## Step-by-Step Solution Let's break down the problem and perform the required scaling analysis. --- ### **1. Write the Energy Equation** For steady-state, axisymmetric, laminar flow of an incompressible, constant-property fluid in a tube, the energy equation (in cylindrical coordinates) is: \[ \rho c_p \left( u \frac{\partial T}{\partial z} + v \frac{\partial T}{\partial r} \right) = k \left( \frac{\partial^2 T}{\partial z^2} + \frac{1}{r} \frac{\partial }{\partial r} \left( r \frac{\partial T}{\partial r} \right) \right) \] where: - \( u \) = axial velocity, - \( v \) = radial velocity (usually much smaller or zero for fully developed flow), - \( T \) = temperature, - \( z \) = axial coordinate, - \( r \) = radial coordinate, - \( \rho \) = density, - \( c_p \) = specific heat, - \( k \) = thermal conductivity. For fully developed flow, \( v \approx \), so the equation simplifies to: \[ \rho c_p u \frac{\partial T}{\partial z} = k \left( \frac{\partial^2 T}{\partial z^2} + \frac{1}{r} \frac{\partial }{\partial r} \left( r \frac{\partial T}{\partial r} \right) \right) \] --- ### **2. Identify Terms to Compare** We want to compare: - **Axial convection**: \( \rho c_p u \frac{\partial T}{\partial z} \) - **Axial conduction**: \( k \frac{\partial^2 T}{\partial z^2} \) - **Radial conduction**: \( k \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) \) The question asks when we can **neglect axial conduction** compared to axial convection. --- ### **3. Scaling Analysis** Let: - Tube diameter = \( D \) - Tube length = \( L \) - Typical velocity = \( U \) - Typical temperature difference = \( \Delta T \) **Order-of-magnitude estimates:** - \( \frac{\partial T}{\partial z} \sim \frac{\Delta T}{L} \) - \( \frac{\partial^2 T}{\partial z^2} \sim \frac{\Delta T}{L^2} \) - \( \frac{\partial^2 T}{\partial r^2} \sim \frac{\Delta T}{D^2} \) **Compare the two terms:** - Axial convection: \( \rho c_p U \frac{\Delta T}{L} \) - Axial conduction: \( k \frac{\Delta T}{L^2} \) **Ratio:** \[ \text{Ratio} = \frac{\text{Axial conduction}}{\text{Axial convection}} = \frac{k \frac{\Delta T}{L^2}}{\rho c_p U \frac{\Delta T}{L}} = \frac{k}{\rho c_p U L} = \frac{1}{Pe} \] where the **Péclet number** is: \[ Pe = \frac{\rho c_p U L}{k} = Re \cdot Pr \] - **Reynolds number:** \( Re = \frac{\rho U D}{\mu} \) - **Prandtl number:** \( Pr = \frac{\mu c_p}{k} \) If \( L \sim D \) (typical for tubes), then \( Pe = Re \cdot Pr \). --- ### **4. Final Condition** - **Axial conduction can be neglected when** \( Pe \gg 1 \), i.e., when **convection dominates over axial conduction**. ### **5. Answer** \[ \boxed{ \text{Axial conduction can be neglected in the energy equation when } Pe = Re \cdot Pr \gg 1. } \] --- #### **Summary Table** | Term | Magnitude | Scaling Ratio | |---------------------|----------------------------------|------------------------| | Axial Convection | \( \rho c_p U \frac{\Delta T}{L} \) | \( \frac{1}{Pe} \) | | Axial Conduction | \( k \frac{\Delta T}{L^2} \) | | > **Neglect axial conduction if** \( Pe = Re \cdot Pr \gg 1 \). --- Let me know if you want to see more about the physical meaning, or how this relates to real engineering applications!