CALCULATION OF TEMPERATURE FIELDS IN A FUEL ROD USING THE METHOD OF ELEMENTARY BALANCES

Transcription

1 С. FOJTÍČEK CALCULATION OF TEMPERATURE FIELDS IN A FUEL ROD USING THE METHOD OF ELEMENTARY BALANCES ŠKODA-Concern NUCLEAR POWER PLANTS DIVISION, INFORMATION CENTRE PLZEŇ-Czechoslovakia

2 ZJE С, Fojtíček CALCULATION OF TEMPERATURE FIELDS IN A FUEL ROD USING THE METHOD OF ELEMENTARY BALANCES ŠKODA CONCERN Nuclear Power Plants Division, friiormotion Centre PLZEČI, CZECHOSLOVAKIA

3 ABSTRACT The report deals with a technique of calculation of temperature fields using the method of elementary balances the progressive development of which in the Nuclear Power Plants Division is supported by a brief description of iiie solution of temperature fields chosen with the aim to show the possibilities of its application in engineering practice. More detailed attention is devoted to the application of this method to the solution of the temperature fields in the region of fuel elenient spacing grids. Printed by Czechoslovak Atomic Energy Commission Centre for Scientific and Technical Information Zbraslav nad Vltavou

4 Introduction It is often necessary in. engineering practice to determine the temperature field for cases which are, due to both the geometry and the boundary conditions, very intricate. Not too long ago the most common way of solution has consisted in a simplification enabling to solve the problem with the use of analytical means. But with more complicated situations the real state of art has been defomied by the simplifications adopted to such an extent as to make the results obtained unsatisfactory. It may be said that from this viewpoint the numerical methods seem to be more suitable. More elaborate techniques accurate enough even for very intricate situations may be used nowadays with the help of computers^ The method of elementary balances belongs to these techniques. The method of elementary balances Its principle consists in the fact that any system may be divided into suitable elements among which heat exchange occurs. Said exchange of heat is not continuous but occurs in such elementary quantities that it is possible to express the heat balance in the system with any accuracy. The temperatures calculated in the central points of individual elements (the other parameters are also concentrated into these points), taken together, represent the temperature field of the system. The temperature gradients for respective directions are exp»ressed by means of the temperatures and mutual distances of the central points of respective elements. So, the temperature distribution among the central points :B linear, and it is the assumption under which the heat flow among adjoining elements is determined. It is necessary to point out that the temperature field is time-dependent and the same is valid for temperature gradients. This problem is associated with the convergence of the method and will receive more detailed attention, * Another fact to be taken into account is that the temperature cf the central point represents a measure of the heat capacity of the whole - 3 -*

5 element and that it is identical with the actual temperature of the central point only subject to certain assumptions. The foregoing considerations may be expressed more precisely in terms of accepted assumptions in the following way; a. A change of the heat capacity of an element is directly proportional to the change of the temperature of its central point. b. The averaged value of heat flow through a certain transfer area during an elementary time interval Zlí" is directly proportional to the temperature gradient for th«=> beginning of Piaid elementary time» interval. c. The temperature gradient in a transfer area is determined under the assumption that the distribution of temperature between two central points is linear. With the foregoing assumptions in view, the solution of the complicated equation of heat conduction may be substituted by the fundamental relationships used in an adequate calculating algorithm. The balance of any element is described by one of the relationships as follows : which expresses the relation between the change of heat capacity and temperature of an element; For о change of heat capacity in an element after expiration of the time interval may be written : The individual components of the heat balance, acting either Individually or in combinations in respective areas of tne element under study, may be expressed, according to circumstances, in the following manner : 4 -

6 a. For the case of heat exchange among elements : л b. For the case of a boundary condition of the second kind : &- r FACjr С For the case of a boundary condition of the third kind d. For the case of heat exchange by radiation : ь~ьгт-(м] глгс e. For the case of heat transmission across a thermal resistance Q^i(^-i)f^ U For the case of generation of heat in an element : The state of every element after the expiration of a time interval Д L is Individually determined from tbe conditions valid for the beginning of this interval. The above mentioned conditions are the temperatures, material properties! boundary conditions, generation of heat etc* It is necessary to add that these quantities may be time-dependent. The time interval entering the above mentioned equations is not arbitrary, tts limitation is due to tbe assumption accepted, according to which the heat flow is steady during The greater is the higher is the error entering the relationship expressing the balance - 5 -

7 of the element under investigation, For certain value of U = A ~Ej the change of element heat exchange would be so high as to shift the corresponding temperature of the element outside the temperature range of the surrounding ones. This, of course, would be in controversy with the second theorem of thermodynamics. The limiting value ^\f *nay be obtained from the following equation In practice it is necessary to solve problems with Дь lesser than A*t~a determined for the highest thermal diffusivity 0/ which will be attained with regard to both the kind of the material (the system may not be homogeneous), and the temperature dependence. An unsuitable choice of AT will cause oscillation of elements temperatures. Decreasing of below the limiting value -4^/ will have a slight influence on the accuracy of the solution. The dependence of the accuracy on Fo which represents the dl chosen, has been evaluated X/ on numerous examples. Concerning the choice of the elements, the dimensions of which enter the equation for the determination of АТ^Г it is a more difficult problem. In dividing the system it is necessary to obtain AT^g ol all elements In proximity to one value. With other than homogeneous systems the size of tine elements is proportional to the temperature conductivity of respective materials. Another viewpoint to be considered in dividing the system is the requirement that the calculated temperature of an element, as a measure of its heat capacity, must be in correspondence with the temperature In its cantral point. The size of tine elements must be therefore chosen in such a way as to obtain sufficient linearity of their temperature fields. Such a requirement needs finer division. The problem of division is associated with the influence of the boundary conditions on the accuracy of the solution, in a case of heat exchange

8 with hirjh heat flows and low thermal conductivities the only regions considerably influenced are the peripheral ones. Taking this into account requires a very fine division. Also the choice of A*C" is influenced in tills esse because the temperatures of the peripheral elements must be in due proportion not only with the temperatures of the adjoining elements, but also with the temperatures of the boundary conditions. With simple situations the relationships for ДТ may be deteraiined taking into consideration the boundary conditions. For instance, in case of a one-dimensional field with a boundary condition of the third kind may be written Fa< J Z Concerning more intricate situations, the ^ 4/ mrry be determined taking into account the second theorem of thermodynamics. If steady, the generation of heat has no influence on the convergence criteria but, if it is a suitable function of temperature, it may have a favourite effect on the choice of Л X* So far the method has been described in connection with the solution of non-stationary temperature fields. It may be stated that similar relationships may be also drawn up for the solution of stationary fields. But in this case the conditions of convergence are different. The advantageous step by step solution of individual elements cannot be used and it is necessary to draw up a matrix. The algorithm of such a solution is rather intricate, especially if the system is not divided regularly, fri using the original non-stationary method the problem has been solved in such a way thai the solution passes into the stationary state. The convergence is considerably speeded up in this case by determining a new temperature field of the system using an extrapolation of three successive iterations of the non-etationary process. This new temperature field serves as a starting state for the next three steps of the non - stationary solution* Steady temperature field is obtained with sufficient accuracy alter several extrapolations.

9 Examples of application The first problem which initiated the application of this method has been the solution of non-stationary temperatures in a steel plate during its rapid cooling. Ir. this case a strong dependence of the material parameters on the temperature has been apparent ( cooling over the recrystailization temperatures ). R should be added that there has been sufficient concordance between the temperatures calculated and rfieasured. A two-dimensional problem ( bending of gas piping caused by a rapid change of gas temperature ) has been verified by a comparison with a solution obtained using an electric analogy, The concordance obtained has been within the range of the accuracy of reading-off. Considered from the viewpoint of the number of elements the greetest problem solved so far by this method has been the determination of a non-etationary three-dimensional field in the intersectional region of two hollow cylinders. The item of the primary interest ha = been the primary piping slide valve during an emergency shut-down of a reactor. This method has also found application ir solving a three-dimensional temperature field around a thermocouple on a fuel rod. The problem is interesting only because of the very small dimensions. The elements are below 1 mm, The task to dete-mine the temperature field in the region of fuel element spacing grid also refers to this report. The two-dimensional temperature field has been obtained from the surface temperatures of the rod obtained by measurement and from known heat input In the axis of the nc*wiomogeneous rod. Surface gradients hove enabled to determine the heat flow into the flowing gas so that it has been possible for the gas temperature against any element to be determined, as well as the temperature difference, the heat transfer Coefficient, and Nu, The heat transfer coefficient has been determined using the relationship

10 Suj А М = /ги, (i s - вс ) Ьп the next stage the obtained value of Nu has been used for several variants ot fuel rods, and. the corresponding temperature field has been determined. * Designation AT-,. elementary time interval А в change of element heat capacity during fc ť density of element material с specific heat of element material /j... thermal conductivity of element" material V &l element volume.. change of element teinperature during Q. individual components involved in the element heat balance t, element temperature t temperature of the adjoining element a P., transfer area 1 distance of the central points q specific heat flowed heat transfer coefficient for convection heat exchange t t к temperature of the environment for convection heat exchange element surface temperature heat tteanbter coefficient for heat exchange by radiation T e /fc/ temperature of the environment for heat exchange by radiation lo, T... element surface temperature / К / к P over-all coefficient of heat transmission

11 w JÍ^ specific heat generation limiting value ol time interval A x Л) i, element dimensions á У d z i a Fo. Bi thermal diffusivity local Fourier's number Bioťs number ~ lo -