## 1 Introduction
Mixing time measurements were made using discoloration and thermal-response techniques, both of them generating reproducible, quantitative data. For some agitators in the real viscous flow region, the mixing time is inversely proportional to the agitator speed. The mixers could be compared in terms of power number and mixing number based on liquid viscosity and density, power consumption, mixing time and vessel diameter. [PERSON] et al. [31] used the conductivity method, thermal method and color addition method to measure mixing time. Experiments were executed in a 396-mm-diameter tank with basket-curve bottom and a 529-mm-diameter tank with plain bottom. They found the most suitable method according to thermal pulse disturbance. The mixing number is the product of mixing time and the rotation speed of the agitator. Neither the temperature and the place of thermal pulse disturbance, nor the location of the measuring point has a significant effect on the mixing time which the dimensionless mixing number is independent of high Reynolds number (Re \(>10^{4}\)) within the turbulence range [31; 32]. The mixing number of the VEGYTERV propeller has been investigated as a function of tank to agitator diameter ratio; [PERSON] and [PERSON] [32] also found the same phenomena of 6-flat-blade turbine agitator. The mixing number depends on the low Reynolds number (Re \(<100\)) of the reduced form [33] in the condition of gate type anchor impellers and multi-paddle agitators. When the mixing time remains the same in the bioreactor at different scales, the P/V\({}_{\rm L}\) increases with two-thirds of the volume of the bioreactor [28]. Because high-volume power inputs are harmful to cells and do not require a constant mixing time to use as a scale up standard for mammalian cell culture of slow biologic reactions [30; 31], which usually introduces concentration gradients forming and longer mixing times in large-scale bioreactors [31; 32]. To improve the transport phenomenon, the dimensionless group is usually utilized in the mixing from small scale to large bioreactors [33; 34]. The dimensional analysis principally presents that the dimensionless mixing number relies on Reynolds number and Froude number. On the other hand, bioreactors installed with baffles to escape vortex formation, and it is reasoned to be a function of Re as the free surface is nearly flat in laminar or transition flow systems [35]. However, for an orbital shaken reactor system acting with a non-flat free surface or with a vortex, Fr must also be considered as it describes the role of fluctuations on the free surface [36]. In stirred-tank bioreactor (STBs) cells, the high maximum power input value occurs at the stirrer tip and the bubble aeration foaming of the medium may result in high shear force. In comparison,as a result of the power input distribution and the generalization of the bubble-free surface aeration, the shearing stress in OSBs is much lower. The mixing principle and the shear issue associated with STRs is well known to engineers [37; 38; 39; 40], but results from studies with STBs cannot be applied to OSBs due to their agitating principle difference. Moreover, considering OSRs are relatively novel, the influence of crucial engineering parameters on bioreactor capability is yet to be well defined. This study aims to perform the mixing of Newtonian liquids in the turbulent regions of batched orbital shaken cylindrical bioreactors, describing the influences of various operating parameters on the power input and mixing time investigation at volumetric scales up to 20 L. Dimensionless Newton numbers and mixing number with regards to power consumption, mixing time, shaking frequency, bioreactor diameter, and fluid physical properties were utilized to determine power input and mixing time, and thereby mixing efficiency.