# The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Its Analytical Solution

#### 2.1. Mathematical Models

_{b}’(x,0), and the temperature at distance x from the boundary at moment t is T

_{b}’(x,t). Under the effect of industrial thermal discharge, the water temperature of the river canal rises instantaneously by ∆T

_{0}and maintains this for a long time; thus, the river channel is considered as a boundary with a constant temperature. The influence of the river channel on the water temperature of the aquifer is illustrated in Figure 3.

_{b}(x,t) is the excess temperature at distance x from the boundary at moment t (°C), i.e., T

_{b}(x,t) = T

_{b}’(x,t) − T

_{b}’(x,0), with the initial temperature as the reference; a is the thermal diffusivity of the aquifer (m

^{2}/d), which is a constant in the homogeneous aquifer; v is the seepage velocity in the confined aquifer (m/d).

_{b}(x,t)/θ(x,t), through which it is converted into a second-order linear partial differential equation.

^{2}t/4a)], we see that

_{b}(x,t) with respect to t is

_{b}(x,t) with respect to x is

_{b}(x,t) with respect to x yields

_{0}/θ(0,t) = ∆T

_{0}· exp (v

^{2}t/4a).

#### 2.2. Theoretical Solution

_{1}and c

_{2}are the constants to be determined.

^{−1}is the inverse Laplace transformation operator; ∗ is the convolution operator.

#### 2.3. Analytical Solution

_{0}·exp (v

^{2}t/4a) into Equation (25), gives

## 3. Materials and Methods

#### 3.1. Example Overview

_{0}= 4 °C.

#### 3.2. Numerical Simulation

## 4. Results

#### 4.1. Analysis of Aquifer Temperature Variation Laws Basing on the Analytical Method

^{−3}m

^{2}/h = 0.0336 m

^{2}/d, in combination with Table 1.

_{0}= 4 °C, i.e., both ∆T

_{0}and a are fixed values for the aquifer. From Equation (26), it can be seen that the excess temperature T

_{b}(x,t) is related to the parameters v, x, and t. Therefore, one of the parameters can be set as a fixed value in turn, and the variation laws of T

_{b}(x,t) caused by the variation of the other two parameters can be studied. It should be noted that Equation (26) contains a parametric variable proper integral, which can be calculated by substituting the corresponding values using WolframAlpha [28]. WolframAlpha is an online computational knowledge engine, which can output integration results and visualize the outcomes based on algorithms and data.

_{b}(x,t) with time at various distances from the river canal at v = 0.1 m/d, T

_{b}(x,t) with time at various flow velocities at x = 1 m, and T

_{b}(x,t) with distance at different flow velocities at t = 20 d.

- (1)
- at the same flow velocity, the further away from the river canal, the slower the aquifer temperature varies;
- (2)
- at the same distance from the river canal, the smaller the aquifer flow velocity, the slower the temperature changes;
- (3)
- at the same time, the larger the aquifer flow velocity, the smaller the temperature changes caused by the increased distance;
- (4)
- at the same distance from the river canal or at the same flow velocity, the aquifer temperature increases with time until it reaches a stable state.

#### 4.2. Analysis of the Boundary Influence Range Basing on Numerical Simulation

#### 4.3. Comparison of the Results of the Analytical and Numerical Methods

_{b}(x,t) at x = 0.5 m and 1.0 m are calculated, and the sum of them and the initial temperature of the aquifer (18 °C) constitute the analytic values of the corresponding aquifer temperature. Table 2 and Table 3, and Figure 9, compare the outcomes of the analytical and numerical simulations, respectively.

## 5. Discussion

#### 5.1. Error Analysis between Analytical Results and Numerical Results

- At the same distance from the boundary, the relative error tends to gradually decrease and then slightly increase with time, and the maximum error of the two methods reaches 10.27% at the virtual observation point 2, which is relatively far from the boundary, under the condition of shorter time (such as 5 days in the example).
- Under the condition of the same time, the closer the distance to the boundary, the relative error tends to decrease.
- When the duration of the boundary effect is relatively short (e.g., t < 15 d), the comparison of the results generally shows that the numerical values are larger than the corresponding analytical values.

#### 5.2. The Impact of ∆T_{0} on T_{b}(x,t)

_{0}was set to a constant value, 4 °C. However, ∆T

_{0}is affected by factors such as the operating conditions of the power plant and the seasons [30]. Therefore, it is necessary to discuss the impact of the temperature variation in the river channel on the aquifer temperature, i.e., the relationship between ∆T

_{0}and T

_{b}(x,t).

^{2}/d, v = 0.1 m/d, and t = 40 d. Then T

_{b}(x,t) is a linear function of ∆T

_{0}with a slope of 0.9888 (Figure 10), according to Equation (26).

#### 5.3. Contributions to the Management of Rivers and Aquifers

## 6. Conclusions

_{0}and remains constant, the groundwater temperature at a certain distance from the river channel varies asymptotically with time. In the initial stage when the variation of groundwater temperature is relatively small, the actual amplitude of groundwater temperature variation is smaller than the linear interpolation value in each calculation element of the numerical method, which leads to the numerical value being larger than the analytical value.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Parameter | Unit | Value |
---|---|---|

volume specific heat of solid skeleton, c_{s} | J·(kg·°C)^{−1} | 1.78 × 10^{3} |

volume specific heat of water, c_{w} | J·(kg·°C)^{−1} | 4.20 × 10^{3} |

density of solid skeleton, ρ_{s} | kg·m^{−3} | 1.50 × 10^{3} |

thermal conductivity of solid skeleton, λ_{s} | W·(m·°C)^{−1} | 1.92 |

thermal conductivity of water, λ_{w} | W·(m·°C)^{−1} | 0.59 |

porosity, n | - | 0.40 |

Time/d | Analytical Value/°C | Numerical Value/°C | Absolute Error/°C | Relative Error */% |
---|---|---|---|---|

5 | 20.7484 | 21.5715 | 0.8231 | 3.97 |

10 | 21.5121 | 21.7456 | 0.2335 | 1.09 |

15 | 21.7722 | 21.8143 | 0.0421 | 0.19 |

20 | 21.8836 | 21.8565 | 0.0271 | 0.12 |

25 | 21.9372 | 21.8830 | 0.0542 | 0.25 |

30 | 21.9648 | 21.9020 | 0.0628 | 0.29 |

40 | 21.9881 | 21.9280 | 0.0601 | 0.27 |

Time/d | Analytical Value/°C | Numerical Value/°C | Absolute Error/°C | Relative Error */% |
---|---|---|---|---|

5 | 19.1558 | 21.1240 | 1.9682 | 10.27 |

10 | 20.5766 | 21.4733 | 0.8967 | 4.36 |

15 | 21.2641 | 21.6146 | 0.3505 | 1.65 |

20 | 21.6022 | 21.7014 | 0.0992 | 0.46 |

25 | 21.7774 | 21.7563 | 0.0211 | 0.10 |

30 | 21.8721 | 21.7959 | 0.0762 | 0.35 |

40 | 21.9553 | 21.8499 | 0.1054 | 0.48 |

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**MDPI and ACS Style**

Wei, T.; Tao, Y.; Ren, H.; Lin, F.
The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel. *Water* **2022**, *14*, 3698.
https://doi.org/10.3390/w14223698

**AMA Style**

Wei T, Tao Y, Ren H, Lin F.
The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel. *Water*. 2022; 14(22):3698.
https://doi.org/10.3390/w14223698

**Chicago/Turabian Style**

Wei, Ting, Yuezan Tao, Honglei Ren, and Fei Lin.
2022. "The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel" *Water* 14, no. 22: 3698.
https://doi.org/10.3390/w14223698