A metal rod of length L and cross-sectional area A converts...

A metal rod of length $L$ and cross-sectional area $A$ converts a large tank of water at temperature $\theta_{0}$ and a small vessel containing mass $m$ of water at initial temperature of $\theta_{1}\left(<\theta_{0}\right)$. If the thermal conductivity of rod is $K$, then the time taken for the temperature of water in smaller vessel to become $\theta_{2}\left(\theta_{1}<\theta_{2}<\theta_{0}\right)$ is : (Given: Specific heat capacity of water is $s$ and all other heat capacities are neglected) (ì) $\bar{i}=\frac{m s}{K A} \operatorname{iog} \frac{\left(\theta_{0}-\theta_{1}\right)}{\left(\theta_{0}-\theta_{2}\right)}$ (b) $T=\frac{L m s}{K A} \log \left(\frac{\theta_{0}-\theta_{1}}{\theta_{0}-\theta_{2}}\right)$ (c) $T=\frac{2 m L s}{K A}\left(\log \frac{\left(\theta_{0}-\theta_{1}\right)}{\theta_{0}-\theta_{2}}\right)$ (d) none of the above

A metal rod of length $L$ and cross-sectional area $A$ converts a large tank of water at temperature $\theta_{0}$ and a small vessel containing mass $m$ of water at initial temperature of $\theta_{1}\left(<\theta_{0}\right)$. If the thermal conductivity of rod is $K$, then the time taken for the temperature of water in smaller vessel to become $\theta_{2}\left(\theta_{1}<\theta_{2}<\theta_{0}\right)$ is :
(Given: Specific heat capacity of water is $s$ and all other heat capacities are neglected)
(ì) $\bar{i}=\frac{m s}{K A} \operatorname{iog} \frac{\left(\theta_{0}-\theta_{1}\right)}{\left(\theta_{0}-\theta_{2}\right)}$
(b) $T=\frac{L m s}{K A} \log \left(\frac{\theta_{0}-\theta_{1}}{\theta_{0}-\theta_{2}}\right)$
(c) $T=\frac{2 m L s}{K A}\left(\log \frac{\left(\theta_{0}-\theta_{1}\right)}{\theta_{0}-\theta_{2}}\right)$
(d) none of the above

Key Concepts

Differential Equation Modeling and Logarithmic Integration

When setting up the energy balance, one often encounters a first-order differential equation that relates the temperature difference between two bodies to the heat transfer rate. Its solution typically involves logarithmic functions, reflecting the exponential approach of the system’s temperature towards equilibrium.

Energy Conservation in Transient Heat Transfer

This concept involves balancing the rate of energy input (or loss) through conduction with the rate at which the thermal energy of the system (in this case, the water) changes. It leads to the formulation of differential equations that describe how temperature evolves with time as heat is transferred.

Thermal Conductivity

This concept defines the intrinsic ability of a material to conduct heat. It is a material-specific property that appears in the rate equation for heat transfer by conduction, determining how effectively thermal energy can be transmitted through the material over a given distance under a temperature gradient.

Fourier’s Law of Heat Conduction

This principle states that the heat flux through a material is proportional to the negative temperature gradient and the area through which the heat flows. It is crucial in deriving expressions for the heat transfer rate along a rod, as it connects the material’s conductivity with the spatial change in temperature.

Specific Heat Capacity

This is the measure of the energy required to raise the temperature of one unit mass of a substance by one degree. In the context of transient heat transfer, it determines the amount of energy needed to change the temperature of the water, linking the heat transfer rate to the rate of temperature change over time.

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