## Step-by-Step Solution for Question 5 ### 5. Describe step by step how to handle: #### a) Negligible Internal Resistance #### b) Heisler Charts --- ### a) **Negligible Internal Resistance (Lumped Capacitance Method)** **Explanation:** When the internal resistance to heat conduction within a solid is negligible compared to the external resistance to heat transfer (convection), the temperature inside the solid is uniform at any instant during the process. This condition is valid when the Biot number (Bi) is less than .1. #### **Steps:** 1. **Check Biot Number (`Bi`):** \[ Bi = \frac{h L_c}{k} \] - \(h\): Heat transfer coefficient (W/m²·K) - \(L_c\): Characteristic length (\(V/A\)), where \(V\) is volume and \(A\) is surface area. - \(k\): Thermal conductivity of the solid (W/m·K) - If \(Bi < .1\), proceed with this method. 2. **Apply Lumped Capacitance Equation:** \[ \frac{T(t) - T_{\infty}}{T_i - T_{\infty}} = \exp\left(-\frac{hA}{\rho V c_p} t\right) \] - \(T(t)\): Temperature at time \(t\) - \(T_{\infty}\): Ambient temperature - \(T_i\): Initial temperature - \(\rho\): Density - \(c_p\): Specific heat - \(V\): Volume - \(A\): Surface area 3. **Solve for Time or Temperature as Needed.** --- ### b) **Heisler Charts (Transient Heat Conduction for Solids)** **Explanation:** Heisler charts are used for transient heat conduction problems in solids (plates, cylinders, spheres) when the Biot number is greater than .1 and the body is large enough for the one-dimensional assumption. #### **Steps:** 1. **Check Biot Number (`Bi`):** - \(Bi > .1\), then internal resistance is significant and Heisler charts are applicable. 2. **Calculate Dimensionless Parameters:** - **Fourier Number (\(Fo\)):** \[ Fo = \frac{\alpha t}{L^2} \] - \(\alpha\): Thermal diffusivity (\(k/\rho c_p\)) - \(t\): Time - \(L\): Half thickness (for slab), radius (for cylinder/sphere) - **Dimensionless Temperature (\(\theta\)):** \[ \theta = \frac{T(x,t) - T_{\infty}}{T_i - T_{\infty}} \] 3. **Select the Shape:** - Choose the appropriate Heisler chart (slab, cylinder, sphere). 4. **Enter the Chart:** - Use your calculated \(Bi\) and \(Fo\) to find or interpolate the value of \(\theta\). 5. **Calculate the Actual Temperature:** - Rearrange the dimensionless equation to solve for \(T(x, t)\). --- ## **Summary Table** | Case | Biot Number Criteria | Main Steps | |-------------------------|---------------------|--------------------------------------------------------| | Lumped Capacitance | \(Bi < .1\) | Use lumped capacitance formula. | | Heisler Charts | \(Bi > .1\) | Use Heisler charts with calculated \(Fo\) and \(Bi\). | --- **In summary:** - For negligible internal resistance, use the lumped capacitance method. - For significant internal resistance, use Heisler charts, following the dimensionless parameters and the right chart for the object shape. Let me know if you need an example for either method!