Intermediate Approach: Simplified Modelling Methods for Temperature and

A wide variety of models for the intermediate approach exist. Figure 5.4 schematically represents the airflow modelling techniques for the intermediate approach [306]. In ASHRAE RP-1222, Chen

29 As mentioned by Griffith and Chen [295], care should be taken to avoid confusing zone with zonal, where the former refers to traditional building zoning (e.g. TRNFLOW) and the later refers to one category of room airflow models.

and Griffith [307] make a distinction between so-called nodal and zonal models, based on how strictly and how resolved the geometry of the control volumes is defined [295,306].

Figure 5.4: Schematic representation of nodal and zonal airflow modelling techniques (based on a figure of Griffith [306]). The zonal models are further subdivided into thermal-zonal, pressure-zonal and

momentum-zonal models. [Blue = distinction made in grid cells, based on expected flow path].

Nodal models consist of a number of pre-defined nodes in a room, which are connected by flow paths. The first model was developed in 1970 by Lebrun [308] who suggested a nodal model which estimated the thermal stratification in a room heated by a radiator under a cold window.

Subsequently, a series of nodal models were developed and nowadays there are numerous such models. Although some of these types of models can predict the temperature stratification with satisfactory precision, the main limitation is that pre-knowledge of the flow pattern is required [309].

Contrary to nodal models which consist of a predefined network, zonal models refer to air models that use a three-dimensional grid to divide the entire room into a system of control volumes and where no assumptions are needed for airflow direction [305]. The air flow between the gridsurfaces of the cells are calculated based on the corresponding conservation equations. The difference with a CFD model is that the cells are much coarser and the conservation equations are simplified. Depending on the simplifications made, the zonal models can be further subdivided into the pressure-zonal, temperature-zonal and momentum-zonal models [306].

Intermediate: Nodal and Zonal approach

Nodal Thermal-zonal

Momentum-zonal Pressure-zonal

Specialized cells

MODELLING INDOOR CLIMATE IN TALL BUILDINGS 119

Figure 5.5: An overview of the different air flow models found in literature (non- exhaustive): narrow scope to buildings.

Pressure-zonal, temperature-zonal and momentum-zonal models

The starting point for these types of zonalmodels are the Navier-Stokes equations which are then simplified. The Navier-Stokes equations (eq. 5.1) include time-dependent equations for momentum and togetherter with the continuity and energy equation they govern the motion of a Newtonian compressible fluid.

ρDv

Dt= −∇p + fvisc+ Fbody (5.1)

Where the left hand-side describes two kinds of acceleration, consisting of time-dependent and advective effects. The right-hand side contains body forces (F), like gravity, and surface forces;

one due to pressure (p) and a second due to viscous stresses (fvisc).

The pressure-zonal models predict the airflow between cells based on pressure differences and specific flow-pressure equations. Pressure-zonal models solve the mass and energy conservation equations, but the momentum equation is not considered [310]. The general idea is that the Navier-Stokes equation is simplified to the Bernoulli equation (eq. (5.2)) and the amount of air exchanged between two zones is calculated bases on an equation, such as the power-law equation (eq. (5.3)), derived from the Bernoulli equation [311]. Over the last decades, a whole

air flow

range of pressure-zonal models were developed. An extended review of pressure-zonal models is provided by Megri and Haghighat [305]. In Figure 5.5 the most important ones are given.

p +1

A common surface between two adjacent cells g gravitational constant [9.81m/s²]

h Height of the cell [m]

Bouia and Dalicieux [31] were the first to use this approach. The original two-dimensional model was later extended to a three-dimensional model by Inard , Dalicieux and Bouia [312]. Two types of cells were defined. First there were the so-called ‘standard cells’. In these cells the flow velocity and thus also the momentum are of low magnitude, allowing for the simplification of the flow being driven by pressure alone. Wurtz [313] improved the method and showed that only the power law (eq. (5.3)) is appropriate to describe air flows in the standard cells [303]. However, when jets, plumes or thermal boundary layers are present, the viscous forces cannot be neglected. One solution was to no longer describe the mass flow in these regions by the standard cells, but by cells with specific equations for jet flow [312] or for thermal plumes [314]. Another solution is to adapt the discharge coefficients [311], as in the work of Teshome and Haghighat [315]. In subsequent years, many researchers developed zonal models. Two well-known models are POMA and CWSZ. POMA (Pressurized zOnal Model with Air-diffuser) was a simplified numerical model developed in 2001 by Haghighat et al. [309] in the framework of Annex 35. CWSZ is a modified version of COMIS to simulate airflow, temperature, and concentration distributions inside a building [316,317].The pressure-based zonal model also has limitations. One criticism is that these models made use of prior knowledge of the rough airflow pattern as expected by the modeller. This holds that the modeller had to define in advance of the simulation study, the type of cell (standard flow, jet flow, plume,…). A change in indoor conditions, e.g. a jet turned off, may change the predefined function of the cell-type [295,318]. A second criticism is that the pressure-based model is unsuccessful in predicting the temperature gradient in large buildings [secondary source: [319]] and in case of forced convection [303,309,320]. In literature, different alternatives are provided for the pressure-based zonal models.

An alternative to the pressure zonal models are the momentum-based zonal models. In contrast to the pressure zonal models, the momentum equation is not neglected. Griffith and Chen [295]

formulated such a model. The velocity and viscosity of the fluid were assumed small enough to treat the fluid as inviscid. The Navier-Stokes equations were reduced to the steady three-dimensional Eüler equation (eq.(5.4)), implying that the frictional forces were neglected. Similar to the model of Griffith and Chen, the VEPZO model [311,318] attempted to avoid the drawbacks of the pressure-zonal models by calculating the acceleration of the airflow between two adjacent

MODELLING INDOOR CLIMATE IN TALL BUILDINGS 121

zones, but compared to the model of Griffith and Chen they also implemented a viscous loss model (eq.(5.5)).

(v∇)v =1

ρ∇P + g (5.4)

v̇ = 1

ρ∇P + ∇(v²) + g∆z

distance between two zones− flosssign(v)v² (5.5)

Another alternative to the pressure zonal models and momentum-zonal models are temperature-zonal models. They use empirical correlations based on temperature differences in combination with mass and energy conservation laws for flows such as jets and plumes [306].

The temperature-zonal model has fewer unknowns and is easier to implement than the pressure-zonal models. It is a good alternative to pressure-pressure-zonal models when only the temperature prediction is required [319]. A widely used temperature-zonal model is based on the model proposed by Togari et al.[66] in 1993 [321–325].

The selected model in this dissertation is also based on the thermal zonal model proposed by Togari et al. [66]. In this model, the airflows along the vertical wall surfaces and supply airstreams are assumed to be the main components of the air movement in a large space. As illustrated in Figure 5.6, this assumption is reasonable for church buildings, where downdraught from cold surfaces, such as tall windows and most tall masonry surface are present [326].

Figure 5.6: Common downdraughts in a church building [326].