### Simulation models

#### Turbulent model

Turbulent flow is required in enclosed PBRs to meet mass transfer requirement and prevent cells from deposition (Perner-Nochta and Posten 2007; Acién Fernández et al. 2013), and an average velocity of 0.5 m s^{−1} is suggested for tubular PBRs (Molina et al. 2001; Perner-Nochta and Posten 2007). The suspension properties are set to the properties of water (Gómez-Pérez et al. 2017). The simulations of turbulent flow in tubes with discrete double inclined ribs have been intensively studied (Zheng et al. 2015; Meng 2003). Among these simulation models, SST *k*–*ω* model has been suggested by Zheng et al. (2015) for this geometry, because it is more reliable than the *k*–*ε* models and standard *k*–*ω* model with a deviation no more than 10% compared with experimental data reported by Meng (2003). Therefore, the SST *k*–*ω* model is selected in this work, and Fluent 14.0 is used to conduct this simulation. The governing equations are:

Continuity equation:

$$ \frac{{\partial u_{i} }}{{\partial x_{i} }} = 0. $$

(1)

Momentum equation:

$$ \rho \frac{{\partial u_{i} u_{j} }}{{\partial x_{j} }} = - \frac{\partial p}{{\partial x_{i} }} + \frac{\partial }{{\partial x_{j} }}\left[ {\mu \left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} - \rho \overline{{u_{i}^{\prime} u_{j}^{\prime} }} } \right)} \right]. $$

(2)

The turbulence kinetic energy equation:

$$ \rho \frac{{\partial ku_{i} }}{{\partial x_{i} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{k} }}} \right)\frac{\partial k}{{\partial x_{j} }}} \right] + \tilde{G}_{k} - Y_{k} + S_{k} . $$

(3)

The specific dissipation rate equation:

$$ \rho \frac{{\partial \omega u_{j} }}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{\omega } }}} \right)\frac{\partial \omega }{{\partial x_{j} }}} \right] + G_{\omega } - Y_{\omega } + D_{\omega } + S_{\omega } . $$

(4)

The constants for this model are:

*σ*_{k,1} = 1.176, *σ*_{ω,1} = 2.0, *σ*_{k,2} = 1.0, *σ*_{ω,2} = 1.168, *α*_{1} = 0.31, *β*_{i,1} = 0.075, *β*_{i,2} = 0.0828.

#### Particle tracking model

Microalgal cell movements are assumed to be the movement of particles, as in previous researches (Gómez-Pérez et al. 2015, 2017; Huang et al. 2014). The discrete random walk model was adopted to calculate the trajectories of particles (Yang et al. 2016; Perner-Nochta and Posten 2007). Particles in fluid are under the governing equation:

$$ \frac{{{\text{d}}\vec{u}_{\text{p}} }}{\partial t} = F_{\text{D}} \left( {\vec{u} - \vec{u}_{\text{p}} } \right) + \frac{{\vec{g}(\rho_{\text{p}} - \rho )}}{{\rho_{\text{p}} }}, $$

(5)

where \( \vec{u} \) is the fluid velocity, \( \vec{u}_{\text{p}} \) the particle velocity, *ρ*_{p} the particle density, *ρ* the fluid density and *F*_{D} the drag force coefficient.

Cells in the particle tracking model are usually assumed to be inert spheres (Pruvost et al. 2008; Gao et al. 2017), which are similar to the tracer used in particle tracking experiments conducted by Luo and Al-Dahhan (2004) and Pruvost et al. (2000). In this work, cells are assumed to be inert particles with a uniform diameter (10 µm) and density (1000 kg m^{−3}) (Moberg et al. 2012; Zhang et al. 2015). The virtual mass force and Saffman’s lift force are neglected (Moberg et al. 2012; Perner-Nochta and Posten 2007). Tube surface is a reflective surface and the outlet is an escape one. Ten seconds are taken to be the maximum tracking time in this work and its independent validation is given in Additional file 1.

#### Light transfer model

The Cornet model is generally more appropriate than the Lambert–Beer model in the condition of high cell density culture (Acie et al. 1997). Thus, the Cornet model is selected to simulate the light profile in tubular PBRs in this work, which is expressed by

$$ \frac{I}{{I_{0} }} = \frac{{4\alpha_{1} }}{{(1 + \alpha_{1} )^{2} \cdot {\text{e}}^{{\alpha_{2} }} - (1 - \alpha_{1} )^{2} \cdot {\text{e}}^{{ - \alpha_{2} }} }}, $$

(6)

$$ \alpha_{1} = \sqrt {E_{\text{a}} /\left( {E_{\text{a}} + E_{\text{s}} } \right)} , $$

(7)

$$ \alpha_{2} = \left( {E_{\text{a}} + E_{\text{s}} } \right) \cdot \alpha_{1} \cdot C_{\text{x}} \cdot L, $$

(8)

$$ L = \sqrt {r^{2} - x^{2} } - y, $$

(9)

where *I*_{0} and *I* are the incident and local light intensity, respectively, *E*_{a} and *E*_{s} are the mass absorption and scattering coefficients of algal cells, *C*_{x} is the biomass concentration and *L* is the light path. The constants in this work are *I*_{0} = 375 μmol m^{−2} s^{−1} (Huang et al. 2014), 800 μmol m^{−2} s^{−1} (Perner-Nochta et al. 2007) and 1200 μmol m^{−2} s^{−1} (Zhang et al. 2013), *E*_{a} = 0.0014 m^{2} g^{−1}, *E*_{s} = 0.9022 m^{2} g^{−1} (for *Chlorella pyrenoidosa*, obtained by nonlinear fitting Huang et al. 2014) and *C*_{x} = 1.3 g L^{−1} (a concentration in Huang et al. 2014).

Light is assumed to be incident along the -*y* direction (Fig. 1a) and transferring forward and backward only (Cheng et al. 2016). Tubes and mixers are assumed to be transparent and have no impact on light transfer (Cheng et al. 2016; Huang et al. 2014; Perner-Nochta and Posten 2007). The light profiles in PBRs calculated by Matlab are shown in Fig. 2.

### Mixer performance

#### Statistic of the L/D cycle

The L/D cycle of individual cells can be calculated by the binary L/D pattern (Perner-Nochta and Posten 2007). The binary L/D pattern takes light field as the combination of the light zone and dark zone and ignores the light gradient within these two zones. The light zone is where the local light intensity is higher than the critical light intensity, and the dark zone is where the local light intensity is lower than the critical one (Perner-Nochta and Posten 2007; Luo and Al-Dahhan 2004). The critical light intensity, separating light and dark zone, is 96.84 μmol m^{−2} s^{−1} (Sorokin 1958; Huang et al. 2014) (Fig. 1).

A complete L/D cycle is defined as (Luo and Al-Dahhan 2004)

$$ t_{\text{c}} = t_{\text{d}} + t_{\text{l}} , $$

(10)

where *t*_{d} is the time that a particle stays in the dark zone and *t*_{l} in the light zone. The L/D cycle frequency is

$$ f = 1/t_{\text{c}} . $$

(11)

An individual particle might experience a number of L/D cycles as it is moving back and forth between the light and dark zones continuously. For every particle, the average duration of L/D cycles is defined as

$$ t_{{{\text{c}},{\text{av}}}}^{\text{ID}} = \frac{{\mathop \sum \nolimits_{1}^{n} t_{\text{c}} }}{n}, $$

(12)

where ID is the serial number of a particle and *n* is the number of L/D cycles of the particle.

The number of the particles should be large enough to ensure the reliability of the L/D frequency results (Huang et al. 2014), because the particle tracking model for an individual particle is based on the Gaussian probability distribution. In this work, 1000 particles are used as recommended by Huang et al. (2014) (the number validation is shown in Additional file 1). The average time of the L/D cycles of a group is (Huang et al. 2014)

$$ t_{{{\text{c}},{\text{av}}}} = \frac{{\mathop \sum \nolimits_{{{\text{ID}} = 1}}^{{{\text{ID}} = N}} t_{{{\text{c}},{\text{av}}}}^{\text{ID}} }}{N}, $$

(13)

where *N* is the number of particles. Therefore, the averaged L/D cycle frequency is

$$ f_{\text{av}} = 1/t_{{{\text{c}},{\text{av}}}} . $$

(14)

#### Pressure drop

The average pressure at a cross section of the tubular PBR is (Gómez-Pérez et al. 2015)

$$ P_{\text{av}} = \frac{{\iint_{S} {P{\text{d}}s}}}{S}, $$

(15)

where *P* is the local pressure and *S* is the area of the cross section.

The pressure drop between the outlet and the surface where particles are released is given by:

$$ \Updelta P = P_{\text{p}} - P_{\text{out}} , $$

(16)

where *P*_{p} is the average pressure at the surface where particles are released and *P*_{out} is the pressure of the outlet.

#### Efficiency of the L/D cycle enhancement

To evaluate the performance of the ribs on the L/D cycle enhancement and energy consumption simultaneously, we defined the efficiency of the L/D cycle enhancement in our previous work (submitted). The efficiency of the L/D cycle enhancement is the ratio of the dimensionless increment of the L/D cycle frequency to dimensionless increment of energy consumption per unit time, that is,

$$ \eta = \frac{{\Updelta f_{\text{av}} /f_{{{\text{av}},0}} }}{{\Updelta E/E_{0} }}, $$

(17)

where *f*_{av,0} is the L/D cycle frequency of a smooth tubular PBR, \( \Updelta f_{\text{av}} = f_{\text{av}} - f_{{{\text{av}},0}} \) is the increment of the L/D cycle frequency of PBRs with discrete double inclined ribs compared with the smooth PBR, \( \Updelta f_{\text{av}} /f_{{{\text{av}},0}} \) is the dimensionless \( \Updelta f_{\text{av}} , \) *E*_{0} is the energy consumption of the smooth PBR, \( \Updelta E = E - E_{0} \) is the increment of energy consumption of PBRs with discrete double inclined ribs compared with the smooth PBR and \( \Updelta E/E_{0} \) is the dimensionless \( \Updelta E. \)

The energy consumption per unit time is (Gómez-Pérez et al. 2015)

$$ E = \varPhi \Updelta P, $$

(18)

where *Φ* is the volume flow rate and \( \Updelta P \) the pressure drop. The diameters and average velocities of all the PBRs investigated in this work are the same, and thus the flow rate is the same. Then, we have \( \Updelta E/E_{0} = ( \Updelta P - \Updelta P_{0} )/\Updelta P_{0} \) and the efficiency of the L/D cycle enhancement can be further expressed as

$$ \eta = \frac{{\Updelta f_{\text{av}} /f_{{{\text{av}},0}} }}{{(\Updelta P - \Updelta P_{0} )/\Updelta P_{0} }}. $$

(19)