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

**Pei En Tham 1, Yan Jer Ng 1, Revathy Sankaran 2, Kuan Shiong Khoo 1, Kit Wayne Chew 3,\*, Yee Jiun Yap 3, Masnindah Malahubban 4, Fitri Abdul Aziz Zakry <sup>4</sup> and Pau Loke Show 1,\***


Received: 12 March 2020; Accepted: 16 March 2020; Published: 25 March 2020

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 *J* is defined as the number of bubbles passing though an area *A* per unit time. Assuming no diffusion, i.e., only convection,

$$
\vec{J} = \sigma \vec{s} \tag{2}
$$

where σ and *s* are the density and velocity of the bubbles respectively. Putting Equation (2) into (1) gives

$$\frac{\partial \sigma}{\partial t} = -\nabla \cdot \overline{\mathcal{J}}.\tag{3}$$

The bubbles are assumed to move into the region R, resulting in a negative ∇·*J*, and the negative sign is to make it positive. To solve Equation (16), Laplace transform is applied to Equation (16), giving

$$f(z)\overline{\sigma}\_z(z,s) + s\overline{\sigma}(z,s) - \sigma(z,0) + F(z)\overline{\sigma}(z,s) = 0$$

$$f(z)\overline{\sigma}\_z(z,s) + [s + F(z)]\overline{\sigma}(z,s) - \sigma(z,0) = 0\tag{17}$$

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

$$f(z)\overline{\sigma}\_{\overline{z}}(z,s) + [s + F(z)]\overline{\sigma}(z,s) = 0\tag{18}$$

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

$$
\overline{\sigma}\_z(z,s) + \frac{s + F(z)}{f(z)} \overline{\sigma}(z,s) = 0 \tag{19}
$$

Multiplying Equation (19) with integrating factor *e <sup>s</sup>*+*F*(*z*) *<sup>f</sup>*(*z*) *dz* yields

$$\frac{d}{dz} \Big[ \epsilon^{\rho(z,s)} \overline{\sigma}(z,s) \Big] = 0 \tag{20}$$

Integrating Equation (20) gives

$$
\overline{\sigma}\_z(z,s) = \mathbb{C}e^{Q(z,s)} \tag{22}
$$


$$q(z,s) = \frac{9s\nu}{8\rho g a^2} \left(\frac{\rho^2 g^2 z^3}{3} - \rho g z^2 P\_{cx} + (P\_{cx})^2 z\right) + \ln\left|\frac{8\rho g a^2}{9\nu \left(\rho g z - P\_{cx}\right)^2}\right|\tag{21}$$

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

$$L^{-1} \left| \overline{\sigma\_z}(z, s) \right| = \sigma\_z(z, t) = \delta \left| t - \frac{9\nu}{8\rho g a^2} \left( \rho g z^2 P\_{ex} - \left( P\_{ex} \right)^2 z - \frac{\rho^2 g^2 z^3}{3} \right) - \ln \left| \frac{8\rho g a^2}{9\nu \left( \rho g z - P\_{ex} \right)^2} \right| \right| \quad \text{(23)}$$

where *L* is the Laplace transform.

#### **Reference**

1. 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. [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/).

MDPI St. Alban-Anlage 66 4052 Basel Switzerland Tel. +41 61 683 77 34 Fax +41 61 302 89 18 www.mdpi.com

*Processes* Editorial Office E-mail: processes@mdpi.com www.mdpi.com/journal/processes

MDPI St. Alban-Anlage 66 4052 Basel Switzerland

Tel: +41 61 683 77 34 Fax: +41 61 302 89 18