Source: Experimental Physics and Engineering Lab, Texas A&M University
Tags: UAE, six-term equation, input, output, generation, consumption, accumulation, conserved quantity, steady state, mass balance, energy balance, bank account example
The UAE in its full form has six terms: initial amount, final amount, input, output, generation, and consumption. Most engineering problems are solved by identifying which of those terms are zero for the system in question and simplifying accordingly. Conserved quantities (mass, energy, momentum) eliminate the generation and consumption terms. Steady-state conditions eliminate accumulation. Combining both gives you Input = Output, which is the simplest and most commonly tested form.
Six-term UAE
The complete form of the Universal Accounting Equation: Final Amount = Initial Amount + Input − Output + Generation − Consumption.
Initial amount
The quantity present in the system at the start of the chosen time period.
Final amount
The quantity present in the system at the end of the chosen time period.
Input
The total quantity that enters the system across its boundary during the time period.
Output
The total quantity that leaves the system across its boundary during the time period.
Generation
The quantity produced (created) inside the system during the time period. Applies to non-conserved quantities only, for example a chemical species formed by a reaction.
Consumption
The quantity destroyed inside the system during the time period. Applies to non-conserved quantities only, for example a reactant used up in a reaction.
Accumulation
The net change in the quantity stored inside the system: Final Amount − Initial Amount. Also equal to (Input − Output) + (Generation − Consumption).
Conserved quantity
A quantity whose total amount in the universe cannot change. Generation and consumption are always zero for conserved quantities.
Steady state
A condition where the system's state does not change over time, so accumulation equals zero (Final Amount = Initial Amount).
The equation in words:
Final = Initial + In − Out + Generated − Consumed
Rearranged to show accumulation explicitly:
(Final − Initial) = (In − Out) + (Generated − Consumed)
Accumulation = Net Input + Net Generation
Every term must refer to the same extensive quantity, the same system boundary, and the same time period. Mixing any of these is the most common source of errors.
Define the system and draw the boundary
Identify the extensive quantity being tracked
Choose the time period
Write out all six terms, assigning known values and marking unknowns
Determine which terms are zero (conserved quantity? steady state? closed system?)
Solve for the unknown
A bank account is a system. Money is the extensive quantity. The boundary is the account itself. The time period is December 2024.
Given values:
Initial amount (balance on 1 December 2024): $478.65
Input (deposits): $2,028.00
Generation (interest earned): $5.34
Output (cheques written): $1,587.65
Output (ATM withdrawals): $450.00
Consumption (service charges): $5.00
Applying the UAE:
Final = Initial + Input − Output + Generation − Consumption
Final = $478.65 + $2,028.00 − ($1,587.65 + $450.00) + $5.34 − $5.00
Final = $478.65 + $2,028.00 − $2,037.65 + $5.34 − $5.00
Final = $469.34
The balance on 1 January 2025 is $469.34.
Note how interest earned counts as generation (money created inside the system by the bank's process) and service charges count as consumption (money destroyed inside the system), while deposits and withdrawals are inputs and outputs crossing the boundary.
For any conserved quantity, the universe neither creates nor destroys it. That means:
Generation = 0 and Consumption = 0
The UAE simplifies to:
Final = Initial + Input − Output
Or equivalently:
Accumulation = Input − Output
The three most commonly encountered conserved quantities in engineering:
Total mass (in non-nuclear systems)
Total energy (first law of thermodynamics)
Momentum (in the absence of external forces)
Individual chemical species are generally not conserved, because reactions generate and consume them. Total mass across all species, however, is conserved.
At steady state, the system's internal state does not change with time. The stored amount stays constant:
Final = Initial, so Accumulation = 0
The UAE becomes:
0 = Input − Output + Generation − Consumption
Output − Input = Generation − Consumption
If the quantity is also conserved (generation and consumption both zero):
0 = Input − Output
Input = Output
This is the simplest possible form of the UAE and the one that appears most often in exam problems. Steady-state, conserved-quantity problems reduce to: whatever goes in must come out.
Condition | Terms that vanish | Simplified UAE |
|---|---|---|
No simplification | None | Final = Initial + In − Out + Gen − Con |
Conserved quantity | Gen = 0, Con = 0 | Final = Initial + In − Out |
Steady state | Accumulation = 0 | 0 = In − Out + Gen − Con |
Conserved + steady state | Gen = 0, Con = 0, Acc = 0 | In = Out |
Closed system | In = 0, Out = 0 | Final = Initial + Gen − Con |
Closed + conserved | In = 0, Out = 0, Gen = 0, Con = 0 | Final = Initial |
Six-term UAE:
Final = Initial + Input − Output + Generation − Consumption
Accumulation form:
Accumulation = Net Input + Net Generation
(Final − Initial) = (In − Out) + (Gen − Con)
Conserved quantity:
Gen = 0, Con = 0
Steady state:
Final = Initial, so Accumulation = 0
Steady state + conserved:
Input = Output
⚠️ The most tested skill is recognising which terms to set to zero. Read the problem statement for clues: "conserved" kills generation and consumption; "steady state" or "does not change with time" kills accumulation; "sealed" or "closed" kills input and output.
⚠️ Interest earned in the bank-account example is generation, not input. It was created inside the system (the account), not transferred in from outside. Service charges are consumption, not output. Getting this classification wrong is a common mistake.
⚠️ Individual chemical species are not conserved even though total mass is. If a problem asks you to track a specific reactant or product, you must keep the generation and consumption terms.
⚠️ Steady state does not mean nothing is happening. It means the internal state is constant. Flows can be large and continuous; they just have to balance so accumulation is zero.
⚠️ Every term in the UAE must use the same units, the same system boundary, and the same time period. A mismatch in any of these three produces a wrong answer that can look plausible.
Q: Write out the six-term version of the UAE.
A: Final Amount = Initial Amount + Input − Output + Generation − Consumption.
Q: In the bank account example, why is interest earned classified as generation rather than input?
A: Interest is created inside the system (the bank account) by the bank's own process. It does not cross the system boundary from outside. Anything produced within the system boundary is generation.
Q: What simplification does a conserved quantity allow, and name three conserved quantities?
A: For a conserved quantity, generation = 0 and consumption = 0, removing two terms from the UAE. Three conserved quantities are total mass, total energy, and momentum.
Q: A pipe carries water at a constant flow rate with no leaks and no chemical reactions. Simplify the UAE for the mass of water in the pipe.
A: Mass is conserved (Gen = 0, Con = 0). Constant flow rate with no change in stored water means steady state (Accumulation = 0). The UAE simplifies to Input = Output, meaning the mass flow rate entering the pipe equals the mass flow rate leaving it.
Q: A batch reactor (sealed vessel) runs a chemical reaction that converts reactant A into product B. The reactor is not at steady state. Write the simplified UAE for species A.
A: The system is closed, so Input = 0 and Output = 0. Species A is consumed by the reaction, so Consumption ≠ 0 (and Generation of A = 0 if A is only consumed). The UAE becomes: Final amount of A = Initial amount of A − Consumption of A.
Q: Why does steady state not imply that generation and consumption are zero?
A: Steady state only means accumulation is zero (the stored amount does not change). A system can still generate and consume a quantity internally, as long as those terms, combined with input and output, balance to produce no net accumulation.
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