How Long Will it Take for All the Ice in Greenland to Melt, if it Were An Ice Cube?
Introduction:
One hot afternoon I was sitting on my desk organizing my books and sipping on a fizzy drink. Soon, I had finished it and all that remained in my glass was the ice cubes. I returned to organizing my books and then went to walk my dog. After returning after a couple of hours, I noticed that the ice had completely melted in my glass. I found this oddly satisfying and wanted to see if I can describe this phenomenon using mathematics. Then I thought what if there I can create a mathematical model that can illustrate this, it can possibly be used to predict the melting of ice and glaciers in the polar caps of the world. I collected secondary data on the ice research in Greenland. Scientists have estimated that the volume of ice, Greenland comprises of, is approximately 3,000,000 〖km〗^3, about 10% of the world’s total volume of ice. Furthermore, many news articles and science publications have mentioned using hyperbolic language about the volumetric loss of ice in last few recent years from ice shelves, sheets and glaciers in Greenland. Originally I wanted to estimate the how long it would take for all the ice in the world to melt, however I couldn’t find substantial data to help proceed with the investigation. As a global citizen and a keen science student I want to investigate further into how much of this is a reality and whether it is genuine global issue to be worried about.
Rationale:
I believe that this exploration will help me improve my skills in implicit differentiation, integration and applications of calculus, as rates of change and proportionality will be thoroughly used throughout the investigation. Also discussing and try to understand the effects of a global issue such as climate change helps me develop in global awareness and my skills in the natural sciences.
I believe that there is other mathematical elements involved in this investigation such as rate of heat transfer, however I’m try to build a mathematical model based purely of differential calculus by collecting my own data. I will be taking into account the effects of other variables such as temperature and pressure, I will attempt to keep them constant in order to make my investigation valid.
Investigation:
I decided to first derive the mathematical model for the melting of an ice cube. I thought that the rate of change in volume must be proportional to the change its change in surface area.
Assuming the cube retained a perfect cubic figure when melting, volume of cube can modelled by V=s^3 where s is the length of the side and V is volume. Surface area of the cube can be modelled as A=〖6s〗^2 where s is the length of the side and A is surface area.
Change in volume (decrease in this case) over change in time can be expressed as dV/dt which is proportional to the decrease in surface area of the ice cube as it melts.
∴ dV/dt∝ -〖6s〗^2
∴ dV/dt= -k(〖6s〗^2) … equation 1
I made a set of completely cubical ice blocks making it in a special mold than the traditional ice tray. I measured the dimensions 4.0 x 4.0 x 4.0cm = 64.0〖cm〗^3 and suspended it with string letting to minimize contact with other surfaces. I conducted this in the fridge at the temperature of 10.0°C which is somewhat close to the atmospheric temperatures of the Arctic in the summer. I let melting ice drip into a container for one hour then I measured the volume of water using a measuring cylinder.
(Set up diagram designed on photoshop by me, 2017)
Analysis:
I found that the volume had reduced to approximately 48 〖cm〗^3
Therefore:
V_0= 64〖cm〗^3
t_0= 0 hours
V_1= 48〖cm〗^3
t_1= 1 hour
It can inferred that the volume decreased by approximately by 1/4 hence:
V_0= 〖s_0〗^3
t_0= 0 hours
V_1= 3/4 s^3
t_1= 1 hour
Since we know that the length of side is also changing in relation to time, we can find the rate of change of s in relation of t.
V=s^3
∴d/dt(V)=〖d/dt(s〗^3)
∴dV/dt=〖d/ds s〗^3
∴dV/dt=〖3s〗^2 ds/dt … Equation 2
However as previously thought of:
dV/dt= -k(〖6s〗^2)
Substituting equation 1 into equation 2:
∴-k(〖6s〗^2 )=〖3s〗^2 ds/dt
∴ds/dt=(-k(〖6s〗^2))/〖3s〗^2
∴ds/dt=-2k … Equation 3
Therefore ds/dt is a constant value, which is the gradient for a linear relationship: The advantage of a linear relationship is that it doesn’t matter how far two points are, the gradient will always remain constant which isn’t true for the parabola where: dV/dt= -k(〖6s〗^2)
The gradient of a straight line is denoted as m in this case:
m= ds/dt=(s_(1-) s_0)/(t_(1-) t_0 )
∴-2k=(s_(1-) s_0)/(t_(1-) t_0 )
Substituting obtained values of t, t_1=1 and t_0=0
∴-2k=(s_(1-) s_0)/(1-0)
∴-2k=s_(1-) s_0 … equation 4
Now we know that
V_0= 〖s_0〗^3
∛(V_0 )= s_0
and
V_1= 〖s_1〗^3
We know that the volume after 1 hour was 3/4 the original volume (V_0)
3/4 V_0= 〖s_1〗^3
∴∛(〖3/4 V〗_0 )= s_1
Now substituting s_0 and s_1 into equation 4
∴-2k=s_(1-) s_0
∴-2k=∛(3/4 V_0 )-∛(V_0 )
Now we have a value for -2k, since we have a linear relationship gradient our linear relationship will be expressed in the form of y=mx+b
Now we can derive this by integrating both sides:
∴ds/dt=-2k
∴ds=-2k dt
∴ ∫▒ds=∫▒〖-2k〗 dt
∴ s=-2kt+C
Now to find the y intercept (C) by substituting terms s_0 and t_0 into our equation:
s=-2kt+C
s_0=-2kt_0+C
∛(V_0 )=-2k(0)+C
C=∛(V_0 )
Therefore putting C into our equation we have:
s=-2kt+∛(V_0 ) …equation 5
Substituting the value of -2k into our equation:
s=(∛(3/4 V_0 )-∛(V_0 ))t+∛(V_0 )
So when the ice cube would have completely been melted s=0,
s=(∛(3/4 V_0 )-∛(V_0 ))t+∛(V_0 )
0=(∛(3/4 V_0 )-∛(V_0 ))t+∛(V_0 )
-∛(V_0 )=(∛(3/4 V_0 )-∛(V_0 ))t
t=-∛(V_0 )/((∛(3/4 V_0 )-∛(V_0 ))) … equation 6
Using equation 6 we can calculate the time it will take for a the 4x4x4 ice cube to melt in the set conditions and assumption I made for my exploration (fridge 10°C, suspension to minimize contact, retention of perfect cube whilst melting)
The initial volume was 64 〖cm〗^3
t=-∛(V_0 )/((∛(3/4 V_0 )-∛(V_0 )))
t=-∛64/((∛(3/4 (64) )-∛64))
t=-4/(((3.63…)-4))
t=-4/((-0.365..))
t=10.9 (3 s.f.)
Graphical Representation of side length (s) decreasing as time (t) is increasing, for given initial volume 64 〖cm〗^3
It would take almost 11 hours for my cube of ice to melt in the atmospheric conditions I have placed it in.
Now to predict the ice melting of glaciers and total ice in the polar caps, we must make similar and assumptions and use different data.
Greenland covers a large volume of the earth’s ice. It had been estimated by the US National snow and Ice Data center that Greenland’s total ice volume is approximately 3 million cubic kilometers in the form or glaciers, ice sheets etc.
In the year 2011-2012, it was estimated that Greenland had lost almost 100 gigatons (approximately 100 〖km〗^3)of ice. This is 0.333% of the total volume of ice in Greenland.
Hence:
V_0= 3,000,000〖km〗^3
t_0= 0 years
V_1= 2,999,900〖km〗^3
t_1= 1 year
V_0= 〖s_0〗^3
∛3,000,000= s_0
V_1= 〖s_1〗^3
∛( 2,999,900)=s_1
Now substituting values of s_0 and s_1 into equation 4:
-2k=s_(1-) s_0
-2k =∛( 2,999,900)- ∛3,000,000
Now putting -2k value into equation 5:
s=(∛( 2,999,900)- ∛3,000,000)t + ∛3,000,000
So when the ice cube would have completely been melted s=0
0=(∛( 2,999,900)- ∛3,000,000)t + ∛3,000,000 t=-∛3,000,000/((∛( 2,999,900)-∛3,000,000))
t= 89,999
If all the ice in Greenland (1/10 of the world) were to melt in the form of a perfect cube with all 6 sides exposed with the conditions being constant (atmospheric temperature) then it would take 89,999 years to melt, regarding the melting rate in 2011 which is apparently the highest ever recorded in mankind. Unfortunately, we would feel the consequences much earlier than 89,999 years as global sea levels will rise and flood countries.
Conclusion and Evaluation:
I have developed a model that has successfully predicted how long it will take an ice cube to melt given that a specific volume loss has been measured over a given time. With my equation, I should be able to predict the initial volume of ice given the time it takes to completely melt. My investigation is completely based off assumptions such as the melting of the cube will retain its shape as it melts. This is highly unlikely since radiation is absorbed disproportionally due to its cubical shape and placement in the fridge. Hence this makes my mathematical model inaccurate. I also assumed that the conditions such as the temperature will be constant in the fridge whilst melting, even though I did not mention the rate of heat transfer in my exploration, it is a factor that needs to be explored to create a more accurate model of the melting of an ice cube.
Predicting the length of time it would take for all the ice in Greenland using my model is very inaccurate since all the ice volume cannot be formed into a cube. Even if it were to form a cube, its nearly impossible to predict the melting as different surface would be exposed to different atmospheric conditions such as the temperature and pressure. Furthermore using the rate of volume in relation to loss in one year is not constant since different factors play a role, e.g. greenhouses gases is propotional to human activity hence, the temperatures of the atmosphere is dependent on human activity. Just because one year the ‘cube’ decreased in height by ∛3,000,000-∛( 2,999,900)) doesn’t mean the next it will be constant as temperatures might rise or decline, hence increasing or decreasing the rate of change of s.
Bibliography:
Hypertextbook.com. (2003). Volume of Earth's Polar Ice Caps – The Physics Factbook. [online] Available at: https://hypertextbook.com/facts/2000/HannaBerenblit.shtml [Accessed 2 Aug. 2017].
Anon, 2009. Conversion factors for ice and water mass and volume. Climate Sanity. Available at: https://climatesanity.wordpress.com/conversion-factors-for-ice-and-water-mass-and-volume/ [Accessed August 4, 2017].
Anon, Desmos Graphing Calculator. Desmos Graphing Calculator. Available at: https://www.desmos.com/calculator [Accessed August 10, 2017].
Greicius, T., 2017. NASA, UCI Reveal New Details of Greenland Ice Loss. NASA. Available at: https://www.nasa.gov/feature/jpl/nasa-uci-reveal-new-details-of-greenland-ice-loss [Accessed August 10, 2017].