulerian-Eulerian multi-fluid model where the RANS standard kmixture model and a multiple reference frame approach were used to model turbulence and impeller rotation respectively. The simulated flowfield was found to be in agreement with the flow pattern of pitched blade axial-flow impellers that was used for mixing. The leaching tank exhibited good “off-bottom suspension” which reveals minimum deposition of gold ore particles on the bottom of the leaching tanks. Simulation results were consistent with experimental results obtained from a radioactive tracer investigation. CFD approach gave a better description of the flow structure and extent of mixing in a leaching tank. Hence it could be a preferred approach for flow system analysis where the cost of experimentation is high.
文摘Removal of condensates from wet steam flow in the last stages of steam turbines significantly promotes stage efficiency and prevents erosion of rotors. In this paper, homogeneous spontaneous condensation in transonic steam flow in the 2-D rotor-tip section of a stage turbine is investigated. Calculated results agree with experimental data reasonably well. On the basis of the above work, a 2-D numerical simulation of wet steam flow in adjacent root sections of a complex steam turbine stage was carded out. Computational results were analyzed and provide insights into effective removal of humidity.
文摘This work applies the Eulerian-Eulerian model in conjunction with the kinetic theory of granular flow and inter-particle collisions to study the characteristics of pressure recovery for a dilute gas-particle suspension flowing through a pipe with a sudden expansion at a relatively higher solid loading.To do so,the numerical procedure was validated against experimental results for relatively small dilute phase flows in a pipe with a sudden expansion;a satisfactory agreement was obtained.Initially,the effect of the two significant numerical parameters namely,the speculairty coefficient and the coefficient of restitution for particle-particle collisions were investigated and then,for fixed combinations of these parameters the effect of particle-phase volume fraction,particle density,particle size,and inlet slip ratio were studied.It was concluded that pressure recovery diminishes as the speculairty coefficient increases and the particle-particle coefficient of restitution decreases.Pressure recovery was found to increase with increasing solid volume fraction for coarse particles whereas,for fine particles,a critical volume fraction is obtained above which pressure recovery reduces.An increase in slip ratio is found to diminish pressure recovery.Similarly,at relatively higher solid volume fractions,a critical diameter was found,above and below which pressure recovery decreases.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11105013,10935003,11275031,11205017,and11075023,the National Basic Research Program of China under Grant No.2013CB834110,the National High-Tech R&D Program(863 Program)under Grant No.2012AA01A303
文摘The validity of single-fluid model in inertial confinement fusion simulations is studied by comparing the results of the multi- and single-fluid models. The multi-fluid model includes the effects of collision and interpenetration between fluid species. By simulating the collision of fluid species, steady-state shock propagation into the thin DT gas and expansion of hohlraum Au wall heated by lasers, the results show that the validity of single-fluid model is strongly dependent on the ratio of the characteristic length of the simulated system to the particle mean free path. When the characteristic length L is one order larger than the mean free path A, the single-fluid model's results are found to be in good agreement with the multi-fluid model's simulations, and the modeling of single-fluid remains valid. If the value of L/A is lower than 10, the interpenetration between fluid species is significant, and the single-fluid simulations show some unphysical results; while the multi-fluid model can describe well the interpenetration and mix phenomena, and give more reasonable results.
基金This work was supported by the National Natural Science Foundation of China(grant No.51876032)Natural Science Foundation of Heilongjiang Province(grant No.ZD2019E002).
文摘In this study,a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed.Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop.One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore,and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations.Predictions are compared with experimental data measured by Lali et al.,where the computed porosities from model B are closer to the measured data than other models.Furthermore,the predicted pressure drop rises as liquid velocity increases,while it decreases with the increase of particle size.Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient,and particle concentration,granular temperature,granular pressure,and granular viscosity go down accordingly.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.51679037,51639003)the National Basic Research Development Program of China(973 Program,Grant No.2013CB036101).
文摘The two-phase micro-bubble flow over an axisymmetric body is investigated using the OpenFOAM framework.The numerical model consists of an Eulerian-Eulerian two-fluid model with closure relationships for the interfacial momentum transfer to capture the multiphase flow,a standard A:-£*model for the continuous phase and one turbulence model inside the OpenFOAM for the dispersed phase.The bubble sizes are calculated based on the solution of the transport equation of the interfacial area density.The simulations in this work are carried out with different air injection rates and different flow velocities.The effects of bubble size on drag reduction are analyzed.The numerical results are compared against some available experiments and other numerical simulations.The numerical results indicate that the airflow rate and air volume fraction within the boundary layer near the body play important roles in micro-bubble drag reduction.The frictional drag reduction effect by micro bubbles is larger for lower water speed,and the presence of the micro bubbles can increase the pressure resistance of the body.Drag reduction rates are generally higher when the bubble diameter is smaller.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19572042).
文摘Bubbles with different sizes have different dynamic and kinetic behavior in a two-phase bubbly flow. A common two-fluid model based on the uniform bubble size assumption is not suitable for a bubbly flow with non-uniform bubble sizes. To deal with non-uniform bubbly flows, a multi-fluid model is established, with which bubbles are divided into several groups according to their sizes and a set of basic equations is derived for each group of bubbles with almost the same size. Through analyzing the bubble-bubble and bubble-pipe wall interactions, two new constitutive laws for the wall-force and pressure difference between the liquid phase and interface are developed to close the averaged basic equations. The respective phase distributions for each group of bubbles measured by a specially designed three-dimensional photographic method are used to check the model. Comparison between model-predicted values and experimental data shows that the model can describe laminar bubbly flow with non-uniform bubble sizes.