This instructable was made as an entry into the Growing Beyond Earth Challenge to be considered under the professional category. It describes a preliminary design methodology that can be followed to help realize a scalable branching fluid network or BFN. The goal of the BFN is to maximize the growth surface area of a given volume while minimizing the cost of transporting/pumping fluid and nutrients to the root system of the plants being grown within that volume. The off the shelf parts required to construct a prototype for growing Outrageous Romaine Lettuce within a 50cm cube that uses this methodology is also outlined.
Background :

One way to get a very high growth surface area for a cubic volume is filling the volume with rows of small tubes that are spaced just enough apart to grow crops in. However, this would require a lot of work to move nutrient rich fluid to the roots of these crops. One large tube would require less work to move the nutrient rich fluid, but would not give as much growth surface area for that volume. In 1926, C.D. Murray was focused on solving a similar problem. He applied the engineering principle of minimum work to the challenges faced by the heart while transporting blood flow through vascular systems, and found a relationship between the radii of blood vessels that minimized the work being done by the heart to transport the blood. If r0 is the radius of the incoming branch and r1,r2,...rn are the radii of branches from a point used for distributing fluid in laminar flow, than work is minimized when the radii are related by the following equation:

r0^3=r1^3+r2^3+...rn^3

This relationship has come to be known as Murray’s Law. You can find his original article from the Proceedings of the National Academy of Sciences here https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1084... The Murray’s Law relationship is used to create a preliminary design methodology that can be followed to help realize a scalable BFN with the goal of maximizing the growth surface area of a given volume while minimizing the cost of transporting/pumping fluid and nutrients to the root system of the plants being grown within that volume.

Supplies

A prototype was designed with the use of the parts listed below. A CAD drawing of the design is presented to show how these assemble together to construct this prototype BFN.

Part (quantity)

3 inch PVC T (2)
3 inch to 2.5 inch bushing (4)
3 inch PVC (Approx. 5 inches)
2.5 inch T (4)
2.5 inch to 2 inch bushing (8)
2 inch T (8)
2 inch PVC (Approx. 5.5 ft)
2 inch plug (8)
LED Grow light strips that are able to be cut. (At least 6ft)
LED drivers and dimmers. (Enough to power at least 9 strips)

Step 1: Prototype Network Design Steps

It was decided to use PVC for the initial proof of concept/ prototyping of this network for multiple reasons.

PVC tubing is readily available in various radii.
Bushings can be used to achieve reductions in radii at branching points along the network.
The use of PVC tubing in Hydroponics and Aeroponics systems is well established and the chances of the material adversely affecting the growth of lettuce are reduced.
The tubing itself has structural rigidity so that the network can hold itself up. While this is not required for systems in zero gravity it helps here on Earth.

The smallest segment where plants will be grown was decided to be a tube with a diameter of 2 inches to allow for sufficient root growth. In order to make use of the most space available, the branches of the tubes diverge at 90 degree angles to one another. Due to material constraints, each branching point was chosen to diverge in two so that T’s could be used. The entrance and exit of the network was placed in the middle of the 50cm cubic volume so that the working fluid could enter and exit the network from a central location. Finally, connecting the the smallest segments to the central point is done step by step. The length of the segments are determined by the volumetric constraints, and the diameter of the pipe was chosen by its closest match to the Murray’s Law relationship. The overall cubic dimensions of this BFN prototype are 45.50cm in length and height, and 46.23cm in width. The resulting geometry of the network creates spaces not only for plant growth but also for lighting, electronic control systems, pumps, reservoirs, and/or fans.

Step 2: How and Where Will the Plants Grow in the BFN?

This branching fluid network or BFN has one single inlet and outlet that can be used to receive and extract nutrient rich fluid from the network. This fluid can be in the form of an air/mist mixture, as is used in aeroponics. In which case, holes would be created for the plant’s root system to communicate with the working fluid. Alternatively the tubes of the BFN themselves can be filled with a substrate and fertilizer mixture, similar to the Veggie vegetable production system currently being used on the International Space Station or ISS. In this case, water would be the only fluid supplied through the inlet or extracted at the outlet. Plants would communicate with the substrate via slits in the tubes and germination wicks again similar to the Veggie vegetable production system. The placement of these holes or slits is determined by identifying open segments within the BFN that meet two constraints. First, these segments would allow the plant sufficient volume to grow without being too close to other plants or segments of the BFN containing components such as reservoirs, pumps, or electronics. Second they would be parallel to another segment of the BFN that does not already have a plant growing. This empty segment is used to attach LED Grow light strips that would be unique to each plant. These light strips both assist in guiding the direction of plant growth, and can be individually tuned to optimize the growth rate of each plant.

Step 3: Areas for Growth and Optimization of the Design

Material:

For the purposes of this design challenge readily available materials were chosen so that a prototype could be constructed. However, these materials also placed limits on the geometry and overall shape of the BFN. With these limitations lifted this design methodology could be used to construct BFNs out of a variety of different materials, making it possible to realize more complex BFN geometries. As an example, the BFN could be constructed using the same TeflonTM coated Kevlar® and Nomex® sewn together in rectangular segments. The size of the segments can be optimized with the Murray’s Law relationship by using the hydraulic diameter of the segment rather than the actual diameter of a tube.

Geometry:

Again for the purposes of this design challenge the material choice was a limiting factor and restricted the geometry of the BFN. Geometric features such as diameter of tubing, taper of tubing along segments, number of branches per node, and branching angles at the nodes are all areas ripe for optimization and could potentially allow for more growth surface area to become available.

Ability to Scale:

One of the advantages of using the Murray’s Law relationship is it lends itself nicely to scale. Once a unit element is designed not only can that element be scaled to fill larger spaces, but the element can be nested within a larger scaled BFN to achieve a fractal like configuration. This could allow for plants with different spatial requirement to be grown on the same network. For example, a scaled up BFN used to feed 4 unit elements is shown in the figure above.