Abstract
This paper reports the design of a vascular structure for an artificial tree, also known as an evaporation-driven porous substrate bioreactor (PSBR), for efficient biofuel production using microalgae. This system consists of multiple vertical ribs, each of which is made of porous membrane and grows algal cells on its surface as a biofilm. Nutrient medium flow through the reactor is driven by evaporation at the terminal end of the porous membrane, and nutrients are delivered from the porous membrane to the cells by diffusion. Flow through the membrane was modeled as a function of the physico-chemical and morphological properties of the membrane, as well as the environmental parameters governing evaporation. It was determined that under typical operating conditions, the evaporative flux from the evaporator region ranged from about 14 to 66 mg/m\(^2\) s. Moreover, there was a membrane pore radius that maximized nutrient medium flow as a result of the competition between capillary, viscous, and gravitational forces. For the range of evaporative fluxes observed in this study, this pore radius was about 10 \(\upmu \)m. Furthermore, a design example is provided for artificial trees made of three different commercially available membrane materials. A design methodology was demonstrated for maximizing photosynthetic productivity by tuning the evaporation-driven flow rate to ensure sufficient nutrient delivery to cells without incurring large evaporative loss rates. It was observed that both the growth rate and the evaporation-driven nutrient delivery rate were directly related to the irradiance in outdoor artificial trees, which provides a passive and efficient nutrient delivery mechanism. It is expected that the design principles along with the physical models governing the fluid flow in these vascular structures will aid researchers in developing novel applications for artificial trees.
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Abbreviations
- \(C\) :
-
Constant of proportionality for calculating nutrient delivery length
- \(D_\mathrm{w,a}\) :
-
Diffusivity of water vapor in air (m\(^2\)/s)
- \(G\) :
-
Irradiance (W/m\(^2\))
- \(Gr\) :
-
Grashof number
- \(h\) :
-
Height (m)
- \(h_\mathrm{c}\) :
-
Convection heat transfer coefficient (W/m\(^2\) K)
- \(h_\mathrm{fg}\) :
-
Heat of vaporization (J/kg)
- \(k\) :
-
Hydraulic permeability (m\(^2\))
- \(k_\omega \) :
-
Mass transfer coefficient (kg/m\(^2\) s)
- \(\dot{m}^{\prime }\) :
-
Mass flow rate per unit length (kg/s m)
- \(\dot{m_\mathrm{e}}^{\prime \prime }\) :
-
Evaporative flux (kg/m\(^2\) s)
- \(P\) :
-
Pressure (Pa)
- \(P_\mathrm{c}\) :
-
Capillary pressure (Pa)
- \(r\) :
-
Pore radius
- \(Re\) :
-
Reynolds number
- \(RH\) :
-
Relative humidity
- \(Sc\) :
-
Schmidt number
- \(Sh\) :
-
Sherwood number
- \(T\) :
-
Temperature (K)
- \(t\) :
-
Rib thickness (m)
- \(v_\mathrm{w}\) :
-
Wind speed (m/s)
- \(\dot{X}_o^{\prime \prime }\) :
-
Areal biomass production rate (kg/m\(^2\) s)
- \(x\) :
-
Distance in the direction of flow (m)
- \(x_\mathrm{c}\) :
-
Critical wetting length (m)
- \(x_\mathrm{ND}\) :
-
Nutrient delivery length (m)
- \(Y_{X/i_\mathrm{L}}\) :
-
Biomass yield based on nutrient \(i\) (kg/kmol)
- \(\alpha \) :
-
Absorptivity
- \(\epsilon \) :
-
Void fraction of porous membrane
- \(\mu \) :
-
Dynamic viscosity (Pa s)
- \(\omega \) :
-
Mass fraction
- \(\rho \) :
-
Mass density (kg/m\(^3\))
- \(\sigma \) :
-
Surface tension (N/m)
- \(\theta \) :
-
Contact angle (\(^{\circ }\))
- \(\infty \) :
-
Refers to ambient
- \(a\) :
-
Refers to air
- \(e\) :
-
Refers to exterior region
- \(f\) :
-
Refers to forced convection
- \(i\) :
-
Refers to interior region
- \(L\) :
-
Refers to limiting nutrient
- \(n\) :
-
Refers to natural convection
- \(r\) :
-
Refers to rib
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Acknowledgments
The authors would like to sincerely thank Dr. Carlos Hidrovo for his helpful discussions. The authors also gratefully acknowledge the financial support of the National Science Foundation (CBET-1125755) making this study possible.
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Murphy, T.E., Fleming, E. & Berberoglu, H. Vascular Structure Design of an Artificial Tree for Microbial Cell Cultivation and Biofuel Production. Transp Porous Med 104, 25–41 (2014). https://doi.org/10.1007/s11242-014-0318-3
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DOI: https://doi.org/10.1007/s11242-014-0318-3