Correction: Tham et al. Recovery of Protein from Dairy Milk Waste Product Using Alcohol-Salt Liquid Biphasic Flotation. Processes 2019, 7, 875

Pei En Tham; Yan Jer Ng; Revathy Sankaran; Kuan Shiong Khoo; Kit Wayne Chew; Yee Jiun Yap; Masnindah Malahubban; Fitri Abdul Aziz Zakry; Pau Loke Show

doi:10.3390/pr8040381

We were not aware of some errors made in the proofreading phase; therefore, we wish to make the following corrections to the mathematical equations in the text.

(1): The bar at some units are placed on the bottom instead of the top, the correct representation for the following equations should be:
Flux $\bar{J}$ is defined as the number of bubbles passing though an area $A$ per unit time. Assuming no diffusion, i.e., only convection,

$\bar{J} = σ \bar{s}$

(2)

where $σ$ and $\bar{s}$ are the density and velocity of the bubbles respectively.
Putting Equation (2) into (1) gives

$\frac{\partial σ}{\partial t} = - \nabla \cdot \bar{J} .$

(3)

The bubbles are assumed to move into the region R, resulting in a negative $\nabla \cdot \bar{J}$ , and the negative sign is to make it positive.
To solve Equation (16), Laplace transform is applied to Equation (16), giving

$f (z) {\bar{σ}}_{z} (z, s) + s \bar{σ} (z, s) - σ (z, 0) + F (z) \bar{σ} (z, s) = 0$

$f (z) {\bar{σ}}_{z} (z, s) + [s + F (z)] \bar{σ} (z, s) - σ (z, 0) = 0$

(17)

Since $σ (z, 0) = 0$ , Equation (17) becomes

$f (z) {\bar{σ}}_{z} (z, s) + [s + F (z)] \bar{σ} (z, s) = 0$

(18)

To solve Equation (18), we divide Equation (18) by $f (z)$ to give

${\bar{σ}}_{z} (z, s) + \frac{s + F (z)}{f (z)} \bar{σ} (z, s) = 0$

(19)

Multiplying Equation (19) with integrating factor $e^{\int \frac{s + F (z)}{f (z)} d z}$ yields

$\frac{d}{d z} [e^{q (z, s)} \bar{σ} (z, s)] = 0$

(20)

Integrating Equation (20) gives

${\bar{σ}}_{z} (z, s) = C e^{Q (z, s)}$

(22)
(2): The cylindrical coordinates $\emptyset$ should have the terms as followed: $\emptyset = \tan^{- 1} \frac{y}{x}$ .
(3): There is an additional ‘ln’ term in both the equations which should not be present. The correct equations are as shown below:

$q (z, s) = \frac{9 s ν}{8 ρ g α^{2}} (\frac{ρ^{2} g^{2} z^{3}}{3} - ρ g z^{2} P_{e x} + {(P_{e x})}^{2} z) + l n | \frac{8 ρ g α^{2}}{9 ν {(ρ g z - P_{e x})}^{2}} |$

(21)

Transforming Equation (22) from the $s$ domain back to the $t$ domain yields

$L^{- 1} {{\bar{σ}}_{z} (z, s)} = σ_{z} (z, t) = δ [t - \frac{9 ν}{8 ρ g α^{2}} (ρ g z^{2} P_{e x} - {(P_{e x})}^{2} z - \frac{ρ^{2} g^{2} z^{3}}{3}) - l n | \frac{8 ρ g α^{2}}{9 ν {(ρ g z - P_{e x})}^{2}} |]$

(23)

where L is the Laplace transform.

Reference

Tham, P.E.; Ng, Y.J.; Sankaran, R.; Khoo, K.S.; Chew, K.W.; Yap, Y.J.; Malahubban, M.; Aziz Zakry, F.A.; Show, P.L. Recovery of Protein from Dairy Milk Waste Product Using Alcohol-Salt Liquid Biphasic Flotation. Processes 2019, 7, 875. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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