# Modelling Air Duct Leakage

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

## Contents:

Fig.1. System scheme

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:

• 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" 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)
• 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 ES-Flo (ES-Flo) /ES ES-Flo (ES-Flo) /ES ES-Flo (ES-Flo) /ES 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 ES-Flo (ES-Flo) /ES ES-Flo (ES-Flo) /ES ES-Flo (ES-Flo) /ES 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