... the cooled surface. The plate has been cooled by 1 and 2 water ... The determination of the heat transfer coefficient (HTC) distribution in the ...... solution to EN 1.4724 steel plate cooling by two water jets. min=1.8, max=6473 kW/m2. Fig. 20.
This is an Author's Accepted Manuscript of an article published by ELSEVIER in International Journal of Heat and Mass Transfer 75( 2014) p. 347-361, available online: http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.078 Dedicated three Dimensional Numerical Models for the Inverse Determination of the Heat Flux and Heat Transfer Coefficient Distributions over the Metal Plate Surface Cooled by Water Z. Malinowski, T. Telejko, B. Hadała, A. Cebo-Rudnicka, A. Szajding Department of Heat Engineering and Environment Protection AGH University of Science and Technology, Mickiewicza 30, 30-059 Krakow, Poland
Key words: three dimensional inverse method, dedicated finite element models, heat transfer coefficient distribution, nonlinear shape functions, water jets cooling
Abstract The inverse method has been developed to determine three dimensional heat flux and heat transfer coefficient distributions in space and time. The numerical tests conducted for simulated temperature sensor indications have shown that the dedicated heat conduction model has to be employed to achieve correct solutions for limited number of temperature sensors. The dedicated three dimensional finite element method based on nonlinear shape functions has been developed to effectively solve the heat conduction problem. The accuracy of 5 finite element models has been compared to analytical solution and to a reference finite element solution. The reduced nonlinear finite element model with 384 degrees of freedom has given in direct simulation of the temperature field errors at a level of 2oC only. Heat transfer boundary condition over the cooled surface has been approximated by serendipity family elements with cubic shape functions. Heat transfer coefficients at surface element nodes have been extended in time of cooling with the parabolic spline functions. Inverse solutions based on the developed three dimensional heat condition and boundary condition models have been obtained without additional regularisation. Solutions have been achieved for measured temperatures as well. Temperature of EN 1.4724 steel plate heated to 900oC and then cooled has been measured by thermocouples located 2 mm below the cooled surface. The plate has been cooled by 1 and 2 water 1
jets. Equations for heat transfer coefficient as functions of dimensionless plate surface temperature have been developed and verified in direct simulations of EN 1.4724 steel cooling.
Nomenclature ATD
average temperature difference between measured and computed temperatures
B
width of plate
c
specific heat
Ckm
heat capacity matrix
Cs(τ)
scaling function defined by Eq. (40)
Dk
heat load vector
DG
average value of the objective function derivatives
E(pi)
objective function defined by Eq. (1)
Fi
cubic shape functions from serendipity family cubic – spline functions
h(x2,x3,τ)
function defining heat transfer coefficient distribution in space and time
havg
average heat transfer coefficient Hermitian shape functions
Kkm
heat conductivity matrix
KT
number of time periods
L
length of plate
Ls
constant equal to 1 or 0
NF
number of form functions
NH
number of optimization parameters
Ni
linear shape functions
NP
number of temperature measurements performed by one sensor
NT
number of temperature sensors
NX
number of surface elements in x3 direction
NY
number of surface elements in x2 direction
o
superscript which denotes quantities computed at the time
pi , pij
unknown parameters to be determined by minimizing the objective function
Pi
function defined by Eq. (8)
Ra
Rayleigh number
qv
heat source heat flux distribution to be determined average heat flux
Q
overall heat transferred from the water cooled surface 2
Se
element surface
Sk
cooling chamber surface
Ss
plate surface
SCj
sensitivity coefficients
T
temperature
Ta
cooling water temperature
Tavg
average temperature sample temperature measured by the sensor m at the time τn
Tk
cooling chamber surface temperature
Tm
unknown parameters at element nodes sample temperature at the location of the sensor m at the time τn
To
initial temperature of plate
Ts
cooled surface temperature
Wj
parabolic-spline functions
x1, x2, x3
Cartesian coordinates
Greek symbols β
dimensionless time for the temperature approximation
εk
emissivity of the cooling chamber surface
εs
emissivity of the plate surface
η
dimensionless time for the heat transfer coefficient approximation
η1, η2, η3
natural coordinates of spline functions
λ
thermal conductivity of steel
λa
thermal conductivity of air
ξ1, ξ2, ξ3
natural coordinates of an element
ρ
density
τ
time
Θ
dimensionless temperature
e
element domain
τ
time increment last time increment
3
1
Introduction
Heat removal technologies are of a great importance in electronic systems, laser systems, power plants [1] and metallurgical processes [2]. Liquid cooling includes several technologies [3] among which jet impingement cooling and spray cooling has increasing interest in industry [4]. Spray and jet impingement cooling offer wide range of heat transfer rates depending on the type of nozzle and flow parameters. A revive of spray cooling heat transfer literature published prior 2006 has been given by J. Kim [5]. Kim has pointed out that the majority of experimental data is limited to the average in space heat flux or heat transfer coefficient. Experimental techniques such as array of microheaters [6], cooling of thin foil [7] or temperature oscillation IR thermography [8] cannot be used for measuring heat transfer coefficient distribution during cooling of metals from temperatures reaching 1000oC. New methods are required to measure local in time and space heat flux or heat transfer coefficient at high temperatures. Continuous casting lines and hot rolling mills are equipped with water cooling systems to keep control of the casted or deformed metal temperature. These systems have a great importance in formation of a proper microstructure and mechanical properties of the product [9]. The desired rate of cooling is achieved by water sprays, water curtains or water jets applied to the hot metal surface [10]. The water flow rate and pressure can be changed in a wide range and it results in a very different heat transfer from the cooled metal to the cooling water. The numerical simulations can be employed to determine suitable cooling rate if thermal boundary conditions are known at the surface of the cooled object [11]. The determination of the heat transfer coefficient (HTC) distribution in the area of water flow would improve numerical simulations significantly [12]. The inverse solution to the heat conduction problem in the cooled plate can be employed to determine heat transfer boundary conditions varying in space and time. Beck [13] has presented an inverse method and successfully determined heat flux variation in time using measured temperature inside the cooled copper block. The solution was based on one dimensional heat conduction problem and has given average value of HTC over the cooled surface. One dimensional analytical inverse solution to the heat conduction problem has been employed by M. Ciofalo at al. [14] to determine heat flux and HTC while spray cooling of copper-beryllium plate. The plate thickness was 1.1 mm. The plate was heated to approximately 300 oC and cooled by spray nozzle. Influence of cooling parameters on the average HTC or heat flux have been presented in [15,16,17,18] for heat treatment of metals and spray cooling using nanofluids . However, convergence problems [19] and question concerning uniqueness of the inverse solutions have been encountered [20]. Regularization methods have been proposed to overcome these difficulties. Limitations of the regularization methods have been addressed in [19,21]. The main problem is that the regularization term may shift the solution from the exact one [22]. Developments in the finite element method and inverse techniques have led to solutions for two dimensional inverse problems [23] for simulated temperature readings. Successful solutions to the HTC variation in time and its distribution over the 4
cylinder circumference have been reported in [24] for water spray cooling of the steel cylinder heated to 884oC and during water jet impingement on the rotating cylinder made of Nickel and heated to 500oC [25]. The most important case is to determine HTC distribution over the cooled surface and the HTC variation in time. In this case heat conduction problem is three dimensional and three dimensional approximation of the HTC is necessary. One of the first solutions to such a problem has been reported in [26] and [27]. Huang and Wang [26] have presented implementation of CFX4.2 commercial software for determining HTC distribution over the cooled surface. The solution was presented only for the simulated temperature sensor indications. Similar solutions to the simulated temperature readings have been reported by Kim et al. [27], Malinowski et al. [28] and Zhou et al. [29]. However, convergence is much easier to achieve for simulated temperature readings, especially if large number of temperature reading points is taken to build the objective function. The inverse solution for the heat transfer coefficient determination during cooling of AISI 304L steel by one axially symmetrical water jet has been presented by Wang et al. [30]. The plate thickness was 25 mm and the plate temperature was measured by four thermocouples located 2.5 mm below the cooled surface. The solution has been obtained from axially symmetrical heat conduction model. Similar inverse solution for a single spray nozzle was proposed in [31]. Inverse solutions based on axially symmetrical models are limited to one spray or jet nozzle only. Inverse solution based on three dimensional heat conduction models are required for heat transfer coefficient determination for plate cooling by a set of water jets, water sprays or water curtains. For example, in hot rolling lines the cooled area may be 2 m wide and up to 100 m long. In such a case it is difficult to adopt small scale laboratory results to industrial applications. Water jets create locally high heat transfer coefficient which can be visually identified as dark spots on the hot red surface. The dark spot increases in time and the maximum heat flux moves over the cooled plate as well. Moreover conductivity of the plate affects this processes and the heat transfer problem becomes three dimensional. In such a case the computation time and the achieved accuracy of the inverse solutions strongly depend on the quality of the heat conduction model and the boundary condition model. In the paper the inverse solutions for the three dimensional heat conduction model and the heat transfer boundary condition approximation in space and time have been proposed. The heat conduction model is based on 3D finite element method with third degree shape functions. It has been shown that the finite element model with the reduced degrees of freedom is well suited for the inverse solutions. Approximation of the heat transfer coefficient over the cooled surface with the cubic shape functions from serendipity family has been proposed. The developed method of the HTC approximation in space and time is characterized by high sensitivity coefficients. The sensitivity coefficients do not depend on the value of the heat transfer coefficient. It makes possible to identify low values of the HTC at the level of radiation and the high HTC typical for critical heat flux.
5
2 2.1
Numerical models Inverse problem formulation
The heat flux distribution at the hot plate surface cooled by water can be determined by minimizing the objective function which defines the difference between measured and calculated temperatures:
(1)
The sample temperature
can be calculated from the finite element solution to the heat conduction
equation:
(2)
The boundary condition at the cooled plate surface shown in Fig. 1 has been expressed as: (3) The HTC distribution over the cooled surface has been approximated by surface elements with cubic shape functions from serendipity family:
(4)
In Eq. (4) Fi are cubic shape functions from serendipity family [32]. For corner nodes: n= 1, 4, 7, 10 and ξ1n=±1 and ξ2n=±1 (5) For side nodes: n= 2, 3, 8, 9 and ξ1n=±1/3 and ξ2n=±1 (6) For side nodes: n= 5, 6, 11, 12 and ξ2n=±1/3 and ξ1n=±1 (7) The cooled surface is divided into rectangular elements. For each element Eq. (4) has been used to approximate the HTC distribution. Functions Pi(τ) describe the heat transfer coefficient variations at nodes of elements in time. At each element node the function Pi(τ) has the following form: 6
(8)
The parabolic-spline functions Wj are given by:
(9)
In Eq. (9)
is dimensionless time (10)
In Eq. (10) k denotes a number of a time period τ (τ1k,τ2k). The time of cooling should be divided in KT periods of time to achieve the required approximation accuracy. The set of unknown parameters pij which have to be determined in the inverse heat conduction model is composed of heat transfer coefficients at nodes of approximation. For KT periods of time and NX×NY surface elements the number of unknowns NH which have to be determined can be calculated from:
(11)
The variable metric method has been employed in minimization of the objective function (1). The variable metric method belongs to a group of methods which require the inverse of Hessian matrix. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [33] has been employed to compute the inverse of the second derivatives matrix. The description of similar numerical procedures for function minimizations can be found for example in [34]. The BFGS formula gives a symmetric positive definite approximation to the inverse of Hessian matrix. The BFGS formula allows for moving pj vector to the next position in a downhill direction. The BFGS algorithm requires the computation of the objective function (1) value and its derivatives:
(12) The sensitivity coefficients in Eq. (12) are numerically estimated: (13)
7
An absolute value of εj=pj·1.E-6 has been utilized in forward estimation of the sensitivity coefficients. Two stopping criteria have been used to terminate the minimization procedure. The first controls a decrease in the objective function value. The average temperature difference (ATD) between indications of temperature sensors and computed temperatures has been defined:
(14)
The second criterion controls an average value of the objective function derivatives with respect to optimization parameters: (15) The minimization procedure is terminated if ATDk-1 – ATD
k
< EPS in the two subsequent
k
minimization steps or if in the last step DG < EPS. In all the tests EPS=1.E-10 have been assumed.
At the lower surface of plate, which is free from water, heat flux has been specified:
(16)
Heat flux at the lower plate surface which is not cooled by water is very small (about 0.5%) compared to heat flux for water cooling. Thus, a possible inaccuracy in emissivity and natural convection determination in Eq. (16) will not have any practical influence on the inverse solution to the HTC at the water cooled surface. In the IHCP solution only part of the plate is considered as shown in Fig. 1. At the side surfaces of the plate zero heat flux boundary conditions have been assumed: (17)
(18)
(19)
(20) It is expected that the heat gains at the side surfaces of the plate neglected in the IHCP will be less than 0.05% of the heat flux for water cooling. 8
In the heat conduction equation (2) qv represents the internal heat source. In case of steel plate cooling heat generation due to phase transformation should be included in the IHCP. However, it would require a separate model for heat generation dedicated for the inverse solution. In the present study EN 1.4724 steel which do not involves phase transformation in the examined temperature range has been selected to avoid the influence of heat generation on the HTC determination. In such a case qv = 0. 2.2
Heat conduction numerical models
If the unknown temperature field T(x1,x2,x3) is approximated by the set of form functions: (21) and the weighting functions are simply the form functions Ni then the integral form of Eq. (2) is given by [32]: (22) For one element matrices K, C and vector D have the following form: (23)
(24)
(25) Parameter Ls equals to 1 if at element surface boundary conditions are specified, otherwise Ls=0. Assuming that temperature varies linearly in time: (26) and introducing dimensionless time: (27) the temperature derivative in Eq. (22) can be replaced by: (28) In case of water cooling the heat load vector D in Eq. (22) varies significantly over the time increment τ and should be replaced by: (29) Assuming Galerkin integration scheme for which β=2/3 after same rearrangements Eq. (22) takes the form:
9
(30)
Several formulations depending on the choice of the weighting and shape functions can be written to define the matrices K, C and vector D. In the developed finite element model the plate shown in Fig.1 is divided into 8 node rectangular prisms presented in Fig. 2. The linear shape functions have been employed to define the prism geometry. In the first finite element model the following linear weighting and shape functions:
(31) have been employed in integrals (23), (24) and (25). In Eq. (31) n is the element node number and ξ1n, ξ2n, ξ3n are values of the natural coordinates at the element node n given in Tab.1. The unknown parameters Tm in Eq. (30) resulting from the first formulation are temperatures at element nodes. In the second finite element model Hermitian interpolation functions Hi are employed in integrals (23), (24) and (25) instead of linear shape functions Ni. In this case NF = 64 in integrals (23), (24) and (25) Hermitian interpolation functions Hi coupled with node temperatures and temperature derivatives have the following form:
for k=1 to 4, l=1 to 4 and m=1 to 4
(32)
The cubic-spline functions Gi have the form [32]:
(33)
In case of Hermitian shape functions the unknown parameters Tm in Eq. (30) are temperatures and their derivatives at element nodes. For one element 64 Tm unknowns can be grouped in a vector:
(34)
10
where n is the node number and η1n, η2n, η3n are node natural coordinates given in Tab. 1. 2.3
Numerical tests of the heat conduction models
The numerical tests of the heat conduction models have been perform for EN 1.4724 steel cooling. The thermophysical properties of EN 1.4724 steel as functions of temperature have been determined based on data published in [35]. Curve fitting have given the equations: Specific heat for 0oC T < 565oC (35) Specific heat for 565oC ≤T ≤ 1000oC (36) Heat conductivity: (37) Density: (38) Eq. (37) and Eq. (38) are valid for temperature of 0oC ≤ T ≤ 1000oC.
In the first series of tests the results of the temperature field computations have been compared with the analytical solution to an infinite plate cooling developed by J. Fourier [36]. The finite element software has been accomplished with the procedure (An-Mo) which calculates plate temperature from J. Fourier solution. It has been assumed that the plate of a thickness of 8 mm is cooled from the uniform temperature of 900oC. It has been assumed that the cooling time is 60 s and at the cooled surface a constant heat transfer coefficient of 5 kW/(m2 K) has been prescribed. Heat flux equal to 11
zero has been specified at the other plate surfaces. The thermophysical properties of EN 1.4724 steel have been calculated at an average temperature of 450oC. Six finite element models have been compared with the analytical solution. The models differ in number of elements, degrees of freedom and the length of the time increment τ. The parameters of the heat conduction models have been given in Tab. 2. In the models: RE-Mo, L1-Mo, and L2-Mo the linear shape functions have been utilized. The nonlinear shape functions have been employed in the models: H1-Mo, H2-Mo and H3Mo. The results of the plate temperature 2 mm below the cooled surface at x1= 6 mm have been presented in Fig. 3. All the models have given good solutions. The details of the finite element models (FEM) accuracy have been given in Tab. 3. The best solution, practically the exact solution of ATD=0.2 and a maximum temperature error of 0.5oC, has given the Re-Mo. The reduced H3-Mo with only two nonlinear elements in the plate thickness has given a maximum temperature error of 1.3 oC for the time increment τ=1 s. The first series of tests has confirmed a good accuracy of all the models for one dimensional heat conduction and for the HTC constant in space and time. The second series of tests has been performed for the HTC variable in space and time. In this case five FEM solutions have been compared with the reference model (Re-Mo). In the direct simulations of temperatures at 35 points located 2 mm below the cooled surface variable in time and space heat transfer coefficient has been assumed:
(39)
In Eq. (39) Cs(τ) is a scaling function which has the form: (40)
The function hm(τ) describes variation of the maximum HTC in time. The cubic-spline functions Gi have been utilized to describe h(τ) variation in time. The cooling time has been divided into two periods. The following parameters: hm(0)=100, hm(30)=10000, hm(60)=1600, ∂hm(30)/∂τ=0, ∂hm(60)/∂τ=45 have been employed in the approximation of hm(τ). The assumed boundary condition given by Eq. (39) describes the heat transfer coefficient distribution over the cooled plate similar to that expected during plate cooling with two water jets. At the beginning of cooling nearly an even HTC distribution similar to film boiling over the cooled surface is modeled. The maximum value of HTC at τ=0 is equal to 100 W/(m2 K). At τ=30 s two maximums of HTC equal to 10160 W/(m2 K) are modeled. The maximums of HTC are located at: (x2=40; x3=25) mm and at (x2=50; x3=75) mm. The HTC distribution over the cooled surface at τ=30 s has been presented in Fig. 4. A critical heat flux is modeled at τ=30 s. At the end of cooling at τ=60 s the HTC drops to a level of 1600 W/(m2 K) typical for natural convection boiling. At the side surfaces of the plate (see Fig. 1) no heat transfer have been
12
assumed. At the bottom surface of the plate which is free from water heat transfer due to natural convection and radiation described by Eq. (16) have been specified. The direct numerical computations have been performed for the plate cooling from a uniform initial temperature of To=900oC. The numerical simulations have given temperature distributions at thermocouples locations presented in Fig. 5. The solutions obtained with the reduced models have been compared to the reference model (Re-Mo) in Fig. 6. The reduced models have given good solutions also for the HTC variable in time and space. The details of the models accuracy have been given in Tab. 3. The lowest error to the reference model has given the H1-Mo with nonlinear shape functions which has 1000 degrees of freedom. In the case of H1-Mo an ATD of 1.4oC and a maximum error of 5.9oC have been obtained. The L1-Mo with linear shape functions and 1540 degrees of freedom has given an ATD of 4oC and a maximum error of 10.8oC. A better accuracy has been obtained in the case of H2-Mo with nonlinear shape functions and only 384 degrees of freedom. In all the cases models with nonlinear shape functions have given a better accuracy for reduced degrees of freedom.
3 3.1
Inverse solutions Simulated temperature sensor indications
The developed 3D finite element models have been tested in inverse solutions. The computed temperature variations at 35 points have been assumed as simulated temperature sensor indications for the inverse determination of the heat transfer coefficient distribution over the cooled plate surface. The Re-Model has been employed in computing the temperature sensors indications. It is considered that the plate sides are: B=90 mm, L=100 mm and the plate thickness is 8 mm. The location of simulated thermocouples at 2 mm below the cooled surface has been presented in Fig. 5. It has been assumed that the plate is made from EN 1.4724 steel. Variable in time and space heat transfer coefficient defined by Eq. (39) has been assumed in the direct simulation of temperatures at 35 points located 2 mm below the cooled surface. The direct numerical computations have been performed for the plate at uniform initial temperature of To=900oC. The boundary conditions at the plate surfaces have been described in the chapter 2.3. In the first series of tests the ability of the reduced finite element models to reflect the specified boundary condition expressed in Eq. (39) has been tested. Six pj parameters are to be determined from the minimum condition of the objective function (1). Initial values of the minimized parameters have been assumed equal to 100. The results of the objective function minimizations have been given in Tab. 4. The variable metric method has been terminated at DG below 1.E-10. The decrease in the ATD was also very low in the range from 0.5E-10 to 0.1E-6. The developed inverse method has reflected the specified boundary condition with the high accuracy, Fig. 7. The highest error to the exact HTC at point P1 is 3.4% and the lowest error is 1%, Tab. 4. The inverse solutions which utilize FEM models with the nonlinear shape functions have 13
given a better accuracy. A local error may not reflect the overall accuracy of the models. For that reason the overall heat Q transferred from the water cooled surface has been calculated: (41) The overall accuracy of the inverse solutions has been compared in Tab. 4. Heat transferred from the water cooled surface obtained from the inverse solutions is nearly equal to the exact one. The overall errors are below 0.2%. In Fig. 8 the distribution of the HTC at the plate surface obtained from H2Model has been shown at τ=30s. Practically a mirror of the specified boundary condition shown in Fig. 4 has been obtained. The computation time (CPU at W3680 3,33 GHz workstation) required to reach stationary value of the objective function has been listed in Tab. 4. A good compromise between the solution accuracy and the computation time gives H2-Model with the nonlinear shape functions and only 384 degrees of freedom. The sensitivity coefficients listed in Tab. 4 are high and ensure a good convergence in the objective function minimization for the specified boundary condition (39). It should be noticed that the inverse solution has been obtained from heat conduction model showing errors listed in Tab. 3 to the simulated temperature histories at the level of measurement errors. In the second series of tests it has been searched for an approximate solution to the specified boundary condition (39). The temperature sensors indications have been obtained from the Re-Model. In the inverse solutions the HTC distribution over the upper surface of the plate has been approximated using surface elements with cubic shape functions from serendipity family. Four elements shown in Fig. 5 have been used over the cooled surface. Since prescribed boundary condition depends on time, the expansion in time of the HTC at nodes of surface elements has been obtained with the parabolic spline-functions. The time of cooling has been divided into KT=3 periods. It has given 231 pi parameters to be determined in the boundary condition model which is expressed in Eq. 4. The parameters of the heat conduction models employed in the inverse solutions have been given in Tab. 2. The parameters of the approximate solutions to the specified boundary condition have been given in Tab. 5. The objective function (1) minimization has been terminated at the average value of the minimized parameters derivatives DG below 1.E-10. The inverse solutions have given temperature variations close to the simulated thermocouples indications. The ATD values listed in Tab. 5 are in the range from 5 to 10oC. The lowest ATD have given L1- and H2-Models with the nonlinear shape functions. However, the computation time per one minimization step is essentially lower in the case of H2-Model. The lowest computation time gives L2-Model but the accuracy of the solution to the HTC variation in time at points: P1, P2 and P3 presented in Fig. 9 is not satisfactory. The accuracy of the H3-Model at points: P1, P2 and P3 is also to low. The overall accuracy at a level of 0.5% to the heat transferred from the cooled surface is very good for all the models, Tab. 5. The sensitivity coefficients given in Tab. 5 are satisfactory with the average value at a level of 1.E-3. The lowest sensitivity at a level of 1.E-7 have only parameters at τ=60 s. It is understandable since the thermocouples are located
14
2 mm below the cooled surface and only backward sensitivity of the parabolic-spline function W3 (Eq.(9)) exist at τ=60 s. Higher order shape functions have given a better convergence of the inverse solution. In the heat conduction H2-Model based on nonlinear shape functions only 2×3×3 elements in x1, x2, x3 direction, respectively have been employed. The inverse solution presented in Fig. 10 has been obtained from the H2-Model. The HTC distribution is very similar to the modeled boundary condition shown in Fig. 4. The solution converged to ATD =5.2oC between simulated sensor indications and computed temperatures. 3.2
Water jets cooling
The plate made of EN 1.4724 steel was heated in the electrical furnace to the uniform temperature of To=900oC. The thickness of the plate was 8 mm. The plate was L=300 mm long and B=210 mm wide. The plate was placed horizontally to the water jets located 390 mm above the plate, approximately at the centre of the plate. The water flow was measured with the turbine flow meter having the accuracy class ±0.5%. The K type thermocouples with 80 μm diameter wires protected by a 500 μm diameter sheath were used. Measurements were recorded by a data acquisition system at high frequency having the accuracy class equal to ±0.2%. The accuracy of the thermocouples was equal to ±0.4% of the measured temperature value. The average temperature measurement error was low at a level of 1.5oC with a maximum deviation of ±15oC. The plate temperature has been measured by thermocouples located 2 mm below the cooled surface. The thermocouples locations have been shown in Fig. 5. The HTC distribution in the inverse solution to the measured temperatures has been approximated by surface elements with cubic shape functions from serendipity family. The H2-Model has been employed in the inverse solutions. 3.2.1
Inverse solution to steel plate cooling by a single water jet
In the case of EN 1.4724 steel plate cooling by a single water jet the initial plate temperature was 895oC and water temperature was 18.5oC. The water flow rate was 0.00308 m3/min. Expansion in time of the HTC at nodes of surface elements has been approximated by 17 elements with parabolic-spline functions. It has given 1155 pi parameters to be determined in the inverse boundary condition model. In the heat conduction model 2×3×3 prism elements in x1, x2, x3 direction, respectively have been employed. The heat conduction model depends on 384 degrees of freedom (Tm parameters in Eq. (30)). In Fig. 11 the picture of the hot plate cooled by one water jet has been presented. The inverse solution to temperature distribution at the plate surfaces after 20 s of cooling has been presented in Fig. 12. The inverse solution converged to ATD=16.4oC between thermocouples indications and computed temperatures. The plate surface temperature below the water jet has dropped to 47oC. At the same time at a distance of 30 mm from the water jet axis the plate surface temperature 15
was higher than 800oC. Temperature of the plate lower surface below the water jet has dropped to 315oC. The temperature difference between upper and lower surface of the plate has reached 270oC. At some distance from the water jet axis the lower surface temperature is the same as the upper surface temperature. It confirms that the boundary condition at the lower surface has been specified correctly, see Eq. (16). The inverse solution has given similar HTC values (about 50 W/(m2 K)) at the upper surface in same places not cooled by water, see Fig. 13. In water cooled area HTC reaches a maximum value of 20 kW/(m2 K). The HTC distributions presented in Fig. 13 have shown that the water jet cooling results in dynamic changes of heat transfer. The heat flux shown in Fig. 14 has reached a maximum value of 2.6MW/m2, while at the lower surface not cooled by water heat flux does not exceed 0.04 MW/m2. The results for plate cooling by a single water jet presented in Fig. 12, Fig. 13 and Fig. 14 have shown that cooling processes is not axially symmetrical even for one nozzle. It can be easily noticed in Fig. 11 which shows picture of the cooling process. The temperature histories at points: P1, P2 and P3, see Fig. 5, have been compared in Fig. 15 with the temperature distributions obtained from the inverse solution. The inverse solution gives good agreement with the measured temperatures over the time of cooling. The HTC distributions in time at points: P1, P2 and P3 have been shown in Fig. 16. The HTC varies significantly in time over the cooled surface. At the stagnation point (approximately at P1) the water jet cools the plate rapidly and the plate surface temperature drops below the saturation temperature. The wetting zone is visible as the dark spot in Fig. 11 and Fig. 12A. The diameter of the wetting zone increases in time and the critical heat flux zone at the border between wetting and non wetting zone moves. Due to that reason a maximum of HTC moves from point P1 to point P3 in time. Since the water flux over the wetting zone decreases as the diameter of the wetting zone grows, the HTC maximum at point P3 is essentially lower from that obtained at point P1. 3.2.2
Inverse solution to steel plate cooling by two water jets
Successful inverse solution to plate cooling by a single water jet has given possibility of further tests for more complicated cooling system. EN 1.4724 steel plate was cooled from the initial temperature of 897oC by two water jets. Water temperature was 18.5oC and the total water flow rate was 0.00627 m3/min. In the inverse solution the same division into elements as it was described for a single water jet has been employed. In Fig. 17 the picture of the plate cooled by two water jets has been presented. The inverse solutions to plate temperature field after 20 s of cooling have been shown in Fig. 18A. The plate cooling by two water jets results in high variations of the plate temperature. In Fig. 18 locations of two water jets can be easily noticed. The cold spots are not perfectly symmetrical and are not in line because of natural water jets movement during free flow of water from the nozzles located 390 mm above the plate. The inverse solution presented in Fig. 18 has converge to ATD = 27.3oC between thermocouples indications and computed temperatures The temperature field at the plate upper surface after 20 s of 16
cooling presented in Fig. 18A indicates that all types of boiling take place at the same time over the cooled surface. It results in rapid changes of HTC presented in Fig. 18B and in heat flux shown in Fig. 19. After 20 s the plate temperature at stagnation points has dropped below 100oC. The HTC field shown in Fig. 18B varies significantly. The low HTC values are observed below the water nozzles and the highest at some distance away from the nozzles line (x2 ~ 45 mm). Such distribution of HTC can be explained by the nature of cooling while free flow of water over plate surface. It is known that water velocity tangent to the plate surface in such type of cooling is the lowest at stagnation zone below the water jets. As the distance grows water accelerates and heat flux shown in Fig. 19 reaches maximum value of about 6 MW/m2 approximately at 20 mm away from the stagnation points. Water jets interactions contribute to grow of turbulence and the highest HTC values are noted where two streams mix, see Fig. 19. Good agreement between measured temperatures and obtained from the inverse solution has been presented in Fig. 20. The variations of HTC in time at points: P1, P2 and P3 have been shown in Fig. 21. At point P1 the HTC reaches a maximum of 20 kW/(m2 K) at 22 s. The HTC maximum at point P3 take place at about 35 s and has a lower value of 16 kW/(m2 K). Due to water jets interactions the difference in the HTC maximum at points P1 and P3 is smaller from that obtained for one water jet. 3.2.3
Heat transfer coefficient as function of plate surface temperature
Inverse solutions have given local heat transfer coefficient and heat flux variations in time. The boundary condition given by Eq. (4) depends on a large number of HTC values which define pi parameters in solved problems of EN 1.4724 steel plate cooling by one and two water jets. However, for practical simulations of cooling problems it is convenient to use heat transfer coefficient as function of cooled surface temperature. Such rearrangement of boundary condition can be obtained after same modification to the finite element model employed in the inverse algorithm. Since the boundary condition has been defined, direct simulation of plate cooling gives a local or an average heat flux for the selected vector of time τi:
(42)
Simultaneously, the average surface temperature can be also calculated from the integral:
(43)
Now, the average heat transfer coefficient can be calculated from the formula:
17
(44) Simple sorting the obtained vector havg (Ti) from the lowest to the highest surface temperature gives the average heat transfer coefficient as a function of surface temperature. In Fig. 22 the average heat flux calculated over the square defined by: La=50 mm, Lb=90 mm and Ba=25 mm, Bb=65 mm for EN 1.4724 steel cooling by 1 water jet has been presented. Heat flux reaches a maximum value of about 1.6 MW/m2 for the surface temperature of 450oC. Wang et al. [30] has obtained the maximum heat flux at a level of 0.7 MW/m2 for AISI 304L steel cooling by 1 water jet. In Wang et al. experiment the water flow rate was 0.015m3/min and the maximum heat flux was observed for surface temperature of about 400oC. The differences may be caused by lower conductivity of AISI 304L steel and/or the solution method. Since the axially symmetrical heat transfer model employed by Wang et al. is not able to reflect local variations of the heat transfer caused by natural water flow. In Fig. 23 the average heat transfer coefficient as function of the dimensionless plate surface temperature Θ=Ts/To for EN 1.4724 steel cooling by a single water jet has been presented. The maximum value of HTC reaches about 9 kW/(m2∙K) for a surface temperature of 105oC. In Wang et al. experiment [30] the maximum HTC was below 5 kW/(m2 K) for the surface temperature below 100oC. It is difficult to find physical explanation for such differences except the solution method. Curve fitting to the data plotted in Fig. 23 has given the average heat transfer coefficient as function of dimensionless surface temperature for EN 1.4724 steel cooling by a single water jet: For 0.25< Θ=Ts/To < 1 (45) For 0.02< Θ=Ts/To < 0.25 (46) The Eq. (45) and (46) are valid for the initial surface temperature of 800oC ≤ T ≤ 900oC and water flow rate of 0.003 m3/min. The developed methodology defined by Eq. (42), Eq. (43) and Eq. (44) has been applied to the inverse solution obtained from EN 1.4724 steel plate cooling by two water jets. The average heat flux calculated over the square defined by: La=10 mm, Lb=90 mm and Ba=25 mm, Bb=65 mm has been presented in Fig. 24. The maximum heat flux has reached 1.6 MW/m2 for the surface temperature of about 550oC. The maximum heat flux has been obtained for higher surface temperature than for a single water jet cooling. In case of two jets high heat flux is nearly stable for temperatures from 600oC to 200oC. In Fig. 25 the average heat transfer coefficient obtained from the heat flux shown in Fig. 24 has been presented. The maximum HTC value has reached about 10 kW/(m2 K) for a surface temperature of 135oC. Curve fitting to the data plotted in Fig. 25 has given the average heat transfer 18
coefficient as function of dimensionless surface temperature for EN 1.4724 steel cooling by two water jets: For 0.258< Θ=Ts/To < 1 (47) For 0.02< Θ=Ts/To < 0.258 (48) The Eq. (47) and (48) are valid for the initial surface temperature of 800oC ≤ T ≤ 900oC and water flow rate of 0.006 m3/min. It should be noticed that in case of EN 1.4724 steel cooling by two water jets the water flux is the same as for the one jet but the heat transfer is higher. It is clear that water jets interaction allows significantly enhancing heat transfer and lowering water consumption.
It is not yet obvious that the developed equations for average heat transfer coefficient as functions of dimensionless plate surface temperature will lead to reasonable results if applied as boundary conditions in the finite element code. The Eq. (45-46) and Eq. (47-48) have been applied in direct simulations of EN 1.4724 steel plate cooling. It should be noticed that boundary conditions in direct simulations are defined as a local heat transfer coefficient for a local surface temperature at Gauss integration points, see Eq. (23) and Eq. (25). It means that boundary condition defined by the HTC is coupled with the computed surface temperature and it is not possible to obtain surface temperature lower than the coolant temperature. Direct implementation of a boundary condition in a form of a heat flux presented in Fig. 22 and Fig. 24 in simulations for slightly different cooling conditions or initial plate temperatures will lead to significant errors in finite element simulations. For that reason it is recommended to convert heat flux to a convective type heat transfer coefficient. The results of simulations have been compared in Fig. 26 to measured data not used in the inverse solutions. The measured temperatures have been obtained in independent cooling tests performed for the same water flow rates. The results shown in Fig. 26 have confirmed that the developed equations for heat transfer coefficient can be employed in direct simulations of steel cooling from initial temperatures of 800oC to 900oC and similar water flow rates.
4
Conclusions
Determination of the heat flux and heat transfer coefficient distributions over the cooled surface and their variation in time is a very important problem in many industrial processes. In case of water cooling of hot metal surfaces heat transfer strongly depends on a cooled surface temperature. The heat conduction process is highly nonlinear and boundary conditions are coupled with the cooled surface temperature. Limited results have been reported in literature for two dimensional inverse solutions to
19
the heat transfer coefficient for measured temperatures. The developed finite element model based on nonlinear shape function is well suited for the inverse determination of the heat flux and heat transfer coefficient variations in space and time. The developed boundary condition model is of a general type and can be employed in identifying heat transfer coefficient or heat flux distributions over the cooled surface in time for all types of boiling, convection and radiation heat transfer processes. Inverse solutions to the heat transfer coefficient based on 35 measurement points have been obtained for a single water jet and for two water jets cooling of EN 1.4724 steel plate heated to about 900oC. Inverse solution to the heat transfer coefficient variation over the cooled surface in time and space depends on a large number of parameters and for that reason cannot be employed in practice. The inverse solution has allowed identifying local heat flux and heat transfer coefficient in time only. For practical simulations of cooling processes boundary conditions which do not depend on time are the most important. Curve fitting to data from the inverse solutions has allowed only determining equations for the average heat transfer coefficient as functions of dimensionless plate surface temperature. Further researches are necessary to develop equations for a local heat transfer coefficient as a function of dimensionless surface temperature and cooling process parameters such as: water flow rate, distance from the jet and jets enragement. Acknowledgements The work has been financed by the Ministry of Science and Higher Education of Poland, Grant No NR15 0020 10.
References [1] E.A. Silk, E.L. Golliher, R.P. Selvam, Spray cooling heat transfer: Technology overview and assessment of future challenges for micro-gravity application, Energy Conversion and Management 49 (2008) 453-468. [2] J. Sengupta, B.G. Thomas and M.A. Wells, The use of water cooling during the continuous casting of steel and aluminum alloys, Metallurgical and Materials Transaction A 36A (2005) 187204. [3] N.I. Kolev, Multiphase flow dynamics 3, Thermal interactions, Springer Verlag, Berlin Heidelberg, 2011. [4] J. Rivallin, S. Viannay, General principles of controlled water cooling for metallurgical on-line hot rolling processes: forced flow and sprayed surfaces with film boiling regime and rewetting phenomena, International Journal of Thermal Sciences. 40 (2001) 263-272.
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[5] J. Kim, Spray cooling heat transfer: The state of the art, International Journal of Heat and Fluid Flow 28 (2007) 753-767. [6] B. Abbasi, J. Kim, A. Marshall, Dynamic pressure based prediction of spray cooling heat transfer coefficients, International Journal of Multiphase Flow 36 (2010) 491-502. [7] S.J. Slayzak, R. Viskanta, F.P. Incropera, Effects of integration between adjacent free surface planer jets on local heat transfer from the impingement surface, International Journal of Heat and Mass Transfer 37 (1994) 269-282. [8] S. Freund, A.G. Pautsch, T.A. Shedd, S. Kabelac, Local heat transfer coefficients in spray cooling systems measured with temperature oscillation IR thermography, International Journal of Heat and Mass Transfer 50 (2007)1953-1962. [9] X. Li, M. Wang, F. Du, A coupled thermal mechanical and microstructural FE model for hot strip continuous rolling process and verification, Materials Science and Engineering A 408 (2005) 3341. [10] J. Horský, M. Raudenský, P. Kotrbáček, Experimental study of long product cooling in hot rolling, Journal of Materials Processing Technology 80-81 (1998) 337-340. [11] C.G. Sun, H.N. Han, J.K. Lee, Y.S. Jin and S.M. Hwang, A finite element model for the prediction of thermal and metallurgical behaviour of strip on run-out table, ISIJ International 42 (2002) 392-400. [12] S. Edalatpour, A. Saboonchi, S. Hassanpour, Effect of phase transformation latent heat on prediction accuracy of strip laminar cooling, Journal of Materials Processing Technology 201 (2011) 1776-1782. [13] J. V. Beck, Nonlinear estimation applied to the nonlinear inverse heat conduction problem, International Journal of Heat and Mass Transfer 13 (1970) 703-716. [14] M. Ciofalo, I. Di Piazza, V. Brucato, Investigation of the cooling of hot walls by liquid water sprays, International Journal of Heat and Mass Transfer 42 (1999) 1157-1175. [15] A. Buczek, T. Telejko, Inverse determination of boundary conditions during boiling water heat transfer in quenching operation, Journal of Materials Processing Technology 155-156 (2004) 1324-1329. [16] H. Bellerova, A.A. Tseng, M. Pohanka, M. Raudensky, Spray cooling by solid jet nozzles using alumina/water nanofluids, International Journal of Thermal Sciences. 62 (2012) 127-137. [17] S. Mitra, S.K. Saha, S. Chakraborty, S. Das, Study on boiling heat transfer of water-TiO2 and water-MWCNT nanofluids based laminar jet impingement on heated steel surface, Applied Thermal Engineering 37 (2012) 353-359. 21
[18] A. Buczek, T. Telejko, Investigation of heat transfer coefficient during quenching in various cooling agents, International Journal of Heat and Fluid Flow 44 (2013) 358–364. [19] J.V. Beck, B. Blackwell, A. Haji-Sheikh, Comparison of some inverse heat conduction methods using experimental data, International Journal of Heat and Mass Transfer 39 (1996) 3649-3657. [20] R. A. Khachfe, Y. Jarny, Determination of heat source and heat transfer coefficient for twodimensional heat flow – numerical and experimental study, International Journal of Heat and Mass Transfer 44 (2001) 1309-1322. [21] J. Su, G. F. Hewitt, Inverse heat conduction problem of estimating time-varying heat transfer coefficient, Numerical Heat Transfer, Part A 45 (2004) 777-789. [22] H. K. Kim, S.I. Oh, Evaluation of heat transfer coefficient during heat treatment by inverse analysis, Journal of Materials Processing Technology, 112 (2001) 157-165. [23] R. A. Khachfe, Y. Jarny, Numerical solution of 2-D nonlinear inverse heat conduction problems using finite-element techniques, Numerical Heat Transfer, Part B 37 (2000) 37-45. [24] M. Raudensky, J. Horsky, A. Tseng and Ch.I. Weng, Heat transfer evaluation of impingement cooling in hot rolling of shaped steels, Steel research 65 (1994) 375-381. [25] F. Volle, D. Maillet, M. Gradeck, A. Kouachi, M. Lebouché, Practical application of inverse heat conduction for wall condition estimation on a rotating cylinder, International Journal of Heat and Mass Transfer 52 (2009) 210-221. [26] C. H. Huang, S. P. Wang, A three dimensional inverse heat condition problem in estimating surface heat flux by conjugate gradient method, International Journal of Heat and Mass Transfer 42 (1999) 3387-3403. [27] S. K. Kim, J.-S. Lee, W. I. Lee, A solution method for a nonlinear three-dimensional inverse heat conduction problem using the sequential gradient method combined with cubic-spline function specification, Numerical Heat Transfer, Part B 43 (2003) 43-61. [28] Z. Malinowski, T. Telejko, B. Hadała, A. Cebo-Rudnicka, Implementation of the axially symmetrical and three dimensional finite element models to the determination of the heat transfer coefficient distribution on the hot plate surface cooled by the water spray nozzle, Key Engineering Materials 504-506 (2012) 1055–1060. [29] J. Zhou, Y. Zhang J.K. Chen, Z.C. Feng, Inverse estimation of front surface temperature of a plate with laser heating and convection-radiation cooling, International Journal of Thermal Sciences. 52 (2012) 22-30.
22
[30] H. Wang, W. Yu, Q. Cai, Experimental study of heat transfer coefficient on hot steel plate during water jet impingement cooling, Journal of Materials Processing Technology 219 (2012) 18251831. [31] A. Cebo-Rudnicka, Z. Malinowski, B. Hadała, T. Telejko, Influence of the sample geometry on the inverse determination of the heat transfer coefficient distribution on the axially symmetrical sample cooled by the water spray, Computer Methods in Materials Science, 13 (2013) 269–275 [32] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method Volume 1: The Basis, Fifth Edition, Butterworth-Heinemann, Linacre House, Jordan Hill, Oxford OX2 8DP, 2000. [33] T. Kręglewski, T. Rogowski, A. Ruszczyński, J. Szymanowski, Metody optymalizacji w języku FORTRAN, PWN, Warszawa, 1984. [34] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77, Second Edition, vol. 1, Cambridge University Press, New York, 1997. [35] A. Goldsmith, T.E. Waterman, H.J. Hirschhorn, Handbook of thermophysical properties of solid materials, vol. 2, Pergamon Press, New York, 1962.
[36] Yunus, A. Çengel, Heat and Mass Transfer: A Practical Approach, Third Edition, McGraw-Hill, New York, 2007.
23
Table 1. Natural coordinates for corresponding nodes of a prism element.
Node number
Linear shape functions
Hermitian shape functions
n
ξ1
ξ2
ξ3
1
-1
-1
-1
0
0
0
2
1
-1
-1
1
0
0
3
1
1
-1
1
1
0
4
-1
1
-1
0
1
0
5
-1
-1
1
0
0
1
6
1
-1
1
1
0
1
7
1
1
1
1
1
1
8
-1
1
1
0
1
1
1
2
3
Table 2. Parameters of the heat conduction models. Model parameter Number of elements in x1 direction Number of elements in x2 direction Number of elements in x3 direction Degrees of freedom Time increment τ, s
Re-Mo 22 18 20 9177 0.05
L1-Mo 13 9 10 1540 0.1
Heat conduction model L2-Mo H1-Mo H2-Mo 9 4 2 5 4 3 5 4 3 360 1000 384 0.5 0.1 0.5
H3-Mo 2 2 2 216 1.
Table 3. Accuracy of the finite element models in forward simulations of a plate cooling. Heat conduction model Re-Mo L1-Mo L2-Mo H1-Mo H2-Mo H3-Mo
Error to the analytical solution ATD +Tmax. -Tmax. o o o C C C 0.5 0 0.2 1.0 0 0.4 2.2 0 1.0 0.6 0 0.3 0.7 -0.2 0.4 1.3 -0.2 0.8
Error to the Re-Model ATD +Tmax. -Tmax. o o o C C C 0 0 0 10.8 -1.3 4.0 36.3 -20.6 11.7 4.4 -5.9 1.4 7.9 -2.3 2.1 25.7 -20.3 7.0
24
Table 4. Parameters of the inverse solutions to the boundary condition which is specified by Eq. (39). Parameter Error to the exact HTC at P1 point at τ=30 s Overall error to the exact Q ATD, oC ATDk-1 – ATD k,oC DG SCmin., K2 m2/W SCmax., K2 m2/W SCavg., K2 m2/W CPU time, min.
Inverse solution obtained from the heat conduction model: L1-Mo L2-Mo H1-Mo H2-Mo H3-Mo +3.4% +3.2% +1.0% +1.5% -2.8% +0.2% 2.21 0.3E-10 0.3E-7 2.76E-2 1.66 0.54 344
+0.2% 10.63 0.5E-10 0.9E-5 2.71E-2 1.64 0.54 23
+0.06% 1.53 0.1E-6 0.3E-10 2.82E-2 1.67 0.55 487
+0.09% 1.57 0.6E-7 0.6E-12 2.79E-2 1.67 0.55 222
-0.1% 6.72 0.2E-6 0.4E-10 2.76E-2 1.66 0.54 125
Table 5. Parameters of the approximate inverse solutions to the boundary condition which is expressed by Eq. (39). Parameter Error to the exact HTC at P1 point at τ=30 s Overall error to the exact Q ATD,oC ATDk-1 – ATD k,oC DG SCmin., K2 m2/W SCmax., K2 m2/W SCavg., K2 m2/W CPU time per 1 minimization step, min.
Inverse solution obtained from the heat conduction model: L1-Mo L2-Mo H1-Mo H2-Mo H3-Mo +8.8% +16.1% -1.5% +3.2% -12.4% +0.1% 5.5 0.5E-9 0.5E-9 8.1E-7 8.5E-3 1.2E-3 145
-0.4% 9.8 0.9E-8 0.8E-9 1.2E-6 7.3E-3 1.3E-3 2
-0.04% 5.3 0.2E-6 0.1E-9 2.7E-7 8.8E-3 1.3E-3 285
-0.3% 5.2 0.1E-6 0.5E-9 3.1E-7 8.6E-3 1.3E-3 23
-0.3% 6.9 0.5E-6 0.3E-9 1.9E-6 8.8E-3 1.3E-3 6
25
390 mm X1
L
X3
3
B 8 mm
00
mm
X2
210 mm
Fig. 1. Schematic for plate cooling and the coordinate system employed in the inverse solution to the 3D heat conduction problem.
6
7 3
2
1 5
3 2
8
1 3
1
4
Node number
2
Fig. 2. Prism with corresponding element nodes numbers and natural coordinates coupled with the linear shape functions ( ξ1,ξ2,ξ3 ) and the Hermitian shape functions ( 1, 2, 3).
26
1000
An-Mo Re-Mo L1-Mo L2-Mo H1-Mo H2-Mo H3-Mo
Temperature, oC
800
600
400
200
0 0
20
40
Time, s
60
Fig. 3. The comparison of temperature distributions at x1=6 mm obtained from six finite element models with the analytical solution (An-Mo) to a plate cooling.
100 90
W
80
m2 K
70
10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0
60
x 3, mm
50 40 30 20 10 0 0
10
20
30
40
50
60
70
80
90
x2 , mm Fig. 4. The heat transfer coefficient distribution over the upper plate surface specified as the boundary condition at τ=30 s, hmin=105 W/(m2 K), hmax=10160 W/(m2 K).
27
100
P3
x3, mm
80
P1
P2
60
40
20
0 0
20
40 60 x2, mm
80
100
Surface elements with cubic shape functions Thermocouples located 2 mm below cooled surface
Fig. 5. Location of the simulated thermocouples 2 mm below the cooled surface and the division of the cooled surface into cubic elements employed in the heat transfer coefficient approximation.
P1, Re-Mo P2, Re-Mo P3, Re-Mo P1, L1-Mo P1, L2-Mo P2, L1-Mo P2, L2-Mo P3, L1-Mo P3, L2-Mo
A)
Temperature, oC
800
600
400
1000
800
600
400
200
200
0
0 0
20
Time, s
40
60
P1, Re-Mo P2, Re-Mo P3, Re-Mo P1, H2-Mo P1, H3-Mo P2, H2-Mo P2, H3-Mo P3, H2-Mo P3, H3-Mo
B)
Temperature, oC
1000
0
20
Time, s
40
60
Fig. 6. Temperature variations in time at points: P1, P2 and P3 compared to the Re-Model. A) Temperature computed with the L1-Mo and L2-Models B) Temperature computed with the H2-Mo and H3-Models
28
12000 10000 8000
Heat transfer coefficient, W/(m2.K)
Heat transfer coefficient, W/(m2.K)
10000
P1, Exact P2, Exact P3, Exact P1, L1-Inv P2, L1-Inv P3, L1-Inv P1, L2-Inv P2, L2-inv P3, L2-Inv
A)
6000 4000 2000
P1, Exact P2, Exact P3, Exact P1, H1-Inv P2, H1-Inv P3, H1-Inv P1, H3-Inv P2, H3-inv P3, H3-Inv
B)
8000
6000
4000
2000
0
0 0
20
Time, s
40
60
0
20
Time, s
40
60
Fig. 7. The reflections of the heat transfer coefficient variations at points: P1, P2 and P3. A) Inverse solutions obtained from the L1- and L2-Models with liner shape functions. B) Inverse solutions obtained from the H1- and H3-Models with nonlinear shape functions
100 90
W
80
m2 K
70
10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0
60
x 3, mm
50 40 30 20 10 0 0
10
20
30
40
50
60
70
80
90
x2 , mm Fig. 8. The mirror of the heat transfer coefficient which has been specified at the upper plate surface at τ=30 s, hmin=106 W/(m2 K), hmax=10282 W/(m2 K). The inverse solution has been obtained from the H2-Model.
29
12000
8000
Heat transfer coefficient, W/(m2.K)
Heat transfer coefficient, W/(m2.K)
10000
P1, Exact P2, Exact P3, Exact P1, L1-Inv P2, L1-Inv P3, L1-Inv P1, L2-Inv P2, L2-inv P3, L2-Inv
A)
4000
P1, Exact P2, Exact P3, Exact P1, H1-Inv P2, H1-Inv P3, H1-Inv P1, H3-Inv P2, H3-inv P3, H3-Inv
B)
8000
6000
4000
2000
0
0 0
20
Time, s
40
60
0
20
Time, s
40
60
Fig. 9. The approximate solutions to the heat transfer coefficient at points: P1, P2 and P3. A) Inverse solutions obtained from the L1- and L2-Models with liner shape functions. B) Inverse solutions obtained from the H1- and H3-Models with nonlinear shape functions.
100 90
W
80
m2 K
70
10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0
60
x 3, mm
50 40 30 20 10 0 0
10
20
30
40
50
60
70
80
90
x2 , mm
Fig. 10. The approximate solution to the specified heat transfer coefficient distribution at τ=30 s; hmin=102 W/(m2 K), hmax=10697 W/(m2 K). The inverse solution has been obtained from the H2Model.
30
Fig. 11. EN 1.4724 steel plate heated in the electrical furnace to the temperature of 900oC and cooled by one water jet.
100
100
90
90 oC
80
oC
80
800
70
700
70
60
600
60
x 3, mm
400
40
300
50
x 3, mm
500
50
40
30
200
30
20
100
20
10 0 0
0
A) 10
20
30
40
50
x 2 , mm
60
70
80
90
800 750 700 650 600 550 500 450 400 350 300
10 0 0
B) 10
20
30
40
50
60
70
80
90
x 2 , mm
Fig. 12 Temperature distributions at the plate upper surface A) and at the plate bottom surface B) after 20 s obtained from the inverse solution to EN 1.4724 steel plate cooling by one water jet. A) ATD=16.4oC, +Tmax.=93oC, -Tmax.=-82oC, Tmin=47oC, Tmax=854 oC; B) Tmin=315oC, Tmax=837 oC
31
100
100
90
90
W/m 2 K
80
60 x 3, mm
50 40 30 20
50
60
10
20
30
40
50
60
70
80
45
50
40 35
40
30
30
25
20
20
10
A)
0 0
55
70
x 3, mm
20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 50
70
10
W/m 2 K
80
15
B)
0 0
90
10
20
30
x 2 , mm
40
50
60
70
80
90
x 2 , mm
Fig. 13. Heat transfer coefficient distributions over the upper surface A) and at the plate bottom surface B) after 20 s obtained from the inverse solution to EN 1.4724 steel plate cooling by one water jet. A) hmin=17, hmax=20386 W/(m2 K); B) hmin=16, hmax=56 W/(m2 K)
100
100
90
90
kW/m 2
kW/m 2 80
80
70 60
40
60
2000
35
50
30
x 3, mm
x 3, mm
50
45
70
2500
1500
40
25
40
1000
30
20
30
15
20
10
500
20 10 0 0
100
10
A) 10
20
30
40
50
60
70
80
0 0
90
x 2 , mm
5
B) 10
20
30
40
50
60
70
80
90
x 2 , mm
Fig. 14. Heat flux distributions over the upper surface A) and at the bottom plate surface B) after 20 s obtained from the inverse solution to EN 1.4724 steel plate cooling by one water jet. A) max=2586
2
kW/m ; B)
min=4.8,
max=45.9
min=5.8,
2
kW/(m K)
32
900
P1, Measurement P2, Measurement P3, Measurement P1, Inv. solution P2, Inv. solution P3, Inv. solution
800
Temperature, oC
700 600 500 400 300 200 100 0 0
10
20
30 40 50 Time of cooling, s
60
70
80
Fig. 15. The measured temperature distributions at points: P1, P2 and P3 compared with the inverse solution. EN 1.4724 steel plate cooled by one water jet.
20000
Heat transfer coefficient , W/m2 K
18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0
10
20
30 40 50 Time of cooling, s
60
70
80
Fig. 16. The inverse solution to the heat transfer coefficient variations at points: P1, P2 and P3 obtained from the H2-Model for EN 1.4724 steel plate cooling by one water jet.
33
Fig. 17. EN 1.4724 steel plate heated in the electrical furnace to the temperature of 900oC and cooled by two water jets. 100
100 90 oC
80
850
70
750
60
650 550
x 3, mm
50
450
40
350 250
30
150
20 10 0 0
50 20
A) 10
20
30
40
50
60
70
x 2 , mm
80
90
W/m 2 K
80 24000 22000 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 50
70 60 50
x 3, mm
90
40 30 20 10 0 0
B) 10
20
30
40
50
60
70
80
90
x 2 , mm
Fig. 18. The temperature field at the plate upper surface A) and the HTC distribution B) after 20 s obtained from the inverse solution to the EN 1.4724 steel plate cooling by two water jets. A) ATD=27.3oC, +Tmax.=114oC, -Tmax.=-92oC,. Tmin=25oC, Tmax=879 oC; B) hmin=10, hmax=26010 W/(m2 K)
34
100 90 kW/m 2 80 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500
70 60 x 3, mm
50 40 30 20 10
A)
0 0
10
20
30
40
50
60
70
80
90
x 2 , mm Fig. 19. The heat flux distribution over the plate upper surface after 20 s obtained from the inverse solution to EN 1.4724 steel plate cooling by two water jets.
min=1.8,
900
kW/m2
P1, Measurement P2, Measurement P3, Measurement P1, Inv. solution P2, Inv. solution P3, Inv. solution
800 700 Temperature, oC
max=6473
600 500 400 300 200 100 0 0
10
20 30 40 Time of cooling, s
50
60
Fig. 20. The measured temperature distributions at points: P1, P2 and P3 compared with the inverse solution. EN 1.4724 steel plate cooled by two water jets.
35
22000
Heat transfer coefficient , W/m2 K
20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0
10
20 30 40 Time of cooling, s
50
60
Fig. 21. The inverse solution to the heat transfer coefficient variations at points: P1, P2 and P3 obtained from the H2-Model for EN 1.4724 steel plate cooling by two water jets.
1.8E+006 1.6E+006 Average heat flux, W/m2
1.4E+006 1.2E+006 1.0E+006 8.0E+005 6.0E+005 4.0E+005 2.0E+005 0.0E+000 0
100 200 300 400 500 600 700 800 900 Surface temperature, oC
Fig. 22. Average heat flux as function of average temperature of the plate surface for EN 1.4724 steel cooling by one water jet. 36
Average heat transfer coefficient , W/m2 K
10000
8000
6000
4000
2000
0 0
0.2 0.4 0.6 0.8 Dimensionless temperature
1
Fig. 23. Average heat transfer coefficient as function of the dimensionless plate surface temperature for EN 1.4724 steel plate cooling by one water jet.
1.8E+006 1.6E+006 Average heat flux, W/m2
1.4E+006 1.2E+006 1.0E+006 8.0E+005 6.0E+005 4.0E+005 2.0E+005 0.0E+000 0
100 200 300 400 500 600 700 800 900 Surface temperature, oC
Fig. 24. Average heat flux as function of the plate surface average temperature for EN 1.4724 steel cooling by two water jets.
37
Average heat transfer coefficient , W/m2 K
12000
10000
8000
6000
4000
2000
0 0
0.2 0.4 0.6 0.8 Dimensionless temperature
1
Fig. 25. Average heat transfer coefficient as function of the dimensionless plate surface temperature for EN 1.4724 steel plate cooling by two water jets.
900 800 700
700
600
600
500 400 300 200
P1, Measurement P2, Measurement P3, Measurement Computed tempe.
800
Temperature, oC
Temperature, oC
900
P1, Measurement P2, Measurement P3, Measurement Computed tempe.
500 400 300 200
100
100 A)
B)
0
0 0
10
20 30 40 Time of cooling, s
50
60
0
10
20 30 40 Time of cooling, s
50
60
Fig. 26. Temperature distributions computed for the heat transfer boundary conditions defined by Eq. 45-46 (A) and Eq. 47-48 (B) compared to measured temperatures during EN 1.4724 steel plate cooling by one water jet (A) and two water jets (B).
38
