ON THE FORMULAS FOR THERMAL ANALYSIS OF LININGS
IN A STEADY-STATE REGIME
N. A. Tyutin
Translated from Novye Ogneupory, No. 9, pp. 14 – 15, September, 2010.
Original article submitted July 19, 2010.
The question of devising formulas for the direct and inverse problems of designing a lining in a steady-state
regime that is in the form of a multi-layer wall is considered. The question involves a refinement of some of
these formulas for the direct problem, a reduction of the formulas of the direct problem to a rational form, and
correction of the conditions under which the formulas in both problems are applied.
Keywords: lining, steady-state regime, direct problem, inverse problem, computational formula
The question of devising formulas for thermal analysis of
a lining in the steady-state regime has been previously con-
sidered by the present author in the article, Direct and In-
verse Problems in Thermal Analysis of a Lining in the
Steady State, published in the journal, Ogneupory i
Tekhnicheskaya Keramika (No. 2, 39 – 41 (2001)). The pres-
ent review of the question derives from the need to refine
certain of the formulas for the direct problem, the reduction
of the formulas of this problem to a rational form, and a cor
rection of the conditions of application of the formulas ob
tained in the previous article for the solution of the direct and
inverse problems for a lining in the form of a multi-layer
The objective of the direct problem is the calculation of
the temperature field of the lining, i.e., the calculation of the
temperatures on the contact surfaces of its adjacent layers or
the surface in contact with the environment, as a function of
the thermal conductivity of the materials of the layers, the
thickness of each of the layers, and the boundary conditions.
The objective of the inverse problem is to calculate the thick
ness of the lining, i.e., the thickness of each of the layers of
the lining, as a function of the thermal conductivity of the
materials of the layers, the temperature field, and the bound
From the foregoing it follows that the layers of a lining
are its computational elements, where the thermal conductiv
ity of the materials of the layers, their thickness, the tempera
tures on their contact surfaces, and the boundary conditions
determine the temperature field and thickness of the lining.
The following heat flux equation is valid for any i-th
layer of a lining:
where q is the heat flux; l
, thermal conductivity of material
of layer; d
, thickness of layer; T
, temperature on the
two contact surfaces of the layer, T
; and i, number of
The thermal conductivity of the material of the i-th layer
in the relatively narrow interval of temperatures at which the
layer is employed is found to be linearly dependent on the
mean temperature of the layer and is expressed by either one
of its equations:
are the thermal conductivity constants.
In solving systems of equations compiled from Eqs. (1), (2),
and (1), (3) with respect to the temperatures of layers T
for only positive results of extraction of roots and with re
placement of the conditions of application of the formulas
Refractories and Industrial Ceramics Vol. 51, No. 5, January, 2011
1083-4877/11/5105-0328 © 2011 Springer Science+Business Media, Inc.
OAO VOSTIO, Ekaterinburg, Russia.