Synchronization of fluid-dynamics related and physiological time scalesand algal biomass production in thin flat-plate bioreactorsAlemayehu Kasahun Gebremariam and Yair Zarmi Citation: J. Appl. Phys. 111, 034904 (2012); doi: 10.1063/1.3678009 View online: http://dx.doi.org/10.1063/1.3678009 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i3 Published by the American Institute of Physics. Related ArticlesA simple method for evaluating the trapping performance of acoustic tweezers Appl. Phys. Lett. 102, 084102 (2013) Review of the effects of additives on biodiesel properties, performance, and emission features J. Renewable Sustainable Energy 5, 012701 (2013) Performance study of a diesel engine by first generation bio-fuel blends with fossil fuel: An experimental study J. Renewable Sustainable Energy 5, 013118 (2013) A dual-polarized broadband planar antenna and channelizing filter bank for millimeter wavelengths Appl. Phys. Lett. 102, 063506 (2013) Multi-fault clustering and diagnosis of gear system mined by spectrum entropy clustering based on higher ordercumulants Rev. Sci. Instrum. 84, 025107 (2013) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Synchronization of fluid-dynamics related and physiological time scalesand algal biomass production in thin flat-plate bioreactors

Alemayehu Kasahun Gebremariam and Yair Zarmia)

Jacob Blaustein Institutes for Desert Research Ben-Gurion University of the Negev Midreshet Ben-Gurion,84990, Israel

(Received 22 October 2011; accepted 7 December 2011; published online 6 February 2012)

Experiments on ultrahigh density unicellular algae cultures in thin flat-plate bioreactors (thickness

�2 cm) indicate that: i) Optimal areal biomass production rates are significantly higher than in

traditional ponds or raceways, ii) productivity grows for radiation levels substantially higher than

one sun; saturation emerging, possibly, at intensities of about four suns, and iii) optimal volumetric

and areal production rates as well as culture densities increase as reactor thickness is reduced.

The observations are reproduced within the framework of a simple model, which takes into account

the random motion of cells across the reactor thickness, and the competing effects of two

physiologically significant time scales. These are TR, the time that elapses from the moment a

reaction center has collected the number of photons required for one photosynthetic cycle until it is

available again for exploiting impinging photons (1–10 ms), and TW, an average of the decay time

characteristic of photon loss processes (several ms to several tens of ms). VC 2012 AmericanInstitute of Physics. [doi:10.1063/1.3678009]

I. INTRODUCTION

A. Algal mass production

Micro-algal mass-production has attracted great interest

since the middle of the twentieth century due to the potential

for the production of valuable materials for the aquaculture,

cosmetic, food and pharmaceutical industries, as a potential

source of proteins, as a photosynthetic gas exchanger for

space travel, as a means for waste-water quality improve-

ment, carbon dioxide fixation, and biomass conversion, and

as a renewable energy source through hydrogen and bio-

diesel production.1–13 A constraining factor has been the

lack of efficient large-scale cultivation techniques. Open and

closed systems have been the two major classes of

bioreactors.

Open bioreactors (outdoor ponds or raceways) suffer

from imprecise control over process parameters, and little or

no control over temperature and incident light intensity. Fur-

thermore, CO2 utilization efficiency is low due to lack of tur-

bulent flow and escape of gases from the culture

medium.2,5,6 Contamination by other micro-organisms

causes a reduction in culture growth rate and product quality.

The low output rate per reactor surface area and high produc-

tion costs make this type of bioreactors uneconomical for

most products, except for high-value compounds. No less

important is the observation, known already for many years,

that the optimal areal productivity in open ponds does not

depend on pond depth.14 These limitations have led to a rise

in interest in enclosed bioreactors, which offer better control

over process variables, greater CO2 utilization efficiency,

and reduced contamination.1–13,15–32 This paper focuses on

flat-plate bioreactors.

B. Thin flat-plate bioreactors—Review of experimentalresults

Figure 1 shows a simplified side view of a thin flat-plate

reactor. The height and width of the reactor are dictated by

the needs of the production plant, but it is just a few cm (typ-

ically, 1–2 cm) thick. Light, either natural or artificial, hits

one or both flat sides of the reactor, its intensity falling off as

it propagates through the culture. Air bubbles, fed at the bot-

tom, rise by buoyancy to the top.

The characteristics of biomass-production of Spirulinaplatensis and Nannochloropsis in thin flat-plate reactors

have been studied in Refs. 21–31. The qualitative features of

the experimental observations were the concurrent rise of the

optimal volumetric and areal production rates and of culture

density at optimum, as reactor thickness was reduced from

20 to about 1 cm. For example, in experiments on Spirulinaplatensis, the optimal dry-weight density grew from about 1

to 20–30 kg m�3, the optimal volumetric production rate

grew from about 10 to about 600 dry weight gr m�3 h�1, and

the optimal production rate per unit reactor surface area

grew from 2 to about 6 dry weight gr m�2 h�1. The low val-

ues, obtained for a 20 cm thick reactor, are comparable to

those obtained in open bioreactors. Finally, when the total

photon flux density on both flat sides of a reactor of 0.75 cm

thickness, was varied from 270 to 8000 lmol m�2 s�1 (about

4 times the photosynthetically active part of the solar flux at

midday), the optimal density rose from about 6 to about 30

dry weight kg m�3, and the optimal volumetric production

rate rose from 100 to 1200 dry weight gr m�3 h�1. Results of

the same characteristics were obtained in the cultivation of

Chlorococcum littorale.32 Most important, the optimal pro-

duction rate grew steadily as the photon flux was increased

up to extremely high values. Saturation seemed to begin set-

ting in only at the highest photon flux.26,27,30,32 Photo-a)Electronic address: zarmi@bgu.ac.il.

0021-8979/2012/111(3)/034904/11/$30.00 VC 2012 American Institute of Physics111, 034904-1

JOURNAL OF APPLIED PHYSICS 111, 034904 (2012)

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inhibition and photo-damage did not seem to have significant

effects even at the highest radiation levels.

C. Modeling biomass production in bioreactors

There is ample literature on modeling algal biomass pro-

duction. Common to many works is the assumption that the

algae cells are exposed to continuous radiation under steady

state conditions. This assumption applies, for example, to a

leaf under sunshine, and is a valid approximation when the

optical depth of the reactor (or shallow pond) is large, e.g.,

in the case of plankton in the ocean (the depth of the

“bioreactor” is, probably, tens of meters, if not more), or to

algae a low-density shallow pond of depth of about 10 cm or

more. First, the time it takes a cell to cross the full length

of the optical depth is then much longer than any physio-

logically significant time scale. Second, for the low culture

densities involved, the attenuation of radiation intensity is

slow. During the time a reaction center completes one

photosynthetic cycle, the cell has moved only a very short

distance. Hence, to a very good approximation, it is

exposed to constant radiation intensity during a cycle.

Such models yield what has been observed experimentally,

namely, that the optimal areal production rate is independ-

ent of the optical depth. This was found already years

ago,14 and, in recent years, in the detailed and thorough

analysis presented in Refs. 33–39 for optical depths in the

excess of about 10 cm.

However, the predictions based on the steady state,

continuous-radiation assumption do not reproduce the exper-

imental observations21–32 of biomass production in thin flat-

plate reactors at ultrahigh culture densities reviewed in Sec. I

B. Reactor thicknesses of less than 10 cm were not modeled

in older works, such as Ref. 14. They have been analyzed,

for example, in Ref. 39. The predictions do not conform to

observations. The volumetric and areal production rates are

found to decrease as the optical depth is reduced from around

10 down to 1 cm. Moreover, the range of culture densities,

over which productivity is significant, is the same as for opti-

cal depths in the excess of 10 cm. The use of light intensities

higher than one sun was not considered.

D. Modeling biomass production in thin flat-platereactors

The discrepancy between predictions based on the

assumption of continuous radiation and experimental obser-

vations in thin bioreactors is a consequence of the fact that

the cells are not exposed to continuous radiation, but to short

light flashes. This is a consequence of the combined effect of

the small optical depth (a few cm), the turbulence induced in

the fluid and the high culture density.

The importance of matching the frequency and duration

of light pulses to the physiological time scales that character-

ize the biomass production has been pointed out in Refs.

1–9, 16–32, 41, 42, 45–53. Fluid-dynamical turbulence is

one possible means for generating exposure of cells to inter-

mittent light flashes. Its role in channel flow was modeled

years ago.54,55 Its role in high-density cultures in tubular col-

lectors was analyzed through detailed numerical models in

Refs. 56–61.

This paper presents a simple model for large-scale

biomass production in ultrahigh density cultures cultivated

in thin flat-plate bioreactors. The focus is on the effect of

turbulence induced in the fluid. By construction, the model

is not meant to reproduce the details of cell physiology.

The need to account for such details is avoided by identi-

fying the major factors that control large-scale biomass

production. The model reproduces the qualitative features

of observed biomass production in such bioreactors. One

message of this work is that bioreactors of small optical

depth (be they flat-plate or tubular ones) seem to probe

time scales of significance in the physiology of micro-

organisms.

Turbulence induced by rising air bubbles triggers a mac-

roscopic diffusion process, random motion of the algae cells,

owing to which cells cross the optical depth of the reactor in

a few tens of milliseconds. In addition, at the ultrahigh cul-

ture densities employed, the attenuation of radiation intensity

is rapid, so that the thickness of the layer near the reactor

wall, across which light intensity is high (the “photic zone”),

is �1 mm. Throughout the rest of the reactor, radiation inten-

sities are very low, mostly even under the compensation

point (radiation intensity at which, under steady state condi-

tions, the rate of photosynthesis is equal to the respiration

rate). Consequently, the cells are exposed to intermittent

light intensity, which may be viewed approximately as short-

duration light flashes.

The combined effect of the random motion and the

high density leads to an increase in productivity in two

ways. First, as cells are exposed to short duration light

flashes, saturation of the photosynthetic process (express-

ing the inability of cells to exploit photons, when they

arrive at too high a rate) is deferred from the commonly

observed low radiation levels of a fraction of one

sun1–8,17,30,44,45,61 to several suns.26,27,30,32 In addition, as

cells, which have just collected enough photons for one

FIG. 1. (Color online) Side view of thin flat plate micro algae photo bio-

reactor, illuminated on both sides.

034904-2 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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photosynthetic cycle, are going through the cycle while

wondering through the reactor, others are collecting pho-

tons in or around the photic zone.

The parameters affecting biomass productivity are pre-

sented in Sec. II. The fluid-related parameters control the

(approximately) exponential extinction of light as it propa-

gates through the dense culture, and the macroscopic diffu-

sion of cells throughout the culture. The latter is

characterized by a diffusion coefficient, D (10–50 cm2/s).

The photosynthetic cycle is treated in a “black box”

approach. It is characterized by two physiological time

scales, which are proffered as the main factors controlling

the production process. They are: TR, the “reaction time”

(1–10 ms), and TW, the “maximum charging waiting time”

(several ms to several tens of ms). TR is the time span,

which elapses from the moment a reaction center has col-

lected enough photons for one photosynthetic cycle,

undergoes the cycle, until it becomes available again to

exploiting the continuously impinging photons. TW is an

average of the longest time span, over which photons col-

lected for the next photosynthetic cycle are not lost to

competing processes. In the absence of detailed knowledge

regarding the physiology of cells, which have been accli-

mated to intermittent, short, light flashes, values for TR

and TW are estimated using information obtained in

experiments on dilute cultures exposed to continuous radi-

ation, and on results concerning loss rates of photons by

the photo-system. The model is described in Sec. III.

Results of its predictions, obtained through a numerical

simulation, are presented in Sec. IV A discussion is pre-

sented in Sec. V.

II. SYSTEM PARAMETERS

A. Light attenuation

Once nutritional requirements are satisfied and environ-

mental conditions are controlled, light is the major limiting

factor in biomass productivity. Efficient utilization of high

light intensities increases the biomass yield;21–32 unless the

intensity is so high that photo-inhibition becomes

important.5,42

For dense cultures, light intensity decreases as a func-

tion of the distance from the irradiated reactor surface into

the culture. This decrease may depend on culture density and

on algae type. (Different species may have different attenua-

tion profiles for different spectra.8,17,22,40) It has a predomi-

nant effect on productivity.

In this paper, the irradiance I at a point within the cul-

ture at a distance x from the illuminated surface is assumed

to approximately obey the Lambert-Beers law

I ¼ I0 exp �l qð Þx½ �: (2.1)

I0 is the intensity on the flat wall, l is the attenuation coeffi-

cient, and q is the culture density. As light is attenuated rap-

idly in a high-density culture, its intensity is appreciable

only in a thin layer close to the irradiated reactor surface.

The thickness of this layer (the “photic” zone”) is of the

order of [1/l(q)] (<1 mm, in high culture densities). To a

good approximation, cells receive sufficient illumination for

photosynthesis only in the photic zone.

There are quite a few empirical relations for the q-de-

pendence of l; most popular among then are the linear, Cor-

net, and hyperbolic function models.17,22,33–40 We assume a

linear relation,

lðqÞ ¼ aq: (2.2)

Equations (2.1) and (2.2) are adopted in the absence of more

detailed information. To begin with, existing data on light

attenuation in cultures pertain to densities substantially lower

than those encountered in the flat plate experiments dis-

cussed here. In addition, the spectral width of the photosyn-

thetically active radiation (PAR) is relatively narrow

(400–700 nm), so that Eqs. (2.1) and (2.2) are expected to

provide reasonable approximations. (Nonlinear correlations

are typically obtained when an effective attenuation coeffi-

cient, averaged over a wide range of wavelengths, is com-

puted.) Lastly, the purpose of this paper is to present

qualitative consequences of the model. Hence, the exact

form of attenuation is not essential, as long as attenuation is

significant and grows with cell density.

For the attenuation of light in the PAR range in culture

densities up to 15 (g l-1), a has been found to vary in the

range 0.9–1.5 l gr�1 cm�1.22,40,62 The error bars are of the

order of 10%–30%. In the numerical simulation results pre-

sented here, the linear value found in Ref. 22 has been

employed,

a ¼ 0:92‘g�1cm�1: (2.3)

However, based on the experimental estimates of a, it may

be as large as 2 l gr�1 cm�1. Increasing a will lead to a

reduction in productivity, and a shift in the optimal density

to lower values.

B. Mixing, bubble-induced turbulence, diffusion, andcrossing time

Next to light, a major factor in biomass production,

external to cell physiology, is mixing. It prevents cell sedi-

mentation or attachment to the reactor wall, and helps

enhance gas exchange. However, its major impact on pro-

ductivity is through its effect on the light regime, a conse-

quence of cell random motion. Due to mixing, cells undergo

through light-dark cycles.

The importance of the light regime has been recognized

years ago,1–11 and demonstrated in many

works.15,16,18–32,41–60 Fast alterations between high-intensity

light flashes and dark periods greatly enhance photosynthesis

efficiency, if the light-dark cycle time is chosen appropri-

ately.2,43 For instance, for a given incoming photon flux den-

sity, higher output biomass-production rates were obtained

with more efficient mixing rates in an ultrahigh density cul-

ture of Spirulina platensis, which was cultivated in a flat-

plate bioreactor.24

In flat-plate bioreactors, rising air bubbles generate tur-

bulence in the fluid. Moving in suspension in the culture, the

cells are carried by turbulent eddies. Consequently, cells

034904-3 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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move rapidly in and out of the photic zone, ensuring that as

many cells as possible receive high light intensity for short

periods of time. Reactor thickness, cell concentration, the

degree of turbulence and light intensity, determine the fre-

quency of the cycles of cell motion between the illuminated

and dark parts of the reactor, as well as the fraction of the

cycle during which a cell is exposed to a high photon flux.

Cell motion can be qualitatively described as a random

walk, characterized by a macroscopic diffusion constant, D.

Estimation of D is based on two areas in fluid dynamics

where “turbulence-induced diffusion” occurs. First, the qual-

itative description of randomly moving eddies has been

developed in the context of turbulent fluid motion (see, e.g.,

Ref. 63 and references therein). Denoting the characteristic

eddy size by l and the average eddy velocity by u0, D is

given by

D ffi lu0=2: (2.4)

In the literature on two-phase, or “bubbly,” flow, bubble-

induced turbulence is characterized by a diffusion constant,

given by a phenomenological relation67–74

D ffi kblu0: (2.5)

In Eq. (2.5), l is the characteristic bubble size, u0 is the aver-

age bubble velocity, b is the void fraction (fraction of fluid

volume occupied by bubbles) and k is an empirical coeffi-

cient of the order of 0.6. Assuming typical a bubble size of

l¼ 0.2 cm and bubble velocity, u0, �30–50 cm/s,23–29 one

finds that D¼ 3–5 cm2/s. For l¼ 0.5 cm, D would be

7.5–12.5 cm2/s.

Denoting the thickness of the reactor by L, a measure of

the average crossing time for a cell is

TCROSS ¼ L2=2D� �

: (2.6)

For L¼ 1 cm, and D¼ 10 cm2/s, TCROSS¼ 50 ms. If the reac-

tor is equally illuminated on both sides, so that, due to sym-

metry, the thickness to be crossed is 0.5 cm, TCROSS is

reduced to 12.5 ms.

C. Physiological time scales

When the light-dark frequency (characterized by an av-

erage crossing time of the order of magnitude of TCROSS) is

of the same order of magnitude as physiological time scales,

which control large-scale biomass production, one expects a

high production rate.

This paper does not deal with the details of the photo-

synthetic process. Rather, the process is viewed as a “black

box,” for the following reasons. First, the goal is to offer a

simple, qualitative picture that provides understanding of the

observed characteristics of large-scale algal biomass produc-

tion, rather than a detailed numerical model. In a detailed

description of the photosynthetic process one must take into

account, for example, that a photosystem I needs two pho-

tons to reduce NADPþ, and a photosystem II needs four

photos to split one molecule of water. These processes occur

on extremely short time scales (microseconds). The time

scales that seem to control observed large-scale biomass pro-

duction are several to several tens of milliseconds. Hence, in

comparison with these long time scales, the net chemical

reactions occur “instantaneously.” It, therefore, makes sense

to distinguish between these short time scales and the overall

“production-cycle” time. It is the latter that affects the mac-

roscopic observations.

Estimates of the overall production-cycle time in contin-

uous illumination experiments vary over a wide range of val-

ues, depending on algae species, culture history (i.e.,

acclimation to growth conditions) and type of bioreactor.

The following are estimates based on dilute culture experi-

ments: 1–15 ms,75,76 7–700 ms,3 50 ms–2 s,2 and 130

ms–4 s.16

In the spirit of the approximation proposed above, the

overall production-cycle time may be written as:

TPR ¼ TCOLL þ TR: (2.7)

TCOLL is the time required to collect the necessary number of

photons (8, if quantum efficiency is accounted for). TCOLL

depends inversely on I, the radiation intensity, to which cells

are exposed. TR, is an effective “reaction” time. It is the time

from the moment a reaction center has absorbed all the

required photons, goes through the (fast!) chemical reactions,

until it is available again for exploiting impinging photons.

There are many works that have collected information

regarding the exposure of algal cells to short duration light

pulses. However, a picture of the characteristics of cells that

have been acclimated to pulse exposure does not exist. It is

still not even clear yet whether the cells are acclimated to the

high irradiance during a pulse or to the time-averaged inten-

sity. Thus, estimates of the time scales offered in the follow-

ing are, at best, rough estimates of yet unknown quantities.

D. Estimating TR

That the time scale, TR, may provide a reasonable quali-

tative description of the production cycle is inferred from in-

formation, collected many years ago, in experiments, in

which dilute algal cultures were exposed to continuous

radiation.1–8,18,30,44,45,61 In plots of the P-E curve (photosyn-

thetic activity rate, measured in terms of, for example, oxy-

gen generation) the production rate increases linearly at low

radiation, and reaches saturation at 20%–25% of one sun in-

tensity. (In the PAR, one sun represents a flux of 2000 lmol

photons m�2 s�1). These observations are readily understood

in terms of the two time scales in Eq. (2.7). At low radiation

levels, photon collection takes a long time, so that TCOLL »

TR, and, hence, TPR � TCOLL. As a result, the production rate

grows linearly with radiation intensity, I. At high irradiance

levels, the photon collection time has becomes so short that

it can be neglected in Eq. (2.7). Equation (2.7) is then

reduced to TPR � TR. Saturation implies that the TR has a

constant average value. The simplest interpretation is that,

although the actual photosynthetic process is very fast, the

reaction center becomes available again for exploiting im-

pinging photons after a much longer time, TR.

The time required to collect the number of photons

needed for a single photosynthetic cycle varies. It depends

034904-4 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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on light intensity and on the chlorophyll antenna absorption

cross-section area. For photosystems I and II, the minimum

achievable chlorophyll antenna absorption cross-section area

for capturing photons is approximately 1–5 nm2.64 This esti-

mate was based on the product of the minimum number of

chlorophyll molecules per photo system, 95 and 37 chloro-

phyll molecules for photo systems I and II, respectively,65

and the absorption cross section of a chlorophyll molecule.41

Assuming the above quoted cross-section area, under 2000

lmol photons m�2 s�1 radiation (PAR in one sun), the num-

ber of quanta absorbed per reaction center per second is of

the order of (1–6)� 103 photons/s.64 To estimate TR, we use

the value of light intensity, at which saturation of the P-E

curve sets in (approx. 20%–25% of one sun). Collecting 8 (6

plus accommodation for quantum efficiency) photons at this

rate yields an upper bound for TR of 3–30 ms. Hence, values

of several ms to a few tens of ms have been used in the pres-

ent study. This range of values is in agreement with the com-

mon understanding that whole photosynthetic process takes

several milliseconds (for a review see Ref. 77).

E. What happens under very low irradiance—Maximum charging waiting time

Experiments under continuous illumination indicate that

the production rate vanishes at very low, but non-vanishing,

radiation levels. The light intensity, at which this happens,

the compensation point, is not known accurately, as there is

ample scatter in production-rate plots at low photon fluxes.

The compensation point seems to vary, depending on growth

conditions and algal species.66 The interpretation given is

that, at this low irradiance, the photosynthetic and respiration

rates are equal, so that the net biomass production vanishes.

However, a totally different scenario emerges in thin

flat-plate reactors as well as in small-diameter tubular reac-

tors at ultrahigh culture densities. In such systems, the cells

wonder most of the time in the bulk of the reactor volume,

where the radiation intensity is extremely low; so low that it

may be under the compensation point. For example, for cul-

ture density of 30 dry-weight gr/l, Eqs. (2.1)–(2.3) yield that

at a distance of 2 mm from the illuminated reactor wall, the

irradiance has fallen to 0.4% of the incoming intensity.

Based of a flux of (1–6)� 103 photons/s per reaction cen-

ter,64 this corresponds to 4–24 photons/s per reaction center.

Hence, collection of, say, 8 photons takes anywhere between

300 ms to 2 s.

Let us assume that a reaction center has already col-

lected, say, 3 of the required photons. Will it “wait” for the

remaining 3 (or 5, if quantum efficiency is accounted for)

indefinitely? Clearly not. What will happen to the photons al-

ready collected? Obviously, unless the remaining photons

are collected within some limited time, the collected ones

will be lost, for example, by deexcitation of electronic levels

or recombination processes, the released energy reradiated or

converted into heat. In summary, if photon collection is too

slow, other processes may compete for the same photons.

The literature on photon energy losses from, e.g., photo-

system II, indicates that there are loss processes that take

nanoseconds, picoseconds, microseconds, millisecond, sec-

onds, and even tens of seconds (for a review, see Ref. 78).

The fastest loss processes occur over time spans much

shorter than the characteristic time span of the photosyn-

thetic process (which is of the order of several ms77); hence,

may be lumped into a factor that determines the efficiency of

the photosynthetic process. The very slow processes take

seconds to tens of seconds. Over such a long of time, a ran-

domly moving cell will have visited the photic zone quite a

few times, and collected the required photons. Hence, it

seems that only loss processes occurring over time scales of

milliseconds will have a predominant effect on large-scale

biomass production. An example for such a process is

delayed chlorophyll a fluorescence.79,80

The time scales for various loss processes may be differ-

ent, and may, perhaps, also have a statistical nature. As the

detailed theory for this scenario does not exist, we propose

to represent the time scales for photon losses by one time

scale, TW, the “maximum photon charging waiting time.” It

represents the (average) time over which a reaction center

does not lose a partial number of photons to other processes.

To estimate TW, would require detailed modeling that is out-

side the scope of this paper. However, a rough estimate of

TW may be obtained as follows.

At irradiance levels, I, above the compensation point,

the net production rate is positive and linear in I. As I is

increased, the photon collection time, TCOLL, becomes

smaller. Hence, the value of TCOLL at the compensation point

provides an estimate of the order of magnitude of TW. To ac-

complish one photosynthetic cycle, the cell needs 8 photons

when quantum efficiency is accounted for. Consequently, the

lower bound for TW is

TW � TCOLL At compensation pointð Þ

¼ 8 photons

Compensation point photon flux on reaction center:

(2.8)

For the green alga Chlorella pyrenoidosa the compensation

point irradiance is about 6 lmol photons m�2 s�l under

light-flash illumination, and about 9 lmol photons m�2 s�l

under continuous LED light.20 Under full sun irradiance

(a PAR flux of 2000 lmol photons m�2 s�l), the estimated

flux is (1–6)� 103 photons/s per reaction center.64 A flux of

6 lmol photons m�2 s�l,20 therefore, corresponds to a flux of

(3–20) photons/s per reaction center. If, as some experiments

seem to indicate, the compensation point is as high as 2% of

one sun, this corresponds to 100 photons/s per reaction cen-

ter. For a photon flux on a reaction center of 3–100 photons/

s, this corresponds to TW� 80 ms–3 s.

The important role played by TW may be seen as fol-

lows. In ultrahigh density cultures, light intensity decreases

exponentially into the depth of the culture. Cells in the inte-

rior of the bioreactor (the majority of cells) are exposed to

extremely low radiation. Hence, these cells require a long

time to collect photons. If their migration through the dark

culture takes longer than TW, then they may lose photons al-

ready collected. Only if they manage to reach the (thin)

photic zone within a time shorter than TW, is the probability

to collect the required number of photons in a sufficiently

034904-5 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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short time high. This explains the advantage of thin reactors.

If the reactor is thick, so that the crossing time [Eq. (2.6)] is

longer than TW, then a sizable fraction of the cells does not

participate in the production process. Thus, TW plays a cru-

cial role in reducing biomass production at high culture den-

sities when reactor thickness is increased.

To attain a significant increase in productivity, the aver-

age crossing time of the reactor width, TCROSS, ought to

become comparable to or shorter than TW. A doubly efficient

way to achieve this goal is exposure of both sides of the reac-

tor to light, as in Fig. 1. For example, in symmetric illumina-

tion, productivity is increased due to the higher photon flux,

as well as due to overcoming the limitation enforced by the

effect of TW: The distance to be crossed by cells is halved,

resulting in reduction of TCROSS by a factor of 4.

III. THE MODEL

The assumptions made in order to ensure simplicity of

the model made in the model are:

1) The culture has been operating for a sufficiently long

time, so that the acclimation process need not be

accounted for.

2) Pigment content is constant.

3) Denoting the position of a cell by x, it executes a lateral

random walk with reflective boundaries over the interval

0� x� L. The random walk is characterized by a diffu-

sion coefficient D.

4) Irradiance attenuation through the culture is determined

by Eqs. (2.1)–(2.3).

5) The number of photons that need to be collected for one

photosynthetic cycle is set at 8 (6þ 2 for quantum

efficiency).

6) The calculation assumes illumination on both sides of the

reactor.

As the cell moves randomly through the reactor, its reac-

tion centers are continuously hit by the photon flux. Owing

to the random motion of the cell, the irradiance, to which

reaction centers are exposed, varies randomly. Once a reac-

tion center is available for processing photons, the number of

photons it collects in a given time span is

n ¼ a

ðt2

t1

I0e�lx tð Þdt=E: (3.1)

where a is the effective area of a reaction center, and E is the

(average) energy of a single photon. If the required number

of photons is not collected within a time span shorter than

TW, then the photons already collected are lost. If 8 photons

are collected within TCOLL� TW, the reaction center enters a

“reaction” phase, during which it does not respond to pho-

tons that continue to hit it. Once this phase is over (its dura-

tion is TR, the “reaction time”), biomass is produced, and the

reaction center returns to photon collection. Cell density is

assumed to be constant in space (due to mixing), and in time

(corresponding to continuous harvesting).

The simulation was written in MATLAB. Time was di-

vided into short time intervals. The length of the time inter-

val, Dt, was selected so that it was appreciably shorter than

all relevant time scales and that statistically significant results

were obtained, in particular, that the statistical samples were

sufficiently large so that smooth steady-state curves were

obtained. The random motion of the cell was generated by a

normalized random number generator. The step, Dx, was gen-

erated by a normal distribution with zero mean, and standard

deviation

r ¼ffiffiffiffiffiffiffiffiffiffiffi2DDtp

(3.2)

T, the total physical operation time of the reactor was chosen

sufficiently long, again, to ensure smooth and statistically

significant results. The main factor in choosing both Dt and

T was the need to ensure a sizable sample also at the highest

densities we considered, for which the rate of production-

cycle completion is low.

The simulation takes one cell and lets it move randomly

through the reactor throughout the operation time, T. Every

time a production cycle is completed by a reaction center, a

counter is increased by 1. The result of one run is N, the total

number of production cycles per reaction center. The flow

chart of the numerical simulation is shown in Appendix A.

A. Calculated quantity - J

The volumetric production rate, R (gr/l s�1) is given by

R ¼ g N=nPRð ÞnCentersnCellsqT

� eq ¼ gnCentersnCells

nPRTJ � eq:

J ¼ qNð Þ (3.3)

In Eq. (3.3), g (gr) is the (average) amount of biomass gener-

ated every nPR photosynthesis cycles, nCenters is the (average)

number of reaction centers per cell, nCells is the (average)

number of cells per 1 gr of dry-weight biomass, q is the dry-

weight density (gr/l), N is the total number of production

cycles per reaction center, and T is the total operation time

of the bioreactor. The subtracted term represents the loss of

biomass owing to respiration, where e (s�1) is the loss rate.

The coefficients multiplying J in Eq. (3.3) amount to a

constant, which may depend on algae species, acclimation

process and other factors. Therefore, in Figs. 2–8 we show J[its units are: (Cycles per reaction center)� (gr/l)] versus q,as a measure of the volumetric production rate. The respira-

tion term has not been included, as it is small, and affects the

production rate only slightly at low densities, and at very

high densities.

B. Very long TW

We first consider the unrealistic limit of TW ! 1, i.e.,

there are no photon loss processes. The purpose is twofold.

First, this limit serves as a test of the credibility of our nu-

merical simulation, because, in two limits, of very low and

of unrealistically high culture densities, the expected depend-

ence of productivity on culture density can be derived inde-

pendently, using simple arguments. Second, the predictions

in this limit accentuate the crucial role played by photon loss

processes, the effect of which is qualitatively described in

terms of the single time scale, TW, when the latter is assigned

034904-6 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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values in the range of milliseconds, as is demonstrated in

Sec. IV.

In the limit of extremely high cell densities, the photic

zone is so narrow that the time spent by cells within that

zone does not allow them to collect the required number of

photons in a single visit. Based on Eqs. (2.1) and (2.2), the

thickness of the photic zone is of the order of

lph � 1= aqð Þ: (3.4)

Hence, the time spent in the in the photic zone is of the order

of

Tph � lph

� �2= 2Dð Þ: (3.5)

For a cell dry-weight density of 50 gr/l, and D¼ 10 cm2 s�1,

one finds Tph � 0.02 ms. Even at a full sun intensity, corre-

sponding to a flux of the order of (1–6)� 103 photons/s (Ref.

64) per reaction center, this does not allow a center to collect

even a single photon. Most of the remainder of the reactor

volume is dark. So dark that irradiance levels are lower even

than the compensation point. As there is no time limitation on

photon collection, a cell may move back and forth for as long

a time as needed until its reaction centers have collected the

required number of photons. Consequently, photon collection

is determined by the average radiation intensity, give by

�I ¼ aI0 1� e�aqL� �

= aqLð Þ� �

: (3.6)

To collect the required number of n photons, n, the col-

lection time, TCOLL, is then given by

TCOLL ¼n�I¼ naqL

aI0 1� e�aqLð Þ : (3.7)

Equation (2.7) for the total production-cycle time, there-

fore, yields

TPR ¼ TR þ TCOLL ¼ TR þnaqL

aI0 1� e�aqLð Þ : (3.8)

The volumetric production rate, R, is proportional to

R / qTPR¼ q

TR þ naqLaI0 1�e�aqLð Þ

!q!1

aI0

naL: (3.9)

Namely, at extremely high densities, R tends to a constant,

which is inversely proportional to reactor thickness, and is

independent of TR.

At very low densities, the radiation intensity throughout

the reactor is approximately uniform. Equation (3.9) is then

replaced by a linear dependence on culture density, given by

R !q!0

qTR þ n= aI0ð Þ½ � : (3.10)

Hence, as expected, low values of TR yield higher pro-

duction rates.

Figure 2 demonstrates that these expectations are indeed

born out for TW¼ 3000 s, D¼ 10 cm2/s and TR¼ 0.1 and

1 ms. The quantity J [see Eq. (3.3)] is proportional to the vol-

umetric production rate. At ultra high densities, the asymp-

totic limit of Eq. (3.9) is attained; the volumetric production

rate is inversely proportional to reactor thickness, so that the

areal production rate does not depend on the thickness, and

becomes independent of TR. At low densities, the lower value

of TR yields higher production rates, as predicted by Eq.

(3.10).

For intermediate densities, neither approximation

described above is valid. However, it is expected that the

density dependence of J will vary smoothly from the low- to

the high-density prediction, which it does. Calculations per-

formed for other values of TR and D yield similar results.

IV. RESULTS

To obtain statistically significant results for all densities

considered (including the highest values) the physical run-

ning time of the production process was fixed at T¼ 2400 s.

For Figs. 2–7, the calculation assumes illumination on both

sides of the reactor with a photon flux of 104 photons/s hit-

ting a reaction center (roughly, the equivalent of the PAR of

1.5 suns64).

When TW, the maximum charging waiting time, has re-

alistic values (several ms to a few tens of ms), at high culture

densities, a cell must reach the photic zone (thin layer of

high radiation intensity near an illuminated wall) by the time

it has completed the previous production cycle, and is ready

again to exploit impinging photons. This means that it is ad-

vantageous to have an average crossing time, TCROSS that is

comparable to the “reaction” time, TR. However, before one

reaches reactor thicknesses sufficiently small to attain this

goal, TCROSS has to be comparable to, or shorter than TW.

Otherwise, many cells will collect only a fraction of the

required 8 photons, and then lose them because they will be

wondering for too much time in the dark part of the reactor

volume. Namely, one expects significant improvement in

productivity when

Tacross� L2= 2Dð Þ� �

�TW : (4.1)

FIG. 2. J of Eq. (3.3) vs cell culture density for different reactor thicknesses.

TW¼ 3000 s, D¼ 10 cm2/s, TR¼ 0.1 ms (full symbols), TR¼ 1 ms (open

symbols).

034904-7 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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Based on experiments, estimates of TW may vary between

about 80 ms and 3 s (see Sec. II E). As accurate information

regarding values of the different parameters does not exist,

we performed simulations for several combinations of pa-

rameters, to see how the motion of cells across the bioreactor

affects productivity. The results are presented for progres-

sively smaller values of TW. Figures 3–7 show J of Eq. (3.3),

as a representative for the volumetric production rate for two

values of TR: 1 and 15 ms. In Figs. 3–5, values of the diffu-

sion coefficient D and of the photon loss time scale, TW,

were chosen, for which one has TCROSS�TW, whereas Figs.

6 and 7 correspond to the case TCROSS> TW. These figures

indicate the following unique features of biomass production

in bioreactors with a small optical depth:

1) The interplay among the three relevant time scales deter-

mines the characteristics of the production process.

2) TW, the time scale for photon losses, plays an important

role. Compared to the unrealistic case of no loss mecha-

nisms (corresponding to extremely large TW), presented

in Fig. 2, when TW is not extremely long, the overall pro-

duction rate is decreased significantly.

3) As cells may lose photons if the required number (8) is

not collected within TW, the reduction of the production

rate at the highest densities leads to the emergence of an

optimal culture density.

4) As expected, when TR (the time a reaction center is not

available for processing impinging photons) is increased,

the production rate is reduced.

The features observed in the experiments reported in

Refs. 22 and 33 are:

1) The optimal density increases as reactor thickness is

reduced.

2) The optimal volumetric production rate grows faster than

(1/L) as L is reduced. Consequently, the optimal produc-

tion rate per unit area grows as reactor thickness is

reduced.

FIG. 3. J of Eq. (3.3) vs culture density for different reactor thicknesses.

TW¼ 80 ms, D¼ 25 cm2/s, TR¼ 15 ms (open symbols), TR¼ 1 ms (full

symbols).

FIG. 4. J of Eq. (3.3) vs culture density for different reactor thicknesses.

TW¼ 80 ms, D¼ 40 cm2/s, TR¼ 15 ms (open symbols), TR¼ 1 ms (full

symbols).

FIG. 5. J of Eq. (3.3) vs culture density for different reactor thicknesses.

TW¼ 50 ms, D¼ 40 cm2/s, TR¼ 15 ms (open symbols), TR¼ 1 ms (full

symbols).

FIG. 6. J of Eq. (3.3) vs culture density for different reactor thicknesses.

TW¼ 15 ms, D¼ 10 cm2/s, TR¼ 15 ms (open symbols), TR¼ 1 ms (full

symbols).

034904-8 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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Both are obtained when TCROSS�TW (Figs. 3–5).

However, these two characteristics are not reproduced

in Figs. 6 and 7, which correspond to the case TCROSS> TW.

The optimal density becomes, essentially, independent of re-

actor thickness; the volumetric production rate is roughly

inversely proportional to reactor thickness L, so that, to a

good approximation, the production rate per unit area

becomes independent of reactor thickness. As TW is reduced,

so does the production rate. The effect is particularly dra-

matic in Fig. 7. Although the diffusion coefficient, D, has a

high value (40 cm2/s), so that the crossing time for a 1 cm

thick reactor is of the order of 10 ms [see Eq. (2.6)], TCROSS

» TW. As a result, the production rates and the optimal den-

sity are all are very low. These results are expected, because

when TCROSS>TW, most cells do reach the photic zone in

sufficiently short a time span in order to collect photons.

A. No saturation at radiation levels far above one sun

In continuous radiation experiments on dilute cultures, it

is invariably found that the photosynthetic rate reaches satu-

ration at radiation levels of the order of 20%–30% of one

sun. It is also quite well understood that saturation is indica-

tive of the existence of TR, the “reaction time scale,” which

does not represent the time scales characteristic of the elec-

tronic energy conversion process (order of microseconds)

but a physiological time scale that characterizes the rate at

which a reaction center completes a photosynthesis cycle

until it is ready again to exploit impinging photons. If all

cells are exposed to high continuous irradiance, say equiva-

lent to one sun, then time is divided into segments of length

TR. At the beginning of each segment photons are collected

in a very short time span, of the order of microseconds or

even less. While the production cycle takes place, the reac-

tion center is not available to processing impinging photons,

until the time TR has passed. This is saturation. In a flat-plate

reactor, saturation is expected to be deferred to higher radia-

tion levels because cells spend a very short time in the photic

zone, so that they are exposed to light flashes, rather than to

continuous irradiation.

The results, shown in Fig. 8, bear out this expectation.

The reactor thickness was L¼ 1 cm, and the “reaction time”

was TR¼ 15 ms. The PAR intensities used were 6, 10, and

30 photons/ms per reaction center, and applied only on one

side of the reactor. (The lowest of the three is roughly equiv-

alent to the PAR of one sun.64) The volumetric production

rate grows with radiation intensity, not exhibiting any signs

of saturation; the optimal rate is still proportional to the radi-

ation intensity.

Figure 8 demonstrates the detrimental effect of the

relative sizes of TW and TR on biomass production. For

TW¼ 50 ms, one has TCROSS � 10 ms « TW, whereas for

TW¼ 5 ms, one has TCROSS � 50 ms » TW. The reduction in

productivity in the latter case is the result of the fact that a

significant fraction of the cells does not reach the photic

zone before TW is over. It migrates through the dark part of

the culture, collecting photons very slowly, and losing them.

V. DISCUSSION

The model proposed here provides a simple description

of biomass production of unicellular algae in high-density

cultures in flat-plate bioreactors. The assumptions on which

the model is based are of two categories: fluid characteristics

and cell characteristics. The motion of the cells across the re-

actor is a random walk, generated owing to bubble-induced

turbulence. The latter is caused by the erratic motion of ris-

ing air bubbles. The attenuation of light through the fluid is

assumed to follow the exponential Lambert-Beers law. The

photosynthetic process is treated as a “black box” character-

ized by two physiological time scales. When reactor thick-

ness is reduced to the order of 1 cm, the time scale for

crossing the reactor thickness by the randomly moving cells

becomes comparable to the physiological time scales. The

interplay among these time scales generates the observed

phenomena. These include the increase of both optimalFIG. 7. J of Eq. (3.2) vs culture density for different reactor thicknesses.

TW¼ 1 ms, D¼ 40 cm2/s, TR¼ 15 ms.

FIG. 8. J of Eq. (3.3) vs culture density for different radiation intensities.

L¼ 1 cm, TR¼ 15 ms; reactor irradiated on one side. TW¼ 50 ms, D¼ 40

cm2/s (full symbols); TW¼ 5 ms, D¼ 10 cm2/s (open symbols).

034904-9 A. K. Gebremariam and Y. Zarmi J. Appl. Phys. 111, 034904 (2012)

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density and of optimal production rate per unit area with

reduction of reactor thickness. The most important character-

istic is the linear increase of production rates with radiation

intensities significantly exceeding one sun, showing no sign

of saturation.

The model is sufficiently flexible to allow for parame-

ter values, which may correspond to algae species different

from the ones for which flat-plate experiments have been

performed, or for new species, the parameters of which

have been affected by genetic engineering. A conclusion

of practical relevance is the advantage in illuminating the

reactor on both sides, as in Fig. 1. Productivity is

increased not only due to the higher photon flux, but also

due to overcoming the limitation enforced by the effect of

TW. For example, in symmetric illumination, the distance

to be crossed by cells is halved, resulting in reduction of

TCROSS by a factor of 4, facilitating implementation of the

requirement of Eq. (4.1).

Very little is known about all the relevant parameters.

The fluid-related processes are: the turbulence-induced diffu-

sion, characterized by the coefficient, D, and radiation

attenuation through the culture. The theoretical basis for the

evaluation of D is based on approximations and on phenome-

nological assumptions. The extent, to which the Lambert-

Beer exponential law provides an accurate description of the

attenuation process, is not known. Assuming Eq. (2.1) for

light attenuation, the coefficient, l, is not known. It is not

known whether l is indeed linear in density, as assumed in

Eq. (2.2). If a linear relation does hold, the value of the coef-

ficient a in Eq. (2.2) is not known precisely. The imprecision

in these factors may be the reason why the optimal densities

obtained in our results are higher than observed densities. In

this paper, a � 1 l gr�1 cm�1 has been used. Values quoted

in the literature for densities appreciably lower from the ones

covered in the flat-plate experiments varied in the range of

1–1.5 l gr�1 cm�1, with rather large standard deviations.

Choosing a � 2 l gr�1 cm�1 would place all relevant den-

sities in our calculations comfortably within the range of

experimentally observed values.

The physiological time scales: the “reaction time,” TR,

and the “maximum-charging waiting time,” TW are also

poorly known. The existence of the “reaction” time scale,

TR, may be inferred with some degree of certitude from

the fact that the photosynthetic rate under continuous illu-

mination of dilute cultures, reaches saturation when the

radiation flux is 20%–30% of one sun. However, whether

vanishing of the biomass production rate at very low (but,

possibly, non-vanishing) radiation intensities can be inter-

preted in terms of another time scale, the “maximum-

charging waiting time,” TW, is definitely an open question.

Even if this turns out to be a reasonable description, the

value of TW is not known at all.

Finally, the fact that the simple model proposed in this

paper has the capability to reproduce the qualitative features

of experimental results calls for a whole series of experi-

ments aimed at determining the relevant parameters for dif-

ferent algae species and (possible) variation in these

parameters when the light regime is changed from continu-

ous irradiation to light flashes.

APPENDIX A

Flow chart of simulation model described in Sec. III.

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