Design a disk centrifuge to separate yeast cells from beer with the following characteristics

Design a disk centrifuge to separate yeast cells from beer with the following characteristics: 1. Flow rate = 400 m³/day 2. Density of beer = 1020 kg/m³ 3. Density of yeast cells = 1075 kg/m³ 4. Average size of yeast cell = 4 µm. It is needed to specify the outside diameter of the disk, inside diameter of the disk, number of disk, rotational speed, Sigma (?) value and angle of the disk with the vertical axis (?). Assume necessary data to design the centrifuge and also provide the justification of the assumptions.

The Correct Answer and Explanation is:

Disk Centrifuge Design for Yeast Separation from Beer

Given Data:

• Flow rate (Q): 400 m³/day = 4.63 L/s
• Beer density (ρf): 1020 kg/m³
• Yeast density (ρp): 1075 kg/m³
• Yeast diameter (dp): 4 µm = 4 × 10⁻⁶ m
• Viscosity of beer (μ): assumed as 1.5 × 10⁻³ Pa·s
• Disk angle (θ): assume 45°
• Acceleration due to gravity (g): 9.81 m/s²

Assumptions and Justifications:

1. Disk Angle (θ) = 45°: Commonly used angle in disk-stack centrifuges for good compromise between sedimentation area and flow resistance.
2. Disk Radii:
   • Outer radius (R₁): 0.20 m
   • Inner radius (R₂): 0.05 m
     These are typical values allowing reasonable residence time and Sigma (Σ) values.
3. Number of Disks (n): 150 – balances sedimentation area and centrifuge size.
4. Rotational Speed (ω): Chosen to provide sufficient centrifugal force. Assume 6000 rpm (628 rad/s).

Calculations:

1. Terminal settling velocity (Stokes’ Law):vt=(ρp−ρf)gdp218μ=(1075−1020)(9.81)(4×10−6)218×1.5×10−3≈3.6×10−8 m/sv_t = \frac{(ρ_p – ρ_f) g d_p^2}{18μ} = \frac{(1075 – 1020)(9.81)(4 \times 10^{-6})^2}{18 \times 1.5 \times 10^{-3}} ≈ 3.6 \times 10^{-8} \, \text{m/s}vt​=18μ(ρp​−ρf​)gdp2​​=18×1.5×10−3(1075−1020)(9.81)(4×10−6)2​≈3.6×10−8m/s

2. Equivalent settling area (Σ):Σ=2πn(R12−R22)sin⁡θ=2π×150(0.202−0.052)sin⁡45°≈35.3 m2Σ = \frac{2π n (R₁^2 – R₂^2)}{\sinθ} = \frac{2π \times 150 (0.20^2 – 0.05^2)}{\sin45°} ≈ 35.3 \, \text{m}^2Σ=sinθ2πn(R12​−R22​)​=sin45°2π×150(0.202−0.052)​≈35.3m2

3. Required Σ for separation:Σrequired=Qvt=4.63×10−33.6×10−8≈128,600 m2Σ_{required} = \frac{Q}{v_t} = \frac{4.63 \times 10^{-3}}{3.6 \times 10^{-8}} ≈ 128,600 \, \text{m}^2Σrequired​=vt​Q​=3.6×10−84.63×10−3​≈128,600m2

4. Speed correction using centrifugal enhancement:
Effective settling velocity increases by factor (ω2r)/g(ω^2 r) / g(ω2r)/g

Use average radius r=(R1+R2)/2=0.125r = (R₁ + R₂)/2 = 0.125r=(R1​+R2​)/2=0.125 mvcentrifugal=vt⋅ω2rg=3.6×10−8⋅(628)2⋅0.1259.81≈1.82×10−3 m/sv_{centrifugal} = v_t \cdot \frac{ω^2 r}{g} = 3.6 \times 10^{-8} \cdot \frac{(628)^2 \cdot 0.125}{9.81} ≈ 1.82 \times 10^{-3} \, \text{m/s}vcentrifugal​=vt​⋅gω2r​=3.6×10−8⋅9.81(628)2⋅0.125​≈1.82×10−3m/sΣnew=Qvcentrifugal=4.63×10−31.82×10−3≈2.54 m2Σ_{new} = \frac{Q}{v_{centrifugal}} = \frac{4.63 \times 10^{-3}}{1.82 \times 10^{-3}} ≈ 2.54 \, \text{m}^2Σnew​=vcentrifugal​Q​=1.82×10−34.63×10−3​≈2.54m2

Our design Σ = 35.3 m² > 2.54 m² ⇒ Sufficient

Final Design Parameters:

• Outer Disk Diameter (D₁): 0.40 m
• Inner Disk Diameter (D₂): 0.10 m
• Number of Disks: 150
• Rotational Speed: 6000 rpm
• Sigma (Σ): 35.3 m²
• Disk Angle with Vertical: 45°

Explanation:

This design outlines a disk centrifuge to separate yeast cells from beer at an industrial flow rate of 400 cubic meters per day. The separation relies on density differences between yeast (1075 kg/m³) and beer (1020 kg/m³), with yeast cells averaging 4 microns in diameter. Due to the small particle size and minimal density difference, gravitational settling is inefficient. A centrifuge increases the settling velocity using centrifugal force, greatly improving separation efficiency.

The settling velocity under gravity was calculated using Stokes’ Law, yielding approximately 3.6 × 10⁻⁸ m/s. This would require an enormous settling area to handle the desired flow rate. Therefore, centrifugal acceleration was considered. By assuming a speed of 6000 rpm and an average disk radius of 0.125 meters, the effective settling velocity increases to approximately 1.82 × 10⁻³ m/s, which significantly reduces the required settling area.

A disk stack with 150 disks, outer diameter 0.40 meters, inner diameter 0.10 meters, and a disk inclination angle of 45 degrees results in a Sigma value of about 35.3 m². This exceeds the required Sigma for effective yeast removal, confirming the design’s suitability. The 45° angle balances effective sedimentation and fluid flow, while the selected diameters are practical for commercial centrifuges. The high number of disks maximizes settling area without expanding the centrifuge footprint excessively.

In conclusion, the proposed centrifuge design effectively leverages centrifugal acceleration to overcome the limitations of gravitational separation. It offers a robust solution for yeast removal in beer processing while maintaining feasible dimensions and operation parameters. This ensures clarity, product quality, and efficient process throughput.