1. Brian Phillippi Daniel Rowe

2. Everyone likes their hot chocolate just right. If it's too hot, you can't enjoy it right away, If it's too cold, you have to warm it up again. With our new and easy program- You can enjoy perfect hot chocolate, every time. H OT C HOCOLATE

3. 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

4. Set Up of the Heat Transfer Problem Finite Difference and Lumped Capacitance in MATLAB Experimental validation

5. Results MATLAB Prediction vs. The Real Thing Starting Temp FDesired Temp FRoom Temp FSpeed m/sActual TimeFinite Differnce TimeLumped Capacitance Time 144.8138.671.405.83 min93.523621.50 138.613571.404.25 min50.303643.50 119.211571.408 min95.723559.83 114.6109.471.4012.15 min130.773501.67 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

6. 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.

7. 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 Input PTemp = 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*Dibott om^2/4*pi*cpwater))*loss; t= t+1; average = (node0+node1+node2+node3+nod e4+node5+node6+node7)/8; end time = 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)

8. Appendix MATLAB Output Welcome to the Hot chocolate perfect temperature calculator Enter the Perfect temperature of your Hot Chocolate in Farenheit - 135 Your perfect temperature in Kelvin is... 330.3722 Enter the starting temp in Farenheit - 138.6 Enter the room temp in Farenheit - 71.4 Starting temperature in Kelvin 332.3722 Room temperature in Kelvin 295.0389 Film temperature in Kelvin = 313.7056 Using the Thermophysical Properities Calculator please enter properties for air at the film temperature - cp = 1008 k =.0273 ro = 1.1158 nu =.00001911 Pr =.705 Starting temperature in Kelvin 332.3722 k for the water the starting temperature - 6528 ro for the water the starting temperature - 1000 cp for the water the starting temperature - 4185 Please enter the diameter of the cup at the top in inches - 3.2 Please enter the height of the cup in inches - 5 Enter speed at which you will be blowing on the hot chocolate in m/s - 0 The time is 3.0178e+003 seconds The average temp of the hot chocolate in Farenheit = 135.0000 Lumped Capacitance method The time is = 2.1861e+005