7.3 Grid creation
The general guideline for sufficient amount of volume cells is given by Nielsen [69] (derived originally from German standard VDI 6019) in Eq. (13).
N=44.4 x103𝑉0,38 Eq.(13)
Where: N is number of necessary grid cells and V is room volume (59.4𝑚2). The resulting number of required grid cells yields to 209 615.
The hexahedral grid was created by use of sweep method with final amount of grid cells of 4 354 482. The maximum skeweness of our model was 0.3 which is considered to be very good according to Nielsen [69].
Average value of skeweness was smaller than 0.00045. This is the foundation for the good CFD model.
The necessary amount of cells in the model also depends on position within the model. The areas with quick changes in velocity, temperature or pressure require more precise mesh with more cells to be able to properly model those high gradients. At least 5-10 grid points are needed in areas where some physical changes take place, for example close to the boundaries. For that reason the grid was refined in CFD model in areas close to the wall surfaces. The typical recommended grid distance in the space of ventilated rooms is 10cm for smaller rooms and 30 cm in bigger rooms [69]. Grid distance in developed CFD model was 2.5 cm, which is much more detailed then required. The main reason for this was to get precise CFD predictions around the occupants, which were the main source of movement in the room.
The independence of the grid was investigated by creating four grids with different cell counts. The investigations started with rather coarse grid with 645,432 cells, which is already three times more compare to the minimum requirement according to Nielsen [69]. Final mesh had 4,354,482 cells, see Table 1. The results for each case were taken after full convergence was achieved, meaning that the solution reached stable conditions and energy and mass were conserved. By comparing different variables at certain locations in models with different amount of grid cells, we can elaborate on grid independence. The deviations in results of calculated temperatures and velocities for horizontal plane at height 1.1 m were minor. Based on this finding the solution can be regarded as grid independent, which is the precondition of successful validation of CFD model.
Table 1: Study of grid independence
Grid 1 Grid 2 Grid 3 Grid 4
Grid cells amount [pcs] 645,432 1,030,544 1,964,820 4,354,482 Average air temperature at 1.1 m [°C] 24.2 24.1 24.2 24.3 Average air temperature at
exhaust [°C] 24.5 24.6 24.6 24.6
Average velocity at 1.1 m [m/s] 0.073 0.069 0.073 0.079
Heat flux imbalance [W] 12.6 2.8 8.7 3.3
Heat flux imbalance [%] 1.2 0.3 0.9 0.3
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7.4 Turbulence modeling
When a diffuse ceiling inlet is used in ventilated room, the air is often moved due to natural convection, especially in close proximity to walls and heat sources. The Boussinesq model was used to account for buoyancy forces. Boussinesq model assumes that the density of fluid is the same throughout the test room, instead of being a function of the temperature. This simplification could be made in our model, since variations in temperature within the investigated domain were rather small. Temperature was then specified as average temperature in the calculated domain. The buoyant model was defined by gravity-acceleration vertical component with value of 9.81 m/𝑠2.
7.4.1 Steady-state turbulence modeling
The steady-state approach to turbulent modeling was taken during numerical simulations in Paper III when wall radiant cooling system has been activated. The semi-empirical two equation k- ξ model was used to account for the effects of the turbulence. Two additional equations k and ξ were solved besides mass, energy and momentum equations (as described in section 7.1). RNG type of k-ξ model was used as it partly accounts for low Reynolds number effects when combined with optimal near wall treatment [70]. The k- ξ model works well in fully developed turbulent flows and is therefore suitable for free stream areas of domain but has its limitations in areas close to the wall regions and very often yields in a high wall shear stress and high heat transfer rates [71]. It is rather important for developed CFD model to solve properly the area close to the wall surfaces because two walls were used for radiant cooling system. As large gradients in momentum and temperature occur in area close to the surfaces, the k-ξ turbulent model needs to be combined with proper near wall treatment. That is a wall function in case of k- ξ turbulent model.
Wall function represents the effect of the wall boundaries by bridging the viscosity affected layers (viscous sub-layer and buffer layer) in calculation and use semi-empirical functions to solve this area [72]. The scalable wall function was applied because it has advantage over other types of wall functions in that there is no limitation on the grid spacing close to the wall surface. The limiting value of 11.067 is used to ensure that the first grid point will always be in the logarithmic profile area [71].
7.4.2 Transient turbulence modeling
The transient approach was taken to model the turbulence in the room during numerical simulations in Paper II. The CFD calculations had difficulties to converge when steady-state turbulence modeling was applied. Large Eddy Simulation method was chosen to account for fluctuation of air in the room. Large Eddy Simulation is using an alternative to the wall function, which is the solution with low Reynolds number turbulence modeling. A fine grid needs to be used in vicinity of wall surface, so the first few nodes can be situated in laminar sub-layer. The grid was refined in developed CFD model in areas close to the wall surfaces in order to allow for proper modeling at this area.
7.5 Porous zone modeling
The pressure drop across a diffuse ceiling inlet was simulated by use of porous zone model in developed CFD model. The diffuse ceiling inlet was modelled in Fluent as fluid domain with added momentum loss equation. The inputs to the porous zone model were based on measured pressure drop across the suspended ceiling. The porosity was set as isotropic with value of 0.15, according to known open area of perforated gypsum plate given by the producer. The momentum loss was defined by viscous and inertial
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resistance coefficients, being 1.306x109 m-2 and 1,185,000 m-1, respectively. Those values were obtained from quadratic regression analysis of measured values of pressure drop and velocity expressed by Eq. (14)
∆p = 35554·v2 + 1181.8·v Eq. (14)
Where: ∆p is measured pressure drop across the porous suspended ceiling and v is velocity of the air flowing through porous suspended ceiling.
7.6 Solver settings
As pressure-velocity coupling the scheme SIMPLE was used, as it is recommended for steady-state calculations [73] Since the flow in the test room and plenum is turbulent, the second order upwind schemes were used for momentum, turbulence and energy spatial discretization as recommended by Fluent manual [73]. The PRESTO! scheme was used for pressure being suitable for modeling of natural convection.
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