Modelling Air Duct Leakage

Jan Hensen, Energy Systems Research Unit, University of Strathclyde, Glasgow

The simple duct system used in practice consists of conditioned space, an adjacent zone, a fixed flow fan, one supply duct and one return duct as is shown in Figure 1. Four test cases are used in the system test. Further, the system assumes no supply and return leaks, no zone pressurization or depressurization, and non neutral conditions.

1 Duct system configuration

Assumptions:

Steady state
Air density = 1.2 kg/m3
Turbulent flow
Outdoor and adjacent zone pressures are assumed to be 101,325 Pa
No stack and height effect
Pressure differences are with respect to the outdoor pressure
No heat transfer at building envelope and fan & AC
Convection boundary condition only at outside surface of a duct
The relationship between pressure and airflow for building envelope, supply and return leaks is

where
m...Airflow rate [kg/s]
C...Flow coefficient
n....Flow exponent (0.65)
ro..Air density [1.2 kg/m3]
dP.Pressure difference [Pa]

1.1 Parameters used for airflow and pressure calculation

Fan

fixed air flow 0.792864 kg/s (1400 cfm)

Supply duct

lenght 20 meters
hydraulic diameter 0.42 meters
cross-section area 0.13935 m²
duct roughness dimension 0.00015 m
turbulent dynamic loss coefficient 0.3 (fitting coefficient) 2 duct fittings, one before supply leak, one after supply leak

Supply Leak

located in middle of supply duct (10 meters from each end)
flow coefficients shown in Table 3.20
flow exponent 0.65

Return duct

lenght 4 meters
hydraulic diameter 0.6 meters
cross-section area 0.28274 m²
duct roughness dimension 0.00015 m
turbulent dynamic loss coefficient 0.1 (fitting coefficient) 2 duct fittings, one before return leak, one after return leak

Return Leak

located in middle of supply duct (2 meters from each end)
flow coefficients shown in Table 3.20
flow exponent 0.65

Test Case Flow Coefficients

Test Case	Flow Coefficients "C"
Test Case	Envelope	Supply Leak	Return Leak
1	0.072	0	0
2	0.072	0.008	0.53
3	0.072	0.019	0.0383
4	0.072	0.019	0.13

Output requirement

Airflows: Supply and return leaks, building infiltration/exfiltration [kg/s (CFM)]
Pressures: Indoor pressure [Pa]

1.2 Parameters used for temperature calculation

Input:

Duct inlet temperature (13°C)
Outdoor temperature (35°C)
Indoor temperature (25°C)
Adjacent zone temperature (30°C)
Suply duct is exposed to outdoor temperature
Return duct is exposed to adjacent zone temperature

Output:

Temperatures at supply register and coil inlet [ C]
Duct conduction loss [W] for supply and return
Duct leakage loss [W] for supply and return
Energy loss due to building infiltration [W]

2. Simulation

The simulation program ESP-r was used for the problem solution.

2.0 Problem description

The model was prepared using the ESP-r program manager and other inbuilt tools. The main problem description file was named "duct" ("duct-1")

2.1 Zone description

Fig.2. Zone scheme

The zone description file defines the conditioned space, envelope constructions and boundary conditions.

The conditioned space is described as cube with dimensions 3x3x3 m and named "zone1"(file "zone1.geo"). All the cube's surfaces were attributed with the multi-layer construction "External wall" from the multi-con data base and bounded with a constant temperature of 35 °C. The size of zone and nature of construction are not important in this example, because the assumptions are steady state, constant temperature and no heat transfer at the building envelope. (for simulation, ideal control within the zone stabilises indoor temperature by heat flux).

2.2 Plant network description

Fig.3. Plant network scheme

The plant network file describes the plant network, and contains all parameters which are important for solving energy flows and temperatures for the plant.

The plant network (Fig. 3) (file "duct.pnf") is composed of 6 components. They are defined by heat flow parameters (mass, specific heat, diameter...) and mass flow parameters (fluid type, diameter, length , roughness...) for every component. Connections are defined between components (connection type 3), through zone (connection type 4) and to adjacent zone (connection type 2), and also containment's of components (Adjacent zone t =30°C, or Outdoor t=35°C).

List of components:

RD1-Return Duct
MB -Mix Box
RD2 -Return Duct
AC -Air-condition Unit
SD1 -Supply Duct
SD2 -Supply Duct

Fan wasn't specified as component because of the assumption of no heat transfer at fan.

Two varations of model were prepared, "duct" (" duct.pnf", duct.ctl") when AC unit was described like component "temperature source" (constant temperature without control), and "duct-1" ("duct-1.pnf", "duct-1.ctl") when AC-unit was described like component "AC unit" (with control for constant outlet temperature).

2.3 Mass flow network description

Fig.4. Mass flow network scheme

The mass flow nodal network description (file) describes building and plant fluid mass flow network. Based on this description mass flow simulation can be pursued in tandem with the energy balance computations.

The mass flow network (Fig. 4) (file "duct.afn") is composed of 8 nodes (N1-N4 between components, N5 in zone1 and outside nodes for Crack, Supply leak and Return leak). The nodes are connected through 8 components, where the type of each component defines a relation between mass flow and pressure difference). Because there are 4 Test Cases, 4 files were prepared, with each file containing different coefficients for leaks:

"duct1.afn" for Test Case 1
"duct2.afn" for Test Case 2
"duct3.afn" for Test Case 3
"duct4.afn" for Test Case 4.

The precedure prior to solving the mass flow was to substitute the relevant file (above) and rename it, duct.afn.

2.4 Control description

All building and plant control details are held in one file - the configuration control file. This holds details on all sensor and actuator locations and defines the time dependent operation of the active controllers which link a sensor and acuator throughout a simulation.

A configuration control file called " duct.ctl" defines a control law (type 1) invoking ideal control of zone temperature. A sensor measures indoor temperature in the conditioned space (Zone 1) and the actuator stabilizes indoor temperature.

Another configuration control file called " duct-1.ctl" defines a control law (type 2) invoking ideal control of zone temperature but with one control loop defining where the sensor measures the temperature in the AC unit outlet and the actuator in the AC unit.

All operations are defined for a single time period (time independent).

2.5 Running the simulation

The integrated simulation was run for the two variations and four test cases. Because there were some problems in results when different time steps (zone, plant) were chosen, the problem was simulated by using many different time steps, finally a 10 zone time steps per hour and 1 plant time step per zone step was chosen, for control type 1 (duct). And a 20 zone time step per hour and 1 plant time step per zone step was chosen, for control type 2 (duct-1).

The simulation period selected was a single day in summer (July 9), the system is steady state (as stated previously) and the results are constant throughout whole period. There was a small oscillation in mass flow through a supply leak in control type 2.

3.Results

The required results were extracted from the result libraries, they are; mass flow through leaks and cracks, pressure in the conditioned space and air temperatures in the plant.

Mass flow was recalculated to CMF (multiply by 1765.75) and as a percentage of fan mass flow (0.792864 kg/s).

Results of ESP-r simulation control type 1 (duct)

Test case	Air Flow ESP-r results 1									Pressure
	out_cr - node5			out_su - node4			out_re - node1			nodle5
	/fan	kg/s	CMF	/fan	kg/s	CMF	/fan	kg/s	CMF	Pa
1	0	2.39E-09	0	0	0	0	0	0	0	-2.8E-12
2	-5.4%	-0.0429	-75.8	-4.8%	-0.0384	-67.7	10.2%	0.0812	143.5	0.4003
3	5.3%	0.0418	73.8	-10.09%	-0.080	-141.2	4.81%	0.0382	67.4	-0.3896
4	-0.014%	-0.00011	-0.195	-10.4%	-0.0822	-145.2	10.4%	0.0827	146.1	4.18E-05

Results of ESP-r simulation control type 1 (duct) are compared to results obtained from the Florida Solar Energy Center.

Test case	Envelope crack			Supply leak			Return leak			Zone Pressure
	Florida results	Difference		Florida results	Difference		Florida results	Difference		Florida results	Difference
	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES
	kg/s	kg/s	%	kg/s	kg/s	%	kg/s	kg/s	%	Pa	Pa	%
1	0	2.4E-09	-	0	0	-	0	0	-	0	-2.8E-12	-
2	-0.04278	-0.00014	0.33%	-0.03813	-0.00023	0.6%	0.080905	0.000365	0.4%	0.389	0.0113	2.8%
3	0.04156	0.00026	0.62%	-0.07971	-0.00028	0.35%	0.038148	2.2E-05	0.06%	-0.372	-0.0176	1.52%
4	-0.00103	0.000915	-826%	-0.08184	-0.00039	0.47%	0.082867	-0.00014	0.17%	0.0013	-0.00126	-301%

Results of ESP-r simulation control type 2 (duct-1)

Test case	Air Flow ESP-r results 2									Pressure
	out_cr - node5			out_su - node4			out_re - node1			nodle5
	/fan	kg/s	CMF	/fan	kg/s	CMF	/fan	kg/s	CMF	Pa
1	0.00%	-1.7E-09	0	0	0	0	0	0	0	1.69E-12
2	-5.41%	-0.0429	-75.8	-4.83%	-0.0383	-67.6	10.25%	0.0813	143.6	0.4005
3	5.29%	0.0420	74.1	-10.28%	-0.0815	-143.9	4.82%	0.0383	67.6	-0.3918
4	-0.06%	-0.00045	-0.8	-10.56%	-0.0837	-147.9	10.4%	0.0828	146.2	0.00036.

Results of ESP-r simulation control type 2 (duct-1) are compared to results obtained from the Florida Solar Energy Center

Test case	Envelope crack			Supply leak			Return leak			Zone Pressure
	Florida results	Difference		Florida results	Difference		Florida results	Difference		Florida results	Difference
	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES	Florida results	ES-Flo	(ES-Flo) /ES
	kg/s	kg/s	%	kg/s	kg/s	%	kg/s	kg/s	%	Pa	Pa	%
1	0	-1.7E-09	-	0	0	-	0	0	-	0	1.7E-12	-
2	-0.0428	-0.00014	0.34%	-0.03813	-0.00016	0.41%	0.08091	0.000395	0.49%	0.389	0.0115	2.87%
3	0.0416	0.00041	0.98%	-0.07971	-0.00179	2.2%	0.03815	0.000132	0.34%	-0.372	-0.0198	5.05%
4	-0.001	0.00057	-127.4%	-0.08184	-0.00191	2.28%	0.08287	-5.7E-05	-0.07%	0.0013	-0.0009	-259%

Resultant temperatures extracted from ESP-r are rounded to the nearest one decimal place. These temperatures are same for all test cases.

Conduction losses were calculated as follows:

where
M...Airflow rate [kg/s]
c...Specific heat of air [1007 J/kg K]
dT..diference of inlet and outlet temperature [K]

Test Case Return Temperature Return Air Flow Return Conduction Loss Return Leak Air Flow Return Air Leak Loss

Duct Inlet Leaking Point AC Inlet

Zone 1 RD1 SD2 SD1 SD2 SD1 SD2 Sum . .

°C °C °C kg/s kg/s W W W kg/s W

1 25 25.04 25.1 0.79286 0.79286 31.9 47.9 79.8 0 0

2 25 25.04 25.1 0.71159 0.79286 28.7 47.9 76.6 0.08127 405.9

3 25 25.04 25.1 0.75469 0.79286 30.4 47.9 78.3 0.03817 190.6

4 25 25.04 25.1 0.71013 0.79286 28.6 47.9 76.5 0.08273 413.2

Test Case Supply Temperature Supply Air Flow Supply Conduction Loss Supply Leak Air Flow Supply Air Leak Loss

AC Outlet Leaking Point Duct Outlet

AC SD1 SD2 SD1 SD2 SD1 SD2 Sum . .

°C °C °C kg/s kg/s W W W kg/s W

1 13 13.4 13.7 0.79286 0.79286 319.4 239.5 558.9 0 0

2 13 13.4 13.7 0.79286 0.7545 319.4 227.9 547.3 0.03836 448.1

3 13 13.4 13.7 0.79286 0.71287 319.4 215.4 534.7 0.07999 934.4

4 13 13.4 13.7 0.79286 0.71063 319.4 214.7 534.0 0.08223 960.5

Test case Indoor Temperature Outdoor Temperature Crack Air Flow Infiltration Loss

°C °C kg/s W

1 25 35 0.000 0.0

2 25 35 -0.04292 -432.2

3 25 35 0.04182 421.1

4 25 35 -0.00011 -1.1

Test Case	Return Temperature			Return Air Flow		Return Conduction Loss			Return Leak Air Flow	Return Air Leak Loss
	Duct Inlet	Leaking Point	AC Inlet	Return Air Flow		Return Conduction Loss			Return Leak Air Flow	Return Air Leak Loss
	Zone 1	RD1	SD2	SD1	SD2	SD1	SD2	Sum	.	.
	°C	°C	°C	kg/s	kg/s	W	W	W	kg/s	W
1	25	25.04	25.1	0.79286	0.79286	31.9	47.9	79.8	0	0
2	25	25.04	25.1	0.71159	0.79286	28.7	47.9	76.6	0.08127	405.9
3	25	25.04	25.1	0.75469	0.79286	30.4	47.9	78.3	0.03817	190.6
4	25	25.04	25.1	0.71013	0.79286	28.6	47.9	76.5	0.08273	413.2

Test Case	Supply Temperature			Supply Air Flow		Supply Conduction Loss			Supply Leak Air Flow	Supply Air Leak Loss
	AC Outlet	Leaking Point	Duct Outlet	Supply Air Flow		Supply Conduction Loss			Supply Leak Air Flow	Supply Air Leak Loss
	AC	SD1	SD2	SD1	SD2	SD1	SD2	Sum	.	.
	°C	°C	°C	kg/s	kg/s	W	W	W	kg/s	W
1	13	13.4	13.7	0.79286	0.79286	319.4	239.5	558.9	0	0
2	13	13.4	13.7	0.79286	0.7545	319.4	227.9	547.3	0.03836	448.1
3	13	13.4	13.7	0.79286	0.71287	319.4	215.4	534.7	0.07999	934.4
4	13	13.4	13.7	0.79286	0.71063	319.4	214.7	534.0	0.08223	960.5

Test case	Indoor Temperature	Outdoor Temperature	Crack Air Flow	Infiltration Loss
Test case	°C	°C	kg/s	W
1	25	35	0.000	0.0
2	25	35	-0.04292	-432.2
3	25	35	0.04182	421.1
4	25	35	-0.00011	-1.1