Calculator Inputs
Example Data Table
This example shows a common step response case for an ideal mixing tank.
| Case | V | q | x(0) | xᵢₙ | k | t | Expected Behavior |
|---|---|---|---|---|---|---|---|
| Slow dilution | 200 | 10 | 0.20 | 1.00 | 0.00 | 30 | Gradual approach to inlet composition |
| Reactive tank | 100 | 12 | 0.10 | 1.00 | 0.05 | 25 | Lower final value due to reaction loss |
| Feed removal | 80 | 8 | 0.90 | 0.00 | 0.02 | 40 | Composition decays toward zero |
Formula Used
This calculator uses the dynamic material balance for a perfectly mixed tank. The governing equation is:
dx/dt = (q / V)(xᵢₙ - x) - kx
For a step change in inlet composition, the exit composition response is:
x(t) = xₛₛ + [x(0) - xₛₛ]e-λt
The combined dynamic rate is:
λ = q / V + k
The steady exit composition is:
xₛₛ = [(q / V)xᵢₙ] / [(q / V) + k]
When feed removal decay is selected, the steady value becomes zero. The response then shows how fast the original composition washes out.
How To Use This Calculator
- Enter the tank volume and volumetric flow rate.
- Add the initial exit composition before the inlet change.
- Enter the new inlet composition after the step change.
- Use zero reaction rate for nonreactive mixing.
- Enter a positive reaction rate for first order consumption.
- Select the evaluation time and table interval.
- Press the calculate button to see x(t) above the form.
- Download the result table as CSV or PDF.
Understanding Exit Composition Response x(t)
Why Dynamic Response Matters
Exit composition response is important in mixing, blending, and reaction systems. A tank rarely changes outlet concentration instantly. Fluid already inside the vessel must first mix with incoming material. This delay creates a smooth response curve. The curve helps engineers estimate product transition time. It also helps operators decide when off-spec material ends.
Role Of Flow And Volume
Tank volume and flow rate control residence time. A large tank with low flow changes slowly. A small tank with high flow changes quickly. The ratio q divided by V is called the dilution rate. A higher dilution rate gives a faster outlet response. This calculator displays residence time and time constant together. Those values make the process behavior easier to compare.
Effect Of Reaction
Some systems include first order reaction or decay. In that case, material disappears while the tank is mixing. Reaction increases the combined response rate. It also lowers the final steady composition. This is useful for reactors, neutralization tanks, and degradation studies. Set reaction rate to zero when the component is conserved.
Using The Time Table
The table shows x(t) at repeated time points. It also shows the remaining gap from the final value. This helps identify when the system is nearly stable. Many users treat three to five time constants as practical settling time. The CSV export is useful for reports. The PDF export helps preserve quick calculation records.
FAQs
1. What does x(t) mean?
x(t) is the exit composition at a selected time. It shows the outlet concentration after mixing, dilution, and optional first order reaction have acted for that time.
2. What is a step inlet change?
A step inlet change means the inlet composition changes suddenly from one value to another. The outlet does not jump instantly because the tank contents mix over time.
3. What should I enter for nonreactive mixing?
Enter zero for the first order reaction rate. The calculator will then model only dilution and mixing inside an ideal well mixed tank.
4. Why is the steady value lower with reaction?
Reaction consumes material while it remains in the tank. This lowers the final exit composition compared with a nonreactive tank using the same inlet composition.
5. What is residence time?
Residence time equals tank volume divided by flow rate. It estimates how long fluid stays in the vessel on average before leaving.
6. Can I use percentage as the composition unit?
Yes. Enter values as percentages and set the unit label to percent. Keep all composition inputs in the same unit system.
7. What does feed removal decay mean?
Feed removal decay models washout after the inlet component becomes zero. The outlet composition gradually falls from the initial value toward zero.
8. Why does the result table use intervals?
Intervals create repeated time points for the response curve. Smaller intervals give more detail, while larger intervals create shorter reports.