The layer water-loss nomograph

The top of a layer loses its sheen while the nozzle is away. By the time the nozzle comes back round, the top is matt and cool to the touch, and the next layer lands on it. The joint that forms there is the weakest place in a printed wall, and its strength follows how much water that top has lost.

Sanjayan et al. (2018) tied the strength of that joint to the moisture on the surface the next layer meets. Keita et al. (2019) measured it against the water the top had lost as a share of the mix water. A loss of 0.1 % left the strength unchanged, and 1 % cut it by about 40 %. They traced the loss to a dry skin on the top, where the cement finds too little water to harden and the paste stays porous. The bond follows a quantity of water, and a quantity is a rate times a time.

The ACI 305 tool reads the rate, the evaporation rate E in kilograms per square metre per hour. This tool multiplies the rate by the layer cycle, the time the top stands open. That product has a name on this tool: the water loss per layer, W, in grams per square metre. It is an upper bound: a wet top loses water as fast as the air can take it, and a drying top loses less. The question at the printer is how long the cycle can be, and the fifth panel answers in minutes.

The nomograph is five panels, read clockwise from the top left, and one construction line threads all five. The top half is the ACI 305 nomograph's top half with its index turned into a measurement. The air moisture level A is the vapour the air already holds, and the moisture deficit D is how much more a saturated top holds. Both are in grams per cubic metre.

Two things differ from the ACI 305 nomograph. The tool draws the air movement rays at the set temperatures, because a warm face drives more of its own air than a cool one. The W axis along the foot is logarithmic and increases leftward. On it the bond threshold, 10 g/m², and the bond limit, 100 g/m², stand a decade apart. The verdict key under the figure names the three tints: Pass below 10, Bond risk 10 to 100, High bond risk 100 and above.

Each panel also draws both thresholds in its own input. A threshold curve is the dashed member of the panel's family that puts the reading exactly on the threshold, with the other four as set. In the layer cycle panel "10 · 6.2 min" says that a cycle longer than about six minutes crosses the bond threshold. Between the curves the panel is tinted with the verdict that value of its input would read. Every curve moves when any input moves, the ambient temperature included. Each panel answers one question: how far its input can move before the verdict changes.

Five inputs, in the order the line reads them. Take the first three as the ACI 305 tool takes them: at the fresh layer, not at the wall of the hall. As the wall rises, the sensor has to rise with it. Air movement is in metres per second, the unit an indoor anemometer reads. The layer cycle is the time from laying a layer at a place to covering it there with the next.

Two questions before you read. Question one: did the surface temperature come off the top of the layer, or is it the printing temperature? Question two: was the cycle timed at the wall, or taken from the perimeter and the print speed? Each answer moves the line.

The rate on this tool is the evaporation from a vertical wet face in a hall, where the air moves at 0.1 to 1 m/s. Menzel (1954) fitted the ACI 305R equation to open water in the open air, and Uno (1998) gave it its closed form. Its air movement term reads as if air at 4 km/h always crossed the face. In a still hall that floor puts the ACI value 25 to 40 % above this model, and from 0.5 m/s up the two draw together. The ACI 305 tool keeps the ACI value. This one reads the hall.

The chain in Figure 04.1 has five steps, M1 to M5. Buck (1981) gives how much vapour saturated air holds at each temperature, and from it come the air moisture level, M1, and the moisture deficit, M2. M3 is the transfer coefficient, the speed at which the air beside the face carries that vapour away, in two parts. Water vapour is lighter than dry air, so the moist air at a wet face rises on its own: that is buoyancy. A warm face adds to it. Churchill and Chu (1975) give that part from the density difference between the hall's air and the air at the face, over the wet height. The wet height is the wall below the fresh layer that is still wet. The laminar plate, the rate for smooth flow along a flat surface, gives the part the hall's ventilation drives. Churchill (1977) joins the two by the cube rule, the cube root of the sum of their cubes, so the larger part leads. M4 is the rate E, and M5 is the loss W: E times the layer cycle.

Neither part can be dropped indoors. In still air the face's own air carries everything: at 25 °C, 55 % and a face at air temperature, the model reads 0.067 kg/m²/h. At 1 m/s the hall's air leads and the face reads 0.192. A face cooler than the hall by more than about 1.5 °C drives its air downward instead. The model reads the size of the density difference, not its sign, so the chain does not change. Where the two cancel the face's own air nearly stops, and the control's floor at 0.10 m/s keeps that dip to about a tenth. ASHRAE Standard 55 treats slower air as still.

The reading is an estimate of the environment, not a measurement of the concrete. It does not replace ACI 305R, the mix design or the engineer's judgement. W is an upper bound: E times the layer cycle, from a top that stays wet for the whole cycle. A cast slab bleeds and keeps a film of water on its face. ACI 308R-23 says the ACI rate matches the concrete only under that film. A printed layer has no film, so its top stays wet from below: capillary flow feeds it as long as the body can supply water (Scherer 1990; Lehmann, Assouline and Or 2008). That lasts until the top's pores empty, at the air-entry value Slowik, Schmidt and Fritzsch (2008) measured. Al-Fadhala and Hover (2001) saw the loss from cast concrete fall away from the free-water rate over hours, not minutes. So over a cycle of minutes the top loses at most W.

The largest doubt is in M3, the transfer coefficient: about a factor of 1.5 on E either way, and most of it is geometry. The model takes the top of the layer to lose water like the face below it. It takes one wet height for the whole wall, and air that moves along the wall at one speed. The top is a strip a few centimetres wide, and air that crosses the wall reaches it differently from air that runs along it. Published measurements are of horizontal surfaces or of faces under a fan, and nobody has measured the vapour's buoyancy at fresh mortar. Shah (2014, 2022), still-air evaporation from pools, is the coarse check on that corner: it catches a factor of two and nothing finer. The wet height is the setting you guess: across its range, 0.1 to 3 m, E moves by more than a factor of 3. The surface temperature is the input the model is steepest in: a top 3 °C under the air loses about a third less. A wet top in moving air runs cooler than the air.

The thresholds carry a different doubt. Keita measured the decade between them, a factor of 10. Where it sits on the W axis is not measured, because the conversion runs through the water per unit of exposed area in Keita's specimens. So both defaults are round numbers, and you can move both. Sanjayan et al. (2018) add a caution. In a mortar that bleeds, the bond did not fall steadily as the top stood open longer, because bleed water kept arriving. A printing mortar bleeds almost nothing, so the model assumes the fall is steady, and Vertico's calibration has to confirm it.

Lengthen the cycle and the line moves left, toward the bond threshold, and a shorter cycle brings it back. The reading is one input to the decision to print. What follows it, the mix, the cover, the curing and the wait, is the engineer's call.

Every number on this page comes from one of these sources or from the model at the worked example. Several figures are quoted as reported, and the originals are listed so that you can check each one.

• Keita, E., Bessaies-Bey, H., Zuo, W., Belin, P. and Roussel, N. (2019). Weak bond strength between successive layers in extrusion-based additive manufacturing: measurement and physical origin. Cement and Concrete Research 123: 105787. The two points that place both thresholds, and the dry skin behind them.
• Sanjayan, J. G., Nematollahi, B., Xia, M. and Marchment, T. (2018). Effect of surface moisture on inter-layer strength of 3D printed concrete. Construction and Building Materials 172: 468-475. Bond follows the moisture on the surface the next layer meets, and does not fall steadily where the mortar bleeds.
• ACI 305R, Guide to Hot Weather Concreting. American Concrete Institute. The rate this model is compared with, and the tool beside this one.
• Menzel, C. A. (1954). Causes and prevention of crack development in plastic concrete. Portland Cement Association. The equation behind the ACI 305 nomograph.
• Uno, P. J. (1998). Plastic shrinkage cracking and evaporation formulas. ACI Materials Journal 95(4): 365-375. The closed form of that equation, and what it was fitted to.
• ACI 308R-23, Guide to Curing Concrete. American Concrete Institute. The ACI rate is evaporativity, the rate from free water, and it stands for the concrete's loss only under bleed water.
• Buck, A. L. (1981). New equations for computing vapor pressure and enhancement factor. Journal of Applied Meteorology 20: 1527-1532. How much vapour saturated air holds at each temperature, in this model.
• Churchill, S. W. and Chu, H. H. S. (1975). Correlating equations for laminar and turbulent free convection from a vertical plate. International Journal of Heat and Mass Transfer 18: 1323-1329. The air the face moves itself.
• Churchill, S. W. (1977). A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE Journal 23(1): 10-16. The cube rule that joins the face's air and the hall's.
• Scherer, G. W. (1990). Theory of drying. Journal of the American Ceramic Society 73(1): 3-14. Capillary flow keeps a drying surface wet.
• Lehmann, P., Assouline, S. and Or, D. (2008). Characteristic lengths affecting evaporative drying of porous media. Physical Review E 77: 056309. How long a porous body can feed its surface.
• Slowik, V., Schmidt, M. and Fritzsch, R. (2008). Capillary pressure in fresh cement-based materials and identification of the air entry value. Cement and Concrete Composites 30(7): 557-565. The air-entry value: the pressure at which the surface pores empty.
• Al-Fadhala, M. and Hover, K. C. (2001). Rapid evaporation from freshly cast concrete and the Gulf environment. Construction and Building Materials 15(1): 1-7. The loss from cast concrete falls away from the free-water rate over hours.
• Shah, M. M. (2014). Methods for calculation of evaporation from swimming pools and other water surfaces. ASHRAE Transactions. And (2022), Science and Technology for the Built Environment. Still-air evaporation from pools, the coarse check on the still-air corner of this model.
• ASHRAE Standard 55. The convention that air below 0.1 m/s is still.