# Dynamics of Water Vapor Content around Isolated Sprinklers: Description and Validation of Model

^{*}

## Abstract

**:**

^{−3}for the area next to the sprinkler, and 1.277 to 3.380 g m

^{−3}for the area 2 m from the sprinkler, under the influence of vapor pressure deficits. The increasing and decreasing rates of the dynamics during and after irrigation were influenced by temperature, relative humidity, and vapor pressure deficits, according to Pearson’s correlations. A period of 2.3 to 4.0 h was required to restore water vapor to the atmospheric level.

## 1. Introduction

## 2. Theoretical Considerations

#### 2.1. Vapor Pressure Deficit

^{−3}) in the following equation [41]:

_{W}is actual vapor pressure (kPa), R

_{W}is a gas constant of water (461.52 J kg

^{−1}K

^{−1}) [41], and T

_{K}is the Kelvin temperature (K).

_{WS}(kPa) is calculated according to the following equation [42]:

_{WS}and P

_{W}is the vapor pressure deficit (VPD; kPa) given by

#### 2.2. Water Vapor Dynamics during Irrigation

_{m}

_{1}is the maximum value of A, a differential equation has been derived to indicate that, at any given time, the increasing rate of water vapor increment is proportional to the difference between the maximum water vapor increment, A

_{m}

_{1}(g m

^{−3}), and the current water vapor increment, A (g m

^{−3}), given by

^{−3}h

^{−1}), and k is the exponential coefficient of water vapor increment (h

^{−1}).

_{m}

_{1}and k, can be determined through experiments.

#### 2.3. Water Vapor Dynamics after Irrigation

^{−3}), t’ is decreasing time duration of A (h), or the duration after irrigation, dA/dt’ is the rate of decrease in water vapor increment (g m

^{−3}h

^{−1}), f(A) is the function in terms of A, and the minus sign indicates the decreasing trend of A. It should be noted that t’ = 0 corresponds to the time when irrigation is completed (defined as t

_{s}) or when A begins to decline.

^{−1}) and s is a parameter of the function.

_{m}

_{2}can be referred to as the carrying capacity [44] of water vapor increment, and it is the theoretical maximum value for the logistic function. A

_{s}is defined as the value of A when irrigation is completed or the initial value when A begins to be reduced just after irrigation. Thus, when t’ = 0 (the time point when irrigation is completed and A begins to reduce), then A = A

_{s}and A

_{s}< A

_{m}

_{2}. When A = A

_{m}

_{2}, the decreasing rate of A should equal 0, and, as a result, f(A

_{m}

_{2}) = 0. To substitute for Equation (7):

_{s.}Thus, the solution of differential Equation (10) results in the following:

_{s}as the time required for the irrigating operation, Equation (11a) can be expressed as follows:

_{s}represents the length of time of irrigation. The parameters A

_{m}

_{2}, A

_{s}, and r can be estimated from experimental data.

_{s}should explain why Equation (5) is equal to Equation (11b), where t = t

_{s}:

#### 2.4. Overall Water Vapor Dynamics

_{s}, reaching a relatively stable level. After the completion of irrigation, the value of A starts to decrease gradually at first; then the deceasing trend continues, and finally gradually stabilizes at zero, which means that water vapor contents of the inner and outer irrigated areas gradually become consistent.

## 3. Materials and Methods

#### 3.1. Experimental Site

#### 3.2. Simulator for Single Sprinkler

^{3}h

^{−1}was used to supply water. Pressure gauges (model YB-150B; ZOHA Co., Xi’an, China) and valves (Φ25 mm ball valve; Huaya Co., Ningjin, China) were installed along pipes to control water pressure. A mesh filter (Modular 100; AZUD Co., Murcia, Spain) was installed to prevent the nozzle from clogging. A pressure regulator (15 PSI; Nelson irrigation Co., Walla Walla, WA, USA) was installed upstream of the test sprinkler to maintain a stable pressure of 105 kPa.

#### 3.3. Water Vapor Measuring System

#### 3.4. Statistics Processing and Analysis

^{−3}). The AH was calculated based on the temperature and relative humidity data collected by the sensors. The relative humidity is defined as the ratio of actual to saturation water vapor pressure at gas temperature:

_{WS}calculated from the temperature (T) data as in Equation (2) was substituted into Equation (15), to compute AH. Then, the AH of position next to sprinkler, and 2 m, 4 m and 6 m away from sprinkler, can be calculated according to data of RH and T measured by corresponding sensors. The AH of surrounding atmosphere can also be calculated according to data of RH and T measured by the sensor located 50 m away from the irrigated field. Then, the water vapor increment caused by irrigation (A) can be calculated as follows:

_{d}represents water vapor increment at d m from sprinkler (g m

^{−3}), AH

_{d}is absolute humidity at d m from sprinkler (g m

^{−3}), and AH

_{sa}is absolute humidity of surrounding atmosphere (g m

^{−3}).

_{m}

_{1}, k, A

_{m}

_{2}, A

_{s}, and r. The fitting significance between developed model and experimental data was judged according to the f-test. The relationships between model parameters and meteorological factors were analyzed by computing Pearson’s correlation coefficients. Model performance was evaluated quantitatively using the Nash–Sutcliffe model efficiency (NSE) coefficient [47], calculated as follows:

_{o,i}is the observed value at time step i, A

_{e,i}is the estimated value at time step i based on the model, and A

_{o}is the mean of observed values. Evaluation criteria for the levels of model performance were in accordance with Ritter and Muñoz-Carpena [48]; the equations were as follows:

_{t}is the number of times that the observation variability was greater than the mean error. RMSE and SD are given, respectively, by

#### 3.5. General Characteristics of the Experiments

## 4. Results and Discussion

#### 4.1. Water Vapor Dynamics

_{m}

_{1}, respectively, among the model parameters. After irrigation, the water vapor increment 2 m from the sprinkler generally changed faster than those next to the sprinkler. This may be because the value of water vapor increment next to the sprinkler was relatively higher than that 2 m from the sprinkler, such that the higher water vapor content would be more likely to diffuse to the surroundings [54] once water vapor stopped being generated (after irrigation).

#### 4.2. Water Vapor Dynamics and Meterorology

_{m}

_{1}(2 m from the sprinkler) declined with reduced VPD, with a Pearson’s correlation coefficient of 0.68 (p < 0.05; Figure 5b). Similarly, at the conclusion of irrigation, the value of A

_{s}(2 m from the sprinkler) was reduced with declining VPD, with a Pearson’s correlation coefficient of 0.73 (p < 0.05; Figure 5b). However, there was no significant relationship between A

_{m}

_{1}(next to the sprinkler) and VPD, or between A

_{s}(next to the sprinkler) and VPD.

_{m}

_{1}and A

_{s}next to the sprinkler are possibly due to intensive water spray and mist diffusion next to the sprinkler, and weak restriction of water vapor content under the meteorological conditions. Nevertheless, the air 2 m from the sprinkler was more influenced by meteorological conditions. As a result, the surrounding meteorological factors had a major influence on the area 2 m from the sprinkler. In this experiment, the VPD and water vapor increment (A

_{m}

_{1}and A

_{s}; 2 m from the sprinkler) were positively and significantly correlated, consistent with previous reports on the relationship between evaporation losses and the VPD [16,21,23,55,56].

#### 4.3. Water Vapor Increment

_{m}

_{1}, A

_{m}

_{2}and A

_{s}(Table 2) reflected the water vapor increment generated by irrigation. The values of A

_{m}

_{1}and A

_{m}

_{2}would probably be unequal, due to experimental errors and measurement fluctuations. Next to the sprinkler, A

_{m}

_{1}ranged from 2.557 to 6.552 g m

^{−3}, A

_{m}

_{2}ranged from 2.769 to 6.769 g m

^{−3}, and A

_{s}ranged from 2.506 to 6.476 g m

^{−3}, with no obvious differences between them. At 2 m from the sprinkler, the value of A

_{m}

_{1}ranged from 1.322 to 3.426 g m

^{−3}, A

_{m}

_{2}ranged from 1.357 to 6.106 g m

^{−3}, and A

_{s}ranged from 1.277 to 3.380 g m

^{−3}. The value of A

_{m}

_{2}was slightly higher than that of A

_{m}

_{1}and the value of A

_{m}

_{1}was slightly higher than that of A

_{s}. Generally, the water vapor increment were higher next to the sprinkler than 2 m from the sprinkler. With the value of A

_{s}as a reference, the maximum amounts of water vapor increment were 2.506 to 6.476 g m

^{−3}and 1.277 to 3.380 g m

^{−3}next to and 2 m from the sprinkler, respectively.

^{−3}with solid sprinkler irrigation during the daytime, which was consistent with our value of A

_{s}2 m from the sprinkler, but lower than the value of A

_{s}next to the sprinkler. This discrepancy may be due to the shorter distance between the observation site (A

_{s}next to the sprinkler) and the sprinkler for sensors in the current experiment, compared with those at the meteorological station in the previous experiment [23].

#### 4.4. Model Validation

_{a}

^{2}) ranged from 0.644 to 0.947 and 0.759 to 0.980 (Table 2), during and after irrigation, respectively, at a significance level of p < 0.001. Therefore, the developed model was a good fit with actual observations and properly reflected the actual dynamics of water vapor increment. Under actual conditions, the meteorological conditions fluctuated and the water vapor of surrounding air was not uniform or stable [59], due to the interference of wind flow [60].

^{−3}for observations next to and 2 m from the sprinkler, respectively. This may be due to a decrease in sensor accuracy when relative humidity is relatively high (i.e., >90%) [66,67], such that we expect more fluctuation and instability in water vapor content measurements at higher levels.

^{−3}. This may be because the observed values did not tend toward zero, due to the interference of meteorological fluctuations and evaporation from soil and vegetation, during the period 2–3 h following the completion of irrigation [16,68]. However, during this period, the trend of water vapor increment estimated by the model was reduced and steadily approached zero.

## 5. Conclusions

_{s}, the maximum amounts of water vapor increment generated by irrigation were 2.506 to 6.476 g m

^{−3}and 1.277 to 3.380 g m

^{−3}for the areas next to and 2 m from the sprinkler, respectively.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Dynamics of water vapor increment during (blue line) and after (red line) irrigation. The shaded area indicates the irrigation periods. A

_{s}is water vapor increment when irrigation is finished, and t

_{s}is the length of time of the irrigation operation.

**Figure 2.**Diagram of the sprinkler simulator for the center pivot irrigation system and water vapor measuring system: (

**a**) side view and (

**b**) top view. The irrigated field was plated with alfalfa (Medicago sativa L.).

**Figure 4.**Observed values and fitting results based on the water vapor dynamics model during irrigation (blue lines) and after irrigation (red lines). A is the water vapor increment generated by irrigation. The shaded area indicates the 1 h irrigation periods.

**Figure 5.**Pearson’s correlation between meteorological conditions and model parameters: (

**a**) next to the sprinkler, and (

**b**) 2 m from the sprinkler. RH, T, and VPD indicate the relative humidity, temperature, and vapor pressure deficits, respectively. The A

_{m}

_{1}, k, A

_{m}

_{2}, A

_{s}and r are model parameters. The color depth of redness indicates the extent of correlation, according to the correlation coefficient values, and the asterisks indicate significance (p < 0.05).

**Figure 6.**Comparison between observed and estimated water vapor increment due to irrigation for all the available tests: next to the sprinkler (

**a**) and 2 m from the sprinkler (

**b**) The diagonal line indicates a 1:1 relationship.

Tests | Date | RH (%) | T (°C) | W (m s^{−1}) | VPD (kPa) |
---|---|---|---|---|---|

1 | 29 August | 44.18 | 31.09 | 0.70 | 2.521 |

2 | 3 September | 43.10 | 31.80 | 0.81 | 2.675 |

3 | 21 September | 42.59 | 28.05 | 0.52 | 2.176 |

4 | 3 October | 41.75 | 24.58 | 0.89 | 1.800 |

5 | 9 October | 20.81 | 19.70 | 0.90 | 1.818 |

6 | 14 October | 48.21 | 21.75 | 0.80 | 1.349 |

7 | 1 November | 31.07 | 17.29 | 0.84 | 1.360 |

Date | Distance (m) | Model: During Irrigation $\mathit{A}(\mathit{t})={\mathit{A}}_{\mathit{m}1}(1-{\mathit{e}}^{-\mathit{k}\mathit{t}})$ 0 < t ≤ t _{s} | Model: After Irrigation $\mathit{A}(\mathit{t})=\frac{{\mathit{A}}_{\mathit{m}2}}{1+\left(\frac{{\mathit{A}}_{\mathit{m}2}}{{\mathit{A}}_{\mathit{s}}}-1\right)\cdot {\mathit{e}}^{\mathit{r}(\mathit{t}-{\mathit{t}}_{\mathit{s}})}}$ t ≥ t _{s} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A_{m1} (g m^{−3}) | k (h^{−1}) | R_{a}^{2} | p | A_{m2} (g m^{−3}) | A_{s} (g m^{−3}) | r (h^{−1}) | t_{s} (h) | R_{a}^{2} | p | ||

29 August | 0 | 2.746 | 17.675 | 0.894 | <0.001 | 2.769 | 2.746 | 4.015 | 1 | 0.960 | <0.001 |

2 | 2.808 | 7.599 | 0.944 | <0.001 | 3.430 | 2.806 | 2.593 | 1 | 0.971 | <0.001 | |

3 September | 0 | 4.096 | 10.103 | 0.858 | <0.001 | 4.129 | 4.096 | 4.174 | 1 | 0.956 | <0.001 |

2 | 3.380 | 10.320 | 0.790 | <0.001 | 3.393 | 3.380 | 6.176 | 1 | 0.901 | <0.001 | |

21 September | 0 | 2.682 | 7.957 | 0.673 | <0.001 | 2.921 | 2.681 | 1.902 | 1 | 0.974 | <0.001 |

2 | 2.609 | 5.300 | 0.756 | <0.001 | 4.868 | 2.596 | 1.765 | 1 | 0.935 | <0.001 | |

3 October | 0 | 6.552 | 4.449 | 0.898 | <0.001 | 6.769 | 6.476 | 3.731 | 1 | 0.966 | <0.001 |

2 | 2.958 | 2.822 | 0.808 | <0.001 | 3.545 | 2.782 | 3.271 | 1 | 0.912 | <0.001 | |

9 October | 0 | 5.842 | 3.403 | 0.868 | <0.001 | 5.768 | 5.648 | 6.460 | 1 | 0.980 | <0.001 |

2 | 3.426 | 3.394 | 0.817 | <0.001 | 3.672 | 3.311 | 4.206 | 1 | 0.952 | <0.001 | |

14 October | 0 | 4.094 | 9.010 | 0.925 | <0.001 | 4.474 | 4.094 | 1.471 | 1 | 0.966 | <0.001 |

2 | 2.029 | 7.209 | 0.644 | <0.001 | 6.106 | 2.028 | 0.672 | 1 | 0.759 | <0.001 | |

1 November | 0 | 2.557 | 3.930 | 0.862 | <0.001 | 2.778 | 2.506 | 1.883 | 1 | 0.969 | <0.001 |

2 | 1.322 | 3.370 | 0.947 | <0.001 | 1.357 | 1.277 | 2.334 | 1 | 0.957 | <0.001 |

^{−3}); t, time from the start of irrigation (h); A

_{m1}, maximum value of water vapor increment during irrigation (g m

^{−3}); k, exponential coefficient of water vapor increment during irrigation (h

^{−1}); R

_{a}

^{2}, adjusted coefficients of determination; P, fitting significance, according to the f-test; A

_{m2}, carrying capacity of the logistic equation, theoretical maximum value of water vapor increment after irrigation (g m

^{−3}); A

_{s}, value of water vapor increment at the conclusion of irrigation (g m

^{−3}); r, exponential coefficient of water vapor increment after irrigation (h

^{−1}); t

_{s}, time length of irrigation operation (h).

Coefficients | Model Performance | |
---|---|---|

Next to the Sprinkler | 2 m from Sprinkler | |

SD | 1.72 | 1.04 |

RMSE | 0.34 | 0.27 |

n_{i} | 4.05 | 2.91 |

Nash—Sutcliffe model efficiency coefficient | 0.961 | 0.934 |

_{i}, number of times that the observations variability is greater than the mean error.

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## Share and Cite

**MDPI and ACS Style**

Jiao, J.; Su, D.; Wang, Y.
Dynamics of Water Vapor Content around Isolated Sprinklers: Description and Validation of Model. *Water* **2017**, *9*, 307.
https://doi.org/10.3390/w9050307

**AMA Style**

Jiao J, Su D, Wang Y.
Dynamics of Water Vapor Content around Isolated Sprinklers: Description and Validation of Model. *Water*. 2017; 9(5):307.
https://doi.org/10.3390/w9050307

**Chicago/Turabian Style**

Jiao, Jian, Derong Su, and Yadong Wang.
2017. "Dynamics of Water Vapor Content around Isolated Sprinklers: Description and Validation of Model" *Water* 9, no. 5: 307.
https://doi.org/10.3390/w9050307