On the theoretical prediction of microalgae growth for parallel flow

The established microalgae growth models are semi-empirical or considerable fitting coefficients exist currently. Therefore, the ability of the model prediction is reduced by the numerous fitting coefficients. Furthermore, the predicted results of the established models are dependent on the size of the photobioreactor (PBR), light intensity, flow and concentration field. The growth mechanism of microalgae has not clearly understood in PBR cultivation. It is difficult to predict the microalgae growth by theoretical methods, owing to the aforementioned factors. We developed an exploratory bridging microalgae growth model to predict the microalgae growth rate in PBRs by using the nondimensional method which is effectively in fluid dynamics and heat transfer. The analytical solution of the growth rate was obtained for the parallel flow. The nondimensional growth rate expressed as function of Reynolds number and Schmidt number, which can be used for arbitrary parallel flow due to the solution was expressed as nondimensional quantities. The theoretically predicted growth rate is compared with the experimentally measured microalgae growth rate on the order of magnitude. The nondimensional method successfully applied to the microalgae growth problem for the first time. The general nondimensional solution can unify the numerous experimental data for different laboratory conditions, and give a direction for the disorder of the microalgae growth problem. The nondimensional solution may be useful to explain the growth mechanism of microalgae and design large-scale PBRs for microalgae biofuel production. The significance of the work is to give a theoretical foundation and methodology of biological theory of microalgae growth.


Introduction
The growth of microalgae is affected by many factors such as light intensity, nutrients, and temperature.
Therefore, the growth of microalgae is an extremely complex interdisciplinary problem [1,2]. This article is the first attempt to use the dimensionless method to study the growth of microalgae for the parallel flow over a plate. The first attempt to use the dimensionless method to unify the chaotic state of microalgae growth research, just as the Reynolds number proposed in the fluid mechanics of the year. The only assumption is that the microalgae concentration satisfies the mass concentration conservation principle.
Enormous microalgae growth models have been established to solve the problem of the prediction of microalgae growth [3][4][5]. The existing microalgae growth models as function of a single substrate [6], such as carbon C, nitrogen N, phosphorus P, or a single light intensity [7], based on this, growth models for a function of multiple factors also proposed [8,9]. Overall, in the growth models of microalgae, the effect of fluid flow and mass concentration diffusion process are not taken into consideration. The growth of microalgae is an extremely complex problem, which affected by many factors, such as fluid flow, nutrient concentration, light intensity/spectral, and PBR size etc.
Moreover, the growth models are semi-empirical or considerable fitting coefficients exist in the theoretical model. The aforementioned reasons cause the models to lose its generality and the prediction -3 -ability of the model is reduced. It is necessary to give a general solution to predict the growth of microalgae by following steps: (1) All the factors affecting the growth of microalgae should be considered; (2) derivation of the governing equations affect the growth of microalgae and dimensionless treatment of the equation to obtain the nondimensional solution. The greatest advantage of the nondimensional solution is its generality, i.e., the solution is independent of the PBR size, fluid type or state, nutrient concentration field and microalgae strains etc. And it is connected the fluid field, nutrient concentration field, and light field which have important effect on the growth of microalgae. In this paper, we will give a nondimensional solution of the problem of microalgae growth prediction for the parallel flow in section 3.
This study specifically aims to derivate the relation between growth rate and the influential factors, including the light, nutrients, fluid flow etc. The analytical solution of the microalgae growth rate was firstly obtained for the parallel flow. The microalgae growth rate was expressed as function of Reynolds number and Schmidt number, which can be used for arbitrary parallel flow due to the solution was expressed as dimensionless quantity. The significance of the work is to give a theoretical foundation and methodology of biological theory of microalgae growth. The results may be useful to explain the growth mechanism of microalgae and design large-scale PBRs for the microalgae cultivation.

Light transfer through microalgae suspensions
Light transfer within absorbing, scattering and non-emitting microalgal suspension in PBRs is governed by the radiative transfer equation (RTE) expressed as [ (1) where I is the radiative intensity in direction s at location r , s is the direction vector, s  ss (2) The asymmetry factor g defined as the mean cosine of the scattering phase function expressed as [10] 4 1 ( , )cos d 4 where  is the angle between the s and   [12].

Nutrient supply of microalgae
Photosynthesis is a universal process through which plants, algae, and some photosynthetic bacteria can store solar energy, electron source and carbon source (mainly CO2) in the form of sugars [16]. The microalgae can produce oxygen, carbohydrates, proteins, and lipids within the cells and some species may produce H2 through the photosynthesis process. The main nutrient elements used in the microalgae culture -5 -include C, N, P, and some minor elements such as potassium, magnesium, calcium, and sulfur etc. [1]. The requirements of microalgae culture can be determined from the elemental composition of the biomass. The nutrient supply should be larger than the nutrient requirements, i.e., under nutrient-sufficient conditions to maximize the productivity of the microalgae cultures.
The mass transfer must occurred which is due to the difference in the concentration of the culture, such as C, N, and P, caused by biochemical process. The transfer rate for mass diffusion is known as Fick's law [17] where the term i  is the generation rate of the mass of species i per unit volume. The concentration field can be obtained by solving the convection diffusion equation, then, to calculate the growth rate of microalgae cells by using the element balance method.

Fluid flow
The microalgae culture are usually in a flow state to prevent the sedimentation of microalgae cells. In addition, in order to ensure light intensity distribution, minimize the gradients of nutrients, and maintain uniform pH, the mixing method is used in PBRs. Culture mixing may be achieved by methods, such as stirring, air bubbling, or liquid circulation, which are known to considerably enhance microalgae productivity [1]. The dynamics of the incompressible flow is governed by the Navier-Stokes equation [18] -6 - which was first proposed by Navier and further completed by Stokes, the first term about time will be disappeared for steady state flow. The vector V represents the fluid velocity,  is the density of the fluid, p is the pressure,  is the kinematic viscosity of the fluid, and F is the mass force. The N-S equation is a second order partial differential equation, which its solution is still an open problem to be solved.
Fortunately, we do not need to solve the N-S equation, the boundary layer form of the N-S equation will be used in the next section. And our aim is to find the relation between the microalgae growth and flow state of the culture.

Assumptions and simplified conditions
The major growth parameters may be obtained by solving the boundary layer equations. Assuming steady, incompressible, laminar flow with constant fluid properties and negligible viscous dissipation, the boundary layer equations can be written as [17] 0 uv xy Assuming the microalgae cells are growth on the surface of the thin layer, the microalgae cells enter the microalgae culture through diffusion process. And the microalgae concentration, N , satisfy the mass concentration conservation principle is assumed. The governing equation of microalgae concentration N can be similarly written as for the boundary layer which N D represent the diffusion coefficient of microalgae cells into the mainstream culture. Solution of these equations is based on the fact that for constant properties in the velocity boundary layer are independent of species concentration. Therefore, we should begin by solving the continuity and velocity boundary layer equations. Once the velocity field has been obtained, solutions to Eq.(10) and Eq. (11), which depend on u and v , may be obtained.

Analytical solution of parallel flow
The solution follows the method defining a stream function ( ) , xy  , and the new dependent and independent variables, ( ) f  and  , respectively, which are defined as [17] u y Using these new variables, we can reduce the partial differential equations to ordinary differential equations.
After some mathematical calculations, the Eq.(9) can be rewritten as by the new variables [17] 32 32 20 The Eq.(15) with appropriate boundary condition can be solved by numerical integration, the important result is as follows [17] 55 which  represents the velocity boundary layer thickness. For the integrity of the results, the local friction coefficient is also given as following [17] -8 - where the , sx  is the shear stress on the plate. From the solution of the velocity boundary layer, the species continuity equation can now be solved. To solve the Eq.(10) we introduce the nondimensional species density [17] , ,, and the boundary condition of the species density for a fixed surface ( ) Making the necessary substitutions, the Eq. (10) can be reduced to [17]  From the foregoing local results, the average boundary layer parameters can be obtained by integrating the local formulas. With the average mass transfer coefficient defined as [17] ,, 0

0.664
If the flow is laminar over the entire surface, the subscript x can be replaced by L .

Radiation energy density within the culture
In the previous, we have already given the energy balance for thermal radiation for an infinitesimal pencil of rays, the radiative energy equation can be obtained from the integration of the radiative transfer equation over all solid angles, which as follows [10] , , , where the G  is the spectral radiation fluence rate, and q  is the spectral radiative heat flux. The self-emission term can be neglected due to the emission wavelength not in the visible region, and when applied to the flow problem the convection term should be added, which can be written as follows we cannot obtain the analytical solutions of the partial differential Eq.(35). However, some simplifications -11 -can be made, the velocity of the fluid is much smaller than the light velocity c , the convection term can be Similarly, we introduce the concept of radiation boundary layer, which is the region that the radiative fluence rate gradients exist, and its thickness is defined as the value for which ( ) For the radiation on direction of x, the boundary layer equation of radiation flux can be written as The same direction of velocity u will be enhanced the light intensity for a fixed point, whereas, the reverse direction of u will recede the light intensity. The effect of flow velocity enhance the light intensity may be very small, however, if this effect makes rational use of us, it will effectively improve the utilization rate of light energy.

Growth rate of microalgae
According to the process of microalgae cells diffusion to the flow from the surface and the microalgae The growth rate can also be expressed from the nutrients substrate analysis, the consumption of nutrient elements is transformed into microalgae biomass through photosynthesis. Therefore, the nutrients supply of the culture must be larger than the nutrients assimilation of microalgae, which follows that [2] , where i Y is the growth yield [2], and subscript i represent the nutrient element, such as C, N, P.
Combining with Eq. (22) From the start point of light supply, the light energy supply must be larger than the needed energy in photosynthesis, which can be expressed as [2] l GA NV Y where l Y is the growth yield for light energy [2]. Combining with Eq.(37), is follows that where the subscript PAR in the integral stands for the photosynthetically active radiation (PAR), which over the spectral region from 400 to 700 nm. The average growth rate can be obtained as following The Eq.(49) and Eq.(52) give the upper limit of the growth rate of microalgae under different perspectives. If we make the two equal, it follows that -14 - The equation gives the connection between fluid flow, mass transfer, and light irradiance. The maximum growth rate of microalgae cultivation may be achieved according to this relation, which points out the quantitative relationship in microalgae cultivation for PBR design. It is indicated that large amount of light irradiance should be accompany with relatively large velocity and mass transfer intensity in order to facilitate the growth of microalgae.

Turbulent flow over an plate
The turbulent flows will be occurred with Reynolds number exceeds the critical Reynolds number, We can see that the local Sherwood number grow more rapidly than laminar boundary layer caused by enhanced mixing of turbulent boundary layer. The average coefficients can be determined for mixed boundary layer conditions with the definition as following [17] ( where it is assumed that transition occurs abruptly at c xx = . The average Sherwood number can be expressed as [17] This relation is valid for mixed boundary layer condition. It is indicated that the more light irradiance the more nutrient supply to promote the growth of microalgae. It is also indicated that different light spectrum will need different nutrient supply. The equation gives the basic quantitative expression for the suitable growth supply of microalgae culture.

Prediction of the solution for parallel flow
In the previous section, we have derived the expression of growth rate of microalgae for parallel flow. -16 -Here, we will further analyze the relation between the Growth number and the Reynolds number, and the growth rate of microalgae varies with the Reynolds number. The calculated results are presented in Fig. 1 and Fig. 2, respectively, for laminar and mixed boundary layer condition by using Eq.(47) and Eq.(61). The critical Reynolds number chooses as 5 Re 5 10 c = for the external parallel flow [17]. The kinematic viscosity is 8.9×10 -6 m 2 ·s -1 , and diffusion coefficient is 1.973×10 -9 m 2 ·s -1 [19]. The velocity is 0.5 m· s -1 , and the length varies from 0.5 to 8.9 m for laminar flow. The ratio R is defined as As shown in Fig. 2, the growth rate decreases with the increase of Reynolds number for the laminar flow, and it has a tendency to increase first and then decrease trend for the mixed boundary layer condition. It is indicated that the low speed flow will be more suitable to the growth of microalgae (Note that this is valid only for laminar flow condition, and the formula given in the paper is no longer applicable for small  The growth rate with respect to the Reynolds number for different diffusion coefficient is shown in Fig.   3. As shown, the growth rate increases with the increase of diffusion coefficient, when the Reynolds number keep invariant. It is in line with our intuition that a large diffusion coefficient corresponds to a large growth rate, which may be due to the relatively strong mass transfer capacity. Therefore, we should improve the mass transfer as much as possible in the culture of microalgae, or choose the medium with strong mass transfer capacity. Fig. 4 shows the growth rate with respect to the Reynolds number for different kinematic viscosity. As shown, the growth rate decreases with the increase of kinematic viscosity of the culture for a fixed Reynolds number. Hence, we should dilute the culture in order to reduce its viscosity in the cultivation of microalgae to improve the productivity of microalgae. It is note worthy that the change of viscosity means Re (10 6 ) D 1 Fig. 3. The growth rate vs the Reynolds number for different diffusion coefficients, the ratio R equals to 1.002, the D1, D2, and D3 equals to 1.973×10 -9 m 2 · s -1 , 1.973×10 -8 m 2 ·s -1 , and 1.973×10 -7 m 2 ·s -1 , respectively.

Experimental data from PBR operation
The growth rate and cultivation parameters are summarized in Table 1. As shown, the range of the growth rate from 0.0465 to 1.752 d -1 for more than 8 species of microalgae. However, the Reynolds number of three cases were obtained for species of Scenedesmus sp. and Chlorella sp., which due to the lack of length and velocity data. The theoretically predicted growth rate (in Fig. 2) is within the range of the experimental growth rate which corresponding to the Reynolds number 8.43×10 4 , 4.49×10 4 , and 5.35×10 4 , respectively. In general, the growth rate predicted by the theory is consistent with the growth rate obtained by the experimental measurement on the order of magnitude. It can be said that the semi-quantitative predictions of microalgal growth rather than accurate predictions achieved success. It is well known that precise theoretical prediction of the growth of microbial life is very difficult or even impossible. Therefore, we are trying to give a new method or a new view for explaining/understanding the growth mechanism of -20 -microalgae in this paper.