Let's solve the problem step-by-step as requested. ## Step 1: **Derive the appropriate conduction equation** **Explanation:** We model the probe as a 1D fin (cylindrical rod) exposed to two different environments: - Length \( L_i \): Exposed to gas at \( T_{gas} \) with convection coefficient \( h_{gas} \) - Length \( L_o \): Exposed to air at \( T_{air} \) with convection coefficient \( h_{air} \) Let \( x = \) at the base (air side), \( x = L_o \) at the duct wall, and \( x = L \) at the tip (gas side). The governing 1D steady-state conduction equation with convection along the side is: \[ \frac{d^2 T}{dx^2} - m^2 (T - T_{\infty}) = \] where \( m^2 = \frac{h P}{k A} \), \( h \) is the local convective coefficient, \( P \) is the perimeter, \( k \) is the thermal conductivity, and \( A \) is the cross-sectional area. - For \( \leq x < L_o \): \( h = h_{air},\; T_{\infty} = T_{air} \) - For \( L_o \leq x \leq L \): \( h = h_{gas},\; T_{\infty} = T_{gas} \) **Boundary Conditions:** - At \( x = \): Insulated or heat flux from the leads (often \( \frac{dT}{dx} = \)) - At \( x = L \): Sensor tip measures \( T_{tip} \) (to be found) --- ## Step 2: **Numerical Solution Setup for Temperature Distribution** **Explanation:** To solve numerically: - Divide the probe into 12 control volumes. - For each node \( i \), apply the energy balance: At interior nodes: \[ \frac{kA}{\Delta x}(T_{i+1} - 2T_i + T_{i-1}) - hP\Delta x (T_i - T_{\infty}) = \] where: - \( \Delta x = \frac{L}{12} \) - For nodes in air: \( h = h_{air},\; T_{\infty} = T_{air} \) - For nodes in gas: \( h = h_{gas},\; T_{\infty} = T_{gas} \) **Numerical Steps:** 1. Discretize the rod into 12 segments. 2. Set up the above equation for each node, adjusting \( h \) and \( T_{\infty} \) depending on the region. 3. Apply boundary conditions. 4. Solve the system of equations for \( T_i \) at each node. --- ## Final Answer - **Step 1:** The governing equation is a 1D steady-state fin equation with piecewise convection boundary (air and gas sides). - **Step 2:** Set up the finite-difference equations for 12 segments, apply boundary conditions, and solve for temperature distribution. **With the temperature profile, you can find \( T_{tip} \) at the last node, and calculate:** \[ T_{err} = T_{tip} - T_{gas} \] for different values of \( L_i/L \). --- **Summary Table for Step-by-Step Solution:** | Step | Action | Explanation | |---------|-----------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------| | Step 1 | Write 1D fin conduction equation with convection | Models heat loss to air and gas, gives basis for temperature profile | | Step 2 | Discretize, apply numerical method (finite difference, 12 volumes) | Solves for temperature at each node, giving temperature at the tip for error calculation | --- **Next Steps:** - Implement the finite difference scheme in code or by hand. - Vary \( L_i/L \) as .25, .5, .75, 1., and plot \( T_{err} \) vs \( L_i/L \). - Check energy balance by ensuring heat lost to air + heat lost to gas = heat conducted through the probe. Let me know if you need the code or example calculations for this setup!