Введение:

В данном проекте мы рассмотрим тепловой калькулятор для зданий, с возможностью посторения графиков.

Теплопроводность – перенос теплоты (или внутренней энергии) при непосредственном соприкосновении тел (или частей одного тела) с различной температурой. Если в твердом теле, неподвижной жидкости или газе температура в различных точках неодинакова, то, как показывает опыт, теплота самопроизвольно переносится от участков тела с более высокой температурой к участкам с более низкой температурой. Внутренний механизм явления теплопроводности объясняется на основе молекулярно-кинетических представлений; перенос энергии при этом осуществляется вследствие теплового движения и энергетического взаимодействия между микрочастицами (молекулами, атомами, электронами), из которых состоит данное тело.

Обзор кода.

Структуры данных: Определяем структуры для хранения ствойств строительных материалов и стен.
Тепловые спротивления: Функция thermal_resistance вычисляет тепловое сопротивление материала на основе его толщины и теплопроводности.
Модель температуры: Функция thermal_model описывает динамику изменения температуры в задании с учетом внешней температуры, телового потока и мощности обогрева.
Симуляция: Функция simulate_bulding запускает симуляцию изменения температуры в здании на заданный период времени.
Анализ эффективности: Функция analyze_efficency рассчитывает потребленную энергию, стоимость и среднюю температуру за период симуляции
Графики и вывод данных: Основная функция run_simulation создает график изменения внутренней температуры.

using DataFrames
using Plots
using DifferentialEquations
using Statistics

# Структуры для описания материалов и их свойств
mutable struct BuildingMaterial
    name::String
    thermal_conductivity::Float64  # Вт/(м·К)
    thickness::Float64            # м
    heat_capacity::Float64       # Дж/(кг·К)
    density::Float64            # кг/м³
end

mutable struct WallStructure
    materials::Vector{BuildingMaterial}
    area::Float64               # м²
    internal_temp::Float64      # начальная внутренняя температура, °C
    external_temp::Float64      # начальная внешняя температура, °C
end

function thermal_resistance(material::BuildingMaterial, area::Float64)
    return material.thickness / (material.thermal_conductivity * area)
end

function temperature_model!(du, u, p, t)
    wall_section, heating_power, external_temp_func = p
    
    T_ext = external_temp_func(t)
    
    # Расчет общего термического сопротивления
    total_resistance = sum(material -> thermal_resistance(material, wall_section.area), wall_section.materials)
    
    # Расчет тепловой массы воздуха
    air_density = 1.225  # кг/м³
    air_specific_heat = 1005.0  # Дж/(кг·К)
    room_height = 2.7  # м
    room_volume = wall_section.area * room_height
    air_thermal_mass = air_density * air_specific_heat * room_volume
    
    # Расчет тепловой массы стен
    wall_thermal_mass = sum(material -> 
        material.heat_capacity * material.density * material.thickness * wall_section.area,
        wall_section.materials)
    
    total_thermal_mass = air_thermal_mass + wall_thermal_mass
    
    # Тепловой поток через стены
    heat_loss = (u[1] - T_ext) / total_resistance
    
    # Изменение температуры
    du[1] = (heating_power - heat_loss) / total_thermal_mass
end

function simulate_building(wall_section::WallStructure, 
                         heating_power::Float64,
                         simulation_time::Float64,
                         external_temp_func::Function)
    u0 = [wall_section.internal_temp]
    p = (wall_section, heating_power, external_temp_func)
    tspan = (0.0, simulation_time)
    
    prob = ODEProblem(temperature_model!, u0, tspan, p)
    sol = solve(prob)
    
    return sol
end

function analyze_efficiency(sol, heating_power, energy_cost)
    time_hours = sol.t[end] / 3600
    energy_consumed = heating_power * time_hours
    
    cost = energy_consumed * energy_cost / 1000
    avg_temp = mean(sol[1,:])
    
    return Dict(
        "energy_consumed_kwh" => energy_consumed / 1000,
        "cost" => cost,
        "average_temperature" => avg_temp
    )
end
# Основная функция для запуска симуляции
function run_simulation()
    # Создаем материалы
    brick = BuildingMaterial(
        "Кирпич",
        0.8,    # теплопроводность
        0.25,   # толщина
        880.0,  # удельная теплоемкость
        1800.0  # плотность
    )
    
    insulation = BuildingMaterial(
        "Минеральная вата",
        0.04,   # теплопроводность
        0.1,    # толщина
        840.0,  # удельная теплоемкость
        100.0   # плотность
    )
    
    # Создаем секцию стены
    wall = WallStructure(
        [brick, insulation],
        20.0,   # площадь стены, м²
        20.0,   # начальная внутренняя температура
        0.0     # начальная внешняя температура
    )
    
    # Функция внешней температуры
    external_temp(t) = 0.0 + 5.0 * sin(2π * t / (24 * 3600))
    
    # Параметры симуляции
    heating_power = 1000.0  # Вт
    simulation_time = 24.0 * 3600.0  # 24 часа в секундах
    energy_cost = 0.15  # Стоимость за кВт·ч
    
    # Запуск симуляции
    sol = simulate_building(wall, heating_power, simulation_time, external_temp)
    efficiency = analyze_efficiency(sol, heating_power, energy_cost)
    
    # Построение графика
    t_hours = sol.t / 3600
    
    p1 = plot(t_hours, sol[1,:], 
        label="Внутренняя температура",
        xlabel="Время (часы)",
        ylabel="Температура (°C)",
        title="Динамика температуры в помещении")
    
    plot!(t_hours, external_temp.(sol.t), 
        label="Внешняя температура",
        linestyle=:dash)
    
    # Вывод результатов
    println("\nРезультаты симуляции за 24 часа:")
    println("Потреблено энергии: $(round(efficiency["energy_consumed_kwh"], digits=2)) кВт·ч")
    println("Стоимость отопления: $(round(efficiency["cost"], digits=2)) руб.")
    println("Средняя температура: $(round(efficiency["average_temperature"], digits=2)) °C")
    
    display(p1)
    
    return sol, efficiency
end

# Запускаем симуляцию
sol, efficiency = run_simulation()

Результаты симуляции за 24 часа:
Потреблено энергии: 24.0 кВт·ч
Стоимость отопления: 3.6 руб.
Средняя температура: 22.29 °C

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UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}(FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Tuple{WallStructure, Float64, var"#external_temp#84"}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Tuple{WallStructure, Float64, var"#external_temp#84"}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Tuple{WallStructure, Float64, var"#external_temp#84"}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Tuple{WallStructure, Float64, var"#external_temp#84"}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}((FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Tuple{WallStructure, Float64, var"#external_temp#84"}, Float64}}(Ptr{Nothing} @0x00007fba521b2040, Ptr{Nothing} @0x00007fbc96e2c0f8, Base.RefValue{SciMLBase.Void{typeof(temperature_model!)}}(SciMLBase.Void{typeof(temperature_model!)}(temperature_model!)), SciMLBase.Void{typeof(temperature_model!)}), FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, 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Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Tuple{WallStructure, Float64, var"#external_temp#84"}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}(Ptr{Nothing} @0x00007fba521b2870, Ptr{Nothing} @0x00007fbc96e2c110, Base.RefValue{SciMLBase.Void{typeof(temperature_model!)}}(SciMLBase.Void{typeof(temperature_model!)}(temperature_model!)), SciMLBase.Void{typeof(temperature_model!)}))), UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing), [26.553988471103402], (WallStructure(BuildingMaterial[BuildingMaterial("Кирпич", 0.8, 0.25, 880.0, 1800.0), BuildingMaterial("Минеральная вата", 0.04, 0.1, 840.0, 100.0)], 20.0, 20.0, 0.0), 1000.0, var"#external_temp#84"())), SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Tuple{WallStructure, Float64, var"#external_temp#84"}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Tuple{WallStructure, Float64, var"#external_temp#84"}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Tuple{WallStructure, Float64, var"#external_temp#84"}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, 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ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}(Ptr{Nothing} @0x00007fba521b2870, Ptr{Nothing} @0x00007fbc96e2c110, Base.RefValue{SciMLBase.Void{typeof(temperature_model!)}}(SciMLBase.Void{typeof(temperature_model!)}(temperature_model!)), SciMLBase.Void{typeof(temperature_model!)}))), UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing), 0.0, (WallStructure(BuildingMaterial[BuildingMaterial("Кирпич", 0.8, 0.25, 880.0, 1800.0), BuildingMaterial("Минеральная вата", 0.04, 0.1, 840.0, 100.0)], 20.0, 20.0, 0.0), 1000.0, var"#external_temp#84"())), [0.0], LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, GenericLUFactorization{RowMaximum}, LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearSolve.InvPreconditioner{Diagonal{Float64, Vector{Float64}}}, Diagonal{Float64, Vector{Float64}}, Float64, Nothing}([6.93902057355323e-310;;], [0.0], [0.0], SciMLBase.NullParameters(), GenericLUFactorization{RowMaximum}(RowMaximum()), LU{Float64, Matrix{Float64}, Vector{Int64}}(Matrix{Float64}(undef, 0, 0), Int64[], 0), true, LinearSolve.InvPreconditioner{Diagonal{Float64, Vector{Float64}}}([0.0;;]), [0.0;;], 1.4901161193847656e-8, 1.4901161193847656e-8, 1, false, LinearSolve.OperatorAssumptions{Nothing}()), FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}([0.0], [0.0], [0.0], [0.0], 1:1, nothing), FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}(nothing, [0.0], [0.0], 0.0), 0.001, Rosenbrock23{1, false, GenericLUFactorization{RowMaximum}, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}(GenericLUFactorization{RowMaximum}(RowMaximum()), OrdinaryDiffEq.DEFAULT_PRECS), nothing)), OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}(-8, 8, Tsit5(stage_limiter! = trivial_limiter!, step_limiter! = trivial_limiter!, thread = static(false)), Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}(nothing, OrdinaryDiffEq.DEFAULT_PRECS), false, 10, 3, 9//10, 9//10, 2, false, 5), 1)), true, 0, DiffEqBase.DEStats
Number of function 1 evaluations:                  51
Number of function 2 evaluations:                  0
Number of W matrix evaluations:                    0
Number of linear solves:                           0
Number of Jacobians created:                       0
Number of nonlinear solver iterations:             0
Number of nonlinear solver convergence failures:   0
Number of rootfind condition calls:                0
Number of accepted steps:                          8
Number of rejected steps:                          0
Maximum eigenvalue recorded:                       0, [1, 1, 1, 1, 1, 1, 1, 1, 1], SciMLBase.ReturnCode.Success), Dict("energy_consumed_kwh" => 24.0, "cost" => 3.6, "average_temperature" => 22.29235062296184))

Визуализация теплопроводности стены:

module HeatTransfer

using Plots
using LinearAlgebra
using DifferentialEquations

# Структура для параметров модели
struct WallParameters
L_brick::Float64 # толщина кирпичной стенки (м)
L_insulation::Float64 # толщина минваты (м)
L::Float64 # общая длина
nx::Int64 # количество точек по x
dx::Float64 # шаг по пространству
x::Array{Float64,1} # массив координат
α_brick::Float64 # температуропроводность кирпича
α_insulation::Float64 # температуропроводность минваты
λ_brick::Float64 # теплопроводность кирпича
λ_insulation::Float64 # теплопроводность минваты
T_inside::Float64 # температура внутри
T_outside::Float64 # температура снаружи
end

# Функция создания параметров
function create_parameters()
L_brick = 0.25 # толщина кирпичной стенки (м)
L_insulation = 0.1 # толщина минваты (м)
L = L_brick + L_insulation
nx = 100
dx = L/nx
x = range(0, L, length=nx) |> collect

return WallParameters(
L_brick,
L_insulation,
L,
nx,
dx,
x,
0.5e-6, # α_brick (м²/с)
0.4e-7, # α_insulation
0.7, # λ_brick (Вт/м·К)
0.04, # λ_insulation
20.0, # T_inside (°C)
-20.0 # T_outside (°C)
)
end

# Функция определения коэффициента температуропроводности
function get_alpha(x, p::WallParameters)
return x <= p.L_brick ? p.α_brick : p.α_insulation
end

function get_lambda(x, p::WallParameters)
return x <= p.L_brick ? p.λ_brick : p.λ_insulation
end

# Начальные условия
function initial_condition(x, p::WallParameters)
return p.T_inside + (p.T_outside - p.T_inside) * x / p.L
end

# Функция для уравнения теплопроводности
function heat_equation!(du, u, p::WallParameters, t)
# Внутренние точки
for i in 2:length(u)-1
x_i = p.x[i]
α = get_alpha(x_i, p)

# На границе материалов используем специальные условия
if abs(x_i - p.L_brick) < p.dx
λ_left = get_lambda(x_i - p.dx/2, p)
λ_right = get_lambda(x_i + p.dx/2, p)
flux_left = λ_left * (u[i] - u[i-1]) / p.dx
flux_right = λ_right * (u[i+1] - u[i]) / p.dx
du[i] = (flux_right - flux_left) / p.dx
else
du[i] = α * (u[i+1] - 2u[i] + u[i-1]) / p.dx^2
end
end

# Граничные условия
du[1] = 0
du[end] = 0
end

function run_simulation()
# Создаем параметры
p = create_parameters()

# Начальные условия
u0 = initial_condition.(p.x, Ref(p))

# Временной интервал (24 часа)
tspan = (0.0, 24.0 * 3600)

# Создание проблемы
prob = ODEProblem(heat_equation!, u0, tspan, p)

# Решение с использованием метода для жёстких систем
sol = solve(prob, TRBDF2(),
saveat=range(0, 24*3600, length=100),
abstol=1e-6,
reltol=1e-6,
maxiters=1e6)

# Подготовка данных для графика
t_hours = sol.t / 3600
temp_matrix = zeros(length(t_hours), length(p.x))
for (i, solution) in enumerate(sol.u)
temp_matrix[i, :] = solution
end

# Создание графика
heatmap = Plots.heatmap(p.x .* 100, t_hours, temp_matrix,
c=:turbo,
xlabel="Толщина стены (см)",
ylabel="Время (часы)",
colorbar_title="Температура (°C)",
right_margin=20Plots.px
)

# Граница материалов
vline!([p.L_brick * 100], color=:white, linestyle=:dash, label="Граница кирпич/минвата")

# Контурные линии
contour!(p.x .* 100, t_hours, temp_matrix,
color=:black,
alpha=0.3,
levels=15,
linewidth=0.5,
label=nothing
)

# Сохранение графика
savefig(heatmap, "wall_heat_conductivity.png")

# Вывод температур
brick_boundary_idx = findlast(x -> x <= p.L_brick, p.x)
println("Температуры в конце симуляции (24 часа):")
println("Внутренняя поверхность: $(round(temp_matrix[end,1], digits=2))°C")
println("Граница кирпич/минвата: $(round(temp_matrix[end,brick_boundary_idx], digits=2))°C")
println("Внешняя поверхность: $(round(temp_matrix[end,end], digits=2))°C")

return heatmap
end

# Запуск симуляции
run_simulation()

end # module

WARNING: replacing module HeatTransfer.

Температуры в конце симуляции (24 часа):
Внутренняя поверхность: 20.0°C
Граница кирпич/минвата: 11.25°C
Внешняя поверхность: -20.0°C

Main.HeatTransfer

Анализ диаграммы.

Анализируя полученную диаграму, можно следать вывод, что толщины нашей изоляции с использованием кирпичей и миниральной ваты с данными характеристиками, позволяет сохранять температуру в 20 градусов без сильных потерь.

Заключение.

Сохранение теплоты в помещение является важной задачей как экономической, так и конструкторской. Используя наш калькулятор мы можем визуально анализировать потери тепла и расчитать стоимость отопления для помещений.

Список использованной литературы:

М.А. Михеев, И.М. Михеева Основы теплопередачи
В.П. Исаченко, В.А. Осипова, А.С. Сукомел Теплопередача

Калькулятор тепловых потерь здания

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