Homework 1
MAE 502
Homework #1
Instructions/Notes:



1
Show all details!
These are individual assignments and absolutely no group collaboration is permitted. Copying
MATLAB codes and scripts from any source, including online mediums, is considered plagiarism and
will not be tolerated.
Submission Deadline in Beachboard: 09/22 (Tuesday, 11:59 pm)
In the figure below, heat is added to the oven as shown through the heating element and is simultaneously
lost to the environment due to temperature drop across the oven walls.
Oven
Heating coil
The variable x denotes the temperature inside the oven, which is assumed to be uniform, while T denotes
the ambient temperature of the air outside to oven. Both are expressed in degrees K . The rate of change of
oven temperature x is proportional to the rate of heat entering the oven from the heating element minus the
rate of heat leaving the oven due to convection and radiation. Empirically, a following non-linear
relationship is observed:
dx
4
4
2
=α (T −x ) + β ( T −x ) + γ u
dt
Here α , β ,∧γ are positive constants that depend on the thermal capacity of the oven. The term α (T −x )
represents heat loss due to convection and β ( T 4 −x 4 ) represents heat loss due to radiation. The input
variable u denotes the current in the heating element. For many manufacturing processes, such as
annealing, the temperature of the oven must be carefully controlled. If the current in the heating element is
kept constant to a range of values, determine the corresponding range of steady state temperatures (
dx /dt=0 ) that would be maintained inside the oven. Use the following data:
Parameter
u
α
β
γ
-
Values
10 A , 15 A , 20 A , 25 A , 30 A , 35 A
−1
50 min
−7
−1
−3
2 ×10 min ° K
−1 −2
800 ° K min A
Use Bisection method, Newton method and Secant method to determine the steady state temperature x
for each value of current to the heating elementu as specified.
Plot u vs . x
Homework 1
MAE 502
-
Comment on your results and discuss your observations.