This study numerically involves the performance of thermal insulation of different types of composite walls and roofs to demonstrate the best model that can be used for energy-efficient building construction in Iraq. The mathematical model is solved by building its code using the Transmission Matrix Method in MATLAB software. The weather data of 21st July 2022 in Baghdad City/Iraq is selected as a test day. The wall types are selected: the first type consists of cement mortar, brick, and gypsum, the second type consists of cement mortar, brick, gypsum, and plaster and the third type consists of cement mortar, brick, gypsum, air cavity, and sheathing timber. The roof types are chosen: the first type consists of reinforced concrete, gypsum, and plaster, and the second type consists of the precast concrete flag, river sand, tar, reinforced concrete, gypsum, and plaster. The obtained solutions are compared with previous studies for the same city but with different types of walls and roofs. The findings display that the second and third types of walls reduce the entry heat flux by 4% and 10% as compared to the first type of wall. Also, the results indicate that the second type of roof reduces the entry heat flux by 21% as compared to the first type of roof. The results confirm that the best models of walls and roofs in Iraq are the third and second types, respectively, as compared to other models and hence, the performance of insulation material strongly depends on the materials used while building them.

Recently, about 40% of energy in the world goes to residential structures, which are responsible for increasing the percentage of carbon dioxide emissions to 30% [

Luo et al. [^{3} in size. Their study investigated the impact of insulation material on reducing energy consumption in buildings. The structure is constructed in one of three distinct layouts. Brick is used in the construction of the walls in the original model. A polystyrene foam with a thickness of four centimeters is used inside the walls and roof as an additional layer of insulation material for the second model. In the third model, the walls are built from two different portions separated by an air gap with 8 cm width. Additionally, a secondary roof is added to this model.

Pekdogan et al. [_{2} emissions and anticipated energy consumption. The value of the overall heat transfer coefficient of the external wall is equal to 0.25 W/m^{2}·°C according to the Silver Standard, whereas it is equal to 0.15 W/m^{2} K according to the Gold Standard. These values are used while calculating the amount of heat loss and heat gain, where the findings are compared. Wei et al. [

Mohammed et al. [

In recent years, however, some researchers have used experiments to test real PCM in warm environments, even though computer models are faster at getting results. Austin et al. [

A complete literature review shows that most previous investigations of heat transfer through either compound walls or roofs are experimentally and numerically performed. Moreover, the literature reveals no numerical investigation about evaluating the amount of heat flux through compound walls or roofs using the transmission matrix method. One of the significant targets of building construction is to obtain and keep the specific thermal conditions inside the building as energy-efficient as possible. The evolution of machine computation has allowed scientists to discover new methods to solve the same problem more accurately and in a shorter time. A z-transform of the heat transmission matrix is necessary to reach a final solution involving the current and previous temperatures and the heat fluxes through walls and roofs. Thus, in this study, the amount of heat flux through three different types of walls and two different types of roofs are numerically evaluated to determine the influences of their layer arrangements and which kind of insulation material is the best for reducing the amount of heat flux. The transmission matrix method and a computer program are used to solve the governing equations. The feature of this method is reducing the computational time of simulations. The test is made on 21st July 2022 in Baghdad/Iraq for 24 h. As a result, the optimum summer construction strategy is used to reduce the building’s peak cooling demand or shift it to a time when the building is not required to use it.

The computer model code for simulating the conduction heat flow through walls and roofs is developed using the transmission matrix method. The transmission matrix method is related to the periodic temperature and heat flux on one side of a homogenous layer to the regular temperature and heat flux on the other side, employing a transmission matrix. Hence, it can be written as [_{o} is the outside surface temperature of the layer (°C), q_{i} is the inside heat flux (W/m^{2}), and q_{o} is the outside heat flux (W/m^{2}). The complex matrix elements A, B, C, and D can be written according to thermal properties of layer as [^{2}/s), ^{3}), c_{p} is the specific heat (J/kg.°C), k is the thermal conductivity (W/m.^{o}C), and j is the complex operator. The overall transmission matrix is obtained by chain multiplying the individual matrices in the order the matrices appear in the composite wall section from outside to inside as follows:

In general, the temperature is presented in the complex form of a Fourier series expression as follows [_{n} and N_{n} are the Fourier coefficients, where their values are calculated. The steady-state portion of heat flux is calculated as follows:
^{2}.°C/W). The value of R is calculated as follows [

The total entry heat flux for

The physical meaning of ^{2}.°C), when determining the heat flow through several materials (multilayer), the overall transmission matrix is the product of the transmission matrix for each material in the order it appears in the composite layer. Assuming the transmission matrix, given in _{i}. The transient heat flux on the interior of the building is obtained as follows:

For the particular case, when the inside–air temperature remains constant concerning time (t_{i} = 0) in

The transmission matrix method calculates a system’s state transition matrix without identifying the roots to ascertain the thermal response factors of a multilayer wall and roof. This approach is straightforward, precise, and suitable for thermal engineering.

Layer | Thermal conductivity (W/m.°C) | Density (kg/m^{3}) |
Specific heat (kJ/kg.°C) | Resistance (m^{2}.°C/W) |
---|---|---|---|---|

Cement mortar | 1.08 | 2105 | 0.85 | 0.018 |

Brick | 0.54 | 1570 | 0.93 | 0.444 |

Gypsum | 0.72 | 1416 | 1.006 | 0.027 |

Plaster | 0.32 | 1068 | 1.115 | 0.006 |

Air | 0.025 | 1.205 | 1.005 | 2 |

Sheathing timber | 0.115 | 500 | 1.6 | 0.043 |

Reinforced concrete | 1.25 | 2290 | 0.92 | 0.16 |

Precast concrete flag | 1.22 | 2270 | 0.896 | 0.04 |

River sand | 0.78 | 1500 | 0.82 | 0.128 |

Tar | 0.24 | 1070 | 1.47 | 0.083 |

Type of wall | Construction | Thickness (cm) |
---|---|---|

Type one | -Cement mortar | 2 |

-Brick | 24 | |

-Gypsum | 2 | |

Type two | -Cement mortar | 2 |

-Brick | 24 | |

-Gypsum | 2 | |

-Plaster | 0.2 | |

Type three | -Cement mortar | 2 |

-Brick | 24 | |

-Gypsum | 2 | |

-Air cavity | 5 | |

-Sheathing timber | 0.5 |

Type of roof | Construction | Thickness (cm) |
---|---|---|

Type one | -Reinforced concrete | 20 |

-Gypsum | 2 | |

-Plaster | 0.2 | |

Type two | -Precast concrete flag | 5 |

-River sand | 10 | |

-Tar | 2 | |

-Reinforced concrete | 20 | |

-Gypsum | 2 | |

-Plaster | 0.2 |

Hours | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

_{out} |
34.41 | 34.06 | 33.9 | 33.66 | 33.41 | 34.61 | 36.18 | 38 | 39.95 | 41.9 | 43.93 | 46.29 |

12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |

_{out} |
47.49 | 48.24 | 49 | 48.24 | 47.49 | 46.29 | 44.73 | 42.9 | 40.95 | 39 | 37.18 | 35.61 |

For thermal resistances in series: the general case of thermal resistances arranged in series is shown in

The Matlab software is used to solve the governing equations, where the solutions are computed with a minimum period of 10–15 min. This time can be problematic for detailed control studies, where the time step of 1 min is used. The algorithm is explained as follows:

Finding the complex matrix elements, A, B, C, and D, using _{j} and the thermal properties k_{j}, ρ_{j}, cp_{j} of the layer j.

Calculating the complex form of a Fourier series expression using

Calculating the exponential coefficient T and phase lag Ф for each harmonic n using

Calculating the steady-state portion of heat flux by

Calculating the total inside heat flux for n harmonics by summating or superimposing the steady-state and transient portions by

Calculating the transient heat flux on the interior of the building by

Since the inside air temperature remains constant over time, the transient heat flux on the interior of the building is evaluated using

It can be noted from the results of

The current solutions are compared with the work of Joudi et al. [

In the current work, the effectiveness of thermal insulation of various types of composite walls and roofs is numerically investigated to display the best model that can be employed for energy building construction in Iraq. The governing equations are solved using the transmission matrix method in MATLAB software. The weather data for 21st July 2022 in Baghdad City/Iraq, was selected as a test day. From the results of this study, the following conclusions are set:

The results indicated that the transmission matrix method has limited input elements. The linear relationship between the temperature and the heat transfer rate at the system’s heat source and environment sides is characterized by a matrix that does not surpass the 2

The findings display that the thermal capacity of the building rises with an increase in the number of insulation layers. Hence, this issue leads to an increase in the thermal resistance of the construction.

The results confirm that the cooling load of a building depends on the types of materials used while building it.

The findings show that the entry heat flux is reduced by 4% (for the second type of wall) and 10% (for the third type of wall) compared to the first type of wall.

The outcomes show that the entry heat flux is reduced by 21% using the second type of roof compared to the first type.

Finally, the results confirm that the third type of wall and the second type of roof represent the best models that can be used in the weather conditions in Iraq compared to the other models.

The current work can be extended for future projects using Phase Change Material (PCM) on walls and roofs. Phase Change Materials (PCM) can efficiently absorb a significant amount of solar energy that falls on the walls or roofs of residential structures. PCM’s high thermal mass allows them to mitigate the impact of considerable temperature variations on the indoor climate of buildings [

Complex matrix element

Thermal diffusivity (m^{2}/sec)

_{p}

Specific heat (J/kg.°C)

Coefficient of elements A, B, C (1/m)

Surface heat transfer coefficient (W/m^{2}.°C)

Complex operator

Thermal conductivity (W/m.°C)

Density (kg/m^{3})

Harmonic frequency (1/sec)

Phase lag

Average temperature differential across layers (°C)

We humbly thank Northern Technical University for their laboratory support for this study.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: study conception and design: Ahmed Mustaffa Saleem, Abdullah A. Badr; data collection: Bahjat Hassan Alyas; analysis and interpretation of results: Omar Rafae Alomar; draft manuscript preparation: Omar Rafae Alomar. All authors reviewed the results and approved the final version of the manuscript.

All data is available on request.

The authors declare that there are no conflicts of interest to report regarding the present study.