The demands on machining accuracy have been increasing lately and therefore research of thermal behaviour of machine tool structures is crucial for successful manufacturing. Generated heat diffuses into the structure of the machine tool components, this process is affected by heat sinks such as heat transfer on the surfaces and cooling systems. Meanwhile the heat warms up the structure of the machine tool and thermal dilatation deforms the structure, which subsequently affects machining accuracy in a negative way. Different systems are used to eliminate the thermal error, but their efficiency corresponds to the quality of the thermal machine tool model. The key problems of machine tool thermal error reduction are not in the thermal model itself, but in the fast, or even real-time, identification of the heat sources and the heat transfer coefficients (HTC) on the surfaces. This paper brings up a new identification method of the HTC, which is based on an analytical description

P. Bárta

University of Queensland eSpace

5
th
 Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Real-time identification method of a heat transfer coefficient 
 
P. Bárta 
 
Faculty of Mechanical Engineering, Czech Technical University in Prague, Czech Republic  
Abstract: The demands on machining accuracy have been increasing lately and therefore research of 
thermal behaviour of machine tool structures is crucial for successful manufacturing. Generated heat 
diffuses into the structure of the machine tool components, this process is affected by heat sinks such 
as heat transfer on the surfaces and cooling systems. Meanwhile the heat warms up the structure of 
the machine tool and thermal dilatation deforms the structure, which subsequently affects machining 
accuracy in a negative way. Different systems are used to eliminate the thermal error, but their 
efficiency corresponds to the quality of the thermal machine tool model. The key problems of machine 
tool thermal error reduction are not in the thermal model itself, but in the fast, or even real-time, 
identification of the heat sources and the heat transfer coefficients (HTC) on the surfaces. This paper 
brings up a new identification method of the HTC, which is based on an analytical description.  
Keywords: heat transfer coefficient, machine tool, modelling, real-time identification, thermal error. 
1 Introduction 
Thermal deformation of machine tool structures can be reduced with a thermally stable construction, 
by cooling the structure to desired temperature or by implementing a correction algorithm in the driving 
system. The success of all three cases depends on a high quality thermal deformation model. The 
thermal deformation model can be based on empiricism, the finite element method or an analytical 
solution. 
Empirical methods are grounded in finding the compensation function, which relates the relative 
thermal displacement to the temperature rise at some points on the structure. Regression techniques 
are used to express the compensation function in the form of polynomial equations of different orders 
or as a series of exponential terms with different time constants. Recently neural method techniques 
were also used. Empirical methods doesn’t include the real base of the heat transfer problem, 
therefore the compensation function is satisfactory only for service conditions used for calibration of 
the empirical model. 
Finite element method (FEM) is very a powerful tool in heat transfer analyses and it respects the real 
essence of the problem. On the other hand FEM is too robust and the solving is time demanding. Thus 
it isn’t suitable for real-time calculations. 
Recently new methods of heat transfer modeling appeared based on the analytical solution of a heat 
conduction equation (Eq. 1) [1, 2, 3]. These methods contain the nature of the heat transfer principles, 
thus the calibration of the empirical parameters is simpler and the model is in addition more reliable 
with untested inputs, because the data is forced to conform to the same principles as the real process. 
At the same time the results are obtained by two orders of magnitude faster [1]. These parameters 
make methods based on the analytical solution very suitable for fast estimation of the heat transfer 
coefficient. 
2 Analytical description 
In order to identify the heat transfer coefficient the transfer function between heat source and its 
temperature response must be found. Such a complex process, as heat transfer in a machine tool 
structure, can be approximated by the analytical solution to a similar, but simpler, phenomenon which 
is physically related [1]. In this case the phenomenon, called fundamental generalized problem (FGP), 
is heat conduction in a thin infinite plate with a convective boundary condition on the face and a 
radially symmetric central ring of heat generation. The heat transfer process is governed by the 
following differential equation: 
t
T
a
trq
T
r
T
rr
T







 1),(1
2
2

  (1) 
  
where  h  ,  is the heat transfer coefficient on the plate surface /W.m
-2
.K
-1
/,   is the material 
thermal conductivity /W.m
-1
.K
-1
/, h is the plate thickness /m/, a is thermal diffusivity /m
2
.s
-1
/ and q is the 
heat released per unit time and volume /W.m
-3
/. The initial plate temperature is zero, 
therefore   00, rT . 
2.1 General solution 
Firstly the term “ T ” is removed by implementing new variable in Eq. (1). 
    atetrtrT   ,,  (2) 
This leads to new formulation of Eq. (1): 
ta
e
trq
rrr
at







 

  1),(1
2
2
 (3) 
This equation can be simplified by the Hankel transformation, where  BrJ
0
 is a zero order Bessel 
function of the first kind: 
     


0
0
,, drtrBrJrtB   (4) 
The Eq. (3) is multiplied by  BrJr
0
  and integrated with respect to r. The integral of the first two terms 
can be found in [4]: 
  













2
0
02
2
1
BdrBrJr
rrr
  (5) 
If the heat source function q is zero everywhere in the plate except for the thin ring at
0
rr  , the 
function can be defined by the space impulse function δ as follows: 
 
 
 
0
0
2
, rr
hr
tq
trq
S 

 

 
(6) 
Then the third term in Eq. (3) is transformed after the substitution of Eq. (6) into Eq. (3). 
      
 
 0 0 0 0
0 0
2 2
at at
S S
q t e q t e
r r r J Br dr J Br
r h h
 

   

 
      
    

 (7) 
The final form of the transformed Eq. (3) is a first order linear differential equation: 
 
 
00
2
2
BrJ
h
eatq
aB
t
at
S 







 (8) 
Homogenous solution, Eq. (9), of this equation is found and substituted into it: 
  taBetBCtB
2
),(,

  (9) 
 
   tsBrJ
h
eatq
e
t
C
at
StaB





 
00
2
2


 (10) 
 
 
     BCdtetq
h
BrJa
tBC
B
t
atB
S



 

0
00
2
2
,


 (11) 
It is clear that CB = 0, when the initial condition is considered. The final solution, Eq. (12), of Eq. (8) is 
transformed by the inverse Hankel transformation, Eq. (13), back to the space domain and finally 
substituted by the Eq. (2), Eq. (14). 
 
 
    




 t
aB
S
taB
deq
h
eBrJa
tB
0
00
2
2
2
, 

  
(12) 
  
       
 22
0 0 0
0 0
,
2
t
B aaB t
S
a
r t B J Br J Br e q e d dB
h
 
  
 


 
         
   
 
 
(13) 
 
     
 
 
 2 2
0 0 0
0 0
,
2
t
B at B a
S
a
T r t B J Br J Br e q e d dB
h
  
 
 

    
         
   
 
 (14) 
Equation (14) represents general analytical solution of heat conduction in a thin infinite plate. 
2.2 Solution for constant heat source 
The heat source qs is considered to be constant during the identification process, qs = Q. Then the 
general solution is expressed: 
 
     
 2
0 0 02
0
, 1
2
B atQ B
T r t J Br J Br e dB
h B

  

   
       
   

 (15) 
      trFrF
h
Q
trT ,0,
2
, 



 (16) 
Where 
     





0
)(
0002
2
, dBeBrJBrJ
B
B
trF
tBa 

 (17) 
3 Identification method 
This method is based on the analytical solution, Eq. (16) and allows identify heat transfer coefficient 
on the surface of a thin plate from temperature response in the vicinity of the defined heat source.  
Let’s assume the plate is heated with known and constant heat source Q. Then the temperature 
response in the distance r=R, as shown in Eq. (16), is a function of time and η, which linearly depends 
on the heat transfer coefficient.  
       ,,,0,,, tRFRFCtRT   (18) 
Where 
     





0
)(
0002
2
,, dBeBrJBRJ
B
B
tRF
tBa 

  (19) 
This leads to an inverse problem, where η must be expressed from Eq. (19) analytically and then 
calculated for each pair of measured temperature and time [T(R,ti), ti] so: 
  
INViiAi
CTtRFt /,,)(   (20) 
Unfortunately the function F is very complex (Figure 1) and it’s very difficult or nearly impossible to 
obtain its analytical solution. Therefore the function F(η,t) is approximated for r = R = 2r0 by the 
function FN(η,t), which can be the problem solve after. The approximation starts with substituting the 
functions F(1,t), identical to F(ηref,t), and F(η*,T0), where η*= η/ηref (Figure 2):   
  3
1
)(,1
1


 

n
tv
u
tUtF
n
j j
j  
(21) 
  3*)(*)(*,
0
0
 

mwsTF
m
j
j
j
  
(22) 
Then: 
        
00
,1*,,1*,)*,( tFtFtFtFtS
iii
   (23) 
can be very well substituted by (Figure 3): 
  
       21*1*)*,(  
iii
tqtptS  (24) 
 
  
 
 
 
Figure 1. Numerical solution of F(t,η) Figure 2. Approximation of F(t,E*), E*=η/ ηref 
 
  
 
 
 
 
 
Figure 3. Function S(E*,t) Figure 4. Scheme of the identification method 
 
Substituting Eq. (24), (21) and (22) into (23) gives: 
       
iii
tUstUtStF  *)*,(*,
0
  (25) 
The approximation becomes simpler and more applicable if the reference temperature is known. 
       
refrefref
tRFRFCtT  ,,,0,   (26) 
This temperature is measured at calibration conditions. Even if these conditions are unknown, i.e. ηref 
is unknown, the resultant η* shows the change of unknown ηref, respectively αref, linearly which is 
sufficient information for the general temperature model.  
Then the temperature difference between reference TRef and measured temperature TM  at unknown 
η* is: 
         *,*,0,,0  tFFtFF
C
TT
refref
Mref


 (27) 
  
Substituting Eq. (25) into (27) results in: 
   0*,*,  StS
C
T

                   →                        21*1* 

 tQtP
C
T  (28) 
Which leads to the quadratic equation: 
        0**
2
 tCtBtA   
(29) 
The solution of Eq. (29) for each t=ti gives the resultant value of the unknown heat transfer coefficient 
on the surface of the plate at the time ti. 
4 Numerical experiments 
The presented method was tested on a FEM model, created in ANSYS (Figure 4).  
4.1 Plate model 
The plate (1600x1600x10 mm) model represents surface of a machine tool with thickness h=10 mm. 
The heat transfer coefficient on the plate surface is  ref = 10 W.m
-2
.K
-1
, the thermal conductivity  =43 
W.m
-1
.K
-1
,the thermal diffusivity a=11.9081x10
-6
m
2
.s
-1
 and the square heat source, with side length 50 
mm, r0=25 mm, Q=10 W is placed in the center. 
The reference temperature was derived 45 mm from the center of the plane or the heat source 
respectively with the heat transfer coefficient on the surface  ref. The function F was approximated for 
the following range: 3200031]*,[ t . Variables P(t) and Q(t) in the Eq. (28) were calculated for 
the range of time from the Eq. (19). 
4.2 Results 
Different heat transfer coefficients were applied on the surface and then the temperature responses 
were derived from the model, see Figure 5. Unknown values η* were calculated from Eq. (29), see 
Figure 6 and Table 1. 
 
   
 
 
 
 
Figure 5. Temperature rise for various η* - plate Figure 6. Identified values η*(t) - plate 
 
 
The table below concludes the results averaged for the time spam (500 - 3200) s.  
 
                                   Table 1. Results 
  
Applied HTC 
/Wm
-2
K
-1
/ 
Calculated HTC 
/Wm
-2
K
-1
/ 
Relative error 
/%/ 
11.7 11.806 0.91 
18.5 18.88 2.05 
21.2 21.142 -2.74 
30 30.894 2.98 
40* 44.011* 10.03* 
                                        *The method was tested for value outside the region of the approximated function F. 
5 Conclusion 
Presented method of heat transfer coefficient identification was tested on a numerical model and the 
applicability was proved, even for the square heat source. The relative error of the method is less than 
3 % and the time needed for the identification is approximately 10 min. If the plate, which can be 
implemented in the structure of the machine tool instead of the real surface, is 1 mm thin its 
temperature response will be one minute, which is sufficient enough for real-time calculations of 
machine tool behaviour. 
Acknowledgement 
This research has been supported by the 1M6840770003 grant of the Ministry of Education of the 
Czech Republic and by the internal grant of the Czech Technical University in Prague – CTU0710812. 
References 
[1]  S. Fraser, M. H. Attia & M. O. M. Osman, “Modelling, Identification and Control of Thermal Deformation of 
Machine Tool Structures, Part 1: Concept of Generalized Modelling”, Journal of Manufacturing Science and 
Engineering, Vol,. 120, pp. 623-631. 
[2]  S. Fraser, M. H. Attia & M. O. M. Osman, “Modelling, Identification and Control of Thermal Deformation of 
Machine Tool Structures, Part 2: Generalized Transfer Functions”, Journal of Manufacturing Science and 
Engineering, Vol,. 120, pp. 632-639. 
[3]  S. Fraser, M. H. Attia & M. O. M. Osman, “Modelling, Identification and Control of Thermal Deformation of 
Machine Tool Structures, Part 3: Real-time Estimation of Heat Sources”, Journal of Manufacturing Science 
and Engineering, Vol,. 121, pp. 501-508. 
[4] I. H. Sneddon, 1972, The Use of Integral Transforms, Mc-Graw-Hill, pp. 534. 
[5]  A. D. Polyanin & A. V. Manzhirov, 1998, Handbook of Integral Equations, CRC Press, pp. 793. 
[6]  K. Rektorys, 1981, Přehled užité matematiky, SNTL – Nakladatelství technické literatury, pp. 1139. 
 


Real-time identification method of a heat transfer coefficient

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Real-time identification method of a heat transfer coefficient

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