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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 intraditional 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 (110 ms), and TW, an average of the decay time

    characteristic of photon loss processes (several ms to several tens of ms). VC 2012 American

    Institute 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.113 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.113,1532 This paper focuses on

    flat-plate bioreactors.

    B. Thin flat-plate bioreactorsReview 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, 12 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 Spirulina

    platensis and Nannochloropsis in thin flat-plate reactors

    have been studied in Refs. 2131. 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 Spirulina

    platensis, the optimal dry-weight density grew from about 1

    to 2030 kg m3

    , the optimal volumetric production rate

    grew from about 10 to about 600 dry weight gr m3 h1, and

    the optimal production rate per unit reactor surface area

    grew from 2 to about 6 dry weight gr m2 h1. The low val-

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

    those obtained in open bioreactors. Finally, when the totalphoton flux density on both flat sides of a reactor of 0.75 cm

    thickness, was varied from 270 to 8000 lmol m2 s1 (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 m3, and the optimal volumetric production

    rate rose from 100 to 1200 dry weight gr m3 h1. 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: [email protected].

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

    JOURNAL OF APPLIED PHYSICS 111, 034904 (2012)

    http://dx.doi.org/10.1063/1.3678009http://dx.doi.org/10.1063/1.3678009http://-/?-http://dx.doi.org/10.1063/1.3678009http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3678009http://-/?-http://dx.doi.org/10.1063/1.3678009http://dx.doi.org/10.1063/1.3678009
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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 yearsago,14 and, in recent years, in the detailed and thorough

    analysis presented in Refs. 3339 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 observations2132 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 theassumption 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.

    19, 1632, 41, 42, 4553. 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. 5661.

    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 r adiation levels of a fraction of one

    sun18,17,30,44,45,61 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 ischaracterized by a diffusion coefficient, D (1050 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

    (110 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 tocompeting processes. In the absence of detailed knowledge

    regarding the physiology of cells, which have been accli-

    mated to intermittent, short, light flashes, values for TRand 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;2132 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 I0 exp l q x : (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)] (

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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 randomwalk, 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 relation6774

    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

    l 0.2 cm and bubble velocity, u0, %3050 cm/s,2329 onefinds that D 35 cm2/s. For l 0.5 cm, D would be7.512.5 cm2/s.

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

    the average crossing time for a cell is

    TCROSS L2=2D

    : (2.6)

    ForL 1 cm, and D 10 cm2/s, TCROSS 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 NADP, and a photosystem II needs fourphotos 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: 115 ms,75,76 7700 ms,3 50 ms2 s,2 and 130

    ms4 s.16

    In the spirit of the approximation proposed above, the

    overall production-cycle time may be written as:

    TPR TCOLL TR: (2.7)

    TCOLL is the time required to collect the necessary number of

    photons (8, if quantum efficiency is accounted for). TCOLLdepends 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.18,18,30,44,45,61 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 understoodin 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 rategrows 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 aconstant 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 15 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 m2

    s1

    radiation (PAR in one sun), the num-

    ber of quanta absorbed per reaction center per second is of

    the order of (16) 103 photons/s.64 To estimate TR, we usethe 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 330 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 irradianceMaximum 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 (16) 103 photons/s per reaction cen-ter,64 this corresponds to 424 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 afew 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

    TWmay 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 forTW is

    TW ! TCOLL At compensation point

    8 photonsCompensation point photon flux on reaction center

    :

    (2.8)

    For the green alga Chlorella pyrenoidosa the compensation

    point irradiance is about 6 lmol photons m2 sl under

    light-flash illumination, and about 9 lmol photons m2 sl

    under continuous LED light.20 Under full sun irradiance

    (a PAR flux of 2000 lmol photons m2 sl), the estimated

    flux is (16) 103 photons/s per reaction center.64 A flux of6 lmol photons m2 sl,20 therefore, corresponds to a flux of

    (320) photons/s per reaction center. If, as some experiments

    seem to indicate, the compensation point is as high as 2% ofone sun, this corresponds to 100 photons/s per reaction cen-

    ter. For a photon flux on a reaction center of 3100 photons/

    s, this corresponds to TW! 80 ms3 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 tobecome 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 ofTCROSS 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 longtime, 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

    0xL. 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 (6 2 for quantumefficiency).

    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 at2

    t1

    I0elx t dt=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 TCOLLTW, the reaction center enters areaction 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 ffiffiffiffiffiffiffiffiffiffiffi

    2DDtp

    (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 flowchart of the numerical simulation is shown in Appendix A.

    A. Calculated quantity - J

    The volumetric production rate, R (gr/l s1) is given by

    R g N=nPR nCentersnCellsqT

    eq gnCentersnCellsnPRT

    J eq:J qN (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 (s1) 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. 28 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

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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= aq : (3.4)

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

    of

    Tph % lph 2= 2D : (3.5)For a cell dry-weight density of 50 gr/l, and D 10 cm2 s1,one finds Tph % 0.02 ms. Even at a full sun intensity, corre-sponding to a flux of the order of (16) 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 aI0 1 eaqL

    = aqL : (3.6)To collect the required number of n photons, n, the col-

    lection time, TCOLL, is then given by

    TCOLL nI naqL

    aI0 1 eaqL : (3.7)

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

    fore, yields

    TPR

    TR

    TCOLL

    TR

    naqL

    aI0 1 eaqL

    : (3.8)

    The volumetric production rate, R, is proportional to

    R / qTPR

    qTR naqLaI0 1eaqL

    q1

    aI0

    naL: (3.9)

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

    which is inversely proportional to reactor thickness, and is

    independentofTR.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 q0

    q

    TR n= aI0 : (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 TW 3000 s, D 10 cm2/s and TR 0.1 and1 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 ofTR. 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 ofJ will vary smoothly from the low- to

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

    formed for other values ofTR 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 T 2400 s.For Figs. 27, 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= 2D

    TW: (4.1)

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

    TW 3000 s, D 10 cm2/s, TR 0.1 ms (full symbols), TR 1 ms (opensymbols).

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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 37 show Jof Eq. (3.3),

    as a representative for the volumetric production rate for two

    values of TR: 1 and 15 ms. In Figs. 35, values of the diffu-

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

    were chosen, for which one has TCROSSTW, whereas Figs.6 and 7 correspond to the case TCROSS> TW. These figures

    indicate the following unique features of biomass productionin 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), presentedin 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.

    TW 80 ms, D 25 cm2/s, TR 15 ms (open symbols), TR 1 ms (fullsymbols).

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

    TW

    80 ms, D

    40 cm

    2/s, TR

    15 ms (open symbols), TR

    1 ms (full

    symbols).

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

    TW 50 ms, D 40 cm2/s, TR 15 ms (open symbols), TR1 ms (fullsymbols).

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

    TW

    15 ms, D

    10 cm

    2/s, TR

    15 ms (open symbols), TR

    1 ms (full

    symbols).

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    Both are obtained when TCROSSTW (Figs. 35).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 insufficiently 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 oreven 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 L 1 cm, and the reaction timewas TR 15 ms. The PAR intensities used were 6, 10, and30 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. ForTW 50 ms, one has TCROSS % 10 ms TW, whereas forTW 5 ms, one has TCROSS % 50 ms TW. The reduction inproductivity 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 whichthe 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 optimal

    FIG. 7. J of Eq. (3.2) vs culture density for different reactor thicknesses.

    TW 1 ms, D 40 cm2

    /s, TR 15 ms.

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

    L 1 cm, TR 15 ms; reactor irradiated on one side. TW 50 ms, D 40cm

    2/s (full symbols); TW 5 ms, D 10 cm2/s (open symbols).

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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 beenperformed, 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 radiationattenuation through the culture. The theoretical basis for the

    evaluation ofD 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 gr1 cm1 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

    11.5 l gr1 cm1, with rather large standard deviations.

    Choosing a % 2 l gr1 cm1 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, whethervanishing 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.

    1J. Myers, J. Gen. Physiol. 29, 429 (1946).

    2B. Kok, Experiments on photosynthesis by Chlorella in flashing light, in

    Algal Cultur., edited by J. S. Burlew (Carnegie Inst., Washington, 1953),

    pp. 6375.3J. N. Phillips and J. Myers, Plant Phsyiol. 29, 152 (1954).4

    H. Markel, Modeling of algal produc tion systems, in Algal Biomass Pro-

    duction and Use, edited by G. Shelef and C. J. Soeder (Elsevier/North Hol-

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    M. A. Borowitzka, J. Biotechnol. 70, 313 (1999).7

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    S. J. Pirt, Y. K. Lee, M. R. Walach, M. W. Pirt, H. H. M. Balyuzi, and

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    J. U. Grobbelaar, J. Appl. Phycol. 6, 331 (1994).19

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