Determination of thermal steady state in the wall with semi Dirichlet boundary conditions

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1 Determination of thermal steady state in the wall with semi Dirichlet boundary conditions Martin Zalesak Abstract An important task in the measurement in a calorimetric chamber is to determine the time in what the steady state thermal conditions in the walls are reached, in order to eliminate errors in measurement of the heat flow due to the thermal accumulative properties of the walls of the chamber There are several ways to get boundary conditions in which the steady state conditions could be reached In this article the process of getting the steady state conditions is studied with the semi Dirichlet boundary conditions The aim is to determine the required time of the process in the relation with the possible measurement error à à Ú (2) Θ, Ú - see (1), dimension, [m], - time, [s, h] The one dimensional case is described in the Fig 1 Keywords Heat transfer, transient thermal conditions, semi Dirichlet boundary conditions T I INTRODUCTION HE common problems in measurements of thermal properties of materials energy properties of equipment are to determine the time in which the thermal steady state of the measuring equipment is reached after the measurement process has been started The same problem occurs in measurement in a calorimetric chamber, the thermal accumulative properties of the chamber walls should be considered The task is to determine the required time of the process in the relation with the possible measurement error In this article the transient process with semi Dirichlet conditions is studied II BASES The principles of heat transfer in the solids are described by the well known Fouriers second order partial differential equation à Ú à (1) Θ - temperature, [ o C], Ú - thermal diffusivity, [m 2 /s] One dimensional heat transfer in the wall, the equation (1) has the form as follows This work was supported by the project CEBIA-TECH NO CZ105/2100/ M Zalesak is with the Tomas Bata University in Zlin, Faculty of Applied Informatics, Department of Automation Control Technologies, Zlin, Czech Republic ( zalesak@faiutbcz) Fig 1 One dimensional case in the heat transfer The solution of (2) derives from both the initial boundary conditions In the relevant case, the semi Dirichlet boundary conditions were considered 0; à à ººº»» º 0; 0, à à ;, à à à - Biot number, [-], - heat transfer coefficient, [W/(m 2 s)], dimension (thickness of the wall), [m], à - thermal conductivity of the wall material, [W/(mK)] (3) ISBN:

2 III THE SOLUTION Equation (10) will be First the substitution was introduced â à à (4) Ù Ù cosh Ú Ù Ú = Equation (2) now has the form â â Ú (5) Ù (14) For the solution of (5) the Laplace transform was used ã â ã Ú â (6) Equation (6) become second order linear differential equation with constant coefficients» Ù â, 0 Ú Ù (7) The boundary conditions will then has the form as (14) could be expressed as a division of the two functions Φ» Ψ» Ù Ù Ú Ú», in (15) could be expressed as»» (15) (16) â0, â ; Ù0,» Ù â, 0; Ù,» 0 (9) The general solution of (7) could be expressed as (8) (17) The function of sinhx, could be expressed as the series Ù cosh Ú sinh Ú (10) sinhx x!! (18) When conditions (8) (9) are considered, the constant,, are as follows The functions Φ», Ψ, in (17) could be expressed as 0; Ù ; 0 Ù cosh Ú sinh Ú (11) (12) Φ» sinh Ú Ú! Ú 1 (19) Ù Ú Ú (13) ISBN:

3 Ψ Ú cosh Ú sinh Ú» cosh Ú sinh (20) Ú ¹ ¹ (28) Roots of (16) must be found to suit the boundary conditions For results ¹ ¹ (29)» 0 roots», should suit the condition sinh Ú 0 (22) It is possible to express sinh Ú º sin Ú Condition (22) will be fulfilled if (23) For» 0 will be ¹ ¹ The LHopital rule could be applied for solution (30) ¹ lim for» 0 1 (30) (31) Using condition (25) by introducing Fourier number, º Ú º Ê (24) Ú º (32) This is valid for all» ¹ could be expressed as» Ú ; º 1, (25) Next task is to get the reverse transform In order to make it, it is possible to use the rule ã! (26) Inversion fiction may be expected as an addition of two functions â â ¹ ¹ (27) ¹ ¹ ¹ Ú 1 sinº Ê 1 ¹ 1 sinº Ê 1 ¹ (33) The final expression for the temperature in the distance (Fig 1) is then as follows â â ¹ ¹ â 1 1 sinº Ê 1 ¹ (34) Deviation from the steady state condition is expressed in the part of (34) by ISBN:

4 â â 1 sinº Ê 1 ¹ (35) V APPLICATION OF THE RESULTS The utilization of the results was studied in some materials conditions, which can occur in practical application Maximal value of the â, in (35) for the certain value of time, is in the distance IV TIME LIMITS TO REACH THE STEADY STATE CONDITIONS Relation of Ø ß shows the Fig 1 Physical parameters of studied materials are stated in the Table 1 0,7 0,6 Fo = f(ε) The task is to find minimal time necessary to reach acceptable steady state conditions it means to evaluate the conditions of â Ú â found from the equation as follows, as a function of time The limit can be Fo 0,5 0,4 0,3 0,2 0,1 Fo â Ú â 1 sinº Ê 1 ¹ ln 2 ¹ Ø ß (36) 1 ln 2 (37) sinº Ê 1 1 (38) For any value of, is then valid the relation â â ln 2 ¹ Ø ß (39) ß is an acceptable deviation from the steady state Now the task is to find the Ø, which will suit the above stated conditions Ø ln ß 0,101 ln2,27 ß (40) 0 Fig 2 Relation Ø ß Table I Physical parameters of studied materials Material 0 0,05 0,1 0,15 0,2 0,25 conductivity ε Density capacity diffusivity λ ρ c a [W/(mK)] [kg/m 3 ] [J/( kgk)] [m 2 /s] Concrete 1, ,41E-07 Foam polystyrol 0, ,35E-07 Foam polyurethan 0, ,68E-07 Hard wood 0, ,66E-07 Brick wall 0, ,63E-07 Plywood 0, ,42E-07 Stone 1, ,94E-07 Clay 1, ,69E-07 Necessary time, left for stabilizing condition with the acceptable errors describe Fig 3 (ß = 0,005) Fig 4 ( ß = 0,01) for different dimensions of the wall, thermal diffusivity of materials Ú It results from the figures that for practical thickness of the wall the time in tens of hours should be left for thermal stabilizing process ISBN:

5 t= f (l) e =0,005 t [h] 200,00 150,00 100,00 50,00 0,00 0,00 0,10 0,20 0,30 0,40 0,50 l[m] a = 1,5E-7 a = 3E-7 a = 5E-7 a = 8E-7 Fig 3 Relation between stabilizing time, the acceptable error ß = 0,005 for different dimensions of the wall, thermal diffusivity of materials Ú t= f (l) e=0,01 t [h] 150,00 100,00 50,00 0,00 0,00 0,10 0,20 0,30 0,40 0,50 l[m] a = 1,5E-7 a = 3E-7 a = 5E-7 a = 8E-7 Fig 4 relation between stabilizing time, the acceptable error ß = 0,01 for different dimensions of the wall, thermal diffusivity of materials Ú It results from the figures that r practical thickness of the wall the time in tens of hours should be left for the thermal stabilizing process REFERENCES [1] M Zalesak, Calorimeter Room Properties Report Unido 45 pp Chulalongkorn University Bangkok, 1991 [2] WM Rohsenow, JPHarnett, YICho, Hbook of Heat Transfer, ch 3,7 3 rd ed McGraw-Hill, 1997 [3] MFCarslaw, JCJaeger, Conduction of Heat in Solids Oxf Un Press, [4] GA Korn,TM Korn, Mathematical Hbook McGraw-Hill, 1968 [5] AVLuikov, Analytical Heat Diffusion Theory Academic Press, New York, 1968 [6] MZalesak, JPostava, Realization of the Non-steady State Measurement Method of Properties of Building Materials, ZTV Academia Prague, 1990 ISBN: