The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

Question

The temperature at the point  in a substance with conductivity  is  Find the rate of heat flow inward across the cylindrical

Quizplus · Accepted Answer

The rate of heat flow can be calculated using Fourier's Law of Heat Conduction:
Q = -kA(dT/dr)
where Q is the rate of heat flow, k is the thermal conductivity, A is the cross-sectional area, and (dT/dr) is the temperature gradient.

Since the cylindrical shape suggests radial symmetry, we can assume that the heat flow is only in the radial direction, and the cross-sectional area A can be taken as a circle with radius r.

Q = -k(2πrL)(dT/dr)

We do not know the radius r or the length L of the cylindrical shape, but we can simplify the equation by assuming that the temperature gradient is constant across the cylindrical shape.

(dT/dr) = (T2 - T1) / (r2 - r1)

where T2 is the temperature at a radius r2 and T1 is the temperature at a smaller radius r1.

Substituting the given values, we get:

(dT/dr) = (11eaa3e5_54b9_758f_9f8f_8727dbd285f1_TB5971_11 - 11eaa3e5_54b9_4e7d_9f8f_0f702c0985b8_TB5971_11) / (r2 - r1)

(dT/dr) = 11eaa3e5_54b9_758f_9f8f_db0494ad0d1c_TB5971_11 / (r2 - r1)

Now, we can substitute this value into the Fourier's Law equation:

Q = -k(2πrL)(11eaa3e5_54b9_758f_9f8f_db0494ad0d1c_TB5971_11 / (r2 - r1))

Since we do not have any information about the dimensions of the cylindrical shape or the thermal conductivity of the substance, we cannot calculate the rate of heat flow explicitly. However, we can compare the answer choices and eliminate those that do not match our equation.

Choice A has an incorrect exponent of conductivity, so it can be eliminated.

Choice C, D, and E are all larger values than B. Since the rate of heat flow is directly proportional to the gradient (dT/dr), a larger value would imply a greater rate of heat flow. However, the smaller value of B matches with the smaller temperature difference and gives a reasonable estimate of the rate of heat flow. Therefore, we can choose B as the best answer choice.

The Temperature at the Point in a Substance with Conductivity