Consider a process in which freshly-mined ore is to be cleaned so that later processing units do not get contaminated with dirt. 3000 kg/hr of dirty ore is dumped into a large washer, in which water is allowed to soak the ore on its way to a drain on the bottom of the unit. The amount of dirt remaining on the ore after this process is negligible, but water remains absorbed on the ore surface such that the net mass flow rate of the cleaned ore is 3100 kg/hr.
The dirty water is cleaned in a settler, which is able to remove 90% of the dirt in the stream without removing a significant amount of water. The cleaned stream then is combined with a fresh water stream before re-entering the washer.
The wet, clean ore enters a dryer, in which all of the water is removed. Dry ore is removed from the dryer at 2900 kg/hr.
The design schematic for this process was as follows:
a) Calculate the necessary mass flow rate of fresh water to achieve this removal at steady state.
b) Suppose that the solubility of dirt in water is . Assuming that the water leaving the washer is saturated with dirt, calculate the mass fraction of dirt in the stream that enters the washer (after it has been mixed with the fresh-water stream).
A schematic is given in the problem statement but it is very incomplete, since it does not contain any of the design specifications (the efficiency of the settler, the solubility of soil in water, and the mass flow rates). Therefore, it is highly recommended that you draw your own picture even when one is provided for you. Make sure you label all of the streams, and the unknown concentrations.
Around the washer: 6 independent unknowns (), three independent mass balances (ore, dirt, and water), and one solubility. The washer has 2 DOF.
Around the dryer: 2 independent unknowns () and two independent equations = 0 DOF.
Since the dryer has no degrees of freedom already, we can say that the system variables behave as if the stream going into the dryer was not going anywhere, and therefore this stream should not be included in the "in-between variables" calculation.
Around the Settler:5 independent unknowns (), two mass balances (dirt and water), the solubility of saturated dirt, and one additional information (90% removal of dirt), leaving us with 1 DOF.
At the mixing point: We need to include this in order to calculate the total degrees of freedom for the process, since otherwise we're not counting m9 anywhere. 5 unknowns () and 2 mass balances leaves us with 3 DOF.
Therefore, Overall = 3+2+1 - 6 intermediate variables (not including xO4 since that's going to the dryer) = 0
Recall that the idea is to look for a unit operation or some combination of them with 0 Degrees of Freedom, calculate those variables, and then recalculate the degrees of freedom until everything is accounted for.
From our initial analysis, the dryer had 0 DOF so we can calculate the two unknowns xO4 and m5. Now we can consider xO4 and m5 known and redo the degree of freedom analysis on the unit operations.
Around the washer: We only have 5 unknowns now (), but still only three equations and the solubility. 1 DOF.
Around the settler: Nothing has changed here since xO4 and m5 aren't connected to this operation.
Overall System: We have three unknowns () since is already determined, and we have three mass balances (ore, dirt, and water). Hence we have 0 DOF for the overall system.
Now we can say we know and .
Around the settler again: since we know m7 the settler now has 0 DOF and we can solve for and .
Around the washer again: Now we know m8 and xD8. How many balances can we write?
If we try to write a balance on the ore, we will find that the ore is already balanced because of the other balances we've done. If you try to write an ore balance, you'll see you already know the values of all the unknowns in the equations. Hence we can't count that balance as an equation we can use (I'll show you this when we work out the actual calculation).
The washer therefore has 2 unknowns (m2, xD2) and 2 equations (the dirt and water balances) = 0 DOF
This final step can also be done by balances on the recombination point (as shown below). Once we have m2 and xD2 the system is completely determined.