Two streams flow into a 500.0-gallon tank. The first stream is 10.0 wt % ethanol and 90.0 wt% hexane (mixture density is rho1 0.

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Two streams flow into a 500.0-gallon tank. The first stream is 10.0 wt % ethanol and 90.0 wt% hexane (mixture density is rho1 0.

680 g/cm3) and the second is 90.0 wt% ethanol, 10.0 wt% hexane (mixture density rho2 0.780 g/cm3). After the tank has been filled, which takes 22.0 minutes, an analysis of its contents determines that it is 40.0 wt% ethanol and 60.0 wt% hexane. Estimate the density of the final mixture and the mass and volumetric flow rates of the two feed streams.

Engineering

1 answer:

grin007 [323] · Answer 1 · 2026-02-23T16:58:48+03:00

Answer:

The density of the resulting mixture is 0.7175 $\frac{g}{cm^3}$ . The mass/volumetric flow rate for stream 1 is 39793.6 $\frac{g}{min}$ /58.52 litre/min. For stream 2, it is 21925.8 $\frac{g}{min}$ /28.11 litre/min.

Explanation:

To begin with, density needs to be determined. You can create a system of equations for this purpose. For stream 1, a combination of 10% by weight ethanol and 90% by weight hexane yields a density of 0.68 $\frac{g}{cm^3}$ . Let "X" represent the density of ethanol and "Y" for hexane:

$0.1*X+0.9*Y= 0.68(Eq. a)$

The same framework is applicable for stream 2, thus:

$0.9*X+0.1*Y= 0.78(Eq. b)$

By solving, we find: X=0.7925 $g/cm^3$ ; Y=0.6675 $g/cm^3$ .

Next, since the final formulation consists of 40% by weight ethanol and 60% by weight hexane, the density of the final blend is:

$\rho_{f}=0.4*0.7925+0.6*0.6675\Rightarrow \rho_{f}=0.7175 \frac{g}{cm^3}$

You are also aware of the tank's volume (500 gallons=1892500 cm^3), its density (0.7175 g/cm^3), and the mixture composition (40% by weight ethanol; 60% hexane). This allows for the calculation of the mass of each component in the tank.

The mass of ethanol is calculated as 0.4*1892500*0.7175 g=543147.5 g (grams).

The mass of hexane is calculated as 0.6*1892500*0.7175 g=814721.25 g (grams).

These masses derive from the two streams, so it is crucial to ascertain the net volume from each stream. The following equations are necessary for this.

$0.1*X*V_1+0.9*X*V_2=543147.5 g$ (Mass of ethanol)

$0.9*Y*V_1+0.1*Y*V_2=814721.25 g$ (Mass of hexane)

On solving, the volume from stream 1 ( $V_1$ ) is 1287528.15 cm^3 and from stream 2 ( $V_2$ ) is 618452.00 cm^3.

The time required to fill the tank is 22 minutes, giving the volumetric flow rates as:

For stream 1: $Q_1=V_1/(22min*1000cm^3/litre)=58.52 litre/min$

For stream 2: $Q_2=V_2/(22min*1000cm^3/litre)=28.11 litre/min$

Finally, by multiplying the volumetric flow rate with the density of each stream, you can find the mass flow rates:

$\dot{m_1}=0.68\frac{g}{cm^3} *58.52 litre/min*1000cm^3/litre=39793.6 \frac{g}{min} \\\dot{m_2}=0.78\frac{g}{cm^3} *28.11 litre/min*1000cm^3/litre=21925.8 \frac{g}{min}$