Understanding Mixture Concentration

When two streams of liquid with different flow rates and concentrations mix, the resulting concentration of the mixture can be determined using the principle of mass balance. This principle states that the total amount of a substance entering a system must equal the total amount leaving the system, assuming no creation or destruction of the substance within the system (like in simple mixing).

Applying Mass Balance to Sewage and River Mixing

Let's define the terms given in the question:

• Cs: Concentration of the substance in the sewage stream.
• Qs: Flow rate of the sewage stream.
• Cr: Concentration of the substance in the river stream.
• Qr: Flow rate of the river stream.
• C: Concentration of the substance in the resulting mixture.

The total amount of the substance entering the mixing point per unit time is the sum of the amount from the sewage and the amount from the river.

• Mass flow rate from sewage = Concentration $\times$ Flow Rate = $C_s \times Q_s$
• Mass flow rate from river = Concentration $\times$ Flow Rate = $C_r \times Q_r$

Total mass flow rate into the mixture = $C_s Q_s + C_r Q_r$

The total volume flow rate of the resulting mixture is the sum of the flow rates of the sewage and the river.

Total volume flow rate of the mixture = $Q_s + Q_r$

The concentration of the resulting mixture (C) is the total mass flow rate divided by the total volume flow rate.

Therefore, the expression for the concentration C of the resulting mixture is:

$\text{C} = \frac{\text{Total mass flow rate}}{\text{Total volume flow rate}}$

$\text{C} = \frac{C_s Q_s + C_r Q_r}{Q_s + Q_r}$

Selecting the Correct Expression

Let's compare our derived expression with the given options:

• Option 1: $\frac{C_s Q_s+C_r Q_r}{Q_s Q_r+C_r Q}$ - This expression does not match our derivation.
• Option 2: $\frac{C_s Q_s}{Q_s+Q_r}$ - This expression assumes the concentration in the river stream (Cr) is zero. It does not account for the substance already present in the river.
• Option 3: $\frac{C_s Q_s}{Q_s Q_r+C_r Q_r}$ - This expression does not match our derivation.
• Option 4: $\frac{C_s Q_s+C_r Q_r}{Q_s+Q_r}$ - This expression exactly matches the one derived using mass balance.

Thus, the correct expression for the concentration of the resulting mixture when sewage mixes with a river stream is given by summing the mass flow rates of the substance from both sources and dividing by the total volume flow rate of the combined streams.