THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING TRANSIENT HEAT TRANSFER IN HEAT EXCHANGERS HAVING ARBITRARY SPACE- AND TIME-DEPENDENT INTERNAL HEAT GENERATION Wen-Jei Yang September, 1962 IP-582

TABLE OF CONTENTS Page LIST OF TABLES................................................... iii ABSTRA CT o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 iv INTRODUCTIONo0...... 0 00 00 00...0..... 0........ o.............0. 1 STATEMENT OF PROBLEMO..... o.... o.. oo.........o............. 3 ANALYSIS.. 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 a.0 0 a 0 00 0 0 0 00 0 0 0 0 0 0 0 0 3 JA CALYSL o OOL o o o o o o o o o o o o 00 o o o o o o o o o o 0 o 0 o o o o o o o o o o o 0 o o o 0o o0 o o o o o SPECIAL CASES o...o...o.......................... o................ 11 1o Zero Solid-Coolant Heat Capacity Ratio o..o........... 11 2o Infinite Heat Transfer Coefficient.oooo 0000000,0.......0 11 35 Infinite Heat Transfer Coefficient and Time-Dependent CONCLUSION.................o o o o o o o o o o o................. 13 COENCLTUSONoooooooooooo o.ooooooooooooooooooooooooooooooo.....o 14 NOMENCLATUREO 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - l0 1 NOMENCLATURE..oo.................................. 14 APPENDICES.oooo...... 00........o00o00....0o0..........00000000... 20 1 Inverse Laplace Transformation of Equation (17) 000oo,00 20 2 Convolution Integral in Equations (26) to (29),........ 23 3 Derivation of the Duhamel's Superposition Integralo.oo. 26 REFERENCES...oo..o....... o.... o....o.... o................. 28 ii

LIST OF TABLES Table Page I Laplace Transforms.................................... 17 II Some Responses of Coolant Temperature for Special Cases ( ), (2) and (3)..o................................... 18 III Table for Derivation of the Duhamel's Superposition Integ ral............................................... 19 iii

ABSTRACT The partial differential equations governing transient heat transfer in a heat exchanger are solved analytically for an arbitrary space- and time-variation in internal. heat generation, which remains space-time separable as expressed by t(x) 0(T) o The heat exchanger consists of thin solid plates through which a coolant flows, The coolant inlet temperature and all physical properties of the solid plate and coolant are taken as constant. The results include the response of the coolant and solid temperatures. Part I of the paper presents the general mathematical analysiso Part II deals with the limiting cases such as (i) zero solid-coolant heat capacity ratio, (ii) infinite heat transfer coefficient, and (iii) infinite heat transfer coefficient and time-dependent coolant velocity, Heat exchanges to which these results apply include the electrical. heater, a chemical reactor in which a chemical reaction occurs within the solid walls and the convection cooled heterogeneous nuclear reactoro iv

Introduction The coupled partial differential equations governing transient heat transfer in heat exchangers having internal heat sources, including nuclear reactors, were solved by many authorso Since a large number of papers and books having important bearing on heat-exchanger dynamics have been revieved in References 1 and 2, only new literature or other pertinent papers will be discussed. Yang(3, 4, 5) has studied the step- and frequency-responses of a single-solid, single-fluid heat exchanger resulting from the following disturbances: (1) the uniformly distributed, internal heat generation in the solid wall, (2) the uniform wall temperature, (3) the uniformly distributed internal heat generation in the flowing fluid, (4) the fluid inlet temperature and (5) the appropriate combinations of (1), (2), (3), and (4). The resonance phenomenon in the frequency response is also physically interpreted. Doggett, et al.,(6' 7) have solved the thin plate problem for axially unreflected and reflected cores with sinusoidal spaceand exponential time-varying power density. Yang(8, 9) has presented a general analysis for the step- and frequency-responses of the solid and coolant temperatures of heat exchangers having sinusoidal space-dependent internal heat generation. Bonilla, et alo(10), present analytical solutions for the response of a nuclear reactor to step change in time with the spatial distribution in power density arbitrary. All these studies just mentioned and References 1 and 2 have assumed that the heat transfer coefficient between the solid and coolant is constant with axial distance -1

and time, The validity of the assumption has been experimentally verified by Yang(ll) for the uniform heat flux distribution and by Hall and Price(l2) for the uniform, exponential and sinusoidal heat flux distributionso Hall and Price have reached a conclusion that the consequences of ignoring the effects of heat flux distribution shape on the heat transfer coefficient are not likely to be large in present design of reactoro However, they may become important in smaller reactors having a smaller length to diameter ratio for the cooling channelso In the present paper the transient heat transfer in heat exchangers having an arbitrary space- and time-dependent internal heat generation is analyzed under the assumption of constant heat transfer coefficiento The following limiting cases are also studied: (1) zero solidcoolant heat capacity ratio, (ii) infinite heat transfer coefficient, and (iii) infinite heat transfer coefficient and time-dependent coolant velocity. The time-dependent rate of heat generation includes (1) arbitrary time rate of change, (2) step-change, (3) sinusoidal-change, and (4) exponential-change for the uniform and sinusoidal space dependency. The arbitrary space-dependent portion of heat generation *(x) in the interval (0, L) is expanded in a Fourier cosine series. Due to the linearity of the governing differential equations, the solution for an arbitrary t(x) is obtained by the superposition of the solutions for the uniform and sinusoidal, space dependency. This linearity of the problem also suggests the convenience of the use of the principle of superposition, in particular Duhamel's Integral, for both time- and space-dependent heat generations.

-3 Statement of Problem The physical system analyzed consists of thin solid plates through which a coolant flows with velocity u o Initially, both the solid and coolant temperatures and the heat generation rate within the solid are at steady states. At zero time, a certain change in heat generation which may be expressed as *(x) 0(T) is introduced, As a consequence of this, a transient process is introduced in the temperatures of both the solid and the coolanto For purposes of mathematical convenience, the following assumptions are imposed: (a) The coolant temperature and velocity are represented by a single value (lumped) at the flow cross section. (b) The solid temperature does not depend on the distance in the traverse direction, which is valid for sufficiently thin plates. (c) The axial conduction is negligible in both coolant and solid and heat flows only to the coolant. This is a reasonable assumption when the Pecket number exceeds 100 (d) The following quantities are constant and uniform throughout: coolant flow area, heat transfer coefficient, inlet coolant temperature, and coolant and solid propertieso Analysis Application of an energy balance to the system produced the following two differential equations to express the transient behavior of the solid and coolanto

-4 P i/ X PxX, T)Vw i G Solid - ( t - t) + (1) h A K T Coolant G - t 1 t + at (2) K aT K ax with the initial and boundary conditions G (x, o) - t (x, o) - (3) t (o, T) = 0 The function Px(x, T) is the arbitrary time- and space-dependent variation of heat generation in the solid and may be expressed as Px(X, T) -- (x) /(T) (4) Equations (1) and (2) are operated on using the Laplace transformation techniqueo The transformed equations in solid and coolant temperatures are integrated with the appropriate initial and boundary conditions as outlined in Reference 9, The results for solid and coolant temperatures are as followsSolid. Mi~(s,)t(x) +_M s)e-B(s)x G(xs) (p p)K(l+/K) + (p)U(l+/Kw)2 [A(x,s) - A(o,s)] (5) (PCp) K'+s/Kw) (pCp) l (1+s/Kw) 2 Coolant M7(s)e-B(s)x xs) M [A(x,s) - A(o,s)] (6) (pCp) u(l+s/Kw)

-5 where K. s1 B(s) -' + - (7) u K 1 + s/K and A(x,s) f- (x)eB(S) Xdx (8) with the integration constant omitted, If t(x), the space-dependent portion of heat generation, is specified, then Equation (8) can be integrated and the transient temperatures of the solid and coolant may be obtained by performing the inverse Laplace transformation on Equations (5) and (6). The theory of Fourier series indicates that *(x), a bounded function continuous or sectionally continuous in the interval (0, L), may be expanded in the sine or cosine series. Let *(x) be expanded in the cosine series as a 00 ~(x) - + Z an cos anX 2 n=ln or ao 00 -+ Z an sin Qn(xo + x) (9) 2 ndl where 2 L (10) O - j )d (10) L an 4- (x) cos nn xdx (11) n L o Q nit (12) and 4n- 2- (1L3) o "(2n+) L

-6 The linearity of the problem suggests that the response to a general transient resulting from px(x, T) may be obtained by superposing the solutions to the volumetric rates of heat generation 2 0(T), a1 0(T) sin Q1 (xo + x), a2 0(T) sin Q2 (xo + x)., ---- an (T) sin on (xo + x), --—. Then Equations (5) and (6) may be written as Solid Q(xs) _ aO o,(x,s, s (1)4) v(s) 2 (s) nl2 (s) Coolant a(x +s) atoan) t(xs)() ^(s):-2 r(s) n —l n (s) where QO(x,s) K M [1-eB(s)x].(~):pC,)Sl~l's/~') + - - - -(16-a) v(s) (pCp)wM(l+s/Kw) (pCp)wu(l+s/Kw)2 B(s) to(x,s) M[l-e-B(s)x] v(s) (pCp) u(l+s/Kw)B(s) n(Xs) M sin 2n(xo+x) M B(s)sinno ~(s) (pCp)wK(l+s/Kw) (pCp)wu(l+s/Kw)2[B2(s)+ns2]i +) - 2ncossn(xo+x)-e ( ) [-cossnxo+B(s)sinnX xo]} (17-a) Tn() (PCp)wu(ls/Kw)[B2(xs)+ n2 {B(s)sin2n(xo+x)-Qncos2n(xo+x) e-B(s) (pC [- csp)u s x+B(s)(s) in2 (17-b) - exB(s)x[-]cos}xo+B(s)sinn xo]} (17-b) for n 1, 2, 3,. —

-7 Equations (14) to (17) represent the response of the transformed temperature at position x to the transformed time rate of change of heat generation /(T), that is, the transfer function between the two physical quantities. Equation (16) refers to the case in which heat generation pX(x, T) is time-dependent but space-independent, and Equation (17) is for the case of a sinusoidal space-dependent variation in heat generation. Therefore the response represented by Equations (14) and (15) may be regarded as the superposition of an infinite number of responses resultao ing from the variations of heat generation, - 0 (T), al (T)sin1Q(xo+x), a2z(T)sinQ2(xo+x), ---- an0( T)sin2n(xo+x)..... The operation of the inverse Laplace transformation on Equations (14) to (17) yields the response of the solid and coolant temperatures in the physical domain of time and space. Due to the nature of the mathematical attack on this problem, solutionsare obtained for G(x,T) and t(x,T) in two time domains O < T < x and T > o The results may be expressed - -U -U as Solid 00 G(x,T) a= Go(xT) + anGn(x,T) (18) 2 n=l Coolant 00 t(X,T) = a to(x,T) + Z antn(x,T) (19) 2 n=l where GO(x,T) and to(x,T) are respectively the transient solid and coolant temperatures resulting from PX(X,T) = ((T), which are given as follows in Reference 1:

-8 (a) 0 < <1 x - G(X,T) t(x,T) M (pCp) (M+l) M (pCp) (M+l) ~T ~ + K(M+l) (_-T) f 0()[l + Me ] I _K(M+1) f 0(~)[1 - e M I a 0 (20) (21) (b) x xb) - _ G(X,T) - M (pCp)w(M+l) Kx K u - -e M T f O()ir*) fT* A(T*-~) 0 K(M+1) 1 -e M - /] d - KI e M Io (2 ^Kx/u K^w) da } (22) t(X, T) M T - - f 0(t)[i - e (pCp)w(M+_) o K(M+l) (-T M I d~ Kx K u - e M - a f' fJ 0 (q)d e M Io (2 Kx/u Kw ) da}(23) 0 0 where T - (M+l)T K -(M+l) t(T) = T ~ (&)d-e M 0( (()e (24) and x T"x = T - - U (25) Gn(XT) and tn(x,T) are respectively the transient solid and coolant temperatures resulting from px(xT) = Z(T)sinn(x +x) o The details of the transform solution are described in Appendix 20

-9 (a) 1 > > 0 - x - ni(X, T) (pCp) * {KKw[Gl(T) + / 2(1 a G2(T) ] * [G3(T). 14b-a2 _ -- G4(T)] 4d-C2 * G6(T)sinmQ(xo+x) * G4(T)cos%(xO+x) 4MRnu _ 4m.- G2(T) 4 (4b-a2) (4d-C2) + G6(T)sinQn(xo+x)} (26) tn (x, T) = ( * {K[G(T) (PCP)w 2(M+)-a G() * [G(T) + 2 aG2(T)] * [G3(T),,/4b-a2 C 2-a - G4(T)]sinQn(xo+x) - M.nU[G1(T) + 4b- G2(T)] * 47-C2 V 4b-a2(27) 4 w,2- G4(T) cosn(xo+x)} (27) 214i-' (b) GnII (X.,T) Z:- nI (X, 7) + (.Tp) K n Kx + (. {KQnue u (pvw + 2-a [GL(T*) + b-a G2(r*)] L 14 4b-a2- 2 * 4d-C2 Kx G4(T*) * G5(T*) cOSQnXo-KKWe U [G (T*) 2(M+l)-a C 2(M+ 1- (T*)] * [G(*) - Cj G4(T*)]*G5(T*)sinflxo} (28) tni(x, T) = tnI(XTr) (T*) +(Pcp)w - Kx * {KQfnue u 2-a [G1(T*) + 4b-a2 G2(T*) ] * [G5(T*) + 2-C 4d -C2 Kx G4(T*)] * G5(T*)cos^nXoKe U [G1(T*) 2(M+1)-a +4b-a2 G2(T*)] * [1 - (C-1)KwG (T*) (2d+C-C2)Kw (2d+C-C )Kw G4(T*)] * G5(T*)sinnxo } d 4d-C2 (29)

where a, b, C and d are functions defined in Equations (46) and (47) and G1(T) ooo G5(T) are functions defined in Table Io The product of transformed functions, such as F1 F2, may be treated by the method of convolution, a technique of the inverse transformation for product of functions, ioe,, T T Gi(T)* Gj(T) = f Gi(S) Gj(T-S)d - Gi(T-S) Gj(t)d5 (30) 0 0 The convolution integral in Equations (26) to (29) are presented in details in Appendix 2. The lineality of the equations and boundary conditions also suggests that the system response 9(x,T) and t(x,T) can be obtained employing Duhamel's superposition integral which is derived in Appendix 35 This technique requires the solution for the response to a time-dependent but space-independent heat generation. The results are as follows: x RXT(,T) = I(O)RT(XT) + RT(X-T) d d (31-a) or, = (X)RT(oT) + /J X() Ta( d (31-b) o dx or, x 6RT(5,T) = r(x)RT(o,T) + Jf (X-T) dS (51-c) 0 o ac where RXT (X,T) and RT (X,T) are the response of coolant or solid temperatures to an arbitrary space- and time-dependent heat generation and to an arbitrary time-dependent but space-independent heat generation respectivelyo

-11 Special Cases lo Zero Solid-Coolant Heat Capacity Ratio If the solid-coolant heat capacity ratio is negligibly small, the energy Equation (1) and (2) reduce to Coolant at 1 at (X)0(T)VW x-+ 3Vu ax u aT pCpVu 2) Solid (33) G = t + t(x) (T)Vw hA 2. Infinite Heat Transfer Coefficient If the heat transfer coefficient from solid to coolant is infinite, the coolant and solid temperatures are always equal. An energy balance on the system leads to the following equationso at M+1 at (X)u( T)Vw ax u aT pCpVu (34) G = t (35) 3. Infinite Heat Transfer Coefficient and Time-Dependent Coolant Velocity If the heat transfer coefficient between the coolant and the solid is infinite and the coolant velocity varies with respect to time, an energy balance gives u(T)- + (M+l)a =1 vT) ax aT pCpV ppp (36) Q = t (37)

-12 Let u(T) = uf(T) (38) and X - -- I f (T) dT (39) M+l o then at t a f(T) at (40) aT aSX T M+l dX Substitution of Equation (40) into Equation (36) gives at + t ()(x)Vw (41) ax u ax pCpVu where v(X) = Xl) (42) f(T) Inspection reveals that Equations (32), (34) and (41) are the first order linear partial differential equations of the same type. Since the same initial and boundary conditions are imposed on each case, the general solutions to Equations (34) and (41) will be in identical form as that of Equation (32). The latter solution may be readily obtained from the general case previously analyzed by merely substituting M = 0 The results in analytical form are presented in Table II for various heat generation conditionso The response to an arbitrary space- and time-dependent heat generation may be expressed in similar forms as Equation (19), where to(X,T) and tn(X,T) refer respectively to the solutions shown in lines 6 and 10 in Table IIo

-135 Conclusion Solutions for transient behaviors of the solid and coolant temperatures resulting from an arbitrary space- and time-dependent variation of heat generation are obtainedo The results for the solid-coolant temperature difference At(x,T) may be formed from G(x,T) - t(x,T) Transient local heat flux may be evaluated by multiplying At(x,T) by the heat transfer coefficient. x Those solutions in the second time domain, T >, might involve two- and three- parameter functions such as 2 4. o 10 in References 1 to 8 as the consequences of the convolution integralso Should 0(T) be a complicated function of time, difficulties might arise from the integration of 1t(T), T2(T), A1(T) and A2(T) o For such case it would be convenient to use the Duhamel's superposition integral for /(T) as described in Appendix 53

NOMENCLATURE A = heat transfer area between the solid and the coolant, ft2o a = function defined by Equation (46). a0 Fourier coefficient defined by Equation (10)o an = Fourier coefficient defined by Equation (ll)o B(s) s + K _ K/u u u 1 + s/Kw b = function defined by Equation (47)0 Cl.ooC4= functions defined by Equation (44)o C = function defined by Equation (46), p Cp = specific heat of coolant, BTU/lbm ~F. CPw = specific heat of solid, BTU/lbm ~Fo D function defined by Equation (48) d = function defined by Equation (48), E = function defined by Equation (47). E = function defined by Equation (49)o Fl(s)oooF6(s), Fj(s) = functions defined in Table I. f( ) = function ofo G1(T)oooG6(T)GGj (T) = functions defined in Table Io Gi(T)*Gj(T) = convolution integral of Gi(T) and Gj(T)o h = heat transfer coefficient between solid and coolant, BTU/hroft2oFo Io = Bessel function of first kind and zeroth order, K = hA/pCpV, reciprocal of coolant time constant, 1/hro Kw = hA/pCpVw), reciprocal of solid time constant, 1/hro L = axial length of heat exchanger, fto M = (pCpV)W/pCpV = K solid-coolant heat capacity ratio, dimensionless. p"(X,T)= * (x)0(T) = volumetric rate of heat generation in the transient state, BTU/hroft3o -14

-15 Rx(xT) = response of coolant or so.lid temperatures to an arbitrary space-dependent but time-independent heat generation, 0Fo RT(xT) = response of coolant or solid temperatures to an arbitrary time-dependent but space-independent heat generation, ~Fo RXT(X T)>- response of coolant or solid temperatures to an arbitrary space- and time-dependent heat generation, ~Fo s - Laplace variable, 1/hro t(x,T) = transient component of coolant temperature,'Fo to(xT) = transient coolant temperature resulting from px(X,T) 0(T)tFo tn(X,T) = transient coolant temperature resulting from pJ(xT) - (T) sin Qn (xo + x), ~Fo u. coolant velocity, ft/hr. V volume of coolant, ft35 Vw = volume of solid, ft3, w reciprocal period of the exponential transient, 1/hro x= axial distance, fto 4n+l X~ = ( 2n)L, Lft. xl, x2, ooo xn = commencing points of spacewise uniform heat generation element for the derivation of the Duhamel's superposition integral, fto aKw CKw = or --- 2 2 = f4b-a2 Kw or, 4d-C2 Kw 2 2 A1(T), A2(T), A3(T) = functions defined by Equations (58), (59) and (60)o G(x,T) = transient component of solid temperature, ~Fo

-16 Go(xT) = transient solid temperature resulting from px(xT) = 0(T), OFo gn(xT) = transient solid temperature resulting from px(xT) = 0() Sin On(xo + x), OFO A(x,s) = function defined by Equation (8), Al(x,s), A2(xs), A3(xs) = functions defined in Table II. X _= function defined by Equation (39)~ = dummy variable. jt(), li(T),oooT4(T) = functions defined by Equations (54), (55), (56) and (57)0 p density of coolant, lbm/ft3o PW = density of solid, lbm/ft3. T = time, hr. ^. = T - x hro f(T) 0(T) = time-dependent portion of heat generation variation, BTU/hrofto t(x) = space-dependent portion of heat generation variation, dimensionless, Qn = n_ L )- = angular frequency, rad/ino (-) = Laplace transformed functiono Subscripts I: refers to 0 < Tu < 1 in general; 0 < - u < 1 for special case (2); -~ x -" " ji-Ml x - O < -u < 1 for special case (3), -x - II: refers to TU > 1 in general; -T u > 1 for special case (2); x M+i x - Au > 1 for special case (3). x

-17 TABLE I Laplace Transforms F(s) G(T) Fl(s) = F2(s) - F3(s) = F4(s) = aKw s + -2 (s + aKw)2 + (b Kw)2 -a4b-a2 ----- Kw 2 2 2 CKw s + - 2. (s + 2K)2 + ( 2 KW)2 (s + Kw) + ( -C2 )2 2 2 aKwT 2 G1(T) = e aKw-r G2(T) = e- CKWT G3(T) = e- CKwr G4(T) = e 2 V 4b-a2 os 2 KwT 2 sin JIBKw cos KwT sin J4d KT 2 Kx /u ) e 1 + s/Kw 5(s) =.... s + Kw F6(s) = s + Kw G5(T) e I-K o(W ) G6(T) = e-W Gj(T) = 0 when T <U Fj(s)e sx... Gj ( _T -) T > _

-18 TABLE II Some Responses of Coolant Temperature for Special Cases (1), (2) and (3) Special Case (1) Special Case (2) Special Case (3) Differential at t1 at, (X)0(T at M+ 1.t *l(X)O(T)VW 6t M 1 6t (X)( O)W (X) Equt +on 1 ~t ~(x)~(vVwo()= Equation TaX -I PCpVu ax u aT pCpVu ox u ax pCpVU Initial and t(x,0) - 0, oQ(,0) - 0 t(x,0) - 0, Q(x,0) - 0 t(x,0) - 0, o(x,0) - O Boundary Conditions t(O,T) - 0 t(O,T) - 0 t(0,') - 0' -- -)V ~ (M~l~s(s)vw ox',3(o1s)1 General Solution t(x,s ) -. 0(s) e u[Al(x,s) - Al(O,s)] s(x,) (= )v e u [2(x, ) - A2(0,s) (x,s) --' [ ) - A 0, pCpVu pCpVU PC 1 (Laplace transform with respect to X) sx (M+l)sx sx Deficition of ex )o4 so Ai(x,s) Al(x,s) = eo-(x)dx A2(x,s) = f e: + (x)dx A3(x,s) = e-*(x)dx Vw T V(W)d VW X (arrx) - t)(X, ) = Y o()dt tj(XT = - 0(t)dt ti(x,X) = ^ o )d arbitrary O(T) oCV Pcp(Ml) or u(X) Vw T T()d vw X teII(XT) V T X(-)d tII(XT) E= pClpV(M( ) T () tII(X',) =p V i ()d *(x) = 1 tC(X,T) CV T tj ( ) VW T tCpV M+i 0(T) -= tj(X,.) t- PV tii(x,T) = VW x fcpv u PCpV u - (x) -1 amplitmude-ratio =2(1 ) amplitde-rato = 2[1-cons. u ] eDX (M+l)0T O(T) - SiXlor X Wx c(T) phase-shift = -tan-1 (1-c0 u) phase-shift = -tanl-1 1-cos(M+l) sin. i sin(M+l) Ax (x) - 1 tI(x,T) V= (e -1) t(x,T) = W (e - l) | =(T) t. eT CCtII(X,T) = /V (ew - )) t (X,T) -C V -' e u ](- I) PipVW pC5V(0+lJw = si (x+) ttI(XT) P=C V (T —)sin (xo+x-tu)dt tj(x,T) = Cp VM+l) o} S(T-t)sinon(Xo+x-u)dt tI(x,X) V- (X-)sinn(xo+x-tu)d | *(x) - sinan(Xo+x) p P o P arbitrary O(T)oro(x) T V V ) or,T) V p- ( 0(T-t)sinn(xo+x-u)dt tII(X,T) = [s 0(T-~)s~ins(xo-iu)d | tC(xx [ f o(X-)sinnn(xo+x-tu)d~ -~t 0f CpV(M+i) [of.(T')sinnn(o~-no)Su)dt) - - - J O(T*-t)sinOn(xo-gu)dS] - O (T*-t)8sinn(X -tu)di] - 7 (X*-s)sinnn(xo-tu)dt] t x,*in (o )' ) cosonn(xo+X-Tu)-cosnG(Xo+X)] t1(x,r) = [coso(oo(xo+x - -1 u)-cosnn(xo+x) ] O(T) 1( t VC(XT) - +x] ) [ - cos-n(Xo+x)] Amplitude ratio 1 - | [)cOsn(xo+x)-cos'xV cosnnxosin (X]2 1-(x' l[cosnnxo-cosnn(o+x)] )(x) = sinX~(oX') +x) | [sin~n(xo+x)-cos ( sinnX)o] - sin ] cosnnxo21 1/2 0(T) - sinLXOT f(- [sinafn(xo+x)-cos [' si[nxon - sin duX coslnXo Phase shift. an-l P cosnn(xo+x)-cos c.- cosnnxo +X)] sinnnxosin' - o, for special case (l) o - (Ml)0. for special case (2) tj(XT)- - (^ {'e in)0n(noon) t(o-,) V V (ecT sin)l n(xo0x) - tao-1.1 n Aplp bi(t.ud'eC rtio [1-(n ru)22 ] *ip[ Ion -O(Xo+x )M+1) ]2 *(x). sinn(xo+x) - tan- lx ] - sint[ (x +x-uT)-tan- - sin)(o. - ) - t l (T) - *e'T t11(nT) - s p e n tII(XT) s lc se i Tas (xo+x) -x =-CpV wa-+( (nu)2 - =)~in -CpV a[w(M+l) ] + (wnu) n - _tan-l L_(T - )esin(nnx-t an' hl - - tean1l ]% -c[ - (M.l) 0)-n Ot] n t' 1 - e 0inot (o+X)

-19 TABLE III Table for Derivation of the Duhamel's Superposition Integral Magnitude of space- Commencing point wise uniform heat of spacewise uni- Effect on the response RX(X,T) generation element form heat genera- at x tion element *(0) 0 *(O)RX(X,7) *(Xi) - (0O) X1 [*(xi) - v(0)] Rx(x-dxT) *(x2) - t(xl) X2 [ -(x2) - (xl)] Rx(x-2dX,T) ~(x3) - 4(x2) x3 [t(x3) - 2(x2)] Rx(x-3dx,T) 1(xn) - f(Xnl) Xn [H(Xn) - f(Xn-l)] Rx(x-ndx,T)

APPENDIX 1 Inverse Laplace Transformation of Equation (17) B2(s) + Q2 which appears in the numerators of Equation (17) may be written as B2s) + 2 KK (1 s )2 (s+Cls3+C2 +C3C C4) ~(s) + =K _ 2(l +s w w (43) in which C1 = 2(K+K2) C2 = (K+K2)2+(Qnu)2 C3 = 2Kw(Qu)2 c4 = (Kwn)2 4 5C4 = (Kwn2 The fourth order factor s4+Cls3+C2s2+C3s+C4 in Equation (43) may be resolved into factors as s4+Cls3+C2s2+Cs+C4 = (s2+aK+b K)(s2+CKws+d ) (45) where a = (M+1) + 21 [(M+1)2M (nu)2 E {(} = ((M+l) T: [(Mj l) _m IK + {b=} D + E + 2DE-(M+)M2( )2 +2D d (46) (47) (48) (49) and D = 1[(M+1)2 + M2( nu)2] 4 K E =4 - )2 -20

-21 The substitution of Equations (7), (43) and (45) into Equation (17) yields -. KK^(s) r2(M+l)-a;(xs) =- (Cs) {[F (s) + 2,/ F2(s)][F3(s) C(Pp() V4bi-an2-() - c F4(s)] F6(s)sinQn(xo+x) J4d-C2 - fnu[ 2 K, 4b-a2 F2(s)][- 2 F4(s)] cosfn(xo+x) K 14d-c2 Kx sx + 2nue e n [Fl(s) + 2-a J/ 4b-a2 F2(s)][ 2 Kw v 4d-C2 F4(s) ]F5(s)coS2nxo Kx sx - e u e u [F1(s) + 2(M+l) -a s/4b -a2 F2(s)][F3(s) C ~ i- 2 F4(s)]Fs(s)sin2nXo + ( sinn(xo+x)} KKw n (50) t(x,s) = ( (pcp)w C \4i-C2 {[Fl(s) + 2(M+l)-a F2(s)][F(s) F4(s)] 4ba s F4(s)] sinQn(xo+x) - Qnu[Fi(s) + 2-a j4b-a2 F2(s)][ 2 K 4da-C2' F4(s)] cos^n(xo+x) +Kx + 2nue u e sx u [Fl(s) + 2-aF2(s)][F(s) J/ 4b-a2 Kx 2-C -- e + F(s) F4(s)] F(s)cosnXo - e u e id-C2 sx " [F1(s) + 2M )-a F2(s) ][l-(C-l)KWF3(s) - (2d+C )Kw () 14b-a2 /4d-C2 F4(s)] w/ 4b- -a x F5(s) sinnX } (51)

-22In Equations (50) and (51) the functions Fl(s).o.F6(s) are Laplace transformed functions in the variables which have corresponding original functions GI(T).ooG6(T) in the variable T obtained by performing an inverse transformation on the function F. The Laplace transformed function F and inverse transformed function G appropriate to Equations (50) and (51) are listed in Table I. The transformed equations in the physical domain of x and T are found to be Equations (26) to (29).

APPENDIX 2 Convolution Integral in Equations (26) to (29) Inspection reveals that Equations (26) to (29) convolution integrals to be discussed in the following, of the convolution integrals as defined by Equation (30) forward. Let aKw CKw = or 2 r 2 a = baKw or d Kw 2 2 consist of the The evaluation is quite straight(52) (53) ad define T il(T) = (e-TsinpT)*O(T) = f O(TZ-)eC sind T (2(T) = (eT cosF)*(T-) = f (-)e-cosd 0 t5(T) = 1 * (T) = j 0(T —)dO o -Kw'r T Kw~ A4(T) = e * O(T) = Jf (T-g)e da 0 Ai(T) = 1ti(T)*G5(T) = l(T)*e-KwTIo(2 J KWT) T -KKx = Jf il(T-f)e Io(2 J Kw-)dg 0T T __ -- & (54) (55) (56) (57) (58) (59) (60),2(T) = t2(T)*G-() = fi 2(T-T)e Kw Io(2 JK Kw)dT = 3* = ( -Kw) A3(T) = 1t3(T)*G5(T) = f J3(Tr-e)e Io(2 - Kw)d5 -23

-24 Since the evaluation of convolution integrals indicates that Gl(T)*G3(T), Gl(T)*G4(r), G2(T)*G3(T) and G2(T)*G4(T) are function of e -TsinBT and e-T cosST, it is found G1 (T)*G53(T) * (T) G1(T)*G4(T)*t(T) G2(T)*G53(T) * (T) G2(T)*G4(T)*(T) j Therefore Gl(f)*G3(T)<*(T)*G5(T Gl(T)*G4(T) * (T)*G5(T G2(T)*G3(T)* (T)*G5(T) G2(T)*G4(T)*(T)*G5(T, I = f[(e TsinpT)* (T),(ee cosPT)* (T)] = f[Tl(T),t2(T)] )' = f[l(T)*G5(T), T 2(T)*G5(T)] = f[Al(T), A2()] (61) (62) ) Similarly one finds that 1 * G1(T) * 0(T) and 1 * G2(T) * 0(T) are function of ti(T), t2(r) and t3(r). Hence l*Gl(T)*((T) )G5() 1} =))*() 1*G2(T)O(T)*G5(T) ()2 = f[AL(T);),((T),94(T)] (63) The convolution integrals of G1(T)*G3(T), G1(T)*G4(T), G2(T)*G3(T) and G2(T)*G4(T) with G6(T) yield G1(T)G3 (T) *G6(T) Gl(T)*G4(T)*G6(T) G2(T)*G3(T)*G6(T) G2(T)*G4(T)*G 6(T) = f[(e-T sinDpT)*G6(T), (e-aTcosPT)*G6(T) = f(e-T inpT e -CT OS, e-KwT ) (64)

-25 Consequently G1(T)*G5 (T)*G6(T)O(T) G1(T)*G4(T)*G6(T)*S(T) G2(7T) *G3 (T)*G6 (T7)* ( T) G2(T)*G4(T)*G6(T) *(T) = f[(e'<TsinpT)*(T), (e-aTcosPT) *(T), e-KwIT(T)] = f[ti(T)Yt2(T),t4(T)] (65) The definition of T4(T) is nothing but G6(T) *(T). Special cases for step change, sinusoidal change and exponential change in 0(T) are respectively given in References 6, 7, 8 and 9.

APPENDIX 3 Derivation of the Duhamel's Superposition Integral The response to an arbitrary space- and time-dependent heat generation *(x)0(r) may be found from the results for an arbitrary time-dependent but space-independent heat generation by the principle of superposition. In terms of differential calculus this principle may be derived as follows. r(x) may be resolved into a number of spacewise uniform heat generation elements, each commencing at a different value of x. These elements are shown in Table IIIo The appropriate value of RxT(x,T) is then the sum of the contributions from the separate cases, each shown in the third column of Table III, 00 RxT(XT) = t(O)RX(XT) + Z RX(XXn T ) [(Xn) -(Xn-l)] (Xn-Xnl) (66) n=l Xn-n-1 In the limit, as the number n of elements becomes infinite, the definition of integral results in RxT(X,T) = (O)RX(x,T) + j Rx(X-ST) - da (67) o d By using the method of integration by parts an alternative form may be obtained as X=x x )Rx(X-),T) RxT(X,T) = *(O)Rx(x,T) + [Rx(x-')T)t( )] - f () X - (68) 0=0. a( -26

-27 Since aRx(X-S,T) _ aRX(x-,T7) (69) ao ax one obtains, x aRx(x~ T ) RX.(x,T) = *(X)RX(O,T) + XJ Rx() x T d (70) This equation can be re-arranged as follows by using its convolutive property: x 6Rx(S,T) RXT(XT) = V(X)Rx(O T) + Of (x- ) a- d (71) Similarly if R (xr), temperature response of the coolant and solid to a unit step change in heat generation having arbitrary x-dependence, is available, the response of coolant or solid temperatures to an arbitrary time-dependent disturbance 0(T) may be obtained by the Duhamel's Integral which results in T dj,(,,) a~ RXT(X,T) = (0)RT(x,9T) + f RT(X,T-s) dE (72) or, in alternative forms RXT(XT) = O(T)RT(9,0) + M(t) aT T-T d (73) 0) + (T- (7 = O(T)R_(x,o) + / O(T —) x.... (74) O aS

REFERENCES 1o Arpaci, V.So, and Clark, JoAo, "Dynamic Response of Heat Exchangers Having Internal Heat Sources -- Part III," Trans. Am. Soc. Mech, Engrso, Sero C J. Heat Transfer, 81, (1959), 253-266. 2. Yang, WoJo, Clark, JoAo, and Arpaci, V.S., "Dynamic Response of Heat Exchangers Having Internal Heat Sources -- Part IV," Trans. Am, Soc. Mecho Engrs., Sero C J. Heat Transfer, 83, (1961), 321-3538 30 Yang, WoJo, "Dynamic Response and Resonance Phenomenon of Single-Solid, Single-Fluid Heat Exchangers -- Part I," Trans. Japan Soco Mech. Engrso, 27, (1961), 1276-1277. 4. Yang, WoJ,, "Dynamic Response and Resonance Phenomenon of Single-Solid, Single-Fluid Heat Exchangers -- Part II," Trans. Japan Soco Mech.Engrs., 27, (1961), 1892-1907o 5o Yang, WoJo, "Dynamic Response and Resonance Phenomenon of Single-Solid, Single-Fluid Heat Exchangers -- Part III," TransoJapan Soc. MechoEngrso, 28, (1962)o 551-558. 60 Doggett, W.oO, and Arnold, EoLo, "Axial Temperature Distribution for a Nuclear Reactor with Sinusoidal Space and Exponential Time-Varying Power Generation," Transo Am, Soco Mech, Engrs., Sero C J, Heat Transfer, 83, (1961), 423-431, 7o Doggett, W.Oo, and Shultz, RoH.,Jro, "Transient Heat Transfer in a Convective Cooled Heterogeneous Nuclear Reactor with Axial Power Density," 1961 Interno Heat Transfer Conf,, Am.Soco Mecho Engrso, III,(1961), 622-6330 8, Yang, W.Jo, "Temperature Response of Nuclear Reactors Having Sinusoidally Space-Dependent Internal Heat Generation," Trans. Japan Soco Mech, Engrs., 28, (1962). 9, Yang, WoJo, "Frequency Response of Heat Exchangers Having Sinusoidally Space-Dependent Internal Heat Generation," Paper No. 62-HT-21 presented at the ASME/AIChE Heat Transfer Conference, August 1962. 10o Bonilla, C.S., Busch, JoS,, Landau, HoGo, and Lynn, LoL., "Formal Heat Transfer Solution," Nuclear Sci. and Engo, 9, (1961),323-331o 11, Yang, WoJo, "The Dynamic Response of Heat Exchangers with Sinusoidal Time Dependent Internal Heat Generation," Ph.D. Thesis, University of Michigan, 1960, 12o Hall, WoBo, and Price, PoH., "The Effect of a Longitudinally Varying Wall Heat Flux on the Heat Transfer Coefficient for Turbulent Flow in a Pipe," 1961 Interno Heat Transfer ConfO, Am. Soc. Mech. Engrs., III, (1961), 607-6135 -28