Let's solve the evaporator mass balance problem step by step as described: --- # **Step 1: Derive Algebraic Expressions for Mass Flows** ## **1.1: Overall Mass Balance** ### **General Mass Balance (Steady-State, No Reaction)** Rate of Accumulation = Rate of Inflow − Rate of Outflow At steady state, accumulation = , so: \[ \text{Inflow} = \text{Outflow} \] ### **Apply to the Evaporator:** - Inlet: Feed \( F \) - Outlets: Product \( P \), Vapor \( V \) \[ F = P + V \] --- ## **1.2: Component (Solute) Mass Balance** Let the solute be salt (non-volatile, only leaves in liquid product). - Feed: \( F x_F \) (where \( x_F \) is the mass fraction of salt in the feed) - Product: \( P x_P \) (where \( x_P \) is the mass fraction of salt in the product) - Vapor: negligible salt (since it only contains solvent) ### **Solute Balance Equation:** \[ \text{Inlet Solute} = \text{Outlet Solute} \] \[ F x_F = P x_P \] --- # **Step 2: Symbolic Solutions** ## **2.1: Product Flow Rate, \( P \)** From the solute balance: \[ F x_F = P x_P \implies P = \frac{F x_F}{x_P} \] --- ## **2.2: Vapor Flow Rate, \( V \)** From overall balance: \[ F = P + V \implies V = F - P \] Substitute \( P \) from above: \[ V = F - \frac{F x_F}{x_P} \] \[ V = F \left(1 - \frac{x_F}{x_P}\right) \] --- # **Summary of Final Expressions** - **Product flow rate:** \[ \boxed{P = \frac{F x_F}{x_P}} \] - **Vapor flow rate:** \[ \boxed{V = F \left(1 - \frac{x_F}{x_P}\right)} \] --- # **Explanation** - **Overall mass balance** accounts for all mass entering and leaving the system. - **Component (solute) balance** applies because only the liquid stream carries the salt; the vapor is pure solvent. - The **product flow rate** is based on how much solute enters (in the feed) and leaves (in the product). - The **vapor flow rate** is simply what's left after subtracting the product from the feed. If you have any specific numbers, you can plug them into these formulas! Let me know if you want a numerical example or further clarification.

# Step-by-Step Solution for the Evaporator Mass Balance Problem ## 1. Overall Mass Balance ### General Mass Balance Equation The general conservation of mass equation is expressed as: \[ \text{Rate of Accumulation} = \text{Rate of Inflow} - \text{Rate of Outflow} + \text{Rate of Generation} - \text{Rate of Consumption} \] In a steady-state system with no chemical reactions, the rate of accumulation is zero. Thus, the equation simplifies to: \[ \text{Rate of Inflow} = \text{Rate of Outflow} \] ### Applying to the Evaporator For the evaporator: - **Inlet**: Feed \( F \) - **Outlets**: Product \( P \) and Vapor \( V \) From this, we can write: \[ F = P + V \] ## 2. Component Mass Balance (Solute) ### Conservation of Solute Let’s focus on the non-volatile solute (salt). The flow rates of solute in each stream can be expressed as: - **Feed stream**: \( F \cdot x_F \) (where \( x_F \) is the mass fraction of salt in the feed) - **Product stream**: \( P \cdot x_P \) (where \( x_P \) is the mass fraction of salt in the product) - **Vapor stream**: Contains negligible salt The conservation equation for the solute is: \[ F \cdot x_F = P \cdot x_P \] ## 3. Symbolic Solutions ### 3.1 Product Flow Rate \( P \) From the solute balance equation, we can express \( P \) as: \[ P = \frac{F \cdot x_F}{x_P} \] ### 3.2 Vapor Flow Rate \( V \) Using the overall mass balance equation, we can express \( V \): \[ V = F - P \] Substituting \( P \) into this equation: \[ V = F - \frac{F \cdot x_F}{x_P} \] This can be simplified to: \[ V = F \left(1 - \frac{x_F}{x_P}\right) \] ## 4. Final Expressions ### Product Flow Rate \( P \) The final expression for the product flow rate is: \[ \boxed{P = \frac{F \cdot x_F}{x_P}} \] ### Vapor Flow Rate \( V \) The final expression for the vapor flow rate is: \[ \boxed{V = F \left(1 - \frac{x_F}{x_P}\right)} \] --- ## Explanation - **Overall Mass Balance**: Ensures that all mass entering the system equals the mass leaving it, which is crucial for maintaining system integrity. - **Component Mass Balance**: Focuses specifically on the solute, recognizing that only the liquid product carries the salt away from the system. - **Derived Formulas**: The formulas for \( P \) and \( V \) help in calculating how much product and vapor can be produced from a given feed based on the concentration of solute. If you have specific values for \( F \), \( x_F \), or \( x_P \), you can substitute them into these equations for numerical results. Let me know if you need further assistance!