THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING FLUCTUATING HEAT TRANSFER IN A SINGLE FLUID HEAT EXCHANGER Vedat S. Arpaci December, 1960 IP-481

,-e r t~r k~ K.( a,

ABSTRACT An unsteady heat transfer analysis is presented for single fluid. heat exchangers having internal heat sources. The transient effect is introduced through fluctuations of the fluid velocity about a mean valueo It is found that the phase angle and amplitude for the fluid and wall temperatures, and the interface heat flux have oscillatory characteristicso These results apply to certain types of nuclear reactors. ii

1. INTRODUCTION Although much work has been published on unsteady heat transfer, it appears to have been confined to problems of the boundary layer type. The customary method of solving these problems is to impose a boundary condition along the fluid-wall interface, and then solve the energy equation of the fluid. The interesting and important problem of how the wall responds to fluctuations of an external fluid flow about a steady mean has apparently remained untreatedo In this paper a start is made to investigate this subjecto The fluid energy equation is simplified by neglecting the viscosity of the fluid. Then the fluid and wall energy equations are coupled along the fluid-wall interface by means of the natural boundary conditionso A specific example, taken from the field of Nuclear Engineering, is used to illustrate the analytical procedureo This example pertains to certain types of nuclear reactors which can be idealized as consisting of three elements. These are a coolant (fluid) which flows coaxially between a moderator and a rod (wall) in which heat is generated. (Figure 1). Because the coolant thickness is much less than the radius of the rod, the radial variations in the coolant velocity and temperature may be neglected. The energy equation for the coolant may then be approximated by a simpler equation. In those instances where the moderator possesses a low thermal, conductivity as compared to the fluid and rod, the heat transfer from the coolant to moderator may be neglected as a further simplicationo The purpose of this paper, under the foregoing assumptions, is to present solutions for the temperature fields of the rod and coolant when the coolant velocity fluctuates about a mean value.

-22o FORMULATION The energy equations for the rod and coolant may be written as: p (Cpg) div (k grad *) + q', () at* D(C T*) P' I PD ) - div (k' grad T*), (2) Dt* where (p,p') are the densities, (g*,T*) the temperatures, (CpCp) the specific heats at constant pressure, and (k,k') the thermal conductivities of the rod. and coolant respectively. The time is t*, and the heat generation per unit volume in the rod is q' o Neglecting radial variations of the velocity and temperature of the coolant, axial conduction in both the rod and coolant, heat transfer from the coolant to the moderator, and. assuming all physical properties are constant, (1) and (2) can be replaced by the following equations: as* 2g = 1 a9G*) + pC =k( Tr. + + (3) pt* or*2 r* r*' (p'C )A(t. - Pk( *(4) where A is the cross-sectional area of the coolant, U the velocity of the collant, P the periphery of the rod, R the radius of the rod, and (r*,x*) are the cylindrical coordinates. This same physical system has been considered recently for a starting type of transient phenomena by introducing a step change in the rate of heat generation within the rod. (1) It will now be assumed that the heat generation is constant with. respect to time and uniform throughout the rod, but the coolant is fluctuating in a simple harmonic motion

-3about a mean value Uo with a small amplitude c and an angular frequency c)* so that U - Uo[l + e exp(iot)l, (5) where c << l By introducing the non-dimensional variables x (x*/R)/Pe', r = (r*/R), t = at*/R2 Pe - aUo/R2, = *(q" R2/k) T -T*/(Xq" R2/k), x = (PR/A)(pCp/A: p)), 3 - *R2/a (6) in which a is the thermal diffusivity of the rod and Pe, the Peclet number, then (3) and (4) become as (a2 + 1,a) i. at ar2 r ar T + [1 + exp(ic.t)] T (8), (8) at ax ar r-L 3o METHOD OF STUDY Considering the steady periodic (or asymptotic) solutions of the rod and coolant temperatures which is the purpose of the present study, it is appropriate to write::' Oo +:9iexp(il)t) (9) T = To + ETlexp(icot), (10) where ~oTo and OQ1T1 satisfy: b20o 1 b o o0 + -I ) + I i (11) r 6r dT _- (g) 9 (12) dx ar r=l

-4and 2(QI 1 91, i ( -r2+ r r) (13) io)Ti + 1 +. - ( 1) (14) dx dx or r i with the boundary conditions: To " O, T1 0", at the inlet of the coolant, (1.5) ^ 0, O, at the center of the rod, (16) or r To = Oo, T1 - 9, along the coolant-rod interface, (17) To and Qo may be obtained readily by solving (11) and (12) subject to the boundary conditions (15), (16), and. (17). The results are: To(x) x/2, (18) 9o(x,r) = x/2 + (1-r2)/4 (19) To determine T1 and Q1, the method is outlined as follows: the appropriate solution of (1.3) satisfying (16) is 1 (r,x) - A(x)Jo(i3/2al0/2r), (20) where A(x) is an integration parameter to be determined, and Jo is the Bessel function of first kind of order zero in the usual notation, Using the Equations (17), (18), and (20), the Equation (14) may be reduced to the following first order, ordinary and non-homogeneous differential equation~ dA [is + (w/i)l/2Jl(i.3/2cl/2)/Jo(i3/2ul/2) ]A = - 1/2 Jo(i3/2I/2)~ dx (21)

-5where J1 is the Bessel function of first kind, of order first. The solution of this equation is: A(x) = B exp {-[ico + (o/i)l/2j(i3/2ml/2)/Jo(i3/2co/2) ]} - 1/2 [iUJo(i3/2l/2) + (cn/i)l/2J1(i3/2Wl/2)] (2) where B is an integration constant to be determinedo Combining Equations (20) and (22), and applying boundary conditions (15) and (17) results in the following: T (x) 1_ 1 - exp {-[ic + (Go/i)l/2Jl(i3/2col/2)/Jo(i3/2ool/2) ]x 1 X 2 ic + (W/i)l/2J1(i3/2al/2)/Jo(i3/2cl/2) (23) and, G1(r,x) = - 1 - xp{-[io + (m/i)l/2Jl(3/2cl/2)/o (i3/2J1/2) ]} 2 icu + (cu/i)l/2J(i.3/2 (Jl/2)/j (i3/2.1/2) Jo (i3/2ul/2r) jl (13/20~./2) For numerical convenience only, the coolant and. rod temperatures may be rearranged. as: T(xt) 21 2 exp(-2ax)-2exp(-ax)cosbx+ l-. 1/ (2 T(x.t) - x [' ] exp[i(rt-T)], (25) 2 2 a2 + b2 and Q(r,x,t) = -x +' (lr2) 2 4 - -(c2d2)1/2 exp(-2ax)-2exp(-ax)cosbx ~ 1 /2 2 a2 + b'2 (26)

-6where a ( (+)2)1/2[(UoVI-VoU] - (UoUI+VoV1) ]/(o2+Vo2), (27) b =- + (c/2)1/2[(UV 1-VoUl) + (UoUi+VoVl)]/ (u02+V2), (28) c = [UOUo(r) + VOVO(r)]/(Uo2 + Vo2), (29) d = [VoUo(r) - UOV(r)]/(Uo2 + Vo2), (30) T= tan-1 b - exp(-ax)(bcosbx + asinbx) (31) a - exp(-ax)(acosbx - bsinbx) 0 - OT - tan-l(d/c), (32) an.d (UoVo) are the real. and imaginary parts of the Bessel function of first kind of order zero with complex argument, (UIV1) the real and imaginary parts of the Bessel function of first kind of order first with. complex argumento (2) It may be noted that U0 Vo U1 and V1 should be calculated for the modulus (oul/2) and the argument (7/4), but Uo(r) and Vo(r) for the modulus (col/2r) and the argument ( t/4) The foregoing procedure completes the required solutions for the temperature distributionso It remains to determine the fluctuating heat transfer from the rod to the coolanto The total heat flux per unit area of the rod is: q* _= -k( ) (33) ar* r*=R In terms of non-dimensional variables Equation (33) becomes q =- ) (34 ) where q = q*/q" R. (35) w'hp-rp~~~~~~~~~~~~~~~~~~{5

-7It follows that q = qo + cql exp(iwt) 9 (36) where Ci o^= I'i n) =. (37) ar r=l 2 = (91-) = 11 - exp{-[io+ (w/i)l/2Jl (i3/2wl/2)/J (i3/2cl/2) ]x} ar r=l 2 1 + (i3/2ol/2)J(i3/2oA/2)/jl(i3/2wl/2) (38) Again, for numerical convenience only, the total heat flux per unit area may be written as: q(x,t) = + ~ [exp(-2ax)-2exp(-ax)cosbx + 1]1/2ep (39 2 2 e2 + f2 exp[i(t-) (39) where e = 1 + (/2)1/2(UoVi-VoUl) + (UoUil+VoVl)]/(U12+V2), (40) f = (a/2)/2 [(UoV1-VoUl.) - (UoUi+VoVi.)]/ (U12Vl2), (41) O - tan-l[f - exp(-ax)(f cosbx + e sinbx)] (42) g- e -exp(-ax)(e cosbx- f sinbx) It will now be convenient to introduce the quasi-steady value of the amplitude of the collant and rod temperatures, and the interface heat fluxo Accordingly, in the Equations (23), (24), and. (38) replacing the Bessel functions and the exponential terms by their series expansions for small values of the argument results in TL(x) Gl(rx) - - I x (43) and ql(x) -1 N(-)exp(iit/2). (44) 9l~x) _P _7

-8The ratio of the amplitude of the fluctuating term to its quasisteady value for the collant and rod temperatures, and the interface heat flux may then be written as follows: ART = [exp(-2ax)-2exp(-ax)cosbx + 1.1/2 (45) I X a2 + b2 ARG = (2+d2)1/2 1 [exp(-2ax)-2exp(-ax)cosbx + 1] / (46) x a2 + b2 and 2 exp(-2ax)-2exp(-ax)cosbx + 11-/2 ARq = e 2+f (47) g i x e2 + f2 where AR stands for the "Amplitude ratio"O Values of.T', g at the center of the rod and q, and ART, ARQ at the center of the rod. and AR q for three different, locations and as functions of the angular frequency, are given in Figures 2 3, 4, 5, 6, and 7. 4o CONCLUSIONS Single fluid heat exchangers having internal heat sources serve as the basis for analysis in this papero The problem is one of unsteady heat transfer wherein the transient phenomena is introduced through fluctuations in the coolant velocityO It is found that the phase lag in the coolant temperature approaches zero for small values of XC and /2 for large values of c, being similar to boundary layer type problemso(3) The behavior of this lag, however, shows a fundamental. difference when compared to boundary layer problemso Phase lags in the latter are monotonic in nature but show an oscillatory character for the situation discussed in this studyo

-9The phase lag in the rod temperature also approaches zero for small values of C, but, unlike the coolant temperature, approaches infinity for large values of u. In regard to the interface heat flux it is found that small values of c cause the phase angle to lead the coolant velocity by jt/2 whereas large values of C produce a lag of it/4o This apparently strange result can be understood if the same angle is evaluated with respect to its quasi-steady valueo If this is done, a phase lag approaching zero for small values of C and 3At/4 for large values of c is obtained. Amplitude ratios for the coolant and wall temperatures and the interface heat flux behave in a similar manner, approaching unity for small values of c, and zero for large values of co Again, in contrast to the monotonic variations in the amplitude ratios of the boundary layer problems, here one finds damped fluctuations in the variation of the amplitude ratioso It would seem pertinent now to mention two possible extentions of this present studyo The first is related to the power generation in. the rod. As is well known, heat generation in a nuclear rod varies cylindrically in the radial direction and hyperbolically in the axial direction rather than being uniform throughout (4) This condition of space dependent power generation causes additional complexities, but could be handled in a manner that is essentially similar to the procedure presented in this papero A second extension of this study regards the introduction of fluctuations in the power generation instead of coolant velocity as was done in the present analysiso The first two approximations of coolant and. rod temperatures would yield the exact solutions, thus, the assumption C << 1 would be unnecessaryo

REFERENCES 1. Arpaci, V. S. "Transient Conduction in Coaxial Cylinders with Relative Motion and Heat Generationo" ASME (Journal of Applied Mechanics), Paper 60-APM-24. 2. "Table of the Bessel Functions Jo(z) and Jl(z) for Complex Arguments." Columbia University Press, 1947. 3. Lighthill, Mo J. "The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocityo" Proceedings of the Royal Societyq Series A, Volo 224, (1954), ppo 1-23, 4. Glasstone, S. and Edlund, Mo C. "The Elements of Nuclear Reactor Theory." New York~ Do Van Nostrand Book Company, (1952) ppo 213o -10

i X ( 0 ~........... —-— ~~....%. %] %~~~~~~~~~~~~~ ~

-12-.r —0 \ j II I'D ci ci ~ 0 rO 4-'0

-13--- - ~0 (0 -— \rd —------—!-, k \\ \ 0-010 3 X ~\\ J J J J g \ NU\\ ////.E" --------------------— ^) %/ —---------- <0 --------------- \ ^,^ --- S. N. —--------- CM b I —-----------------------— ^ —-------— ^0~~~~~~ CM ^- ^ CM ^ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C Ei ~ Is El 6 e~~~~~~~~~~~~ ^9,<~~~~~~~~~~~~~~~~~~

-1172 _____ _ _~-14TT / 2 0 2 4 6 8 10 12 14 16 18 Figue. P 25n Figure 4. Phase in Interface Heat Flux versus Angular Velocity

-151.0 0.9 X =0.25 X =0.50 0.8 \ \// X =1.00 0.7 k- 0.5 0.4 0.3 Q2 0.I o L —------- I TZ —--- 0 0 2 4 6 8 10 12 14 16 18 CAJ - Figure 5. Amplitude Ratio for Coolant Temperature versus Angular Velocity (r=O)

-161.0.4.... 0.9 0.8 X\ -/-X=0.25 - \ \//-=0:o.50 s A r V y (.00 0.7 t 0.6 0.5 0.4 0.3 0.2 0 2 4 6 8 10 12 14 16 18 Figure 6. Amplitude Ratio for Rod Temperature versus Angular versus Angular Velocity (r'O)

-171.0 0.9 0.8 0.7 X= 0.25 0.6 \ X=0.50 X: 1.00 <st 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 W) Figure 7. Amplitude Ratio for Interface Heat Flux versus Angular Velocity

UNIVERSITY OF MICHIGAN 3 9015 02499 5410l 3 9015 02499 5410