Review on buoyancy-driven natural ventilation in an enclosure with various types of heat sources

. This paper reviews indoor heat convection and buoyancy-driven natural ventilation in enclosed space with heat sources in different forms such as point, line, plane, volume or combination of them. The indoor thermal flow is driven by these heat sources and accumulated in the enclosure. Thermal plume evolves based on its dynamic law above heat sources and is conversely affected by the thermal environment and geometric structure. Therefore, the dynamic of thermal-driven flows and the restriction by the thermal environment and geometric framework are both of interest in the field of indoor heat convection and buoyancy-driven natural ventilation. Based on this fact, the indoor thermal convection can be divided into two basic components which are buoyant plume above the heat source and indoor thermal stratification flow. Research and analysis on these laws and restriction are of significance in not only the advances in building thermal environment technology but also further cognition and effective solutions for current engineering practice.


Introduction
Buoyancy-driven natural ventilation is a field of intense research in both non-industrial and industrial buildings.Indoor thermal flow driven by thermal plumes will be restricted by the geometric framework of buildings and significant thermal stratification can be observed in the enclosure in the vertical direction, which has been gradually found and recognized in the engineering research in the past fifty years [1][2][3][4][5][6][7][8][9][10].A comprehensive consideration on indoor heat convection focused on thermal-driven natural ventilation is of engineering and economic significance to create a suitable environment for occupants and equipment.
The typical model of indoor heat convection is the buoyant plume flow above a localized heat source.Considerable work has been done to investigate the single thermal plume rising from one single heat source [12][13][14].Research on the near-field of a plume is a field of intense research [15][16][17][18][19] as an effort to obtain a comprehensive understanding on the thermal plume development and thermal-driven natural ventilation in the ventilation design for buildings.In the 1960s, scholars in the field of aerodynamics defined the interface height of thermal stratification to characterize the indoor stratified flow.The relationship between the thermal stratification height and the properties of building natural ventilation has been analyzed [2,3,11,20,21].However, the above conclusion is based on the analytical method of point origin, which weakened the accuracy of the model in the practical applications.
Generally, the induced air flow and distribution in nonisothermal room is driven by indoor heat convection and gravitational flow.Under the pure-convection indoor environment, the heat distribution factor or ventilation temperature efficiency can be expressed with a single variable function of thermal stratification height.where m is the heat distribution factor, i t (ºC) is the air temperature in the occupied zone, ow t (ºC) is the outdoor design air temperature for ventilation, e t (ºC) is the exhaust air temperature.The scholars of former Soviet Union [22] first proposed the concept of heat distribution factor.A series of research has been carried out to calculate the heat distribution factor or to provide the approximate estimation.
where 1 m is the coefficient related to the ratio of heat-covered area to the total ground area, 2 m is the coefficient related to the height of heat source, 3 m is the coefficient related to the ratio of radiation heat r Q to the total heat release t Q .In the design process of indoor buoyancy-driven natural ventilation, the heat distribution factor m or the exhaust air temperature plays an important role.Regarding the enclosure containing multiple heat sources, more accurate estimation on the heat distribution factor is an important issue to determine air flow rate more accurately.

Point heat source
Bases on the assumption of self-similar solution by Morton et al. [1], the analytical solution can be deduced to describe the model mathematically.Besides, the temperature and vertical velocity distribution in the radial direction in a turbulent plume are assumed to be Gauss distribution [1,2]: where the Z-axis is positive upwards, r is a radial coordinate, z u (m/s) and z t (℃) are the mean velocity and temperature on the axis respectively, 0 t (℃) is the temperature outside the plume and assumed to be constant outside the plume.m and p are the velocity empirical factor and temperature empirical factor respectively.Based on conservation of volume, momentum and energy, the solutions of this analytical model are as the followings [23]: where Z G (m 3 /s) is the volume airflow rate, 0 B (m 4 /s 3 ) denotes the buoyancy strength of the source, Z is the vertical co-ordinate, ' g (m/s 2 ) is the reduced acceleration of gravity, g (m/s 2 ) is the acceleration of gravity, 0 U (kg/m 3 ) is a reference density.Similar conclusion was drawn by Morton [1]: The formula to calculate ' g is the same on the power of main variables.Z G can also be calculated by the following equation: Based on equations ( 6) and (11), the expression to calculate the volume flux of a point plume can be finally shown as: where C is a constant, which is the function of the velocity empirical factor (m), the temperature empirical factor ( p ), or the entrainment constant for the plume D .

Line, plane and volume heat source
Similar to the analytical model of a point heat source, the solutions of thermal plume induced by a line heat source can be obtained by defining L Q (W/m) as heat strength per length [23]: In the case of a plane heat source, the heated air above the source rises and the surrounding air flows toward the center, making the boundary layer gradually thickened from the edge to the center.Above the center of the plane heat source, upward plume flow is produced because of the separation of boundary layer induced by buoyancy.For the thermal plume of a circular plane, the volume rate Z G can be obtained: For the thermal plume of a rectangular plane, the volume rate Z G can be obtained: Zhao [23] provided an empirical formula to calculate the volume rate of a volume heat source: is the roof or ceiling area of the volume heat source, S (m 2 ) is the total surface area of the volume heat source, s h (m) is the height of the volume heat source.

Indoor thermal stratification 3.1 Emptying-filling box model
The emptying filling box model can provide a simple and useful guideline for ventilation designs and strengthen the concept of turbulent buoyancy stratification in the field of engineering analysis.
Thermal stratification will be produced by buoyant plumes in enclosed spaces.In 1962, Shebie Lev defined the height of thermal stratification by matching the room natural ventilation rate with the volume flow rate of a buoyant plume [25].Linden, et al. [3]  point source in an enclosure which is connected to a homogenous environment via a top opening and a bottom opening.After a single point buoyancy source injects buoyancy fluid for some time, a steady-state interface forms in this room.The dimensionless height [ of the steady-state interface is given by: 18) where A (m 2 ) is the "effective" area of the top and bottom openings of the enclosure, and H (m) is the height difference between the top and bottom openings.The constant C could be calculated by: where D is the plume entrainment constant.It shows that the dimensionless thermal stratification height of the naturally ventilated enclosure is only a function of architecture geometric parameter.The strength of the source and the floor area has no influence on the thermal stratification height.Non-adiabatic walls were studied in the emptyingfilling box model by Lane-Serff and Sandbachand [26], and the effect of heat transfers occurring at the top and bottom walls on the thermal stratification was investigated.Cases of two interacting plumes in the enclosure were also carried to study the effects of the distance and the ration of the heat source strength on the thermal stratification [24,27].A rising plume restricted by a side wall or corner was investigate to study effects of the distance between the heat source and the wall on the thermal stratification [28].

Thermal stratification height and neutral level
Consider a case of an enclosure with inlets at the bottom and outlets at the roof level and that the indoor air temperature is warmer than the outside environment, namely out in

U U
. In this case, there exists a horizontal level where the indoor hydrostatic pressure equals to that of the external environment, which is defined as the neutral level.Therefore, above the neutral level the hydrostatic pressure difference between the indoor and the external environment is positive and indoor air flow is driven out, and below the neutral level the opposite air flow from the external environment to the indoor happens.An expression of Equation ( 20) is provided to show the expression to calculate the thermal stratification height and neutral level.Z (m) is the height shown in Fig. 1.
Fig. 1 Displacement flow in an enclosure with inlets at the bottom and outlets at the roof level
Analytical methods are based on fundamental theory of fluid mechanics and thermodynamics.Essential assumptions and preconditions are made in order to obtain an exact function relation among some key flow parameters.
Experimental models can be divided into small-scale and full-scale.The objective of experimental model, either small-scale or full-scale, was not to directly study the detail of flow dynamic.Experimental models were mainly used to provide important benchmark for analytical or numerical models, especially CFD models in recent years.More details are provided by the validated analytical or CFD models in the study of indoor buoyancy-driven natural ventilation in real buildings.
Numerical simulations are mainly referred to CFD models.CFD model has been increasingly used in evaluating building ventilation performance and aiding in the improvement of ventilation system [44,45].The selection of a turbulence model for the prediction of room airflow and temperature is of importance for numerical simulation work [46].Fig. 2 shows the CFD results of two interacting plumes in a naturally building.The results show that driven by the interacting plumes a thermal stratification forms.

Conclusions
This paper presented an overview of indoor heat convection and buoyancy-driven natural ventilation in different enclosures with heat sources in different forms.There has not introduced a mature design method handling with buoyancy-driven natural ventilation in general buildings with various types of heat sources.Basic components, buoyant plume above the heat source and indoor thermal stratification flow in buildings with heat sources in different forms are discussed.Methods for predicting indoor buoyancy-driven natural ventilation are also proposed.For an enclosure containing multiple heat sources, according to the form of heat sources (point, surface or volume) and whether they are interacting or restricted, calculation processes can be classified into different strategies to make full use of the building's thermal stratification and indoor and outdoor environmental conditions.In this way can we extend the classic "emptying-filling box model" of a point heat source to the general conditions in real buildings, to simultaneously achieve the effective use of building energy and improve the indoor environment.

Fig. 2
Fig.2 CFD results of two interacting plumes in a naturally ventilated building.