C is for Cookie (Not Specific Heat) Ken Langley and Robert Klaus Prepared for ME 340: Heat Transfer Winter Semester 2010 Prepared for ME 340: Heat Transfer Winter Semester 2010 Modeling Cookie Cooling with MatLab

Introduction What makes the perfect cookie? What makes the perfect cookie? – Soft, warm, chewy center – Outside that is firm enough to pick up How can you know when to eat the cookie after it comes out of the oven? How can you know when to eat the cookie after it comes out of the oven?

Objectives Determine how a cookie’s internal temperature changes over time to identify how long it will take to perfectly cool a cookie. Determine how a cookie’s internal temperature changes over time to identify how long it will take to perfectly cool a cookie. Create a MatLab code that will visually show how the internal temperature of the cookie is changing. Create a MatLab code that will visually show how the internal temperature of the cookie is changing. EAT COOKIES!!!! EAT COOKIES!!!!

Heat Transfer Principles To model 1D Transient Conduction we used the Explicit Finite-Difference Method. The cookie is modeled as a flat plate with convection on both sides.

Solution Cookie Dough Properties ρ = kg/m 3 k =.405 W/(m*K) c p = 2940 J/(kg*k) T i = 175 ºC T f = 60 ºC T ∞ = 27 ºC Cookie Thickness = m Number of Divisions = 20 Case I Properties h = 10 W/(m 2 *K) Time Step = 1 s Case I Properties h = 100 W/(m 2 *K) Time Step = 0.1 s

Conclusions & Recommendations From our simulation we realized that the convective heat coefficient (h) is the most important parameter in cooling a cookie. We recommend that if you want to eat a perfect cookie with a soft, warm center and a firm exterior you must use forced convection with a convective heat coefficient of at least 100 W/(m 2 *K).

Appendix %File: ME340_FiniteDifference.m %By: Robert Klaus and Kenneth Langley %This program uses the 1-D Transient Finite Difference Method to compute %the transient heat conduction in a flat plate. clc clear all clf %Thermophysical properties of cookie dough %k=0.405 W/m*K %rho= kg/m^3 %cp=2940 J/kg*K cookie = input('Do you want to model a cookie (1=yes)? '); if(cookie == 1) k = 0.405; rho = ; cp = 2940; else %obtain user input k= input('Enter thermal conductivity of plate [k(W/m*K)]:'); rho= input('Enter density of plate (kg/m^3):'); cp= input('Enter specific heat of plate (J/kg*K):'); end h= input('Enter convection coefficient of surrounding fluid [h(W/m^2*K)]:'); delta_t= input('Enter time step (s):'); L= input('Enter length of plate (m)]:'); M= input('Enter number of length divisions:'); Ti= input('Enter intial temperature of plate (degrees C):'); Tinf= input('Enter temperature of surrounding fluid (degrees C):'); Tf= input('Enter final temperature (degrees C):'); %Calculate delta_x, Biot and Fourier Numbers, and alpha alpha=k/(rho*cp); delta_x=L/M; Fo= alpha*delta_t/(delta_x)^2; Bi= h*delta_x/k; if(Fo>0.5) disp('Problem is unstable, delta x is too small'); disp('The new value of delta x is '); disp(delta_x); delta_x= sqrt(alpha*delta_t/0.4); Fo= 0.4; M= floor(L/delta_x); end %Initialize Temperature and Length Matrcies x=0:delta_x:L; T=zeros(15000,M+1); T(1,:)=Ti; %Calculate temperatures until middle node reaches Tf m=floor(M/2); j=2; while T(j-1,m)>Tf, %Calculate the temperature at the boundary nodes T(j,1)=2*Fo*(T(j-1,2)+Bi*Tinf)+(1-2*Fo-2*Bi*Fo)*T(j-1,1); T(j,M+1)=2*Fo*(T(j-1,M)+Bi*Tinf)+(1-2*Fo-2*Bi*Fo)*T(j-1,M+1); %Calculate the temperature at the interior nodes for i=2:M, T(j,i)=Fo*(T(j-1,i+1)+T(j-1,i-1))+(1-2*Fo)*T(j-1,i); end j=j+1; end d=size(T,1); %Plot results stepping through time figure(1) fig = figure(1); set(fig,'DoubleBuffer','on'); set(gca,'xlim',[0 L],'ylim',[Tinf Ti],'nextplot','replace','Visible','on'); hold on fps = 10; aviobj = avifile('Temp_dist_ANIMATION.avi','fps',fps,'quality',100); for i=1:100:j; plot(x,T(i,:)); hold on set(gca,'DrawMode','fast') frame = getframe(fig); aviobj = addframe(aviobj,frame); end aviobj = close(aviobj); disp('Movie Finished');

References Kulacki, FA, Kennedy, SC. “Measurement of the Thermo-Physical Properties of Common Cookie Dough.” Journal of Food Sciences. Vol. 43(2), pp