Bria

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pi Daniel Row

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Everyone likes their hot chocolate Everyone likes their hot chocolate just right. If it's too hot, you can't just right. If it's too hot, you can't enjoy it right away, If it's too cold, enjoy it right away, If it's too cold,

you have to warm it up again. you have to warm it up again. With our new and easy program- With our new and easy program-

You can enjoy perfect hot You can enjoy perfect hot chocolate, every time. chocolate, every time.

HOT CHOCOLATE

How long should I wait Until I can drink my

chocolate?

• Can we predict what effects the time?• Do the dimensions of my mug matter?• Does the initial temperature matter?• Does the amount of hot chocolate matter?

The Hot Chocolate Problem

Set Up of the Heat Transfer Problem

Finite Difference and Lumped Capacitance in

MATLAB

Experimental validation

ResultsMATLAB

Prediction vs.

The Real Thing

Starting Temp F Desired Temp F Room Temp F Speed m/s Actual Time Finite Differnce Time Lumped Capacitance Time144.8 138.6 71.4 0 5.83 min 93.52 3621.50138.6 135 71.4 0 4.25 min 50.30 3643.50119.2 115 71.4 0 8 min 95.72 3559.83114.6 109.4 71.4 0 12.15 min 130.77 3501.67

Assumptions Made•Mug is perfectly insulated•Volume for each node is at a constant diameter•Experimental measurements were in the middle of mug•We did not account for free convection flow

Conclusion

=+• Your favorite temperature can be predicted… but not with the assumptions we made.• If you want to drink it right away…• Pour the water at a temperature as close to your favorite temperature as possible.•Pour only a little hot chocolate.•Get a short mug with a big diameter.

Appendix%Brian Phillippi and Daniel Rowe%ME 340 Project%Vladimir Soloviev%The Perfect Hot Chocolate

clc;

disp('Welcome to the Hot chocolate perfect temperature calculator')%User InputPTemp = input('Enter the Perfect temperature of your Hot Chocolate in Farenheit - ');

PKel = (PTemp + 459.67)*5/9;

disp('Your perfect temperature in Kelvin is...')disp(PKel)

Tstart = input('Enter the starting temp in Farenheit - ');Tstart = (Tstart + 459.67)*5/9;Troom = input('Enter the room temp in Farenheit - ');Troom = (Troom + 459.67)*5/9;

disp('Starting temperature in Kelvin')disp(Tstart)disp('Room temperature in Kelvin')disp(Troom)

Tfilm = (Tstart + Troom)/2;disp('Film temperature in Kelvin = ')disp(Tfilm)

disp('Using the Thermophysical Properities Calculator please enter properties')disp('for air at the film temperature -')

cp = input('cp = ');k = input('k = ');ro = input('ro = ');nu = input('nu = ');Pr = input('Pr = ');

kwater = input('k for the water the starting temperature - ');rowater = input('ro for the water the starting temperature - ');cpwater = input('cp for the water the starting temperature - ');

alpha = k/(ro*cp);alphawater = kwater/(rowater*cpwater);

alpha = k/(ro*cp);alphawater = kwater/(rowater*cpwater);

Di = input('Please enter the diameter of the cup at the top in inches - ');Height = input('Please enter the height of the cup in inches - ');

Di = Di*2.54/100;Height = Height*2.54/100;

Lc = (pi*Di^2/4)/(pi*Di);

V = input('Enter speed at which you will be blowing on the hot chocolate in m/s - ');if V >0; Rel = V*Di/nu; Nul = .664*Rel^.5*Pr^(1/3); h = Nul*k/Di; else Ral = 9.8*Tfilm^(-1)*(Tstart-Troom)*Lc^3/(nu*alpha); Nul= .54*Ral^.25; h = Nul*k/Lc;end

nodes = 8;delx = Height/7;delt = .01; %s

Fo = alphawater*delt/(delx^2);Bi = h*delx/kwater;

node0 = Tstart;node1 = Tstart;node2 = Tstart;node3 = Tstart;node4 = Tstart;node5 = Tstart;node6 = Tstart;node7 = Tstart;

t=0;average= Tstart;loss=1;

while average > PKel node0 = (2*Fo*(node1 + Bi*Troom) + ( 1- 2*Fo -2*Bi*Fo)*node0)*loss; node1 = (Fo*(node0 + node2) + (1 - 2*Fo)*node1)*loss; node2 = (Fo*(node1 + node3) + (1 - 2*Fo)*node2)*loss; node3 = (Fo*(node2 + node4) + (1 - 2*Fo)*node3)*loss; node4 = (Fo*(node3 + node5) + (1 - 2*Fo)*node4)*loss; node5 = (Fo*(node4 + node6) + (1 - 2*Fo)*node5)*loss; node6 = (Fo*(node5 + node7) + (1 - 2*Fo)*node6)*loss; node7 = (node7+2*kwater*(node6-node7)*delt/(rowater*delx*Dibottom^2/4*pi*cpwater))*loss; t= t+1; average = (node0+node1+node2+node3+node4+node5+node6+node7)/8;

endtime = t*delt;disp('The time is ') disp(time); disp(' seconds')averageinF = 9/5*average-459.67;disp('The average temp of the hot chocolate in Farenheit = ')disp(averageinF)

Matlab Code

disp('Lumped Capacitance method')timelump= rowater*cpwater*Height/h*log(abs((Troom-Tstart/Troom-PKel)));disp('The time is = ')disp(timelump)

Appendix

MATLAB Output