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The thermal conductivity of a solid depends upon the solid’s temperature as k = aT = + b where a and b are constants. The temperature in a planar layer of this solid as it conducts heat is given by
- a)aT + b = x + C2
- b)aT + b = C1x2 + C2
- c)aT2 + bT = C1x + C2
- d)None of them
Correct answer is option 'D'. Can you explain this answer?
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The given expression for the thermal conductivity of a solid, k = aT + b, suggests that the thermal conductivity (k) depends linearly on the temperature (T) of the solid. Here, a and b are constants.
To determine the temperature in a planar layer of this solid as it conducts heat, we need to solve the heat conduction equation, which is given by:
q = -kA(dT/dx)
where q is the heat flux, k is the thermal conductivity, A is the cross-sectional area, and (dT/dx) represents the temperature gradient. In this case, we are interested in the temperature variation along the x-direction in a planar layer of the solid.
To solve this equation, we need to integrate it. However, since the thermal conductivity depends on temperature, we need to rearrange the equation to isolate dT/dx. The rearranged equation becomes:
(dT/dx) = -q / (kA)
Now, substituting the expression for k from the given information, we have:
(dT/dx) = -q / [(aT + b)A]
Integrating both sides of the equation, we get:
∫(dT/dx) dx = ∫[-q / [(aT + b)A]] dx
This simplifies to:
∆T = -q / [aA] ln|aT + b| + C
where ∆T represents the change in temperature, C is the constant of integration, and ln represents the natural logarithm.
From the given answer choices, none of them match this derived equation. Therefore, the correct answer is option 'D' (None of them).
In conclusion, the temperature in a planar layer of the solid as it conducts heat cannot be accurately represented by any of the provided answer choices.
To determine the temperature in a planar layer of this solid as it conducts heat, we need to solve the heat conduction equation, which is given by:
q = -kA(dT/dx)
where q is the heat flux, k is the thermal conductivity, A is the cross-sectional area, and (dT/dx) represents the temperature gradient. In this case, we are interested in the temperature variation along the x-direction in a planar layer of the solid.
To solve this equation, we need to integrate it. However, since the thermal conductivity depends on temperature, we need to rearrange the equation to isolate dT/dx. The rearranged equation becomes:
(dT/dx) = -q / (kA)
Now, substituting the expression for k from the given information, we have:
(dT/dx) = -q / [(aT + b)A]
Integrating both sides of the equation, we get:
∫(dT/dx) dx = ∫[-q / [(aT + b)A]] dx
This simplifies to:
∆T = -q / [aA] ln|aT + b| + C
where ∆T represents the change in temperature, C is the constant of integration, and ln represents the natural logarithm.
From the given answer choices, none of them match this derived equation. Therefore, the correct answer is option 'D' (None of them).
In conclusion, the temperature in a planar layer of the solid as it conducts heat cannot be accurately represented by any of the provided answer choices.
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The thermal conductivity of a solid depends upon the solid’s temperature as k = aT = + b where a and b are constants. The temperature in a planar layer of this solid as it conducts heat is given bya) aT + b = x + C2b) aT + b = C1x2 + C2c) aT2 + bT = C1x + C2d) None of themCorrect answer is option 'D'. Can you explain this answer?
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