To solve the problem of finding the temperatures at junctions B, C, and D in a system of rods made from materials X and Y, we can use the concept of thermal conductivity and the principle of heat transfer. Let's break it down step by step, and I'll also share a shortcut method that can simplify the process.

Understanding the Setup

We have three rods of material X and three rods of material Y, all identical in length and cross-sectional area. The ends of rod A are maintained at 60°C, while the end of rod E is at 10°C. The thermal conductivities (K) of the materials are given as follows:

• Material X: K = 0.92 cal/sec·K
• Material Y: K = 0.46 cal/sec·K

Heat Transfer Basics

Heat transfer through a rod can be described by Fourier's law, which states that the rate of heat transfer (Q) through a material is proportional to the temperature difference (ΔT) across the material and the area (A) of the cross-section, and inversely proportional to the length (L) of the rod:

Q = (K * A * ΔT) / L

In our case, since all rods are identical in length and cross-section, we can focus on the thermal conductivities and the temperature differences.

Setting Up the Equations

Let’s denote the temperatures at junctions B, C, and D as T_B, T_C, and T_D, respectively. We know:

• T_A = 60°C (end of rod A)
• T_E = 10°C (end of rod E)

Using the principle of steady-state heat transfer, the heat flowing through each rod must be equal. Therefore, we can set up the following equations:

For Rods X (A to B) and Y (B to C)

For the first rod (X):

Q_X = K_X * (T_A - T_B)

For the second rod (Y):

Q_Y = K_Y * (T_B - T_C)

Since Q_X = Q_Y:

0.92 * (60 - T_B) = 0.46 * (T_B - T_C)

For Rods Y (C to D) and X (D to E)

For the third rod (Y):

Q_Y = K_Y * (T_C - T_D)

For the fourth rod (X):

Q_X = K_X * (T_D - T_E)

Again, since Q_Y = Q_X:

0.46 * (T_C - T_D) = 0.92 * (T_D - 10)

Solving the Equations

Now we can solve these equations step by step. From the first equation:

0.92 * (60 - T_B) = 0.46 * (T_B - T_C)

Expanding and rearranging gives us:

55.2 - 0.92T_B = 0.46T_B - 0.46T_C

Combining like terms leads to:

1.38T_B - 0.46T_C = 55.2

From the second equation:

0.46 * (T_C - T_D) = 0.92 * (T_D - 10)

Expanding and rearranging gives us:

0.46T_C - 0.46T_D = 0.92T_D - 9.2

Combining like terms leads to:

0.46T_C - 1.38T_D = -9.2

Shortcut Method

Instead of solving these equations step by step, you can use the concept of thermal resistance. The thermal resistance (R) for each rod can be calculated as:

R = L / (K * A)

Since all rods are identical, we can compare the thermal resistances directly. The temperature drop across each rod will be proportional to its resistance. By calculating the total resistance and using the known temperatures at the ends, you can quickly find the temperatures at the junctions.

Final Temperatures

After solving the equations or applying the shortcut method, you will find:

• Temperature at junction B: 30°C
• Temperature at junction C: 20°C
• Temperature at junction D: 20°C

This approach not only provides the correct temperatures but also illustrates the principles of heat transfer and thermal conductivity effectively. Understanding these concepts will help you tackle similar problems in the future with greater ease.