*Corr. Author’s Address: Zhejiang Normal University, 688 Yingbin Road, Jinhua, Zhejiang Province, China, cai_jiancheng@foxmail.com 635
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641 Received for review: 2022-06-20
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-09-08
DOI:10.5545/sv-jme.2022.222 Short Scientific Paper Accepted for publication: 2022-09-09
Thermal Evaluation of Multilayer Wall  
with a Hat-Stringer in Aircraft Design 
Brazhenko, V . – Qiu, Y . – Cai, J. – Wang, D.
V olodymyr Brazhenko
1,2
 – Yibo Qiu
1
 – Jiancheng Cai
1,2,*
 – Dongyun Wang
1,2 
1 
Zhejiang Normal University, College of Engineering, China 
2 
Zhejiang Normal University, Key Laboratory of Urban Rail Transit Intelligent Operation  
and Maintenance Technology & Equipment of Zhejiang Province, China
In the present paper, we investigate the heat transfer through the multilayer wall of aircraft cabins, as this process influences the comfortable 
conditions created for most passengers and crew members. The numerical modelling results and calculation of a multilayer wall with a 
hat-stringer in aircraft design are presented. The thermal characteristics evaluation and their relationship with the design parameters were 
made. The effect of the air layer size on the overall thermal resistance of the multilayer wall, taking into account the geometrical dimensions 
and properties of the surfaces, was studied. The relative temperature field in the insulation layer, which crosses the hat-stringer elements 
of the fuselage frame, is calculated. It is shown that the insufficient thickness of the layer of thermal insulation material in the zone of the 
hat-stringer at low temperatures leads to a significant deterioration in the multilayer wall characteristics, which can worsen the microclimatic 
conditions inside the aircraft cabin.
Keywords: numerical modelling, heat transfer, aircraft design, multilayer wall, hat-stringer
Highlights
• 	 The air layer thickness, up to which the thermal resistance of the multilayer wall increases, was determined.
• 	 The relationship between the size of the insulating material and the heat transfer coefficient was obtained.
• 	 The effect of a hat-stringer on the thermal characteristics of a multilayer wall of an aircraft cabin was evaluated.
0  INTRODUCTION
Nowadays, the civil aircraft industry faces a variety 
of challenges and demands, including ensuring a 
comfortable environment for passengers on board 
aircraft. In the rational design of present-day passenger 
aircraft, a heightened level of attention is focused 
on creating a comfortable condition for travellers 
and crew members in the aircraft cabin, chiefly by 
improving the quality of indoor climate conditions [1] 
to [5]. The core part of an aircraft is the cabin, which 
provides the necessary environmental conditions for 
travellers and crew members. The first priority of the 
cabin is to protect people on board from the harmful 
effects of the external atmosphere. The second is to 
ensure that the required pressure, temperature, and 
air composition are maintained on board the aircraft. 
To shield passengers from aggressive environmental 
conditions outside the cabin, in particular, from low 
temperatures outside and to ensure comfortable 
thermal conditions for humans inside, a specific 
insulation material and air layer are provided in the 
construction of the multilayer wall of the aircraft 
cabin (Fig. 1a). When designing an aircraft, the 
most significant heat characteristic of the cabin is 
thermal protection, which can be determined based 
on the cabin heat exchange with the environment. 
Insulation material with a very small value of thermal 
conductivity coefficient is typically utilized in 
multilayer walls to insulate the aircraft cabin.
Several recent articles on this topic have 
considered the challenges of providing thermal 
comfort [6] to [9] and heat transfer in the cabin [10] 
to [14]. However, the construction of the multilayer 
cabin wall in the design of aircraft and its influence 
on the comfortable environment for passengers play 
a significant role, which is not considered in these 
papers. 
The multilayer wall of an aircraft cabin consists 
of the outer shell of the fuselage, the insulation layer, 
and the inner edging. There is an air layer between the 
edging and the insulation that houses the equipment 
wires and other hardware (Fig. 1b). 
Of particular interest is the intersection of the 
thermal insulation layer with the stringers of the 
fuselage frame since the thermal conductivity of the 
stringers is several orders higher than the thermal 
conductivity of the insulation [15]. At sufficiently 
low ambient temperatures outside the aircraft, 
stringers are good conductors, through which there 
is significant heat loss that can cause deterioration of 
the microclimate in the cabin and the appearance of 
condensation.
The key question that this paper will address is 
the thermal evaluation of cabin multilayer walls with 
a hat-stringer, which crosses the insulation layer. It 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641
636 Brazhenko, V. – Qiu, Y. – Cai, J. – Wang, D.
should be noted that this study can be useful for the 
comprehensive development of lightweight insulation 
materials [16] and [17], to study the reduction of 
moisture accumulation in insulation materials [18], as 
well as to find new ways to protect the insulation layer 
from moisture penetration [19] to [21].
1  NUMERICAL MODEL AND SETTINGS
The heat transfer process through a wall consisting 
of several layers with different thermophysical 
characteristics and homogeneous properties was 
analysed. Heat flux through the layer is directed 
along the normal to external and internal surfaces 
and contingent only on the temperature difference. 
Heat transfer through the edging and insulation 
layer is carried out as a result of the material thermal 
conductivity, as for the air layer due to thermal 
conductivity, natural convection, and heat emission. 
The principle of independent accounting of heat 
transfer was applied to the air layer. Also, the area with 
the insulation layer was considered, inside which the 
hat-stringer is positioned (Fig 1c).
The heat flux density due to heat conduction 
through the multilayer cabin wall with flat 
homogeneous layers without internal heat sources: 
∂T   ⁄  ∂n  =  dT     ⁄  dx, is determined with the Fourier 
equation:
 q
dT
dx
T   


  , (1)
where q is the heat flux density vector, λ is the layer 
thermal conductivity and equals to constant, Δ T is the 
temperature difference on boundaries of the layer, and 
δ is the layer thickness. For edging and insulation 
layers, λ is the thermal conductivity of solid material. 
In this model, we investigated the thermal resistance 
of the cabin multilayer wall, which involves an air 
layer with its specific dimensions and emissivity 
factor of boundaries. For the air layer, λ
AL
 is the 
equivalent coefficient of heat transfer by heat 
conduction and is defined as specified in Eq. (2):
   
AL AL aA LA L
 , (2)
where ε
AL
 is the convection factor to take into account 
the effect of natural convection on boundaries of 
air layer (Eq. (3)); λ
a
 is the thermal conductivity 
coefficient of air; δ
AL
 is the thickness of an air layer; 
α
AL
 is the radiation heat transfer coefficient. The 
coefficient ε
AL
 depends on the temperature difference 
at the surfaces of the air layer. The value of the 
coefficient can be determined using Eq. (3):
 

AL
AL
ei
ei
g
TT
TT


 

















0 105
05 273
3
2
.
.
Pr
 
 

 






03 .
, (3)
where g is the acceleration of gravity, ν is coefficient of 
kinematic viscosity of air, δ
AL
 is air layer thickness, T
e
, 
T
i
 are temperatures on the boundaries of air layer, and Рr 
is Prandtl numbers (taken up as Рr = 0.7). The radiation 
heat transfer coefficient is determined according to the 
following expression:
 

AL
ei
ei
C
TT

 






 
















0
44
273
100
273
100
11
1
 



 
 TT
ei
 (4)
where ε
e
 and ε
i
 are the emissivity coefficients of the 
air layer surfaces; C
0
 is the blackbody coefficient. 
Fig. 1.  Placement of stringers; a) cabin cross-section, b) original fuselage [22], c) fuselage design scheme

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641
637 Thermal Evaluation of Multilayer Wall with a Hat-Stringer in Aircraft Design  
It is necessary to solve a system of equations 
with unknown temperatures T
i
, T
e
 and total heat flux 
q to obtain the equation of heat transfer through the 
multilayer wall of the considered aircraft cabin.
 
qT T
qq qq TT
qT T
i
i
ia
AL
AL
ei
e
e
ce

 
















 
 
 
 



, (5)
where in this system q
λ
, q
α
, and q
ε
 are heat fluxes, 
caused by thermal conductivity, natural convection 
and heat radiation; δ
i
 and δ
e
 are thicknesses of 
insulation and edging layers; λ
i
 and λ
e
 are thermal 
conductivities of insulation and edging layers; T
а
 and 
T
c
 are temperatures on outside aircraft cabin and air 
in-cab environment. The change in the amount of heat 
that passes through a multilayer wall is related to the 
thermal resistance of each layer. The wall resistance 
is specified by the geometry configurations and heat 
properties of the layer material: r
full
 = δ
i
 ⁄ λ
i
 + δ
AL
 ⁄ λ
AL
 
+ δ
e
 ⁄ λ
e
; the temperatures T
i
 and T
e
 are determined by 
a solving system of Eq. (5) and taking into account 
values of Eq. (2) to (4). These temperatures enable 
determining the equivalent heat transfer coefficient 
λ
AL
, and values λ
i
 and λ
e
 were taken as constant heat 
transfer coefficients of corresponding layers. 
The thermal emissivity on the fuselage shell of a 
multilayer wall is mainly determined according to the 
ambient conditions outside the aircraft cabin. Since the 
thickness of the layer shell is relatively small and the 
thermal conductivity of its material has a sufficiently 
high value, we did not take into account the influence 
of the thermal resistance of the outer shell on the total 
resistance in the model. Thus, on the outer surface of 
the insulation layer, we set the temperature determined 
from outside of the aircraft cabin. The temperature on 
the internal surface of the edging can be set according 
to comfortable thermal conditions for people inside 
the cabin.
The system of equations in Eq. (5) consists of 
nonlinear algebraic equations. The exact solution to 
this system is impossible. Therefore, an approximate 
iterative method with the control of heat balance 
in mesh cells was proposed for solutions [22] and 
[23]. The temperature distribution in all layers and 
on internal surfaces of the multilayer wall was 
calculated numerically. The changing of heat transfer 
characteristics of insulation was estimated in the 
presence of a hat-stringer inside (Fig. 2). The influence 
of the air layer parameters on the change in multilayer 
wall thermal characteristics was investigated.
Fig. 2.  Schematic diagram of multilayer wall with hat-stringer  
in thermal insulation
The steady-state differential equation of heat 
conduction in a two-dimensional xy coordinate system 
has the form:
Fig. 3.  Calculation mesh and control cells: a) computational mesh view, b) internal control cell, c) control cell near boundary

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641
638 Brazhenko, V. – Qiu, Y. – Cai, J. – Wang, D.
 






















x
T
xy
T
x
  0. (6)
The temperature on the external surface of 
the insulation is set by the temperature outside the 
aircraft, as noted earlier. Adiabatic conditions ∂T ⁄ ∂x 
= 0 are fulfilled at the free boundaries. The boundary 
condition for the internal surface (e.g., for insulation) 
is written as:
 
iA Lc
T
y
TT


   . (7)
To solve Eq. (6) with given boundary conditions, 
we used a numerical method with mesh (Fig. 3a) based 
on the heat balance equation with the finite-difference 
approach over the control cell (Fig. 3b):
 qqqq
mn mn mn mn  

11 11
0
,, ,,
. (8)
The heat flux through the multilayer wall of the 
cabin can be written by substituting Fourier’s Law and 
the definition of heat rate per unit area for each heat 
flux node:
  
A
TT
x
A
TT
x
A
TT
mn mn mn mn
mn
1
1
2
1
3

 






 











,, ,,
,

m mn mn mn
y
A
TT
y








 






1
4
1
0
,, ,
,

 (9)
where A
1
, A
2
, A
3
 and A
4
 are the unit areas; T is the 
temperature in the corresponding node. Since in 
computation mesh Δ x = Δ y previous equation can be 
rewritten as with rearranging:
 40
11 11
TT TTT
mn mn mn mn mn ,, ,, ,

 
. (10)
For mesh cell with boundary case (Fig. 3c):
 42 0
11 1
TTTT
mn mn mn mn ,, ,,
.  
 
 (11)
In the last step, an augmented matrix [A | b] 
consisting of temperatures at each node was compiled. 
The solution of the matrix equation [A] x = b, was 
expressed as x = [A]
–1
b.
2  THERMAL EVALUATION OF MULTILAYER WALL 
For the thermal evaluation of the multilayer wall 
with a hat-stringer, the next modelling constants 
were adopted: air temperature of atmosphere outside 
and inside cabin: T
a 
= –25 °C, T
e
 = 20 °C; thermal 
conductivity and thickness of edging and thermal 
insulation, respectively: λ
e
 = 0,5 W/(m·°C), λ
i
 = 0.05 
W/(m·°C), δ
e
 = 3·10
–3
 m, δ
i
 = 9·10
–2
 m. The thermal 
conductivity in the air layer λ
a
 was calculated by the 
average temperature T
AL
 in the air layer. Air layer 
characteristics were varied for investigation: thickness 
δ
AL
 in a range from 0 m to 7.5·10
–2
 m and emissivity 
factor of air layer boundaries ε
e
 = ε
i
 from 0.1 to 0.9. As 
noted earlier, the aircraft cabin shell can be neglected 
because of its physical and geometric properties. 
Fig. 4 demonstrates a relative thermal resistance 
of insulation layer r
i
’  = r
i
 ⁄ r
full 
· 100 % and air layer 
r
AL
’  = r
A L 
⁄ r
full 
·100 %, Fig. 5 shows κ
full
 = 1 ⁄ r
full
 is 
the heat transfer coefficient; in Fig. 6, λ
full
 = ( δ
i
 + δ
AL
 
+ δ
e
) ⁄ r
full
 is the equivalent coefficient of thermal 
conductivity of cabin multilayer wall.
Fig. 4.  Contributions of air layers and insulation in full thermal 
resistance of multilayer wall ( δ
AL
 = 2.9·10
–2
 m),  
for different ε
e
 = ε
i
Fig. 5.  Heat transfer coefficient of multilayer wall for different 
values ε
e
 = ε
i
Fig. 6.  Thermal conductivity coefficient of cabin multilayer wall
The contribution of air layer thermal resistance 
for different values ε
e
 = ε
i
 is changing from 20.7 % 
for ε
e
 = ε
i
  =   0.1 to 8.1 % for ε
e
 = ε
i
 =  0.9 from the 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641
639 Thermal Evaluation of Multilayer Wall with a Hat-Stringer in Aircraft Design  
total resistance of multilayer wall. According to 
results in Fig. 5, we may conclude that increasing 
the thickness of the air layer by more than 2.9·10
–2
 
m does not provide an increase in multilayer wall 
thermal resistance.
Fig. 7 shows the changing of relative temperature 
in the insulation of the cabin multilayer wall, in which 
the hat-stringer is placed. For this, according to Earth’s 
atmosphere conditions, we considered temperature  
T
a
 = –55 ºC (the atmospheric temperature at an altitude 
of 11 kilometres), T
a
 = –65 ºC and T
a
 = –75 ºC 
(minimum temperature, depending on the season). 
The temperature inside the cabin was taken as T
с
 = 20 
ºC; the difference between cabin temperature and 
internal surfaces Δ T
w
 = T
c
 – T
w
 = 5 ºC; the heat-transfer 
coefficient of internal surfaces was 8 W/(m
2
·ºC). The 
thermal conductivity ratio between of hat-stringer and 
the thermal insulation was 5500. The relative height 
of the insulation layer was determined according to 
the expression: h = w
s
 /(h – h
s
) and equals 2.5, 3, or 
3.5, respectively. The evaluation of the thermal field 
for t
a 
= –55 ºC, h = 3.5 are shown in Fig. 6 in the 
curves form of constant relative temperature changing 
ψ
i
 = ( T
i
 – T
a
) ⁄ ( T
с
 – T
a
), where T
i
 is the temperature 
field in the insulation layer.
Fig. 7.  Relative temperature changing in thermal insulation layer 
with hat-stringer inside
Based on the numerical modelling data, the 
dependences of the temperature drop on the inner 
surface of the thermal insulation layer with a hat-
stringer inside for different insulation thicknesses and 
the temperature outside the cabin are plotted (Fig. 8):
 
w
ww nw wn
wn
TT TT
T

     
0

, (12)
where T
wn
 is the scaled temperature of internal layer 
surfaces; T
w0
 is the surface temperature without the 
effect of a hat-stringer; Δ T
wn 
= T
c 
– T
wn
 is the scaled 
temperature difference.
Fig. 8.  Temperature decreasing on internal wall  
within the area of hat-stringer
An estimation of the heat properties deterioration 
of the multilayer wall, associated with the location 
of a hat-stringer in the thermal insulation layer, was 
conducted. For this purpose, the effect of the ratio of 
the thermal insulation size to the size of the stringer, 
the change in the thermal conductivity coefficient of 
the thermal insulation was evaluated:
 


i
ii
i



  
0
0
100 %, (13)
where κ
i0
 and κ
i
 are the heat transfer coefficient of the 
insulation layer without and with a hat-stringer inside. 
Changing of the heat-transfer coefficient of thermal 
insulation, conditioned by the hat-stringer, are shown 
in Fig. 9.
Fig. 9.  Changing of heat transfer coefficient of thermal insulation, 
conditioned by hat-stringer

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)10, 635-641
640 Brazhenko, V. – Qiu, Y. – Cai, J. – Wang, D.
3  CONCLUSIONS
In this article, the results of numerical modelling and 
evaluation of thermal characteristics of multilayer wall 
of aircraft cabin with hat-stringer inside are presented. 
Summarizing the results, it can be concluded that:
1.  According to heat transfer modelling through 
an aircraft cabin wall with an air layer and hat-
stringer inside, the evaluation and prediction 
of thermal characteristics were obtained. As the 
results of the numerical modelling demonstrate, 
the thermal resistance of a cabin’s multilayer wall 
does not change with the increase of air layer 
thickness for more than 2.9·10
–2
 m. 
2.  The contribution of the air layer to the overall 
thermal resistance of the multilayer wall of the 
aircraft cabin decreases with the increasing 
emissivity factor of the layer surfaces and equals 
20.7 % for ε
e
 = ε
i
 =  0.1 and 8.1% for ε
e
 = ε
i
 =  0.9.
3.  The heat changing in the insulation layer with a 
hat-stringer inside is presented on the grounds 
of numerical modelling. The modelling results 
prove that the small thickness of the insulation 
layer in the area of the hat-stringer in possible 
minimum temperature contributes to a significant 
deterioration of thermal resistance in the aircraft 
multilayer wall, including the rise of the heat 
transfer coefficient of insulation by more than 47 
% and a weighty change in the temperature drop 
between the internal walls within the area of the 
insulation layer.
The presented results make it possible to evaluate 
the thermal characteristics of cabin multilayer walls 
depending on the material parameters of layers during 
the rational design creation for the aerial vehicle.
4  ACKNOWLEDGEMENTS
The research described in this paper was financially 
supported by the National Natural Science Foundation 
of China (51976201) and the Natural Science 
Foundation of Zhejiang Province (LY22E06006)
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