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Three rods of Copper, Brass and Steel are welded together to form a Y-shaped structure. Area of cross section of each rod $4c{m^2}$. End of copper rod is maintained at${100^0}C$ whereas the ends of brass and steel are kept at ${0^0}C$. Lengths of the copper, brass and steel rods are $46,13,12cm$ respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are $0.92,0.26,0.16CGS$ units respectively. Rate of heat flow through copper rod is:

a. $4.8cal/s$
b. $6.0cal/s$
c. $1.2cal/s$
d. $2.4cal/s$

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Last updated date: 17th Jun 2024
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Answer
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Hint: Diffusing thermal energy within one material or between materials in contact is called thermal conductivity. In thermal conduction, the heat directly is proportional to the area of cross section. Apply the formula to determine the thermal conductivity. Then using that determines the rate of flow through the copper rod.

Formula used:
$\dfrac{{\vartriangle Q}}{{\vartriangle T}} = \dfrac{{KA\left( {{T_1} - {T_2}} \right)}}{x}$
$K,A,Q$ are the coefficient of thermal conductivity, area of cross section and the heat transferred.

Complete step by step answer:
Diffusing thermal energy within one material or between materials in contact is called thermal conductivity.
As the temperature increases in an object then the kinetic energy also increases.
This results in the collisions between molecules distributions. The main mode of heat transfer in between the solids is conduction.
In thermal conduction, the heat directly is proportional to the area of cross section. It is proportional to the change in temperature.
Greater the molecular agitation greater is the heat conduction.

A substance is said to be a good conductor of heat if the value of coefficient of thermal conductivity is large. Silver is the best conductor of heat among all the materials.
Gases are less efficient conductors than the liquid. Conductor increases when the resistance of the body decreases. Conductors are the main mode of heat transfer in case of solids.
$ \Rightarrow \dfrac{{\vartriangle Q}}{{\vartriangle T}} = \dfrac{{KA\left( {{T_1} - {T_2}} \right)}}{x}$

According the thermal conductivity,
$ \Rightarrow \dfrac{{d{Q_1}}}{{dt}} = \dfrac{{d{Q_2}}}{{dt}} = \dfrac{{d{Q_2}}}{{dt}} + \dfrac{{d{Q_3}}}{{dt}}$
Putting the values and we get,
$ \Rightarrow \dfrac{{0.92\left( {100 - T} \right)}}{{46}} = \dfrac{{0.26\left( {T - 0} \right)}}{{13}} + \dfrac{{0.12\left( {T - 0} \right)}}{{12}}$
Since, $T = {40^0}C$
$ \Rightarrow \dfrac{{d{Q_1}}}{{dt}} = \dfrac{{0.92 \times 4\left( {100 - 40} \right)}}{{40}} = 4.8cal/s$

Hence, the correct answer is option (A).

Note: For heat exchangers, the thermal conductors are very essential and they allow heat to be exchanged between liquids without mixing them.
The heat produced is transferred by thermal conduction. A substance is said to be a good conductor of heat if the value of coefficient of thermal conductivity is large.