Thermal transmission of the fluent water

Abstract

We study the thermal transmission of the fluent water whit steady velocity. First we divide the water into lots of little grids in two-dimensional space and study the temperature distribution of the fluent water. To make accurate numerical calculation, we then regard the grid in fluent water as an open system and establish discretions equations of the different situation. Finally, we calculate relevant date by the iterative method of Gauss-Seidel and get the temperature distribution in space. Introduction The condition of the motionless water can be calculate by the traditional numerical calculation, but the thermal transmission of fluent water is too complex to the traditional calculation[1]. In this article we extend traditional model and assume that each grid is an open system, and we analyze the thermal transmission of the grid in open system at the base of the laws of thermodynamics. Analysis of the thermal transmission To make clear the thermal transmission of the fluent water, we divide the water into myriad little grids and assume that every grid is an open system[2], then we analyze the inner grids. Since the hot water is flowing at a constant speed, we can study the thermal transmission of the fluent water referring the conservation of the fluent water. We first analyze the inner grids in new two-dimensional, the fluency of the water in each grid is showed in flowing picture: Fig. 1 The situation that water flowing in the open system As is showed on Fig. 1, we know that much thermal is transferred from one grid to its next grid. Since the velocity of the water is stable, we can calculate the numeric of the thermal transferred by the adjacent grids with the temperature of the border[3].With the basis of the laws of thermodynamics, we could analyze the thermal transmission in the open system: 2nd Workshop on Advanced Research and Technology in Industry Applications (WARTIA 2016) © 2016. The authors Published by Atlantis Press 591 Fig. 2 Thermal transmission on the open system As we can see, the thermal conduction of the new grid with the grids along the width and the evaporation thermal loss averaged to every grid has no change. And the thermal conduction along the length is neglected since the water is fluent on this direction. What’s more, there is considerable heat transferred by the flow of the water from the previous grid. In a similar way, the grid loss some heat with the drainage to the next grid. Numerical Calculation Table 1 Symbol table Symbol Meaning Value/Units Qin The extra heat from the outside world kJ Q’m,n the extra heat from the flowing water of grid (m,n) from grid (m-1,n). kJ λ The ability of conducting heat W/(m·K) t0 The temperature of environment °C tm,n The temperature of the grid (m,n) °C ρ The density of the water kg/m c Specific heat capacity of the water J/(kg·K), To make sure the numeric of the in Q , we assume that the temperature of the water is constant when flowing, thus we can know that the heat transferred by the fluent water is determined by the temperature of the previous grid[4]. ' , 1, , 1, , ( ) ( ) m n m n m n m n m n Q c V t t c x y t t ρ ρ − − = − = ∆ ∆ − (1) Then we can get the whole change of the heat from the flowing water ingrid (m,n): ' ' , 1, 1, 1, , ( 2 ) m n m n m n m n m n Q Q Q c x y t t t ρ + − + ∆ = − = ∆ ∆ + − (2) Since the temperature distribution is stable, the temperature of every grid doesn’t change with time, we can establish the following equation of the inner grid: