1. MTRL 361
Project 5: Leaching of PbS Concentrate
Group 10
Muhammad Harith Mohd Fauzi
Jaehyuk Shim 32739112
Muhammad Arshad Hassni 24477119

2. Question 1
In this section, we are to determine the reaction rate constant kt with a given data of time (t) and
the fraction of PbS leached (X). It is evident that the leaching of PbS will increase over time and
the rate expression for leaching is:
Correspondingly, the time it takes to fully leach PbS is:
 D= average particle diameter, which is 76.4 microns (given in problem statement)
 kt = the reaction rate constant
The rate expression (1) can be rearranged in a different form to accommodate linear leaching as
the reaction rate constant kt is constant.
(2)
Taking our initial data (Table 1), we can plot a graph of 1 – (1-X)1/3 as a function of t (time)
which is shown in Graph 1.
Table 1: Data for the fraction of Pb leached over time
Time (min) Fraction Pb Leached (X) 1-(1-X)^(1/3)
2.5 0.032 0.0107825
5 0.21 0.0755665
10 0.54 0.2280557
15 0.71 0.3380894
20 0.82 0.4353784
30 0.95 0.6315969

3. Graph 1: Plot of 1 – (1-X)1/3 as a function of time, from the rate expression.
The linear line of the plot produces a slope of 0.0201 min-1, which is the (1/τ) term of the
equation (2).
Therefore,
Slope =
1
τ
=
2𝑘𝑡
𝐷
𝐤𝐭 =
D
2τ
The results of our calculations are shown below:
Table 2: Calculation of kt
Slope 0.0226 min-1
τ 44.248 min
do 76.4 μm
kt 0.863 μm/min
y = 0.0226x - 0.0235
R² = 0.991
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25 30 35
1-(1-X)^(1/3)
Time (min)
1-(1-X)^(1/3) vs time

4. Question 2
In this section, we will aim to find a continuous Particle Size Distribution (PSD) function using a
Rosin-Rammler distribution to the classifier data.
Defining the Tyler mesh size as variable D and f(D) as weight % retained (given data), the
cumulative distribution function F(D) can be described as:
After that, we need to define some numbers that characterize the PSD.
 The mean diameter µ:
 The variance σ:
 The coefficient of variation CV2
Given the classifier data for the Pb concentrate, we can obtain the numbers defined above using
the following calculation steps.
First we solve for f(Di, average) using values from the classifier data,
f(D1, average) = f(
0+29
2
) = 𝑓(14.5) =
0.21
29−0
= 0.00724 µ−1
f(D2,average) = f(33) =
0.091
37−29
= 0.0114 µ−1
and similarly up to D7
Using these seven discrete values of f(D), we can now evaluate for F(D).

5. = 0.21 = ∆F1
= 0.301
where ∆F2 = f(33)(37-29) = 0.091
With F(D), we can get the mean particle size µ.
*we will use the summation to approximate the integral
µ = 14.5(0.21) + 33(0.091)+ …
Finally we can calculate CV2,
= (
𝟏𝟒. 𝟓
µ
− 𝟏)
𝟐
𝟎. 𝟐𝟏 + (
𝟑𝟑
µ
− 𝟏)
𝟐
𝟎. 𝟎𝟗𝟏 + ⋯
A summary of our results is shown below on Table 3.
Table 3: Summary of calculations using the classifier data
Tyler
Mesh Size
D (μm)
Weight %
Retained
f(Di,average) ∆Fi F(D) Di,average∆Fi
(Di,average/μ
-1)^2∆Fi
0 21 0 0
29 9.1 0.0072413 0.21 0.21 3.045 0.1338483
37 15.6 0.0113750 0.091 0.301 3.003 0.0266425
53 18.3 0.0097500 0.156 0.457 7.02 0.0218451
74 17.5 0.0087142 0.183 0.640 11.6205 0.0025025
105 13.2 0.0056451 0.175 0.815 15.6625 0.0104726
149 5.2 0.0030000 0.132 0.947 16.764 0.0774763
420 0 0.0001918 0.052 0.999 14.794 0.4544924
sum 71.909 0.7272795

6. Therefore,
µ= 71.909
CV2 = 0.727795
Now we can apply the Rosin-Rammler distribution.
where D* is the normalizing particle size and m is the Rosin-Rammler coefficient .
We can obtain m by using our CV2 value,
and then with Goal Seek
With m, we can calculate the normalizing size D*:
Now the Rosin-Rammler distribution can be applied:

7. We then plot F(D) for the classifier data and Rosin-Rammler distribution as a function of D.
Graph 2: Comparison of the continuous PSD function and the Rosin-Rammler distriubution
Question 3
In this section, we will plot the normalized residence time distribution (Eθ) against
dimensionless time θ, using the following procedure.
The distribution of residence times is called the Exit Age Distribution or the E-curve. We can
define E as the mass fraction of the particles entering the system at some time =0 that exit during
time t ± dt/2 divided by t.
Therefore, E(t) can be deduced as:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300 400 500
F(D)
D (μm)
F(D) vs D
Rosin-Rammler
Classifier Data

8. At this point, we will introduce some new terms.
 The mean residence time:
 Non-dimensional time:
where Q is the volumetric flow (m3/s) and V is the tank volume (m3).
The Residence Time Distribution functions for the entire system:
where N is the number of equal-sized tanks
The exit-age distribution function can also be given in nondimensional time:
Using these results we were able to plot the results for 1, 10, 50, 100 equal size tanks.

9. Graph 3: Plot of normalized residence time versus dimensionless time for 1, 10, 50, 100 tanks.
Question 4
For a single tank RTD at 40˚C, we are to generate a plot of the fraction of Pb unleached versus
the average residence time (0 -100 minutes) for Pb particle sizes of 20, 120 and 225 µm.
For a particle diameter of D and residence time t, the fraction unreacted is X:
For RTD effects the equation can be re-written as:
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
E(θ)
θ
E(θ) vs θ
1 Tank
10 Tank
50 Tank
100 Tank

10. *N is simply 1 (single tank)
In order to integrate this equation, we must use the 2-point Gauss-Legendre method and write a
macro to repeat the process several times.
Using the macro, we are able to obtain plots for the 20, 125 and 225 µm particle sizes shown
below.
Graph 4: Fraction unleached vs. average residence time for 20 µm particle size
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
FractionUnleached
Residence Time (mins)
Fraction Unleached vs Average Residence Time

11. Graph 5: Fraction unleached vs. average residence time for 125µm particle size
Graph 6: Fraction unleached vs. average residence time for 225µm particle size
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
FractionUnleached
Residence Time (mins)
Fraction Unleached vs Average Residence Time
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
FractionUnleached
Residence Time (mins)
Fraction Unleached vs Average Residence Time

12. Table 4: Summary of time required to achieve >97.5% Pb extraction
Particle size (µm) Time required to achieve > 97.5% Pb extraction (mins)
20 ~116
125 ~711
225 ~1426
As we can see from the results, the larger the particle size, the longer the time it takes to achieve
more than 97.5% Pb extraction. This is observed due to the relation 𝜏 𝐷 =
𝐷
2𝑘 𝑡
that defines the
time required to fully leach the particle of diameter D that affected by the reaction-rate effects.
From the graph 4, it is observed that the plot demonstrates that the fraction unreacted is larger
compared to the graph 5 and 6 as these processes are governed by:
Question 5
By using the Rosin-Rammler PSD now, we are to plot fraction unleached as a function of
residence time once again. Taking into account of our previous Rosin-Rammler methods (part 2),
we can generate our plot by integrating the formula:
Again, the 2-point Gauss-Legendre method was used for the integration of this formula on macro.
The plot of our values shown in Graph 7:

13. Graph 7: Fraction unleached vs. residence time using the Rosin-Remmler PSD
Our results indicate that it takes approximately 35 minutes to achieve greater than 98% Pb
extraction.
Question 6
Taking into account both the PSD and the RTD functions, we will examine a system using 1
MFR and 10 MFR’s in series. Some of the operating conditions are temperature at 40˚C, 7200
m3/hr slurry feed and will specifically examine the condition at 96% conversion.
Combining PSD and RTD effects,
The inner integral (related to time) will be integrated and solved through the 2-point-Gauss
Legendre method.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30 35 40
FractionUnleached
Residence Time (mins)
Fraction Unleached vs Residence Time

14. The outer integral (diameter) will be solved using Simpson 1/3 Rule.
Table 5: Summary of time required to handle slurry feed at 96% conversion
No of mixed flow tanks (MFR) Time required to handle slurry feed at 96% conversion (mins)
1 ~256
10 ~35
As we can see from the table, 10 mixed flow tanks (MFR) in series gives a shorter time required
to handle 7200 m3/hr slurry feed at 96% conversion.
For N equal size MFR’s in series the average residence time can be calculated for either the
entire system for each tank within the system:
It is observed that the volume for a single big tank is much larger compared to the total volume
mixed flow tanks as the E function is reduced to (1/t)*exp(-t/𝑡̅)

15. The exit-age distribution function for the ideal MFR’s series has demonstrates that as N increases,
the time required decreases. In the limit N→ ∞, the RTD approaches ideal plug flow like having
the uniform residence time. Thus we achieve the objective of multiple tanks in series of having a
more uniform RTD that results with the reduction of total tank volume required to achieve some
specified fractional conversion.
Other than that, we have found another explanation for this phenomenon:
RTD effects are taken into account when this formula:
This formula relates the number of tank, N to the time required to handle the slurry feed at 96%
conversion. After solving the integral, time required is inversely proportional to the number of
tank. To simplify:
1 − 𝑋
𝑡
≈ 𝑁 ∗ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
However, this relation only applies for the mono-sized particle of diameter D.