A centrifugal cream separator receives $1000\text{ kg/h}$ of whole milk containing $4.0\%$ fat. It separates it into cream containing $40.0\%$ fat and skim milk containing $0.05\%$ fat. The mass flow rate of cream produced in $\text{kg/h}$ is ________ (round off to 1 decimal place).
Answer Key:
98.9
Step-by-Step Solution & Explanation
1. Total mass balance: $M = C + S = 1000\text{ kg/h} \implies S = 1000 - C$.
2. Fat mass balance:
$M \times x_M = C \times x_C + S \times x_S$
$1000 \times 0.04 = C \times 0.40 + (1000 - C) \times 0.0005$
$40.0 = 0.40 C + 0.50 - 0.0005 C$
$40.0 - 0.50 = (0.40 - 0.0005) C$
$39.50 = 0.3995 C \implies C = \frac{39.50}{0.3995} \approx 98.874\text{ kg/h} \approx 98.9\text{ kg/h}$