In the square frame of side l of metallic rods- corners A and C are maintained at T1 and T2 respectively- The rate of heat flow from A to C is - If A and D are instead maintained at T1 and T2 respectively- the total rate of heat flow is x-3- Find - x-

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Partially Filled Dielectrics

In the square...

Question

In the square frame of side $l$ of metallic rods, corners $A$ and $C$ are maintained at $T_{1}$ and $T_{2}$ respectively. The rate of heat flow from $A$ to $C$ is $ω$ . If $A$ and $D$ are instead maintained at $T_{1}$ and $T_{2}$ respectively, the total rate of heat flow is $(\frac{x}{3}) ω$ . Find ' $x$ '.

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Solution

Thermal Resistance $R = \frac{L}{K A}$ where $L$ is the length of conductor, $K$ is the thermal conductivity and $A$ is the cross-sectional area.
If A and C are maintained at temp $T_{1}$ and $T_{2}$ respectively, we can consider $R_{A B}$ and $R_{B C}$ as being in series.
$R_{A C - u p p e r} = R_{A B} + R_{B C} = \frac{L}{K A} + \frac{L}{K A} = 2 \frac{L}{K A}$
$R_{A C - l o w e r} = R_{A D} + R_{D C} = \frac{L}{K A} + \frac{L}{K A} = 2 \frac{L}{K A}$
Both $R_{A C - u p p e r}$ and $R_{A C - l o w e r}$ are in parallel.
$∴ \frac{1}{R_{A C}} = \frac{1}{R_{A C - u p p e r}} + \frac{1}{R_{A C - l o w e r}}$
$∴ R_{A C} = \frac{L}{K A}$
$(\frac{d Q}{d t})_{A C} = \frac{T_{2} - T_{1}}{R_{A C}} = (T_{2} - T_{1}) \frac{K A}{L} = ω$ as given
When A and D are maintained at $T_{1}$ and $T_{2}$ respectively.
$R_{A B}$ , $R_{B C}$ and $R_{C D}$ will be in series.
$∴ \frac{1}{(R_{A D})_{e q}} = (R_{A B} + R_{B C} + R_{C D}) | | R_{A D}$
$∴ \frac{1}{(R_{A D})_{e q}} = (\frac{L}{K A} + \frac{L}{K A} + \frac{L}{K A}) | | (\frac{L}{K A})$
$∴ (R_{A D})_{e q} = \frac{\frac{3 L}{K A} \times \frac{L}{K A}}{\frac{4 L}{K A}} = \frac{3 L}{4 K A}$
$(\frac{d Q}{d t})_{A D} = \frac{T_{2} - T_{1}}{(R_{A D})_{e q}} = \frac{T_{2} - T_{1}}{\frac{3 L}{4 K A}} = \frac{4}{3} ω$

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In the square frame of side l of metallic rods, corners A and C are maintained at T1 and T2 respectively. The rate of heat flow from A to C is ω. If A and D are instead maintained at T1 and T2 respectively, the total rate of heat flow is (x3)ω. Find 'x'.