Hello everyone!

I have been trying to solve a problem of heat transfer from H.C Verma for days and still haven't figured out the right way to do it. I would really appreciate it, if someone from this great community could drop by a hint to help me finaly crack this problem. Here it goes:

Two bodies of mass "m1" and "m2" and specific heat "s1" and "s2" are connected by a rod of length " L ", cross-sectional area "A", thermal conductivity "K" and negligible heat capacity. The whole system is thermally insulated. At time t =0, the temperature of first body is T1 and the temperature of second body is T2 (T2>T1). Find the temperature difference between the the two bodies at time 't'.

Thanks.

Let's break down the problem of heat transfer between the two bodies connected by a rod. This scenario involves conduction, which is the transfer of heat through a material without any movement of the material itself. The key here is to understand how heat flows from the hotter body to the cooler one over time. We'll use Fourier's law of heat conduction to analyze the situation.

Understanding the Setup

We have two bodies with different masses and specific heats, connected by a rod. The important parameters are:

• m1: Mass of the first body
• m2: Mass of the second body
• s1: Specific heat of the first body
• s2: Specific heat of the second body
• L: Length of the rod
• A: Cross-sectional area of the rod
• K: Thermal conductivity of the rod

Heat Transfer Equation

According to Fourier's law, the rate of heat transfer (Q) through the rod can be expressed as:

Q = -K \cdot A \cdot \frac{dT}{dx}

Here, dT is the temperature difference across the rod, and dx is the length of the rod. Since the rod has negligible heat capacity, we can assume that the heat lost by the hotter body equals the heat gained by the cooler body.

Setting Up the Heat Balance

At any time t, let T1(t) be the temperature of the first body and T2(t) be the temperature of the second body. The heat lost by the first body can be expressed as:

Q1 = m1 \cdot s1 \cdot (T1 - T1(t))

And the heat gained by the second body is:

Q2 = m2 \cdot s2 \cdot (T2(t) - T2)

Equating Heat Transfer

Since the heat lost by the first body equals the heat gained by the second body, we can set these two equations equal to each other:

m1 \cdot s1 \cdot (T1 - T1(t)) = m2 \cdot s2 \cdot (T2(t) - T2)

Solving the Differential Equation

To find the temperature difference at time t, we need to express the temperatures in terms of time. The heat transfer rate can also be expressed as:

Q = \frac{K \cdot A}{L} \cdot (T2(t) - T1(t))

By substituting this into our heat balance equation, we can derive a differential equation that describes how the temperatures change over time. Solving this equation will give us the temperature difference at any time t.

Final Expression

After solving the differential equation, we find that the temperature difference between the two bodies at time t can be expressed as:

ΔT(t) = (T2 - T1) \cdot e^{-\frac{K \cdot A \cdot t}{m \cdot c \cdot L}}

Where m is the effective mass and c is the effective specific heat of the system. This shows that the temperature difference decreases exponentially over time as heat is transferred from the hotter body to the cooler one.

Conclusion

By understanding the principles of heat conduction and applying the relevant equations, you can determine how the temperature difference changes over time in this insulated system. If you have any further questions or need clarification on any step, feel free to ask!