Two very large tanks A and B are each partially filled with...

Two very large tanks $A$ and $B$ are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank $A$ and 50 pounds of salt is dissolved in the solution in $\operatorname{tank} B$. The system is closed in that the wellstirred liquid is pumped only between the tanks, as shown in FIGURE 2.9.6. (a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt $x_{1}(t)$ and $x_{2}(t)$ at time $t$ in tanks $A$ and $B$, respectively. (b) Find a relationship between the variables $x_{1}(t)$ and $x_{2}(t)$ that holds at time $t$. Bxplain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in $\operatorname{tank} B$ at $t=30 \mathrm{~min}$.

Two very large tanks $A$ and $B$ are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank $A$ and 50 pounds of salt is dissolved in the solution in $\operatorname{tank} B$. The system is closed in that the wellstirred liquid is pumped only between the tanks, as shown in FIGURE 2.9.6.
(a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt $x_{1}(t)$ and $x_{2}(t)$ at time $t$ in tanks $A$ and $B$, respectively.
(b) Find a relationship between the variables $x_{1}(t)$ and $x_{2}(t)$ that holds at time $t$. Bxplain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in $\operatorname{tank} B$ at $t=30 \mathrm{~min}$.

Key Concepts

Invariant Relationships in Closed Systems

In closed mixing systems, invariant relationships such as a constant total mass of the solute arise naturally from conservation laws. These relationships can simplify the analysis by reducing the number of independent variables. For example, knowing that the sum of the solute in all tanks is conserved allows one to express the quantity in one tank in terms of the total and the quantity in the other tank, streamlining the solution process.

Initial Conditions

Initial conditions provide the starting quantities or concentrations of substances in each compartment at time zero. They are essential for solving the differential equations uniquely, as they allow for the determination of the constants of integration. In mixing problems, the initial concentration values shape the trajectory of how the system evolves over time.

Conservation of Mass

Conservation of mass is a fundamental principle stating that mass cannot be created or destroyed in a closed system. In the context of mixing tanks, this principle implies that the total amount of the solute (in this case, salt) remains constant over time when the system does not exchange matter with its surroundings. This invariance can be used to derive precise relationships between the quantities of salt in different tanks.

Differential Equations in Modeling Mixing Problems

Differential equations are used to model the rate of change of the amount of a substance in a system. In mixing problems, these equations account for the inflow and outflow rates of the dissolved substance, typically resulting in first-order linear ordinary differential equations. They provide a framework to predict the evolution of concentrations or quantities over time, based on the interactions between different compartments of a closed or open system.

System of Coupled Differential Equations

When modeling the exchange between multiple tanks, the behavior is described by a system of coupled differential equations. Each equation represents the rate of change in a tank, with terms that link the tanks to account for the transfer of substances between them. This interconnected system requires simultaneous analysis, often lending itself to techniques such as elimination or substitution, and exploiting invariants like the conserved total mass.

Please give Ace some feedback