Absolutely! Let's break down **System 5** step by step and clarify what is being asked. --- # **Problem Breakdown** ## **Given Data** - **Cucumber preserved in brine:** Brine concentration = **24% NaCl** - **Cucumber initial composition:** - NaCl: **5% (wet basis)** - Moisture: **87% (wet basis)** - **Biot number (Bi_m):** High, **> 100** (means internal mass transfer controls) - **NaCl mass diffusivity in cucumber:** \( D_{NaCl,\text{cucumber}} = 1.7 \times 10^{-9}\ \text{m}^2/\text{s} \) - **Cucumber size:** - Average diameter = **1.6 cm** - Length = **8 cm** - **Equilibrium distribution coefficient:** - \( K = 1.23\ \text{kg brine}/\text{kg cucumber} \) - **Brine tank:** 20 L brine + 500 cucumbers - **NaCl diffusivity in brine:** \( D_{AB} = 2.4 \times 10^{-9}\ \text{m}^2/\text{s} \) ## **Tasks** 1. **Calculate salt (NaCl) concentration at midpoint of cucumber** after: - 1 h, 5 h, 12 h, 48 h, 72 h 2. **Plot NaCl concentration vs. time** and discuss relationship. 3. **How much salt will the brine lose** initially, for 20 L brine and 500 cucumbers? --- # **Step 1: Physical Model** Since Biot number is very high (\( Bi_m > 100 \)), **internal diffusion** is the rate-controlling step. Surface concentration of NaCl in cucumber quickly reaches equilibrium with brine, so we analyze **diffusion inside a cylinder**. --- ## **Step 2: Geometry** - **Cucumber as a cylinder:** - Radius \( r = \frac{1.6\,\text{cm}}{2} = .8\,\text{cm} = .008\,\text{m} \) - Length \( L = 8\,\text{cm} = .08\,\text{m} \) - However, for midpoint concentration, assume 1D radial diffusion if length ≫ radius. --- ## **Step 3: Diffusion Equation (Fick's Law)** For a sphere or cylinder with constant surface concentration, the unsteady-state solution gives the concentration at the center as a function of time. For a sphere (similar for cylinder if length ≫ radius): \[ \frac{C_{mid}(t) - C_{surf}}{C_{init} - C_{surf}} = \frac{6}{\pi^2} \sum_{n=1}^{\infty} \frac{1}{n^2} \exp\left(-D_{NaCl} n^2 \pi^2 t / r^2\right) \] - \( C_{mid}(t) \): NaCl concentration at midpoint at time \( t \) - \( C_{surf} \): Surface concentration (equilibrium with brine) - \( C_{init} \): Initial concentration in cucumber - \( D_{NaCl} \): Diffusivity of NaCl in cucumber - \( r \): Radius of cucumber For practical calculations, first term (\( n=1 \)) is often enough for early times. --- ## **Step 4: Surface Concentration (Equilibrium Condition)** **Surface concentration** of NaCl in cucumber is set by equilibrium with brine: \[ \frac{C_{brine,surf}}{C_{cucumber,surf}} = K \] Given \( K = 1.23 \). If brine is 24% NaCl, at equilibrium: \[ C_{cucumber,surf} = \frac{C_{brine,surf}}{K} \] You can convert % to kg/kg as needed. --- ## **Step 5: Plug in Values** - \( D_{NaCl} = 1.7 \times 10^{-9}\ \text{m}^2/\text{s} \) - \( r = .008\,\text{m} \) Calculate at \( t = 1, 5, 12, 48, 72 \) h \( \rightarrow \) convert hours to seconds. --- ## **Step 6: Brine Salt Loss Calculation** - Find **total NaCl transferred**: Multiply average concentration change in cucumber by total cucumber mass. - 20 L brine (\(\approx 20\,\text{kg}\)), 500 cucumbers. - **Mass balance**: Salt lost by brine = Salt gained by cucumbers. --- # **Summary of Steps** 1. **Find equilibrium surface concentration** using partition coefficient. 2. **Use unsteady-state diffusion equation** to find midpoint concentration at each time. 3. **Plot concentration vs. time**. 4. **Calculate total salt transferred** using mass balance. --- # **Next Steps** If you want, I can show you the calculations for a specific time point or help you set up the equations in Excel or Python for plotting. Let me know how you'd like to proceed!