Night cooling by hybrid ventilation – analytical predictions of the purge time

Abstract

Many low-energy buildings rely on a night purge strategy during periods of excessively warm weather in order to flush out excess heat and thereby help prevent overheating on the following day. A typical purge utilises either a mechanical or a natural displacement ventilation strategy. However, we investigate purging by the simultaneous application of natural and mechanical ventilation, i.e. using a hybrid strategy. Hybrid ventilation is widely promoted as a means of overcoming the inherent limitations of both natural and mechanical ventilation, although relatively little is understood about its operation on a fundamental level. Such was the lack of knowledge on hybrid ventilation that even the time taken to complete a purge was not known at the outset of this work. Through the development of a mathematical model, supported by a series of laboratory experiments, we show that the time taken to purge a room using hybrid ventilation can be established analytically, and is controlled by two ratios: (i) the flow rate of the mechanical supply (the ‘forcing’) relative to the initial flow rate through the vents in the absence of mechanical forcing, and (ii) the ratio of the floor-level and ceiling-level vent areas. The analytical predictions show close agreement with our measurements. Supplementing a natural displacement ventilation strategy with even a modest mechanical supply can reduce significantly the time taken to purge warm air from a room. Some of the implications of our findings for preventing overheating in buildings by increasing the efficacy of a night cooling strategy are discussed. Practical Application This research is directly applicable to practitioners involved in the first-order design of hybrid ventilation systems – particularly designs which incorporate a ‘night purge’ regime. Governing equations are derived and the key parameters controlling the rate of purging of warm air are revealed, allowing informed first-order design decisions to be made without the need for expensive and time-consuming computational simulations.

I. Introduction

Effective strategies for purge ventilation are crucial for maintaining a comfortable internal environment in many buildings during the summer months, particularly those incorporating phase change materials^1,2 or with high levels of exposed thermal mass.^3,4

In general terms, purge ventilation refers to the flushing of excessively warm, stale or polluted air from a room through the introduction of air from the external environment. For a successful purge, the external air should be cool and/or fresh compared with the internal air, so that purging improves the quality of the internal environment. Such improvements have been shown to enhance the health and well-being of occupants, ⁵ and to increase their productivity. ⁶ The purging of excessively warm air typically takes place at night – and hence is commonly referred to as a night purge – in order to take advantage of the lower external night time air temperatures and thereby maximise the cooling effect achieved during the purge.

Studies from around the world have shown that effective night cooling strategies that rely on the purging of warm air from buildings can reduce the amount of mechanical cooling energy required on the following day to maintain the thermal comfort of occupants. For example, Piselli et al. ⁷ study the simulated performance of typical mid-rise apartment buildings in different Italian cities and show that it is possible to reduce the cooling energy requirement of these buildings by between 22% and 60% through the use of phase change materials and a natural night cooling strategy. Shaviv et al. ⁸ demonstrate that even in a hot and humid climate, reductions in peak internal air temperature of 3-6°C are achievable in a “heavy constructed building”, i.e. a building with significant thermal mass, through the use of a natural night cooling ventilation strategy.

Historically, designers have relied on either natural or fully mechanical ventilation strategies as the means to purge excess heat from a room. Hybrid ventilation offers an alternative approach, with a well-designed hybrid system being perceived to embody the best elements of both natural and mechanical ventilation in terms of energy use, ventilation control, occupant comfort, and cost. ⁹ Despite the potential benefits, a survey of the literature reveals that little is known about the use of hybrid ventilation systems to purge air from a room – hence the focus of this work being on the the use of hybrid ventilation to implement a purge.

Hybrid purge ventilation is defined herein as the flushing of warm air from a room by the simultaneous use of natural displacement ventilation and a constant mechanical supply of cooler air from the external environment. This definition is fundamentally different to that of a ‘mixed-mode’ ventilation strategy. In a typical ‘mixed-mode’ strategy, ventilation is provided by either a mechanical system or via natural ventilation, with the choice of mechanical or natural ventilation provision being made based on measurements of the internal and external conditions. For example, a ‘mixed-mode’ strategy may utilise a mechanical ventilation system during the winter months and a solely natural ventilation strategy during the summer months. For the purposes of clarity, we refer to mechanical ventilation as being the provision of an unvarying volume flow rate of external air to a room using a fan or pump. Natural ventilation refers to the provision of external air to a room using the buoyancy force generated by differences in air temperature between the internal and external environments, i.e. by the ‘stack effect’. As there is no guarantee of an overnight breeze, our focus is on understanding a hybrid purge in the absence of wind. We consider the purge to be complete if all of the warm air in the room at the start of the purge has been replaced by air at the external air temperature.

The capability to perform rapid predictions, as would, for example, be afforded by an analytical description of hybrid purge ventilation, is essential to facilitate and expedite preliminary design. Moreover, analytical predictions can assist those using computational fluid dynamics (CFD) to design and study hybrid purge systems. For example, analytical predictions can provide initial estimates for the areas of the vents and the strength of the mechanical forcing (m³s⁻¹) required to complete a purge in a given time period, thereby streamlining the CFD design process by reducing the number of iterations which must be computed before a final design solution is reached. Moreover, as we shall establish in §II.B, the development of analytical solutions for hybrid purge ventilation reveals the key parameters that control the purge and their interdependence – this knowledge offering the designer new insights that would not be as readily achievable using other approaches. Despite the potential benefits that a hybrid strategy may offer, based on a review of the available literature it is evident that the purging of excess heat from rooms by hybrid ventilation has not been previously modelled theoretically or considered in any detail by experiment or in numerical simulations. This provides the motivation for the current study.

Based on our survey of the open literature, we have identified a number of significant gaps in the current understanding of hybrid purge ventilation. These concern:

(1) How the flow rate of the mechanical supply and the areas of the vents influence the rate of purging (§II.B.6);

(2) How the rate at which excess heat is removed varies during a purge (§II.B.1) and therefore what controls the time taken to complete a hybrid purge (§II.B.7);

(3) Whether a potentially undesirable reversal in the direction of air flow through the floor-level vent occurs during a purge. Indeed, in §II.B.4 we demonstrate that, somewhat counter-intuitively, the direction of flow will reverse before the purge is complete and there will always be part, or all, of a hybrid purge which takes place in this ‘Reverse Flow’ regime; and

(4) How to guide a practitioner in the design of a hybrid purge ventilation strategy, given, to our knowledge, there is no existing guidance available.

Without bridging these gaps in knowledge, practitioners have no bespoke analytical framework on which to even begin designing a hybrid purge ventilation system, and little or no insight into whether their design will be efficient or even operate as intended. Herein we develop a mathematical model for predicting the purging of excess heat from a room by means of hybrid ventilation and, in doing so, are able to throw considerable light on these concerns. The primary purpose of the model we develop is to offer designers a rapid predictive capability for assessing how the time taken to complete a hybrid purge varies with the area of vents, the flow rate of the mechanical supply (sometimes also referred to as the ‘strength’ or ‘duty’), and the geometry of the room. The model also offers new insights into hybrid flows and addresses a number of fundamental questions. For example, a practitioner might reasonably enquire whether the purge time is halved if the flow rate of the mechanical supply is doubled, or whether the relative sizing of floor-level and ceiling-level vents influences the time taken to purge the room if the combined area of these openings is fixed. We address these and other questions that are likely to be of interest to a designer.

Figure 1 depicts the essential geometry investigated, in which a typical box-shaped room is to be purged of warm air by means of a mechanical supply of cool external air at floor level operating in tandem with a natural displacement ventilation strategy – the latter achieved by means of stack-driven flow through horizontal vents in the floor and ceiling, again connecting the interior to the same cooler external environment.

Figure 1.

Sketches of the basic room geometry under consideration. (a) The positions of the natural vents and the mechanical supply; (b) the direction of airflow through each vent and the thermal stratification in the forward flow (FF) regime; and (c) the reverse flow (RF) regime. The flow of warm/cool air is indicated by the red/blue arrows respectively.

The mathematical model we develop for hybrid purge ventilation follows a tried-and-tested approach – an approach based on solving the underlying equations of fluid mechanics which describe the bulk motion of air by the natural and mechanical components that comprise the hybrid system. For natural ventilation, mathematical models developed following this approach now form the bulk of the relevant first-order design guidance written for practitioners, for example CIBSE’s AM10 ¹⁰ and ASHRAE’s Design Guide for Natural Ventilation ¹¹ . Working within the same framework of standard assumptions and justifiable simplifications that underpin this guidance for natural ventilation, in §II.B.3–II.B.7 we develop a suite of analytical solutions for hybrid purge ventilation, thereby facilitating rapid ‘hand’ calculations at the initial design stage. To our knowledge, the work presented herein is the first to mathematically model time-dependent hybrid ventilation, the previous theoretical works on hybrid ventilation focusing exclusively on the prediction of steady flows^12–14. The aspects of these works on steady flows and of others pertinent to our theoretical developments are outlined below.

A number of studies have succeeded, through the use of simplified mathematical models, to enhance our fundamental understanding of purging by natural ventilation. For example, for a room with vents at high and low level, Linden et al. ¹⁵ predict the time taken to purge the room with the assumption that a displacement flow is maintained, and Wise and Hunt ¹⁶ quantify how this purge time is extended given that Hunt and Coffey ¹⁷ show that such a displacement flow must transition to an (unbalanced) exchange flow before the purge is completed. For solely high-level vents, Epstein ¹⁸ established the rate at which the air cools by balanced exchange flow in a room with a single high-level opening, and Kuesters and Woods ¹⁹ examine the formation and evolution of stratification during a purge facilitated by two or more high-level vents. Hunt and Linden ²⁰ extend our predictive capability still further by developing a mathematical model for the stack-and-wind driven purging of air from a room. Notably, the wind influences the external pressures on a façade, thereby modifying the pressure differences that are established across the vents compared with a solely stack-driven flow. By contrast, for hybrid ventilation, the addition of a mechanical supply to an otherwise solely stack-driven flow increases the internal pressures relative to the those externally. Thus, a hybrid purge cannot be modelled by simply replacing the terms describing the wind-driven component of a stack-and-wind driven purge in the theoretical model of Hunt and Linden ²⁰ with terms that describe the mechanical component of the hybrid system. This then provides further motivation for the current work.

The remainder of this paper is laid out as follows. Section II begins by outlining, with justification, the simplifying assumptions that underpin the development of the mathematical model for hybrid purge ventilation. The limiting case of a hybrid purge for which the strength of the mechanical supply is zero, i.e. a natural ventilation purge, is then discussed (§II.A) – this discussion being necessary to establish a reference flow rate and purge time with which to scale the results for the general hybrid case. The governing equations are then developed for the time-varying flow rates through each vent for hybrid purge ventilation (§II.B.1). After non-dimensionalisation (§II.B.2), these equations are solved to yield analytical solutions for the flow rates in the ‘forward flow’ and ‘reverse flow’ regimes (§II.B.3 and §II.B.5). Given our focus is to derive expressions for the time taken to purge a room using a hybrid system, we proceed by considering the onset of flow reversal (§II.B.4) rather than dwelling on the nuances of the time-varying flow rate through each vent – an in-depth analysis of the flow rates during the hybrid purge is the focus of a complementary paper by Waterson & Hunt ²¹ . Equations describing the time rate of change of the depth of the layer of warm air are then derived and solved analytically (§II.B.6). These solutions reveal the controlling parameters and their interdependence. A solution of immediate practical interest to practitioners is then explored, specifically the time taken to complete a hybrid purge, i.e. the ‘hybrid purge time’ (§II.B.7). The details of a complementary experimental campaign are outlined in (§III) before theoretical and experimental results are presented (§IV). Excellent agreement is found between the theoretical and experimental results. An analytical approximation of the overall purge time is then derived (§V) and example calculations are presented for the purging of a lecture room (§VI). Concluding remarks are made (§VII), highlighting some of the key implications of our results for hybrid ventilation design.

II. Mathematical model for a hybrid purge

To gain a first insight into hybrid night purging, we consider the purging of heat from a single room, of floor area S, rather than from interlinked rooms in a building. Lynch & Hunt ²² highlight the considerable added complexities which arise when purging heat naturally from multi-storey atrium buildings due to the interconnected nature of the geometry and unavoidable mixing. Consequently, a single room represents a logical starting point for the current study. Given our focus on a purge at night, the room is considered to be unoccupied and without any active internal heat sources (e.g. lights, computer equipment, radiators or underfloor heating) or solar heat gains.

Figure 2 depicts the basic room geometry considered. The layer of warm air has a temperature T _in and depth d, and the cooler external air temperature is T _ex . As the timescale for purging by natural displacement ventilation (the worst-case scenario for this hybrid model) is generally significantly shorter than the timescale over which there is a significant variation in external air temperature, the temperature of the external air is considered to be constant for the entire duration of the purge. The difference in temperature between the warm internal air and the cooler external air, ΔT = T _in − T _ex is more conveniently expressed in terms of the buoyancy (or ‘reduced gravity’) of the warm air relative to the cooler external air:

$g^{\prime }=g\left(\frac{T_{in}-T_{ex}}{T_{ex}}\right),$

(1)

where g is the acceleration due to gravity. Alternatively, g′ = g(ρ _ex − ρ _in )/ρ _ex , where ρ _ex is the density of the external air, and ρ _in is the density of the layer of warm air. The initial conditions we consider are:

$g^{\prime }=g_0^{\prime }\hspace{1em}\text{and}\hspace{1em}d=d_0\text{at}t=0,$

(2)

where t is the time in seconds from the start of the purge and the subscript ‘0’ indicates values at the instant the purge starts.

Figure 2.

Annotated sketch of a typical room with floor area, S. A vertical section is highlighted, with red shading indicating warm air to be purged, and blue shading the cooler air at the external air temperature.

As the buoyancy driven pressure difference driving the flow is a function of the depth and buoyancy of the layer of warm air, we find that our selection of d and g′ as the key variables allows for a clearer understanding of the purge behaviour relative to that gained on selecting, say, h, H, T _in and T _ex . Indeed, it is worth noting that the room height, H, is not a key parameter in the theoretical description of a purge given that it serves only to limit the maximum depth that the warm air layer can initially take.

Consistent with the notation adopted by Connick & Hunt, ¹² quantities relating to the vents are given the subscripts (⋅)₁ for the floor-level vent and (⋅)₂ for the ceiling-level vent, and those relating to the mechanical system are identified by the subscript (⋅) _F which reads ‘forced’. Accordingly, a₁ denotes the area of the floor-level vent and a₂ the area of the corresponding ceiling-level vent. As is typical of the situation in practice, the areas of the vents in the floor and ceiling are considered to be small in comparison to the floor area, i.e. {a₁, a₂}≪ S. The instantaneous natural ventilation flow rates are denoted Q₁(t) and Q₂(t), and the mechanical, or ‘forced’, supply flow rate is denoted Q _F . As anticipated in practice, the mechanical supply is maintained at a fixed rate. Figure 3 illustrates this notation and shows the directions of flow for the distinct flow regimes anticipated for the hybrid purge.

Figure 3.

Hybrid purging. A timeline with associated schematics of a typical hybrid purge showing a vertical section through the room modelled in §II at times t < t _rev , t = t _rev , and t > t _rev , where t = 0 represents the instant the purge commences. For t < t _rev , the direction of airflow through the vents is characteristic of the FF-regime. At t = t _rev , the time of transition between the flow regimes, there is no airflow through the floor-level vent. For t > t _rev , the direction of airflow is characteristic of the RF-regime. To track the progress of the purge, d(t) denotes the depth of the layer of warm air as a function of the time since the start of the purge.

In line with previous first-order mathematical models that consider natural (e.g. Linden et al. ¹⁵ ) and hybrid ventilation (e.g. Connick and Hunt, ¹² Hunt, Carroll and Waterson ¹³ ), we have opted to depict a box-shaped room and anticipate that it is these geometries that will be of primary interest to practitioners, particularly those considering ventilation strategies for modern buildings. However, our model is not restricted to box-like rooms because the primary constraint we impose is that the horizontal cross-sectional area of the room, S, does not vary with height – restrictions on how small this cross section can be are discussed in §II.B.1.

The purge is initiated by the simultaneous activation of the mechanical supply and the opening of the vents in the floor and ceiling. The inflow of cooler external air at low level displaces the warm air from the room via the vent in the ceiling. By assuming that there is no mixing between the upper layer of warm air and the cooler air entering through either the mechanical supply or the vent at floor level, and that there is unidirectional flow through the ceiling-level opening, it follows that a stable two-layer stratification is established (warm air above cool) and maintained as the purge progresses. The formation of a two-layer stratification is a standard assumption in models seeking to predict the purge time for natural displacement ventilation. ¹⁵ Hunt and Coffey ¹⁷ establish that the ratio of the inertial force to the gravitational force of the inflow at the height of the interface, expressed as the Froude number Fr _i , must be less than 0.67 (Fr _i < 0.67) for interfacial mixing to be avoided. It is assumed that the mechanical supply is fitted with an appropriate diffuser, if required, such that this Froude number criteria is satisfied. Wise & Hunt ²³ show that bidirectional exchange flow occurs if the Froude number at the ceiling-level opening Fr₂ falls below a critical value of Fr _crit = 0.491c₂ for a circular opening and Fr _crit = 0.375c₂ for a square opening. Thus, we restrict the applicability of our model to Fr _i < 0.67 and Fr₂ > Fr _crit (see Appendix B for further details).

Buildings built to modern standards, or those that have recently been refurbished, will generally have high levels of insulation specified. For example, in the UK, Part L of the building regulations^24,25 stipulates maximum U-values of 0.16, 0.26, and 0.18 Wm⁻²K⁻¹ for roofs, walls and floors, respectively. Thus, we make the simplifying assumption that there is no heat exchange with the fabric of the room over the duration of the purge and thus the temperature of the air to be purged remains constant at temperature T _in .

A. The limiting case of a natural ventilation purge

Although the existing literature does not address hybrid purging, the limiting case of a hybrid purge, namely, if the strength of the mechanical supply is reduced to zero can be assessed on examining a purge by natural displacement ventilation. ¹⁵

Given Q _F ≡ 0 for a natural purge, with reference to Figure 3, Q₁ = Q₂ = Q _N where Q _N is the natural ventilation flow rate, the subscript N reading ‘natural’. It is readily shown that this flow rate is initially:

$Q_{N,0}=A*\sqrt{g_0^{\prime }{d}_0},$

(3)

where the subscript ‘0’ refers to values at time t = 0. Following Coffey and Hunt ²⁶ and Hunt and Linden ²⁰ , the overall effective opening area, A*, is:

$A*=\sqrt{\frac{2{\left(a_1{c}_1\right)}^2{\left(a_2{c}_2\right)}^2}{{\left(a_1{c}_1\right)}^2+{\left(a_2{c}_2\right)}^2}},$

(4)

where c₁ is the loss coefficient associated with the flow through the floor-level opening and c₂ that associated with the ceiling-level opening ¹ . The symbol * is used throughout to read ‘effective’ and the product of a and c is referred to as the effective area of the individual vent, hence:

$a_1*=a_1{c}_1\hspace{1em}\text{and}\hspace{1em}{a}_2*=a_2{c}_2,$

(5)

are the effective areas of the floor-level and ceiling-level openings respectively.

Based on the assumptions of no mixing (see Appendix B) and that displacement flow is maintained ² , Linden et al. ¹⁵ predict that the variation in layer depth with time is:

$d={\left(\sqrt{d_0}-\frac{A*\sqrt{g^{\prime }}t}{2S}\right)}^2,$

(6)

and therefore the time taken to completely purge the layer of warm air from the room, hereafter referred to as the ‘natural purge time’, is:

$t_{NP}=\frac{2S}{A*}\sqrt{\frac{d_0}{g_0^{\prime }}}=\frac{2S{d}_0}{Q_{N,0}}.$

(7)

The initial natural ventilation flow rate, Q_N,0, and the natural purge time, t _NP , as defined in (3) and (7) respectively, are used to scale the hybrid flow rates and purge times derived later in this paper. Scaling in this way facilitates intuitive comparisons between our predictions for a hybrid purge and those for a natural purge. Such comparisons between the natural and hybrid strategies are important given that a hybrid ventilation strategy is often intended to offer enhanced performance over a natural strategy. However, we establish herein that this ‘enhancement’ is not as a designer may have anticipated – indeed we show that on enhancing the rate of purging by increasing the mechanical supply the contribution of the natural (stack-driven) component is diminished.

B. Model development for a hybrid ventilation purge

B.1 Flow rates through the vents

In order to establish analytical predictions of the required purge time for a hybrid strategy, we must first derive relationships for the flow rates in the hybrid purge. Specifically, we need to quantify Q₁(t) and Q₂(t), for both the forward flow (FF) and reverse flow (RF) regimes, in terms of the mechanical supply flow rate Q _F , the area of the vents, and the difference in air temperature between the room and the exterior, the latter represented by $g_0^{\prime }$ , (1). Positive values of Q₁ and Q₂ indicate that the directions of airflow match those shown in Figure 3 for the flow regime in question, i.e. a positive value of Q₁ indicates inflow in the FF-regime and outflow in the RF-regime.

For hydrostatic distributions of air pressure internally and externally (given that {a₁, a₂}≪ S, the vertical acceleration of the rising thermal interface is small in comparison to the acceleration due to gravity) the driving pressures across the vents at floor level (ΔP₁) and ceiling level (ΔP₂) are:

$\begin{array}{ll}\mathrm{\Delta }{P}_1& =\{\begin{array}{ll}{\rho }_{ex}{g}_0^{\prime }\left(d-d_{npl}\right)& FF-\text{regime}\\ {\rho }_{ex}{g}_0^{\prime }\left(d_{npl}-d\right)& RF-\text{regime}\end{array}\end{array}$

(8)

$\begin{array}{ll}\mathrm{\Delta }{P}_2& ={\rho }_{ex}{g}_0^{\prime }{d}_{npl} FF-\text{and}RF-\text{regimes},\end{array}$

(9)

where d _npl denotes the vertical distance between the ceiling and the neutral pressure level, i.e. the vertical elevation at which the internal and external pressures are equal. Combining (8) and (9) to eliminate d _npl yields:

${\rho }_{ex}{g}_0^{\prime }d=\{\begin{array}{ll} \mathrm{\Delta }{P}_1+\mathrm{\Delta }{P}_2& FF-\text{regime}\\ -\mathrm{\Delta }{P}_1+\mathrm{\Delta }{P}_2& RF-\text{regime}.\end{array}$

(10)

Applying the steady form of the Bernoulli equation (the flow is assumed to be quasi-steady given that the vertical acceleration of the interface is small in comparison to g), the difference in pressure across the vents may be expressed as:

$\mathrm{\Delta }{P}_1=\frac{Q_1^2}{2{a*}_1^2}{\rho }_{ex}\hspace{1em}\text{and}\hspace{1em}\mathrm{\Delta }{P}_2=\frac{Q_2^2}{2{a*}_2^2}{\rho }_{in}.$

(11)

Combining (10) and (11) and applying the Boussinesq approximation ²⁸ (i.e. setting ρ _in /ρ _ex = 1 in the inertial term $\left({\rho }_{in}/{\rho }_{ex}\right)\times \left(Q_2^2/\left(2{a*}_2^2\right)\right)$ justified on the grounds that the differences in density are small for a typical room ventilation scenario) then enables the term g′d, which represents the square of a buoyancy-induced velocity, to be expressed in terms of the effective areas and the flow rates through each vent:

$g_0^{\prime }d=\{\begin{array}{ll}\frac{1}{2}{\left(\frac{Q_1}{{a*}_1}\right)}^2+\frac{1}{2}{\left(\frac{Q_2}{{a*}_2}\right)}^2& FF-\text{regime}\\ -\frac{1}{2}{\left(\frac{Q_1}{{a*}_1}\right)}^2+\frac{1}{2}{\left(\frac{Q_2}{{a*}_2}\right)}^2& RF-\text{regime}.\end{array}$

(12)

Considering a control volume coincident with the perimeter of the room, conservation of volume for an incompressible flow requires:

$FF-\text{regime}: Q_2=Q_1+Q_F$

(13)

$RF-\text{regime}: Q_2=Q_F-Q_1.$

(14)

As evident from (13) and (14), Q₁ = Q₂ only if there is zero flow rate from the mechanical supply (Q _F = 0) and the purge is in the FF-regime. Combining (12) with (13) or (14), depending on the flow regime, and eliminating Q₂ yields the following quadratic polynomials for Q₁:

$\begin{array}{l}FF-\text{regime}:\\ Q_1^2\left(\frac{{a*}_1^2+{a*}_2^2}{2{a*}_1^2{a*}_2^2}\right)+Q_1\left(\frac{Q_F}{{a*}_2^2}\right)+\frac{Q_F^2}{2{a*}_2^2}=g_0^{\prime }d\end{array}$

(15)

$\begin{array}{l}RF-\text{regime}:\\ Q_1^2\left(\frac{{a*}_1^2-{a*}_2^2}{2{a*}_1^2{a*}_2^2}\right)-Q_1\left(\frac{Q_F}{{a*}_2^2}\right)+\frac{Q_F^2}{2{a*}_2^2}=g_0^{\prime }d\end{array}$

(16)

Note the changes of sign in the quadratic and linear terms in (16) relative to those in (15). Applying the appropriate volume conservation equation, (13) or (14), to (15) or (16) yields quadratic polynomials for Q₂:

$\begin{array}{l}FF-\text{regime}:\\ Q_2^2\left(\frac{{a*}_1^2+{a*}_2^2}{2{a*}_1^2{a*}_2^2}\right)-Q_2\left(\frac{Q_F}{{a*}_1^2}\right)+\frac{Q_F^2}{2{a*}_1^2}=g_0^{\prime }d\end{array}$

(17)

$\begin{array}{l}RF-\text{regime}:\\ Q_2^2\left(\frac{{a*}_1^2-{a*}_2^2}{2{a*}_1^2{a*}_2^2}\right)+Q_2\left(\frac{Q_F}{{a*}_1^2}\right)-\frac{Q_F^2}{2{a*}_1^2}=g_0^{\prime }d.\end{array}$

(18)

The expressions (15-18) govern the behaviour of the hybrid system, and they reveal how the instantaneous flow rates through the vents are related to the effective areas of the vents, the mechanical forcing, the buoyancy of the warm internal air, and the instantaneous depth of the layer of warm air.

Setting Q _F = 0 in the equations for Q₁ and Q₂ in the FF-regime, (23) and (25), yields: $Q_1=Q_2=A*\sqrt{g_0^{\prime }d}$ . As anticipated, this is equivalent to the natural flow rate derived by Linden et al. ¹⁵ of $Q_N=A*\sqrt{g_0^{\prime }\left(H-h\right)}$ given in their notation H is the room height and h is the height of the thermal interface above floor level such that d = H − h.

B.2 Non-dimensionalisation & governing parameters

Regarding the non-dimensionalisation of the governing equations and the initial conditions, d₀ is the characteristic length scale and ${g\prime }_0$ the characteristic buoyancy. Thus, the dimensionless depth and buoyancy of the layer of warm air are introduced as:

$\widehat{d}=\frac{d}{d_0}\hspace{1em}\text{and}\hspace{1em}{\widehat{g}}^{\prime }=\frac{g^{\prime }}{g_0^{\prime }}.$

(19)

Consistent with (19), the time since the start of the purge is scaled using the natural purge time, t _NP defined in (7) to give the dimensionless time:

$\widehat{t}=\frac{t}{t_{NP}}.$

(20)

In dimensionless form, the initial conditions become:

${\widehat{g}}^{\prime }=1\hspace{1em}\text{and}\hspace{1em}\widehat{d}=1\hspace{1em}\text{at}\hspace{0.17em}\widehat{t}=0.$

(21)

Thus, $\widehat{d}=1$ at the start of the purge and $\widehat{d}=0$ at completion, i.e. when all of the warm air has been purged from the room.

Scaling d as in (19), rather than using, for example $\sqrt{a_2}$ or $\sqrt{a_1}$ as the sole characteristic length scale throughout, stems from a desire to make comparisons with the natural purge, whilst also facilitating a comparison of the rate of purging for hybrid designs with different opening areas. From (20), the dimensionless time is expressed as multiples of the natural purge time. For example, if a particular hybrid purge takes a time $\widehat{t}=0.5$ , then this shows that the addition of the mechanical supply allows the purge to be completed in half of the time of a solely natural purge. This comparison is useful because it allows the validity of the perception that hybrid ventilation offers enhanced performance compared to a purely natural ventilation strategy to be assessed.

Defining:

$R*=\frac{a_1{c}_1}{a_2{c}_2}, \mathcal{A}=\frac{1}{{R*}^2+1}, \mathcal{B}=\frac{1}{{R*}^2-1},$

(22)

the dimensionless governing equations for the flow rate through the floor-level vent, (15) and (16), reduce to:

$\begin{array}{l}FF-\text{regime}:\\ {\widehat{Q}}_1^2+2\mathcal{A}{R*}^2{\widehat{Q}}_F{\widehat{Q}}_1+\mathcal{A}{R*}^2{\widehat{Q}}_F^2-{\widehat{g}}^{\prime }\widehat{d}=0\end{array}$

(23)

$\begin{array}{l}RF-\text{regime}:\\ \frac{\mathcal{A}}{\mathcal{B}}{\widehat{Q}}_1^2-2\mathcal{A}{R*}^2{\widehat{Q}}_F{\widehat{Q}}_1+\mathcal{A}{R*}^2{\widehat{Q}}_F^2-{\widehat{g}}^{\prime }\widehat{d}=0,\end{array}$

(24)

and those for the ceiling-level vent, (17) and (18), reduce to:

$\begin{array}{ll}FF-\text{regime}:& {\widehat{Q}}_2^2-2\mathcal{A}{\widehat{Q}}_F{\widehat{Q}}_2+\mathcal{A}{\widehat{Q}}_F^2-{\widehat{g}}^{\prime }\widehat{d}=0\end{array}$

(25)

$\begin{array}{ll}RF-\text{regime}:& \frac{\mathcal{A}}{\mathcal{B}}{\widehat{Q}}_2^2+2\mathcal{A}{\widehat{Q}}_F{\widehat{Q}}_2-\mathcal{A}{\widehat{Q}}_F^2-{\widehat{g}}^{\prime }\widehat{d}=0,\end{array}$

(26)

where

${\widehat{Q}}_1=\frac{Q_1}{Q_{N,0}},{\widehat{Q}}_2=\frac{Q_2}{Q_{N,0}},$

(27)

and the dimensionless strength of the mechanical forcing is:

${\widehat{Q}}_F=\frac{Q_F}{Q_{N,0}}.$

(28)

The value of ${\widehat{Q}}_F$ is shown to be crucial in governing the transition between flow regimes (§II.B.4), the purge time (§II.B.6), and the flow rates through the vents in both the FF-regime (§II.B.3) and the RF-regime (§II.B.5).

Scaling Q _F using the initial natural ventilation flow rate Q_N,0 offers a straightforward and intuitive means of assessing the relative strengths of the mechanical and natural components of a hybrid purge system. For example, if ${\widehat{Q}}_F\ll 1$ , the initial behaviour of the purge is dominated by the natural ventilation, whereas if ${\widehat{Q}}_F\gg 1$ then the natural ventilation component is negligible in comparison to the strength of the mechanical supply.

If there is no mechanical supply, ${\widehat{Q}}_1={\widehat{Q}}_2=1$ at the start of the purge. Therefore, if the addition of the mechanical supply yields ${\widehat{Q}}_1{|}_{\widehat{t}=0}>1$ or ${\widehat{Q}}_2{|}_{\widehat{t}=0}>1$ then the mechanical system assists the natural flow through that opening, whereas if ${\widehat{Q}}_1{|}_{\widehat{t}=0}<1$ or ${\widehat{Q}}_2{|}_{\widehat{t}=0}<1$ then the mechanical system opposes the natural flow through that opening.

The parameter R* is introduced, (22), in line with the existing body of work on hybrid ventilation.^12,13 This ratio of the effective opening areas in the floor and ceiling is typically in the region 0.5 < R* <2.0 for a room served by a natural ventilation system. ¹²

B.3 Solutions for forward flow

For the FF-regime, the solutions of (23) and (25) may be expressed as:

$\begin{array}{ll}{\widehat{Q}}_1& =-\mathcal{A}{R*}^2{\widehat{Q}}_F+\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{FF}}\end{array}$

(29)

$\begin{array}{ll}{\widehat{Q}}_2& =\mathcal{A}{\widehat{Q}}_F+\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{FF}},\end{array}$

(30)

where ${\widehat{d}}_{FF}$ is a virtual dimensionless depth:

${\widehat{d}}_{FF}=\widehat{d}-{\mathcal{A}}^2{R*}^2\frac{{\widehat{Q}}_F^2}{\widehat{g\prime }}.$

(31)

The second roots of the quadratics (23) and (25) are discarded given these yield non-physical predictions which do not match the natural case for ${\widehat{Q}}_F=0$ . Similarly, in the equations describing the flow rates in the RF-regime (§II.B.5), the second roots are discarded given these predict flow rates at the end of the purge (i.e. when the stack-driven component has reduced to zero) which do not match those of a mechanical purge.

B.4 Onset of flow reversal

In physical terms, a reversal in the direction of flow through the floor-level vent (from inflow to outflow) occurs if the increase in internal pressure generated by the mechanical supply exceeds the buoyancy-induced pressure differences – for a purely natural purge, the latter drives an inflow through the floor-level vent. It is this reversal in the direction of flow through the floor-level vent which characterises the reverse flow regime. In this RF-regime, air enters the room only via the mechanical supply, i.e. there is no inflow through either vent (see Figure 3).

The depth of the layer of warm air at the instant t = t _rev that the RF-regime commences is denoted d _rev , where ‘rev’ reads ‘reversal’. In non-dimensional form:

${\widehat{d}}_{rev}=\frac{d_{rev}}{d_0}.$

(32)

At the instant of transition between the FF- and RF-regimes, depicted in Figure 3, there is no pressure difference across the floor-level vent and consequently no flow of air (Q₁ = 0). This absence of airflow through the floor-level vent was observed in our laboratory experiments, see §III. Substituting Q₁ = 0, into the volume conservation equation for either the FF-regime (13) or the RF-regime (14) yields:

$Q_2=Q_F\hspace{0.17em}\Rightarrow \hspace{0.17em}{\widehat{Q}}_2={\widehat{Q}}_F.$

(33)

This expression of volume conservation is identical to that which would arise for a solely mechanical system supplying a flow rate Q _F . Thus, the natural component does not contribute to the flow rate through the ceiling-level vent at the onset of flow reversal.

An expression for ${\widehat{d}}_{rev}$ can be derived by substituting ${\widehat{Q}}_2={\widehat{Q}}_F$ from (33), into (30):

${\widehat{d}}_{rev}=\mathcal{A}{R*}^2\frac{{\widehat{Q}}_F^2}{{\widehat{g}}^{\prime }}.$

(34)

This relationship is valid only for ${\widehat{Q}}_F>0$ given that there can be no reversal in flow direction for the limiting natural ventilation case considered in §II.A. If flow reversal commences at t = 0 then d _rev = d₀, so that ${\widehat{d}}_{rev}=1$ , and if it commences for t > 0, the layer depth d _rev < d₀, so that ${\widehat{d}}_{rev}<1$ .

From (34), it is clear that increasing ${\widehat{Q}}_F$ increases ${\widehat{d}}_{rev}$ . It follows that if the mechanical forcing exceeds a critical value, ${\widehat{Q}}_{F,crit}$ , for which ${\widehat{d}}_{rev}=1$ , then a RF-regime is established from the start of the purge. The critical value ${\widehat{Q}}_{F,crit}$ can therefore be regarded as the minimum flow rate for a mechanical supply to ensure the purge commences in the RF-regime and thus guarantees that there will be no inflow through the floor-level vent. To derive ${\widehat{Q}}_{F,crit}$ , we substitute ${\widehat{d}}_{rev}=1$ into (34) and rearrange to yield:

${\widehat{Q}}_{F,crit}=\sqrt{\frac{1}{{R*}^2}+1}.$

(35)

A reduction in ${\widehat{Q}}_{F,crit}$ can be achieved by increasing the area of the floor-level vent relative to the area of the ceiling-level vent so that R* increases – with the limit being that as R* → ∞, ${\widehat{Q}}_F\to 1$ . Thus, a hybrid purge with ${\widehat{Q}}_F<1$ must commence in the FF-regime, irrespective of the value of R*.

B.5 Solutions for reverse flow

The solutions of (24) and (26) for the RF-regime are more complex than for the FF-regime, as they take a different form depending on the value of R* relative to unity. The solutions are:

$\begin{array}{ll}{\widehat{Q}}_1& =\{\begin{array}{ll}\frac{{\widehat{Q}}_F}{2}-\frac{{\widehat{g}}^{\prime }\widehat{d}}{{\widehat{Q}}_F}& \hspace{0.17em}\text{for}R*=1\\ \mathcal{B}{R*}^2{\widehat{Q}}_F+\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{RF}}& \hspace{0.17em}\text{for}R*<1\\ \mathcal{B}{R*}^2{\widehat{Q}}_F-\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{RF}}& \hspace{0.17em}\text{for}R*>1\end{array}\end{array}$

(36)

$\begin{array}{ll}{\widehat{Q}}_2& =\{\begin{array}{ll}\frac{{\widehat{Q}}_F}{2}+\frac{{\widehat{g}}^{\prime }\widehat{d}}{{\widehat{Q}}_F}& \hspace{0.17em}\text{for}R*=1\\ -\mathcal{B}{\widehat{Q}}_F-\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{RF}}& \hspace{0.17em}\text{for}R*<1\\ -\mathcal{B}{\widehat{Q}}_F+\sqrt{{\widehat{g}}^{\prime }{\widehat{d}}_{RF}}& \hspace{0.17em}\text{for}R*>1,\end{array}\end{array}$

(37)

where

${\widehat{d}}_{RF}=\frac{\mathcal{B}}{\mathcal{A}}\widehat{d}+{\mathcal{B}}^2{R*}^2\frac{{\widehat{Q}}_F^2}{{\widehat{g}}^{\prime }}.$

(38)

B.6 Rate of purging by hybrid ventilation

The rate at which warm air is purged from the room, and thus the time-varying depth of the layer of warm air, can now be established mathematically from the solutions to ${\widehat{Q}}_1$ and ${\widehat{Q}}_2$ in the FF- and RF-regimes derived in §II.B.3 and §II.B.5. Applying the principle of volume conservation to a control volume coincident with the perimeter of the layer of warm air yields:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(Sd\right)=-Q_2\hspace{0.17em}\Rightarrow \hspace{0.17em}\frac{\mathrm{d}}{\mathrm{d}\widehat{t}}\left(\widehat{d}\right)=-2{\widehat{Q}}_2.$

(39)

Defining:

$X=-\frac{1}{\mathcal{A}}\left(\frac{{\widehat{Q}}_{2,0}}{{\widehat{Q}}_F}\right)\exp \left[\frac{1}{\mathcal{A}}\left(\frac{\widehat{t}}{{\widehat{Q}}_F}-\frac{{\widehat{Q}}_{2,0}}{{\widehat{Q}}_F}\right)\right]$

(40)

$\begin{array}{l}Y=\frac{1}{\mathcal{B}}\left(\frac{{\widehat{Q}}_{2,rev}}{{\widehat{Q}}_F}\right)\exp [-\frac{1}{\mathcal{A}{\widehat{Q}}_F}\left(\widehat{t}-{\widehat{t}}_{rev}\right)\\ +\frac{1}{\mathcal{B}}\left(\frac{{\widehat{Q}}_{2,rev}}{{\widehat{Q}}_F}\right)],\end{array}$

(41)

where X and Y are arguments of the Lambert W-function W, ²⁹ on substituting the expressions for ${\widehat{Q}}_2$ from the FF-regime (30) and the RF-regime (37) into (39), and integrating, we obtain:

$\begin{array}{l}\begin{array}{l}FF-\text{regime}:\\ \widehat{d}\left(\widehat{t}\right)={\mathcal{A}}^2{\widehat{Q}}_F^2\left(W{\left(X\right)}^2+2W\left(X\right)+\frac{1}{\mathcal{A}}\right)\end{array}\\ \begin{array}{l}RF-\text{regime},R*\ne 1:\\ \widehat{d}\left(\widehat{t}\right)=\mathcal{A}\mathcal{B}{\widehat{Q}}_F^2\left(W{\left(Y\right)}^2+2W\left(Y\right)-\frac{1}{\mathcal{B}}\right)\end{array}\\ \begin{array}{l}RF-\text{regime},R*=1\text{and}{\widehat{Q}}_F<{\widehat{Q}}_{F,crit}:\\ \widehat{d}\left(\widehat{t}\right)=-\frac{{\widehat{Q}}_F^2}{2}+\left({\widehat{d}}_{rev}+\frac{{\widehat{Q}}_F^2}{2}\right)\\ \exp \left[\frac{-2}{{\widehat{Q}}_F}\left(\widehat{t}-{\widehat{t}}_{rev}\right)\right]\end{array}\\ \begin{array}{l}RF-\text{regime},R*=1\hspace{1em}\text{and}\hspace{1em}{\widehat{Q}}_F\ge {\widehat{Q}}_{F,crit}:\\ \widehat{d}\left(\widehat{t}\right)=-\frac{{\widehat{Q}}_F^2}{2}+\left(1+\frac{{\widehat{Q}}_F^2}{2}\right)\exp \left[\frac{-2\widehat{t}}{{\widehat{Q}}_F}\right].\end{array}\end{array}$

(42)

In (40), ${\widehat{Q}}_{2,0}$ and ${\widehat{Q}}_{2,rev}$ are defined:

$\begin{array}{l}\text{for}\hspace{1em}{\widehat{Q}}_F<{\widehat{Q}}_{F,crit}:\\ {\widehat{Q}}_{2,0}=\mathcal{A}{\widehat{Q}}_F+\sqrt{1-{\mathcal{A}}^2{R*}^2{\widehat{Q}}_F^2}\end{array}$

(43)

$\begin{array}{l}{\widehat{Q}}_{2,rev}={\widehat{Q}}_F\end{array}$

(44)

$\begin{array}{l}\text{for}\hspace{1em}{\widehat{Q}}_F\ge {\widehat{Q}}_{F,crit}:\\ {\widehat{Q}}_{2,rev}=\{\begin{array}{ll}\frac{{\widehat{Q}}_F}{2}+\frac{1}{{\widehat{Q}}_F}\hspace{0.17em}& R*=1\\ -\mathcal{B}{\widehat{Q}}_F-\sqrt{{\mathcal{B}}^2{R*}^2{\widehat{Q}}_F^2+\frac{\mathcal{B}}{\mathcal{A}}}& R*<1\\ -\mathcal{B}{\widehat{Q}}_F+\sqrt{{\mathcal{B}}^2{R*}^2{\widehat{Q}}_F^2+\frac{\mathcal{B}}{\mathcal{A}}}& R*>1.\end{array}\end{array}$

(45)

Physically, ${\widehat{Q}}_{2,0}$ is the initial dimensionless flow rate through the ceiling-level vent for ${\widehat{Q}}_F<{\widehat{Q}}_{F,crit}$ and ${\widehat{Q}}_{2,rev}$ is the dimensionless flow rate through the ceiling-level vent at the instant of transition to the RF-regime – for ${\widehat{Q}}_F\ge {\widehat{Q}}_{F,crit}$ , the purge commences in the RF-regime and ${\widehat{Q}}_{2,rev}$ in (45) is equal to the initial flow rate.

As we seek the hybrid purge time, (42) can be rearranged to instead give $\widehat{t}$ as a function of $\widehat{d}$ :

$\begin{array}{l}FF-\text{regime}:\\ \widehat{t}\left(\widehat{d}\right)={\widehat{Q}}_{2,0}-{\widehat{Q}}_2\left(\widehat{d}\right){|}_{{\widehat{g}}^{\prime }=1}+\mathcal{A}{\widehat{Q}}_F\hspace{0.17em}\ln \left(\frac{{\widehat{Q}}_2\left(\widehat{d}\right){|}_{{\widehat{g}}^{\prime }=1}}{{\widehat{Q}}_{2,0}}\right)\\ RF-\text{regime}:\\ \widehat{t}\left(\widehat{d}\right)={\widehat{t}}_{rev}+\frac{\mathcal{A}}{\mathcal{B}}\left({\widehat{Q}}_{2,rev}-{\widehat{Q}}_2\left(\widehat{d}\right){|}_{{\widehat{g}}^{\prime }=1}\right)+\\ \mathcal{A}{\widehat{Q}}_F\hspace{0.17em}\ln \left(\frac{{\widehat{Q}}_{2,rev}}{{\widehat{Q}}_2\left(\widehat{d}\right){|}_{{\widehat{g}}^{\prime }=1}}\right),\end{array}$

(46)

where

$\begin{array}{l}\text{for}\hspace{1em}{\widehat{Q}}_F<{\widehat{Q}}_{F,crit}:\\ {\widehat{t}}_{rev}={\widehat{Q}}_{2,0}-{\widehat{Q}}_F+\mathcal{A}{\widehat{Q}}_F\hspace{0.17em}\ln \left(\frac{{\widehat{Q}}_F}{{\widehat{Q}}_{2,0}}\right)\\ \text{for}\hspace{1em}{\widehat{Q}}_F\ge {\widehat{Q}}_{F,crit}:\\ {\widehat{t}}_{rev}=0\end{array}$

(47)

B.7 Time to complete a hybrid purge

The dimensionless time, ${\widehat{t}}_{HP}$ , taken for a hybrid system to purge the entire volume of warm air from the room, i.e. the value of $\widehat{t}$ for which $\widehat{d}=0$ , hereafter referred to as the ‘hybrid purge time’ is derived by substituting $\widehat{d}=0$ into (46):

${\widehat{t}}_{HP}=\underset{FF-}{\underbrace{{\widehat{t}}_{rev}}}+\underset{RF-\text{regime}}{\underbrace{\frac{\mathcal{A}}{\mathcal{B}}\left({\widehat{Q}}_{2,rev}-{\widehat{Q}}_{2,f}\right)+\mathcal{A}{\widehat{Q}}_F\hspace{0.17em}\ln \left(\frac{{\widehat{Q}}_{2,rev}}{{\widehat{Q}}_{2,f}}\right)}},$

(48)

where ${\widehat{Q}}_{2,rev}$ is given by (44) or (45), ${\widehat{t}}_{rev}$ by (47), and

${\widehat{Q}}_{2,f}=\mathcal{C}{\widehat{Q}}_F\text{where}\mathcal{C}=\frac{1}{R*+1}.$

(49)

The ‘f’ in ${\widehat{Q}}_{2,f}$ reads ‘final’, i.e. ${\widehat{Q}}_{2,f}$ is the value of ${\widehat{Q}}_2$ evaluated at $\widehat{d}=0$ . The dimensional form of the hybrid purge time, t _HP (in seconds), is then simply the product of (48) and (7):

$t_{HP}={\widehat{t}}_{HP}{t}_{NP}.$

(50)

III. Laboratory experiments

In order to assess the accuracy with which our theoretical approach models a hybrid purge, complementary laboratory experiments were conducted for a wide range of R* and ${\widehat{Q}}_F$ values. As predicted theoretically in the absence of mixing, a well-defined steady density interface was observed during each purge. Measurements were taken of the time-varying depth of the layer being purged and the layer depth at the onset of flow reversal. Good agreement was observed between the experimental and theoretical results (§IV). In §III.A, numerical values are given to three significant figures (3.s.f.) unless stated otherwise.

Before commencing experiments on the hybrid purge, independent experiments were conducted (see Appendix A) to deduce the value of the discharge coefficient for the openings in the laboratory model of the room – this value being required to compare the theoretical predictions to the experimental results in §IV. Based on the data gathered, c₁ = c₂ = 0.669 ± 0.005. This value, lying between the accepted value of 0.61 for a sharp-edged opening ²⁷ and 0.8 for a ‘short tube’, ³⁰ was as anticipated given the aspect ratio of the opening.

A. Setup

Experiments were conducted at laboratory scale, with the setup depicted in Figure 4. The room being purged was represented by an acrylic box with a uniform rectangular cross-section (internal dimensions: 30.0 cm (height) × 30.0 cm (length) × 40.0 cm (width)) suspended within a larger acrylic visualisation tank representing the external environment (internal dimensions: 75.0 cm (height) × 100 cm (length) × 100 cm (width)). Using an established approach, dense saline solution and fresh water were used as the analogue of warm and cool air respectively. This choice is routinely used to investigate the purging of warm air from buildings, for example.^{15,20,26,31–33} The relatively large density differences which are possible between saline and fresh water allow for approximate dynamic similarity with the flows expected in the purging of rooms in buildings. The box was filled with a saline solution made by dissolving sodium chloride crystals in fresh water drawn from the same mains supply as the water that filled the visualisation tank. The densities of the freshwater and saltwater solutions were measured using an Anton Paar DMA 5000 M density meter, with the corresponding calculated values of g′ = g(ρ _saline − ρ _fresh )/ρ _fresh being in the range: 5.40 ≤ g′ ≤ 29.2 ± 0.0585 cm s⁻². Initial layer depths were in the range 17.9 ≤ d₀ ≤ 25.9 ± 0.125 cm.

Figure 4.

Annotated sketch of the experimental setup – elevation view.

Because the (dense) saline solution was used in the laboratory experiments to represent the (buoyant) warm air, the experimental orientation is vertically inverted compared to that of a room filled with warm air. This inversion does not unduly alter the dynamics of the purge. In the experiments, the dense layer of saline solution was purged out of the base of the box, whereas in a room the buoyant layer of warm air is purged out of a vent in the ceiling. The Ismatec MCP-Z Process gear pump, which represented the mechanical supply, was connected to the top of the box and provided a mechanical supply rate of fresh water from the visualisation tank equivalent to $0.147\le {\widehat{Q}}_F\le 2.12\pm 0.00681$ over the range of experiments.

In order to replicate the vents in the floor and ceiling of the room, circular openings of diameter 3.00 cm and 5.00 cm were revealed as required in the top and base of the box through the removal of acrylic bungs. For a subset of the experiments, smaller circular openings of diameter 2.00 cm were used. The range of R* and A* values explored experimentally were: 0.500 ≤ R* ≤2.00 ± 0.0835 and 2.11 ≤ A* ≤5.95 ± 4.03 × 10⁻¹⁰ cm².

For the purposes of tracking the density interface, the (relatively) dense saline solution was dyed using methylene blue dye. A light box, comprising an array of fluorescent tubes in an opal acrylic box, was used to blacklight the box with uniform diffuse lighting. The purge was recorded using a JAI SP-5000M-USB CCD (charge-coupled device) camera fitted with either a 35 mm or 75 mm fixed lens. The camera was aligned vertically with the base of the box in order to have a clear view of the interface, even as the purge was nearing completion. A frame rate of at least 3.125 frames per second was used to ensure that the behaviour around the instant of transition between the FF- and RF-regimes was captured. Image processing of the recorded experiments allowed the time-varying position of the density interface, and thereby the depth of the layer of dense fluid, to be established.

The direction of flow through the opening in the top of the box (equivalent to the floor-level vent in the room orientation) was monitored by tracking the motion of releases of neutrally buoyant dye injected (horizontally) into the opening. For experiments which commenced in the FF-regime, a change from an inflow to an outflow of dye indicated that there had been a transition to the RF-regime.

IV. Results and discussion

A. Flow reversal

Figure 5 plots the variation of ${\widehat{d}}_{rev}$ with ${\widehat{Q}}_F$ for R* = {0.5, 1, 2}. Evidently, ${\widehat{d}}_{rev}$ increases as ${\widehat{Q}}_F$ increases up to a maximum value of ${\widehat{d}}_{rev}=1$ . A higher value of R* results in a steeper curve and hence the maximum value of ${\widehat{d}}_{rev}=1$ is reached at a lower value of ${\widehat{Q}}_F$ . There is good agreement between the theoretical (solid lines) and experimental results (circles).

Figure 5.

Variation of the fractional warm air layer depth at the onset of flow reversal ${\widehat{d}}_{rev}$ with dimensionless mechanical forcing ${\widehat{Q}}_F$ . Solid lines show the analytical predictions from (34) and circles show the experimental results – green R* = 0.5, blue R* = 1.0, red R* = 2.0. Measurement errors in ${\widehat{d}}_{rev}$ and ${\widehat{Q}}_F$ were estimated to be ± 0.0138 and ±0.00681, respectively.

The implications of these results for the room ventilation scenario are that the depth of the layer of warm air at the onset of flow reversal increases as the strength of the mechanical forcing increases or as the area of the floor-level vent increases relative to the area of the ceiling-level vent. In other words, increasing ${\widehat{Q}}_F$ or R* allows the RF-regime to commence when there is a deeper layer of warm air and thus a greater natural driving force. From a practical perspective, the depth of warm air that remains at the instant of flow reversal increases as ${\widehat{Q}}_F$ or R* increases. While the influence of ${\widehat{Q}}_F$ on the onset of reversal could have been anticipated, the level of control over ${\widehat{d}}_{rev}$ offered by changes in R* alone (noting that A* is a function of R*, such that changing R* whilst keeping ${\widehat{Q}}_F$ , Q _F , ${g\prime }_0$ , and d₀ constant requires a change in the value of both ${a*}_1$ and ${a*}_2$ ) is perhaps unexpected. For example with ${\widehat{Q}}_F=1$ , ${\widehat{d}}_{rev}=0.2$ for R* = 0.5 and ${\widehat{d}}_{rev}=0.8$ for R* = 2.0.

B. Rate of purging

The variation of the layer depth $\widehat{d}$ with time $\widehat{t}$ from (42) or (46) is plotted in Figure 6 alongside our experimental results. For clarity, only a subset of the experimental results are shown, with the results chosen to span the range of ${\widehat{Q}}_F$ values investigated. The left column shows the theoretical predictions, the central column shows the experimental results and the right column shows the experimental results combined with the theoretical predictions. The layer depth decreases with time and the rate of purging increasing as the mechanical forcing ${\widehat{Q}}_F$ increases. Comparing the results for the hybrid purges to those of the theoretical natural purge (dash-dot line) reveals that a hybrid approach does indeed yield a more rapid purge. Comparing Figure 6(c) with Figure 6(a), note that decreasing R* enhances the rate of purging. This comparison also reveals that the increase in the rate of purging for the hybrid strategy above that of a solely natural purge is less pronounced for larger values of R*. In particular, for ${\widehat{Q}}_F\le 1$ , the experimental results for R* = 2 (Figure 6(c)) could be misinterpreted, without knowledge of the theoretical model, as showing that the rate of purging is independent of ${\widehat{Q}}_F$ . In contrast, there is a marked increase in the rate of purging as ${\widehat{Q}}_F$ increases for R* = 0.5 (Figure 6(a)) and R* = 1 (Figure 6(b)).

Figure 6.

Variation in the warm air layer depth with time. (a) R* = 0.5, (b) R* = 1.0, (c) R* = 2.0. Left – theoretical predictions, centre – experimental results, right – combined theoretical predictions and experimental results. Analytical predictions for the FF- and RF-regimes from (46) are shown by solid lines and dashed lines respectively. The dashdot lines show the idealised natural purge from (6). The analytical predictions for flow reversal from (34) are shown by triangles, and the experimental results are shown by filled circles for the FF-regime and hollow circles for the RF-regime. Measurement errors in $\widehat{d}$ and $\widehat{t}$ were estimated to be ± 0.0138 and ±0.000442, respectively. For clarity, arrows highlight the theoretical predictions for the onset of reversal on the rightmost plot of (c).

Qualitatively, the right column of Figure 6 highlights that there is generally very good agreement between the experimental measurements and theoretical predictions for the wide range of R* and ${\widehat{Q}}_F$ values examined experimentally (0.5 ≤ R* ≤2 and $0.15\le {\widehat{Q}}_F\le 2.12$ ). This agreement was also investigated quantitatively, with the R-squared value being greater than 0.991 for all values of ${\widehat{Q}}_F$ and R* investigated. There is, however, some divergence between the experimental results and the theoretical predictions for $\widehat{d}\lesssim 0.1$ , whereby the rate of draining observed in the experiments decreased more rapidly than predicted by our theoretical model. In the experiments, a ‘drawing down’ of the density interface was observed towards the end of each purge when only a shallow layer of the saline solution remained in the box. This ‘draw down’ regime is beyond the scope of our simplified theoretical model and was identified as the cause of the marked decrease in the rate of purging during the experiments. Whilst there are a number of works which consider the criteria governing the onset of draw down (e.g. the work of Lubin and Springer ³⁴ ), we know of no works quantifying the rate of outflow within this regime – this is the focus of ongoing research.

The results in Figure 6 have a number of practical implications for the hybrid purging of a room. Firstly, a hybrid purge can offer a significant improvement in purge time compared with a solely natural purge (for which ${\widehat{t}}_{NP}=1$ ). Also, as the depth of the layer of warm air decreases, the natural component decreases and therefore the rate at which warm air is purged from the room decreases. This decrease in purge rate does, however, become less significant for higher mechanical flow rates and lower values of R*. Choosing a lower value of R* can help expedite a hybrid purge, but note that the limit on how small R* can be (in order that a displacement flow is maintained) is given by the solution of:

$\frac{{R*}_{\mathit{min}}^2}{\left({R*}_{\mathit{min}}^2+1\right){\left({R*}_{\mathit{min}}+1\right)}^2}=\frac{{0.491}^2}{2{\widehat{Q}}_F^2}{\left(\frac{a_2}{d_0^2}\right)}^{1/2},$

(51)

see (63) in Appendix B.

Figure 7 shows theoretical predictions of the warm air layer depth alongside images taken from a typical experiment at times $\widehat{t}=\left\{0,0.2,0.4,0.6,0.8,1\right\}$ . The closeness of the agreement between the theoretical predictions and experimental observations is representative of the other experiments. We anticipate that a visual comparison of the theoretical and experimental layer depths shown in each snapshot in Figure 7 will give practitioners confidence in the use of our theoretical model for making first-order predictions for a hybrid purge.

Figure 7.

Theoretical predictions superimposed onto images from one of the laboratory experiments at times $\widehat{t}=\left\{0,0.2,0.4,0.6,0.8,1.0\right\}$ . For the experiment shown, ${\widehat{Q}}_F=0.26$ and R* = 2. The experimental images have been rotated by 180^◦ in order to be consistent with the ‘real-world’ orientation of warm air being purged from a room.

In order to highlight the differences between natural and hybrid purges, Figure 8 shows vertical sections through an example room at the following six ‘snapshots’ in time during the purge, $\widehat{t}=\left\{0,0.2,0.4,0.6,0.8,1\right\}$ , with the theoretically-derived layer depths calculated using (46) and the flow rates calculated using (29), (30), (36), and (37). Comparing the top row to the lower rows reveals that adopting a hybrid strategy can expedite the purging of a room compared with a solely natural approach, as in each hybrid case the (red) layer of warm air is purged sooner than in the natural case. Figure 8 also highlights the role of R* on the hybrid purge time, with the a lower R* value expediting the purge – compare purges for R* = 0.5 (row 4) with R* = 2.0 (row 2).

Figure 8.

Diagrams showing the theoretically derived layer depths and flow rates (the length of each arrow is proportional to the volume flow rate – a longer arrow indicating a higher flow rate) for four purges at six snapshots in time (left to right) to illustrate the role of R* at a fixed ${\widehat{Q}}_F\left(=1\right)$ . Warm air is shaded in red and cool air in blue, with the layer depths calculated using $\widehat{d}={\left(1-\widehat{t}\right)}^2$ (the dimensionless form of (6)) for the natural purge and (46) for the hybrid purges. Row 1: a natural purge. Rows 2-4: hybrid purges for R* = {2.0, 1.0, 0.5}. Arrows indicate the direction of airflow through each opening. Flow rates are calculated from (30), (36), and (37).

C. Time to complete a hybrid purge

In light of the draw down regime observed experimentally for shallow layer depths, Figure 9 plots the experimental results and theoretical predictions for the variation in the time taken to purge 90% of the initial layer depth $\widehat{t}{|}_{\widehat{d}=0.1}$ with ${\widehat{Q}}_F$ for R* = {0.5, 1.0, 2.0}, rather than the overall hybrid purge time. Increasing ${\widehat{Q}}_F$ or decreasing R* decreases the time required to purge. The rate at which $\widehat{t}{|}_{\widehat{d}=0.1}$ decreases with increases in ${\widehat{Q}}_F$ is largest for ${\widehat{Q}}_F\lesssim 0.5$ , with the relative benefit to the pure time of further strengthening the mechanical forcing decreasing as ${\widehat{Q}}_F$ increases.

Figure 9.

Variation in the time to purge 90% of the layer of warm air from (46) with mechanical flow rate ${\widehat{Q}}_F$ . Solid lines show the analytical predictions from (34) and circles show the experimental results – green R* = 0.5, blue R* = 1.0, red R* = 2.0. Measurement errors in ${\widehat{Q}}_F$ and $\widehat{t}$ were estimated to be ± 0.00681 and ±0.000442, respectively.

In practical terms, the addition of a floor-level mechanical supply to a room previously ventilated solely using natural displacement ventilation (i.e. using a hybrid strategy) can significantly reduce the time required to purge a given proportion of the layer of warm air, especially in a room with a low R* value (compare $\widehat{t}{|}_{\widehat{d}=0.1}$ at ${\widehat{Q}}_F=0$ to the values for $\widehat{Q}>0$ ). Comparing the 90% purge time at ${\widehat{Q}}_F=0.5$ and ${\widehat{Q}}_F=1.0$ reveals, in answer to the question posed in the Introduction, that the purge time is not halved if the mechanical forcing is doubled – the same is true for the theoretical prediction of the overall purge time shown in Figure 10.

Figure 10.

The variation in the hybrid purge time ${\widehat{t}}_{HP}$ with mechanical flow rate ${\widehat{Q}}_F$ : solid lines show the full solution (48); dashed lines show the approximation (52) to the full solution.

V. Approximating the hybrid purge time

With a view to improving the accessibility of our work, we now present an approximation for the hybrid purge time which avoids the use of logarithmic functions. This approximation shows good agreement with the full solution given in (48) for $0\le {\widehat{Q}}_F<2$ , and 0.25 < R* <3, whilst being easier to compute by hand. Thus we envisage that it may be of value to practitioners for use in rapid preliminary design calculations.

A ‘curve-fitting’ analysis shows that the hybrid purge time can be approximated by:

${\widehat{t}}_{HP}\approx \frac{1}{\alpha {\widehat{Q}}_F+1}+\beta {\widehat{Q}}_F+\gamma {\widehat{Q}}_F^2,$

(52)

where the coefficients α, β and γ depend on the value of R*, and can be read from Appendix C. Figure 10 shows both the analytical solution (48) and the approximation (52) for the hybrid purge time, and highlights, by the closeness of the solid and dashed lines, that there is good agreement for the values of ${\widehat{Q}}_F$ and R* which we anticipate will span the range of typical designs (0.25 < R* <3 and ${\widehat{Q}}_F<2$ ).

In addition to the approximation above, we show in Appendix D that assuming the mechanical forcing dominates throughout the purge yields the upper bound estimate for the hybrid purge time:

${\widehat{t}}_{HP}=\frac{1}{2\mathcal{C}{\widehat{Q}}_F}\text{for}{\widehat{Q}}_F\gtrsim 6.$

(53)

VI. Example calculations

In order to acknowledge the wider applicability of our model for hybrid purging, the chosen example purge takes place when the room is unoccupied during the day, rather than at night. All calculated values are given to three significant figures, unless stated otherwise.

The scenario: Air at 25°C (=298.15 K) is to be purged from a modern well-insulated box-shaped lecture room with a floor area of S = 80 m² and a floor-to-ceiling height of H = 4 m. The external air temperature is 15°C (=288.15 K). The room must be purged in the hour prior to the first lecture of the day and, on occasion, in the short break of circa 30 minutes between lectures, without relying on any wind assistance.

Solution procedure: As a starting point, we first seek a natural ventilation design that achieves the desired purge time of circa 60 minutes – this requires us simply to establish the required value $A*={A*}_r$ for the combined effective area of the upper and lower vents. We then investigate whether the required purge time of 30 minutes can be achieved using a hybrid strategy comprising the vents of area ${A*}_r$ and a mechanical supply. Three strengths of mechanical supply are investigated, equivalent to supplementing the initial natural ventilation flow rate by 50%, 100% and 150%, namely, ${\widehat{Q}}_F=\left\{0.5,1.0,1.5\right\}$ . These values were chosen on the basis that the very premise of hybrid ventilation is to combine the benefits of natural and mechanical ventilation – it would of course, be straightforward to design for Q _F ≫ 1 but this is unlikely to be practical, particularly given the associated energy demands. Given we have shown that the hybrid purge time may vary strongly with the ratio of the upper and lower vent areas, the variation in the purge time is assessed for three realistic ratios, namely, R* = {0.5, 1.0, 2.0}. Prior to developing the solution below, it is worthy of note that:(i) In order to achieve these values of R* whilst ensuring $A*={A*}_r$ , from (4) the effective areas of the lower and upper vents must satisfy:

$a_1{c}_1=A*\sqrt{\frac{1}{2\mathcal{A}}}\hspace{1em}\text{and}\hspace{1em}{a}_2{c}_2=A*\sqrt{\frac{1}{2\mathcal{A}{R*}^2}},$

(54)

(ii) the effective area may be expressed in terms of the known quantities that characterise the example problem, namely, on combining (3) and (7):

$A*=\frac{2S{d}_0}{t_{NP}\sqrt{g_0^{\prime }{d}_0}}.$

(55)

In what follows we take g = 9.81 ms⁻².

The natural solution: From (1), the initial reduced gravity of the warm air is:

$g_0^{\prime }=\frac{9.81 {\text{ms}}^{-2}\left(298.15 \mathrm{K}-288.15 \mathrm{K}\right)}{288.15 \mathrm{K}}=0.340 {\text{ms}}^{-2}.$

Substituting ${g\prime }_0=0.340\hspace{0.5em}{\text{ms}}^{-2}$ into (55) together with S = 80 m², d₀ = 4 m, and t _NP = 3600 s gives the effective area required to achieve a 60 minute purge:

${A*}_r=\frac{2\times 80\hspace{0.5em}{\mathrm{m}}^2\times 4\hspace{0.5em}\mathrm{m}}{3600\hspace{0.5em}\mathrm{s}\sqrt{0.340\hspace{0.5em}{\text{ms}}^{-2}\times 4\hspace{0.5em}\mathrm{m}}}=0.152\hspace{0.5em}{\mathrm{m}}^2.$

From (3) the initial natural flow rate for ${A*}_r=0.152\hspace{0.5em}{\mathrm{m}}^2$ is:

$Q_{N,0}=0.152\hspace{0.5em}{\mathrm{m}}^2\sqrt{0.340\hspace{0.5em}{\text{ms}}^{-2}\times 4\hspace{0.5em}\mathrm{m}}=0.178\hspace{0.5em}{\mathrm{m}}^3{\mathrm{s}}^{-1}.$

Table 1 summarises the inputs and outputs for the natural purge solution.

Table 1.

A summary of the values defined by our chosen scenario, and those calculated from the natural purge solution: $g_0^{\prime }$ from (1), $A_r^{*}$ from (55), and Q_N,0 from (3).

t NP (minutes)	d0 (m)	S (m2)	T in (oC)	T out (oC)	${g\prime }_0$ (ms−2)	${A*}_r$ (m2)	QN,0 (m3s−1)
60	4	80	25	15	0.340	0.152	0.178

We proceed by calculating the hybrid purge times for ${A*}_r=0.152\hspace{0.5em}{\mathrm{m}}^2$ , t _NP = 60 minutes, and our chosen dimensionless values of R* = {0.5, 1.0, 2.0} and ${\widehat{Q}}_F=\left\{0.5,1.0,1.5\right\}$ . All input values for the hybrid purge are summarised in Table 2. Full workings are presented for both the full and approximate hybrid purge times, from (48) and (52), for R* = 0.5 and ${\widehat{Q}}_F=0.5$ , with the complete results for all of our chosen R* and ${\widehat{Q}}_F$ values shown in Table 3.

Table 2.

A summary of the inputs used to calculate the full and approximate hybrid purge times from (48) and (52). $\mathcal{A}$ and $\mathcal{B}$ are calculated from (22), $\mathcal{C}$ from (49), $a_1*$ and $a_2*$ from (54) and Q _F from (28). The value of $\mathcal{B}$ is not required for R* = 1, see (36) and (37). Q _F values are calculated from ${\widehat{Q}}_F$ and Q_N,0 = 0.178 from Table 1.

t NP (minutes)	R* (−)	$\mathcal{A}$ (−)	$\mathcal{B}$ (−)	$\mathcal{C}$ (−)	$a_1*$ (m2)	$a_2*$ (m2)	${\widehat{Q}}_F$ (−)	Q F (m3s−1)	Constraint (51)
	0.5	4/5	−4/3	2/3	0.121	0.242			✓ satisfied
60	1.0	1/2	N/A	1/2	0.153	0.153	{0.5,1.0,1.5}	{0.0888,0.178,0.266}	✓ satisfied
	2.0	1/5	1/3	1/3	0.242	0.121			✓ satisfied

Table 3.

Full and approximate hybrid purge times, from (48) and (52), for R* = {0.5, 1.0, 2.0} and ${\widehat{Q}}_F=\left\{0.5,1.0,1.5\right\}$ . The values of α, β and γ were read from Appendix C. The final column shows the percentage difference between the hybrid (full form) and natural purge times.

		α	β	γ	t HP (minutes) – approx. Form			t HP (minutes) – full form			(t HP − t NP )/t NP (%)
					${\widehat{Q}}_F$			${\widehat{Q}}_F$			${\widehat{Q}}_F$
					0.5	1.0	1.5	0.5	1.0	1.5	0.5	1.0	1.5
R*	0.5	2.40	0.175	−0.060	31.6	24.5	20.7	32.2	24.5	20.5	−46.3	−59.2	−65.8
	1	1.65	0.275	−0.098	39.7	33.3	28.8	40.0	33.4	28.6	−33.3	−44.3	−52.3
	2	0.85	0.26	−0.088	48.6	42.8	37.9	48.8	42.9	37.5	−18.7	−28.5	−37.5

The hybrid approximation: To calculate our approximate solution for the hybrid purge time, the values of α, β and γ must first be read from Appendix C and then substituted into (52). For R* = 0.5: α ≈ 2.4; β ≈ 0.175; and γ ≈ − 0.06. Thus for ${\widehat{Q}}_F=0.5$ , (52) yields:

$\begin{array}{l}{\widehat{t}}_{HP}\approx \frac{1}{\left(2.4\times 0.5\right)+1}+\left(0.175\times 0.5\right)-\left(0.06\times {\left(0.5\right)}^2\right)\\ \approx 0.527.\end{array}$

Multiplying by t _NP gives a purge time of:

$t_{HP}\approx 0.527\times 60 \text{minutes}=\underset{¯}{31.6 \text{minutes}}.$

The full hybrid solution: To apply the full solution for the hybrid purge time (48), the values of ${\widehat{Q}}_{2,0}$ , ${\widehat{Q}}_{2,rev}$ , ${\widehat{Q}}_{2,f}$ , and ${\widehat{t}}_{rev}$ must first be calculated. First, the value of ${\widehat{Q}}_{F,crit}$ is calculated from (35) in order to ascertain whether the purge commences in the FF- or RF-regime:

${\widehat{Q}}_{F,crit}=\sqrt{\frac{1}{0.5^2}+1}=2.24.$

As ${\widehat{Q}}_F<{\widehat{Q}}_{F,crit}\hspace{0.5em}\left(0.5<2.24\right)$ , the purge commences in the FF-regime. Second, the coefficients $\mathcal{A}$ , $\mathcal{B}$ and $\mathcal{C}$ are calculated using (22) and (49) to yield

$\begin{array}{l}\mathcal{A}=\frac{1}{0.5^2+1}=\frac{4}{5},\mathcal{B}=\frac{1}{0.5^2-1}=-\frac{4}{3},\\ \mathcal{C}=\frac{1}{0.5+1}=\frac{2}{3}.\end{array}$

Third, we must establish whether or not a classic displacement flow will be maintained for the flow(s) under consideration, as only then can we apply the solutions developed in §II.B.7. This requires the following two calculations: (i) we can readily establish from (51) in Appendix B that unidirectional outflow through the top opening is achieved and maintained:

$\begin{array}{ll}0.5^2\times \frac{4}{5}\times {\left(\frac{2}{3}\right)}^2& >\frac{{0.491}^2}{2\times 0.5^2}{\left(\frac{0.240/0.61}{4^2}\right)}^{1/2}\\ 0.0889& >0.0757\hspace{0.17em}\Rightarrow \hspace{0.17em}\text{unidirectional} \text{flow};\end{array}$

(ii) similarly, we can readily establish that, without intervention, some mixing is to be expected between the inflowing fresh air through the floor-level vent and the warm air layer given that, from (58) in Appendix B

$\begin{array}{l}{\widehat{Q}}_{1,0}=-\left(\frac{4}{5}\times 0.5^2\times 0.5\right)+\sqrt{1-\left(\frac{4}{5}\times 0.5^2\times 0.5^2\right)}\\ =0.875,\end{array}$

and, from (59) in Appendix B

$\begin{array}{l}0.875\times {\pi }^{1/4}\times \sqrt{2\times \frac{4}{5}}\times {0.61}^{5/4}{\left(\frac{4^2}{0.120}\right)}^{1/4}\nless 0.67\\ \text{and}\hspace{1em}0.5\ngeqq 2.24\end{array}$

The installation of an appropriate diffuser over the lower vent would prevent such mixing – this pragmatic approach has been assumed here. Thus, we proceed by applying the analytical solutions developed herein.

Given the purge commences in the FF-regime (as ${\widehat{Q}}_F<{\widehat{Q}}_{F,crit}$ ), the values of ${\widehat{Q}}_{2,0}$ , ${\widehat{Q}}_{2,rev}$ , ${\widehat{Q}}_{2,f}$ , and ${\widehat{t}}_{rev}$ can be calculated from (43), (44) or (45), (49), and (47), respectively, as follows:

$\{\begin{array}{l}{\widehat{Q}}_{2,0}=\left(0.5\times \frac{4}{5}\right)+\sqrt{1-\left({\left(\frac{4}{5}\right)}^2\times 0.5^2\times 0.5^2\right)}\\ =1.38\\ {\widehat{Q}}_{2,rev}=0.5\\ {\widehat{Q}}_{2,f}=\frac{2}{3}\times 0.5=\frac{1}{3}\\ {\widehat{t}}_{rev}=\left(1.38-0.5\right)+\left(\frac{4}{5}\times 0.5\right)\ln \left(\frac{0.5}{1.38}\right)=0.474.\end{array}$

Substituting these values into (48) yields the dimensionless hybrid purge time:

$\begin{array}{l}{\widehat{t}}_{HP}=0.474+\left(-\frac{3}{4}\times \frac{4}{5}\right)\left(0.5-\frac{1}{3}\right)+\\ \left(0.8\times 0.5\right)\ln \left(\frac{3}{1}\times 0.5\right)=0.536.\end{array}$

Therefore,

$t_{HP}=0.536\times 60\hspace{0.5em}\text{minutes}=\underset{¯}{32.2\hspace{0.5em}\text{minutes}}.$

With a percentage difference of less than 2%, this example highlights the close agreement between the approximate and the full solutions, as well as highlighting that, with the addition of a modest mechanical supply flow rate of ${\widehat{Q}}_F=0.5$ , it is possible to almost half the time required to purge the room – in this case from circa 60 minutes to just over 30 minutes.

After following the same calculation process for the other R* and ${\widehat{Q}}_F$ values (R* = {0.5, 1.0, 2.0} and ${\widehat{Q}}_F=\left\{0.5,1.0,1.5\right\}$ ), we now present the full and approximate hybrid purge times in Figure 11(a) and Table 3. As ${\widehat{Q}}_F$ increases, the purge time decreases from the 60 minutes of the solely natural purge, with the fastest purge for each value of ${\widehat{Q}}_F$ being achieved for R* = 0.5. Indeed, Figure 11(a) highlights the sensitivity of the purge time to R*. For example, on reducing R* from 2.0 to 0.5 the purging time reduces by 17 minutes for ${\widehat{Q}}_F=0.5$ . Note that this dependence on R* is unique to the hybrid purge – the natural purge time being independent of R* so long as the value of A* remains constant. The choices a designer of a hybrid ventilation system makes in terms of: (i) the distribution of a given effective opening area between vents at ceiling-level and floor-level (i.e. the value of R*), and (ii) the strength of the mechanical forcing, can therefore have a significant influence on the purge time.

Figure 11.

(a) Bar chart comparing the natural purge time to the full and approximate hybrid purge times from (48) and (52). The bars show the full solution and the dashed lines the approximate purge time, the two are almost graphically indistinguishable. (b) Graphic showing, for $A_r^{*}=0.152{\mathrm{m}}^2$ , the variation in $a_1^{*}$ and $a_2^{*}$ with R* from (54) – the area of each circle is proportional to the value of the effective area of that vent ( $a_1^{*}$ or $a_2^{*}$ ).

Using Figure 11(a) or Table 3, a hybrid purge time of under 30 minutes is achieved for: ${\widehat{Q}}_F=1.0$ and R* = 0.5, or ${\widehat{Q}}_F=1.5$ and R* = 1.0 ( ${\widehat{Q}}_F>1.5$ to achieve the desired purge time for R* = 2.0). Using the weakest mechanical forcing which still achieves the desired purge time is beneficial in terms of the mechanical energy use, size, and upfront cost of the mechanical supply, hence a lower R* value is preferable if using a hybrid system which incorporates a mechanical supply ³ . To achieve t _HP < 30 minutes we would therefore select ${\widehat{Q}}_F=1.0$ and R* = 0.5 such that Q _F = 0.178 m³s⁻¹, ${a*}_1=0.120\hspace{0.17em}{\mathrm{m}}^2$ and ${a*}_2=0.240\hspace{0.17em}{\mathrm{m}}^2$ .

While the results in Figure 11(a) may suggest that an R* value lower than 0.5 would be preferable, note that Figure 11(b) highlights that the sum of the effective areas for R* = 0.5 and R* = 2.0 exceeds that for R* = 1. This increase in the sum of the vent areas becomes more pronounced as R* decreases further, for example, from (54), if R* = 0.5 then a₁c₁ + a₂c₂ = 0.360 m², whereas if R* = 0.25 then a₁c₁ + a₂c₂ = 0.555 m² – the latter a 53.7% increase from R* = 0.5 and an 82.2% increase from R* = 1. Thus for R* values further from unity, the sum of the effective vent areas must increase in order to ensure that $A*={A*}_r$ , and this may not be practical. In addition, (51) limits how small R* can be in order that displacement flow is maintained. If the criteria in (51) is not satisfied then a bidirectional exchange flow will occur at the ceiling-level vent, and such exchange flows have been shown to prolong a natural purge. ¹⁶

VII. Conclusions

Following within the simplifying framework of assumptions that underpin the current industry guidance on the calculation of natural ventilation flows, we have developed analytical solutions, supported by corresponding laboratory experiments, for predicting the transient flows associated with the purging of excess heat from a room using a hybrid ventilation strategy. The analytical solutions we derive, including for the time taken to complete a purge, offer practitioners the capability to expedite the first-order design of a hybrid system. Given the absence of any previous mathematical modelling on hybrid purge flows, we focused on an idealised scenario in which the building fabric was highly insulating and there was no reliance on wind to assist the purge. The hybrid strategy considered comprised the simultaneous use of a mechanical supply at floor level combined with stack-driven natural displacement ventilation via vents in the floor and ceiling.

A priori, one may have reasonably anticipated that a mathematical model for a hybrid purge would require no more than a straightforward extension to the theory that underpins a natural purge – and that there would be no change in the basic behaviour of the flow, other than for the rate of purging to increase. However, to the contrary, our work has shown that the behaviour is highly nuanced, the hybrid purge typically taking place across two distinct flow regimes with the time taken to complete the purge being the summation of the elapsed time in each. The rate of purging is indeed enhanced, with the ever-decreasing depth of the layer of warm air varying from a quadratic dependence in time for a natural purge, to a linear dependence for a mechanically-dominated hybrid purge.

Our approach reveals that the time to complete a purge depends on two ratios, (i) the strength of the mechanical supply relative to the natural stack-driven component $\left({\widehat{Q}}_F\right)$ and (ii) the relative areas of the floor-level and ceiling-level vents (R*). If the aforementioned mechanical forcing ratio is below a critical value, the purge will commence in the forward flow regime before transitioning to the reverse flow regime before all of the excess warm air has been purged. Conversely, if this ratio exceeds the critical value, the reverse flow regime is established and maintained throughout the entire purge. A potential advantage of the purge being maintained solely in the reverse flow regime is in preventing the ingress of external pollutants through the vents.

In our model, we restrict the vigour of the inflow in order to prevent interfacial mixing and restrict the area of the ceiling-level opening in order to prevent the onset of unbalanced exchange flows. These behaviours are the focus of ongoing research, and are likely to prolong a hybrid purge, as well as influencing the spatial distribution of internal heat during the purge.