a thin copper rod, 10 m in length, is heated at its midpoint and the ends are he
ID: 2867591 • Letter: A
Question
a thin copper rod, 10 m in length, is heated at its midpoint and the ends are held at a constant temperature of 0o. When the temperature reaches equilibrium, the temperature profile is given by T(x)=100x(10-x), where 0 ? x ? 10 m is the position along the rod. The heat flux at a point on the rod equals -kT' (x), where k is a constant. If the heat flux is positive at a point, heat moves in the positive x-direction at that point, and if the heat flux is negative, heat moves in the negative x-direction.
a. With k=1, what is the heat flux at x=1? at x=9?
b. for what values of x is the heat flux negative?
(a) 0 ? x ? 10
(b) 1 ? x ? 9
(c) 5 ? x ? 10
(d) 0 ? x ? 5
for what values of x is the heat flux Positive?
(a) 5 ? x ? 10
(b) 0 ? x ? 10
(c) 1 ? x ? 9
(d) 0 ? x ? 5
c. explain the statement that heat flows out of the rod at its ends.
(a) T'(1)=800 < 0 and T'(9) = -800 > 0
(b) -T'(1)=800 < 0 and -T'(9) = -800 > 0
(c) -T'(1)=0 and -T'(9)=0
Explanation / Answer
T(x) = 100x (10-x)
T(x) = 1000x - 100x2
Differentiate both sides
T' (x) = 1000 - 200x
Heat Flux = -kT' (x)
= - k (1000 - 200x)
To find the value of heat flux at k=1 and x=1, let's substitute these values in the above expression.
Heat Flux = -1(1000 - 200 * 1)
Heat Flux = -1(1000 - 200) = -800
At k=1, x=9
Heat Flux = -1(1000 - 200*9) = -1(1000 - 1800) = 800
To figure out for what values of x, heat flux is positive or negative, we need to set the expressions for heat flux >0 and <0 respectively.
Heat Flux = -k(1000 - 200x)
For positive heat flux
-k(1000 - 200x) > 0
Divide both sides by -k
1000 - 200x < 0
1000 < 200x
Divide both sides by 200
5 < x
Therefore, Heat Flux is positive for 5 < x < = 10
For negative heat flux
-k(1000 - 200x) < 0
Divide both sides by -k
1000 - 200x > 0
1000 > 200x
Divide both sides by 200
5 > x
So, Heat Flux is negative for 0 < = x < 5
We have already been found T' (x) = 1000 - 200x
T' (1) = 1000 - 200(1) = 1000 - 200 = 800
-T' (1) = -800 < 0
T' (9) = 1000 - 200(9) = 1000 - 1800 = -800
-T' (9) = 800 > 0
So we can see that the Heat Flux is negative at x=1 and positive at x=9. It means heat is flowing out of the rod at it's ends.
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