Calculation of the temperature field of a flat radiation heat exchanger
Determination of heat exchanger parameters, design scheme, initial and boundary conditions
In this example, the solution of the heat conduction equation by the Euler method will be demonstrated.
Task: Calculate the temperature field for a flat radiation heat exchanger.
Given:
The flat radiation heat exchanger is 20 cm long and 1.25 cm thick.
Initial conditions:
The vacuum,
the integral absorption capacity of the body ,
spectral absorption capacity of the body ,
the temperature at each point of the heat exchanger at the initial time .
Boundary conditions:
temperature of the coolant .
Assumptions:
heat transfer along the liquid flow is not taken into account, and therefore the width of the heat exchanger is assumed to be 1 cm.,
The radiation from only one side of the heat exchanger is taken into account, since the other side absorbs the heat re-reflected from the cooled spacecraft.
The general view of the heat exchanger is shown in the figure:
The classical equation of thermal conductivity has been modified to take into account radiant heat exchange with the environment, which is described by the Stefan-Boltzmann law.
Applying the Euler method to this equation, we obtained:
where:
(J/kg*K) is the heat capacity of aluminum,
kg is the mass of the calculated heat exchanger section.,
W/(m*K) is the coefficient of thermal conductivity,
m2 - the area of the calculated plot,
m is the length of the calculated section of the heat exchanger,
- the integral absorption capacity of the body,
W/(m2*K4);
- the area of the midsection of the calculated area (projection of the area of the site perpendicular to the direction of solar radiation)
- spectral absorption capacity of the body,
W/m2 is the intensity of solar radiation,
c is the calculation step.
The location of the calculated segments is shown in the figure:
Determination of initial conditions, parameters of the heat exchanger and the environment
Determination of heat exchanger and environmental parameters:
# @markdown #### Выбор материала теплообменника с определённой теплоёмкостью (по порядку: алюминий, железо, золото)
c = "903.7" # @param ["903.7", "450.0", "130.0"]
c = parse(Float64, c)
m = 0.003375
A = 235.9
F = 0.00025
d = 0.01
E = 0.90
S = 0.000000056704
Fm = F*sin(67*pi/180)
L = 0.14
# @markdown #### Определение теплового потока от солнца (солнечная сторона/тень)
q = "1400.0" # @param ["1400.0", "0.0"]
q = parse(Float64, q)
# @markdown #### Определение шага расчёта по времени
h = 0.1 # @param {type:"number"}
Calculation and visualization of data
Plotting the time decision for segments 1 to 10 (from the left edge to the middle of the heat exchanger):
using Plots
# Определение начальных условий
T0 = 293.15
T = fill(293.0, (1080,20)); #определение массива для температуры
Q3 = fill(0.094, (1080,20)); #определение массива для теплового потока, излучаемого теплообменником
Q4 = fill(0.0451047, (1080,20)); #определение массива для теплового потока, излучаемого Солнцем
# Расчёт температур и тепловых потоков в каждый момент времени
# для всех отрезков теплообменника:
for i in 1:1079 #цикл для расчёта по времени
# @markdown #### Температура теплоносителя слева:
T[i,1] = 290.06 # @param {type:"slider", min:250, max:350, step:0.01}
# @markdown #### Температура теплоносителя справа:
T[i,20] = 293.13 # @param {type:"slider", min:250, max:350, step:0.01}
for j in 2:19 #цикл для расчёта по координате
T[1,j] = T0 #определение начального условия
T[i+1,j] = T[i,j] + h*(
(1/(c*m))*
(A*F*(T[i,j-1]-T[i,j])/d #тепловой поток, передаваемый от левого отрезка к текущему с помощью теплопроводности
+ A*F*(T[i,j+1]-T[i,j])/d #тепловой поток, передаваемый от правого отрезка к текущему с помощью теплопроводности
- E*S*T[i,j]*T[i,j]*T[i,j]*T[i,j]*F #тепловой поток, излучаемый текущим отрезком теплообменника
+ L*q*Fm) #тепловой поток от Солнца, поглощаемый текущим отрезком теплообменника
)
Q3[i,j] = E*S*T[i,j]*T[i,j]*T[i,j]*T[i,j]*F #заполнение массива для излучаемого теплового потока
Q4[i,j] = L*q*Fm #заполнение массива для поглощаемого теплового потока
end
end
T[end,1] = T[end-1,1]
Legend = ["1 отрезок" "2 отрезок" "3 отрезок" "4 отрезок" "5 отрезок" "6 отрезок" "7 отрезок" "8 отрезок" "9 отрезок" "10 отрезок"]
# построение графиков для отрезков с 11 по 20 не имеет смысла, так как они геометрически симметричны с отрезками с 1 по 10
Plots.plot(T[:,1:10], xlabel="Время", ylabel="Температура", label=Legend) # , ylims=(lim_low,lim_high)
Conclusions:
In this example, the calculation of the temperature field of a flat heat exchanger was demonstrated. From the calculation of the ratio of radiated and absorbed heat fluxes, it can be seen that the heat exchanger actually absorbs the heat brought by the coolant and radiates it into the environment. In turn, depending on the coordinate, the temperature values were obtained at each time point. Thus, the problem of calculating the temperature field in a flat heat exchanger was solved.