expression for the minimum attainable value for the product 1/2 * * AE  under the given Reynolds number. 4.5 Concluding remarks As exemplified by the water-water counter-flow heat exchanger, the present work shows that there exists an optimal duct aspect ratio for heat exchangers under the fixed Reynolds number and mass velocity when the entransy dissipation number is taken as the performance evaluation criterion. Furthermore, the formula for the optimal duct aspect ratio was obtained analytically. Under constraints of the fixed heat transfer area (or duct volume) and Reynolds number, it was shown that there is an optimal dimensionless mass velocity; for which an analytical expression was also given. The results indicated that to reduce irreversible dissipations in heat exchangers, largest-possible heat transfer areas and lowest- possible mass velocities should be adopted. This conclusion is in agreement with numerical results obtained by design optimization of the shell-and-tube heat exchanger based on the entransy dissipation number as the objective function (Guo et al., 2010). From the results obtained in this study, it can be seen that the traditional heat exchanger design optimizations based on total cost as an objective function usually sacrifice heat exchanger performance. This issue has been demonstrated by numerical results (Guo et al., 2009). Guo et al. (2009) found that a little improvement in heat exchanger performance can lead to large gains in terms of energy saving and environmental protection. Hence, in heat exchanger design, reduction in total cost and improvement in heat exchanger performance should be treated equally. The present work will be useful to drive new research in this direction. 5. Acknowledgements The support of our research by National Basic Research Program of China (Project No. 2007CB206900) is greatly appreciated. 6. 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Exergy, an International Journal, Vol.1, pp. 278-294. Ziman, J. M. (1956). The general variational principle of transport theory. Can J Phys, Vol. 34, pp.1256-1273 15 Inverse Space Marching Method for Determining Temperature and Stress Distributions in Pressure Components Jan Taler 1 , Bohdan Węglowski 1 , Tomasz Sobota 1 , Magdalena Jaremkiewicz 1 and Dawid Taler 2 1 Cracow University of Technology 2 University of Science and Technology Poland 1. Introduction Thermal stresses can limit the heating and cooling rates of temperature changes. The largest absolute value of thermal stresses appears at the inner surface. Direct measurements of these stresses are very difficult to take, since the inner surface is in contact with water or steam under high pressure. For that reason, thermal stresses are calculated in an indirect way based on measured temperatures at selected points, located on an outer thermally insulated surface of a pressure element. First, time-space temperature distribution in pressure element is determined using the inverse space marching method. High thermal stresses often occur in partially filled horizontal vessels. During operation under transient conditions, for example, during power plant start-up and shut-down, there are significant temperature differences over the circumference of the horizontal pressure vessels (Fetköter et al., 2001; Rop, 2010). This phenomenon is caused by the different heat transfer coefficients in the water and steam spaces. This takes place in large steam generator drums, superheater headers and steam pipelines. High thermal stresses caused by nonuniform temperature distribution on vessel circumference also occur in emergency situations such as fire of partially filled fuel tanks. The upper part of the horizontal vessel is heated much faster than the lower part filled with liquid. Similar phenomenon occurs in inlet nozzles in PWR nuclear reactor, at which high temperature differences on the circumference of the feed water nozzles are observed. The study presents an analysis of transient temperature and stress distribution in a cylindrical pressure component during start-up of the steam boiler and shut-down operations. Thermal stresses are determined indirectly on the basis of measured temperature values at selected points on the outer surface of a pressure element. Having determined transient temperature distribution in the entire component, thermal stresses are determined using the finite element method. Measured pressure changes are used to calculate pressure caused stresses. The calculated temperature histories were compared with the experimental data at selected interior points. The presented method of thermal stress control was applied in a few large conventional power plants. It can also be used successfully in nuclear power plants. The developed method for monitoring thermal stresses and pressure-caused stresses is also suitable for Developments in Heat Transfer 274 nuclear power plants, since it does not require drilling holes for sensors in pressure element walls. Measurements conducted over the last few years in power plants demonstrate that the presented method of stress monitoring can be applied in systems for automatic power boiler start-up operations and in systems for monitoring the fatigue and creep usage factor of pressure components. 2. Present methods used to determine temperature and thermal stress transient in boiler pressure elements The simplest and most frequently used method for reducing thermal stresses in pressure elements is to limit heating and cooling rates. The allowable rates of fluid temperature changes can be determined using German Boiler Codes TRD 301 (TRD 301, 2001) or European Standard EN 12952-3 (EN 12952-3, 2001). The allowable heating rate of temperature changes is determined from the following condition (EN 12952-3, 2001; TRD 301, 2001) ( ) 2 min 2 om T mTwf ppd vs sa ααφφσ − += . (1) The maximum allowable cooling rate is calculated in a similar way: ( ) 2 max 2 om T mTwf ppd vs sa ααφφσ − += . (2) In Equations (1-2), the following nomenclature is used: α = k/(c·ρ) – thermal diffusivity, m 2 /s, r in , r out – inner and outer radius, m, d m = r out + r in – mean diameter, m, p – absolute pressure, MPa, p o – ambient pressure, MPa, s = r out - r in – thickness of cylindrical element, m, v T – rate of temperature changes of fluid or pressure element wall, K/s, α m – pressure caused stress intensity factor, α T – thermal stress intensity factor, σ min – allowable stress during start-up (heating), MPa, σ max – allowable stress during shutdown (cooling), MPa. Coefficients φ w and φ f are defined as follows 1 w E β φ ν = − , (3) ( ) ( ) () () 22 4 2 2 31 14ln 811 f uu uu uu φ −−− = −− , (4) where: E – Young's modulus, MPa, β – linear coefficient of thermal expansion, 1/K, ν – Poisson’s ratio, u = r out /r in – ratio of outer to inner surface radius. Equations (1-2) can also be used to calculate the maximum total stress at hole edges. The heating or cooling rate of temperature changes in pressure elements v T can be calculated using the moving average filter (Taler, 1995) ( ) 43 2 1 01234 1 63 42 117 162 693 177 162 117 42 63 , i Tiiii tt iiii df vffff dt t ff f f f −− − − = ++++ == −++ + + Δ ++ + + − (5) where f i are medium or wall temperatures at nine successive time points with Δt time step. Inverse Space Marching Method for Determining Temperature and Stress Distributions in Pressure Components 275 Equations (1-2) and (5) are not only valid for a quasi-steady state (Taler, 1995), but also when the temperature change rate v T is the function of time: v T = v T (t). For pressure components with complex geometry, the stress value in the stress concentration areas can be calculated using the finite element method (FEM). By determining the so called influence function with the use of the FEM, one is subsequently able to carry out an on-line stress calculation with known heat transfer coefficient and fluid temperature transient (Taler et al., 2002). This chapter formulates the problem of determining the transient temperature in a pressure element as an inverse transient heat conduction problem. The temperature distribution is determined on the basis of temperature histories measured at selected points at the outer insulated surface of a pressure component. After determining transient temperature distribution in the entire pressure component, thermal stresses are computed using the FEM. Inverse problem is solved using the finite volume method (FVM). Thermal and pressure caused stresses are calculated using the FEM. 3. Mathematical formulation of inverse problem In the following, two dimensional inverse heat conduction problem (IHCP) will be solved (Fig. 1). The analyzed domain is divided into two subdomains: direct and inverse. Boundary and initial conditions are known for the direct region so that the transient temperature distribution is obtained for the solution of the boundary-initial problem. The temperature distribution on the inner closed surface S m which is located inside the analyzed area (Fig. 1) is known from measurements. Based on the solution of the direct problem the heat flux on the boundary S m can be evaluated. Thus, the two boundary conditions are known on the surface S m : ( ) ( ) ,, m S Tst fst= , (6) () , m S T k q st n ∂ −= ∂ , (7) while on the inner surface S in of the body, the temperature and heat flux are unknown. In order to evaluate the transient temperature distribution in the inverse region, this region is divided into control volumes (Fig. 2). The method marches in space towards the inner surface of the body S by using the energy balance equations for the finite volumes placed on the boundary S m to determine the temperatures in adjacent nodes. In this way of proceeding the time derivatives of the measured temperature changes have to be calculated. The accurate calculation of the time derivatives of the measured temperature histories is difficult since the measured temperature values are burdened with random measurement errors. Thus, time-temperature charts have to be smoothed before evaluating the time derivative. In the present section this was achieved by using the local polynomial approximation. The successive nine temperature data points were approximated using the polynomial of the 3rd degree and then the derivatives in the middle of each interval (time coordinate of the point 5) were calculated. The space-marching method will be illustrated by an example showing the evaluation of the temperature distribution in the cylindrical wall using the temperature measurement points Developments in Heat Transfer 276 equally distributed on the surface S m (Fig. 2). It is assumed that the temperature and heat flux distributions are known on the surface S m from temperature measurements at points 16- 22 and from the solution of the direct problem. Fig. 1. The analyzed body divided into the inverse and direct regions Fig. 2. Division of the inverse region into control volumes Inverse Space Marching Method for Determining Temperature and Stress Distributions in Pressure Components 277 For the ideally insulated outer surface the heat fluxes q 16 -q 22 and their time derivatives are equal to zero. The heat balance equation for the finite volume with the node (i, j) is as follows (Fig. 3): ()() ()( ) ()( ) ()() () 22 , ,, ,, ,1, 1, , , ,1, ,1, 1, , , 1 , , , ,1 , 1 22 2 22 22 2 ij ij ij ij ij ij i j ij ij ij ij i j ij ij i j ij ij ij ij ij ij i dT rr rr cTT dt kT kT TT r r r kT kT kT kT TT TT r rr rr kT kT r ϕρ ϕ ϕ ϕ + + −+ −+ − ⎡⎤ ΔΔ ⎛⎞⎛⎞ +−− Δ = ⎢⎥ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ⎢⎥ ⎣⎦ + − Δ ⎛⎞ =+Δ + ⎜⎟ Δ ⎝⎠ ++ −− Δ ⎛⎞ +−Δ +Δ + ⎜⎟ ΔΔ ⎝⎠ + +Δ () ,1 , , , 2 j ij ij ij TT r ϕ − − Δ (8) where: c – specific heat, J/(kg·K), ρ – density, kg/m 3 , k – thermal conductivity, W/(m·K), t – time, s, T – temperature, °C, Δr – space step in radial direction, m, Δφ – angular step, rad. Transforming the heat balance Equation (8) for T i-1,j , we obtain: ()() ()( ) ()( ) ()( ) () () ()() ()( ) 22 ,, ,, , 1, , ,1, , 2 , ,1, ,1, ,1 , 1, , 2 , ,1, ,1, ,, 22 2 2 22 ij ij ij ij ij ij ij ij i j ij ij ij i j ij ij ij ij ij ij ij ij i j ij i j ij ij rr rr cT T dT TTr r dt kT kT r r r kT kT kT kT TT r TT rr r kT kT kT kT rr ρ ϕ − − ++ + + −− ΔΔ ⎛⎞⎛⎞ +−− ⎜⎟⎜⎟ ⎝⎠⎝⎠ =+Δ ⋅ ⋅ − Δ + − Δ + ++ − Δ −⋅ ⋅−−⋅ ⋅ ΔΔ Δ ++ −− () ()() ()( ) 2 ,1 , ,1 , 2 , ,1, , . 2 ij ij ij ij ij ij i j ij kT kT TT r r r kT kT r ϕ − − − − + − Δ −⋅ ⋅ Δ Δ + − (9) Since Eq. (9) is nonlinear, the fixed-point iterative technique is used to determine the temperature T i,j − 1 : () ()() () () ( ) ()( ) () () ( ) () () ()() () () ( ) 22 ,, ,, , 1 , 1, , 1, , 2 , ,1, ,1, ,1 , 1, , 2 , ,, 1, 1, ,, 22 2 2 22 ij ij ij ij ij n ij ij n ij ij ij ij ij i j ij ij i j i j ij ij nn ij ij ij ij ij ij ij rr rr cT T dT TTr r dt kT kT r r r kT kT kT kT TT r TT rr r kT kT kT kT rr ρ ϕ + − − ++ + + −− ΔΔ ⎛⎞⎛⎞ +−− ⎜⎟⎜⎟ ⎝⎠⎝⎠ =+Δ − Δ + − Δ + ++ − Δ −−− ΔΔ Δ ++ −− () ()() () () ( ) 2 ,1 , ,1 , 2 , , 1, , . 2 ij ij ij ij n ij ij ij ij kT kT TT r r r kT kT r ϕ − − − − + − Δ − Δ Δ + − (10) Developments in Heat Transfer 278 Fig. 3. Space marching in the inverse region The iteration process continuous until the condition () () 1 1, 1, nn ij ij TT ε + −− − ≤ is satisfied. Symbol n denotes the iteration number and ε is some small number (tolerance), for example, ε = 0.0001 K. If the thermal properties are constant, the iterations are not required. To determine the wall temperature at the node 1 (Fig. 2) we proceed as follows. First the heat balance equations are written for the nodes 16 to 22 from which the temperature at the nodes 9 to 13 is determined. Then the heat balance equations are set for the nodes 9 to 13 from which the temperature at the nodes 4 to 6 is evaluated. From the heat equations for the nodes 4, 5 and 6, we can determine the temperature at node 1 located at the inner surface of the pressure component. Based on the measured temperatures at the nodes 17 to 23, the temperature at the node 2 is calculated in similar way. Using the measured temperature at nodes 18 to 24 the temperature at the node 3 is estimated. Repeating the procedure described above for all the nodes, located at the outer surface, at which the wall temperature is measured, the temperature at the nodes situated at the inner component surface are determined. After determining the temperature distribution at the whole cross-section at the time point t, the temperature distribution at time t + Δt is computed. After calculating the temperature distribution the thermal stresses were calculating using the FEM. The finite element mesh is constructed so that the FEM nodes are coincident with the nodes used in the finite volume method. Random measurement errors of the temperature f j (t) have great influence on the estimated temperature and thermal stress distributions. If the temperature data are burdened with random errors, least squares smoothing is used to reduce the effect of the measurement errors on the calculated time derivatives d f j /dt or dT i,j /dt. The Gram orthogonal polynomials were used for smoothing the measured time-temperature history f j (t) and estimated temperatures T i − 1,j (t) (Taler, 1995). For linear IHCP, when thermal properties are temperature independent, node temperatures can be expressed in explicit form (Taler & Zima, 1999; Taler et al., 1999). In the following subsection the linear IHCP will be presented in detail. [...]... change of heat transfer coefficient occurs, the grid of control volumes should be made finer in this area  286 Developments in Heat Transfer Fig 10 Comparison of heat transfer coefficients determined from solution of inverse heat conduction problem at point 6 (hs = 2000 W/(m2·K)) and a point 2 (hw = 1000 W/(m2·K)) (see Fig 9): 1 and 2 – given (exact) and calculated value of heat transfer coefficient in steam... correct determining permits for the proper selection of heat transfer area during designing of heat exchangers and calculation of the fluids outlet temperature A lot efforts have been made during experimental investigations of pressure drop and heat transfer in different types of heat exchangers to obtain proper heat transfer correlation formulas 2.1 The Wilson plot technique to determine heat transfer. .. the determination of the temperature and stress distributions at the wall crosssection including the inner and outer surfaces of the steam header using the inverse heat conduction methods Inverse Space Marching Method for Determining Temperature and Stress Distributions in Pressure Components 287 Fig 11 View of the experimental installation for testing the computer system for on-line monitoring thick-wall... measurement points The smoothed value of measured temperature (approximating polynomial) and its time derivatives are determined only in the middle of the interval, i.e in time : ti + 5(Δt), where ti is time coordinate of the first point in the analyzed time interval: ti, ti +10(Δt) Inverse Space Marching Method for Determining Temperature and Stress Distributions in Pressure Components 285 4.3 Calculation... 3T11 − 2T12 + T13 ⎟ − r4 k 2 2 ⎝ ⎠ ⎦ 2 ( ) In order to determine the heat flux in the node on the inner surface it is indispensable to know the temperature at two adjacent points also on the inner surface Assuming that the temperatures in nodes 2 and 3 have been determined in a similar way as for node 1, we can define the heat flux in node 2 from the equation of heat balance for this node: cρ T − T2 Δr... cooling liquid, cp1 – average specific heat of cooling liquid, and T1inlet, T1outlet, are inlet and outlet temperatures of cooling liquid, respectively, yields to 296 Developments in Heat Transfer Rtotal = ΔTlm m1c p 1 (T1outlet − T1inlet ) (7) As the constants C1 and C2 are determined from straight–line approximation of measured data, to evaluation, for a given mass flow rate, the internal heat transfer. .. and exponents in correlation formulas for Nusselt number This method enables to obtain values of heat transfer coefficient on both sides of the barrier simultaneously without earlier indirect calculations of the overall heat transfer 3.1 Mathematical formulation of the inverse problem The issue consisting of simultaneous determining of the heat transfer coefficient on the cooling and heating liquid is... Köln – Berlin, S 143- 185 , ISBN 9 783 452273925 16 Experimental Prediction of Heat Transfer Correlations in Heat Exchangers Tomasz Sobota Cracow University of Technology Poland 1 Introduction Heat exchangers is a broad term related to devices designed for exchanging heat between two or more fluids with different temperatures In most cases, the fluids are separated by a heat- transfer surface Heat exchangers... and internal heat transfer coefficient Eq (8) : ho = 1 Ao (C1 − Rwall ) (8) The original Wilson plot technique depends on the knowledge of the overall thermal resistance, that involves to remain of one fluid flow rate constant and varying flow rate of the another fluid Approach of Wilson plot technique to determine constant in heat transfer correlation formula for helically coiled tube -in- tube heat. .. division into elementary cells has been shown in Fig 8a (a) (b) Fig 8 Division of drum wall into control volumes: (a) calculation pattern in inverse problem to determine temperature at point 1 at the inner surface on the basis of temperature measurement at five points: 9 ÷ 13 at the surface; (b) half of drum cross-section divided into control volumes when solving the inverse problem Temperature is determined . Generation in Momentum and Heat Transfer, Proceedings of the 14 th International Heat Transfer Conference, IHTC14-233 48, Washington, DC, USA, August 8- 13, 2010 Jaynes, E. T. (1 980 ). The minimum. abrupt change of heat transfer coefficient occurs, the grid of control volumes should be made finer in this area. Developments in Heat Transfer 286 Fig. 10. Comparison of heat transfer coefficients. the interval, i.e. in time : t i + 5(Δt), where t i is time coordinate of the first point in the analyzed time interval: t i , t i +10(Δt). Inverse Space Marching Method for Determining