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Analytical solutions for open-channel temperature response to unsteady thermaldischarge and boundary heating

 

 

Consider a river flow: its upsteady temperature changes with time T_up=T^bar_0+T_0 sin(omega_0 t+alpha), and it is heated over the distance of the flow with temperature T_bd=T^bar_0+T_0 sin(omega _0 t+alpha).

 

This a simplified model and can be used as a first-hand estimate of temperature in engineering problems. The tempertaure is derived as (Tang and Keen 2009):

in which

 

t -- time

s -- distance in flow direction

T^bar_0 -- average upstream temperature

T_0 -- upsteam temerpature amplitude

omega_0 -- frequency of upsteam temperature

alpha -- a phase

T^bar_1 -- average heating temperature

T_1 -- heating temerpature amplitude

omega_1 -- frequency of heating

U --- river flow speed

K -- heat transfer coefficient

 

and theda_1=tan^{-1}(omega_1/K), L_T=U/K. An example of the solution is plotted in Figure 1.

 

Figure 1. Eample solutions with

,

 

 

Input your parameters and get a solution:

 

 

T^bar_1 (C):

T_1 (C):

T^bar_0 (C):

T_0 (C):

omega_1, omega_0 (1/s):

phase (radian):

K^bar (Wm/(m^2C)):

U (m/s):

D (kg/m^3):

Cp (J/(kgC)):

h0 (m):

num_t:

num_x (m):

icnw:

dtime (s):

s (m):

 

Note: K=K^bar/(D Cp h0), K^bar is the air–water exchange coefficient, Cp the heat capacity of water, and h0 the water depth. These are the parameters to visulize the solution (you may use the default values): num_t is the number of time step for computation, num_x the grid spacing, icnw - inetrval of time steps, and dtime the time step.

 

For details and more solutions for different upstream and heating temperature, see Tang and Keen (2009).

 

 

Reference

 

H. S. Tang and T. R. Keen, Analytical solutions for open-channel temperature response to unsteady thermal discharge and boundary heating, ASCE J. Hydraulic Eng., 135(4), 2009, 327-332. DOI:10.1061/
ASCE0733-9429(2009)135:4
327

 

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