Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.


c. At what time does the mass of sugar reach 95% of its steady-state level?

Verified step by step guidance

1. Define the variable \( m(t) \) as the mass of sugar (in grams) in the tank at time \( t \) minutes. Since the tank volume is constant at 100 L, the concentration in the tank at time \( t \) is \( \frac{m(t)}{100} \) gm/L.
2. Set up the differential equation for the mass of sugar in the tank. The rate of change of mass \( \frac{dm}{dt} \) equals the rate of sugar entering minus the rate of sugar leaving: \[ \frac{dm}{dt} = (\text{inflow rate}) \times (\text{inflow concentration}) - (\text{outflow rate}) \times (\text{tank concentration}) \] Substitute the given values: \[ \frac{dm}{dt} = 0.5 \times 20 - 0.5 \times \frac{m(t)}{100} \]
3. Simplify the differential equation to standard linear form: \[ \frac{dm}{dt} = 10 - \frac{m(t)}{200} \]
4. Solve this first-order linear differential equation with the initial condition \( m(0) = 0 \) (since the tank starts with pure water). The solution will be of the form: \[ m(t) = \text{steady-state mass} \times \left(1 - e^{-kt}\right) \] where \( k \) is a positive constant related to the outflow rate and volume.
5. Determine the steady-state mass by setting \( \frac{dm}{dt} = 0 \) and solve for \( m \). Then, find the time \( t \) when \( m(t) \) reaches 95% of this steady-state value by solving: \[ m(t) = 0.95 \times m_{steady-state} \] Use the expression for \( m(t) \) to isolate \( t \).

Key Concepts

First-Order Linear Differential Equations

This problem involves modeling the change in sugar mass over time using a first-order linear differential equation. The rate of change depends on the inflow concentration and the outflow rate, leading to an equation that can be solved to find the sugar mass as a function of time.

Steady-State Concentration

The steady-state occurs when the amount of sugar in the tank no longer changes, meaning inflow and outflow rates balance. Calculating this steady-state value provides a reference point to determine when the system reaches a certain percentage of this equilibrium.

Exponential Approach to Equilibrium

The solution to the differential equation shows that the sugar mass approaches steady-state exponentially over time. Understanding this exponential behavior allows us to calculate the time required for the mass to reach a specific fraction (e.g., 95%) of the steady-state value.