Question

Scale-up with Constant Power/Volume or Maximum Shear Rate A protein precipitation reactor with a mechanical agitator is being scaled up from volume V1 to volume V2 by two different methods: keeping power per unit volume (P/V) constant, or keeping the maximum shear rate (?max) constant. (a) For the scaleup keeping P/V constant, determine the ratio of ?max at volume V2 to ?max at volume V1 as a function of V1 and V2: ? ? max max , ( ) ( ) = ( ) 2 1 1 2 f V V (b) If constant ?max is being used for scaleup, determine the ratio of P/V at volume V2 to P/V at volume V1 as a function of V1 and V2: P V P V g V V ( ) ( ) = (

Short answer

The ratios are expressed through two functions of V1 and V2. For the first part of the question where the power to volume ratio is constant, the function is \( f(V1, V2) = V2/V1 \). For the second part, where the maximum shear rate is constant, the function is \( g(V1, V2) = V1/V2 \).

Step-by-step solution

Analyzing the first situation

In this step, the goal is to determine the ratio of the maximum shear rate (\( \max \)) at volume V2 to the maximum shear rate (\( \max \)) at volume V1, keeping power per unit volume (P/V) constant. With the P/V constant, we apply the principle of proportionality to determine this ratio. As the power per unit volume is constant, the shear rates are directly proportional to the reactor volumes, so \( \max 2 / \max 1 = V2 / V1 \).

Analyzing the second situation

In this second step, the aim is to determine the ratio of P/V at volume V2 to P/V at volume V1, keeping the maximum shear rate constant. Here, volume and power per volume ratio are inversely proportional, when \( \max \) is constant. Therefore, the equation would be \( (P//V)_2 / (P//V)_1 = V1 / V2 \).

Expressing the result as functions of V1 and V2

Following from step 1 and 2, the mathematical expression for f(V1, V2) is \( f(V1, V2) = V2/V1 \) and for g(V1, V2), it is \( g(V1, V2) = V1/V2 \).

Key concepts

Protein Precipitation

Protein precipitation is a common technique used in biochemistry and biochemical engineering to separate proteins from a solution. This process works by reducing the protein's solubility, thereby causing them to form a solid. Here’s how it generally happens:

• Altering the ionic strength, pH, or temperature of the solution can promote precipitation.
• Adding salts, like ammonium sulfate, can drive the proteins out of the solution.
• Organic solvents, such as ethanol or acetone, may also be used to precipitate proteins by disrupting water-protein interactions.

This technique is essential for purifying proteins and preparing them for analysis or further use. In biochemical engineering, controlling the conditions for protein precipitation is crucial, especially when scaling up processes from lab to industrial production, as the efficiency and yield can be markedly impacted.

Mechanical Agitation

Mechanical agitation involves using mechanical devices to stir or mix components within a reactor or vessel. This is fundamental in ensuring uniform distribution and effective mixing of reactants, products, or even heat. Common devices used include impellers and stirrers, which can vary widely in design to suit different applications.

Mechanical agitation is important in processes like protein precipitation to ensure even exposure of all particles to the conditions that promote precipitation. Effective agitation can prevent the excessive growth of precipitate particles, which can otherwise lead to issues like clogging or uneven mixing. Adjusting the speed and design of the agitators is crucial when scaling up operations, as different volumes can react differently based on the mixing dynamics.

Shear Rate

The shear rate, in the context of mechanical agitation in biochemical processes, refers to the rate at which adjacent layers of fluid move with respect to each other. It's typically measured in reciprocal seconds ( ^-1).

• High shear rates can lead to better mixing and more efficient reactors.
• However, excessive shear can also damage sensitive components like delicate protein structures.
• In scaling up, maintaining the appropriate shear rate is crucial to preserving process integrity and product quality.

The shear rate can be controlled by adjusting the speed of agitation and selecting appropriate mixing devices, especially when moving from smaller lab-scale reactors to larger industrial ones.

Power Per Unit Volume

Power per unit volume (P/V) is a measure of the energy applied by mixing or agitation devices per unit volume of fluid within a reactor. It’s a critical parameter in scale-up operations as it impacts both mixing efficiency and process economics. Here are some key points:

• In laboratory settings, high P/V ensures thorough mixing and effective reactions.
• When scaling up, maintaining a constant P/V ensures that the mixing remains effective relative to the size of the vessel.
• Both too low and too high a P/V can lead to inefficiencies—too low can mean inadequate mixing, while too high can mean wasted energy or damaged materials.

By carefully managing P/V, engineers can ensure that scaling up processes maintains efficiency, preventing issues during larger-scale production.