Problem: A researcher measured temperature at various locations in a 1-m-long solid rod of constant cross-section. The cross-section of the rod is square with each side being 1 cm. From the measurements, the researcher established that the temperature of the rod is changing as T(x) = To + Ax² + Bx, where x is the distance from the left end of the rod expressed in m, To = 301.23 K, A = 20.34 K/m², and B = 33.1 K/m. Assuming that the rod can be treated to go through one-dimensional heat transfer along the positive x direction, what is the conductive heat transfer rate along x direction at 0.5 m from the left end of the rod (i.e., at x = 0.5 m)? The thermal conductivity of the rod is 26.7 W/m.K. Assume that the rod is insulated from the surrounding (i.e., there is no convection or radiation with the surrounding).
Problem: A researcher measured temperature at various locations in a 1-m-long solid rod of constant cross-section. The cross-section of the rod is square with each side being 1 cm. From the measurements, the researcher established that the temperature of the rod is changing as T(x) = To + Ax² + Bx, where x is the distance from the left end of the rod expressed in m, To = 301.23 K, A = 20.34 K/m², and B = 33.1 K/m. Assuming that the rod can be treated to go through one-dimensional heat transfer along the positive x direction, what is the conductive heat transfer rate along x direction at 0.5 m from the left end of the rod (i.e., at x = 0.5 m)? The thermal conductivity of the rod is 26.7 W/m.K. Assume that the rod is insulated from the surrounding (i.e., there is no convection or radiation with the surrounding).
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Transcribed Image Text:Problem: A researcher measured temperature at various locations in a 1-m-long solid rod of constant
cross-section. The cross-section of the rod is square with each side being 1 cm. From the measurements,
the researcher established that the temperature of the rod is changing as T(x) = To + Ax² + Bx, where
x is the distance from the left end of the rod expressed in m, To = 301.23 K, A = 20.34 K/m², and
B = 33.1 K/m. Assuming that the rod can be treated to go through one-dimensional heat transfer along the
positive direction, what is the conductive heat transfer rate along x direction at 0.5 m from the left end
of the rod (i.e., at x = 0.5 m)? The thermal conductivity of the rod is 26.7 W/m.K. Assume that the rod is
insulated from the surrounding (i.e., there is no convection or radiation with the surrounding).
Solution: Since this is one-dimensional heat transfer along x, the conduction heat transfer rate is
dT
dx
q" = -k
Using the expression for temperature obtained by the researcher, = 2Ax + B.
Plugging this back in,
Putting x = 0.5,
q" = -k (2Ax + B)
dT
dx
q"
So, the heat transfer rate at x = 0.5 m is 1426.85 W along the positive x direction.
= -1426.85 W
(1)
(2)
(3)
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