2,*📷1 and📷Christopher McHardy

1

LSTME Busan, Busan 46742, Korea

2

Technische Universität Berlin, Chair of Food Biotechnology and Food Process Engineering, 14195 Berlin, Germany

*

Author to whom correspondence should be addressed.

Energies 2022, 15(11), 3966; https://doi.org/10.3390/en15113966

Received: 30 March 2022/ Revised: 19 May 2022/ Accepted: 25 May 2022/ Published: 27 May 2022

(This article belongs to the Special Issue Conception, Modelling, Control, and Intensification of Photobioreactors Applied to the Valorization of Microalgae)

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Abstract Computational Fluid Dynamics (CFD) have been frequently applied to model the growth conditions in photobioreactors, which are affected in a complex way by multiple, interacting physical processes. We review common photobioreactor types and discuss the processes occurring therein as well as how these processes have been considered in previous CFD models. The analysis reveals that CFD models of photobioreactors do often not consider state-of-the-art modeling approaches. As a comprehensive photobioreactor model consists of several sub-models, we review the most relevant models for the simulation of fluid flows, light propagation, heat and mass transfer and growth kinetics as well as state-of-the-art models for turbulence and interphase forces, revealing their strength and deficiencies. In addition, we review the population balance equation, breakage and coalescence models and discretization methods since the predicted bubble size distribution critically depends on them. This comprehensive overview of the available models provides a unique toolbox for generating CFD models of photobioreactors. Directions future research should take are also discussed, mainly consisting of an extensive experimental validation of the single models for specific photobioreactor geometries, as well as more complete and sophisticated integrated models by virtue of the constant increase of the computational capacity.photobioreactor; microalgae; computational fluid dynamics; growth kinetics; light transfer; multiphase flow; turbulence; population balance modeling; mass transfer; heat transferKeywords:

1. Introduction In the 21st century, the major challenge for mankind is to reshape societies and their economy such that the challenges of a growing world population, increasing demand for food protein, global warming, and declining fossil and non-fossil raw materials can be overcome. This requires a bio-economy being based on the utilization of sustainable resources, of which microalgae may be part of [1] as they can be cultivated in photobioreactors with high area yields and productivity [2,3].However, a major restriction for growing microalgae phototrophically is that the dry biomass concentration in photobioreactors is typically restricted to a few g/L. This is mainly due to the limitation of cell growth by light, but also due to an insufficient supply of carbon dioxide, insufficient mixing, or limited heat transfer into the culture [4,5,6,7,8,9]. The cell growth kinetics are determined by all of these factors, and their interaction is a major reason for the complexity of photobioreactors. Among the mentioned factors, the distribution of light and its spectrum are the most important ones [10,11]. Under the presumption that light propagation is by far the fastest process in photobioreactors, the light intensity field determines the local supply of energy for growth, while the local energy demand is defined by the cell concentration and their trajectories in the flow, i.e., their light history. This point of view considers that the energy demand of individual cells can either be governed by local steady states of the photo- and biochemical reactions involved in photosynthesis, or by the dynamics of these reactions which depend on the light history of single cells [12,13]. Accordingly, knowledge about the flow field in a photobioreactor might be important to achieve optimized growth conditions. A related aspect in this context is the relevance of transferring heat and CO2 into or out of the culture, whereas both processes are strongly affected by the flow. However, assessing flow conditions can be complex as the flow in many photobioreactors is governed by multiple interactions between the flowing liquid and a dispersed gas phase [14,15,16].Numerical simulation is a suitable tool for investigating systems like photobioreactors, whose performance is governed by the simultaneously occurrence of the mentioned physical and chemical processes, which all affect the biomass growth. Compared to experiments, simulations offer the advantage of being less expensive and that they can deliver information about the importance of single factors by separating their effects from each other [13]. Moreover, particularly Computational Fluid Dynamics (CFD) allows for investigating local phenomena, performing design studies or design optimization as well as assessing detailed information of flows in general. Accordingly, CFD models of photobioreactors have been developed by a number of researchers, e.g., [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] and we will review these models in detail in Section 3. However, investigations based on numerical simulations require valid models, which are challenging to formulate for complex systems such as photobioreactors. This is due to the complex interactions between the involved physical phenomena, the need for their in-depth understanding, and their non-linear coupling to the kinetics of biomass growth [8,14]. As we show in Section 3 of this review, an additional aspect to consider is that multiple modeling approaches and techniques exist for simulating the different phenomena in photobioreactors. This brings with it the challenge of formulating an appropriate set of equations that describes the system of interest with sufficient accuracy. Thereby, models must be considered as an abstract representation of the real system, thus, neglecting and/or simplifying parts of reality. However, the predictive power of a CFD model depends decisively on the selection of the considered phenomena as well as on the selected modeling approach and the concrete submodels [46,47,48]. As the modeling of photobioreactors requires deep knowledge regarding multiple physical, chemical, and biological relationships, it is likely that deviations can be found between the state-of-the-art in CFD simulation of the photobioreactor community and the state-of-the-art CFD modeling in the fluid dynamics community. This seems to be true especially with regard of the modeling of complex multiphase flows, which is a topic being particularly important for the CFD modeling of the common types of pneumatically agitated photobioreactors.With this review, we aim to provide a comprehensive overview of the CFD modeling of photobioreactors. The review is organized as follows: in Section 2 we present common reactor types and the respective operating conditions, which define the processes to be modeled and possible simplifications. We focus this presentation on four major phenomena being relevant for the reaction environment in photobioreactors: light transfer, hydrodynamics, mass transfer and heat transfer. Following on from that, in Section 3 we review how these processes have been considered in published CFD models of photobioreactors. For this, we review the submodels researchers have chosen for the phenomena of light transfer, fluid flow, mass transfer and heat transfer and compare them with state-of-the-art modeling approaches, showing that the convergence of the different scientific disciplines is far from being achieved. In Section 4, we provide an overview of state-of-the-art modeling approaches for all relevant phenomena with a focus on a fluid mechanical perspective. We summarize the available methodological frameworks and physical models for the modeling of light transfer, single-phase and multiphase flows, mass transfer and heat transfer. In addition, we sum up the most common approaches to include microalgae growth kinetics into CFD models. Therefore, in this section, we provide a comprehensive overview of the available physical and biological models and methodological approaches being relevant as a toolbox for generating CFD models of the most common photobioreactor types. We close with a discussion on the topic and future research needs in Section 5, before concluding in Section 6.

2. Photobioreactors and Relevant Phenomena Therein 2.1. Photobioreactor Types As a result of the intensive research activities on the design and optimization of photobioreactors, various configurations have been developed and can be found in practice. Desired properties of all types of photobioreactors are a large surface-to-volume ratio for illuminating the culture and the possibility to supply CO2 for photosynthesis. Presenting the entire diversity of designs is beyond the scope of this review and we point the interested reader to specific review papers [14,15,16,49,50,51,52]. Instead, we restrict ourselves to the photobioreactor types which are most important in practice.A common way of classification is to differentiate between open and closed reactors [5,9]. This classification can be extended by considering the type of fluids in the illuminated part of the reactor, which leads to the classification of typical photobioreactors as being depicted in Figure 1. As the focus of this review is on Computational Fluid Dynamics (CFD), we only consider common reactors for submerged cultures and alternative reactor types, e.g., [53,54,55], are omitted.📷Figure 1.Classification of common photobioreactor types. Top left: Open Raceway Pond ([56], published under CC-BY-SA-4.0 license). Top right: Tubular photobioreactor PBR 4000G, IGV Biotech ([57], published under CC BY-SA 3.0 license). Bottom right: Flat-panel Airlift photobioreactor, Bubble Column photobioreactor, Airlift photobioreactor (own work).The typical design variant of open reactors are Open Raceway Ponds (ORP), see Figure 1, top left. They are characterized by a liquid depth between 20 and 30 cm [58]. The flow in ORP is created by a rotating paddle wheel with a typical flow velocity between 20 and 30 cm s−1 [58,59]. Consequently, the corresponding values of the Reynolds number 𝑅𝑒=𝑢𝐿/𝜈

range roughly between 104 and 105

and the flow is turbulent. We consider ORP as single-phase reactors even though they have a free surface and carbon dioxide is supplied by sparging CO2-enriched air into the liquid at certain spots in the reactor. However, this occurs locally and no significant effects of gas sparging on the distribution of light, the flow, mixing or heat transfer must be expected. The assumption of single-phase flow is in line with most models for ORP [22,23,32,33,34,35,43].

Among closed photobioreactors, one finds a much larger variety of designs, see Figure 1, top right and bottom right. Tubular photobioreactors consist of vertically or horizontally aligned tubes having diameters in the order of 5 cm to ensure a large surface-to-volume ratio [14]. Experimental design variants include helical tubular reactors [60] or tubes with static mixers [24,61]. Centrifugal pumps are used to create the a turbulent flow in the tubes, with values of the Reynolds number between 10,000 and 50,000 [14]. For the supply of CO2 and out-gassing of O2, one usually finds a connected bubble column or airlift, whose volume is small compared to the tubular solar receiver.

There is a large variety of photobioreactor types where the flow is created pneumatically by sparging gas into the liquid. Among these, the most common designs are Flat-panel airlift (FPA) photobioreactors, bubble column photobioreactors and airlift photobioreactors, see Figure 1, bottom right. Numerous design variants exist for all basic configurations, see e.g., [62,63] or [14,15,16,49,50,51,52]. FPA reactors are characterized by a thickness of 2–3 cm, while bubble column and airlift reactors can have diameters between 5 and 20 cm or even larger. The sparging of the gas in any kind of these reactors can occur via dip tubes, perforated pipes, ring spargers or porous plates [64]. The sparger design as well as the gas flow rate determine the bubble size distribution, the gas hold-up, gas-liquid mass transfer and the overall flow field.

Typical values of the gas flow rate in pneumatically agitated photobioreactors range between 0.5 and 2.5 vvm (gas volume per liquid volume and minute) [65]. Another characteristic operation parameter is the gas superficial velocity 𝑢𝑔=𝑉𝑔˙/𝐴0

which relates the gas volume flow rate 𝑉𝑔˙ to the cross-section 𝐴0 of the reactor and typically ranges between 10−4 and 10−1 m s−1 [44,66,67,68,69]. Referring to the flow maps provided by Shah et al. [70] and Zhang et al. [71], one can expect gas-liquid flow in the (pseudo-) homogeneous regime, which is characterized by a narrow bubble size distribution and a small impact of bubble break-up and coalescence. However, one should note that this depends very much on the sparger geometry, and very different bubble size distributions can result at similar gas flow rates for different gas distributors [72]. The bubble shape can be estimated from the Eötvös (𝐸𝑜=𝑔Δ𝜌𝑑2/𝜎), Reynolds (𝑅𝑒=𝜌𝑑𝑢/𝜇) and Morton (𝑀𝑜=𝑔𝜇4Δ𝜌/𝜌2𝜎3

) numbers [73]. This shows a relation between the bubble shape and bubble size, which both affect the interphase forces in two-phase flow.

Finally, it should be stated that little knowledge is available on how the presence of algae cells and their increasing concentration during the cultivation affect the liquid properties and the flow field. Manjrekar et al. [74] estimated an increase of the gas-hold up in microalgae cultures compared to water. Ojha and Al-Dahhan [75] found that the gas hold-up was significantly reduced at high biomass concentration, which they related to a growth-associated increase of the mixture viscosity at constant surface tension. The increase of the dynamic viscosity at higher cell concentration was also reported by other authors [76]. However, as reviewed by Besagni et al. [64], controversial results concerning the effects of viscosity on the bubble column operation characteristics are reported in the literature and further research about this issue will be necessary in future

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Let me give a few ideas on how to do it. I reckon that it would help you.

Higbie's Penetration Theory equation:

• J is the flux of the diffusing species through the porous medium.
• D is the diffusion coefficient of the species in the medium.
• δ is the thickness of the diffusion boundary layer.
• Cs is the concentration of the species at the surface of the porous medium.
• dC/dx is the concentration gradient of the species with respect to distance.

Simple Simulation:

# Parameters

D = 0.001 # Diffusion coefficient

Cs = 1.0 # Concentration at the surface

L = 1.0 # Length of the porous medium

N = 100 # Number of spatial points

dx = L / N # Spatial step size

delta = 0.1 * dx # Thickness of diffusion boundary layer

# Initialize concentration field

C = np.zeros(N+1)

C[0] = Cs # Concentration at the surface

# Time parameters

dt = 0.001 # Time step size

t_end = 1.0

num_steps = int(t_end / dt)

# Time-stepping loop

for step in range(num_steps):

# Calculate concentration gradient using finite difference

dC_dx = np.diff(C) / dx

# Update concentration using Higbie's Penetration Theory equation

flux = D / delta * Cs * dC_dx

C[1:-1] += flux * dt / dx

# Plot the concentration profile

x = np.linspace(0, L, N+1)

plt.plot(x, C)

plt.xlabel('Distance')

plt.ylabel('Concentration')

plt.title('Concentration Profile')

plt.grid(True)

plt.show()