SIMPLE EXPLICIT EXPRESSIONS FOR CALCULATION OF THE HEISLER-GROBER CHARTS. M.M. Yovanovich. Microelectronics Heat Transfer Laboratory

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1 SIMPLE EXPLICIT EXPRESSIONS FOR CALCULATION OF THE HEISLER-GROBER CHARTS M.M. Yovanovich Microelectronics Heat Transfer Laborator Department of Mechanical Engineering Universit ofwaterloo Waterloo Ontario Canada Abstract Simple single term approimations for the Heisler cooling charts the Grober fractional energ loss are presented for the plate innite circular clinder sphere. These solutions are accurate to within % of the series solutions provided the dimensionless time Fo is greater than some critical value Fo c which lies in the range: 0:80:4. Simple eplicit epressions are provided for the accurate calculation of the rst eigenvalue for all values of the Biot number. Polnomial epressions are presented for the accurate calculation of the roots of the Bessel functions of the rst kind of orders zero one. Epressions are developed for the accurate computation of the Bessel functions of the rst kind of orders zero one. Simple accurate solutions are proposed for calculating the dimensionless temperature the heat loss fraction for nite circular clinders rectangular parallelopipeds innite rectangular bars. Maple V R3 worksheets are given for the accurate calculation of the dimensionless temperature dimensionless heat loss for the innite plate innite circular clinder the sphere. Nomenclature A n Fourier coecients for dimensionless temperature B n Fourier coecients for heat loss fraction Bi Biot number; Bi hlk C ;C correlation coecients c p specic heat at constant pressure; JkgK D diameter of clinder or sphere; m Fo Fourier number; Fo tl h heat transfer coecient; Wm K J 0 () ;J () Bessel functions rst kind orders 0 L some characteristic bod dimension N number of panels in trapezoidal approimation n constant power energ loss; J initial internal energ c p V i ; J R radius of circular clinder S total active surface area; m S c ;S p ;S s spatial functions for clinder plate sphere T (;Fo) temperature of bod; K T f uid temperature; K t time; s Copright c996 M.M. Yovanovich. Published b the American Institute of Aeronautics Astronautics Inc. with permission. Professor Director Fellow AIAA V total volume; m 3 X; Y; Z half-dimensions of cuboid rectangular bar; m ; ; z Cartesian coordinates Greek Smbols thermal diusivit; kc p ; m s 0 ; parameters in modied Stokes approimation n nth root of characteristic equations n;c nth root for innite clinder n;p nth root for innite plate n;s nth root for sphere st root for 0 <Bi< ;0 st root for Bi! 0 ; st root for Bi! dimensionless temperature; i temperature ecess; T (;Fo) T f ; K i initial temperature ecess i T (;0) T f ; K mass densit; kgm 3 dimensionless position within an bod Subscripts c innite clinder cp nite clinder i initial value

2 p innite plate ; ; z innite plates along ;;zcoordinates innite rectangular bar z cuboid 0 at ver small time the rst root or rst eigenvalue at ver large time Introduction One-dimensional transient conduction solutions inside plates innite circular clinders spheres are presented in all heat transfer tets. The governing equations for the three classical geometries are given in Cartesian circular clinder spherical coordinates. The initial boundar conditions are specied. The dimensionless temperature histor () which is a function of three dimensionless parameters: position () time (Fo) boundar condition (Bi) is presented in smbolic form for the three geometries. All tets present the respective solutions in graphical form frequentl called the Heisler cooling charts (for the temperature at the centerline (plate) or the origin (clinder sphere) as a function of Fo Bi. Auiliar charts are available for all o-center or o-origin points 0 <. In addition Heisler charts are presented for small time Fo< 0:. Grober et al. introduced the charts for the total heat loss fraction for the three geometries. These charts are presented in great detail in Liukov 3 Grigull Sner 4 all heat transfer tets. Heisler Liukov 3 Grigull Sner 4 discuss the fact that the temperature heat loss fraction charts can be computed with acceptable accurac using the leading term in the respective series solutions. The leading term can be used for all values of Bi provided Fo Fo c where according to Heisler the critical Fourier number is approimatel equal to Fo c 0:4; 0:; 0:8 for the innite plate innite circular clinder sphere respectivel. For Fo< Fo c more terms in the series solutions are required to give acceptable accurac. Heisler also noted that more than 80% of the total cooling time is accounted for with the single term solution. Liukov 3 reports an earl attempt to provide approimations for the calculation of the rst roots (eigenvalues) of the characteristic equations for the three geometries. He showed graphicall that when ln ( ; ) is plotted against ln (Bi) over the range: 3 ln (Bi) 3 that the points were close to a straight line. A log-linear t of the data for the three geometries leads to the following correlation equation: ; q+c Bi C () where ; is the value of at Bi. The values of ; are: ; :404855::; for the innite plate the innite circular clinder the sphere respectivel. Liukov 3 reports the correlation coecients for the plate: C :4;C :0 the clinder: C :45;C :04 the sphere: C :70;C :07. This correlation equation for the rst eigenvalues gives acceptable values for Bi 00; otherwise the errors are much greater than %. The largest errors occur when Bi 0:. It is unknown what errors are introduced into the computation of the Fourier coecients A B b the use of this correlation equation. Chen Kuo 5 applied the heat balance integral method to obtain approimate solutions for the innite plate the innite circular clinder. These equations which are reported in Chapman 6 can be evaluated b means of programmable calculators the are said to be accurate provided Fo> Fo c. Since these equations are length involved the will not be presented here. Some of the recentl published heat transfer tets: Chapman 6 Bejan 7 Holman 8 Incropera DeWitt 9 Mills 0 White recognize that the series solutions converge to the leading term for long times i.e. Fo > 0: with errors of about %. The present tables for the roots (eigenvalues) of the corresponding characteristic equations the Fourier coecients: A B that appear in the dimensionless temperature heat loss fraction epressions. For values of Bi not given in the tables it is necessar to emplo interpolation methods to use the tabulated values. Once the eigenvalues are known the evaluation of the Bessel functions J 0 () J () that appear in the characteristic equation for the circular clinder the Fourier-Bessel coecients that appear in the temperature heat loss epressions must be considered. These calculations are tedious prone to errors unnecessar with the availabilit of programmable calculators computers. There is therefore a need to develop simple accurate equations for the computation of the rst root of the characteristic equations for the three geometries. Secondl there is a need to develop simple relationships for the accurate calculation of the Bessel functions that appear in the solutions for the innite circular clinder. These relationships will then be used to develop simple accurate relationships for the accurate calculation of the dimensionless temperature the heat loss fraction for the three geometries for Fo Fo c. Finall b means of superposition epressions will be developed for calculating the dimensionless temperature within composite bodies such as cuboids (Fig. ) innite rectangular bars (Fig. ) nite circular clinders (Fig. 3). Also b means of the Langston relationships for the determination of the heat loss fraction for composite bodies a method will be proposed for the simple but accurate calculation of the heat loss fraction from: cuboids innite rectangular bars innite plates nite innitel long circular clinders.

3 Heisler Dimensionless Temperature Charts X Y z 0 Y X Z Z The Heisler dimensionless temperature charts that appear in all heat transfer tets were developed for the classic geometries: plate innite circular clinder sphere. Since the dimensionless temperature depends on three dimensionless parameters: Bi; F o; where Bi is the Biot number Fo is the Fourier number is the dimensionless position it is not possible to show the temperature at an point within the solid at an arbitrar time. The dimensionless temperature charts are based on the general solution: Fig. The cuboid X A n ep nfo S ( n ) () n X + Y where A n are the temperature Fourier coecients that are functions of the boundar condition through Bi the initial condition n are the eigenvalues which are the positive roots of the characteristic equation S ( n )is the position function. For the three geometries the position function has the forms: 0 S p cos ( n ) (3) X Y - Fig. The innite rectangular bar R z r 0 Z Z Fig. 3 The nite circular clinder S c J 0 ( n ) (4) S s sin ( n) ( n ) Fourier Coecients for Temperature (5) The Fourier coecients for temperature for the three geometries have the following forms: or A n;p sin n n + sin n cos n (6) Bi Bi + A n;p () n+ n (7) n Bi + Bi + n or A n;c A n;c J ( n ) n J 0 ( n )+J ( n ) (8) Bi J 0 ( n ) n + Bi n [+nbi ] J ( n ) (9) 3

4 A n;s (sin n n cos n ) n sin n cos n (0) A n;s () n+ Bi h n +(Bi ) i n + Bi Bi () The eigenvalues that appear in the above relationships are the positive roots of the characteristic equations which have the following forms for the three geometries: Analsis of the Solutions Bi! 0 sin Bicos () J () BiJ 0 () (3) ( Bi) sin cos (4) For these three geometries for all dimensionlesss time Fo > 0 as Bi! 0 the rst Fourier coecient A! all other Fourier coecients A n! 0 for n. The rst root of the three characteristic equations approaches zero in the following manner: ;p! p Bi (5) ;c! p Bi (6) ;s! p 3Bi (7) the respective dimensionless temperature solutions become: p e BiFo (8) c e BiFo (9) s e 3BiFo (0) which are seen to be particular cases of the general lumped parameter solution: e c hs p V t () where S is the total active heat transfer surface of the geometr V is its volume. Bi! 0 Limit At this limit the eigenvalues for the three geometries go to the following relationships: n;c 6 4 n;p (n ) ; n () ; n ; ; 3::: 5 6 (3) with (4n + ). The above relationship is a modication of the Stokes approimation (Abramowitz Stegun 3 ). It gives acceptable values of the roots of J () 0 for all values n. The largest error is approimatel 0:005% when n the error is much smaller for all n. There is no simple relationship for the eigenvalues n;p for n. The eigenvalues are the roots of cos () sin () 0 (4) Numerical methods are required to nd the roots which lie in the interval: (n ) ; (n ). The rst nine roots are approimatel: 0 4: : : : : : : : For ver large values of n the roots approach the value n! (n ). Bi! Limit At this limit the roots (eigenvalues) of the characteristic equations are given b the following relationships: n (n ) n n ; ; 3::: (5) ; n ; ; 3::: (6) with 0 (4n ). The above relationship is a modication of the Stokes approimation (Abramowitz Stegun 3 ). It gives acceptable values of the roots of 4

5 J 0 () 0 for all values n. The largest error is approimatel 0:004% when n the error is much smaller for all n. n n n ; ; 3; ::: (7) The Fourier coecients for temperature are determined b means of the following epressions: A n () n+ 4 (n ) A n n J ( n ) n ; ; 3::: (8) n ; ; 3::: (9) A n () n+ n ; ; 3::: (30) Heat Loss Fraction Charts The heat loss fraction where c p V i is the initial total internal energ depends on the boundar condition parameter Bi the dimensionless time Fo in the following wa: X n B n ep n Fo (3) The Fourier coecients B n are given b the following relationships for the three geometries: B n;p A n;p sin n n B n;c A n;c J ( n ) n B n;s n Bi (3) n Bi + Bi + n n 6Bi 4Bi n + Bi (33) n + Bi Bi (34) Analsis of the Heat Loss Coecients The heat loss coecients B n have particular values in the two limits: Bi! 0 Bi!. In the rst limit all Fourier coecients for heat loss fraction are equal to zero for n. In the second limit the are given b: B n;p 8 (n ) n ; ; 3::: (35) B n;c 4 n n ; ; 3::: (36) where n are the roots of J 0 ( n ) 0 which are given above. B n;s 6 (n) n ; ; 3::: (37) Clearl the heat loss coecients are easil computed as Bi! therefore the heat loss fraction can be determined without dicult. Numerical Solutions Accurate numerical results for the three geometries can be obtained easil b means of a Computer Algebra Sstem such asmaple 4 MathCAD 5 Mathematica 6 or MATLAB 7. The proposed solutions procedure can also be implemented in spreadsheets. Maple 4 VR3worksheets for the plate innite circular clinder the sphere are presented in the Appendi. For each geometr the input parameters are: Bi; F o; ; N where N is the number of terms in the partial sum. It is recommended that N 5 although it can be set to an integer value N>. In each worksheet the rst ve inputs are: i) the denition of the characteristic equation ii) denition of the Fourier coecient for temperature iii) denition of the Fourier coecient for heat loss fraction iv) denition of the dimensionless temperature v) definition of the heat loss fraction. The net ve inputs create lists for the N: i) eigenvalues ii) coecients A n iii) coecients B n iv) dimensionless temperature terms n v) heat loss fraction terms ;n. The last two inputs give the dimensionless temperature the heat loss fraction for the given values of Bi;Fo. Approimations of Bessel Functions There are several methods that can be used to compute the Bessel functions: J 0 () J (). There are polnomial approimations (Abramowitz Stegun 3 ) available for all positive values of for both Bessel functions. These polnomial approimations are based on man terms that require space the are somewhat dif- cult to implement in spreadsheets or in a programmable calculator. The following epression which is based on the application of the trapezoidal rule to the integral form of J () for arbitrar order was developed b means of the Maple function trapezoid which is found in the student package: J () N X cos () + N + N N i cos sin i N i N (38) 5

6 where N is the number of panels is the order of the Bessel function. From this general epression one can develop epressions for both J 0 () J () for dierent ranges of for dierent accurac. For the range: 0 :4048 that is applicable for the rst term for the entire range of Bi one can use 6 panels in the above epression to obtain the following epressions which provide accurate values of the two Bessel functions: J 0 () 6 J () 6 " " 5 X i + 5X i cos sin 6 i # # cos sin 6 i 6 i Eplicit Solutions of Characteristic Equations (39) (40) The observations regarding the relationship of the rst root of the the three characteristic equations with respect to Bi reported b Liukov 3 the results of the analsis of the theoretical solutions presented above suggests that it is possible to use the asmptotic values: ;0 ; corresponding to Bi! 0 Bi!respectivel to develop interpolation functions for which will be accurate for all values of Bi. Plotting the ratio ; versus Bi gives a function that various smoothl between the asmptote: ; ;0 for Bi! 0 the other asmptote is equal to for Bi!. Based on the above observations the eplicit solutions (rst eigenvalues) of the characteristic equations for the three geometries can be written in the general form: ; + ; ;0 n n (4) The form is based on the method rst proposed b Churchill Usagi 4. The approimate eplicit solution alwas gives ver accurate values for ver small ver large values of Bi independent of the value of the parameter n. To obtain accurate values for intermediate values of Bi: (0:5 Bi 5) it is necessar to nd appropriate values of n. The values: (n :39;n :38;n :34) for the plate clinder sphere respectivel provide values of which dier b less than 0:4 % from the eact values of. This accurac is acceptable for most applications. To develop more accurate solutions for the intermediate range it ma be necessar to nd relationships between the parameter n Bi for each geometr. Temperature Heat Loss Fraction for Composite Geometries The basic solutions given above can be used to develop solutions for composite geometries such as cuboids nite circular clinders with convection cooling at all boundar surfaces. Since the cuboid solution is based on the superposition of three plate solutions it reduces to the innite rectangular bar solution the innite plate solution. The dimensionless temperature heat loss fraction solutions for the cuboid the nite circular clinder will be presented in the following sections. For the general case of a cuboid (Fig ): X X; Y Y;Z z Z that is cooled at its perpendicular faces: X Y z Z through uniform heat transfer coecients: h ;h ;h z there are three Biot numbers to consider: Bi ;Bi ;Bi z. The cooling of a cuboid is also characterized b three Fourier numbers: Fo ;Fo ;Fo z. Dimensionless Temperature for Cuboids The dimensionless temperature at an point within the cuboid for arbitrar time can be obtained b the means of the product of the solutions for three innite plates: z (; ; z; t) ;p ;p z;p (4) where the basic innite plate solution given b Eq. () is used three times: ;p A ; ep ; Fo cos (; X) (43) ;p A ; ep ;Fo cos (; Y ) (44) z;p A ;z ep ;z Fo z cos (;z zz) (45) The corresponding eigenvalues: ; ; ; ; ;z are dependent on the respective Biot numbers: Bi ;Bi ;Bi z. The Fourier coecients: A ; ;A ; ;A ;z are determined according to Eqs. (6) or (7). The eigenvalues: ; ; ; ; ;z are calculated b means of the general eplicit relationship developed for. Rectangular s The dimensionless temperature heat loss fraction from rectangular plates or bars (Fig. ): X X; Y Y is a special case of the cuboid solution. Here two Biot numbers: Bi ;Bi two Fourier numbers: Fo ;Fo are required to characterize its cooling. Dimensionless Temperature of Rectangular s The dimensionless temperature at an point within innite rectangular plates for arbitrar time can be obtained b the means of the product of the solutions for two innite plates: (; ; t) ;p ;p (46) 6

7 where the basic innite plate solution given b Eq. () is used two times: ;p A ; ep ;Fo cos (; X) (47) ;p A ; ep ;Fo cos (; Y ) (48) Heat Loss Fraction for Cuboids The heat loss fraction can be determined b means of the relationship developed b Langston : + + (49) z z where c p 8XY Z i. " Heat Loss Fraction from Rectangular s The heat loss fraction from a rectangular plate: X X; Y Y is a special case of the cuboid solution. Here two Biot numbers: Bi ;Bi two Fourier numbers: Fo ;Fo are required to characterize its cooling. The heat loss fraction is obtained b following relationship(langston ): + # (50) where c p 4XY i. In the above relationships the component heat loss fractions are obtained b means of the following epressions: z B ; ep ; Fo B ; ep ; Fo B ;z ep ;zfo z (5) (5) (53) The corresponding eigenvalues: ; ; ; ; ;z are dependent on the respective Biot numbers: Bi ;Bi ;Bi z. Finite s The dimensionless temperature the heat loss fraction from a nite circular clinder (Fig. 3): 0 r R; X X is based on the superposition of the innite circular clinder innite plate solutions. Here two Biot numbers: Bi p h Xk; Bi c h r Rk two Fourier numbers: Fo p tx ;Fo c tr are required to characterize its cooling. The heat transfer coecients are identical over the two ends: X but dierent from the side heat transfer coecient h r. Dimensionless Temperature for Finite s The dimensionless temperature at an point within the nite circular clinder for arbitrar time can be obtained through the product of the solutions for the innite circular clinder the innite plate: where cp (r;z;t) c p (54) c A ;c ep ;c Fo c J0 ( ;c rr) (55) p A ;p ep ;pfo p cos (;p X) (56) Heat Loss Fraction for Finite s The heat loss fraction is obtained b following relationship (Langston ): + (57) cp c p c p where c p XR i. In the above epressions the component heat loss fractions are obtained b means of the following epressions: B ;c ep ;cfo c (58) c p B ;p ep ;p Fo p (59) The corresponding eigenvalues: ;c ; ;p are dependent on the respective Biot numbers: Bi c ;Bi p. Summar Simple eplicit accurate epressions were developed for the calculation of the rst roots of the characteristic equations for innite plates innite circular clinders spheres for all values of the Biot number. Accurate polnomial epressions which are modications of the Stokes approimations for the roots of the Bessel functions: J 0 () 0 J () 0 are proposed. Simple epressions based on the application of the trapezoidal rule to the integral form of the Bessel function of the rst kind arbitrar order are presented. These epressions are eped in terms of the trigonometric functions which are easil computed in spreadsheets with programmable calculators. Maple V R3 worksheets are 7

8 presented in the appendi for accurate calculation of dimensionless temperature dimensionless heat loss for the plates clinders spheres for all values of the Biot number an value of the Fourier number provided Fo 0:0. Simple single term epressions are given for the accurate calculation of the dimensionless temperature the dimensionless heat loss fraction for innite plates innite clinders spheres. These single term epressions are used to develop epressions based on superposition the method of Langston for the calculation of dimensionless temperature the dimensionless heat loss fraction of bodies such nite circular clinders cuboids innite rectangular bars. The proposed epressions are simple accurate provided Fo Fo c. The should replace the tabular method currentl presented in all heat transfer tets. The tabular method eectivel replaces the Heisler Grober charts. Acknowledgements The author acknowledges the continued support of the Canadian Natural Sciences Engineering Research Council under grant A7445. Also the assistance of P. Teertstra Y. Muzchka in the preparation of the gures plots is greatl appreciated. References Heisler M.P. Temperature Charts for Induction Heating Constant-Temperature Trans. ASME Vol pp Grober H. Erk S. Grigull U. Fundamentals of Heat Transfer McGraw-Hill Book Compan Inc. New York Liukov A.V. Analtical Heat Diusion Theor Academic Press New York Grigull U. Sner H. Heat Conduction Hemisphere Publishing Corporation New York Chen R.Y. Kuo T.L. \Closed Form Solutions for Constant Temperature Heating of Solids" Mech. Eng. News Vol. 6 No. Feb. 979 p Chapman A.J. Fundamentals of Heat Transfer Macmillan Publishing Compan New York Bejan A. Heat Transfer John Wile & Sons New York Holman J.P. Heat Transfer Seventh Edition McGraw- Hill Publishing Compan New York Incropera F.C. DeWitt D.P. Introduction to Heat Transfer 3rd ed. John Wile & Sons Mills A.F. Heat Transfer Richard D. Irwin Inc. 99. White F.M. Heat Mass Transfer Addison- Wesle Reading MA Langston L.S. Heat Transfer from Multidimensional Objects Using One-Dimensional Solutions for Heat Loss Int. J. Heat Mass Transfer Vol pp Abramowitz M. Stegun I.A. 965 Hbook of Mathematical Functions with Formulas Graphs Mathematical Tables Dover Publications Inc. New York. 4 Maple Waterloo Maple Software Waterloo ON Canada. 5 MathCAD MathSoft Inc. 6 Mathematica Wolfram Research Inc. Champaign IL. 7 MATLAB The MATH WORKS Inc. Natick MA. 8 Churchill S.W. Usagi R. A General Epression for the Correlation of Rates of Transfer Other Phenomena AIChE J. Vol pp Appendi Maple V R3 Worksheets The Maple 4 worksheets for the innite plates circular clinders the sphere are presented here. The are valid for an value of for all values of Bi. The worksheets can hle small dimensionless times: Fo> 0:0 b increasing the number of terms in the summation. The input parameters for each worksheet are: Bi; F o; ; N where N is the number of terms in the summation. It is recommended that the number of terms should be limited to N 5 or less for most problems. Maple Worksheet for s restart: case: (Bi 0.3 Fo zeta 0 N 5): ce: *sin() - Bi*cos() 0: A: *sin()/( + sin()*cos()): B: A*sin()/: phi: A*ep(- ^*Fo)*cos(*zeta): _i: B*ep(- ^*Fo): vals: [seq(fsolve(subs(case ce) j*pi..(j + /)*Pi j 0..rhs(case[4]))]: As: evalf([seq(subs( vals[j] A) j..nops(vals))]): Bs: evalf([seq(subs( vals[j] B) j..nops(vals))]): phis: 8

9 [evalf(seq(subs(case vals[j] phi) j..nops(vals)))]: _is: [evalf(seq(subs(case vals[j] _i) j..nops(vals)))]: plate_temp: evalf(convert([seq(phis[j] j..nops(vals))] `+`) 4); plate_heat_loss: evalf( - convert([seq(_is[j] j..nops(vals))] `+`) 4); Maple Worksheet for Clinders restart: case: (Bi 0.3 Fo zeta 0 N 5): ce: *BesselJ( ) - Bi*BesselJ(0 ) 0: A: *BesselJ( )/(*(BesselJ(0 )^ + BesselJ( )^)): B: A**BesselJ( )/: phi: A*ep(- ^*Fo)*BesselJ(0 *zeta): _i: B*ep(- ^*Fo): vals: [seq(fsolve(subs(case ce) (j - )*Pi..j*Pi) j 0..rhs(case[4]))]: As: evalf([seq(subs( vals[j] A) j..nops(vals))]): Bs: evalf([seq(subs( vals[j] B) j..nops(vals))]): phis: [evalf(seq(subs(case vals[j] phi) j..nops(vals)))]: _is: [evalf(seq(subs(case vals[j] _i) j..nops(vals)))]: clinder_temp: evalf(convert([seq(phis[j] j..nops(vals))] `+`) 4); clinder_heat_loss: evalf( - convert([seq(_is[j] j..nops(vals))] `+`) 4); Maple Worksheet for s. #Bi cannot be set to zeta cannot be set to 0. #For Bi put Bi for zeta 0 #put zeta restart: case: (Bi 0.3 Fo zeta 0 N 5): ce: *cos() - ( - Bi)*sin() 0: A: *(sin() - *cos())/( - sin()*cos()): B: A*3*(sin() - *cos())/^3: phi: A*ep(- ^*Fo)*sin(*zeta)/(*zeta): _i: B*ep(- ^*Fo): vals: [seq(fsolve(subs(case ce) (j - )*Pi..j*Pi) j 0..rhs(case[4]))]: As: evalf([seq(subs( vals[j] A) j..nops(vals))]): Bs: evalf([seq(subs( vals[j] B) j..nops(vals))]): phis: [evalf(seq(subs(case vals[j] phi) j..nops(vals)))]: _is: [evalf(seq(subs(case vals[j] _i) j..nops(vals)))]: sphere_temp: evalf(convert([seq(phis[j] j..nops(vals))] `+`) 4); sphere_heat_loss: evalf( - convert([seq(_is[j] j..nops(vals))] `+`) 4); 9