The original SHB method works by applying a known amount of heat to a small segment of the stem from a thin flexible heater that surrounds the stem and is itself encircled by foam insulation and measuring a temperature difference above and below the heater (Figure 1). In steady state, heat input from the heater is balanced by the heat fluxes out of the stem, i.e., inputs = outputs as originally given by [6] . Following is a description on the SHB to calculate T_crop and for additional details the reader is referred to [6] - [24] .

The power of the heater (P) in Equation (1) is calculated from Ohm’s Law given the input emf voltage (E) (V) and the resistance (R) of the heater (Ohms) as,

Conductive fluxes above (q_u) and below (q_d) the heater in W are calculated by applying Fourier’s Law:

where K_st is the thermal conductivity of the stem (W/m・˚C), A is the cross sectional area of the stem (m²), and dT_u,d/dx are the measured temperature gradients (˚C/m) above (dT_u) and below (dT_d) the heater. Temperature differences, dT_u,d are measured with a pair of thermocouples separated by a distance dx (m). In Equation (3) a value of 0.422 W/(m・˚C) was used for K_st as suggested by several authors [7] [18] - [22] .

The outward radial flow (q_r) in W is calculated with,

where K_sh is the sheath conductance (W/V) and E is the output of the thermopiles located in the thin wrapper encircling the heater. The value of K_sh is obtained when the sap flow rate F is assumed to be ≈ 0, i.e., at dawn, several hours before sunrise and is calculated by an algorithm provided by the manufacturer of the sensors.

The remaining term in Equation (1) is the convective heat flux carried by the mass transport of the sap (q_c) in W and this value is calculated from the measured rise in temperature,

where C_p is the volumetric heat capacity of water (J/kg・˚C), F is the sap flow rate in the stem (kg/s), T_u and T_d are the mean temperatures of sap flowing out of and into the system, respectively (˚C). The value of F is equivalent to T_crop.

Substituting Equations (3)-(5) into Equation (1), assuming that S ≈ 0 and solving for the sap flux F (kg/s) gives,

This equation is used in the original sap flow sensors [23] [24] to calculate T_crop with an algorithm provided by the manufacturer. The user has the option to specify, if so needed, values for K_st and K_sh. Examples on the measurement of F on different plant species, e.g., grapes is given by [8] , olives by [15] , cotton by [17] [19] , soybean by [20] [22] , and in trees by [14] [16] .

The new EXO-Skin^TM Sap Flow Sensor [25] is also based on SHB; however, the placement and the number of thermocouples used to measure dT_u and dT_d was modified resulting in a new energy balance equation. In Figure 2, several thermocouples threaded with constantan copper wire used in the new sensor design are shown [25] . This is an example of EXO-Skin^TM Sap Flow model SGEX-13 (Dynamax Inc., Houston, TX) designed to accommodate plant stems with a 12 - 16 mm diameter.

In the new sensor design, the placement of the thermocouples, above and below the heater, is such that pairs of thermocouples are wired together, in parallel, resulting in one reading that causes a difference of +0.5 to 1.0˚C in the measurement of dT_u,d when compared to the original sensor [23] [24] . In this specific model, the upper thermocouple pair is 0.013 m above the heater and the lower pair is 0.018 m below the heater (Figure 2) and these distances vary according to the size (model) of the sensor used. In the new sensor design, the measurement of a single and averaged value of dT_u,d eliminates one channel compared to the original design that uses two channels to record dT_u,d. A thermopile with multiple junctions is wired around the circumference of the heater and this is used to calculate q_r as also done in the original sensor [23] [24] .

The fundamental difference on the design of the new sap flow sensor is that dT_u,d is measured by one reading; thus, eliminating the need to calculate q_u and q_d individually.

Further, it is assumed that all energy losses by conduction are grouped into a single value of q_c, which is calculated from the radial thermopile that represents all heat conduction in and out of the stem. In preliminary tests it was shown that axial (q_a) heat loss was ~10% to 20% of the radial (q_r) heat loss, and that when the two variables were combined into a single variable (q_f), i.e., q_f = q_a + q_r, it resulted in a valid energy balance, i.e., inputs = outputs. Thus, the energy balance as given in Equation (1) reduces to:

The sheath conductivity (K_sh) is again calculated assuming F ≈ 0 and thus calculated as,

In the new sensor, heat from conductivity is all derived from radial flow including any increase in K_sh due to the grouping of q_a and q_r into a single variable. The end result of these modifications is that the value of K_sh calculated with Equation (8) is 10% to 20% higher than the value calculated from Equation (4) that does include radial flow. By design in the new sensor the higher K_sh means that the vertical (axial) heat conduction q_a is included in q_c, resulting in a representative conductance term in the energy balance. Therefore, dT (˚C) is calculated from the average temperature (A_h and B_h) measured with the two thermocouples (type T-thermocouple, 0.040 mV/˚C) as,

Thus the value of F is found from substituting into Equation (7) and rearranging terms as,

The new sensors require less wiring and copper cable connectors, and because dT is measured from an average temperature the number of data-logging channels is reduced by 25% when compared to the original design. In addition, the calculation of F is simplified eliminating the area of the stem (A), stem’s thermal conductivity (K_st), see Equation (6), and specific sensor-constants that are required as input when using the original design [23] [24] .

Our experiments were done in a greenhouse located at the facilities of the Cropping Systems Research Laboratory, USDA-ARS, Lubbock TX (33.59˚N, 101.89˚W, and average elevation of 960 m above sea level). Cotton (FiberMax®, 91880B2F, Bayer Crop Science, Research Triangle Park, NC) plants were grown in individual 11-liter (3 gallons) pots containing a commercial potting mixture. Cottonseeds were planted in early June 2014 and the test-period where T_crop measured with EXO-Skin^TM sensors [25] and using lysimeters were compared was for an 8-d period, from the 23 July to 30 July 2014, day of year (DOY) 204 - 211.

Measurements of T_crop with the EXO-Skin^TM stem flow gauge sensors [25] started when the cotton stem diameter, as measured with a caliper, were > 9 mm. For this purpose we used two models of the EXO-Skin^TM sensors, i.e., SGEX-10 and SGEX-13. The SGEX-10 model was used on stem diameters that ranged between 9.5 and 13 mm, and the SGEX-13 model accommodated stem diameters between 12 and 16 mm. We started with eight sensors numbered 1 - 8; however, sensor #2, a 13-mm sensor, malfunctioned and was not used. The installation was done following the recommend procedure given by the manufacturer and care was followed to ensure that the wrapper of each EXO-Skin^TM sensor was in close thermal contact to the stem of the cotton plant. The outputs of the sensors were recorded with a datalogger (CR-3000, Campbell Scientific, Logan UT) with a built-in sap flow algorithm, i.e., Equation (10), provided by the manufacturer (Dynamax, Houston, TX). Measurements started on DOY 204 and ended on DOY 211. Output from each sensor was recorded on a 10-minute interval and transpiration values were calculated on an hourly basis for the 8-d measurement period.

To generate a range of transpiration values the potted cotton plants used in our experiments were individually watered at different rates between seedling-emergence and time that T_crop was measured with the stem flow gauge sensors and lysimeters. Each potted plant had a drip emitter placed on the surface of the container. Further, during the 8-d measurement period the amount of water applied to each cotton plant was varied. At the time when T_crop was measured with the stem flow gauge sensors the surface of each pot was covered with a plastic film to minimize evaporation of water from the potting mixture.

On each plant with an EXO-Skin^TM stem flow gauge sensor [25] , the changes in mass as a function time were measured gravimetrically by weighing the potted plant with load cells, i.e., using a lysimeter. The output from each lysimeter was recorded on a datalogger (CR-1000, Campbell Scientific, Logan UT) on a 5-minute time interval. We made sure that the time-stamp of the datalogger used with the EXO-Skin^TM stem flow gauge sensors and the datalogger used with the lysimeters were synchronized. We used s- beam tension and compression type load cells (LSB302, Futek Advanced Sensor Technology Inc., Irvine, CA) and each load cell had a maximum weighing capacity of 23 kg. Each load cell was calibrated and provided a continuous record in mV that was converted to changes in mass in kg. The calibration was done under laboratory conditions by placing certified calibration weights on the load cell plate and registering the corresponding output (mV). This calibration indicated that a change in mass of 18.15 g/d was equivalent to an evaporative water loss of 1.0 mm/d. Additional detail on the calibration and load cells used as lysimeters is given by [5] .

To evaluate the EXO-Skin^TM sensors, we plotted the hourly and daily values of T_crop measured with the new sensors [25] as a function of the corresponding value measured with the lysimeters. This analysis was done for each of the seven sensors and for the pooled hourly and daily values of T_crop. In this comparison we did linear regression analysis and tested if the slope was significantly different than 1, i.e., slope ¹ 1, and if the intercept was significantly different than 0, i.e., intercept ¹ 0. Both of these t-tests were done with a p-value of > 0.05. A description of the procedures used is given by [26] . This statistical comparison was made for both hourly and daily values of T_crop for the test-period from DOY 204 to 211.

At the end of the test-period, on DOY 212, each cotton plant was harvested and the leaf area (LA) of each plant was measured using a leaf area meter (LI-3100C, Li-Cor, Lincoln, NE). The measurement of LA was used to convert T_crop measured with the EXO-Skin^TM sensors [25] , see Equation (10), in kg/s to volume of water per unit LA in mm/h. In this calculation we assumed a density of water of 1000 kg/m³. In addition, ancillary measurements of the environmental conditions inside the greenhouse were made. These included air temperature and humidity (model HMP50-L, Campbell Scientific Inc., Logan, UT) and shortwave irradiance (model LI-200C, Li-Cor, Lincoln, NE) connected to a datalogger (CR-1000, Campbell Scientific, Logan UT) and all variables were measured at a 1 Hz frequency.

As examples, hourly values of T_crop measured with EXO-Skin^TM stem flow gauge sensors and lysimeters, all as a function of DOY, across the 8-d measurement period are given in Figure 3. For this purpose we selected plants 3, 4, 7 and 8 for hourly values shown in Figure 3. To obtain a range of T_crop water was applied and withheld at different rates during the test period and as expected T_crop varied by the amount of water in each container. To illustrate the different patterns of hourly T_crop we selected four cotton plants shown in Figure 3.

In plant #3, water was added at nighttime, from DOY 204 to 208, withheld on DOY 209 and again watered on DOY 210 (Figure 3(a)). These applications of water resulted in an increase of the hourly maximum values of T_crop. For example, on DOY 204 and 205 the maximum T_crop was ~80 g/h, on DOY 206 the maximum T_crop increased to 96 g/h, on DOY 207 - 209, T_crop maximum was ~130 g/h, declining to 80 g/h as a result of no water being applied on the previous night and increasing to 140 g/h on DOY 211 when it was re-watered (Figure 3(a)). Cotton plant #4 was watered daily, from DOY 204 to 211; however, this was the smallest plant in our experiments and the maximum

hourly value of T_crop was ~25 g/h on the first five days and increased to ~35 g/h on the last two days (Figure 3(b)). Cotton plant #7 (Figure 3(c)) and #8 (Figure 3(d)) were watered daily from DOY 204 to 207 and water was withheld thereafter for the next three days and watered again on DOY 210. On both cotton plants, the EXO-Skin^TM stem flow gauge sensor captured this watering regime. On cotton plant #7 the maximum T_crop on DOY 204 - 205 was ~120 g/h, declining to 100 g/h on DOY 206, increasing to 150 g/h on DOY 207, and thereafter declining as water was withheld (Figure 3(c)). On DOY 208 the maximum T_crop was 95 g/h and thereafter declined to ~20 g/h (DOY 209 - 210) and increased to 40 g/h on DOY 211 (Figure 3(c)). A similar pattern was measured on cotton plant #8 (Figure 3(d)) where the hourly value of maximum T_crop increased from 70 g/h to 90 g/h during DOY 204 to 208, declined to 60 g/h the next two days and then increased to ~90 g/h on DOY 211 (Figure 3(d)). The hourly values of T_crop measured with the EXO-Skin^TM stem flow gauge sensors and shown in Figure 3 are typical of values measured with the original sensor on potted plants in greenhouse experiments [27] - [29] and follow the hourly pattern of short-wave irradiance measured in the greenhouse.

The integrated hourly values of cotton T_crop shown in Figure 3 are plotted as daily values in Figure 4.

In general, the daily values of T_crop measured with the EXO-Skin^TM stem flow gauge sensors are in agreement with values measured with the lysimeters (Figure 4). However, there were discrepancies on several days and these varied by sensor. For example, on plant 3 (Figure 4(a)), the sensors underestimated the daily value of T_crop by 210 g (18%) on DOY 207, 321 g (27%) on DOY 208, and overestimated T_crop by 91 g (19%) on DOY 210. On plant #4, the stem flow gauge sensors underestimated, on average by 40 g (17%), the daily value of T_crop compared to the lysimeters (Figure 4(b)) with a similar result in plant #7 (Figure 4(c)). However, on plant #8 (Figure 4(d)) the daily values of T_crop measured with the stem flow gauges and lysimeters agreed within ±15 g, equivalent to ±4%, of each other.

A comparison of hourly values of T_crop measured with the EXO-Skin^TM stem flow gauge sensors as a function of hourly values of T_crop measured with the lysimeter are given in Figure 5. This comparison is a linear regression analysis for the four sensors shown in Figure 3 and Figure 4.

As expected and based on the comparison of hourly T_crop values shown in Figure 3 and for daily values shown in Figure 4, there is more scatter of values for measurements on plant #3 (Figure 5(a)) and on plant #4 (Figure 5(b)), compared to the other

two plants (Figure 5(c) and Figure 5(d)). Nevertheless, the slopes for both regressions are not statistically different than 1 (p > 0.05) and the intercepts are not statistically different than 0 (p > 0.05). The regression analysis for the other two plants also yielded a slope not different than 1 and intercept not different than 0 for plant #7 (Figure 5(c)) and plant #8 (Figure 5(d)).

Pooling all the hourly T_crop measurements obtained across the 8-d period resulted in 1404 paired measurements of EXO-Skin^TM stem flow gauge and lysimeter measurements and the linear regression of these measurements is given in Figure 6. Similarly, pooling the integrated daily values of T_crop resulted in 56-paired measurements and the linear regression of these measurements is given in Figure 7.

Regression analysis on the pooled measurements of hourly and daily values of T_crop indicated that the slopes of the lines were not different than 1 (p > 0.05) and intercepts were not different than 0 (p > 0.05) as shown in Figure 6 and Figure 7, respectively.

The measured values of LA for each cotton plant on DOY 212 were 1.2 m² for plant #1, 1.4 m² for plant #3, 1.6 m² for plant #4, 1.2 m² for plant #5, 1.4 m² for plant #6, 0.6 m² for plant #7, and 1.1 m² for plant #8. These values of LA are typical for potted plants grown in 11-liter containers. Values of LA were used to convert hourly values of T_crop obtained with Equation (10) in kg/s to daily values in mm/d. As examples, the calculated daily values of T_crop in mm/d for plant #7 and plant #8 are shown in Figure 8. In these calculations the measured LA on DOY 212 was linearly extrapolated to DOY 204 assuming a 5% change in LA over the 8-d period (Figure 8).

These results (Figure 6 and Figure 7) showed that the modifications introduced in the design of the EXO-Skin^TM stem flow gauge sensors (Figure 2) did not affect the accuracy of the measurement of T_crop. Therefore, we conclude that the new sensors correctly measured the T_crop of potted cotton plants in this greenhouse experiment. However, these results are for relatively small cotton plants and thus the low values of daily T_crop i.e., the maximum value of T_crop was 1.7 mm/d on DOY 204 and 207 on plant #7 (Figure 8). All other daily values of T_crop were < 0.9 mm/d and the pattern across the 8-d measurement period was similar to plant #8 (Figure 8).

The daily values of T_crop in mm/d shown in Figure 8 are low compared to field grown cotton plants and are typical of cotton in the seedling growth stage [5] [17] . In a semiarid climate daily values of T_crop for irrigated cotton are on average 8 mm/d and may exceed 11 mm/d [17] . Nevertheless, these results have encouraged us to continue with field trials to further evaluate the EXO-Skin^TM stem flow gauges and this will be the subject of an upcoming paper.