C is for Cookie Not Specific Heat Modeling

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

Introduction • 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? ?

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

Heat Transfer Principles • To model 1 D 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 = 1252. 3 kg/m 3 k =. 405 W/(m*K) cp = 2940 J/(kg*k) Ti = 175 ºC Tf = 60 ºC T∞ = 27 ºC Cookie Thickness = 0. 015 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: ME 340_Finite. Difference. 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=1252. 3 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 = 1252. 3; 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=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, 'Double. Buffer', '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, 'Draw. Mode', '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. 380 -384. 1978.