In this section the simulation results are presented that are obtained for various values of ΔT. These simulations demonstrate that this new method is useful for modelling non-equilibrium and non-equilibrium phase transitions as well. Consequently an non-equilibrium equation-of-state type model can be thought as a special case of the hysteresis model, by applying ΔT =0K.
The transient behaviour of the model system is analysed by applying a space and time varying excitation. The outer wall of the tube is increasingly cooled to the middle of the tube and then the temperature difference gradually decreased to the end of the tube. This effect can be achieved for example with a sinusoidal varying heat transfer coefficientα0
( )
z , α0( )
z =αmaxsin(π z L), where z is the tangential space coordinate, αmaxis the maximum value of the heat transfer coefficient. Besides the space variation, the heat transfer coefficient has been increased and decreased artificially in time whereas the outer temperature has been set to a constant level( ) ( )( )
The dimension of t time was set to second in the simulations, but it has no meaning since the other parameters have distorted values. Therefore in the following the dimension of the time will not be indicated. The time dependent variation of αmax is shown in Fig. 56.
Equilibrium and non-equilibrium phase transitions are allowed to occur in the different test cases. Therefore the results obtained by a simple equation-of-state model (first test case) can be compared to models of hysteresis phase transformations. In this study two examples of hysteresis are presented. In the second test case only a narrow hysteresis band is introduced. This example can represent an increased version of the equation-of-state model, in which the hysteresis is used due to avoid considerable condensation or evaporation to occur before the saturation temperature is reached. In this way effects arising from the symmetrically smoothed jump function, that are not thermodynamically justified, can be prevented. The third test case shows an example for the
supersaturation-superheating process with hysteresis calculated from saturation temperature according to (44). The main hysteresis curves applied in the examples are similar to hysteresis curves in Fig. 53. Initial and boundary conditions as well as the numerical model correspond to the model above.
0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time index αmax10-2 (W/m2 K)
Fig. 56. The time dependent variation of the maximum heat transfer coefficient αmax
Transient simulation has been performed in the same time intervals, t=0L300s, for all test cases. In this study we focus on the temperature induced hysteresis of the vapour mass fraction as the order parameter. The surface-plots of the axially symmetric 2D temperature fields are shown in left column of Fig. 57 and the corresponding vapour mass fractions as order parameter φ are depicted in the right column in this figure. In the temperature surfaces the colours represent the temperature, where blue corresponds to the cold and red to the hot regions respectively, and the arrows show the velocity field at ten equally distanced cross-sections. It can be seen that temperature change is very different for the tree test cases, depending from the selected ΔT. The largest differences are in the dynamics of cool areas appearing near the wall of the tube. The same outer excitation provide more effective heating–cooling dynamics in the case of
K
=10
ΔT . The fields of the order parameter have different colour palette, blue correspond to the low vapour content and pink to the pure vapour phase. The water mass fraction, (1−φ) distributions are well observable in the order parameter field with its blue shades. The most diffuse concentration fields are obtained in the first case study, where no hysteresis has been assumed. The second case study with ΔT =4K corresponds to the hysteretic version of the case study one. It seems that the water phase components are concentrated near the wall in this case. In the third case study, with hysteresis parameter of ΔT =10, the condensation of the vapour phase begins later in time and the smallest water content has been arrived at the same time and same excitations. Contrary, the temperature decreases was the largest in this case. In Fig. 58 the time evolution of the temperature and the mass fraction can be seen in certain points of the flow fields at the cross-section of z=3.5D. The coordinates of radial locations are given in the figure legends. It can be seen that in the equilibrium approximation
constant during the increasing - decreasing cycles of the excitation but the values of vapour mass fraction decrease continuously. The largest is the hysteresis loop, the largest are the temperature variations and less are the condensation rates during the same time intervals.
Fig. 57. Time evolution of the temperature (left) and the vapour mass fraction (right) over the flow field.
Arrows correspond to the velocity. Times slices at t = 50, 100, 150, 200, 250 and 300. Top: without hysteresis, middle: ΔT =4K, bottom: ΔT =10K. The colour bar at bottom left represents the
distribution of the heat transfer coefficients
0 50 100 150 200 250 300
Fig. 58. Time evolution of the temperature and the order parameter close to the wall, at axial location D
.
z=35 , radial locations close to the wall, r =0.09K0.1m, with a) – b) ΔT =0K; c) – d) ΔT =4K and e) – f) ΔT =10K.
Temperature dependent minor loops at several locations inside the tube are shown in Fig. 59. The radial distances are signed in the figure legends. Near the wall the largest loops can be observed, that are shown in Fig. 60 and Fig. 61 for the two test cases. The hysteresis loops in the left hand side of the figures represent the variation of the order parameter with temperature (Fig. 60a and Fig. 61a), and the corresponding density variations are shown in the right hand sides (Fig. 60b and Fig. 61b). It can be seen that by providing intensive cooling-heating across the wall, the minor loops become larger near the wall.
-6 -4 -2 0 2 4 6
Fig. 59. Minor hysteresis loops corresponding to the time-evolution of the mass fraction as order parameter, farther from the wall (see Fig. 54 for geometry), at axial coordinate z=2.5D.
Fig. 60. a) Minor hysteresis loops corresponding to the time-evolution of the mass fraction as order parameter, close to the wall (see Fig. 54 for geometry), b) The average density variation follows the
hysteresis of the mass fraction, at axial coordinate z=3.5D with ΔT =4K
The calculated pressure fields at time step t=300 are presented in Fig. 62 for all the three analyzed test cases. Inside the diffuse interface a pressure ‘hump’ can be observed.
The hump is a manifestation of the capillary stress as a result of the applied interfacial momentum source term due to surface tension in the governing equations (see Appendix C). 0The pressure humps are observable in the pressure fields near the wall, which are well presented in the cross-sections profiles in Fig. 63 as well. If the two phases are fluids, this pressure hump would act to keep the fluids from mixing.
Increasing differences between densities on the two sides of the interface increases the magnitude of the hump, indicating that in the presence of a normal flow, a hump would be present even in the absence of surface tension. The smoothness of the pressure humps depend on the slope of the diffuse interface, therefore the sharpest pressure humps are provided by the case study of ΔT =4K.
-6 -4 -2 0 2 4 6 100
101 102
T - Ts (K) ρ (kg/m3)
z = 0.7
r=0.09 r=0.092 r=0.094 r=0.096 r=0.098 r=0.1
-6 -4 -2 0 2 4 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T - Ts (K)
φ
z = 0.7
r=0.090 r=0.092 r=0.094 r=0.096 r=0.098 r=0.1
a) b)
Fig. 61. a) Minor hysteresis loops corresponding to the time-evolution of the mass fraction as order parameter, close to the wall (see Fig. 54 for geometry), b) The average density variation follows the
hysteresis of the mass fraction, at axial coordinate z =3.5D with ΔT =10K
Fig. 62. Pressure fields at time t=300. a) Without hysteresis, b) ΔT =4K, c) ΔT =10K
The time evolution of the diffuse interface between the two phases can be characterized by the evolution of the liquid volume fraction, that can be seen in left column of Fig. 63, for three different ΔT. The sharpest interfaces have been provided by simulation with ΔT =4K. The highest values of liquid volume fractions comparing to the other two simulations correspond to the better phase separation observed in this model simulation.
Fig. 63. Volume fractions and pressure profiles at cross-section z=2.5D, time t = 50, 100, 150, 200, 250 and 300. Top: without hysteresis, middle: ΔT =4K, bottom: ΔT =10K
2.6 Summary and Conclusion
In the first part of this study I have reviewed the numerical models of the vapour liquid phase transitions that can be used in homogeneous two-phase models. The equation-of-state type models are the numerically easiest approach to describe the phase transition phenomena but they have some thermodynamically not supported simplifications. I have concluded that these models are not able to predict the experimentally observed meta-stable states and therefore overestimate the rate of condensate formation and the resulted heat transfer coefficient. These models allow considerable amount of phase transformations before the local pressure or temperature reaches the saturation
conditions due to numerical reasons. The application of kinetic type phase transition models is restricted to that cases in which the free energy functions and the relaxation time can be defined in advance.
To increase the equation-of-state type equilibrium models by enabling occurrence of metastable states I introduced a new, phenomenological vapour-liquid phase transition model. The model is similar to the equation-of-state type models since it is a function of an intensive state variable (temperature), but in this case this function is hysteretic. The proposed hysteresis function can be easier parameterized than the kinetic type phase transition models. The model can be very useful in several engineering applications, in which an expert could estimate the assumable degree of supersaturation/superheating.
The hysteresis model of the thermally induced phase transformation has been derived from the statistics of the model fluid consists of bistable, constant-mass clusters of molecules. To overcome the flowing system caused memory handling problems of classical Preisach-type hysteresis models, I have used a PDE based hysteresis operator to describe the phase change. To ensure that unstable conditions could be always avoided, I have proposed a saturation temperature dependent upper limit of the allowable supersaturation.
In order to verify the effectiveness of the model, I have implemented it into a 2D finite element model of a transient non-isothermal two-phase flow in a horizontal tube. I have compared two hysteresis test examples with different degrees of supersaturation enabled with a case study of a model without hysteresis. In the presented case studies the film condensation is the assumable phenomenon. The time evolution of the temperature and therefore the presents of condensation are strongly influenced by the supersaturation allowed. The sharpness of the diffuse interface between the phases as well as the thickness of the liquid film evolved depends on the selected model parameter. A model with narrow hysteresis of the order parameter does not mean really non-equilibrium phase transition; this model can be considered as an improvement of a simple equation-of-state type model. The broader hysteresis of the order parameter describes the supersaturation and superheating occurrences in the phase transition. Introducing the proposed hysteresis model of the vapour-liquid phase transition can improve other macro scale heat transfer models as well, that are associated with boiling and condensation phenomena and have been treated until now as virtually isothermal heat transfer processes.
3 Visualisation and Inverse Analysis of IR Images of a Free Turbulent Steam Jet
3.1 Introduction
In this experiment I investigated time series based infrared (IR) images of a free turbulent steam jet. With this study my aim is to verify that thermograms are suitable not only for visualization of the turbulent flow field (after applying correspondent image filters) but to determine the liquid water content in the jet. A study of steam jets is of practical importance in numerous industrial applications. Correct prediction of moisture content in steam flows is both scientifically interesting and of engineering importance. For example: condensing flows of moist air or combustion products, aerosol formation in mixing processes, aerodynamic tests in cryogenic wind tunnels, steam turbines and expansion in vapour nozzles and so on.
Thermography is commonly used in surface flow visualization, but few researchers have visualized the free fluid flow field using IR imaging. The detector signal of the camera is proportional to the IR radiation of the condensed water droplets in the optically thin steam jet. Thermograms provided by the signal processing system of the camera show only fictitious temperature distributions since the detected radiation intensity depends not only on the temperature but also on the concentration of the dispersed radiating components and on the thickness of the jet [95]. In the instantaneous images the jet appears as a semitransparent, diffuse cloud, which only after contrast stretching can be inspected visually, however raw images contain considerable noise as well. Removing the high-frequency noise from the image requires a special smoothing technique, since the observed turbulent phenomenon also contains high frequency components, and its smoothing is not desirable. While commercial image processing systems are unable to further improve these images, I introduced successfully the previously developed fuzzy-rule based diffusion filter on instantaneous images.
Flow visualization is an important tool for investigating turbulent flow [38]. I verified with a special wavelet thresholding technique that large coherent structures can be extracted from the instantaneous images, i.e. the thermograms of the steam jet are useful for visualizing fluid motions. Summation of time series images approximates the time-averaged flow. To determine the minimum required number of images, I introduced an energy content based calculation to prove that the jet properties calculated from averaged images show good correlation with experimental data that can be found in literature.
Estimating the liquid water content from the measured radiation intensity requires a so called inverse analysis. The inverse analysis of radiation in a participating medium has a broad range of engineering applications, for example, remote sensing of the atmosphere,
determining the radiative properties of the medium, and predicting temperature distribution in a flame, and so on. In this study I developed a new iterative inverse radiative transfer algorithm for simultaneous estimation of temperature and absorption coefficient profiles in jet cross-sections. To calculate the optical properties of condensed droplets in my study, a parameterization of the optical properties of spheres in terms of the physical quantities is applied according to [89]. In this way the liquid water content can be estimated. I validated the proposed inverse algorithm by test data obtained from a finite element simulation of the jet.
The corresponding literature had to be reviewed from different aspects according to the complex nature of the presented problem. The first aspect of review concerned the literature of fluid flow measurements and analyses in which infrared thermography has been applied. The second aspect is a survey of the literature of wavelet analyses of 2D images of turbulent phenomena. (Images were obtained through various types of sensing.) The third aspect is the review of solutions of inverse radiation problems related to IR measurements. The fourth aspect concerned with the measurement and simulation of condensing steam jets and some papers from the atmospheric science that describe the properties of water droplets dispersed in the air and that are used as references in the present study. Since even the short presentation of the studied literature needs more space than is available here, I have moved the overview of the literature to Appendix D. In the following I cite the corresponding references with a short description, where needed, of the explanations.