What Is a Mixing Problem?
A mixing problem (or tank problem) tracks the amount of a dissolved substance, classically salt in a brine tank, as fluid flows in and out. The governing principle is a balance law:
dA/dt = (rate of salt in) − (rate of salt out) = rincin − rout·(A/V)
where A is the amount of salt, V is the tank volume, and the outflow carries salt at the current tank concentration A/V. The subtlety that makes this a genuine differential equation is that the outflow concentration A/V changes as the salt amount changes, so the loss term depends on the unknown A, creating a first-order linear equation. When the inflow and outflow rates are equal the volume stays constant and the equation has constant coefficients; when they differ the volume changes with time, making the coefficient time-dependent. This calculator sets up the balance equation from your flow rates and concentrations, solves it (exactly for constant volume, numerically for changing volume), reports the steady-state salt level, and graphs the salt amount approaching its equilibrium.
How to Use the Mixing Problems Calculator
Enter the tank's initial volume, the inflow rate and its salt concentration, the outflow rate, and the starting amount of salt. The steps set up the balance equation, identify whether the volume is constant or changing, solve accordingly, report the steady-state amount and the time scale to approach it, and forecast the salt amount at several times. The graph shows salt versus time approaching (or, for a draining tank, evolving toward) its limiting behavior. This is a classic application of the linear first-order ODE calculator, closely related to Newton's law of cooling (both are relaxation-to-equilibrium models), and solved by the same techniques as separable equations.
Worked Example
A 100 L tank of pure water (no salt) receives brine at 5 L/min containing 2 kg/L of salt, and well-mixed solution drains at 5 L/min. Since inflow equals outflow, the volume stays 100 L, and the equation is
dA/dt = 5·2 − 5·(A/100) = 10 − 0.05A
a linear equation with constant coefficients. Its solution with A(0) = 0 is A(t) = 200(1 − e−0.05t). The salt rises from zero and levels off at the steady state of 200 kg, which makes sense: at steady state the tank concentration equals the inflow concentration (200 kg in 100 L is 2 kg/L, matching the incoming brine), so salt enters and leaves at the same rate and A stops changing. The half-way point to steady state is reached at t = ln(2)/0.05 ≈ 13.9 minutes. The counterintuitive lesson is that the final salt level depends only on the inflow concentration and tank volume, not on the starting amount, any initial salt content converges to the same 200 kg.
Why the Concentration Term Makes It a Differential Equation
The heart of every mixing problem is that the rate of salt leaving depends on the current concentration, which depends on the current amount of salt, the very quantity being solved for. This self-reference is what turns a simple bookkeeping problem into a differential equation. The salt leaves at rate rout times the concentration A/V, so the outflow term is routA/V, proportional to the present salt amount A. This creates a first-order linear equation dA/dt + (rout/V)A = rincin, which for constant volume has constant coefficients and is solved by the integrating-factor method (or recognized as a relaxation-to-equilibrium model). The steady state, where dA/dt = 0, occurs when the inflow rate of salt exactly balances the outflow rate, giving A* = (rincinV)/rout, which for equal flow rates simplifies to cinV (the tank reaches the inflow concentration).
Common Mistakes to Avoid
- Using the wrong outflow concentration. The salt leaves at the current tank concentration A/V, not the inflow concentration. This feedback term, proportional to the unknown A, is what makes it a differential equation; using a fixed concentration misses the dynamics.
- Ignoring changing volume. When inflow and outflow rates differ, the volume V(t) changes with time, so the concentration denominator is not constant. Treating V as fixed when it is not gives wrong results.
- Confusing amount with concentration. A is the amount of salt (kg); the concentration is A/V (kg/L). The equation tracks the amount, and the concentration is derived from it. Mixing up the two scrambles the setup.
- Forgetting the tank can empty. If outflow exceeds inflow, the volume decreases and the tank eventually empties, ending the model. The equation is only valid while V > 0.
- Assuming instantaneous mixing is always valid. The model assumes the tank is perfectly and instantly mixed, so the outflow concentration equals the average tank concentration. Poorly mixed tanks with concentration gradients need more elaborate models.
Real-World Applications
Mixing problems, despite their simple brine-tank framing, model an enormous range of real processes involving the flow of a substance through a reservoir. In chemical engineering, the continuously stirred tank reactor (CSTR) is the workhorse of the process industries, and its concentration dynamics are exactly a mixing problem, so designing reactors, predicting product concentrations, and controlling chemical processes all rest on this mathematics. Environmental engineering uses mixing models for pollutant concentration in lakes, rivers, and groundwater: how long a contaminant persists in a body of water as clean water flows through, or how quickly a pollutant reaches dangerous levels, is a tank problem, informing cleanup strategies and discharge regulations.
Frequently Asked Questions
What is a mixing problem?
A model tracking the amount of a dissolved substance (like salt) in a tank as fluid flows in and out. The balance law dA/dt = rate in − rate out becomes a differential equation because the outflow carries salt at the current tank concentration, which depends on the unknown amount A.
Why is the outflow term proportional to A?
Because the draining fluid carries salt at the current tank concentration A/V, so the salt leaving per unit time is rout·(A/V), proportional to the present amount A. This feedback, the loss depending on the unknown quantity, is exactly what makes the problem a differential equation rather than simple arithmetic.
What is the steady state?
The salt amount where inflow and outflow balance (dA/dt = 0), so A stops changing. For equal flow rates it equals the inflow concentration times the volume, meaning the tank reaches the incoming concentration. The system approaches this steady state exponentially, regardless of the starting amount.
Does the initial salt amount affect the final concentration?
No. For constant volume, the tank always converges to the same steady state (inflow concentration times volume), whatever the starting amount. The initial condition affects how the salt evolves toward steady state, but not the final level, a consequence of the exponential relaxation.
What happens when inflow and outflow rates differ?
The volume changes linearly with time, V(t) = V₀ + (rin − rout)t, making the concentration denominator time-dependent. The equation becomes variable-coefficient (still linear), and the behavior is power-law rather than pure exponential. A tank with more outflow than inflow eventually empties.
How fast does the tank reach steady state?
For constant volume, the time constant is V/rout (volume over outflow rate), and the system is essentially at steady state after about five time constants. The half-way time is ln(2) times the time constant. Larger tanks or slower flows take longer to equilibrate.
How is this related to Newton's law of cooling?
Both are linear first-order relaxation models: a quantity driven toward equilibrium at a rate proportional to its displacement from equilibrium, giving exponential approach. Cooling relaxes temperature toward ambient; mixing relaxes salt toward the steady-state amount. The mathematics and exponential behavior are identical.
What assumption does "well-mixed" make?
That the tank contents are uniformly and instantly mixed, so the concentration is the same everywhere and the outflow carries exactly the average tank concentration. Real tanks with imperfect mixing have concentration gradients, requiring more complex models; the well-mixed assumption is the standard idealization.
How do mixing problems apply to medicine?
Pharmacokinetics treats the body (or a compartment) as a tank: a drug enters by dosing and leaves by metabolism and excretion at a rate proportional to its concentration. This mixing equation predicts blood drug levels over time, determines steady-state concentrations for repeated dosing, and guides safe and effective dosing schedules.
What is a continuously stirred tank reactor?
A CSTR is a chemical reactor modeled exactly as a mixing problem: reactants flow in, products flow out, and the well-mixed contents react at the current concentration. Its concentration dynamics follow these equations, making the mixing model foundational to chemical process design and control in the process industries.