Source: Experimental Physics and Engineering Lab, Texas A&M University
Tags: UAE, universal accounting equation, extensive quantity, intensive quantity, system boundary, open system, closed system, isolated system, steady state, engineering accounting, mass balance, energy balance
The Universal Accounting Equation (UAE) is engineering's general-purpose bookkeeping framework: it tracks any measurable quantity (mass, energy, charge, money) flowing into, out of, and accumulating within a defined system. Before you can apply it, you need to know which quantities scale with system size (extensive) versus which do not (intensive), and how to draw a proper system boundary.
Universal Accounting Equation (UAE)
A single equation that accounts for every extensive quantity in any engineering system by tracking initial amount, final amount, inputs, outputs, generation, and consumption over a chosen time period.
Extensive quantity
A measurable property that depends on the scale (size) of the system. If you double the system, the value doubles.
Intensive quantity
A measurable property that does not depend on the scale of the system. Doubling the system leaves it unchanged.
System
A subset of the universe, defined by the engineer, with a boundary chosen so that specific processes can be monitored and evaluated.
System boundary
The closed surface that separates the system from its surroundings. Must be fully enclosed, can be any shape, and can be rigid or flexible.
Closed system
A system whose boundary permits no mass transfer in or out.
Isolated system
A system whose boundary permits no transfer of mass, energy, or momentum in any direction.
Open system
A system whose boundary may allow mass (and typically energy and momentum) to cross.
Homogeneous system
A system in which properties are uniform throughout.
Steady system (steady-state system)
A system whose state does not change over time, even though flows may still be moving through it.
Unsteady system
A system whose state changes with time in some measurable way.
The distinction matters because the UAE only tracks extensive quantities. If a quantity doubles when you double the system size, it is extensive; if it stays the same, it is intensive.
Extensive examples: mass, moles, volume, area, energy, charge, enthalpy
Intensive examples: temperature, pressure, density, viscosity, colour, melting point, boiling point
A derived quantity can behave differently from its components. Density is mass divided by volume (both extensive), yet the ratio itself is intensive, because it does not change when you scale the system up or down.
A useful rule of thumb: extensive quantities can be counted or accumulated; intensive quantities describe a condition or state.
Consider a laboratory reactor running at 500 K, 300 kPa, with 0.5 L volume, 300 g of reactant, and a catalyst concentration of 50 g/L (25 g of catalyst).
When this process is scaled to a plant processing 50 tonnes of reactant per day:
Temperature (intensive) stays at 500 K
Pressure (intensive) stays at 300 kPa
Catalyst concentration (intensive, derived) stays at 50 g/L
Reactant amount (extensive) scales up enormously
Reactor volume (extensive) scales up
Catalyst amount (extensive) scales up
The intensive quantities carry over directly from the lab. The extensive quantities must be recalculated for the new scale.
A system is whatever region of the universe you choose to analyse. The boundary is the surface you draw around it. Three rules govern that boundary:
It must be a closed surface (fully encloses the system, no gaps)
It can be any shape, and it can be rigid (defining a fixed volume) or flexible (tracking a moving object)
It cannot be redefined partway through a calculation, though the quantities inside it are free to change
Choosing where to draw the boundary is an engineering decision. A well-chosen boundary simplifies the problem; a poorly chosen one can make it unsolvable.
Systems are classified by what their boundary allows to cross and by whether their internal state changes over time.
By boundary permeability:
Closed system: no mass crosses the boundary (energy may still cross)
Isolated system: nothing crosses the boundary, neither mass, energy, nor momentum
Open system: mass may cross the boundary (most real engineering systems fall here)
Non-isolated system: at least one of mass, energy, or momentum can cross
By internal behaviour:
Steady (steady-state): the system's state does not change over time
Unsteady: the system's state changes over time in some way
Homogeneous: properties are uniform throughout the system
A system can combine categories. A sealed thermos flask is approximately closed and isolated. A river section with constant flow is open and steady. A batch reactor heating up is closed and unsteady.
The scaling test (conceptual):
If System × 2 → Quantity × 2, the quantity is extensive. If System × 2 → Quantity unchanged, the quantity is intensive.
No numerical formula here, but this mental model is the fastest way to classify any quantity you encounter on an exam.
⚠️ The UAE applies only to extensive quantities. If you try to "balance" temperature or pressure across a system boundary, you are using the wrong framework.
⚠️ Density looks like it should be extensive (it involves mass and volume), but it is intensive because it is a ratio of two extensive quantities. This is a common exam trap.
⚠️ A closed system is not the same as an isolated system. Closed blocks mass transfer only; isolated blocks mass, energy, and momentum. Mixing these up loses marks.
⚠️ The system boundary must not change during a calculation. You can track changing quantities inside the boundary, but you cannot move the boundary itself mid-problem.
Q: What is the key difference between an extensive and an intensive quantity?
A: An extensive quantity depends on the size (scale) of the system and changes when the system is scaled up or down. An intensive quantity is independent of system size and remains the same regardless of scale.
Q: Classify the following as extensive or intensive: enthalpy, boiling point, charge, viscosity.
A: Enthalpy is extensive. Boiling point is intensive. Charge is extensive. Viscosity is intensive.
Q: Why is density considered intensive even though it is derived from mass and volume, which are both extensive?
A: Density is a ratio (mass/volume). When the system is scaled up, both mass and volume increase proportionally, so the ratio stays the same. Any ratio of two extensive quantities behaves as an intensive quantity.
Q: What three rules must a system boundary satisfy?
A: It must be a closed surface with no gaps. It can be any shape (rigid or flexible). It cannot be redefined during a calculation.
Q: A sealed flask of gas is heated on a hot plate. Is this system closed, isolated, or open? Explain.
A: It is closed. Mass cannot leave or enter the sealed flask, but energy (heat) crosses the boundary from the hot plate. Because energy crosses, it is not isolated.
Q: Can a steady-state system have non-zero inputs and outputs?
A: Yes. Steady state means the system's internal state does not change over time, but flows can still pass through. Input and output can both be non-zero as long as they balance (along with any generation and consumption) so that accumulation is zero.
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