91Ó°ÊÓ

Ninety kilograms of sodium nitrate is dissolved in \(110 \mathrm{kg}\) of water. When the dissolution is complete (at time \(t=0\) ), pure water is fed to the tank at a constant rate \(\dot{m}(\mathrm{kg} / \mathrm{min}),\) and solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed. (a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value. (b) Write a balance on sodium nitrate, letting \(x(t, \dot{m})\) equal the mass fraction of \(\mathrm{NaNO}_{3}\) in the tank and outlet stream. Convert the balance into an equation for \(d x / d t\) and provide an initial condition. (c) On a single graph of \(x\) versus \(t,\) sketch the shapes of the plots you would expect to obtain for \(\dot{m}=50 \mathrm{kg} / \mathrm{min}, 100 \mathrm{kg} / \mathrm{min},\) and \(200 \mathrm{kg} / \mathrm{min} .\) (Don't do any calculations.) Explain your reason- ing, using the equation of Part (b) in your explanation. (d) Separate variables and integrate the balance to obtain an expression for \(x(t, \dot{m})\). Check your solution. Then generate the plots of \(x\) versus \(t\) for \(\dot{m}=50 \mathrm{kg} / \mathrm{min}, 100 \mathrm{kg} / \mathrm{min},\) and \(200 \mathrm{kg} / \mathrm{min}\) and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.) (e) If \(\dot{m}=100 \mathrm{kg} / \mathrm{min}\), how long will it take to flush out \(90 \%\) of the sodium nitrate originally in the tank? How long to flush out 99\%? 99.9\%? (f) The stream of water enters the tank at a point near the top, and the exit pipe from the tank is located on the opposite side toward the bottom. One day the plant technician forgot to turn on the mixing impeller in the tank. On the same chart, sketch the shapes of the plots of \(x\) versus \(t\) you would expect to see with the impeller on and off, clearly showing the differences between the two curves at small values and large values of \(t .\) Explain your reasoning.

Step by step solution

01

Establishing Total Mass Balance

The total mass in the tank is made up of the initial mass plus any mass that enters minus any mass that leaves the tank. This gives us M(t) = Mi + ∫_{0}^{t} m_dot dt - ∫_{0}^{t} m_dot dt. Simplifying gives M(t) = Mi since the quantities that are added and removed are equal.

02

Sodium Nitrate Balance

The mass balance for sodium nitrate can be represented by an equation such as: N(t) = Ni + ∫_{0}^{t} ω * m_dot dt - ∫_{0}^{t} (x * m_dot) dt. ω represents the mass fraction of NaNO3 in the feed. As pure water is being fed to the tank, ω is 0. So the equation simplifies to: N(t) = Ni - ∫_{0}^{t} (x * m_dot) dt. Differentiating with respect to 't' gives: dN/dt = - x*m_dot, which can be rearranged to find dx/dt.

03

Illustrating the Sodium Nitrate Balance Over Time

Without specific values for the curves, for a lower feed rate, we expect a slow decrease of sodium nitrate concentration. For a higher feed rate, the concentration decreases faster. As the rate increases, the curve becomes steeper since the water enters and leaves the tank faster, replacing the sodium nitrate with pure water. Analyzing the derivative, dx/dt = - (m_dot / Mi) * x, it can be inferred that the graph must decrease exponentially because dx/dt is directly proportional to x.

04

Compute x(t, m_dot) Expression

By rearranging the equation dx/dt = - m_dot/Mi * x and integrating, we get: x(t, m_dot) = x(0, m_dot) * exp(- m_dot/Mi * t), where x(0, m_dot) = Ni/Mi is the initial mass fraction.

05

Computing Required Time

The equation from Step 4 can be used to compute the time necessary to flush out a certain fraction of the sodium nitrate. It has to be set equal to (1 - desired fraction) and then solve for 't'.

06

Effect of Mixing on Sodium Nitrate Removal

If the mixing impeller was off, the rate at which sodium nitrate is washed out would be slower. This is because without mixing, the water flowing in mainly interacts with water near the inlet and not much with the sodium nitrate solution set at the bottom. The curve for this would therefore be below the curve with the impeller on.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Balance Equation

In chemical engineering, a mass balance equation helps us understand how the mass of a system changes over time. It's crucial for determining how reactants and products are distributed.
The mass balance is based on the principle of conservation of mass, and for this exercise, we consider a tank initially containing a sodium nitrate solution.
The important takeaway is that when the inlet and outlet flows are equal, the total mass remains constant. This means if you add water and draw the same amount of solution out simultaneously, the total mass inside the tank doesn't change.

• Initial Mass (Mi): The mass of sodium nitrate and water initially in the tank.
• Mass Flow Rate (m): The rate at which water is added and solution removed.
• Time (t): The period over which the process occurs.

To prove this, we set up a mass balance as: \( M(t) = M_i + \int_{0}^{t} \dot{m} \, dt - \int_{0}^{t} \dot{m} \, dt \) simplifies to \( M(t) = M_i \), showing total mass stays constant as water entering equals solution leaving.

Sodium Nitrate Concentration

Determining sodium nitrate concentration within the mixture is essential for describing the chemical process over time. This could be visualized through how the concentration changes as water flows in and solution flows out.
Sodium nitrate concentration is crucial because it determines the purity and usability of the solution in various processes.
In this scenario, the mass fraction equation involves calculating the amount of sodium nitrate compared to the total mixture.

• Initial Sodium Nitrate Mass (Ni): How much sodium nitrate was there initially.
• Mass Fraction (x): The proportion of sodium nitrate in the mixture.

If no sodium nitrate enters with water, meaning pure water is used, the balance equation is \( N(t) = N_i - \int_{0}^{t} (x \cdot \dot{m}) \, dt \). Differentiating gives \( \frac{dN}{dt} = -x \cdot \dot{m}\), meaning \( \frac{dx}{dt} = - \frac{\dot{m}}{M_i} \cdot x\), indicating an exponential decrease in concentration.

Differential Equations

Understanding the role of differential equations is critical when looking at how variables change with respect to time in chemical processes.
In our scenario, differential equations help track the sodium nitrate mass fraction as it exits the tank. The rate of change of this fraction illustrates how quickly the solute is replaced with water.
For sodium nitrate, the differential equation derived is \( \frac{dx}{dt} = - \frac{\dot{m}}{M_i} \cdot x \). Here, the equation signifies that the decrease rate of sodium nitrate is proportional to its current concentration. This type of equation generally results in an exponential decay, common in chemical processes of dilution.

Graphical Analysis

Graphical analysis involves visually comparing how sodium nitrate concentration decreases over time with varying flow rates.
By plotting concentration (\( x \) versus time (\( t \)), we can better understand the process behavior for different conditions.
A steeper graph indicates quicker depletion of sodium nitrate, while a flatter one shows slower depletion.
No calculations are required to see this intuitive pattern. As \( \dot{m} \) increases (e.g., 50 kg/min, 100 kg/min, 200 kg/min), the influence of rapidly incoming and outgoing streams is visible on a graph:

• Slow feed results in a gently sloping graph, gradual decrease in concentration.
• Fast feed shows a steep drop, swift change in concentration.

Observing these concepts graphically allows easy visualization of the effect flow rates have on mixing efficiency, and it's a common method engineers use to analyze and communicate chemical processes.

Mixing Impeller Effect

The mixing impeller, when operational, significantly changes the efficiency of decreasing the sodium nitrate concentration in the solution.
An impeller ensures a homogeneous mix, meaning solution properties are uniform throughout, leading to predictable and regular depletion patterns.
However, without it, the water influent is less effective at thoroughly mixing, creating a layered concentration:

• With the impeller on, mixing ensures a smooth, predictable concentration decrease over time.
• Without mixing, the graph may show irregularities, slower decrease initially due to poor mixing.

Therefore, the impeller plays a pivotal role in process optimization, aiding in faster removal of sodium nitrate and resulting in a more uniform solution composition over the entire volume of the tank.