A cylinder with a piston is filled with ideal gas. The gas temperature is held at 450 K. The cylinder is heated by an oven through a square metal rod connected between the oven and the cylinder. The rod has sides d = 2 cm and length L = 2.3 m.
(a) If the oven is held at 850 K, and the heat current conducted through the metal rod is 23 W, find the thermal conductivity of the metal in the rod.
k = W/m K
(b) For a larger cylinder at 450 K, a heat current of 57 W is needed. If we use the same metal rod to conduct heat, find the oven temperature in Kelvin that will maintain the necessary heat current.
T = K
(a) To find the thermal conductivity (k) of the metal in the rod, we can use Fourier's law of heat conduction:
Q = -kA(dT/dx)
where Q is the heat current, k is the thermal conductivity, A is the cross-sectional area of the rod, dT/dx is the temperature gradient along the rod.
In this case, we are given the heat current (Q = 23 W) and the temperature difference (dT = 850 K - 450 K = 400 K) between the oven and the cylinder. We also know the length (L = 2.3 m) and the cross-sectional area (A = d^2) of the rod, where d = 2 cm = 0.02 m.
Substituting these values into Fourier's law, we get:
23 = -k(0.02^2)(400/2.3)
Simplifying the equation:
23 = -0.0173913 k
Dividing both sides by -0.0173913:
k = 23 / -0.0173913
So, the thermal conductivity of the metal in the rod is approximately k = -1323.38 W/m K.
(b) To find the oven temperature (T) that will maintain the necessary heat current (Q = 57 W), we can rearrange the equation again to solve for T:
Q = -kA(dT/dx)
57 = -k(0.02^2)(450 - T)/2.3
Simplifying and rearranging the equation:
0.05 k = T - 450
Substituting the value of k (obtained from part a) as -1323.38:
0.05 (-1323.38) = T - 450
-66.17 = T - 450
T = -66.17 + 450
T = 383.83 K
So, to maintain the necessary heat current of 57 W, the oven temperature must be approximately 383.83 K.