# Experimental Study of Thermal Buoyancy in the Cavity of Ventilated Roofs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Pitched Wooden Roofs

_{2}-emissions from buildings and the favourable carbon footprint of wood make wooden roofs attractive to use in a growing number of buildings. The use of pitched wooden roofs on larger building structures with longer roof spans has been introduced, further underlining the complexity as well as the need for ventilation.

#### 1.2. Driving Forces

#### 1.3. Previous Research

#### 1.4. Knowledge Gap

#### 1.5. Objectives and Scope

- How are the temperature conditions in the air cavity related to the air cavity design?
- How is the airflow through the cavity influenced by the air cavity design?
- To what degree may thermal buoyancy drive airflow in the air cavity?

## 2. Theoretical Framework

^{3}/s) is the airflow, ΔP (Pa) is the driving force and S (Pa/(m

^{3}/s)) is the flow resistance. The airflow may be further described by established principles from fluid mechanics, as studied by Kronvall [38]. The fluid mechanics theory applied in the present study is valid if incompressible and laminar flow is assumed. Given the physical conditions present in building applications, negligible error is associated with assuming that air and water vapour are ideal gases, hence incompressible [39]. The Reynolds number, Re (-), used to determine the flow regime in the air cavity, is given by Equation (2) [40].

_{avg}(m/s) is the average velocity of the airflow, D

_{h}(m) is the hydraulic diameter of the cavity and ν (m

^{2}/s) is the kinematic viscosity of the air. Given laminar flow in wide rectangular cavities, u

_{avg}may be approximated to 0.67⋅u

_{max}, where u

_{max}is the maximum velocity of the airflow [41]. Below a critical value of the Reynolds number, Re

_{crit}, the airflow is laminar. Re

_{crit}typically appears in the interval 2000–2500 [38]. The transitional flow regime describes the changeover from laminar to turbulent flow, where fully turbulent flow often occurs at Re > 4000.

_{T}(Pa), is given by Equation (3). Normally, the temperature, and consequently the density, of the air in the cavity varies. In this case, the driving force must be determined by integration from the inlet to the outlet, as given by Equation (4) [42]. If the air cavity can be divided into a finite number of sections, each with an assumed constant temperature, the driving force may be calculated with Equation (5).

_{a}and ρ

_{cavity}(kg/m

^{3}) are the densities of the surrounding air and cavity air, respectively, g (m

^{2}/s) is the gravitational acceleration, and H [m] is the height difference between the inlet and outlet of the air cavity. The density of humid air is given by Equations (6) and (7) [43].

_{a}(Pa) is the partial pressure of dry air (101,325 Pa at ground level), R

_{a}(J/kg K) is the specific gas constant for dry air (287.058 J/kg⋅K), T and T

_{c}are the temperature in K and °C, respectively, p

_{v}(Pa) is the pressure of water vapour, R

_{v}(J/kg⋅K) is the specific gas constant for water vapour (461.495 J/kg⋅K), and RH (-) is the relative humidity.

_{0}, which is an upper limit of the temperature in the cavity.

## 3. Methods

#### 3.1. Laboratory Model

#### 3.2. Experimental Procedure

#### 3.3. Test Setups

#### 3.4. Validation of Experimental Test Regime

## 4. Experimental Results

#### 4.1. Temperature Conditions

#### 4.2. Air Velocity and Flow Conditions

^{−1}) in the interval 30–400 h

^{−1}.

#### 4.3. Driving Force

_{T}) at different roof inclinations, air cavity heights and applied heating power was determined by Equation (5). The calculation takes into account the density decreases of the cavity air as it moves upwards in the cavity. The results are presented in Figure 10. Figure 11a shows the relationship between the thermal driving force and the air cavity height at a roof inclination of 30°. Thermal driving force and airflow rate in the air cavity are presented in Figure 11b. The latter diagram includes results for air cavity heights of 23 mm, 36 mm, 48 mm and 70 mm at all roof inclinations and applied heating powers.

## 5. Analysis of Measurements

#### 5.1. Temperature Conditions

#### 5.1.1. Temperature Profile along the Cavity

#### 5.1.2. Temperature Profile across the Cavity

#### 5.1.3. Influence of Air Cavity Design and Heat Power Level

#### 5.1.4. Heat Power Level Related to Local Climate

#### 5.2. Air Velocity and Flow Conditions

#### 5.2.1. Flow Characteristics

_{avg}, the approximation u

_{avg}≈ 0.67⋅u

_{max}was used. This is valid for wide rectangular ducts [41]. Given a more quadratic cavity, the factor multiplied with u

_{max}will be smaller, and found in the interval between 0.5 and 0.67. Consequently, the Reynolds number may be overestimated.

#### 5.2.2. Influence of Air Cavity Design and Heat Power Level

#### 5.3. Driving Force

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Temperatures (°C) (y-axis), in the cavity air as a function of distance (m) from the cavity inlet (x-axis). Each plot represents a given air cavity height (h) and roof inclination (θ).

**Figure A2.**Temperatures (°C) (x-axis), as function of distance from the bottom unheated surface (m) (y-axis). Each plot represents a given air cavity height (h) and roof inclination (θ).

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**Figure 1.**(

**a**) Top aluminium lid showing the arrangement of XPS and heating foil on the inside of the air cavity; (

**b**) complete test model, at 0° inclination; (

**c**) inclined model with plastic foil cover, at 45° inclination.

**Figure 2.**Position of the thermocouples in the air cavity (T = top, A = air, B = bottom, h = height of the air cavity, H = height difference between the cavity inlet and outlet). Lengths between measuring points are given in millimetres.

**Figure 3.**(

**a**) Validation of the experimental method, presented as a comparison of measured air velocity in initial test run and a control test run for test setup h = 48 mm, θ = 15°; (

**b**) Comparison of methods utilized for smoke production, at test setup h = 48 mm, θ = 15°. The mean is calculated as the average of the five measurements.

**Figure 4.**Analysis of thermocouple position on the heating foil at different levels of heating power. Placement completely on, partially on, and between the electrically conducting heating elements was tested in the middle of the cavity. Placement on the edges of the heating foil width was also included.

**Figure 5.**Temperatures (°C) in the cavity air as a function of distance (m) from the cavity inlet. Each plot represents a given air cavity height (h) and roof inclination (θ).

**Figure 6.**Temperatures (°C) as a function of distance from the bottom unheated surface (m). Each plot represents a given air cavity height (h) and roof inclination (θ).

**Figure 7.**Measured air velocities at different roof inclinations, air cavity heights and applied heating power. Note that measured air velocity is the assumed maximum velocity through the air cavity.

**Figure 8.**(

**a**) Measured air velocity in relation to the air cavity height at a roof inclination of 30°; (

**b**) Calculated airflow rate presented in relation to the air cavity height at a roof inclination of 30°.

**Figure 9.**Reynolds number, Re, as a function of roof inclination. The shaded area represents possible beginning of the transitional flow region.

**Figure 10.**Thermal driving force at different roof inclinations, air cavity heights and applied heating power.

**Figure 11.**(

**a**) Driving force as a function of cavity height at a roof inclination of 30°; (

**b**) airflow rate and driving force for different air cavity heights. Measurements at all roof inclinations are included in the diagram.

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**MDPI and ACS Style**

Bunkholt, N.S.; Säwén, T.; Stockhaus, M.; Kvande, T.; Gullbrekken, L.; Wahlgren, P.; Lohne, J. Experimental Study of Thermal Buoyancy in the Cavity of Ventilated Roofs. *Buildings* **2020**, *10*, 8.
https://doi.org/10.3390/buildings10010008

**AMA Style**

Bunkholt NS, Säwén T, Stockhaus M, Kvande T, Gullbrekken L, Wahlgren P, Lohne J. Experimental Study of Thermal Buoyancy in the Cavity of Ventilated Roofs. *Buildings*. 2020; 10(1):8.
https://doi.org/10.3390/buildings10010008

**Chicago/Turabian Style**

Bunkholt, Nora Schjøth, Toivo Säwén, Martina Stockhaus, Tore Kvande, Lars Gullbrekken, Paula Wahlgren, and Jardar Lohne. 2020. "Experimental Study of Thermal Buoyancy in the Cavity of Ventilated Roofs" *Buildings* 10, no. 1: 8.
https://doi.org/10.3390/buildings10010008