Three rods of material X and three rods of material Y are...

Three rods of material $X$ and three rods of material $Y$ are connected as shown in figure. All are identical in length and cross-sectional area. if end $A$ is maintained at $60^{\circ} \mathrm{C}$, end $E$ at $10^{\circ} \mathrm{C}$, thermal conductivity of $X$ is $0.92 \mathrm{cal} / \mathrm{sec}-\mathrm{cm}-{ }^{\circ} \mathrm{C}$ and that of $Y$ is $0.46 \mathrm{cal} / \mathrm{sec}-\mathrm{cm}-{ }^{\circ} \mathrm{C}$ them find the temperature of junctions $B, C, D:$ (a) $20^{\circ} \mathrm{C}, 30^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}$ (b) $30^{\circ} \mathrm{C}, 20^{\circ}, 20^{\circ} \mathrm{C}$ (c) $20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}, 30^{\circ} \mathrm{C}$ (d) $20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}$

Three rods of material $X$ and three rods of material $Y$ are connected as shown in figure. All are identical in length and cross-sectional area. if end $A$ is maintained at $60^{\circ} \mathrm{C}$, end $E$ at $10^{\circ} \mathrm{C}$, thermal conductivity of $X$ is $0.92 \mathrm{cal} / \mathrm{sec}-\mathrm{cm}-{ }^{\circ} \mathrm{C}$ and that of $Y$ is $0.46 \mathrm{cal} / \mathrm{sec}-\mathrm{cm}-{ }^{\circ} \mathrm{C}$
them find the temperature of junctions $B, C, D:$
(a) $20^{\circ} \mathrm{C}, 30^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}$
(b) $30^{\circ} \mathrm{C}, 20^{\circ}, 20^{\circ} \mathrm{C}$
(c) $20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}, 30^{\circ} \mathrm{C}$
(d) $20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law states that the rate of heat transfer through a material is proportional to the temperature gradient and the area through which the heat flows. This principle is fundamental in determining how heat is conducted through different materials and is used to establish relationships between heat flux, thermal conductivity, cross-sectional area, and the temperature difference across a material.

Thermal Resistance

Thermal resistance is a measure of a material's resistance to heat flow. It combines the effects of material properties, such as thermal conductivity, with geometric factors like length and cross-sectional area. In complex configurations, thermal resistances can be combined in series or parallel to analyze the overall heat transfer, much like resistors in an electrical circuit.

Steady-State Heat Conduction

In steady-state heat conduction, the temperature at every point in the system remains constant over time because the heat entering any section of the system equals the heat leaving it. This condition allows for the application of energy balance principles to determine unknown temperatures and heat fluxes in interconnected systems.

Thermal Network Analysis

Thermal network analysis involves modeling a system of heat-conducting elements as a network of thermal resistances. This approach is analogous to analyzing electrical circuits using Kirchhoff's laws and enables the determination of junction temperatures and heat flow paths by setting up and solving systems of equations based on the conservation of energy.

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