Question

Problem 3 A portable thermocouple is used to measure the air temperature inside of a sauna. The sauna is unbearably hot and a person can only stay for 20 seconds, during which the temperature reading keeps on changing and a steady temperature is not reached. Leaving the thermocouple reader in the sauna for extended period of time has been deemed too risky for an electronics device Fortunately, a transient heat transfer model can be used to extract the temperature in the sauna 10 15 seconds [0 5 [25.2 35.9 45.5 53.1 59.3 degC Time Temperature According to a transient heat transfer analysis model (with lumped capacitance), the following relationship exists hA T-T +(T, -T )exp- I, where t time (s) T temperature (C) air temperature (C) ? primary unknown Ti = initial temperature (C) (= temperature at time zero) h convection coefficient (W/m2K) secondary unknown As-surface area of thermocouple junction (assuming sphere, As-?D2) D = thermocouple junction diameter : 0.001 (m) thermocouple junction density-8900 (kg/m3) V thermocouple junction volume (assuming sphere, V D3/6) c specific heat of thermocouple junction 380 (J/kgK) Using a nonlinear least squares method, fit the t vs. T data with the above equation, which calculates T as a function of t with two parameters: air temperature (T) and convection coefficient (h)-these are the regression coefficients What are the air temperature (T and the convection coefficient (h)? Plot the data points and the curve-fit smooth line as an overlay figure Note: It is important to use reasonable initial guesses for the temperature and the convection coefficient

Answer 1

%display A A (1) %display h h-x(2) held on %plot the smooth curve plot(tEE(t),' hold off 1K-

Sample output: Non linear least square method 65 60 45 40 35 30 10 12 14 16 18 20 0

%declare t

t=[0,5,10,15,20];

%declare T

T=[25.2,35.9,45.5,53.1,59.3];

%plot the given points

plot(t,T,'ro')

%print title

title('Non linear least square method');

%given function

FF=@(x,xdata) x(1)+(25.2-x(1))*exp(-((x(2)*3.14*0.001^2)/(8900*3.14*((0.001^3)/6)*380))*xdata);

%initial guess

x0=[1 1];

%call method

x=lsqcurvefit(FF,x0,t,T);

%display A

A=x(1)

%display h

h=x(2)

hold on

%plot the smooth curve

plot(t,FF(x,t),'*--');

hold off