The rate of cooling of a body can be expressed as dT dt = -k(T-Ta) where T = temperature of the body (°C), t = time, Ta = temperature of the surrounding medium (°C), and k = a proportionality constant (per minute). Thus, this equation, called the Newton's law of cooling specifies that the rate of cooling is proportional to the difference in the temperature of the body and of the surrounding medium (the ambient temperature). 1) Design an experiment to determine the value of k of a cup of hot coffee experiencing a natural cooling for one hour. Utilize numerical differentiation of 0(h²) order to determine at each value of time. Plot versus (T- Ta) and employ a suitable dt dt regression to evaluate k. Evaluate the equation obtained by regression method if it best fit the data.
The rate of cooling of a body can be expressed as dT dt = -k(T-Ta) where T = temperature of the body (°C), t = time, Ta = temperature of the surrounding medium (°C), and k = a proportionality constant (per minute). Thus, this equation, called the Newton's law of cooling specifies that the rate of cooling is proportional to the difference in the temperature of the body and of the surrounding medium (the ambient temperature). 1) Design an experiment to determine the value of k of a cup of hot coffee experiencing a natural cooling for one hour. Utilize numerical differentiation of 0(h²) order to determine at each value of time. Plot versus (T- Ta) and employ a suitable dt dt regression to evaluate k. Evaluate the equation obtained by regression method if it best fit the data.
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Please answer both q1 and q2
Transcribed Image Text:The rate of cooling of a body can be expressed as
dT
= -k(T – Ta)
dt
where T = temperature of the body (°C), t = time, Ta = temperature of the surrounding
medium (°C), and k = a proportionality constant (per minute). Thus, this equation, called the
Newton's law of cooling specifies that the rate of cooling is proportional to the difference in
the temperature of the body and of the surrounding medium (the ambient temperature).
1) Design an experiment to determine the value of k of a cup of hot coffee experiencing a
natural cooling for one hour. Utilize numerical differentiation of 0(h?) order to
determine
at each value of time. Plot versus (T – Ta) and employ a suitable
regression to evaluate k. Evaluate the equation obtained by regression method if it best
fit the data.
2) Based on the value of k determined from the experiment, estimate the temperature of
the cup of coffee at time, t = 30 min using the second-order Runge-Kutta with Heun,
Midpoint and Ralston's method. Use the same initial temperature, Tinitial from part (1)
with a step size of 5 min. How accurate is your answers compared to the measured
temperature from the experiment?
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