o tra e c ła e and

Article history:

Received 7 January 2013

Received in revised form 25 March 2014

Accepted 27 March 2014

Keywords:

Three dimensional inverse method processes [2]. Liquid cooling includes several technologies [3] 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 rolling mills are equipped with water cooling systems to keep conre. These systems er microstructure e desired ins or wa ow rate an sure can be changed in a wide range and it results in a very ent heat transfer from the cooled metal to the cooling wat numerical simulations can be employed to determine s 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. ⇑ Corresponding author. Tel.: +48 606405098.

E-mail addresses: malinows@agh.edu.pl (Z. Malinowski), telejko@agh.edu.pl (T.

Telejko), Beata.Hadala@agh.edu.pl (B. Hadała), cebo@agh.edu.pl (A. Cebo-Rudnicka),

Artur.Szajding@agh.edu.pl (A. Szajding).

International Journal of Heat and Mass Transfer 75 (2014) 347–361

Contents lists availab

International Journal of H .eamong 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 Kim [5]. Kim has trol of the casted or deformed metal temperatu have a great importance in formation of a prop and mechanical properties of the product [9]. Th cooling is achieved by water sprays, water curta applied to the hot metal surface [10]. The water flhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.078 0017-9310/ 2014 Elsevier Ltd. All rights reserved.rate of ter jets d presdifferer. The uitable1. Introduction

Heat removal technologies are of a great importance in electronic systems, laser systems, power plants [1] and metallurgical measuring heat transfer coefficient distribution during cooling of metals from temperatures reaching 1000 C. New methods are required to measure local in time and space heat flux or heat transfer coefficient at high temperatures. Continuous casting lines and hotDedicated finite element models

Heat transfer coefficient distribution

Nonlinear shape functions

Water jets coolingThe 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 2 C 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 regularization. Solutions have been achieved for measured temperatures as well. Temperature of EN 1.4724 steel plate heated to 900 C 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 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. 2014 Elsevier Ltd. All rights reserved.a r t i c l e i n f o a b s t r a c tDedicated three dimensional numerical m determination of the heat flux and heat distributions over the metal plate surfac

Zbigniew Malinowski ⇑, Tadeusz Telejko, Beata Hada

Department of Heat Engineering and Environment Protection, AGH University of Scienc journal homepage: wwwdels for the inverse nsfer coefficient ooled by water , Agnieszka Cebo-Rudnicka, Artur Szajding

Technology, Mickiewicza 30, 30-059 Krakow, Poland le at ScienceDirect eat and Mass Transfer l sevier .com/locate / i jhmt of HNomenclature

ATD average temperature difference between measured and computed temperatures

B width of plate c specific heat

Ckm heat capacity matrix

Cs(s) 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

Gj cubic – spline functions h(x2,x3, s) function defining heat transfer coefficient distribution in space and time havg average heat transfer coefficient

Hi 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 348 Z. Malinowski et al. / International JournalBeck [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 Ciofalo et 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 C and cooled by spray nozzle. Influence of cooling parameters on the average