Thermal Mug: Mug Project

The following situation is proposed, concerning the cooling of a liquid (in this case water) using ice.  

“Warm water (an idealization of beer or soda) is poured into a container (an idealization of a chilled mug) that has been placed in the freezer. The system (i.e. mug plus water) is allowed to reach thermal equilibrium.”   -learning4doing.com

When creating a math model to predict the final temperature of the water several steps should be taken.  The first step is defining the situation so that I understand clearly what the real problem is.  The second is defining in concrete terms what my goal is, i.e. what exactly is the math model supposed to do.  Once a goal is established I can move on to generating ideas on how to solve this problem.  In this particular situation I see that the law of conservation of energy can be applied.  This gives me the tools I need to create an algebraic equation to reach my goal.  The following pictures are of my hand-written math model.  

After creating my math model and programming my equation into Matlab, I found that my answers were not at all satisfactory.  Somewhere in my math model I made a mistake.  But that is the exciting part, because it is usually in our mistakes that we learn the most.  I will have to revisit this problem in a later post after I have found my mistakes.  My current Matlab script function is as follows:

 function finaltemp1

%finaltemp1 determines the final temperature of water in a glass after a

%certain amount of ice has been melted in it (this function assumes that

%the glass with water is a closed system)

Sw = 4.18; % specific heat of water (in Joules/grams*Kelvin)

Si = 2.09; % specific heat of ice (in Joules/grams*Kelvin)

Hfus = 333.9; % heat of fusion of ice (in kJ/kg)

syms m1 t1 m2 mi ti; 

ti = input('Enter initial temperature of ice (in Kelvin):'); 

mi = input('Enter mass of ice (in grams):');

t1 = input('Enter temperature of water before ice is added (in Kelvin):');

m1 = input('Enter mass of water before ice added (in grams):');

m2 = m1 +mi;

T2 = (Sw*m1*t1 - (Si*mi*(273.15 - ti) + (Hfus*mi)/1000 + Sw*mi*273.15)/(Sw*(m1+m2)));

T2c = T2 - 273.15;

disp('final temperature of water (in celcius)')

disp(T2c)

end