Properties of stirred tank solutions


b. Verify that M(0) = M₀

Verified step by step guidance

1. Identify the function M(t) that describes the amount of substance in the stirred tank at time t. This function is usually given or derived from the problem context.
2. Recall that M(0) represents the initial amount of substance in the tank at time t = 0.
3. Substitute t = 0 into the function M(t) to find M(0). This means replacing every occurrence of t in the expression with 0.
4. Simplify the resulting expression after substitution to see if it equals M₀, the initial amount given in the problem.
5. If the simplified expression equals M₀, then you have verified that M(0) = M₀, confirming the initial condition of the solution.

Key Concepts

Initial Condition in Differential Equations

The initial condition specifies the value of the function at the starting point, often time zero. Verifying M(0) = M₀ ensures the solution matches the known initial state, which is crucial for the correctness of models describing physical systems like stirred tanks.

Mass Balance in Stirred Tank Systems

Mass balance involves accounting for the mass entering, leaving, and accumulating in the tank. Understanding this principle helps formulate the differential equation governing M(t), the mass in the tank, and ensures the model reflects physical reality.

Solving and Verifying Solutions of Differential Equations

After solving the differential equation for M(t), verifying M(0) = M₀ confirms the solution satisfies the initial condition. This step validates the mathematical model and ensures consistency between the solution and the physical system.