International Journal of Heat and Mass Transfer, 2015
The closed-form solution of differential approximation (DA) for radiative transfer in a planar, s... more The closed-form solution of differential approximation (DA) for radiative transfer in a planar, scattering, refractive medium can be derived when the medium is in radiative equilibrium or purely scattering. In this work, linearly and exponentially spatial variations of refractive index are considered. The results obtained by the DA agree well with those obtained by the discrete ordinates method (DOM) and the Monte Carlo method (MCM) for the optically thick cases and the discrepancy between the DA results and the accurate numerical solutions obtained by the DOM and the MCM increases as the optical thickness decreases. The discrepancy between the results obtained by the DA and by the DOM also increases as the gradient of the linear refractive index increases. Besides, the influence of the scattering albedo and the scattering phase function coefficient on the accuracy of the analytical DA solution is less noticeable. Since the internal reflection due to the exponential decay of refractive index increases with the variation range of refractive index, the dimensionless radiative flux decreases with the increase of variation range of refractive index in a purely scattering medium. The present results show that the effect of the internal reflection is stronger in a thinner medium with an exponentially decaying refractive index.
Journal of Materials Processing and Manufacturing Science
A heat transfer model is developed as a furnace-design aid to predict skidmark formation on a ste... more A heat transfer model is developed as a furnace-design aid to predict skidmark formation on a steel slab in a reheating furnace. Quasi-steady two-dimensional heat transfer transverse to the marching direction of the slab in the furnace is considered. By taking account of the heat exchange between the slab and its surroundings in the furnace, including radiant heat transfer among the slab, the skids, the hot gases and the furnace walls; conduction interaction between the slab and the skids; and the gas convection heat transfer in the furnace, the model can predict the temperature distributions in the slab and the skids throughout the reheating process. At the exit of the soaking zone of the furnace, the extent of the skidmark formed on the slab can then be evaluated quantitatively based on the predicted temperature distribution. A comparison is made between the predictions of the present model and the data from an in situ measurement in the furnace, and a reasonable agreement is foun...
ABSTRACT This work applies the discrete-ordinate method (DOM) to study azimuthally dependent radi... more ABSTRACT This work applies the discrete-ordinate method (DOM) to study azimuthally dependent radiative transfer in a two-dimensional cylindrical medium with spatially varying properties. Product angular quadratures are selected to generate discrete-ordinate approximations. The validity of the present DOM scheme is examined by comparing the present results with the results available in the literature. The effects of the optical thickness and the spatially varying extinction coefficient, scattering albedo and coefficients of the phase function are investigated.
Journal of Quantitative Spectroscopy and Radiative Transfer, 2002
First, we apply the modiÿed di erential approximation (MDA) suggested by Chandrasekhar to transie... more First, we apply the modiÿed di erential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P 1=3 approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modiÿed P 1=3 approximation (MP 1=3 A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP 1=3 A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP 1=3 A are more accurate than those obtained by the MDA for most of the cases considered, because the MP 1=3 A corrects the propagation speed of the transmitted radiation.
International Journal of Heat and Mass Transfer, 2009
ABSTRACT In this work, a Monte Carlo method is developed to simulate transient radiative transfer... more ABSTRACT In this work, a Monte Carlo method is developed to simulate transient radiative transfer in a refractive planar medium exposed to a collimated pulse irradiation. The time of flight in closed form is derived for a medium with a linearly varying refractive index. The time-resolved reflectance and transmittance of the slab are obtained by tracing photon bundles and calculating the time of flight. There is a very satisfying correspondence between the present results and the discrete ordinates solutions. The magnitude of numerical uncertainty decreases with the increase of bundles. To simulate transient radiative transfer in an optically thick medium, we need a large number of bundles. The reflectance decreases with the increase of the positive gradient of the refractive index considered. The appearance instant of the transmittance peak postpones as the refractive index increases. Influence of the optical thickness, the scattering albedo and the anisotropically scattering coefficient on the time-resolved reflectance and transmittance is also investigated.
International Journal of Heat and Mass Transfer, 2012
In this work, we derive the integral equations of radiative transfer in terms of intensity moment... more In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n 0) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n 0 for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.
International Journal of Heat and Mass Transfer, 2010
The characteristics of the transient and polarization must be considered for a complete and corre... more The characteristics of the transient and polarization must be considered for a complete and correct description of short-pulse laser transfer in a scattering medium. A Monte Carlo (MC) method combined with a time shift and superposition principle is developed to simulate transient vector (polarized) radiative transfer in a scattering medium. The transient vector radiative transfer matrix (TVRTM) is defined to describe the transient polarization behavior of short-pulse laser propagating in the scattering medium. According to the definition of reflectivity, a new criterion of reflection at Fresnel surface is presented. In order to improve the computational efficiency and accuracy, a time shift and superposition principle is applied to the MC model for transient vector radiative transfer. The results for transient scalar radiative transfer and steady-state vector radiative transfer are compared with those in published literatures, respectively, and an excellent agreement between them is observed, which validates the correctness of the present model. Finally, transient radiative transfer is simulated considering the polarization effect of short-pulse laser in a scattering medium, and the distributions of Stokes vector in angular and temporal space are presented.
International Journal of Heat and Mass Transfer, 1989
... Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC... more ... Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC 27695-7910, USA and BRIAN VICK Mechanical ... MacCormack's predictor-corrector method is used to perform the numerical computations for the hyperbolic equation, while a ...
International Communications in Heat and Mass Transfer, 1985
An exact solution of the Stokes' problem to magnetohydrocynamic for the flow past an infinite ver... more An exact solution of the Stokes' problem to magnetohydrocynamic for the flow past an infinite vertical oscillating plate, is presented. The induced magnetic field is assumed negligible and the effects of free convection and mass transfer is taken into account. It's observed that the skin friction decreases with increasing Sc (Schmidt number), but increases with increasing Gr (Grashof number) or Gm (Modified Grashof number).
This work considers transient conductive and radiative heat transfer in a two-dimensional, cylind... more This work considers transient conductive and radiative heat transfer in a two-dimensional, cylindrical, scattering medium heated or cooled by internal heat source or boundary surface. A finite difference scheme is employed for handling the energy storage and the heat diffusion by conduction, while a discrete-ordinate method is used to analyze the radiative heat transfer. The effects of various parameters, including
A novel passive micromixer with rapid mixing and low pressure drop has been successfully demonstr... more A novel passive micromixer with rapid mixing and low pressure drop has been successfully demonstrated by simulations and micro-fabrication. The structure of this micro-mixer contains one main channel, two sub-channels for injecting the other specie and some baffles in the main channel to form mixing chambers. Adding baffles to the main channel will produce the recirculation to increase the contact
An oblique flat surface model is used lo simulate the rotating roll. The flow regions proposed by... more An oblique flat surface model is used lo simulate the rotating roll. The flow regions proposed by Landau and Levich is utilized in the analysis. The inertia force terms are included and a more realistic 2-dimensional boundary conditions for the free surface is used in this paper. Also, the solution is obtained by utilizing a refinement of integral method. The results are compared with available experimental data. It is shown that the prediction is sufficiently accurate over a wide range of capillary number, and better than the other existing results. The film thickness increase with respect to increasing capillary number up to a certain higher values, from then this tendency slows down and even decreases due to inertia effects
This study aims to investigate the mixing behaviors of deionized (DI) water and carboxymethyl cel... more This study aims to investigate the mixing behaviors of deionized (DI) water and carboxymethyl cellulose (CMC) solutions in C-shaped serpentine micromixers over a wide range of flow conditions. The flow of CMC solutions exhibits shear-thinning behaviors. Numerical simulations are performed to investigate the effects of the mean flow speed, fluid properties and geometry parameters on flow and mixing in the micromixers with the serpentine channel of the same overall channel length. From the results, we can find the following trends. When convection dominates fluid mixing, the curvature-induced vortices enhance fluid mixing effectively. The mixing efficiency of a micromixer consisting of semicircular C-shaped repeating units with a smaller centerline radius is better than that of a micromixer consisting of major segment repeating units with a larger centerline radius. The viscosity of DI water is less than the overall average apparent viscosity of CMC solutions, and so the effect of cur...
In this work, we examine fluid mixing in a full three-stream mixing channel with longitudinal vor... more In this work, we examine fluid mixing in a full three-stream mixing channel with longitudinal vortex generators (LVGs) built on the channel bottom by numerical simulation and experiment. The effects of the asymmetrical arrangement and the attack angle of the LVGs on fluid mixing are investigated. The results show that the micromixer with LVGs at a small asymmetry index (defined by the ratio of the distance from the center plane of the gap between the winglets to the center plane of the main channel to the width of the main channel) is superior to the micromixer with symmetric LVGs and that with LVGs at a large asymmetry index. The micromixer using five mixing modules of the LVGs with an attack angle between 16.5 degrees and 22.5 degrees can achieve excellent mixing over a wide range of Reynolds numbers. Here, we call a section of channel with two pairs of staggered asymmetrical LVGs a mixing module. Besides, the micromixer with LVGs at a small attack angle is more efficient than tha...
In this work, we perform numerical simulation of fluid<br> mixing in a floor-grooved micro-... more In this work, we perform numerical simulation of fluid<br> mixing in a floor-grooved micro-channel with wavy sidewalls which<br> may impose perturbation on the helical flow induced by the slanted<br> grooves on the channel floor. The perturbation is caused by separation<br> vortices in the recesses of the wavy-walled channel as the Reynolds<br> number is large enough. The results show that the effects of the wavy<br> sidewalls of the present micromixer on the enhancement of fluid<br> mixing increase with the increase of Reynolds number. The degree of<br> mixing increases with the increase of the corrugation angle, until the<br> angle is greater than 45 degrees. Besides, the pumping pressure of the<br> micromixer increases with the increase of the corrugation angle<br> monotonically. Therefore, we would suggest setting the corrugation<br> angle of the wavy sidewalls to be 45 degrees.
A numerical study on the heat transfer in the thermal barrier coatings and the substrates of a pa... more A numerical study on the heat transfer in the thermal barrier coatings and the substrates of a parallel-plate enclosure is carried out. Some of the thermal barrier coatings, such as ceramics, are semitransparent and are of interest for high-temperature applications where radiation effects are significant. The radiative transfer equations and the energy equations are solved by using the discrete ordinates method and the finite difference method. Illustrative results are presented for temperature distributions in the coatings and the opaque walls under various heating conditions. The results show that the temperature distribution is more uniform in the interior portion of each coating away from its boundary for the case with a larger average of varying refractive index and a positive gradient of refractive index enhances radiative transfer to the substrates.
International Journal of Heat and Mass Transfer, 2015
The closed-form solution of differential approximation (DA) for radiative transfer in a planar, s... more The closed-form solution of differential approximation (DA) for radiative transfer in a planar, scattering, refractive medium can be derived when the medium is in radiative equilibrium or purely scattering. In this work, linearly and exponentially spatial variations of refractive index are considered. The results obtained by the DA agree well with those obtained by the discrete ordinates method (DOM) and the Monte Carlo method (MCM) for the optically thick cases and the discrepancy between the DA results and the accurate numerical solutions obtained by the DOM and the MCM increases as the optical thickness decreases. The discrepancy between the results obtained by the DA and by the DOM also increases as the gradient of the linear refractive index increases. Besides, the influence of the scattering albedo and the scattering phase function coefficient on the accuracy of the analytical DA solution is less noticeable. Since the internal reflection due to the exponential decay of refractive index increases with the variation range of refractive index, the dimensionless radiative flux decreases with the increase of variation range of refractive index in a purely scattering medium. The present results show that the effect of the internal reflection is stronger in a thinner medium with an exponentially decaying refractive index.
Journal of Materials Processing and Manufacturing Science
A heat transfer model is developed as a furnace-design aid to predict skidmark formation on a ste... more A heat transfer model is developed as a furnace-design aid to predict skidmark formation on a steel slab in a reheating furnace. Quasi-steady two-dimensional heat transfer transverse to the marching direction of the slab in the furnace is considered. By taking account of the heat exchange between the slab and its surroundings in the furnace, including radiant heat transfer among the slab, the skids, the hot gases and the furnace walls; conduction interaction between the slab and the skids; and the gas convection heat transfer in the furnace, the model can predict the temperature distributions in the slab and the skids throughout the reheating process. At the exit of the soaking zone of the furnace, the extent of the skidmark formed on the slab can then be evaluated quantitatively based on the predicted temperature distribution. A comparison is made between the predictions of the present model and the data from an in situ measurement in the furnace, and a reasonable agreement is foun...
ABSTRACT This work applies the discrete-ordinate method (DOM) to study azimuthally dependent radi... more ABSTRACT This work applies the discrete-ordinate method (DOM) to study azimuthally dependent radiative transfer in a two-dimensional cylindrical medium with spatially varying properties. Product angular quadratures are selected to generate discrete-ordinate approximations. The validity of the present DOM scheme is examined by comparing the present results with the results available in the literature. The effects of the optical thickness and the spatially varying extinction coefficient, scattering albedo and coefficients of the phase function are investigated.
Journal of Quantitative Spectroscopy and Radiative Transfer, 2002
First, we apply the modiÿed di erential approximation (MDA) suggested by Chandrasekhar to transie... more First, we apply the modiÿed di erential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P 1=3 approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modiÿed P 1=3 approximation (MP 1=3 A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP 1=3 A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP 1=3 A are more accurate than those obtained by the MDA for most of the cases considered, because the MP 1=3 A corrects the propagation speed of the transmitted radiation.
International Journal of Heat and Mass Transfer, 2009
ABSTRACT In this work, a Monte Carlo method is developed to simulate transient radiative transfer... more ABSTRACT In this work, a Monte Carlo method is developed to simulate transient radiative transfer in a refractive planar medium exposed to a collimated pulse irradiation. The time of flight in closed form is derived for a medium with a linearly varying refractive index. The time-resolved reflectance and transmittance of the slab are obtained by tracing photon bundles and calculating the time of flight. There is a very satisfying correspondence between the present results and the discrete ordinates solutions. The magnitude of numerical uncertainty decreases with the increase of bundles. To simulate transient radiative transfer in an optically thick medium, we need a large number of bundles. The reflectance decreases with the increase of the positive gradient of the refractive index considered. The appearance instant of the transmittance peak postpones as the refractive index increases. Influence of the optical thickness, the scattering albedo and the anisotropically scattering coefficient on the time-resolved reflectance and transmittance is also investigated.
International Journal of Heat and Mass Transfer, 2012
In this work, we derive the integral equations of radiative transfer in terms of intensity moment... more In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n 0) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n 0 for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.
International Journal of Heat and Mass Transfer, 2010
The characteristics of the transient and polarization must be considered for a complete and corre... more The characteristics of the transient and polarization must be considered for a complete and correct description of short-pulse laser transfer in a scattering medium. A Monte Carlo (MC) method combined with a time shift and superposition principle is developed to simulate transient vector (polarized) radiative transfer in a scattering medium. The transient vector radiative transfer matrix (TVRTM) is defined to describe the transient polarization behavior of short-pulse laser propagating in the scattering medium. According to the definition of reflectivity, a new criterion of reflection at Fresnel surface is presented. In order to improve the computational efficiency and accuracy, a time shift and superposition principle is applied to the MC model for transient vector radiative transfer. The results for transient scalar radiative transfer and steady-state vector radiative transfer are compared with those in published literatures, respectively, and an excellent agreement between them is observed, which validates the correctness of the present model. Finally, transient radiative transfer is simulated considering the polarization effect of short-pulse laser in a scattering medium, and the distributions of Stokes vector in angular and temporal space are presented.
International Journal of Heat and Mass Transfer, 1989
... Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC... more ... Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC 27695-7910, USA and BRIAN VICK Mechanical ... MacCormack&amp;amp;#x27;s predictor-corrector method is used to perform the numerical computations for the hyperbolic equation, while a ...
International Communications in Heat and Mass Transfer, 1985
An exact solution of the Stokes' problem to magnetohydrocynamic for the flow past an infinite ver... more An exact solution of the Stokes' problem to magnetohydrocynamic for the flow past an infinite vertical oscillating plate, is presented. The induced magnetic field is assumed negligible and the effects of free convection and mass transfer is taken into account. It's observed that the skin friction decreases with increasing Sc (Schmidt number), but increases with increasing Gr (Grashof number) or Gm (Modified Grashof number).
This work considers transient conductive and radiative heat transfer in a two-dimensional, cylind... more This work considers transient conductive and radiative heat transfer in a two-dimensional, cylindrical, scattering medium heated or cooled by internal heat source or boundary surface. A finite difference scheme is employed for handling the energy storage and the heat diffusion by conduction, while a discrete-ordinate method is used to analyze the radiative heat transfer. The effects of various parameters, including
A novel passive micromixer with rapid mixing and low pressure drop has been successfully demonstr... more A novel passive micromixer with rapid mixing and low pressure drop has been successfully demonstrated by simulations and micro-fabrication. The structure of this micro-mixer contains one main channel, two sub-channels for injecting the other specie and some baffles in the main channel to form mixing chambers. Adding baffles to the main channel will produce the recirculation to increase the contact
An oblique flat surface model is used lo simulate the rotating roll. The flow regions proposed by... more An oblique flat surface model is used lo simulate the rotating roll. The flow regions proposed by Landau and Levich is utilized in the analysis. The inertia force terms are included and a more realistic 2-dimensional boundary conditions for the free surface is used in this paper. Also, the solution is obtained by utilizing a refinement of integral method. The results are compared with available experimental data. It is shown that the prediction is sufficiently accurate over a wide range of capillary number, and better than the other existing results. The film thickness increase with respect to increasing capillary number up to a certain higher values, from then this tendency slows down and even decreases due to inertia effects
This study aims to investigate the mixing behaviors of deionized (DI) water and carboxymethyl cel... more This study aims to investigate the mixing behaviors of deionized (DI) water and carboxymethyl cellulose (CMC) solutions in C-shaped serpentine micromixers over a wide range of flow conditions. The flow of CMC solutions exhibits shear-thinning behaviors. Numerical simulations are performed to investigate the effects of the mean flow speed, fluid properties and geometry parameters on flow and mixing in the micromixers with the serpentine channel of the same overall channel length. From the results, we can find the following trends. When convection dominates fluid mixing, the curvature-induced vortices enhance fluid mixing effectively. The mixing efficiency of a micromixer consisting of semicircular C-shaped repeating units with a smaller centerline radius is better than that of a micromixer consisting of major segment repeating units with a larger centerline radius. The viscosity of DI water is less than the overall average apparent viscosity of CMC solutions, and so the effect of cur...
In this work, we examine fluid mixing in a full three-stream mixing channel with longitudinal vor... more In this work, we examine fluid mixing in a full three-stream mixing channel with longitudinal vortex generators (LVGs) built on the channel bottom by numerical simulation and experiment. The effects of the asymmetrical arrangement and the attack angle of the LVGs on fluid mixing are investigated. The results show that the micromixer with LVGs at a small asymmetry index (defined by the ratio of the distance from the center plane of the gap between the winglets to the center plane of the main channel to the width of the main channel) is superior to the micromixer with symmetric LVGs and that with LVGs at a large asymmetry index. The micromixer using five mixing modules of the LVGs with an attack angle between 16.5 degrees and 22.5 degrees can achieve excellent mixing over a wide range of Reynolds numbers. Here, we call a section of channel with two pairs of staggered asymmetrical LVGs a mixing module. Besides, the micromixer with LVGs at a small attack angle is more efficient than tha...
In this work, we perform numerical simulation of fluid<br> mixing in a floor-grooved micro-... more In this work, we perform numerical simulation of fluid<br> mixing in a floor-grooved micro-channel with wavy sidewalls which<br> may impose perturbation on the helical flow induced by the slanted<br> grooves on the channel floor. The perturbation is caused by separation<br> vortices in the recesses of the wavy-walled channel as the Reynolds<br> number is large enough. The results show that the effects of the wavy<br> sidewalls of the present micromixer on the enhancement of fluid<br> mixing increase with the increase of Reynolds number. The degree of<br> mixing increases with the increase of the corrugation angle, until the<br> angle is greater than 45 degrees. Besides, the pumping pressure of the<br> micromixer increases with the increase of the corrugation angle<br> monotonically. Therefore, we would suggest setting the corrugation<br> angle of the wavy sidewalls to be 45 degrees.
A numerical study on the heat transfer in the thermal barrier coatings and the substrates of a pa... more A numerical study on the heat transfer in the thermal barrier coatings and the substrates of a parallel-plate enclosure is carried out. Some of the thermal barrier coatings, such as ceramics, are semitransparent and are of interest for high-temperature applications where radiation effects are significant. The radiative transfer equations and the energy equations are solved by using the discrete ordinates method and the finite difference method. Illustrative results are presented for temperature distributions in the coatings and the opaque walls under various heating conditions. The results show that the temperature distribution is more uniform in the interior portion of each coating away from its boundary for the case with a larger average of varying refractive index and a positive gradient of refractive index enhances radiative transfer to the substrates.
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Papers by Chih-yang Wu