Sure, let's break this down step by step in a simple, human way. --- **Question Recap** We have a pipe made of **two different materials** (like two layers), each with its **own thickness and thermal conductivity**. Hot fluid is **inside** (temperature \(T_i\)), and cooler fluid is **outside** (temperature \(T_o\)). - **Inner radius:** \(R_i\) - **Outer radius:** \(R_o\) We need to **find the formula for heat flow** from inside to outside. --- ## **Step 1: Visualizing the Pipe** Imagine the pipe is like an onion with two layers: - **Layer 1:** From \(R_i\) to \(R_1\), conductivity = \(k_1\) - **Layer 2:** From \(R_1\) to \(R_o\), conductivity = \(k_2\) So, the pipe has: - **Inner surface:** at \(R_i\) (temperature = \(T_i\)) - **Boundary between materials:** at \(R_1\) - **Outer surface:** at \(R_o\) (temperature = \(T_o\)) --- ## **Step 2: Formula for Heat Conduction Through a Cylinder** For one layer, the heat flow rate (Q) is: \[ Q = \frac{2\pi L k (T_\text{inside} - T_\text{outside})}{\ln(R_\text{outside}/R_\text{inside})} \] Where: - \(L\) = length of the pipe - \(k\) = thermal conductivity - \(T\) = temperature - \(R\) = radius --- ## **Step 3: For Two Layers (Composite Wall)** Think of the two layers as **two resistances in series** (like two resistors in a row). **Thermal resistance for a cylinder:** \[ R_{th} = \frac{\ln(R_\text{outside}/R_\text{inside})}{2\pi L k} \] So: - **Layer 1:** \(R_{th,1} = \frac{\ln(R_1/R_i)}{2\pi L k_1}\) - **Layer 2:** \(R_{th,2} = \frac{\ln(R_o/R_1)}{2\pi L k_2}\) **Total resistance:** \[ R_{th, \text{total}} = R_{th,1} + R_{th,2} \] --- ## **Step 4: Final Heat Flow Expression** Now, total heat flow is: \[ Q = \frac{T_i - T_o}{R_{th, \text{total}}} \] Substitute the resistances: \[ Q = \frac{T_i - T_o}{\frac{\ln(R_1/R_i)}{2\pi L k_1} + \frac{\ln(R_o/R_1)}{2\pi L k_2}} \] Or, combine into one fraction: \[ Q = \frac{2\pi L (T_i - T_o)} {\frac{\ln(R_1/R_i)}{k_1} + \frac{\ln(R_o/R_1)}{k_2}} \] --- ## **In Short: Final Answer** \[ \boxed{ Q = \frac{2\pi L (T_i - T_o)} {\frac{\ln(R_1/R_i)}{k_1} + \frac{\ln(R_o/R_1)}{k_2}} } \] - \(T_i\): Inside fluid temperature - \(T_o\): Outside fluid temperature - \(R_i\): Inner radius - \(R_1\): Radius between two layers - \(R_o\): Outer radius - \(k_1\): Conductivity of inner material - \(k_2\): Conductivity of outer material - \(L\): Length of pipe --- ### **In Simple Words:** You add the "difficulty" for heat to pass through each layer (thermal resistance), then use the total to find how much heat moves from inside to outside. Let me know if you want this with a real example or a diagram!