7 Implementation of Developed Models and Results
7.1 General Approach
7.1.2 Convergence
Convergence was assessed based on the level of residual drop, monitoring outlet values and water evaporation rate. Residuals decrease below 10−3 is assumed as satisfactory and by monitoring outlet values and water evaporation rate is confirmed that the solution reached steady state solution.
48 7.1.3 Radiation Influence
In 7.1.1 is stated that radiation is not neglected due to its significance to temperature field distribution. For purpose of this work, this assumption was confirmed by comparing temperature fields of simulations with and without radiation of identical conditions - Figure 14 and Figure 15 respectively. Titles of Figure 14, Figure 15 and Figure 16 suggest type of particular regime and boundary conditions, those terms are explained in 7.1.5.
The result of the comparison can be seen in Figure 16, from which might be concluded that the temperature field of simulation with radiation is significantly warmer. Apart the effect of warmer temperature field should be noted that the character of temperature field is different when the radiation is not considered – top area of the temperature field is significantly colder.
Later in the text are shown experimentally measured data (Figure 28, Figure 31, Figure 34, Figure 42, Figure 45 and Figure 48) which confirms that the top area is influenced by the radiative heat transfer.
7.1.4 Mesh Generation
Mesh was created using available meshers in STAR-CCM+. Polyhedral cells were chosen since they provide a balanced solution for complex mesh generation problems and polyhedral mesh contains approximately five times fewer cells than tetrahedral mesh [13].
Prism layers were created in order to accurately simulate the boundary layer. For heat transfer involved problems might be a definition of boundary layer crucial. Volume mesh above the water film is refined to enhance better accuracy.
For both models different meshes were used. The setting of each of them is described in 7.2.1 and 7.3.1.
7.1.5 Boundary Conditions
Figure 17 describes parts of the computational domain and Table 1 how boundary conditions are defined within the computational domain.
Table 1 – Definition of boundary conditions.
Part of the computational domain Boundary condition definition
• Inlet Mass flow inlet
• Walls Non-adiabatic walls
• Fluid film Wall of static temperature (Post-processing Model) / Wall of static temperature and species source (Fick’s Law Model)
• Outlet Pressure outlet
49
Figure 14 – Temperature field of simulation with radiation for boundary conditions BC2 (Post-processing Models).
Figure 15 - Temperature field of simulation without radiation for boundary conditions BC2 (Post-processing Models).
Figure 16 – Simulation with and without radiation temperature field difference for boundary condition BC2 (Post-processing Models).
50 As it is stated in Table 1 a heat transfer through walls is considered. The test rig is manufactured from different materials; therefore, the heat transfer coefficient distribution is not constant. The distribution of heat transfer coefficient along the test rig shows Figure 18.
Figure 17 – Parts of the computational domain.
Figure 18 – Heat transfer coefficient along the test rig.
7.1.6 Initialization
Simulations are initialized to ambient conditions. That means properties of a continuum in the domain are for zero iteration same as ambient.
51 7.1.7 Results Comparison
Simulation results are compared with the experimental measurements from which are the boundary conditions determined. The validity of simulations is confirmed based on a comparison of water evaporation rate, temperature field and outlet temperature in the case of the Post-processing Models. Additionally, the outlet humidity is compared in case of the Fick’s Law Model. In the case of Post-processing Models comparing the outlet humidity is not possible since the fluid film is assumed only as a heat source.
Table 2 shows boundary conditions and experimental data for three data sets. These data sets were used to compare simulation results of water evaporation rate, outlet temperature and outlet humidity to experimental data.
Table 2 - Three data sets of experimental data determining boundary conditions for simulations and data used for simulation validity assessment based on water evaporation rate, outlet temperature and outlet humidity
comparison.
Parameters / Data set BC1 BC2 BC3
Boundary conditions
Reynolds number 6430.72 6290.36 6412.09
Grashof number 4.08E+09 3.26E+09 2.79E+09
Richardson number 98.73 82.49 67.97
Rayleigh number 5.33E+09 4.26E+09 3.65E+09
Prandtl number 1.3061 1.3074 1.3076
Schmidt number 0.4898 0.5086 0.5200
𝑃𝑎𝑚𝑏 [Pa] 98490 99200 99200 temperature. Dimensionless temperature is in this work defined as follows
𝜃 = 𝑇 − 𝑇𝑆
𝑇∞− 𝑇𝑆 . (7.1)
Outlet water vapour mass fraction 𝜔𝑉𝑜𝑢𝑡 represents the outlet humidity. Water evaporation rate is quantified as a permil of the inlet mass flow rate of the dry air, this relates the evaporative process with the intensity of the inlet flow.
52 Table 3 and Table 4 shows boundary conditions for simulations used for temperature fields comparison. In Post-processing Models is the water film assumed only as a heat source unlike to the Fick’s Law Model, where the fluid film is defined as heat and species source. The same approach was also followed in experimental measurement – two regimes of temperature measurement were proceeded: “dry” and “moist”. In the case of dry regime (Table 4), there is only heat transfer considered unlike to moist regime (Table 3), where heat and mass transfer took place simultaneously.
Temperature fields are compared in the centre longitudinal plane of height H, which corresponds to the mixing area height, and of length L, which corresponds to the fluid film length. In the case of simulation, temperature fields of three planes are averaged to cover the geometrical characteristic of temperature sensors used during experimental measurement.
Figure 19 illustrates described approach.
Figure 19 – Position of three longitudinal temperature fields in the simulation.
In tables Table 2, Table 3 and Table 4 are the boundary conditions mostly defined using dimensionless numbers (Re, Gr, Ri, Ra, Pr and Sc). For Re and Gr is characteristic length L considered as the length of the fluid film. For Gr 𝑇𝑠 and 𝑇∞ represent water film and ambient temperatures, respectively.
Another parameter used to define boundary conditions is ambient pressure 𝑝𝑎𝑡𝑚. Inlet vapour mass fraction 𝜔𝑉𝑖𝑛represents the inlet humidity. In STAR-CCM+ the species ratio is defined using the vapour mass fraction, therefore it is convenient to employ its usage instead of specific humidity. 𝑚𝑖𝑛̇ is the inlet mass flow of dry air and Δ𝑇 is defined as Δ𝑇 = 𝑇𝑆− 𝑇∞ and represents amount of temperature potential. For all boundary conditions listed in tables Table 2, Table 3 and Table 4 are valid assumptions described in 5.1.1.
53
Table 3 - Three data sets of experimental data (moist regime) determining boundary conditions for simulations used for simulation validity assessment based temperature field comparison.
Parameters / Data set BC4 BC5 BC6
Boundary conditions
Reynolds number 6690.88 6544.46 6549.31
Grashof number 5.46E+09 5.19E+09 6.37E+09
Richardson number 122.04 121.12 148.57
Rayleigh number 7.21E+09 6.84E+09 8.41E+09
Prandtl number 1.3198 1.3188 1.3208
Schmidt number 0.4802 0.4855 0.4687
𝑃𝑎𝑚𝑏 [Pa] 99790 101000 100000
𝜔𝑉𝑖𝑛 0.0052 0.0057 0.0046
𝑚𝑖𝑛̇ [kg/s] 0.0108 0.0106 0.0105
Δ𝑇 [°C] 36.02 34.13 40.12
Table 4 - Three data sets of experimental data (dry regime) determining boundary conditions for simulations used for simulation validity assessment based on temperature fields comparison.
Parameters / Data set BC7 BC8 BC9
Boundary conditions
Reynolds number 6689.15 6603.00 6723.52
Grashof number 5.44E+09 5.26E+09 6.25E+09
Richardson number 121.57 120.66 138.28
Rayleigh number 7.18E+09 7.18E+09 8.26E+09
Prandtl number 1.3198 1.3194 1.3218
Schmidt number 0.4805 0.4841 0.4699
𝑃𝑎𝑚𝑏 [Pa] 99790 101000 100000
𝜔𝑉𝑖𝑛 0.0052 0.0054 0.0040
𝑚𝑖𝑛̇ [kg/s] 0.0108 0.0107 0.0108
Δ𝑇 [°C] 35.92 34.68 39.75
7.2 Fick’s Law Model
Following part (7.2.1 to 7.2.6) is describing the implementation of the Fick’s Law Model into STAR-CCM+. Chapter 4.10 describes the Fick’s Law Model from the theoretical side, recalling this chapter might be principles of model simplified as:
• The effect of natural convection buoyancy forces is activated using implementation of the Boussinesq approximation into the momentum equation as a momentum source.
54 Boussinesq approximation considers buoyancy forces caused by temperature and solutal differences.
• Water film is defined as heat and species source, in other words, both, heat and mass transfer is simulated.
• Water evaporation rate is evaluated using the implemented field function of STAR-CCM+.
7.2.1 Mesh Generation
Table 5 lists parameters of the mesh which was generated in STAR-CCM+ and used during validity assessment of Fick’s Law Model. Values of other parameters which are not mentioned remain set as default.
Table 5 – Settings of used mesh (Fick’s Law Model).
Parameters Values
• Base Size 0.014 m
• Target Surface Size 80 % of base size
• Minimum Surface Size 10 % of base size
• Surface Growth Rate 1.3
• Number of Prism Layers 8
• Prism Layer Stretching 1.1
• Prism Layer Total Thickness 0.038 m
• Wake Refinement (above fluid film) 50 % of base size
Figure 20 – Water evaporation rate dependency on a number of cells (Fick’s Law Model).
55 The solution mesh dependency test was proceeded. Three meshes with different base size were generated. The dependency of water evaporation rate on a number of cells can be seen in Figure 20, from which results that mesh with 641 349 cells provides efficient convergence.
7.2.2 Boundary Conditions
In the case of Fick’s Law Model, the water film area is defined in STAR-CCM+ as a species source and it is considered as an air-water film interface. The water vapour mass fraction on the air-water film interface 𝜔𝑉 is calculated according to (6.6). Temperature on the interface is assumed same as temperature of the plate which is heating up the water film.
To be able to define the species source in STAR-CCM+, equations (6.6) and (6.7) needs to be reproduced using user defined field functions. Other boundary conditions are defined according to chapter 7.1.
7.2.3 Simulation Approach
Table 6 describes the simulation setting for Fick’s Law Model. Constant density model is used as an Equation of state. Because of using the constant density model, the Boussinesq approximation needs to be involved to simulate the natural convection.
Table 6 – Simulation setting (Fick’s Law Model).
Models Parameters
• Three dimensional
• Steady
• Multi-component gas Gas Components: H2O, Air
• Non-reacting
• Segregated flow (Gradients, Segregated Species)
Convection: 2nd-order (Gradient Method:
Hybrid Gauss-LSQ, Limiter Method:
Venkatakrishnan, Custom Accuracy Level Selector: 2.0, Verbose: False; Convection:
2nd-order)
• Constant density
• Turbulent (Reynolds-Averaged Navier-Stokes)
• k-𝜺 Turbulence (Realizable k-𝜀 Two-Layer, Exact Wall Distance, Two Layer All y+ Treatment, Exact Wall Treatment)
(Convection: 2nd-order, Curvature Correction Option: Off, Two-Layer Type:
Buoyancy Driven(Xu))
• Gravity
• Segregated Fluid Temperature Convection: 2nd-order
• Radiation
• Surface-to-Surface Radiation (View Factors Calculator)
• Gray Thermal Radiation Radiation Temperature: 300.0 K
56 In STAR-CCM+ the Boussinesq model involving the effects of buoyancy force is available.
This model covers only temperature differences, though, and it is applicable to single component gas only. Therefore, momentum source corresponding to (6.5) was developed using C programming. Momentum source is treated as a User coded field function in STAR-CCM+ and the code might be seen in Appendix A.
Implementation of above mentioned momentum source into STAR-CCM+ does not require any knowledge of C programing language from the user when User Field Functions are created according to Table 7.
Table 7 – User field functions definition.
User field functions Description
• TRef Scalar field function defining the reference temperature [K]. Ambient temperature should be considered as a reference.
• MwRef Scalar field function defining the reference mass fraction of H2O. The ambient mass fraction of H2O should be considered as a reference.
7.2.4 Continuum Definition
Continuum is assumed as a multi-component gas of Air and H2O and related mass fractions define the species ratio.
7.2.5 Convergence
Figure 21 presents residual dependency on iteration in the case of Fick’s Law Model.
Residuals decrease satisfies convergence requirement defined in 7.1.2.
Figure 21 – Residuals dependency on iteration (Fick’s Law Model).
Figure 22 shows outlet values and water evaporation rate dependency on iteration, which confirms that simulation reached steady state solution.
57
Figure 22 – Monitoring of outlet values and water evaporation rate dependency on iteration (Fick’s Law Model).
7.2.6 Results
Dependencies presented in Figure 23, Figure 25 and Figure 26 are within ranges of Δ𝑇 ∈ < 20; 34 > °C, 𝑚𝑖𝑛̇ ∈ < 0.0104; 0.0106 > and 𝜔𝑉𝑖𝑛 ∈< 0.0125; 0.0134 > assumed as linearly dependent.
From the comparison of experimental data and simulation results of water evaporation rate (Figure 23) might be concluded that the Fick’s Law Model deviation from the experiment is within 10%.
Comparison of simulation results and experimental data of outlet humidity dependency on inlet humidity (Figure 25) shows good agreement and simulation results deviation is within the range of 10%.
Simulation results of water evaporation rate and outlet humidity are higher for all boundary conditions compared to the experimental data. This results from the fact, that the temperature on air-water film interface is assumed same as the temperature of aluminium plate heating up the water film. In reality, the temperature at air-water film interface is slightly lower due to the process of vaporisation. This is illustrated in Figure 24.
Figure 26 compares the outlet temperature dependency on the temperature difference.
Outlet temperature is in dimensionless form. For boundary conditions BC1 and BC2, the simulation results and experimental data are within 20% deviation. In the case of boundary condition BC3, the deviation of simulation result from experiment data is higher than 20%.
58
Figure 23 – Water evaporation rate dependency on temperature difference (𝑇𝑆− 𝑇𝑎𝑚𝑏) (Fick’s Law Model).
Figure 29, Figure 32 and Figure 35 show differences of temperature fields experimentally measured and simulated. After comparing experimentally measured and simulated temperature fields for each boundary conditions, it might be concluded that the differences are within an acceptable error and distribution of temperature fields are in good agreement. In top and bottom areas simulation and experiment are slightly different. Simulation is warmer in the top area and colder in bottom area for all three boundary conditions. The warmer area is most likely caused by overestimated value of emissivity and colder area by underestimated value of reflectivity of water film area in simulation settings.
Figure 24 – Scheme of temperature distribution on a plate, water film and air-water film interface.
59
Figure 25 – Outlet vapour mass fraction dependency on an inlet vapour mass fraction (Fick’s Law Model).
Figure 26 – Outlet temperature dependency on temperature difference (𝑇𝑆− 𝑇𝑎𝑚𝑏) (Fick’s Law Model).
60
Figure 27 – Simulation temperature field for boundary conditions BC4 (Fick’s Law Model).
Figure 28 - Experiment temperature field for ambient conditions BC4.
Figure 29 – Simulation (Fick’s Law Model) and experiment temperature fields difference for boundary/ambient conditions BC4.
61
Figure 30 - Simulation temperature field for boundary conditions BC5 (Fick’s Law Model).
Figure 31 - Experiment temperature field for ambient conditions BC5.
Figure 32 – Simulation (Fick’s Law Model) and experiment temperature fields difference for boundary/ambient conditions BC5.
62
Figure 33 - Simulation temperature field for boundary conditions BC6 (Fick’s Law Model).
Figure 34 - Experiment temperature field for ambient conditions BC6.
Figure 35 – Simulation (Fick’s Law Model) and experiment temperature fields difference for boundary/ambient conditions BC6.
63
7.3 Post-processing Models
Following part (7.3.1 to 7.3.6) is describing the implementation of the Post-processing Models into STAR-CCM+. It should be remained that Post-processing Models are divided into two sub-approaches: Heat Transfer Analogy Based Model and Lewis Factor Analogy Based Model. In chapter 6.2 were presented the Post-processing Models principles from the theoretical side. Main features of this approach might be recalled as follows:
• The buoyancy caused by natural convection is simulated via defining the user defined equation of state (STAR-CCM+: User Defined EOS). The equation of state employs density variation due to the temperature difference, therefore the advection of the bulk flow is involved. To achieve this, equation (5.1) is adapted.
• Water film is defined as a heat source. Only heat transfer is simulated.
• Water evaporation rate evaluation is based on analogies presented in 6.2.1 and 6.2.2.
7.3.1 Mesh Generation
Table 8 describes parameters of mesh which was generated in STAR-CCM+ and used for final evaluation of Post-processing Models. Values of other parameters which are not mentioned in this table remain set as default.
Table 8 – Setting of used mesh (Post-Processing Models).
Parameters Values
• Prism Layer Total Thickness 0.05175 m
• Wake Refinement (above fluid film) 30 % of base size
Since evaluation of the heat transfer coefficient is very important for Post-processing Models, the mesh used is much finer than in the case of the Fick’s Law Model. Because of the turbulent modelling involved, there is an additional requirement to size the extrusion layer to achieve 𝑦+≈ 1. The solution dependency on mesh was analysed. Four meshes with different base size were generated. Dependency of water evaporation rate on number of cells can be seen on Figure 36, from which results that mesh with 1 861 742 cells provides convergence.
64 7.3.2 Boundary Conditions
The water film area is defined as a wall of a given temperature. The temperature is same as the temperature of the plate which is heating up the fluid film. Other boundary conditions are defined in the same way as it is described in 7.1.
Figure 36 – Water evaporation rate dependency on a number of cells (Post-processing Models).
7.3.3 Simulation Approach
Table 9 lists the simulation setting in the case of Post-processing Models. Instead of Constant Density model User Defined EOS model is used. User Defined EOS model allows the user to define arbitrary equation of state of the gas and in the case of Post-processing Models, the equation of state was defined as (5.1). When using the equation (5.1) it is necessary to define the water vapour mass fraction 𝜔𝑉. The value of water vapour mass fraction 𝜔𝑉 in (5.1) is defined as equivalent to inlet mass fraction of water vapour 𝜔𝑉𝑖𝑛 corresponding each boundary condition.
It should be noted that other properties of a single-component gas are defined in the same way as density. In other words, single-component gas properties are defined for a mixture of inlet humidity. This is done since values of density, specific heat capacity and dynamic viscosity influence evaluation of heat transfer coefficient.
The assumption, that the medium in computational domain is incompressible, reduces the overall computational time. This assumption is applicable since differences of absolute pressure are very small.
65 Evaluation of evaporation rate is employed using two sub-approaches Heat Transfer Analogy Based Model and Lewis Factor Analogy Based Model implemented via User Defined Field Functions. Their principles were presented in chapters 6.2.1 and 6.2.2.
Table 9 – Simulation setting (Post-processing Models).
Models Parameters
• Three dimensional
• Steady
• Gas Gas Component: Air
• Non-reacting
• Segregated flow (Gradients) Convection: 2nd-order (Gradient Method:
Hybrid Gauss-LSQ, Limiter Method:
Venkatakrishnan, Custom Accuracy Level Selector: 2.0, Verbose: False; Convection:
2nd-order)
• User Defined EOS Compressible: deactivated
• Turbulent (Reynolds-Averaged Navier-Stokes)
• k-𝜺 Turbulence (Realizable k-𝜀 Two-Layer, Exact Wall Distance, Two Layer All y+ Treatment, Exact Wall Treatment)
(Convection: 2nd-order, Curvature Correction Option: Off, Two-Layer Type:
Buoyancy Driven(Xu))
• Gravity
• Segregated Fluid Temperature Convection: 2nd-order
• Radiation
• Surface-to-Surface Radiation (View Factors Calculator)
• Gray Thermal Radiation Radiation Temperature: 300.0 K
7.3.4 Continuum Definition
Continuum is assumed as a single-component gas, which properties are defined as it would be a mixture of dry air and water vapour of inlet humidity.
7.3.5 Convergence
Figure 37 presents residual dependency on iteration in the case of Post-processing Models.
Residuals decrease satisfies convergence requirement defined in 7.1.2.
Figure 38 shows outlet values and water evaporation rate dependency on iteration, which confirms that simulation reached steady state solution.
66
Figure 37 - Residuals dependency on iteration (Post-processing Models).
Figure 38 - Monitoring of outlet values and water evaporation rate dependency on iteration (Post-processing Models).
7.3.6 Results
Dependencies presented in figures Figure 39 and Figure 40 are for ranges of Δ𝑇 ∈ <
20; 34 > °C, 𝑚𝑖𝑛̇ ∈< 0.0104; 0.0108 > and 𝜔𝑉𝑖𝑛 ∈ < 0.0125; 0.0134 > assumed as linearly dependent.
Figure 39 compares results of experimental measurement and results of Post-processing Models (Heat Transfer Analogy Based Model and Lewis Factor Analogy Based Model). It can be seen that results of Post-processing Models are within an acceptable difference from the experiment.
67 Comparing both sub-approaches, the Lewis Factor Analogy Based Model differs less from the experiment.
Figure 39 – Water evaporation rate dependency on temperature difference (Post-processing models).
Figure 40 presents the outlet temperature dependency on the temperature difference.
Simulation results compared to experimental data are in good agreement. Because of the satisfactory agreement, it might be stated that the evaluation of outlet temperature is not significantly influenced by the assumption of the water film only as a heat source. Assuming the water film area only as a heat source means, that the thermal conductivity of the continuum is constant, which in this case does not influence the evaluation of the outlet temperature.
Figure 41, Figure 44 and Figure 47 show longitudinal temperature fields evaluated by simulation in the case of Post-processing Models. Figure 42, Figure 45 and Figure 48 present longitudinal temperature fields experimentally measured. Simulated and experimentally measured temperature fields are compared in figures Figure 43, Figure 46 and Figure 49. From
Figure 41, Figure 44 and Figure 47 show longitudinal temperature fields evaluated by simulation in the case of Post-processing Models. Figure 42, Figure 45 and Figure 48 present longitudinal temperature fields experimentally measured. Simulated and experimentally measured temperature fields are compared in figures Figure 43, Figure 46 and Figure 49. From