BPS5234A Chap.1 Governing Equations for Building Airflow
Governing Equations for Building Airflow
The balance statements under every simulation
Every airflow result in this course is the numerical solution of a few conservation laws. Mass is neither created nor destroyed, so the air entering a room must leave it or accumulate. Momentum changes only when pressure, viscous stress or a body force such as gravity acts.
Energy is conserved, so heat carried in by supply air, conducted through walls or released by occupants appears as a temperature field.
Species such as carbon dioxide, water vapour and volatile organic compounds are carried and diffused by the air and added or removed at their sources.
Why buildings need the coupling between them
In most building flows density can be treated as constant, which keeps the continuity equation simple. Temperature still matters because warm air rises.
The Boussinesq approximation reconciles the two by allowing density to vary only where it multiplies gravity, so the energy equation feeds a buoyancy force into the momentum equations.
That single link is the difference between a model that can show a stack effect, a thermal plume or displacement ventilation and one that cannot.
One template, many physics
The lecture then shows that every equation has the same anatomy: storage, transport by the flow, diffusion and a source. Only the transported variable, its diffusion coefficient and its source change.
Seeing the template turns equation choice into a design decision, and the course states that decision as a rule: solve only what the governing physics requires.
Isothermal wind studies need flow alone; heated rooms add energy; air quality adds species; rain and aerosols add a particle phase.
What to be able to do
Expect to name the equations a given study needs and justify leaving others out, explain the Boussinesq coupling in words, and recognise the term in the generic equation that carries a particular effect.
These are short-answer skills, and they underpin the justification section of every assignment report.
What this chapter covers
- 01
Continuity in integral and differential form
- 02
The continuum assumption and the Knudsen number test
- 03
Momentum, pressure gradients and body forces
- 04
Energy in temperature form and buoyancy coupling
- 05
Species and humidity transport with sources
- 06
The generic transport equation and equation selection
Worked example · free
Checking that air behaves as a continuum near a louvre
- 1Write the Knudsen number as mean free path over the characteristic length.
- 2Substitute consistent units and evaluate it.
- 1Compare with the continuum threshold and conclude.
Key terms
- Continuity Equation
- The statement of mass conservation: whatever builds up inside a volume must have flowed in across its faces.
- Knudsen Number
- The ratio of molecular mean free path to a characteristic length, used to confirm that air can be treated as a continuum.
- Body Force
- A force acting on the whole volume of fluid, such as gravity, buoyancy or a fan, rather than on its surface.
- Fick's Law
- The rule that a species diffuses down its concentration gradient at a rate set by a diffusion coefficient.
- Source Term
- The part of a transport equation that adds or removes the transported quantity, such as occupant heat or a pollutant emission.
- Generic Transport Equation
- The single balance of storage, convection, diffusion and source from which every governing equation in the course can be written.
Governing Equations for Building Airflow FAQ
Why does a building simulation treat air as incompressible when its density changes with temperature?
The density changes in rooms and streets are only a few per cent, too small to alter mass conservation noticeably. They are kept where they matter, in the gravity term of momentum, through the Boussinesq approximation, so buoyant plumes and stratification still appear while the continuity equation stays simple.
When can the energy equation be switched off?
Only when temperature differences cannot drive or modify the flow, as in a neutral wind study around buildings without heated surfaces. The urban array lab switches gravity off for exactly that reason. Rooms with occupants, equipment, solar gain or fire need energy solved.
What does the generic transport equation add beyond the individual equations?
It shows that heat, momentum, pollutants and turbulence quantities obey the same balance with different coefficients. That is why a single finite volume solver can handle all of them, and why switching physics on or off is a configuration decision rather than a new method.
How is humidity modelled?
As one more species: the water vapour mass fraction is carried by the flow, diffused, and given sources at occupants, wet surfaces, humidifiers and dehumidifiers, with condensation or evaporation at boundaries. It matters for comfort in warm, humid climates.
Is the integral or the differential form the one a solver uses?
A finite volume solver is built on the integral balance applied to each cell, while the differential form describes local behaviour at a point. Both express the same law; the integral form also remains meaningful across discontinuities.
Assessment move
For every working example you meet, write one line listing the equations it needs and one line naming any equation it could drop and why. Doing this before looking at the set-up sheet trains the selection rule that the reports and quizzes keep testing.
