Question

In: Physics

According to Newton’s law of cooling, the temperature of a body changes at a rate pro-...



According to Newton’s law of cooling, the temperature of a body changes at a rate pro-
portional to the difference between its temperature and that of the surrounding medium (the ambient
temperature),
dT
dt = −k(T − Ta) with T(0) = T0
where T is the temperature of the body in oC, t is time in min, k = 0.019 min−1
is the proportion-
ality constant, and Ta = 20 oC is the ambient temperature. The analytical solution for this ordinary
differential equation (ODE) is:
T(t) = Ta + (T0 − Ta)e
−kt
The Euler’s numerical method for solving the ODE assumes that the first derivative is approximated
using the finite difference equation:
dT
dt ≈
∆T
∆t
=
Ti+1 − Ti
∆t
Suppose that a cup of coffee originally has a temperature of 70 oC. Solve the ODE
(a) Using Eulers method for the temperatures from t = 0 to 20 min using a step size of 5 min.
(b) Plot the results obtained in parts (a) together with the analytical solution given above

Solutions

Expert Solution

actually in this problem we are ananlysing the problem of body after every 5 min and applying Euler Equation

step size = 5 min =t

According to gven Euler equation here

dT/dt =T/t =    -k(T -Ta)

Initial Temperature of cup of tea = Tinitial = 70 oC

Let T1 ,T2 ,T3 ,T4    be the temperature of body after first , 2nd, 3rd and 4th step size

given Ta =20 oC and k = 0.019 min^-1

after first step size of 5 min

T / t   = T / 5 = - k ( T1 - Ta )      and     [    T1   = Tinitial - T    ]

T1 = 65.66 oC , T 2 = 61.69 oC , T3 = 58.07 oC , T4 = 54. 77 oC


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