A perfectly mixed tank contains one hundred liters of solution and twenty grams of dissolved material. Inflow is two liters per minute at three grams per liter; outflow is two liters per minute. Find the incoming mass rate before considering any calculus. Multiplying concentration by volume flow gives six grams per minute.
Let Q(t) be the amount of material in grams. The outgoing concentration is Q/100 grams per liter because the tank is assumed perfectly mixed. The outflow therefore removes Q/50 grams per minute. This is a synthetic balance model; instantaneous mixing and constant flow are assumptions.
Predict
At the initial state, does the amount increase or decrease? The incoming concentration is three, whereas the tank concentration is only one fifth. Predict the sign of Q′ and the eventual equilibrium amount. Explain why using the incoming concentration for the outgoing stream would erase the very change we are modeling.
Experiment
Use amount change equals input minus output: Q′=6−Q/50. Initially this is 6−20/50=28/5 grams per minute. Solving the linear equation gives Q=300−280e^(−t/50).
Quantity
Initial value
Long-time limit
Volume, liters
100
100
Amount, grams
20
300
Concentration, grams/liter
1/5
3
Net mass rate, grams/minute
28/5
0
Check the formula at zero and differentiate it. The derivative is (28/5)e^(−t/50), matching 6−Q/50. Notice that the equilibrium concentration matches the inlet concentration while the equilibrium amount also depends on tank volume. Doubling the volume would not double the target concentration.
Understand
When inflow and outflow differ, volume changes. With inflow a and outflow b in liters per minute, V(t)=V₀+(a−b)t until a physical boundary is reached. For inlet concentration c, the amount equation is Q′=ac−bQ/V(t). Every term has mass-per-time units. Using Q/V₀ throughout would silently impose constant volume despite unequal flows.
For V₀=50, a=3, b=1, and c=2, the model becomes V=50+2t and Q′=6−Q/(50+2t). At five minutes the volume is sixty liters. The coefficient depends on time, but this is still a first-order linear equation. The integrating factor is proportional to √(50+2t), since the integral of 1/(50+2t) is one half its logarithm.
A model also needs a valid operating interval. If a finite tank fills, its equations must change at capacity. If a tank starts with thirty liters and loses two net liters per minute, it empties at fifteen minutes. Continuing the same formula into negative volume has no physical meaning. The singular concentration expression warns that the mathematical assumptions have failed at the boundary.
Perfect mixing may be a poor approximation when there are layers or transport delays. No algebraic accuracy can repair an incorrect mixing assumption. Measurement and model checks must address those mechanisms separately.
Apply
Construct a balance for a new inlet concentration while keeping the flow rates fixed. Annotate every symbol with units, and predict the initial direction using concentrations before solving. Then compare amount and concentration: with changing volume, an increasing amount does not necessarily mean an increasing concentration.
Derive concentration dynamics from Q=VC using the product rule, Q′=V′C+VC′. Substituting the balance gives VC′=a(c−C). This cancellation is a useful independent check, but it still relies on perfect mixing and the stated volume equation.
Continue
Next approximate trajectories numerically. Keep the balance equation and its units visible: a small numerical step addresses discretization error, while incorrect flow or mixing assumptions require a different model.
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Practice and reasoning
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A tank holds 100 L with 20 g solute. Inflow is 2 L/min at 3 g/L, outflow 2 L/min. Find the instantaneous solute rate in g/min.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A tank starts with 40 L, receives 2 L/min and loses 5 L/min. Find its volume after 6 minutes, before emptying.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Under perfect mixing, solute outflow rate is:
For the 100 L tank with inflow and outflow 2 L/min, input concentration 3 g/L and Q(0)=20 g, solve for Q(t).
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A tank starts with 50 L, receives 3 L/min at 2 g/L and loses 1 L/min. Form the ODE for solute amount Q(t) and state a physical limitation.
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A perfectly mixed tank starts with 30 L, has inflow 1 L/min and outflow 3 L/min. Why is a balance with denominator 30−2t not a physical model past t=15?
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Reflect and continue
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