Problem 32 - Calculus for Biology and Medicine

Question

Filling Box Models In Problem 10 of Section \(8.3\) we analyzed the concentration in a tank whose volume changes over time because the inflows and outflows are not matched. For such a tank it can be shown that if the concentration in the inflow is \(C_{I}\), and the inflow and outflow rates are respectively \(q_{\text {in }}\) and \(q_{\text {out }}\), then both concentration, \(C(t)\), and volume of water in the tank, \(V(t)\), vary with time and can be modeled by a pair of differential equations: $$ \frac{d}{d t}(C V)=q_{\text {in }} C_{I}-q_{\text {out }} C $$ and $$ \frac{d V}{d t}=q_{\text {in }}-q_{\text {out }} . $$ (a) Show that the differential equations (8.96) imply that $$ \left(\left(q_{\mathrm{in}}-q_{\mathrm{out}}\right) t+V_{0}\right) \frac{d C}{d t}+q_{\mathrm{in}} C=q_{\mathrm{in}} C_{I} $$ where \(V_{0}\) is the initial volume of water in the tank. (b) Assuming that \(q_{\text {in }}=2, q_{\text {out }}=1, V_{0}=1, C(0)=0\), and \(C_{l}=1\), solve \((8.97)\) using integrating factors to find \(C(t)\). (c) Assuming that \(q_{\text {in }}=1, q_{\text {out }}=2, V_{0}=1, C(0)=0\), and \(C_{I}=1\), solve \((8.97)\) using integrating factors to find \(C(t) .\) What does your model predict will occur when \(t=1 ?\) Explain whether this answer makes sense given that \(V(1)=0\).

Step-by-Step Solution

Answer

(a) Derived equation holds. (b) \(C(t) = 1 - \frac{1}{t+1}\). (c) \(C(t) = \ln(t+1)\); at \(t=1\): predicts well mathematically, but physically volume should not sustain a solution if zero.

Step 1: Substitute Volume Rate Equation

Start with the differential equation \(\frac{d}{d t}(C V)=q_{\text {in }} C_{I}-q_{\text {out }} C\) and use the volume rate equation \(\frac{d V}{d t}=q_{\text {in }}-q_{\text {out }}\). Integrate \(\frac{d V}{d t}\) to express \(V(t)\) as \(V(t) = (q_{\text{in}} - q_{\text{out}})t + V_0\).

Step 2: Differentiate and Combine Equations

Convert \(\frac{d}{dt}(CV)\) using the product rule: \(C\frac{dV}{dt} + V\frac{dC}{dt}\). Substitute \(\frac{dV}{dt}=q_{\text {in}}-q_{\text {out}}\) and \(V(t)\) expression: \[ \left((q_{\text{in}}-q_{\text {out}})t + V_0\right) \frac{dC}{dt} + q_{\text {in}} C = q_{\text {in}} C_{I} \].

Step 3: Apply Integrating Factor for (b)

For \(q_{\text{in }}=2, q_{\text{out }}=1\), calculate the integrating factor \(\mu(t) = e^{\int (q_{\text{in}} - q_{\text{out}}) V^{-1}(t) dt}\). Simplify values: integrating factor \(\mu(t) = (t+1)\). Solve using substitution into the modified equation to obtain \(C(t) = 1 - \frac{1}{t+1}\).

Step 4: Simplify and Solve Part (b)

Apply given conditions \(C(0) = 0\) to find the constant. Integrate and rearrange to solve for \(C(t)\): \[C(t) = 1 - \frac{1}{t+1}\].

Step 5: Apply Integrating Factor for (c)

For \(q_{\text{in }}=1, q_{\text{out }}=2\), calculate the integrating factor \(\mu(t) = e^{\int (q_{\text{in}} - q_{\text{out}}) V^{-1}(t) dt}\). With values, integrating factor is \(\mu(t) = \frac{1}{t+1}\). Solve the DE related to \((t + 1) \frac{dC}{dt} + C = 1\).

Step 6: Solve Part (c) and Analyze

Integrate using the integrating factor and initial condition \(C(0) = 0\) to find \(C(t) = \ln(t+1)\). When \(t=1\), \(C(1) = \ln(2)\). Analyze: \(V(t) = (1 - 2)t + 1\) implies \(V(1) = 0\), meaning no volume; however, \(C(t)\) defined in terms of \(\ln(t+1)\) doesn't initially specify empty volume. Indicates discrepancy in physical interpretation if volume is zero.

Key Concepts

Integrating Factor

An integrating factor is a mathematical function used to solve certain types of differential equations, particularly first-order linear ordinary differential equations. The idea is to make the equation integrable, allowing us to solve it more easily. This method is especially useful for equations of the form \(\frac{dC}{dt} + P(t)C = Q(t)\), where \(P(t)\) and \(Q(t)\) are functions of time.
To determine the integrating factor, \(\mu(t)\), we calculate it using the expression:

• \(\mu(t) = e^{\int P(t) dt}\)

The beauty of the integrating factor is in how it transforms the original differential equation. Once we multiply all terms of the differential equation by \(\mu(t)\), it becomes possible to express the left-hand side of the equation as the derivative of the product of \(\mu(t)\) and \(C(t)\).
This transformation makes the equation straightforward to integrate. Once integrated, we rearrange the expression to solve for \(C(t)\), our desired concentration function. In the context of the exercise, knowing this technique is key to finding \(C(t)\) in situations where the inflow and outflow rates of the tank cause the volume to change.

Volume Rate

Volume rate is the rate at which fluid enters or exits a tank, and it affects the tank's volume over time. In this exercise, the tank's volume changes due to different inflow \((q_{\text{in}})\) and outflow \((q_{\text{out}})\) rates.
The differential equation \(\frac{dV}{dt} = q_{\text{in}} - q_{\text{out}}\) helps us understand how the volume \(V(t)\) changes over time. By integrating this equation, we get the expression:

• \(V(t) = (q_{\text{in}} - q_{\text{out}})t + V_0\)

where \(V_0\) is the initial volume at time zero. If inflow and outflow rates are equal, the volume remains constant. Otherwise, the volume changes linearly over time.
In part (b) and (c) of the problem, different values for \(q_{\text{in}}\) and \(q_{\text{out}}\) demonstrate how volume influences the concentration over time. It's essential to keep track of these rates because they directly impact the concentration \(C(t)\) by affecting the \(V(t)\) function in the differential equations we solve.

Concentration Model

The concentration model describes how the concentration of a substance changes in a tank when the volume also changes over time. This model involves setting up and solving differential equations to reflect the concentration dynamics due to inflow and outflow.
Starting with the equation \( \frac{d}{dt}(C V) = q_{\text{in}}C_I - q_{\text{out}}C \), where \(C_I\) is the concentration in the inflow, we break it down using product rule differentiation. The focus is to express this in a form that allows using integrating factors:

• \(\left((q_{\text{in}} - q_{\text{out}})t + V_0\right) \frac{dC}{dt} + q_{\text{in}} C = q_{\text{in}} C_I\)

Knowing the initial conditions and using integrating factors, as demonstrated in parts (b) and (c), we solve for \(C(t)\). This provides us a function showing how concentration evolves given specific initial conditions and volume changes.
This model is fundamental when dealing with dynamic systems, as it offers a realistic representation of how concentration varies with changing volumes and flow rates. Understanding this model allows us to predict and analyze concentration behavior in practical situations, like chemical mixing or liquid processing scenarios.