*Corr. Author’s Address: Birla Institute of Technology, Mesra, Department of Mechanical Engineerng. Mesra, Ranchi 835215, India, skdhiman@bitmesra.ac.in 342
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354 Received for review: 2023-11-18
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-04-10
DOI:10.5545/sv-jme.2023.864 Original Scientific Paper Accepted for publication: 2024-04-25
Estimation of Surface Temperature and Heat Flux over a Hollow 
Cylinder and Slab using an Inverse Heat Conduction Approach
Roy, A.D. – Dhiman, S.K.
Apoorva Deep Roy – Sushil Kumar Dhiman
*
Department of Mechanical Engineering, Birla Institute of Technology, India
A non-iterative approximation of the inverse heat conduction problem has been developed through energy balance between the control 
volumes. The heat flow domain along the surface normal was divided into three control volumes and studied in three cases, specifically on 
a heated hollow metallic cylinder cooled in crossflow of air, a heating flat plate in still air, and a heated flat plate when cooled in still air with 
random air blows. The time derivatives of temperature and heat flux estimates at the surface appearing in derived equations were evaluated 
by approximate expressions. Both estimates were obtained from the derived equations by simultaneous measurement of temperature-time 
histories at two locations: one at the inner surface and the other anywhere within the control volume along the surface normal. The deviation 
was checked by real-time simultaneous measurement of temperature-time history at the outer surface along the normal surface. Comparison 
between the estimated surface temperature-time history using the derived equations and the real-time measurement showed deviations 
within 0.5 % for the cylinder, within 0.03 % for the plate in still air and within 0.5 % when air blows were given. Heat fluxes estimated using 
these time histories were correspondingly arrived at consistent and close approximations for all cases. Estimates from the derived equations 
were compared with reported equations from the literature, and a wind tunnel experiment for the validation of circumferential distribution of 
heat transfer was conducted, for which they also showed reasonably good agreements. 
Keywords: surface temperature and heat flux, inverse heat conduction, energy balance approach, hollow cylinder and flat plate, derived 
equations
Highlights
• 	 A non-iterative approximation of the inverse heat conduction problem has been developed through energy balance between 
the control volumes.
• 	 Three cases were studied: on a heated hollow metallic cylinder cooled in crossflow of air, a heating flat plate in still air, and a 
heated flat plate when cooled in still air with random air blows.
• 	 Estimates of surface temperature and heat flux were obtained by derived equations.
• 	 The developed methodology was tested experimentally and validated from the reported literature.
0  INTRODUCTION
In many engineering applications, such as rocket 
nozzles, boiler tubes, nuclear reactors, etc., where 
high temperatures and pressures persist, a direct 
measurement of surface temperature and heat flux 
may lead to the burning of sensors as well as create 
disturbances in fluid flow structures. To handle such 
situations, the sensors were placed in some interior 
positions within the solid to capture temperature 
histories so that the corresponding surface temperature 
histories and heat flux could be estimated. Such 
a problem (or approach) is referred to as inverse 
heat conduction problem (IHCP) or inverse heat 
conduction approach. 
Hollow cylinders have been extensively used 
in the aforementioned industries under various 
types of heat exchangers, including boiler tubes, 
reheater tubes, superheater tubes, etc. In engineering 
applications, such as rocket nozzles and hard material 
machining, etc., slabs are used for which IHCP’s 
solution techniques are utilized to estimate the surface 
temperature and heat flux.  
The solution of an IHCP is studied in domains of 
direct and inverse problems. The direct problem uses 
the known boundary conditions, and the variation 
within the solid is investigated. However, under an 
inverse problem, which is mathematically ill-posed in 
nature, only one boundary condition is known, and it 
requires some special techniques of solutions. Here, the 
prime importance is given to errors in measurement. 
The measurement error should be minimized to obtain 
close approximations of the estimates. Many iterative 
or non-iterative techniques have been presented to 
minimize the errors in measurements.
Taler [1] utilized a singular value decomposition 
method and the Levenburg-Marquardt method (LMM) 
to solve IHCP and reported that the singular value 
decomposition method takes less computational time. 
Golsorkhi and Tehrani [2] utilized LMM and a finite 
volume method (FEM); Cuadrado et al. [3] proposed 
a non-iterative approach, whereas Bohacek et al. [4] 
utilized minimizing the Least Squares norm between 
experimental and simulated data to obtain a solution 
of an IHCP.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
343 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
Felde and Reti [5] presented an iterative inverse 
algorithm to solve IHCP for evaluation of the hardening 
performance of cooling media. Charifi and Zegadi [6] 
utilised an inverse global descent method based on a 
finite difference method to solve an inverse problem 
for spherical geometry solidification. Gostimirović et 
al. [7] used a numerical method with finite differences 
in an implicit form to predict grinding wheel and 
workpiece interface surface temperature as well as 
heat flux by measuring time temperature history at 
the interior surface. Hribersek et al. [8] determined the 
surface heat transfer coefficient for nitrogen in liquid 
and gaseous phases on an Inconel 718 workpiece by 
numerical simulation.
Stolz [9] solved an IHCP for shapes like slabs, 
cylinders, and spheres kept at different thermal 
conditions in a fluid medium by the principle of 
superposition. The formulation was done by measuring 
transient temperature in proximity to the outer surface 
of the solid. To obtain an IHCP solution, Frank [10] 
applied the least-squares method on experimentally 
determined temperatures. Sparrow et al. [11] presented 
an integral technique coupled with finite difference 
approximation to obtain the solution of an IHCP by 
the measurement of transient temperature within the 
solid of different shapes. Beck and Wolf [12] and Beck 
[13] analysed the non-linear IHCP and solved it using 
the finite difference method (FDM).
Imber and Khan [14] presented an approximation 
method for the solution of IHCP consisting 
of a polynomial function, and they implanted 
thermocouples to measure the transient temperature 
inside the solid. Space-marching methods have 
been utilized to obtain the solution of IHCP [15] and 
[16]. Li and Yan [17] estimated heat flux by using 
the conjugate gradient method. Masanori et al. [18] 
utilized the Laplace transform technique, considering 
cylindrical, spherical, and rectangular coordinates by 
measuring the transient temperature at two interior 
locations within a body for solving IHCP. Chen et al. 
[19] presented a solution of IHCP on a circular pipe 
with turbulent fluid flow by measuring the temperature 
within the fluid at different locations. 
Jang et al. [20] formulated an IHCP solution 
method by using a finite-element scheme and an input 
estimation method on the hollow cylinders. Chen et 
al. [21] carried out an analysis on a circular pipe with 
laminar fluid flow and solved the IHCP using the 
matrix system form of governing equations. 
Wikstrom et al. [22] estimated surface heat flux 
and temperature by recording transient temperatures 
at three different locations in a thick slab by using 
the Crank-Nicholson implicit scheme. Dennis and 
Dulikravich [23] solved a three-dimensional IHCP by 
using FEM. Kim and Lee [24] suggested a maximum 
entropy method to get a solution for an IHCP. The 
space marching method has been utilized to predict 
the control rod stem surface temperature [25]. 
  Dhiman and Prasad [26] used LMM coupled 
with Duhamel’s theorem, whereas Singh et al. [27] 
utilised LMM coupled with truncated Fourier series 
for solving IHCP. Cebula et al. [28] used the finite 
element-finite volume method to obtain the solution 
of an IHCP. Duda and Konieczny [29] proposed a new 
method to solve IHCP for a complex configuration. 
Mehta [30] estimated the outer surface heat flux of 
a rocket nozzle by using FDM. Da Silva et al. [31] 
estimated the heat transfer coefficient in a stacked 
microchip by using fundamental solutions. Shahnazari 
et al. [32] and Matsevityi et al. [33] formulated a 
hybrid method. 
Seryakov [34] presented a mathematical 
formulation, Gomez et al. [35] presented a virtual 
experiment approach, whereas Leu and Tu [36] 
proposed a hybrid exact solution method to solve 
IHCP. Cialkowski et al. [37] solved an IHCP to obtain 
the surface temperature on a gas turbine blade coated 
with ceramics. A literature review was presented by 
Roy and Dhiman [38] to discuss the various methods 
of finding solutions for IHCP.
The brief literature review presented above 
clearly indicates that the profound utilization of IHCP 
in numerous research studies has been carried out in 
previous decades. The technological developments 
in recent decades have been a result of the search for 
alternative possible methodologies or mathematical 
approaches that closely estimate the unknown 
parameter. Hence, there is always a scope for the 
development of novel mathematical approaches 
which, if available, can be readily utilized to 
accomplish the tasks. The research gap in the present 
approach highlights the use of the heat flow domain 
fragmented into several control volumes and analysis 
of their energy balance equations, which is a novel 
work in the IHCP approach. 
The advantages of using the present methodology 
can be highlighted as an inverse approach that 
provides a very close-exact solution with simplicity 
in equations, based on a non-iterative approach of 
approximating the estimates and being able to treat 
linear inverse heat conduction problems with data 
containing random errors. Typically, while solving 
IHCPs, errors in the temperature data affect the 
accuracy of the difference approximation of time 
derivatives, which leads to large errors. The present 
technique does not involve sequential stepping 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
344 Roy, A.D. – Dhiman, S.K.
forward in time and overcomes the above-mentioned 
drawback. It is also not necessary under the presented 
technique to compute all the nodal temperatures at 
each time step. In contrast to most previous IHCP 
solution methods, this method does not require any 
information about the initial temperature distribution 
in the solid. The surface heat flux at a specific time 
can be directly calculated from the transient measured 
temperature inside a solid without step-by-step 
computation in the time domain.
The present paper aims to develop a methodology 
based on a non-iterative approach of approximating 
the estimates through IHCP. To validate the derived 
estimate equations, experiments are performed on a 
heated hollow metallic cylinder cooled in crossflow of 
air, a heating flat plate in still air, and a heating flat 
plate with random air blows when cooled in still air. 
The surface temperature and heat flux were estimated 
by acquiring time histories of temperature at some 
interior location of the solid by means of a T-type 
thermocouple in all the cases. 
The estimated results were compared with the 
measured real-time surface temperature and calculated 
surface heat flux. The derived equation outcomes were 
also compared with the results of the exact solution 
from the reported literature for the flat plate. 
1  PROBLEM STATEMENT AND ANALYSIS
Consider a hollow metallic cylinder initially heated 
with a constant heat flux condition, assuming that the 
entire wall of the cylinder is at a uniform temperature. 
Here, “initially” refers to the condition when no 
external flow of fluid is given across the cylinder, 
while the “constant heat flux” refers to the heat flux 
provided to the cylinder is the same all the time, i.e., 
initially as well as when its external surface is exposed 
to the fluid flow. 
Fig. 1. Schematic view of a heated hollow cylinder in the flow of air
Referring to Fig. 1 as per the cylindrical 
coordinate system where a hollow cylinder has been 
shown with inner and outer radii r
in
 and r
out
 and an 
intermediate radius r
M
. The region between r
in 
and r
M
 
represents the direct region, while the region between 
r
M
 and r
out
 represents the inverse region. 
Referring to Fig.2, a plan of inverse analysis in a 
quadrant of the cylinder is considered. Three control 
volumes (cv) cv
1
, cv
2
, and cv
3
, named radially inwards, 
have been considered in the Inverse region of step size 
Δr.  Each of these control volumes have been provided 
a central node (L = 0, 1, 2, …, M–3, M–2, M–1, M, M+1) 
by considering node M–2 at r
out 
(at L = M–2 = 1 if 
r
out
 = r
1
), node M at r
M
 (or at L = M = 3 if r
M
 = r
3
), node 
M–1 at r
M–1 
(or at L = M–1 = 2 if r
M–1
 = r
2
), node M+1 
at r
M+1 
(or at L = M+1 = 4 if r
M+1
 = r
4
) and node F at 
r
in
. Of all these nodes, node F and node M represent 
the real-time measurement locations, and node M–2 
is the real-time estimated location. To complete the 
solution, a fictitious node M–3 with step size Δr has 
been assumed beyond r
(M–2)
.
In addition, the corresponding radii of outer 
control surfaces r'
(M–2)
, r'
(M–1)
,
 
r'
M
, and r'
(M+1)
 have 
been considered, for instance, r'
M
 = r
M
 + (Δr/2). 
1.1  Problem Statement
A heated hollow cylinder of isotropic material having 
no internal heat generation, when subjected to an 
external flow of fluid of constant fluid properties, 
undergoes unsteady temperature variation at any point 
on its surface. Hence, a one-dimensional mathematical 
statement of the problem can be represented in 
cylindrical coordinates as [39]:
 








 
2
2
11
0
T
r r
T
r
T
t
rr rt
M out

,
. fora nd (1)
The heat conduction in the r-direction through the 
direct region provides the heat flux q
M
 to the Inverse 
region, which is expressed as [39]:
 
qt K
Trt
r
rr
ttjN
MM
j
 
 











,
,
,, ,..., .
at
for times 12 (2)
We may write the Newton’s law for convective 
heat transfer in terms of convective coefficient as [39]: 
 
ht
qt
Tt T
rr
ttj
out
j
 

 











1
1
12
at
for times
,
,, ,...,N N. (3)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
345 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
Here, T
∞
 is the ambient temperature and the 
temperature T
1
(t) as well as the heat flux q
1
(t) at 
a surface node are the unknown parameters. The 
unknowns T
1
(t) and q
1
(t) will be solved by the 
analysis of the energy balance approach among the 
control volumes.
1.2  Energy Balance Between Control Volumes
Consider (r
L
, θ
J
) as a one-dimensional line of analysis, 
where suffix J shows the azimuthal angle in terms 
of degrees of rotation in an anticlockwise direction, 
J = 0, 1, 2, …, J. For the present case, consider (r
L
, θ
0
) 
as the line of analysis. The outer control surface of 
cv
1
 is so assumed that the node M-2 lies at r
out
. The 
connectivity between all the cv’s is so provided that 
the inner control surface of the preceding cv merges 
with the outer control surface of the succeeding cv. 
First, the energy balance is applied at cv
3
. To initiate 
the analysis, a prerequisite of known temperature-
time history is needed at r
F
 and r
M
. It provides the 
heat flux q
M
 to the inverse region, which subsequently 
determines temperatures at the nodes M–1,
 
M–2, …, 1, 
where node M–2 is the desired location of estimates at 
the surface.
To address circumferential issues, highly 
responsive thermocouples may be fixed below the 
outer surface at radius r
M
 of the metallic cylinder 
at different intervals along the azimuthal direction, 
which may be used for the measurement of transient 
temperature. Another method involves testing 
the cylinder with only two thermocouples fixed 
along the radial direction below the outer surface. 
In this method, the cylinder is rotated at different 
intervals along the azimuthal direction and transient 
temperature measurements are taken.
1.3  Heat Balance Equation
Heat accumulated within the 
 cv = (Heat input to cv) – (Heat exit from cv).
At node M: 
 



rr rC
f
t
Kr
r
Tf
Kr
r
Tf
MM
MM
MM
M
MM
''
'
'
 


 
 




1
1
1
1
2
2

, (4)
Fig. 2.  Plan of Inverse analysis in a cylinder

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
346 Roy, A.D. – Dhiman, S.K.
At node M–1:



rr rC
T
t
Kr
r
fT
Kr
r
TT
MM
MM
MM
M
MM
''
'
'





 


 

1
1
1
1
21
2
2

  , (5)
At node M–2:



rr rC
T
t
Kr
r
TT
Kr
r
T
MM
MM
MM
M
M
''
'
'





 


 

21
21
12
2
2
2

3 31
 

T
M
. (6)
Using Fourier’s law of heat conduction and the 
boundary conditions for t > 0 at r = r
M 
and
 
at r = r
M–2
, 
Eqs. (7) and (8) for temperature can be represented.
 TT
qr
K
MM
M


11
2 
, (7)
 TT
qr
K
MM
M



31
2
2 
. (8)
Hence, the modified form of the equation at node 
M on substitution of T
M+1
 is 
Tf
r
f
t
r
rr
qr
K
MM
MM
MM
M




 


 

1
2
1
1
1
2
2 

'
''
, (9)
which in turn substituted in Eq. (10) at node M–1 that 
is modified as 
Tf
rr
r
r f
t
r f
t
MM
MM
M
MM




 





 


 




2
1
1
24
2
2
2
4
''
'

 
 








 








r
rr
r
K
q
t
qr r
Kr r
M
MM
M MM
MM
1
1
3
1
1
2
'
''
'
''


 





, (10)
The substitution of T
M–1
, T
M–2
 and T
M–3
 in the 
equation at node M–2 provides heat flux q
M–2
 at node 
M–2, which is expressed as
    
q
Kr
r
rr
f
r
f
t
q
Kr
r
MM M
MM
MM
 

 









2
1
21 2
1
4
2
4



'
''
'



r rr
T
r
T
t
MM
MM
'' 











1
2
2
2
2

. (11)
Finally, considering five nodes from the surface at 
L = 0, 1, 2, 3 and 4, as shown in Fig. 2, by considering 
node 1 at r
out 
(at L = M–2 = 1 if r
out
 = r
1
), node M at r
M
 (at 
L = M = 3 if r
M
 = r
3
), node M–1 at r
M–1 
(at L = M–1 = 2 if 
r
M-1
 = r
2
), node M+1 at r
M+1 
(at L = M+1 = 4 if r
M+1
 = r
4
) 
and node 0 as fictitious node after node 1. The above 
equations, which represent estimates of temperature 
and heat flux at node L = M–2 = 1, will be expressed 
as the surface node, which can be rewritten as follows:
 
Tf
rr
r
r f
t
r f
t
r
r
13
23
2
2
3
4
2
2
3
2
4
4

 





 


 






''
'
'

 
'' '
'
''
34
3
3 34
34
2







 



 





r
r
K
q
t
qr r
Kr r


, (12)
 
q
Kr
r
rr
f
r
f
t
q
Kr
r
rr
1
1
12
3
2
3
3 4
34
4
2
4
 














'
''
'
''


 2 2
1
2
1
T
r
T
t
 







. (13)
In a similar way, a heated flat plate of isotropic 
material having no internal heat generation, when 
subjected to an external flow of fluid of constant 
fluid properties, undergoes unsteady temperature 
variation at any point on its surface. Hence, a one-
dimensional mathematical statement of the problem 
can be represented in cartesian coordinates by 
considering cylindrical coordinates within the limits 
of all radii tending to infinity. So, the equations for 
estimates of temperature and heat flux obtained in 
cylindrical coordinates can be modified, as shown 
below, assuming r'
(M–2)
 = r'
(M–1)
 = r'
M
 = r'
(M+1)
 = r', 
for which Fig. 2 can be referred. The derivatives in 
the equations have been approximated by the finite 
difference approach.
 
Tf
r
f
t
r
f
t
r
K
q
t
qr
K
MM
MM
M
M


 


 


 



2
24
2
2
2
3
2
2
2


 

,, (14)
q
K
r
fT
Kr f
t
T
t
q
MM M
MM
M 

  












22
2
2 


.(15)
2  EXPERIMENTAL APPROACH FOR VALIDATION
Validation of the derived equations was conducted 
using a hollow cylinder initially heated and a flat 
plate initially unheated. An experimental model of the 
cylinder shown in Fig. 3 was placed in a test section 
of the subsonic wind tunnel with dimensions of 0.3 
m × 0.3 m and 0.6 m long, as shown in Fig. 4. After 
initial heating of the hollow cylinder by means of 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
347 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
inbuilt heating element, it was allowed to cool under 
the flow of air at ambient condition. The model of a 
flat plate was kept in an insulated chamber, which was 
subjected to heating by means of a hot plate followed 
by intermittent blow of air.
a)
b)
Fig. 3.  Experimental models a) real experimental tool; b) a scheme
Both models were made of aluminium (K = 200 
W/(mK)), thermal diffusivity (α = 7.4 × 10
–5
 m
2
/s) in 
which three T-type thermocouples of size 0.1 mm were 
fixed in the material. Of the three thermocouples, two 
were placed at the in and out extreme surfaces while 
one was placed at the centre of the wall thickness 
along the line of surface normal. Hence, all three 
thermocouples lie in a straight line. The dimensions 
of the cylindrical model were ID = 22 mm, OD = 31.6 
mm and height = 125 mm, and that of the flat plate 
model was length = 125 mm, width = 50 mm, and 
thickness = 5 mm.
The aluminium cylinder was fixed using a 
Teflon plug in the test section. A heating cartridge 
was pasted on one part of the plug, which fits into 
the aluminium cylinder. The other part of both the 
plugs was fitted into aluminium cylinders in such 
a way that the complete assembly is 0.3m in height. 
Thus, a high aspect ratio (H/D = 9.49) and a very low 
blockage ratio (D/W = 0.105) were obtained in the test 
section. Therefore, the dimension of the test cylinder 
is selected based on a high aspect ratio and very low 
blockage ratio, whereas the dimension of the plate is 
selected arbitrarily to suit available lab equipment.
As shown in Fig. 3, the surface with heat input 
is treated as F, which lies in the direct region, and the 
surface with heat exit is treated as 1, which is in the 
inverse region, and an intermediate layer is considered 
at the centre of the wall thickness at M which is the 
interface of the direct and inverse region. At all these 
locations, the measurement of temperature vs time 
history was conducted; however, the thermocouple 
Fig. 4.  Experimental subsonic wind tunnel

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
348 Roy, A.D. – Dhiman, S.K.
at 1 has been used for the comparison with the 
temperature-time history obtained by the derived 
equation. NI PCIe6321 card with signal conditioner 
has been utilised for temperature acquisition, for 
which a LabVIEW program was written.
3  RESULTS AND DISCUSSION
3.1  Heated Hollow Metallic Cylinder Cooled in Crossflow 
of Air
Fig. 5 illustrates the time history of temperature 
measured at node 1, i.e., T(1), when an initially heated 
cylinder was allowed to cool via air flowing in a 
subsonic wind tunnel.
Fig. 5.  Comparison of measured and estimated temperature  
vs time history for initially heated hollow cylinder  
placed in the flow of air
Simultaneously measured temperature histories 
at nodes F and M, i.e., T(F) and T(M), are also 
represented, which were utilised to estimate a 
simultaneous time history of temperature at node 
1, i.e., T*
1 
(estimated surface temperature in ºC 
by derived equation, i.e., T
1
), which is obtained 
from derived Eq. (12) of temperature. The standard 
deviation σ of T*
1 
and T(1) was then estimated and 
subtracted from the T*
1
; thereby, the estimated 
temperature-time history at node 1 as T#(1) has been 
illustrated in the figure.
Hence, the empirical form of temperature 
estimate at the surface can be represented as:
 T#(1) = T
*
1
 – σ. (16)
It is hereby made clear that the σ must be  
evaluated only once when the uncertainty has to be 
evaluated. Fig. 6 illustrates the percentage deviation 
between T#(1) and measured T(1), which was 
consequently found to be less than 0.5 %. The heat 
fluxes were then estimated using the derived Eq. (13) 
for the case of cylinder, which can be depicted in Fig. 
7.
Fig. 6.  Percentage deviation between measured temperature 
history T(1) and estimated temperature history T#(1) for initially 
heated hollow cylinder placed in the flow of air
Fig. 7.  Comparison of heat fluxes using derived equation by means 
o f	measured	as	w ell	as	es timat ed	˝t em perature	–	time	his t o r y	f o r	
initially heated hollow cylinder placed in the flow of air
The presented methodology explains that T(F) 
is the temperature of the inner hot surface, which 
is ideally at an unchanged temperature. The wall 
material (metallic aluminium) will have extremely 
low temperature gradients. T(1) is the temperature 
at the outer surface, which is treated as an interface 
of internal and surface resistances. This surface 
is always exposed to the flow of ambient air. As a 
result, the surface temperature drops drastically with 
the airflow and raises temperature gradients along 
the F–M–1 direction. The combined effect of inner 
hotness and outer cooling initially results in very 
low temperature gradients followed by gradually 
increasing temperature gradients along F–M–1. From 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
349 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
considering measured temperatures T(1) represented 
by q(1). Both exhibit remarkably good consistency 
with negligible deviations.
Fig. 9.  Percentage deviation between measured temperature 
history T(1) and estimated temperature history T#(1) for flat plate 
heated in an ambient environment of atmospheric condition
Fig. 10.  Comparison of heat fluxes using derived equation 
by means of measured as well as estimated temperature vs 
time history for flat plate heated in an ambient environment of 
atmospheric condition
3.3  A Heated Flat Plate when Cooled in Still Air with 
Random Air Blows
Fig. 11 illustrates the time history of temperature 
measured at node 1, i.e., T(1), simultaneous measured 
temperature histories at nodes F and M, i.e., T(F) and 
T(M), which were utilised to estimate a simultaneous 
time history of temperature at node 1, i.e. T*
1
 
and subsequently T#(1) using Eqs. (14) and (16), 
respectively, when one surface of a flat plate was 
heated with a consistent heating source, and the other 
surface was exposed to an ambient environment of 
atmospheric condition, which was given random air 
blow at 35 °C. Simultaneously measured temperature 
histories at nodes F and M, i.e., T(F) and T(M), are 
also represented, which were utilized to estimate a 
the experiments, a noticeable temperature gradient 
was observed approximately at the middle plane, 
which was why the M was taken as the centre of F–1. 
Hence, the T(F) and T(M) have a small difference, 
while T(M) and T(1) have a large difference.It 
illustrates the heat fluxes estimated using temperature 
estimates T#(1) obtained from Eq. (16) represented by 
q#(1) and measured temperatures T(1)
 
represented by 
q(1). The estimated heat flux q#(1) shows reasonably 
good consistency with heat flux q(1); however, the 
consistent deviation between them is due to the 
temperature estimate’s deviation between T#(1) and 
T(1)
 
as of approximation errors that will always exist 
because T#(1) is estimates and not the exact. 
3.2  A Heated Flat Plate in Still Air
Fig. 8 illustrates the time history of temperature 
measured at node 1, i.e., T(1), simultaneous measured 
temperature histories at nodes F and M, i.e., T(F) and 
T(M), which were utilised to estimate a simultaneous 
time history of temperature at node 1, i.e., T*
1
 
and subsequently T#(1) using Eqs. (14) and (16), 
respectively, when one surface of a flat plate was 
heated with a consistent heating source, and the other 
surface was exposed to an ambient environment of 
atmospheric condition.
Fig. 8.  Comparison of measured and estimated temperature  
vs time history for flat plate heated in an ambient environment  
of atmospheric condition
The percentage deviation of T#(1) with measured 
T(1) was consequently found to be less than 0.03 %, 
as shown in Fig. 9.
Fig. 10 illustrates the comparative variations 
between estimated surface heat flux obtained by 
using the derived Eq. (15) considering temperature 
estimates T#(1) of Fig. 8 represented by q#(1) and by 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
350 Roy, A.D. – Dhiman, S.K.
corresponding time history of temperature at node 1, 
i.e. T*
1
, by means of presented derived Eq. (14) of 
temperature. 
Fig. 11.  Comparison of measured and estimated temperature  
vs time history for initially heated flat plate given random air blows
The percentage deviation of T#(1) with measured 
T(1) was consequently found for the initially heated 
flat plate given with random air blows, showing 
less than 0.01 % when no blow was given and it is 
less than 0.5 % when a blow was given, which can 
be depicted respectively in Figs. 12 13, showing the 
comparative variations between heat fluxes obtained 
by using derived Eq. (15) considering T#(1) and 
T(1) i.e., by means of estimated as well as measured 
temperature vs time history, respectively, for the 
initially heated flat plate given random air blows. 
q#(1) exhibits remarkably good consistency with q(1) 
with negligible deviations.
Fig. 12.  Percentage deviation between measured temperature 
history T(1) and estimated temperature history T#(1) for initially 
heated flat plate given random air blows
Fig. 13.  Comparison of heat fluxes using derived equation by 
means of measured as well as estimated temperature vs time 
history for initially heated flat plate given random air blows
4  VALIDATIONS WITH THE EARLIER  
PUBLISHED LITERATURE
A comparison of derived equations in the present 
study was conducted with the reported literature by 
Burggraaf [40] for both estimated temperature and 
heat flux at the surface, which can be observed in Figs. 
14 to 16. Burggraaf [40] has given an exact solution 
for IHCP in various geometrical shapes. For the plane 
slab, the equation for the estimated temperature at the 
surface is expressed as:
  
Tx tT t
n
x T
t
x
K
qt
n
n
n
n
,
!
   




















 


 0
1
2
0
0
1
2 

 
 





















n
n
n
n
n
x q
t
1
2
0
1
21 !
,


 (17)
where x, T(x,t), T
0
(t) and q
0
(t) represent respectively 
the thickness of the inverse region, the estimated 
temperature of the outer surface, the measured 
temperature at some inner location and the heat flux 
at the measured location. For n = 2, i.e., considering 
the first two terms of summation terms, the equation 
is represented as:
Tx tT t
x T
t
x T
t
x
K
qt
,    















  
0
2
0
4
2
2
0
2
0
1
2
1
24
1
 
6 6
1
120
2
0
4
2
2
0
2
x q
t
x q
t  















, (18)
Fig. 14 shows a comparison of estimated 
temperatures using derived Eq. (14) by means of 
measured temperature-time history between the 
present study and Burggraaf [40] for an initially heated 
flat plate given random air blows. The comparison 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
351 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
depicts consistent and close approximations 
throughout the time history.
Fig. 14.  Comparison of estimated temperatures using derived 
equation between the present study and Burggraaf for initially 
heated flat plate given random air blows
The heat flux equation reported by Burggraaf 
[40] is also shown by Eq. (19) and for n = 2, i.e., 
considering the first two terms of summation terms, 
the equation is represented by Eq. (20), where q
s
(t) 
is the estimated heat flux at the outer surface. The 
estimated heat flux in the present study was compared 
with Burggraaf [40] using the same measured 
temperatures in both equations.
qt c
n
T
t
qt
n
n
n
n
n
n





 








 








1
2
1
0
0
1
1
21
1
2
!

 









!
,


2
1
0
n
n
n
q
t
 (19)
 
qt qt cx
T
t
cx
x T
t
x q
t
x q
S
  










0
0
2 2
0
2
2
0
4
2
2
1
6
1
2
1
24


 
0 0
2
t
.
 (20)
Comparisons of heat fluxes at the outer surface 
using derived Eq. (15) in the present study by means 
of measured as well as estimated temperature vs time 
history with Burggraaf [40] for the flat plate heated 
in an ambient environment of atmospheric condition 
shown in Fig. 15 as well as for initially heated flat 
plate given random air blows shown in Fig. 16. Both 
figures show reasonably good and consistent data 
of heat flux with the time history and establishes a 
favourable validation. 
In Fig. 15, the difference between the results of 
the present study and the Burggraaf [40] results have 
increased over time as the third derivative of the 
measured temperature f
M 
only in the present developed 
solution Eq. (15), which is apparent from simultaneous 
observation of Eqs. (14), (15) and (20). In Eq. (14), 
the second derivative of f
M
 exists, and further putting 
T
M-2 
from Eq. (14) in Eq. (15), there is an increase in 
the derivative of f
M
 from the second derivative to the 
third derivative due to the differentiation of term T
M-2
.
 
The presence of the third derivative term in Eq. (15) 
more accurately estimates the surface heat flux with 
time.
Fig. 15.  Comparison of heat fluxes at the outer surface using the 
derived equation in the present study with Burggraaf for flat plate 
heated in an ambient environment of atmospheric condition
 
Fig. 16.  Comparison of heat fluxes at the outer surface using the 
derived equation in the present study with Burggraaf for initially 
heated flat plate given random air blows
5  CIRCUMFERENTIAL DISTRIBUTION  
OF NUSSELT NUMBER IN A CYLINDER
An experiment was conducted to determine the 
circumferential distribution of the Nusselt number 
around the cylinder. The cylinder was a three-piece 
model assembled as a single cylinder. Of the three, 
two were Teflon cylinders of 22 mm ID, 31.9 mm 
OD and 135 mm height, while the test cylinder is of 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
352 Roy, A.D. – Dhiman, S.K.
copper having the same ID and OD as that of Teflon 
cylinders while the height was kept 30 mm. The 
copper test cylinder was fixed between the two Teflon 
cylinders so as to restrict the longitudinal flow of heat. 
Hence, variation of temperature takes place in radial 
and azimuthal (circumferential) direction only. The 
experimental setup and the cylinder in the wind tunnel 
test section (size: 0.3 m × 0.3 m × 0.6 m) have been 
shown in Fig. 4) while the original photographs and 
the schematic diagram of the test cylinder model have 
been shown in Fig. 17. The cartridge heating was done 
with a DC power supply. The cylinder was heated 
to 20 ºC above the ambient temperature. For every 
reading at a 10º interval in the azimuthal direction 
from the front stagnation point, the cylinder was 
rotated by 10º. The Nusselt number (Nu) distribution 
thus obtained is represented in Fig. 18. The flow 
Reynolds number based on the diameter was kept at 
35,000. The obtained Nu distribution was compared 
with the Nu distribution of Tsutsui and Igarashi [41], 
who conducted the experiments at Reynolds numbers 
of 31,000 and 41,000, as shown in Fig. 19. The Nu 
distribution obtained from the present methodology 
was well in agreement with those of Tsutsui and 
Igarashi [41].
    
Fig. 17.  Photographs of a cylinder in the test section of the wind 
tunnel
Fig. 18.  Schematic diagram of the cylinder assembly
6  CONCLUSIONS
A novel approach developed to estimate the surface 
temperature and surface heat flux has been explained 
in the present paper. The approach is based on the 
inverse heat conduction method, where the heat flow 
domain is divided into some control volumes, and the 
energy balance between them is analysed to arrive at 
equations for temperature and heat flux estimations. 
The derived equations were tested by conducting 
experiments that show very close approximations with 
respect to the exact data. All the estimates lie within 
0.5 % deviations from the exact data, which is very 
close data; however, the flat plate smooth cooling or 
heating shows extremely close data within 0.03 %. 
To address the circumferential variation of the heat 
transfer, an experimental approach of validation was 
conducted in a wind tunnel for a crossflow of air 
with a cylinder under constant heat flux conditions 
that validates well with the reported literature data. 
Hence, the proposed methodology of the inverse heat 
conduction approach is valid and recommended to be 
used.
Fig. 19.  Nusselt number distribution over a single cylinder 
obtained from the present methodology and its comparison with 
the Nu distribution of Tsutsui and Igarashi [41]
7  ACKNOWLEDGEMENTS
The authors express their gratitude for the facility 
provided by the Department of Mechanical 
Engineering, B.I.T. Mesra, Ranchi, India.
8  NOMENCLATURES
C  specific heat, [J/kg°C] 
cv  control volume,
f
M
  measured transient temperature at node 3,
  [
o
C]
h(t)  heat transfer coefficient, [W/(m
2
K)]

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 342-354
353 Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach
k  thermal conductivity, [W/(m°C)]
ρ  density, [kg/m
3
]
q
M
(t) calculated heat flux at sensor location 
  on time t, [W/m
2
]
q
M
(t) estimated surface heat flux at time t, [W/m
2
]
r  radius, [m]
Δr  thickness of each control volume,
T
∞
  measured ambient temperature, [°C]
T*
1
  estimated surface temperature by derived
  equation, [°C]
T
1  
measured real time surface transient 
  temperature, [°C]
T
#
1 
 estimated surface temperature, [°C]
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