How to Solve a Mass Balance Problem: A Step-by-Step Guide for Homework

[decatte]

Homework Statement

A tank used to prepare 10% salt solution was shut down for the day. The next day, you were told that the effluent concentration should be 5 wt %. The tank volume is 1000 m^3, and it is completely filled with 10% solution. You decide to run pure water and remove the contents at the same rate until the concentration drops from 10% to 5% and readjust the new steady state flow rates of salt and water. How long it will take to decrease the concentration in the tank if you feed water at 100 lt/hour?

Homework Equations

Mass Balance Equation: dm(total)/dt=dm(1)/dt + dm(2)/dt + ...

The Attempt at a Solution

Firstly, it is not a steady state respect to salt, salt concentration is changing with time.

I tried to write mass balance (difference) equation respect to salt:

dm(salt)/dt= 0 - ?

Now, 0 (zero) because salt is not being added to system, conversely, it's concentration is lowerin'.

Please guys, help me. It's very important.

Thanks from now!

 

[nate808]

Thank you for your question. Based on the information provided, we can approach this problem using a mass balance equation. As you mentioned, the mass balance equation is given by:

dm(total)/dt=dm(1)/dt + dm(2)/dt + ...

Where dm(total)/dt represents the change in total mass in the tank over time, and dm(1)/dt and dm(2)/dt represent the inflow and outflow rates of the first and second components (in this case, salt and water).

In this problem, we are interested in determining the time it takes for the concentration of salt to decrease from 10% to 5%. Let's assume that the initial concentration of salt is denoted as c1 and the final concentration is denoted as c2. We can then write the mass balance equation as:

dm(total)/dt = dm(salt)/dt + dm(water)/dt

Where dm(salt)/dt represents the change in mass of salt over time, and dm(water)/dt represents the change in mass of water over time.

Since we are removing water from the tank at a constant rate of 100 lt/hour, we can write dm(water)/dt as -100 lt/hour. This negative sign indicates that water is being removed from the tank.

Next, we need to determine the value of dm(salt)/dt. We can do this by using the following formula:

dm(salt)/dt = c1 * dm(water)/dt - c2 * dm(water)/dt

Substituting the values of c1 (10%) and c2 (5%), we get:

dm(salt)/dt = 0.1 * (-100) - 0.05 * (-100) = -5 kg/hour

This negative value indicates that salt is also being removed from the tank, as the concentration of salt decreases from 10% to 5%.

Now, we can plug in the values of dm(total)/dt, dm(salt)/dt, and dm(water)/dt into the mass balance equation and solve for time (t):

dm(total)/dt = dm(salt)/dt + dm(water)/dt

t = dm(total)/ (dm(salt)/dt + dm(water)/dt)

Substituting the values, we get:

t = 1000 m^3 / (-5 kg/hour - 100 lt/hour)

 

[Blueshift5]

it is important to approach mass balance problems systematically and accurately. Here is a step-by-step guide to solving this particular problem:

Step 1: Identify the variables and their units.

In this problem, we have the following variables:

- Salt concentration (wt %)
- Tank volume (m^3)
- Flow rate of water (lt/hour)

Step 2: Write the mass balance equation.

The mass balance equation for this problem can be written as:

dm(salt)/dt = dm(in)/dt - dm(out)/dt

Where dm(salt)/dt is the rate of change of salt concentration in the tank, dm(in)/dt is the rate at which salt is being added to the tank, and dm(out)/dt is the rate at which salt is being removed from the tank.

Step 3: Determine the initial and final conditions.

The initial condition is that the tank is completely filled with 10% salt solution. The final condition is that the salt concentration should be 5 wt %.

Step 4: Calculate the initial and final amounts of salt in the tank.

The initial amount of salt in the tank can be calculated by multiplying the initial salt concentration (10 wt %) by the tank volume (1000 m^3). This gives us 100,000 kg of salt.

The final amount of salt in the tank should be half of the initial amount (since the concentration is being reduced from 10% to 5%). Therefore, the final amount of salt in the tank should be 50,000 kg.

Step 5: Calculate the rate of change of salt concentration.

We know that the rate of change of salt concentration (dm(salt)/dt) is equal to the difference between the rate of salt being added and the rate of salt being removed from the tank. Since we want to decrease the concentration, the rate of salt being removed should be greater than the rate of salt being added. Therefore, we can write:

dm(salt)/dt = 0 - dm(out)/dt

Step 6: Calculate the rate of salt being removed from the tank.

We can use the mass balance equation to calculate the rate of salt being removed from the tank. Since we know that the final amount of salt in the tank should be 50,000 kg, we can write:

dm(out)/dt = (100,000 kg - 50,000 kg)/t

Where t is the time taken to