In a blending tank, two streams containing salt and water are combined. The first stream has a salt concentration of x1 and a mass flow rate of w1, while the second stream has a sa... In a blending tank, two streams containing salt and water are combined. The first stream has a salt concentration of x1 and a mass flow rate of w1, while the second stream has a salt concentration x2 and a mass flow rate w2. The exiting stream has concentration x and a mass flow rate of w. Additionally, heat q is added to maintain uniform temperature.
Understand the Problem
The question is asking us to analyze a blending tank scenario with two streams of saltwater being mixed. It involves determining the concentration and flow rate of the resulting mixture based on the given concentrations and flow rates of the input streams.
Answer
The concentration of the mixture is given by $C_{mix} = \frac{C_1 \times Q_1 + C_2 \times Q_2}{Q_1 + Q_2}$.
Answer for screen readers
The concentration of the resulting mixture is given by the equation: $$ C_{mix} = \frac{C_1 \times Q_1 + C_2 \times Q_2}{Q_1 + Q_2} $$
Steps to Solve
- Identify the Input Streams We have two streams of saltwater with different concentrations and flow rates. Let's define them as:
- Stream 1 has a flow rate of $Q_1$ and a concentration of $C_1$ (in kg/m³).
- Stream 2 has a flow rate of $Q_2$ and a concentration of $C_2$ (in kg/m³).
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Calculate the Total Flow Rate To find the total flow rate of the mixture exiting the tank, we will sum the flow rates of both streams: $$ Q_{total} = Q_1 + Q_2 $$
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Calculate the Mass of Salt in Each Stream The mass of salt in each stream can be calculated as follows:
- For Stream 1, the mass of salt is: $$ M_1 = C_1 \times Q_1 $$
- For Stream 2, the mass of salt is: $$ M_2 = C_2 \times Q_2 $$
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Calculate the Total Mass of Salt Now, sum the mass of salt from both streams: $$ M_{total} = M_1 + M_2 $$
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Determine the Concentration of the Mixture Finally, to find the concentration of the resulting mixture, we divide the total mass of salt by the total flow rate: $$ C_{mix} = \frac{M_{total}}{Q_{total}} $$
The concentration of the resulting mixture is given by the equation: $$ C_{mix} = \frac{C_1 \times Q_1 + C_2 \times Q_2}{Q_1 + Q_2} $$
More Information
This formula calculates the concentration of the mixture based on the flow rates and concentrations of the individual streams. It's commonly used in chemical engineering to ensure proper mixing ratios.
Tips
- Not aligning units: Ensure that all flow rates and concentrations are in compatible units to avoid errors.
- Forgetting to sum flow rates: Remember to add the flow rates correctly in the total calculation.
