Tutorial 1 Calculating Flow
Rate Through Openings:
-The Equations
[See
Disclaimer and Important Notes]
This tutorial covers:
- Background;
- Volume Flow vs. Mass Flow:
- The 'Crack Flow' Power Law Equation;
- The 'Flat Plate' Orifice Equation;
- Making an Orifice Equation look like a Power Law Equation;
- Changing to Mass Flow;
- Quadratic Flow Representation;
- The Next Step.
Background
The air flow equation is the building block of air
infiltration and ventilation calculations. Every opening in the building
fabric must be accurately represented by such an equation. Essentially
there are two types of opening. The first are the random gaps and cracks
that appear at construction joints or through porous material. These are
ill defined and their geometry is rarely directly measurable. The second
are essentially 'purpose provided' openings such as vents and windows.
These are usually clearly defined in terms of geometry, location and flow
characteristics. Openings in the former category require either
measurement or implied knowledge about their characteristics. The
necessary flow characteristics of the second type of opening can be
inferred from the geometry of the opening and, therefore are easier to
determine. The mathematical representation of these two types of opening
is described in this tutorial. It is also shown that, by algebraic
manipulation, both types of opening can be represented by an equation of
identical structure. This considerable eases the calculation of total air
flow into and out of a space.
Volume vs. Mass Flow
Flow rate can be expressed either in terms of 'volume'
flow (e.g. m3/s, l/s, cfm) or 'mass' flow (e.g. kg/s). Most often
fan capacity is expressed as a volume flow ,whereas calculations involving
heat loss, for example, require that flow be expressed as a mass flow. In
fundamental terms, it is more accurate to express flow rate as a mass
flow. This is because, for a given mass of air, its volume varies with
temperature. The Law of Continuity demands that the mass flow rate of air
entering a building must match the mass leaving the building. Thus, in
theory, a balance in mass flow does not necessarily mean that there is a
balance in volume flow. Notwithstanding this situation, our air
infiltration tutorials will be based on volume flow analysis. This is
because for 'mild' climate temperature differences, where temperature
differences between inside and outside peak at no more than 25 - 30C for a
few hours at most in the year, any error will be marginal and because the
'iterative' approach that will be introduced to calculate infiltration and
ventilation rate is much more stable for volume flow. Any error could also
be minimised by selecting an air density (to convert to mass flow) that is
midway between the outside and inside air temperature. However, switching
to mass flow is very straightforward (as presented at the end of this
tutorial). Also 'multi-zone' models such as Contam96 are structured in
mass flow and can be downloaded and used for precision analyses if needed.
Cracks and Gaps
Air flow through general cracks and gaps in the building
fabric is a function the size and structure of the opening and the
pressure difference acting across it. The simplest representation of a
crack is presented in the equation below.
This equation gives the volume flow rate (e.g. m3/s).
'C' is defined as the 'Flow Coefficient'. This is related
to the size and structure of the opening. Typical flow coefficients are
given in the AIVC Numerical Data Guide.
'n' is the flow exponent and indicates the degree of
turbulence. An 'n' value of 0.5 represents fully turbulent flow and '1.0'
represents fully laminar flow. The typical 'n' value for whole buildings
is 0.66.
'Orifice' Openings
Clearly defined openings such as vents or windows are
frequently represented as 'flat plate orifices' in which the volume flow
is represented in the equation:
Thus flow can be determined directly from a geometric
analysis of the opening. In the absence of other information, the
'discharge coefficient' is usually based on a value of 0.61. Air
flow through an orifice is assumed to be turbulent, thus the flow exponent
'n' = 0.5.
Making an Orifice Flow Equation Look Like a Crack Flow
Equation
For ease of infiltration and ventilation calculations it
is important that the above two equations take on the same form. This is
achieved by making the orifice equation look like a power law crack flow
equation. To achieve this:
Exercise: Manipulate the orifice flow equation to achieve
the above 'C' and 'n' values.
Changing to Mass Flow
Volumetric flow is changed to mass flow by multiplying by
the air density. Since density is dependent on air temperature, it is not
a constant. Therefore, if this approach is used, it needs to be expressed
as a function of temperature. This will be covered in a future
tutorial.
Quadratic Formulation of the Air Flow Equation
Some authorities prefer to express air flow in the form of
a quadratic equation of the form:
This
is perceived to be dimensionally correct since the laminar and turbulent
components of flow are separated. In essence, all the arguments and
processes applied to the Power Law approach can be applied to the
quadratic approach. In fact the single zone model, to be developed in
these tutorials, will run just as easily in quadratic form. Again, at a
later date, an equivalent analysis will be attempted.
The Next Steps
So far we have established the Power law Equation of flow
through an opening as a building block towards infiltration and
ventilation analysis. We have also shown that the flow characteristics of
purpose provided opening can be approximated by the geometry of the
opening. We now have to:
-
Introduce a method to determine the pressure
difference across openings;
-
Establish a 'flow network' that represents the
openings in the building;
-
Find 'missing' data (primarily the 'C' and 'n' values
of cracks and gaps).
-
Develop a mathematical model that enables us to
calculate the total flow into and out of the building and to calculate
the flow rate and flow direction through individual openings.
-
Incorporate mechanical ventilation systems.
-
Introduce the concept of multi-zone or multi - room
networks.
These steps will provide you with a basic design and
ventilation evaluation tool that can be quickly established to solve 95%
of pre-design and basic design steps. It will also give you the skills to
apply more complex modelling tools that are available through the
internet.
This completes Tutorial 1
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