General material balance: five terms, and the zeros are the assumptions

Scale

The four fixed panels share one scale, 0.0 to 24.0 kg/h, and it does not move. The live chart above rescales with what you type.

Basis

One species, on a mass basis, over one hour of running at the moment described

Assumptions

The boundary is the vessel wall and the two stream faces; the reaction figure is the rate at the instant described, not an average over a shift

Model version

1.0

The formula, in words

• Five terms, and only five. Whatever you are counting, over whatever boundary, over whatever interval, what entered plus what was made equals what left plus what was used plus what is still inside. There is no sixth place for a quantity to go.
• The zeros are where the model lives. A working balance usually writes three of the five as zero. Each of those zeros is a claim about the vessel, the reaction or the boundary, and none of them is visible in the arithmetic once it is written down.
• The residual is the check, and it is not the answer. A positive residual means the left side is larger. A residual of zero says the five figures reconcile; it does not say the boundary, the basis or the interval was the right choice.
• The rearranged form is the same equation. Most books write the accumulation term on its own. It is this page's equation with that term moved. The page keeps the two sided form because the two sides are the two columns of the chart.

A worked example

The vessel

A small continuous vessel has one feed stream in and one product stream out. It is well mixed and runs at constant temperature. Inside it, a dissolved reagent $R$ rearranges into a dissolved product $P$. The boundary is the vessel wall and the two stream faces. The basis is mass, the clock is a rate, and every figure is kilograms an hour.

The feed is $120.0 \mathrm{kg/h}$: $30.0 \mathrm{kg/h}$ of $R$, $0.0 \mathrm{kg/h}$ of $P$ and $90.0 \mathrm{kg/h}$ of water. At the moment described, the reaction uses $22.0 \mathrm{kg/h}$ of $R$ and makes $22.0 \mathrm{kg/h}$ of $P$. The rearrangement moves no atoms out of the vessel, so what it uses and makes are the same mass.

Case A, steady

The product stream leaves at $120.0 \mathrm{kg/h}$ carrying $8.0 \mathrm{kg/h}$ $R$, $22.0 \mathrm{kg/h}$ $P$ and $90.0 \mathrm{kg/h}$ water. Check the stream: $8.0 + 22.0 + 90.0 = 120.0\ \mathrm{kg/h}$.

For $P$, the left side is $0.0 + 22.0 = 22.0$. The right side is $22.0 + 0.0 + 0.0 = 22.0$, so the residual is $22.0 - 22.0 = 0.0$. For $R$, the left side is $30.0 + 0.0 = 30.0$. The right side is $8.0 + 22.0 + 0.0 = 30.0$, so the residual is $30.0 - 30.0 = 0.0$. Total mass has 120.0 on both sides and a residual of 0.0.

On the $P$ ledger, input is named Closed boundary because no $P$ crosses the inlet face even though the face is wide open to the mixture. Consumption is named No reaction because nothing inside destroys $P$. Accumulation is named Steady state.

On the $R$ ledger, generation is named No reaction because nothing inside makes $R$, and accumulation is named Steady state. On the total mass ledger, generation and consumption are both named No reaction. Here the name is a fact about arithmetic rather than this vessel, because a reaction rearranges what is already inside and creates no mass.

Three of the $P$ ledger's five trays are zero. Nothing about this vessel appears in that ledger except the number 22.0, twice. The rest of it is three claims.

Case B, accumulating

The product pump is slowed to $96.0 \mathrm{kg/h}$, which is 0.8 of the feed. The vessel is well mixed, so every outlet figure is 0.8 of Case A: $0.8 \times 8.0 = 6.4$ $R$, $0.8 \times 22.0 = 17.6$ $P$ and $0.8 \times 90.0 = 72.0$ water. Check the stream: $6.4 + 17.6 + 72.0 = 96.0\ \mathrm{kg/h}$.

The reaction rate is the instantaneous rate at the moment described. The vessel is filling, so its composition is moving, and the balance is written at one instant rather than over a shift.

For $P$, $0.0 + 22.0 = 17.6 + 0.0 + A$, so $A = 22.0 - 17.6 = 4.4$ and the residual is $22.0 - (17.6 + 0.0 + 4.4) = 0.0$. For $R$, $30.0 + 0.0 = 6.4 + 22.0 + A$, so $A = 30.0 - 28.4 = 1.6$ and the residual is $30.0 - 30.0 = 0.0$. Water has $A = 90.0 - 72.0 = 18.0$. Total mass has $A = 120.0 - 96.0 = 24.0$ and a residual of 0.0.

Only two trays changed name between the cases. In Case A accumulation was a zero carrying Steady state. In Case B it is $4.4 \mathrm{kg/h}$ and carries nothing, because a term that is not zero needs no name.

Case C, the ledger that closes without balancing

Use the same $P$ ledger but set output to $20.0 \mathrm{kg/h}$. Input is 0.0, generation is 22.0, output is 20.0, consumption is 0.0 and accumulation is 0.0. The left side is 22.0, the right side is 20.0, and the residual is $2.0 \mathrm{kg/h}$.

All three zeros are named, so the ledger closes to $2.0 \mathrm{kg/h}$. That state is not an error: $2 \mathrm{kg/h}$ of $P$ is either a stream missing from the ledger or a figure that is not what it says, and the page names both possibilities and chooses neither.

Check it yourself

The subject note says why the generation tray is zero on a total mass ledger for every process ever run, and the ledger will still close on it if you name it, because the page records claims rather than grading them. On a mole basis the same tray is a live term, because a reaction that turns two molecules into three adds a mole without adding a gram.

What this does not model

Energy in any form. This is a material balance, and a vessel that closes on mass can be running away on temperature. Reaction stoichiometry and extent, which decide what the generation and consumption figures actually are. Degrees of freedom, so the page will happily hold five figures that over specify the vessel. Multiple units, recycles and purges, where the boundary changes which streams exist. Phase equilibrium and any composition that depends on it. Every property a real calculation needs: density, molecular weight and heat capacity. Time varying inputs, and the difference between an instantaneous rate and an average over a shift, which this page states and does not compute. Measurement error, which is the most common reason a real ledger closes to something other than zero and the one the residual cannot tell apart from a missing stream.