Double Declining Depreciation Method: First Order Kinetic Equation
Plants, machines, and equipment are considered long-lived assets. They are long-lived assets because the benefits provided by these assets go beyond one accounting period. However, these assets depreciate in value as they are utilized.
Depreciation of a long-lived asset is shown as an expense in the income statement. Usually, depreciation is inversely proportional to the life of the asset.
An engineer or analyst assumes the life of a machine and depreciate the value of the machine, uniformly, over the life of the machine. This method of depreciation is called Straight Line Depreciation method. In this method, a fixed value is expensed every year till the machine’s worth reaches its salvage value.
Straight Line Depreciation Expense = (Cost – Residual Life)/Useful Life
Few firms use Accelerated Depreciation method. In the accelerated depreciation mode, a machine depreciates more in its initial years than its later years. The higher degree of depreciation results in higher expenses in the income statement. It reduces Earning before Interest and Tax (EBIT), Earning before Tax (EBT) and Earning after Tax (EAT).
One may ask the benefit of showing accelerated depreciation in the income statement. The benefit is that high depreciation, caused by selecting accelerated instead of straight line depreciation, provides a ‘Tax Shield’, using the logic, lower income will have lower taxes.
Double Declining or Diminishing Balance is one of the Accelerated Depreciation method.
The formula for DDB depreciation is –
DDB Depreciation Expense = 2*(Cost – Accumulated Depreciation)/Useful Life …… Eq. 1
If we pay attention to the above formula, it is nothing but a First Order Kinetic Equation. The general form of a first order kinetic equation is –
A = A0 × e -kt…… Eq. 2
Where t = time in years; A0 = initial cost of the machine; and A = residual value of the machine after year t.
On comparing Eq 1 and 2, we can easily deduce that
k = ln [Useful Life/(Useful Life – 2)]
Formula shown in Equation 2 is not given in any of the CFA exam study books. This formula can reduce the steps to calculate the worth of a machine saving time in the exam.
SAG Mill Grinding Balls
Few points on Grinding Balls –
- In the mineral processing industry, manpower is the biggest operating expenditure followed by electric power. Grinding balls is the third largest cost.
- Throughput (Tons per Hour) is directly proportional to % Ball Charge.
- Grinding balls are made of steel. Grinding balls break the ore by impact and attrition.
- When carbon in iron is less than 2%, the alloy is called steel. When it is more than 2%, it is called cast iron.
- Grinding balls have unique hardness profiles, where the hardness from ball surface to ball centre drops.
- Carbon, Manganese, Sulphur, Phosphorous, Silicon, Nickel, Chromium, Molybdenum and Copper are the main alloying elements in the grinding ball.
- Among all the elements, carbon predominantly contributes to the hardness (up to 0.9%).
- Total wt% of alloying elements in the grinding balls is usually less than 5%.
- Surface of a grinding ball has around 90% tempered martensite, 8% retained austenite and 1% bainite. Center of a ball has 75% tempered martensite, 20% bainite & 5% retained austenite.
- Higher wt% of martensite on surface and lower wt% of martensite at center make the surface hard and center soft.
To be continued…
PE Exam Pass Rate – Metallurgical & Materials Discipline
I took PE exam in October 2011. My discipline was Metallurgical and Materials engineering. At that time, I was looking for historical pass rate in my discipline but there was no such information readily available.
After hours of search on internet, I am able to collect the historical pass rates for PE exam. Hope it helps someone out there –
PE Pass Rates in Metallurgical & Materials Engineering Exam –
| Year | First Time Takers | Repeat Takers |
| Pass Rate | Pass Rate | |
| 2015 | 83% | 55% |
| 2014 | 47% | 25% |
| 2013 | 62% | 0% |
| 2012 | 56% | 22% |
| 2011 | 53% | 46% |
| 2010 | 64% | 50% |
| 2009 | 56% | 33% |
| 2008 | 56% | 56% |
| 2007 | 52% | 45% |
Sources –
2015 – http://ncees.org/exams/pe-exam/
2014 – https://engineers.texas.gov/exam_stats/ppbydis_10_14.html
2013 – http://www.testmasters.com/pe/faq
2012 – http://www.engineeringdesignresources.com/category/pe-exam-study-guide/
2007 to 2011 – http://www.eng-tips.com/viewthread.cfm?qid=300642
Magical Squares
Finally, got the opportunity to build a simple software program called ‘Magical Square’. The software was built using Java programming language. Microsoft Expression Encoder 4 was used to capture the video of GUI. Background music is from the song – Rainy Day Games.
Below is the link to the YouTube video –
https://www.youtube.com/watch?v=YQgahHua-sE
Enjoy !

Rittinger’s Theory of Grinding
P. R. Rittinger proposed his theory of grinding in 1867. It was one of the oldest and widely accepted theories on grinding in his time. He proposed that the work required to break a particle is proportional to the new surface area created.
Mill Sound
Historically, load cells and conductivity probes were/are being used to measure the load inside the tumbling mills. Starting from 1980s, mass transducers were introduced for load level sensing which proved to be more effective. The biggest problem with the mass transducers was that it could not be retrofitted. Meanwhile, several research groups and mining companies tried to explore the acoustic emission (mill sound) emanating from the tumbling mills to understand the load level. Some of the studies conducted on mill sound are summarized below.
Research done by J.L. Watson et al. –
- Steel-steel collisions produce sound at higher frequencies compared to the rock-rock or rock-steel collisions. Steel-steel collisions can be found above 2000 Hz.
- At higher frequencies (suitable to detect steel-steel collision) larger size balls makes less noise compared to smaller size balls.
- Higher mill speed makes more noise compared to the lower mill speed as long as mill speed is below its critical value.
- Increasing the ball charge level increases the mill sound up to a certain level before it starts to level off.
- Pulp viscosity has an inverse relation with the mill sound i.e. higher the viscosity, lower the sound. Pulp viscosity is a function of pulp density as well as particle size distribution in the pulp.
- Higher circulating load indicates a longer grinding time which means finer particles in the pulp and changes in the viscosity of the pulp as well as sound.
Research done by Klimpel et al. –
- As the pulp density inside the mill increases, the rheology of the pulp changes from dilatant to pseudoplastic.
- The change from dilatant to pseudoplastic condition can be identified by the increment in the mill sound.
- Rocks are ground effectively in the pseudoplastic conditions.
- Lower viscosity in the pseudoplastic region increases the grinding efficiency. Lower viscosity is attained with the help of grinding additives. Change in the viscosity by grinding additives can be observed with the help of mill sound.
Implementation of mill sound at Saaiplaas mill –
Anglo-American’s SAG mill at Saaiplaas (South Africa) was subject to several testing with respect to mill sound. Saaiplaas’ SAG mill was 14 feet in diameter and 24 feet in length. The grinding circuit had a SAG mill making a close loop with the hydrocyclones. Saaiplaas SAG was bidirectional in rotation and had two microphones on each side of the mill. The microphones were positioned above and below the normal point of impact.
Research done at Saaiplaas indicated that as the load level rises in the mill, an upper microphone reads higher signals compared to the lower microphone, and as the mill unloads the lower microphone reads higher. Saaiplaas also found a strong correlation between the signals from the lower microphone and the mill pulp density as long as the point of impact is above the lower microphone and the pulp density is low.
Implementation of mill sound at PTFI –
PTFI grinding circuit consist of a variable speed SAG in a closed circuit with a pebble crusher and ball mills. The SAG mill is 34.5’ in diameter and 15.5’ in length. Initially, PTFI used mill sound, bearing pressure and load cells to sense the mill load level. Their studies revealed that the mill sound is more sensitive to load level sensing compared to bearing pressure, power, torque, amps and mill weight. Mill sound along with the speed of the mill plays a vital role in their control strategy.
Research on mill sound at JKMRC –
Studies done by JKMRC group indicated that a grinding mill generates noise at different frequencies. These frequencies are caused by the different type of grinding action. Noise being generated at the lower frequencies (around 100 Hz) are likely to be from attrition grinding while noise from higher frequencies are of impact grinding (300 Hz and above). Noise generated by steel to steel impact is at higher frequencies than the rock to rock impact confirming the results from Watson et al. JKMRC also mentioned that even if the operating conditions are stable, the type of collisions are likely to be ever-changing in nature causing the sound signals to change.
Solid in Liquid
Solid in liquid is usually found in one of the three forms –
- Solution
- Colloidal Dispersion
- Suspension
These three forms are defined by the particle size of solid in liquid. Lately, I saw a figure in a presentation where additional information on particle visibility, separation methodology etc. were given. Below is the table describing the three categories –

As can be seen depending on the sizes of the particles, different techniques can be used to separate the particles.
Rec-Cal Calculator
While working for Endako Mine, I realized that there is serious need for a simple, user friendly, stand-alone recovery calculator for supervisors and non-metallurgists. I envisioned a stand alone calculator in Java GUI. After leaving Endako, I got the opportunity to build one. Later, I donated this calculator to Endako for use (free of charge).
Below is the screen shot of the calculator. It is called Rec-Cal –

This is a custom-made calculator for molybdenite but it can be easily modified for any other metal e.g. copper, zinc, nickel etc.
Please leave a comment if you want a free copy (modified as per your system) of this software.
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