# Step-by-Step Solution Let's answer both parts: --- ## 1. **How long does sterilization take when the water temperature is 90°C?** ### **Given Data** - **Initial pecan temperature(T_i))**: 10 - **Water temperature(T_\infty\))**: 90°C - **Sterilization temperature (\(T_s\))**: 67.5°C (surface must reach this) - **Diameter of pecan (\(D\))**: 6 mm = .006 m - **Radius (\(r_\))**: \(.003\) m - **Heat transfer coefficient (\(h\))**: 250 W/m²·K - **Thermal conductivity (\(k\))**: .25 W/m·K - **Density (\(\rho\))**: 500 kg/m³ - **Specific heat (\(c_p\))**: 200 J/kg·K --- ### **Step 1: Lumped Capacitance Check** Calculate Biot number (\(Bi\)) to check if lumped capacitance method is valid: \[ Bi = \frac{h r_}{k} \] Plug in values: \[ Bi = \frac{250 \times .003}{.25} = 3 \] - Since \(Bi > .1\), **lumped capacitance is NOT valid**. Surface and center temperatures will differ. --- ### **Step 2: Use Heisler Solution for Sphere (Surface Temperature)** Since \(Bi > .1\), use Heisler chart/solution for surface temperature of a sphere in convection. #### **Dimensionless Temperatures** \[ \theta = \frac{T(r, t) - T_\infty}{T_i - T_\infty} \] We need when **surface** (\(r = r_\)) reaches \(T_s\): \[ \theta_s = \frac{T_s - T_\infty}{T_i - T_\infty} = \frac{67.5 - 90}{10 - 90} = \frac{-22.5}{-80} = .28125 \] --- #### **Heisler Solution for Sphere (Surface)** \[ \theta_s = \frac{T(r_, t) - T_\infty}{T_i - T_\infty} = \frac{1}{Bi + 1} + \frac{2}{Bi + 1} \sum_{n=1}^{\infty} \left[ \frac{\sin \lambda_n}{\lambda_n + \sin \lambda_n \cos \lambda_n} \exp(-\lambda_n^2 Fo) \right] \] But for surface, the solution simplifies to: \[ \theta_s = \frac{1}{Bi + 1} + \frac{2 Bi}{(Bi+1)^2} \exp\left(-\frac{(Bi+1)^2 Fo}{Bi}\right) \] But the **first term approximation** is often used if Bi > .1, and the chart/Heisler solution for sphere is used. --- #### **Calculate Fourier Number (\(Fo\))** \[ Fo = \frac{\alpha t}{r_^2} \] Where: \[ \alpha = \frac{k}{\rho c_p} = \frac{.25}{500 \times 200} = \frac{.25}{1,000,000} = 2.5 \times 10^{-7}\ \text{m}^2/\text{s} \] --- #### **Find \(Fo\) at required \(\theta_s\) and \(Bi\)** - For \(Bi = 3\), \(\theta_s = .281\). - Use **Heisler chart** or table for sphere, surface, with these values. From standard Heisler charts (see Incropera & DeWitt, or "Fundamentals of Heat and Mass Transfer"), for \(Bi = 3\): - For \(\theta_s = .28\), \(Fo \approx .30\) --- #### **Calculate Time** \[ Fo = \frac{\alpha t}{r_^2} \implies t = \frac{Fo \cdot r_^2}{\alpha} \] Plug in values: - \(Fo = .30\) - \(r_ = .003\) m - \(\alpha = 2.5 \times 10^{-7}\) \[ t = \frac{.30 \times (.003)^2}{2.5 \times 10^{-7}} = \frac{.30 \times 9 \times 10^{-6}}{2.5 \times 10^{-7}} = \frac{2.7 \times 10^{-6}}{2.5 \times 10^{-7}} = 10.8\ \text{seconds} \] --- ### **Final Answer — Part 1** **The surface of the pecan will reach the sterilization temperature of 67.5°C after approximately 11 seconds.** --- ## 2. **Why is the surface temperature different from the water temperature? (Physical Reasoning)** - **Heat Transfer Resistance:** There is a resistance to heat flow from the water to the pecan surface (convection) and from the surface into the pecan interior (conduction). - **Finite Heat Transfer Rate:** The heat transfer coefficient (\(h\)) limits how quickly energy can flow from the hot water to the pecan's surface. - **Thermal Inertia:** The pecan starts at a lower temperature, so heat must first flow into the surface, then conduct inward. - **Transient Heating:** At any instant before thermal equilibrium, the surface is still absorbing heat, so its temperature is between the water temperature and pecan's interior temperature. - **Biot Number Effect:** Since \(Bi > .1\), there is a significant temperature gradient between the surface and center, so the surface does not instantly jump to the water temperature. **Summary Table:** | Location | Temperature during heating | |------------------|--------------------------| | Water (\(T_\infty\)) | 90°C | | Pecan Surface (\(T_s\)) | Rises from 10°C toward 90°C, reaches 67.5°C after ~11 s | | Pecan Center | Rises more slowly | --- ## **Summary** 1. **Sterilization time:** About **11 seconds** for the pecan surface to reach 67.5°C. 2. **Surface ≠ Water Temperature:** Because of finite heat transfer (convection and conduction), the surface heats up gradually and always lags behind the water temperature during the heating process. --- **Reference:** "Fundamentals of Heat and Mass Transfer" by Incropera & DeWitt (Heisler charts for sphere, surface temperature, Biot number, Fourier number)