A Study of Air Movement in the Cavity of a Brick Veneer Wall

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1 Christopher Hannan, M.A.Sc. student a, Dominique Derome, Associate professor a Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Canada c_hannan@alcor.concordia.ca Abstract As part of a project on moisture transfer in walls with porous cladding that have been wetted by rain water and subsequently exposed to solar radiation, this article aims to discuss the movement of air within the cavity between the cladding and backwall. The effect of wind pressure notwithstanding, convective movement of air in the cavity is the result of the increased temperature of the cladding due to solar radiation. The main component of the project is the development of a control volume model that calculates air movement driven by the pressure gradient between the top and bottom of the cavity, as well as heat transfer by conduction, convection and radiation. Ongoing and future advances of the model are discussed, including the development of a profile of vapor pressures throughout the wall assembly. The model is validated by comparison of its results with the results of analytical models taken from literature. Finally, the experimental setup to be used to further confirm the previous findings is presented. Nomenclature Symbols : A elemental surface, m² C coefficient matrix L length, m Nu Nusselt number P pressure, Pa R known value matrix R 1 thermal resistance between the cavity and the exterior, m². C/W R 2 thermal resistance between the cavity and the interior, m². C/W S storage term T temperature, C Y aerodynamic losses in the cavity c specific heat, J/kg C d hydraulic diameter, m g vapor flow rate, kg / m² h heat transfer coefficient, W/ m² C k thermal conductivity, W/m C q heat flow rate, W/ m² t time, s v speed, m/s x x position (horizontal), m y y position (vertical), m z z position (depth), m Greek Letters : σ Stephen-Boltzman constant, 5.67 x 10-8 W/m 2 K 4 δ permeability, kg/s/m/pa ε emissivity ξ moisture capacity, kg/kg ρ density, kg/m 3 θ temperature, C φ relative humidity, % 1

2 Indices / Exponants : top extremity a air avg average cd conduction cv convection e exterior i time step in x direction j time step in y direction L section length m mass r radiation s surface sat saturation t-1 previous time step v vapor 1. Introduction Air movement within brick wall cavities can be caused by a temperature gradient between the top and bottom of the cavity. Wind pressures are not considered in this paper. This article presents a model that determines how the cavity air is affected by the exposure of the brick cladding to solar radiation and rain. Air movement is caused by the pressure difference resulting from the aforementioned temperature gradient, as well as the effects of thermal convection, conduction and radiation on the wall assembly. The main equations describing vapor flow through the assembly, while not yet fully implemented into the model, are also discussed. The model results will be validated by comparing them to analytical equations developed by Hens [1]. Finally, an experimental setup to measure the air movement in the cavity to further confirm the model results will be presented. 2. Model Parameters and Assumptions The computational domain of the model is composed of a brick wall with a height of 3 meters and a width of 1 meter. Inside the cladding, a 25 mm air cavity is followed by a back wall composed of fiberboard sheathing, wood studs, insulation and interior gypsum board. This wall assembly is typical of one-storey buildings in Canada, such as bungalows. The computational domain is subdivided into 100 nodes where the temperature is calculated over a period of 48 hours. Eventually, the vapor pressure at each node will also be calculated in an analogous process. The model is developed in two dimensions, and uses control volume theory for the flows of heat and moisture. The selected climate is Montreal, Canada during the month of July, when solar radiation is usually the most elevated of the year. The exterior air temperature is assumed to follow a sinusoidal curve that varies from a low of 15 C overnight to 26 C during the afternoon. The interior temperature is maintained to be constant at 23 C. Figure 1 demonstrates the configuration and specifications of the model. 2

3 Figure 1: Configuration of Nodal Network (not to scale) 3. Equations The equations used to calculate the nodal temperatures are taken from literature. They represent the heat transfer between each node. Most equations are derived from the theory of conduction, convection and radiation. However, in the cavity, the effects of the mass transport of the air also affect heat transfer. These are the result of the air displacement caused by the temperature gradient. The equations describing vapor pressure are equally taken from literature, and most are analogous to those from heat transfer. These are presented for information purposes only as they have not been fully implemented into the model. 3.1 Thermal Conduction Two equations are used to describe heat transfer by thermal conduction; both of which are derived from Fourier s Law. Equation 1 is for transfer in the x-direction, while equation 2 is for transfer in the y-direction. q q kδyδz = ( Ti 1 T ) Δx (1) kδxδz = ( Tj 1 T ) Δy (2) cdx, i i cdy, j j 3

4 3.2 Thermal Convection Equations describing thermal convection fall into two categories, those for the interior and exterior walls, and those for the air movement within the cavity. In the first case, it is common to combine heat transfer by convection and long-wave radiation exchanges with the interior and exterior surroundings. This phenomenon is calculated via equation 3, in which the values of h cv are assumed by Hutcheon [2] to be 8.3 and 34 W/m 2 K for interior and exterior facades, respectively, under normal conditions. q cv, i cv ( s i / e = h A T T ) (3) In the second case, the heat transfer coefficient h cv must be determined by the Dittus- Boehler correlation, as shown in equation 4. The Nusselt Number is dependant upon the velocity of the air in the cavity, which will be calculated lower. Nu L L h cv = (4) ka 3.3 Thermal Radiation Two sources of heat transfer by thermal radiation must be considered, solar radiation and radiation exchange between the two surfaces of the cavity. Solar radiation only affects the nodes of the exterior surface of the brick. The calculation method for solar radiation was elaborated by Athienitis [3] and assumes a south-facing wall, where solar energy is maximal at above 1000 W/m 2 at noon. The profiles of the magnitude of solar radiation, as well as the outdoor temperature, can be seen in Figure Solar Rad. (W/m²) Outdoor Temperature Solar Radiation Temperature ( C) time (hours) Figure 2: Solar Radiation and Outdoor Temperature Profiles over 48 hours 4

5 Within the cavity, the radiation exchange between the interior surface of the brick and the fiberboard are significant contributors to the heat transfer in the system. In order to calculate this phenomenon, equation 5, developed by Hutcheon, may be used. Due to the complexity of manipulating temperatures of the fourth power, equation 6 presents a linearized version of equation A σ ( Ti + 1 Ti 1) q r, i = (5) ε ε 1 2 r, i = r i+ 1 i q h A ( T T 1) (6) The radiative heat transfer coefficient h r from equation 6 may be calculated via equation 7, which makes use of the overall average temperature in the cavity. 4 σ T h = avg r (7) ε ε Thermal Mass Transfer To determine the precise effects of the heated air passing over the interior surface of the brick on the overall temperature profile in the cavity, it is necessary to first calculate the air speed in the cavity. To accomplish this, equation 8 is presented. It is derived from the textbook of Hens. Δ P = Y (8) 2 a v a In order to calculate the overall average cavity air speed, the air pressure difference between the entrance and exit of the path of the air, as well as aerodynamic losses (expressed as Y), must be determined. Figure 3 shows the trajectory of the air through the cavity and the weepholes at the top and bottom of the brick cladding. 5

6 Figure 3: Path of Air through the Wall Assembly The calculation of the air pressure at the top and bottom of the path is made possible by determining the air temperature at these two locations. At first, these temperatures are assumed, however as the model iterates, the values become increasingly precise. Once the temperatures are stable, the pressure difference may be determined by the ideal gas law. According to Hens, two categories of losses exist within the air flow path: local and friction losses. Local losses are due to air passing into and out of a tube, as well as when the air flow is deviated by a predefined angle. Fiction losses are due to the roughness of the surfaces over which the air passes. In order to accurately tabulate these losses, the air flow path must be subdivided into nine distinct sections, as can be seen in Figure 3. The losses can then be calculated via tables available in Hens textbook. Once the average air velocity in the cavity has been determined, the heat transfer by mass transport may follow by the use of equation 9, defined by Janssens [4]. This heat transfer applies only to the nodes in the air cavity. The nodes used to define the air pressure difference are located at the top and bottom of the exterior brick surface at the entrances of the weepholes. q m, j = a a H a( j 1 j ρ c d v T T ) (9) 6

7 3.5 Calculation of Nodal Temperatures Having calculated all of the heat flows within the nodal network, the full temperature matrix may now be considered. To accomplish this, heat balance equations must be developed for each node of the network, as can be seen in equation 10. As the model is in transient state, the energy stored by each building material within the control volume is incorporated into the right-hand side of the equation. By convention, it is understood that all heat moves toward the direction of the node under consideration. q ρiciδxδy ( Ti 1, Ti ) + qcv, i( Ti 1, Ti ) +... = ( Ti Ti,( t 1) ) Δt cd, i (10) Since all of the heat flow components defined in equations 1, 2, 3, 6 and 9 are functions of two temperatures, each node may have its own corresponding row in the final temperature matrix by redistributing equation 10. Equation 11 is thus obtained. All coefficients of unknown temperatures are placed in the [C] matrix, while all coefficients of the known interior or exterior temperatures, of the previous time step or of solar radiation are placed in the [R] matrix. [ ]{} T [ R] C = (11) By multiplying the [R] matrix by the inverse of the [C] matrix, the complete profile of the 100 nodal temperatures may be obtained for a given time step. 3.6 Vapor Transfer Equations A future development of the model will involve the calculation of a vapor pressure profile for the nodal network. Moisture transfer is considered to be analogous to heat transfer [4], as heat transfer variables are replaced by the corresponding variables for vapor transfer. It must be noted that, unlike heat transfer, some components, such as vapor permeability, are functions of the relative humidity at each node of the cavity. As the brick will be initially considered to be saturated by rain water to a relative humidity of 99.93%, the storage of moisture is considered first. Equation 12 describes this phenomenon. The moisture capacity term, ξ i (φ ), can be defined as the slope of the sorption isotherm for the material in which the node is found. The sorption isotherms and their derivatives are adapted from the work of Kumaran [5]. ρiξi ( φ) ΔxΔy S( φ ) = ( Pi Pi,( t 1) ) (12) Δt P v, sat The moisture stored by the brick may be diffused based on the permeability of the materials and their vapor pressure. From thermal conduction equations 1 and 2, vapor diffusion in the x and y directions can be defined by equations 13 and 14. 7

8 g g δ ( φ) ΔyΔz ( Pi Δx δ ( φ) ΔxΔz ( Pj Δy v, i = 1 i P) v, j = 1 j P ) (13) (14) Vapor flow by convection, conversely, can be defined based on equation 3 to form equation 15. This only applies to nodes bordering the cavity and outside surface. The mass transfer coefficient is defined by a variation of the Chilton-Colburn analogy [6] which is described by equation 16. g h = h A P P ) (15) cv, i m ( s i / e δ ( P, T ) h a cv m = (16) ka In equation 16, the diffusion of vapor in dry air was defined by Schirmer (1938) and can be defined by equation Patm Ti ( T, P) = Pi δ (17) Mass transport of vapor in the cavity is analogous to equation 9, forming equation 18. The termξ a refers to the moisture capacity term for air, which is assumed to be constant at 6.1 x 10-6 kg/kg/pa. g ρ ξ v P P ) (18) m, j = a a a ( j 1 j Finally, the moisture transfer flows from equations 13, 14, 15 and 18 can be summed similarly as in equation 10, and then equated to the storage term which is also analogous to heat transfer. The vapor pressure profile at all 100 nodes may then be calculated by Gaussian elimination as was explained for the temperature profile. 4. Results and Observations As the model produces a temperature profile for all nodes of the wall assembly, as well as an air velocity profile for various heights within the cavity, it is relevant to examine these results separately. The validation of the results is discussed in section Temperature Profile As of this writing, the model calculates the temperature profile for all of the nodes over the 48 hour period at 10 second intervals. Figure 4 demonstrates the temperature profile at the four surfaces of the system, namely at the inside and outside brick surfaces, on the 8

9 cavity side of the fiberboard and at the inside of the gypsum board. All shown temperature profiles are taken at a height of 1.5 m. As can be seen in Figure 4, the outside surface of the brick undergoes a significant temperature increase due to its exposure to the solar radiation. It is also relevant to note that the inside brick surface (facing the air cavity) will be affected by the solar radiation; the air passing over this surface will thus be indirectly heated by the solar radiation. On a final note, the inside gypsum surface temperature will be almost constant, as it converges to the room temperature of 23 C. Temperature ( C) Ext Brick Int Brick Cavity Int Fibrebrd Ext Gypsum time (hours) Figure 4: Temperature profile ( C) at height 1.5 m at the four surfaces and in the cavity The temperature profile of the air in the cavity as a function of height can be seen in Figure 5. For simplicity, only the maximum curves are shown for the first day of measurement. The maximum temperature in the cavity of 27.5 C is found at the top of the cavity (3.00 m). This result is found to be at approximately 15:00, not coincidentally when the solar radiation also is at its maximum. This would seem to indicate that the heating of the brick by the sun has a direct impact not only on the velocity of the air in the cavity, but also on its temperature. A conclusion that can be drawn is that the air passing through the cavity receives the heat from the brick but does not transfer that heat at the same rate to the back wall. It is therefore reasonable to assume that the cavity air evacuates a significant quantity of heat out of the system. Figure 4 demonstrates this further as the temperature of the fiberboard does not change significantly after contact with the cavity air. In section 5, the temperature profile will be validated by comparison to analytical equations developed by Hens. 9

10 27.6 Temperature ( C) m m time (hours) Figure 5: Maximum air temperatures ( C) at 0.33 m intervals from the bottom to the top of the cavity during the first day (model results) 4.2 Air Velocity Profile The velocity of the air within the cavity, calculated previously, is also a good indicator of the effect of the air movement on the temperature profile of the cavity. Figure 6 illustrates the air velocity profile for heights of m, m, m and m in the cavity. In order to properly validate these results, testing on an experimental setup was performed, and this will be discussed further in section 5. In order for the results to be comparable to those in the laboratory, the outdoor temperature was increased and now varies from 20 C to 23 C, the indoor temperature was decreased to 16 C, and the magnitude of the solar radiation was raised so as to produce a curve similar to that of Figure 2 but with a maximum of 1700 W/m 2. The conditions were modified to match the experimental setup used for validation. As can be seen in Figure 6, the air velocity in the cavity increases upwards. This trend closely follows the magnitude of the solar radiation. However, as the brick cools, the velocity decreases and eventually moves in the opposite direction during the overnight hours, albeit at a slower flow rate. During exposure to the solar radiation, the air velocity at the various locations achieves peaks ranging from 0.18 to 0.25 m/s. Also of note is the apparent acceleration of the velocity in relation to the height of the cavity. 10

11 m 0.15 Air Velocity (m/s) m time (hours) Figure 6: Air velocity profile (m/s) for heights of m, m, m and m in the cavity (model results, temperature and solar radiation for validation setup conditions) 5. Validation The model results obtained are validated by two different methods. The cavity node temperature profile is compared to a temperature profile obtained separately by analytical equations. The velocity profile is compared with results obtained experimentally. 5.1 Temperature Profile Validation Hens has developed a series of analytical equations to calculate the gradient of temperature for the cavity air. Equation 19 displays the final results of his derivation, where term b 1 is calculated via equations 20 through 23. z θ cav = θcav, ( θcav, θcav,0) exp b1 (19) ρacadhva b1 = hcv( 2 C1 C2) (20) 1 hcv hcv + hr + + hrhcv R C 2 = 1 (21) D 11

12 1 h hcv + hr + R C 1 = 2 D 1 D = hcv + h r + h R 1 cv + cv h h r + h cv r + 1 R 2 - h 2 r (22) (23) Figure 7 demonstrates the profile of the temperature as a function of the cavity height obtained from Hens equation. While the shape of the curves is different, the trends observed in Figure 5 are quite similar. The maximal values are obtained at the same time, 15:00, while the temperature at the base of the cavity is the same at 26.2 C. The maximum temperature is slightly lower, at 27.3 C from 27.5 C, but this remains within the margin of error. It is therefore reasonable to conclude that the model results are quite comparable to those obtained analytically Temperature ( C) m m time (hours) Figure 7: Maximum temperatures ( C) at 0.33 m intervals from the bottom to the top of the cavity during the first day (analytical results from Hens) 5.2 Air Velocity Profile Validation An experimental setup has been constructed to aid in the calculation and later visualization of the air velocity in the cavity. The setup can be seen in Figure 8. The interior backwall is composed of 39 mm by 140 mm wood studs with glass fiber insulation in between. The studs are sandwiched by fiberboard panels to the outside and gypsum board to the inside. The cladding is composed of 90 mm thick brick veneer, and the setup allows for an air cavity of variable thickness. On the inside of the wall assembly, a test hut will maintain the interior environmental conditions at a constant temperature of 16ºC, as discussed previously. For the purposes of this experiment, the 12

13 cavity has been maintained at an approximate thickness of 25 mm, although a slight incline of the backwall has proven in practice to cause small errors. The cavity is enclosed on the sides by a polyethylene barrier (with small holes punctured to allow for the velocity measurements), while the top and bottom are open to the surroundings. A uni-directional wind velocity meter with accuracy of ±0.015 m/s is used to determine the upward velocity in the cavity at various heights. It should be noted that small lateral velocities in the magnitude of 0.01 to 0.02 m/s were detected indicating a mild amount of turbulence that can also affect results. Figure 8: Experimental setup allowing the visualization of air movement in the cavity and simulation of solar radiation on the wall assembly by infrared lamps The brick cladding is exposed to an array of infrared lamps that will simulate uniform illumination from the sun, as can be seen in the second photo of Figure 8. The exterior surface of the brick can therefore be adjusted from 30ºC to over 50ºC. Figure 9 demonstrates the experimental results, while Figure 10 shows a close-up view of the model results during the corresponding period of testing. While the inside brick surface temperatures are almost identical, there are small discrepancies for the velocity measurements. While it is apparent that no clear trends can be observed from the experimental velocity measurements due to the errors previously described, the magnitude of the results can be taken between 0.20 m/s and 0.32 m/s once they have reached steady-state. This range is slightly above the 0.18 to 0.25 m/s determined by the model, but it does indicate a relative coherence between the two methods. 13

14 Temperature (ºC) Inside Brick Surface Temperatures 1.20 m 2.10 m 0.30 m Recorded Air Velocities Air Velocity (m/s) :00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 time of day T top T mid T bot 0.0 m 0.6 m 1.2 m 1.8 m 2.4 m Figure 9: Air velocity recorded at heights of 0.0m, 0.6 m, 1.2 m, 1.8 m and 2.4 m in the cavity, compared to the top, middle and bottom inside brick surface temperatures (experimental results) Inside Brick Surface Temperatures 0.50 Temperature ºC m Air Velocity (m/s) Air Velocities m :00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 time of day Figure 10: Air velocity profile for heights of m, m, m and m in the cavity, compared to the top and bottom inside brick surface temperatures (model results) 6. Future Work 6.1 Incorporation of Moisture Transfer into Model As seen in this text in equations 12 to 18, the equations required to implement moisture transfer are analogous to those used for heat transfer. Their incorporation into the model is ongoing with results to be obtained in the very near future. 14

15 6.2 Validation by Particle Image Velocimetry The wall assembly described in section 5.2 is built to allow the movement of the air to be visualized through a series of windows and mirrors. Visualization of the air movement within the cavity is made possible by Particle Image Velocimetry, or PIV. PIV is a technique that has been developed to measure the instantaneous velocity of a fluid in a given area. The air in the cavity is impregnated with predetermined particles, such as oil droplets, that become suspended within the air flow. The flow field is then illuminated by a laser, and a high-frequency camera registers the position of the particles through a fiberglass window placed perpendicularly to the cavity at selected time intervals. The displacement of a given oil droplet between each photograph is then found, from which the velocity of the air may be determined. At first, PIV will be used to study the magnitude and direction of the air movement globally within the cavity. However, local tendencies will later be incorporated into the study. 6.3 Validation of Mass Transfer Coefficients A wind tunnel is presently under construction as the central part of an experiment to assist in the validation of the methods used to determine convective mass transfer coefficients. A current photograph and drawing of the setup can be seen in Figure 11. A layer of brick will be placed below an air stream having the same thickness as the cavity, 25 mm. Three evenly spaced moisture-laden bricks will be left to move freely, and will be placed on load cells. As the air velocity, temperature and relative humidity conditions in the closed loop are controlled, it will be possible to determine the rate of moisture loss from the brick and thus the mass transfer coefficient may be obtained. While the wind tunnel fan will produce the velocity in the system, a contraction will induce the flow into the horizontal cavity. Since the three test bricks will presumably be subject to flow in different phases of development, it will also be possible to determine the effect that this phenomena will have on the coefficient. The convective mass transfer coefficients obtained may then be used to further validate the model vapor transfer results. 15

16 Figure 11: Wind tunnel experimental setup 7. Conclusion A model has been developed to simulate air movement within the cavity of a brick veneer wall assembly. The heat transfer and temperature profile for a wall assembly exposed to solar radiation has also been determined. A vapor pressure profile will also soon be calculated. It has been concluded that the velocity and the temperature of the air in the cavity are greatly influenced by the magnitude of the solar radiation to which the exterior of the assembly is exposed. It is also clear that the cavity air, under the influence of the stack effect and driven by an air pressure differential, evacuates an important amount of heat out of the system before it reaches the back wall. The results and observations of the temperature profile have been validated by comparison to analytical equations. The velocity profile has been favorably compared to experimental results obtained in a laboratory setup. Future validation experiments are also planned using particle image velocimetry for the velocity profile, and a wind-tunnel experiment for the convective mass transfer coefficients of the forthcoming vapor pressure profile. Finally, it is conceivable that the effects of wind will be modeled to further improve the accuracy and applicability of the model. References [1] HENS, H., Building Physics, Wiley Publishing (2006). [2] HUTCHEON, N.B. and HANDEGORD, G.O.P., Building Science for a Cold Climate, IRC (1983). [3] ATHIENITIS, A., Building Thermal Analysis, MathCAD Electronic Books, Mathsoft Inc, (1993). [4] JANSSENS, A., Reliable Control of Interstitial Condensation in Lightweight Roof Systems, Université Catholique de Leuven (1998). [5] KUMARAN, M., A Thermal and Moisture Transport Database for Common Building and Insulating Materials, Final Report from ASHRAE Research Project 1018-RP (2002). [6] CHILTON, A., COLBURN, A.P., Mass Transfer (Absorption) Coefficients, Industrial and Engineering Chemistry 26 (1934). 16