A trial for theoretical prediction of microalgae growth for parallel flow
- 1 -
A trial for theoretical prediction of microalgae growth for parallel flow
C. Y. Ma* School of Mechanical and Electrical Engineering, Nanchang University, 999 Xuefu Avenue, Nanchang 330031, People’s Republic of China *Corresponding author: [email protected]
Abstract
Many models established for solving the problem of the prediction of microalgae growth. However, the models are semi-empirical or considerable fitting coefficients exist in the theoretical model. Therefore, the prediction ability of the model is reduced by the fitting coefficients. The growth mechanism of microalgae is not clearly understood until now, and the growth state is related to the microalgae strains. The above reasons conducted the problem of microalgae growth is much difficult in theoretical prediction. Furthermore, the predicted results of the established models are dependent on the size of the photobioreactor (PBR), light intensity, flow field, and concentration field. Therefore, the growth rate of the dependent variable is the function of independent variables including nutrients concentration, light intensity, flow field, PBR size, temperature, pH. The experimental works are 10 for each independent variable selects 10 values of 6 variables which can not be accomplished. The dimensionless method maybe provide a way to solve the problem. In this paper, the analytical solution of the growth rate was obtained for the parallel flow. The dimensionless growth rate expressed as function of Reynolds number and Schmidt number, which can be used for arbitrary parallel flow due to the parameters are expressed as dimensionless quantity. The solution of growth rate was used to predict the experimentally measured data. The results show that the theoretically predicted growth rate is consistent with the experimentally measured growth rate of microalgae on the order - 2 - of magnitude. These results will be useful in the design and operation of PBRs for biofuel production. Keywords:
Microalgae growth; Dimensionless method; Parallel flow; Photobioreactors; Biofuel
1. Introduction
The microalgae cultivation have many potential applications, such as animal feed, food, health drinks, nutraceutical, pharmaceutical, wastewater treatment, and biofuels production [1, 2]. The world energy consumption is predicted to increase by 50% from 2005 to 2030 [3]. It is crucial to find renewable and clean fuels to instead the traditional fossil fuels. Microalgae is considered to be the most promising alternative resources for biofuel production. Ullah et al. reported that biofuels with the potential to meet 50% of the world energy consumption while reducing the carbon emissions from fossil fuels simultaneously [4]. Photosynthesis microalgae for biofuel production are typically cultivated in open ponds or enclosed photobioreactors (PBRs) [5]. The photosynthesis process is affected by light distribution, nutrients supply (C, N, P), flow of fluid, PBR type, temperature, and pH in the cultivation. For example, the light utilization efficiency of these PBRs significantly affects the overall biofuel production process for the autotrophic microalgae [3]. Indeed, photosynthesis process of microalgae require an optimum irradiance to achieve their maximum biomass production rate. The microalgae will behave light inhibition for excessive irradiance, opposite, limited light irradiance will cause biomass productivity decrease [6]. In addition, microalgae increase their photoprotective carotenoid concentration in response to excessive irradiance and reducing the amount of photosynthetic carotenoid through the xanthophyll cycle [7, 8], while they increase their pigments concentration in limited light irradiance [9]. Thus, careful light transfer analysis must be conducted to design and optimize the light distribution in PBRs to make them efficiently. The other conditions also exhibit important influence on the growth of microalgae, the nutrients, such as the nitrogen also exist an optimal - 3 - concentration [6]. Besides, the effects of different nitrogen source on the growth of microalgae are also different [10]. The effect of cultivation conditions on the growth of microalgae is like the “Barrel principle”, i.e., the maximum growth rate can be only achieved when the overall conditions at optimal. More detailed and comprehensive information about the growth of microalgae maybe be find in literatures [6, 11]. There are many models established for solving the problem of the prediction of microalgae growth [12-14]. However, the models are semi-empirical or considerable fitting coefficients exist in the theoretical model. The above reasons cause the models to loose its generality and the prediction ability of the model is reduced. It is necessary to give a general model to predict the growth of microalgae. The starting point is all the factors affecting the growth of microalgae are considered. Derivation of the governing equations affecting the growth of microalgae and dimensionless treatment of the equation to obtain the dimensionless solution. The greatest advantage of the dimensionless solution is its generality, i.e., the solution is independent of the PBR size, fluid type, flow sate, and microalgae strains etc. In this paper, we will give a dimensionless solution of the problem of prediction of microalgae growth for the parallel flow in section 3. This study specifically aims to derivate the relation between growth rate and the influence factors, including the light, nutrients, flow state etc. Understanding how the growth rate of microalgae affected by the cultivation condition. The results maybe used to design PBRs to improve the production of microalgae.
2. Problem statement and methods
Light transfer within absorbing, scattering and non-emitting microalgal suspension in PBRs is governed by the radiative transfer equation (RTE) expressed as [15] ( , ) ( , ) ( , ) ( )d4 s I I I s r s r s r s s s (1) - 4 - where I is the radiative intensity in direction s at location r , s is the direction vector, s is the scattering coefficient, a s is the extinction coefficient, a is the absorption coefficient, ( , ) s s is the scattering phase function and is the solid angle. The scattering phase function ( , ) s s represents the probability that the radiation transfer in the solid angle d around the direction s scattered into the solid angle d around the direction s and is normalized such that s s (2) The asymmetry factor g defined as the mean cosine of the scattering phase function expressed as [15] g s s (3) where is the angle between the s and s . It was observed the microalgal suspensions featured strongly forward scattering, and its g value around 0.97. Bidigare et al. used a predictive method to determine the spectral absorption coefficient , a by expressing it as [16] , ,1 na i ii Ea c (7) where , i Ea (m ·kg -1 ) is the in vivo pigment specific spectral absorption cross-section of pigment i , and i c (kg·m -3 ) is the mass concentration of the cell’s pigment i . The specific absorption cross-section of Chl a, b and c, and -carotene have been reported by Bidigare et al. in the spectral region 400 to 750 nm [17]. Besides, the temporal evolution spectral absorption and scattering cross-sections were experimentally measured in Refs. [9, 18, 19]. Photosynthesis is a worldwide process which plants, algae, and some photosynthetic bacteria can store solar energy, electron source and carbon source (mainly CO ) in the form of sugars [3]. The microalgae can produce oxygen, carbohydrates, proteins, and lipids within the cells, some species may produce H through - 5 - the photosynthesis process. The main nutrient elements used in the microalgae culture include carbon C, nitrogen N, phosphorus P, and some minor elements such as potassium, magnesium, calcium, and sulfur etc. [6]. 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 excess 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, that caused by biochemical process. The transfer rate for mass diffusion is known as Fick’s law [20] i ij i D j (2) where the subscript i represent species i in a mixture of i and j , here, the j maybe monocomponent or multicomponent. The quantity i j is the diffusive mass flux of species i . And ij D is defined as the mass diffusion coefficient, i stands for the mass concentration of species i of the culture. From the species conservation, the convection diffusion equation can be expressed as [20] i i i i iij i u v Dt x y x y (3) 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. 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 maintaining uniform pH the mixing method is used in PBRs. Culture mixing maybe achieved by methods, such as stirring, air bubbling, or liquid circulation, which is known to considerably enhance microalgae productivity [6]. The - 6 - dynamics of the incompressible flow is governed by the Navier-Stokes equation [21] V V V p V Ft (4) which was first proposed by Navier and further completed by Stokes, the first term about time will be disappeared for steady state flow. 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.
3. Laminar flow over a plate
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 [20] u vx y (5) u u uu vx y y (6) i i iij u v Dx y y (7) Assuming the microalgae cells are growth on the surface of the thin layer, the microalgae cells enter the microalgae culture through diffusion process. The governing equation of microalgae concentration N can be similarly written as for the boundary layer N N N Nu v Dx y y (8) - 7 - 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.(7) and Eq.(8), which depend on u and v may be obtained. The solution follows the method defining a stream function , x y , and the new dependent and independent variables, f and , respectively, which defined as [20] u y and v x (9) f u x u (10) y u x (11) Using these new variables, we can reduce the partial differential equations to ordinary differential equations. After some mathematical calculations, the Eq.(6) can be rewritten as by the new variables [20] d f d ffd d (12) The Eq.(12) with appropriate boundary condition can be solved by numerical integration, the important results as follows [20] x xu x Re (13) which represent the velocity boundary layer thickness. For the integrity of the results, the local friction coefficient is also given as following [20] - 8 - , 1 2, 2 s xf x x C Reu (14) From the solution of the velocity boundary layer, the species continuity equation can now be solved. To solve the Eq.(7) we introduce the dimensionless species density [20] ,, , i i si i i s (15) and the boundary condition of the species density for a fixed surface i and i (16) Making the necessary substitutions, the Eq.(7) can be reduce to [20] i i d dSc fd d (17) For the Schmidt number Sc , the results of the surface species concentration gradient can be correlated by the following relation [20] i d Scd (18) The local mass transfer convection coefficient defined as ,, , i sm i s i nh (19) which , i s n is the mass transfer rate of species i , using the Fick’s law , i s n can be expressed as , 0 ii s ij y n D y (20) and combining this with Eq.(19), we obtain
0, , ij i ym i s i
D yh (21) Using the definition of i , it follows that , ,, , 0 i i s im iji s i y h D y (22) - 9 - and with using the variable , can be rewritten as im ij duh D x d (23) It follows that the local Sherwood number is of the form [20] , 1 2 1 3 m xx xij h xSh Re ScD (24) for the Schmidt number Sc . From the solution to Eq.(17), the ratio of the velocity to concentration boundary layer thickness is [20] c Sc (25) where c is the boundary layer thickness of the concentration, and is given by Eq.(13). The boundary layer equation of microalgae concentration, Eq.(8) is of the same form as the species concentration boundary layer equation Eq.(7), with N D replacing ij D . Introducing a dimensionless microalgae concentration s s N N N N N , and the appropriate boundary conditions are N and N . We also see that the microalgae concentration boundary conditions are of the same form as the species boundary conditions. Therefore, the mass and microalgae cells transfer analogy can be applied since the differential equation and boundary conditions for the microalgae concentration are of the same form as for species concentration. Hence, with reference to the solution of the species equation , 1 2 1 3 N xNx x NN h xSh Re ScD (26) By analogy to Eq.(25), it also follows that the ratio of boundary layer thickness is NN Sc (27) where N is the boundary layer thickness of the microalgae concentration. The foregoing results can be used to compute average laminar boundary layer parameters for x L , where L is the distance from the leading edge at which transition begins. The three boundary layers experience nearly identical growth for - 10 - values of Sc and N Sc close to unity, from Eq.(25) and Eq.(27). 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 [20] , ,0 xm x m x h h dxx (28) for an dimensional parallel flow. Integrating and substituting from Eq.(24) and Eq.(26), respectively, it follows that , , m x m x h h . Thus, we obtain [20] , 1 2 1 3 m xx xij h xSh Re ScD (29) Similarly, employing the analogy of mass and microalgae cells transfer, it follows that , 1 2 1 3 N xNx x NN h xSh Re ScD (30) If the flow is laminar over the entire surface, the subscript x can be replaced by L . 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 [15] , , , a b a q I G (31) 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 , a G Gu vq Gc x c y (32) we can not obtain the analytical solutions of the partial differential Eq.(32). However, some simplifications can be made, the velocity of the fluid is much smaller than the light velocity c , the convection term can be - 11 - neglected in Eq.(32), the radiative heat flux term can be simplified as dG dy due to the scattering phase function of microalgae exhibited strongly forward scattering feature for radiation on direction of y , it follows that , a dG Gdy (33) The analytical solution can be obtained as , a G G y G y (34) where the dimensionless y y s , and , a s is the optical thickness, s is the optical path length. Similarly, we introduce the concept of radiation boundary layer, which is the region that the radiative fluence rate gradients exist, and its thickness defined as the value for which , , , s s G G G G . For the radiation on direction of x , the boundary layer equation of radiation flux can be written as , a dGu Gc dx (35) If the velocity u u y , which maybe a reasonable assumption for the parallel flow, the solution can be expressed as , a uG x G x c (36) The same direction of velocity u will be enhance 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 maybe very small, however, if this effect makes rational use of us, it will effectively improve the utilization rate of light energy. According to the process of microalgae cells diffusion to the flow from the surface and the microalgae cells concentration increase rate in the parallel flow, it follows that - 12 - , , | c M s M sS V V V V d dNN n dS NdV dV NdVdt dt (37) where the is the average growth rate of microalgae culture, the c V is the volume of constant concentration of microalgae, and | c V V represent the total volume subtract the volume c V . From the definition, the microalgae cells transfer rate can also be expressed as , M s s NS
N N N h dS (38) where the subscript M emphasize the microalgae. Combining the Eq. (37) and Eq.(38), it follows that BL s NSV N N h dSNdV (39) Using the relation of local and average mass transfer coefficient, the Eq.(39) can be rewritten as BL N sV h N N SNdV (40) For an one dimensional parallel flow, the integral can be approximated by the following N s LN h N N LN L (41) which assumed that the concentration of boundary layer is slightly difference with the concentration of mainstream. By multiplying variable x and dividing N D on both sides of Eq.(41), and combine with Eq.(26), it follows that L sNN xN
N Nx ShD N (42) Substituting the expression of LN L N
L Sc into Eq.(42), and defining the dimensionless number x N Mg xL D which may be temporally called ‘Microalgae growth number’, we obtain s Nx x L N N NMg Sh ScN (43) Combining the Eq. (30) we obtain, sx x L N N NMg Re ScN (44) - 13 - The Microalgae growth number is linearly proportional to the Reynolds number from Eq.(44). The growth rate can also 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 larger than the nutrients assimilation of microalgae, which follows that [22] , i s i n A NV Y (45) where i Y is the growth yield [22], and subscript i represent the nutrient element, such as C, N, P. Combining with Eq.(19) and Eq.(29), we can obtain ij i i s i x D AY Re Scx NV (46) The average growth rate can be determined from Eq.(46) by an integration over the plate iij i s i L AYD Re ScNVL (47) From the start point of light supply, the light energy supply must larger than the needed energy in photosynthesis, which can be expressed as [22] l GA NV Y (48) where l Y is the growth yield for light energy [22]. Combining with Eq.(34), is follows that PAR l AY G y dNV (49) 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 PAR l GAY dNV (50) The Eq.(46) and Eq.(49) give the upper limit of the growth rate of microalgae under different perspectives. If we make the two equal, it follows that ij i i s i x l D Y Re Sc Y G y dx (51) The equation gives the connection between fluid flow, mass transfer, and light irradiance. The maximum - 14 - growth rate of microalgae cultivation may be achieved according to this relation, which point 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. The turbulent flows will be occurred with Reynolds number exceed the critical Reynolds number, , x c Re . By employing the Chilton-Colburn analogy, the local Sherwood number for turbulent flow is [20] * 4 5 1 3 x x x Sh St Re Sc Re Sc (52) with the condition 0.6 < Sc < 3000. And the local Sherwood number for microalgae is * 4 5 1 3 Nx N x N x N
Sh St Re Sc Re Sc (53) 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 [20] c c x LL l tx h h x h xL (54) where it is assumed that transition occurs abruptly at c x x . The average Sherwood number can be expressed as [20] L L
Sh Re C Sc (55) and the average Sherwood number for microalgae is
NL L N
Sh Re C Sc (56) where the C = 871 for a transition Reynolds number of x c Re [20]. The average Microalgae growth number for mixed boundary layer is - 15 - s NL L L N N NMg Sh ScN (57) Substituting Eq.(56) into Eq.(57), we can obtain sL L L N N NMg Re C ScN (58) The growth number of mixed boundary layer increase faster than the growth number of laminar boundary layer. This implies that the growth of microalgae will benefit from the enhanced mixing. The average growth rate based on the nutrient elements balance for mixed boundary layer is i ij i s i L AY D Re C ScNVL (59) This gives the upper limit of the growth rate at the mixed boundary layer condition. It is also revealed the quantitative relation for growth rate with the flow velocity and mass transfer. Combining with the Eq.(50), the light, flow, and mass transfer formula can be obtained i ij i s i L l Y D GRe C Sc Y dL (60) 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.
4. Results and discussions
In the previous section, we have derivated the expression of growth rate of microalgae for parallel flow. 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 - 16 - calculated results are presented in Fig. 1 and Fig. 2, respectively, for laminar and mixed boundary layer condition by using Eq.(44) and Eq.(58). The critical Reynolds number chooses as Re 5 10 c for the external parallel flow [20]. The kinematic viscosity 8.9×10 -6 m ·s -1 , diffusion coefficient 1.973×10 -9 m ·s -1 [23]. The velocity 0.5 m·s -1 , and the length varied from 0.5 to 8.9 m for laminar flow. The ratio R defined as s N N . As shown in Fig. 1, the varying trend abruptly changed at the point of critical Reynolds number, which due to the transition from laminar flow to turbulent flow. The Growth number increases with the increase of Reynolds number for different ratios of R . As shown in Fig. 2, the growth rate decreases with the increase of Reynolds number for the laminar flow, and its have a tendency to increase first and then decrease trend for the mixed boundary layer condition. It is indicated that the low speed flow will 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 Reynolds number). The extreme point of the Reynolds number which to achieve the maximum growth rate for mixed boundary layer can be calculated by the derivation of Eq.(58) with respect to length L , and make it equal to zero follows that L Re C (61) here, the L Re when C get the value 871. That is to say, when the Reynolds number takes , the growth rate reaches the maximum for the mixed boundary layer in this paper. We can easily obtain the Reynolds number that maximizes the growth rate according to Eq.(61). Therefore, according to the kinematic viscosity and characteristic length, we can determine the optimum flow velocity. - 17 - R R M g ( ) Re ( ) Fig. 1.
The Growth number vs the Reynolds number, the ratio R , R , and R equals to 1.001, 1.002, and 1.005, respectively. G r o w t h r a t e , d Re ( ) R G r o w t h r a t e , d Re ( ) R R Fig. 2.
The growth rate vs the Reynolds number, the ratio R , R , and R equals to 1.001, 1.002, and 1.005, respectively. - 18 - The growth rate with respect to the Reynolds number for different diffusion coefficient is shown in Fig. 3. As shown, the growth rate increase 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 decrease 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 also should note that the change of viscosity means the change of velocity and characteristic length under the condition of the Reynolds number is constant. G r o w t h r a t e , d Re ( ) D D G r o w t h r a t e , d Re ( ) D Fig. 3 . The growth rate vs the Reynolds number for different diffusion coefficients, the ratio R equals to 1.002, the D , D , and D equals to 1.973×10 -9 m ·s -1 , 1.973×10 -8 m ·s -1 , and 1.973×10 -7 m ·s -1 , respectively. - 19 - G r o w t h r a t e , d Re ( ) G r o w t h r a t e , d Re ( ) Fig. 4.
The growth rate vs the Reynolds number for different kinematic viscosity, the ratio R equals to 1.002, the , , and equals to 3.5 × -6 m ·s -1 , 8.9 × -6 m ·s -1 , and 1.7 × -5 m ·s -1 , respectively. 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. The Reynolds number was calculated merely for species of Scenedesmus 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 corresponds to the Reynolds number 8.43×10 . In addition, in general, the growth rate predicted by the theory is consistent with the growth rate obtained by the experiment on the order of magnitude. Of course, the theoretical calculation method given here needs more experimental data to verify and modify. We are just trying to give a new method or a new view for calculating the growth rate of microalgae. - 20 - Table 1
The growth rate of microalgae. Microalgae strains Velocity, v (m·s -1 ) Length, L (m) Reynolds number, Re Growth rate, (d -1 ) Source Chlorella spp.
Scenedesmus sp. C. vulgaris and
Nannochloropsis sp.
Haematococcus pluvialis m Zhang et al. [27]
Chrysophycea m Malve et al. [28]
Chlamydomonas acidophila m Spijkerman et al. [29]
Chlorella sp. , Synechocystis sp. PCC 6803 , and
Tetraselmis suecica m He et al. [30] m Maximum growth rate.
5. Conclusions
This study derivate the analytical solution of the growth rate of microalgae for a parallel flow, which considered influence factors of light, nutrients, and flow state. The solution gives comprehensive understanding of how the growth rate affected by the influence factors. The dimensionless growth rate of microalgae expressed as function of Reynolds number and Schmidt number, which can be used for arbitrary parallel flow due to the solution is expressed in dimensionless form. From the solution, we can see that lager diffusion coefficient will promote the growth of mciroalgae, which may also explain why microalgae growth faster in solid medium. The growth rate solution can predict the experimentally measured growth rate on the order of magnitude. These results may be useful in the design and operation of PBRs for biofuel production. - 21 -
Acknowledgements
This study was mainly completed at Nanchang University. Thank you for the environment provided by the Nanchang University, and the support of my family.
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