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Chapter 5: Problem 4

The volume of a steady-state crystallizer is 85,000 L, and the solids fraction in the unit and the exiting stream is 0.35 ; that is, there are \(0.35 \mathrm{kg}\) of crystals per \(\mathrm{kg}\) of slurry (the crystal-solution mixture). The density of the solution is \(1.1 \mathrm{g} / \mathrm{mL}\) and that of the crystals is \(2.3 \mathrm{g} / \mathrm{mL}\). The production rate of crystals from the crystallizer is \(19.5 \mathrm{kg}\) crystals/min. Estimate the volumetric flow rate of slurry from the crystallizer and the drawdown time of the crystallizer (the time it would take to empty the crystallizer if the feed were discontinued).

Short Answer

Expert verified
The volumetric flow rate of slurry from the crystallizer is approximately 55.71 L/min and the drawdown time of the crystallizer is approximately 758.99 minutes.

Step by step solution

01

Calculate Total Mass of Slurry in Crystallizer

Given the volume of the crystallizer and the solids fraction, the total mass of the slurry in the crystallizer can be found using the formula: masses of crystals = volume * solids_fraction / (1 - solids_fraction). Substituting the given values, we get: \(masses\_of\_crystal \ = \ 85000 * 0.35 / (1 - 0.35) = 46538.46 KG\)
02

Calculate Total Volume of Slurry in Crystallizer

The volume of the slurry in the crystallizer can be found using the formula: total volume (= total mass / density). Given the density of the solution and the total mass we calculated, we substitute these values into the formula to obtain the total volume. \(total\_volume = masses\_of\_crystal / density\). So, \( total\_volume = 46538.46 / 1.1 = 42307.69 L\)
03

Calculate Volumetric Flow Rate of Slurry From Crystallizer

Knowing the production rate and the solids fraction, the volumetric flow rate can be calculated by the formula: \(volumetric\_flow\_rate = production\_rate / solids\_fraction\). Here, \(volumetric\_flow\_rate = 19.5 / 0.35 = 55.71 L/min\)
04

Calculate Drawdown Time of the Crystallizer

The drawdown time is the time it would take to empty the crystallizer if the feed were discontinued. It can be estimated using the formula: drawdown time \(= total\_volume / volumetric\_flow\_rate\). So, the drawdown time = 42307.69 / 55.71 = 758.99 min

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chemical Process Calculations
Chemical process calculations are crucial for designing and operating engineering systems, particularly within chemical production environments. In a chemical process, understanding the relation between quantities like mass, volume, concentration, and flow rates enables engineers to determine the efficiency and productivity of a process.

For instance, with a steady-state crystallizer, one would often need to calculate parameters such as the total mass of the slurry (a mix of solids and fluid), the volumetric flow rate (how much volume of slurry is produced over time), and the distribution of solids within the slurry (solids fraction). These calculations involve balancing mass input with output, applying conservation principles, and employing material density information to translate between mass and volume measures.
Steady-State Crystallizer
A steady-state crystallizer is a specialized device used in the chemical industry to form solid crystals from a solution or melt which is in equilibrium, meaning that the rate at which crystals form is equal to the rate at which they exit the system. This balance is essential for continuous operation without substantial variation in the crystal size and population within the crystallizer.

The concept of 'steady state' implies that the system's properties do not change with time, which simplifies calculations and process control. Engineers must take into account the properties of both the crystalline product and the remaining solution, the kinetics of crystal formation and growth, as well as the thermodynamic conditions that favor crystallization.
Volumetric Flow Rate
The volumetric flow rate is a fundamental parameter in chemical engineering, representing the volume of fluid that passes through a given surface per time unit. It is commonly measured in units such as liters per minute (L/min) or cubic meters per second (m鲁/s).

Knowing the volumetric flow rate of slurry exiting the crystallizer is essential to process control and optimization because it helps determine how much product is being produced over time. When calculating the volumetric flow rate in the context of a crystallization process, engineers use the solids fraction and production rate of crystals to establish an accurate flow rate for the slurry.
Solids Fraction
Solids fraction is a term used to describe the proportion of solid material in a multiphase mixture, relative to the total mass. In the context of crystallization, it typically refers to the ratio of the mass of crystals to the mass of the entire slurry. This ratio is dimensionless and can range from 0 (no solids) to 1 (fully solid).

Understanding the solids fraction is critical for designing separation processes, where one may need to separate the crystalline material from the liquid. It impacts various process parameters including viscosity, flow behavior, and sedimentation rates. As we have seen in our exercise, the solids fraction directly affects the mass balance and contributes to determining the volumetric flow rate and the drawdown time of the crystallizer.

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Most popular questions from this chapter

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