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Ind. Eng. Chem. Res. 2010, 49, 5006–5011
Periodic Thermal Behavior of Porous Media under Oscillating Flow Conditions Margaritis Kostoglou* Department of Chemical Technology, School of Chemistry, Aristotle UniVersity, UniV. Box 116, 541 24 Thessaloniki, Greece
In the present work an analytical study of the heat transfer in an oscillating flow through a porous medium is performed using a two equation model. The previous analysis performed [Byun et al. Int. J. Heat Mass Transfer 2006, 49, 5081-5085] is extended to include an arbitrary velocity magnitude and arbitrary oscillation pattern of the flow. Using asymptotic analysis with respect to several dimensionless parameters, the limits of validity of the analytical solution based on the previous approach [Byun et al. Int. J. Heat Mass Transfer 2006, 49, 5081-5085] are assessed. A new analytical solution is derived for the case of thermal equilibrium between solid and fluid. Several asymptotic results existing in the literature [Byun et al. Int. J. Heat Mass Transfer 2006, 49, 5081-5085; Klein and Eigenberger Int. J. Heat Mass Transfer 2001, 44, 3535-3563] are compared with the findings of the present work. This work completes the analytical study of the particular mathematical problem, previously advanced [Byun et al. Int. J. Heat Mass Transfer 2006, 49, 5081-5085; Klein and Eigenberger Int. J. Heat Mass Transfer 2001, 44, 3535-3563]. 1. Introduction Porous media are used extensively as devices for transporting and storing thermal energy. The porous media used for this purpose may be either unstructured (e.g., metallic wire mesh operating at temperatures smaller than 1200 °C) or highly structured (e.g., ceramic monoliths capable to operate at higher temperatures). There are two modes of operating the above-mentioned devices. In the first mode, the flow direction is unchanged and the heat transfer is achieved through the switching of the inlet stream temperature. In the second mode, not only the temperature but also the flow direction and the entrance point of the inlet stream are altered in time. In this second case, the flow direction switching is achieved while retaining fixed the porous matrix by either altering externally the flow (e.g., periodically operating monoliths) or rotating the porous matrix (e.g., rotary heat exchangers). The development of mathematical modeling theory for the above type of devices has followed two parallel ways: Numerical techniques proved capable to solve accurately the most complex problems including all the geometrical and operating details in more than one spatial dimension. On the other hand, analytical solutions of simplified models have allowed more straightforward physical interpretation of the results and they have been more convenient to apply.1 A review on analytical and numerical solutions of the mathematical problem of regenerative heat exchangers can be found in the literature.2 A table is given therein with references to existing solutions with the corresponding assumptions for their derivation. Extensive reviews on the subject can be found also3 where the first mode of operation of the heat storing device is studied analytically using Laplace transform and perturbation expansion with respect to solid conductivity. This method has been extended for a rotary heat exchanger.4 An analysis of the problem for a structured porous medium in two spatial dimensions can be found in another publication.5 This specific problem has been recently solved numerically.6 * To whom correspondence should be addressed. E-mail: kostoglu@ chem.auth.gr. Tel.: +30-2310997767. Fax: +30-2310997759.
Very recently, approximate solutions for the periodic part of the temperature variation for the second mode of operation of the particular type of heat exchangers were derived. These solutions are based on the linearity of the solid temperature profile which is well-known in the literature.7 The purpose of the present work is to extend the analysis made1 and to show that its validity is not general but it depends on the specific problem parameter values. In addition, all the possible analytical/ asymptotic solutions to the particular mathematical problem are derived and presented. It is emphasized that the particular mathematical model for heat transfer in generalized porous medium used here has been employed extensively in the literature concerning modeling catalytic converters used by the automotive industry8,9 (even considering flow reversal10). The structure of the present work is the following: At first, the one-dimensional problem is presented and dimensionless parameters are derived. Then, the approximate solution for the periodic component of the temperature profile is derived under conditions much more general than in previous work.1 Finally, several asymptotic solutions are developed showing clearly that the approximate technique1 is valid only in a restricted region of the parameter space. 2. Problem Formulation Let us assume a slab of porous medium (structured or unstructured) with thickness L and an infinitely large cross section. The gas flow through the porous medium oscillates back and forth according to a prescribed periodic function of time F(t). The gas at the one side of the porous medium has a low temperature T1 whereas the gas at the other side has a high temperature T2. The energy equation for each phase can be written as follows: fluid εFfcpf
∂Tf ∂Tf ∂2Tf + cpfG ) kf 2 + hR(Ts - Tf) ∂t ∂x ∂x
10.1021/ie9014638 2010 American Chemical Society Published on Web 04/19/2010
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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010
solid
solid ∂Ts ∂2Ts (1 - ε)Fscps ) ks 2 + hR(Tf - Ts) ∂t ∂x
(2)
The mass flow rate of the gas is given as G ) GoF(t) where the function F(t) is periodic with a zero average value, a period P, and a maximum value equal to 1 (which makes Go equal to the maximum mass flow rate). In addition, this function must be antisymmetric with respect to the points where it takes zero values. Here, the analysis refers to the interval [0, P] so F(t) is chosen to be an odd function for the clarity of presentation. Correspondingly, in previous work1 where the results are presented in the interval [-P/2, P/2], F(t) was chosen to be an even function. The two approaches are entirely equivalent differing only by a time shift. Regarding the absence of the fluid flow equations from the presented model, it is noted that given the macroscopic symmetries of the problem and applying the volume average technique the only equation needed is the continuity equation which degenerates to a uniform value of mass flow rate through the porous medium provided that the residence time in this (equal to LF/Go) is much smaller than the reversal period P. The momentum balance degenerates to a relation for the pressure drop in the porous medium which is not related to the heat transfer problem examined here so the fluid momentum conservation problem is irrelevant to the present work. The boundary conditions of the above system of partial differential equations are (no heat losses from the solid to the environment):
( ) ∂Ts ∂x
)
x)0
( ) ∂Ts ∂x
x)L
)0
(3)
cpfG(T1 - Tf) ) -kf
∂Tf ∂x
at x ) 0 if G > 0
(4)
cpfG(Tf - T2) ) -kf
∂Tf ∂x
at x ) L if G < 0
(5)
T - T1 T - T1 t x js ) s jf ) f , τ) , T , T , L P T2 - T1 T2 - T1 εFfL (1 - ε)FscpsL hRL , A) , B) N) Gocpf PGo PGocpf GocpfL GocpsL Pef ) , Pes ) kf ks
(6)
The dimensionless form of the mathematical problem is fluid A
2j jf jf ∂T ∂T 1 ∂T f js - T j f) + F(τ) ) + N(T ∂τ ∂xj Pef ∂xj2
B
2j js ∂T 1 ∂T s jf - T j s) ) + N(T ∂τ Pes ∂xj2
Boundary conditions: js ∂T
( ) ( ) ∂xj
jx)0
js ∂T ∂xj
)
jf ∂T jf ) 1 T Pef ∂xj
jx)1
)0
at jx ) 0 if F(τ) > 0
jf ∂T jf ) 1 1-T Pef ∂xj
at jx ) 0 if G < 0
(8)
(9)
(10)
(11)
The physical significance of the parameters Pef and Pes which are the effective Peclet numbers of the fluid and solid phase, respectively, is obvious. The parameter A stands for the thermal inertia of the fluid phase whereas the parameter B is for the thermal inertia of the solid phase. The parameter N is the socalled number of transfer units which describes the intensity of heat transfer between the gas and solid phase. The solution of the above system of equations for any initial condition exhibits, after an initial transient, a purely periodic behavior with period P. The initial transient can last from few periods to many thousands of periods depending on the problem parameters. The final periodic state is of interest in practice. Henceforth, the overbars on the dimensionless variables will be omitted for clarity since only dimensionless variables will appear. The axial thermal dispersion in the fluid can be neglected as it is always the case in the corresponding studies in literature so Pef ) ∞. The remaining problem is examined in ref 2 for A ) 0 and a square waveform for F(τ) i.e. F ) 1 for 0 < τ < 0.5 F ) -1 for 0.5 < τ < 1. The case of Pes ) ∞ with F(τ) ) sin(2πτ) and A ) 2 is examined also therein.1 Here we will try to reexamine the approach of ref 1 extending it for an arbitrary value of A and an arbitrary shape of F(τ) (but still obeying the requirements mentioned above). In addition, the validity limits of the approach in terms of the parameters B and N will be considered. 3. Approximate Solution for the Periodic Terms
The basic feature of the above mathematical problem is that not only the form of the boundary condition for Tf but also the location of its application changes periodically in time. The following nondimensionalization which is somewhat different from those used in other works,1,2 is introduced: jx )
5007
(7)
Any odd periodic function of time can be written as m
F(τ) )
∑ f sin(i2πτ) i
(12)
i)1
According to ref 1, the solid and fluid temperature can be decomposed into a time averaged term which is linear in x with a slope γ (at least far from the two edges) and a periodic time dependent term (with zero time average value) which does not depend on x (spatially uniform). This means that Tf ) C + γx + θf(τ)
(13a)
Ts ) C + γx + θs(τ)
(13b)
where C and γ are constants (absorbed in the nondimensionalization in ref 1). The use of the same time average for Tg and Ts is not an assumption but it results directly from eq 8 by taking integrals with respect to time over one period. The periodic functions θf and θs are of interest next. Without loosing generality, it can be assumed that
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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010 m
∑ a cos(i2πτ) + b sin(i2πτ)
θf(τ) )
i
(14a)
i
i)1 m
θs(τ) )
∑ c cos(i2πτ) + d sin(i2πτ) i
(14b)
i
i)1
Substituting eqs 14 into eqs 13 and the resulting equations in eqs 7 and 8, performing the differentiations, and then equating the coefficients of the similar trigonometric terms leads to the following series of 4 × 4 linear systems of equations for the coefficients ai, bi, ci, di (i ) 1,2, ..., m). -i2πAai + γfi ) N(di - bi)
(15a)
i2πAbi ) N(ci - ai)
(15b)
-i2πBci ) N(bi - di)
(15c)
i2πBdi ) N(ai - ci)
(15d)
The above system can be solved explicitly to give
Figure 1. Oscillating temperature profiles along the porous slab for the case of thermal equilibrium with Xo ) 0.4.
γfi Bci + ai ) A 2πiA
(16a)
-N (γfi - 2πi(A + B)ci) (2πi)2A2
(16b)
N (γfi - 2πi(A + B)ci) di ) (2πi)2AB
(16c)
bi )
Appendix A. At the end, the common temperature T ) Tg ) Ts is given from the solution of the equation: (A + B)
∂T ∂T + F(τ) )0 ∂τ ∂x
This equation can be solved using the method of characteristics in order to take the following periodic solution:
where ci )
γfi 2
(2πi) AB
-1
[N((2πiA)
-1
-1
-1
+ (2πiB) ) + N ((2πiA)
T)x+
+
(2πiB)-1)-1] (16d)
From the above expressions, it is evident that θs and θf are proportional to γ. For m ) 1 and f1 ) 1, the solution for the case studied in ref 1 (given there only in the complex number domain) can be found explicitly. It is noted that the main assumption implied by the expressions 13 is that there is no periodic contributions with position dependent amplitude in the temperature profile. This assumption is not compatible to a time dependency of the slope γ as it will be shown here: T ) C + γ(τ)x + θ(τ) ) C + γavex + (γ(τ) - γave)x + θ(τ) ) C + γavex + Θ(x,τ) (17)
Apparently, relations 13 are not valid in case of a time dependent slope, since their last periodic term should have a spatial dependence also. 4. Asymptotic Solutions Before proceeding, it is noted that the parameters N, B are equivalent to the parameters S, K of ref 1 and to NTU, σΓ of ref 2. The reference value of time used for nondimensionalization in ref 1 is τ ) 1/2π. At first, the case of thermal equilibrium between solid and gas will be analyzed as the simplest one since then the temperatures of gas and solid coincide. According to ref 1, the condition of thermal equilibrium is N/B . 1. In this limit, the problem can be seriously simplified as it is shown in
(18)
Xo - X(τ) 2
(19)
where X(τ) ) Xo )
1 A+B
1 A+B
∫ F(t) dt τ
0
∫
1/2
0
F(t) dt
(20a) (20b)
and for x < X(τ): T ) 0, for x > 1 - Xo + X(τ): T ) 1. The structure of the solution is the following. The spatial profile consists of three parts: two constant temperature parts at the edges corresponding to the two boundary conditions and a linear part with slope equal to 1 (see Figure 1 for Xo ) 0.4). It is noted that the shape of the temperature profile is independent from the function F(τ) which determines only the left and right motion of the profile. In the region Xo < x < 1 Xo, the temperature profile is always linear with a constant slope so the approximating method of the previous section is valid and the corresponding solution (16) is the same with the exact one. This is not true for x outside this interval. Interestingly enough, as Xo increases the range of validity of the approximating technique decreases and degenerates to the single point x ) 1/2 at Xo ) 1/2. For smaller values of Xo, the slope of the T profile is no more constant in time for any value of x. Instead there are periods of slope equal to 0 and to 1. Clearly, the solution (16) is not valid. A further transition occurs at Xo ) 1. The solution for Xo > 1 degenerates to T ) 0 for 0 < τ < 1/2 and T ) 1 for 1/2 < τ < 1. The time averaged part of the solution is simply T ) 1/2.
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After the study of the asymptotic case N/B . 1, we will proceed in examining another asymptotic case; the one with B . 1, A ) 0 considered previously2 for the case of a square wave F(τ). Extending the asymptotic technique for the limit B f ∞ developed2 for a general form of F(τ) results in the following governing equations for the gas and solid temperature: Ts(x) ) F(τ)
∫
1
0
Tf(x,y) dy
∂Tf(x,τ) ) N(Ts(x) - Tf(x,τ)) ∂x
(23)
(24)
The arguments of the functions are shown in order to display the structure of the problem. According to these equations, the solid temperature is time independent and it is in fact equal to the time average of the local gas temperature. The gas temperature evolves along the slab exchanging progressively heat with the solid by a rate dictated by the transient function F(τ). The only case for which a closed solution is possible is the square wave F(τ) for which the absolute magnitude of F is always the same (only the sign changes). The solution for this case is Figure 2. Periodic component of the temperature θs(τ) at the point x ) 1/2 versus transformed time for the case of thermal equilibrium.
Let us try now to solve eq 18 (valid for N/B . 1) using the approximation technique based on eqs 13. A straightforward substitution and solution leads to γ θs(τ) ) A+B
∫ F(y) dy
(21)
This equation is a generalization of the two asymptotic cases III and IV presented in ref 1. Therein A ) 2 and the two cases B . 1 and B , 1 are examined separately (i.e., the one or the other term of the denominator A + B being prevalent). Such a separation is not needed but instead the requirement that B > 1/2 must be added in order for eq 21 to be valid at least at one position x. For the sinusoidal F(τ), this yields θs ) θf )
γ cos(2πτ) (A + B)2π
(22)
The same result is obtained by simplifying appropriately the general solution (16). The above expression is also compatible with the one given in ref 2 derived for A ) 0, large B, and F(τ) of a square wave shape but without restriction on the ratio N/B. Equation 21 shows how to extend the solution given there2 for nonzero A and arbitrary F(τ) but under the additional condition of N/B . 1 which (since Xo > 1/2 w B > 1/2π for the solution to be valid) means that N . 1. In Figure 2, the function θs(τ) at x ) 1/2 is shown during the first half of the period versus the transformed time Χ(τ)/Xo (the second half can be found by invoking symmetry with respect to vertical axes). It is apparent that for Xo ) 1/2 the solution (21) is valid, but for Xo > 1/2, it cannot follow the exact behavior of θ(τ) shown in Figure 2.
N 1 + x N+2 N+2
(25)
N 1 1 + x+ F(τ) N+2 N+2 N+2
(26)
Ts ) Tf )
The slope γ is constant and equal to N/(N + 2) for both the gas and solid temperatures. The only time dependence is a periodic shift for the gas temperature due to the change of the flow direction. It must be noted that the analytical solution of this particular case in the presence of a finite solid Peclet number was derived in ref 2 but the simplification to take the above result for infinite Pes is not trivial. The effect of conduction comes not from the equation itself, since the linear temperature profile fulfills the conduction term, but from the boundary conditions. The finite heat fluxes at the edge of the slab predicted by the linear profile must be replaced with the zero flux boundary conditions. This is a typical singular boundary value problem with respect to the parameter 1/Pes. There are two boundary layers at the edges of the slab with width proportional to 1/(Pes)1/2. There the outer solution (eqs 25 and 26) matches the inner solution which fulfills the boundary conditions of zero flux. The important point not explicitly mentioned in ref 2 is that the first-order expansion with respect to 1/B has no dependency on x which means that it fulfills identically the conduction term and the no-flux boundary condition so this term is valid for all values of Pes. For the case of arbitrary F(τ), an exact solution of eqs 23 and 24 is not possible but asymptotic results in the limits of small and large N can be derived (see Appendix B). In the limit of large N, the approximating technique is valid far from the boundaries. The region of validity decreases as N decreases, and as it is shown in the appendix that the approximate solution (16) is not valid in the limit of small N. This means that the asymptotic solution presented in ref 1 (N , 1, B . 1) does not hold in any case.
Table 1. Summary of the Recent Analytical Solutions for the Temperature Profiles in Porous Media under Oscillating Flow Conditions ref
A
B
1 2 present present
2 0
.1
N
F(τ)
Pes
solution type
0
.1 or .B
simple harmonic square wave arbitrary arbitrary
periodic component asymptotic exact periodic component
0 0
comments no reference to the limitations of the approach limitations with respect to the magnitude of B and N are indicated
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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010
It is noted that in some cases asymptotic solutions can be served as something more than benchmarks for the numerical solutions. For example, in the present problem, for large values of the parameters, discontinuities may appear as it is shown in Figure 1. General purpose numerical methods will have problems to represent the discontinuities due to wellknown numerical dispersion and diffusion errors. Examining the character of the expected solution by asymptotic means helps to the choice of appropriate numerical methods modified to account for the expected discontinuities.
Substituting the expansions for the temperature in (A.1) leads to: A
∂Tfo ∂Tf1 ∂Tf1 ∂Tfo + F(τ) + εA + εF(τ) ) Nε(Ts1 - Tf1) ∂τ ∂x ∂τ ∂x (A.4)
The key point in the present procedure is that Nε ) B which may be a finite term even if N/B goes to infinity, so it must be considered as a zeroth-order term with respect to ε. The corresponding zerothorder expansion of (A.4) takes the form:
5. Conclusions In the present work, the mathematical problem of heat transfer in a porous medium under oscillating flow conditions is analyzed using analytical and asymptotic tools. Useful comparisons are made with results obtained previously by other authors.1,2 In particular (i) The approximate technique introduced previously1 for the isolation of periodic components of the temperature field is extended for the case of the dimensionless parameter A * 2 and for an arbitrary flow forcing function F(τ). (ii) An additional region of the parameter space leading to gas-solid thermal equilibrium is shown to exist. The exact solution of the problem in the case of thermal equilibrium is derived. (iii) By invoking several asymptotic results derived here, it is shown that the technique referred to in conclusion (i) is valid only in a restricted spatial domain which decreases as B and N decrease, until it vanishes completely. (iv) As regards the several asymptotic cases derived in ref 1, it is shown that cases III and IV can be reduced to a single one for thermal equilibrium for arbitrary A and arbitrary forcing function. On the other hand, the asymptotic case I lies outside the domain of validity of the solution technique and therefore it does not refer to an actual solution of the problem. The conditions for the existing closed form solutions (present and older ones) to the mathematical problem of thermal behavior of porous media under oscillating flow are displayed in Table 1. It is believed that the new results given in the present work improves significantly the comprehension of the structure of the solutions to the particular mathematical problem. Appendix A: On the Thermal Equilibrium Conditions Let us assume the two phase mathematical problem without the conduction terms: A
∂Tf ∂Tf + F(τ) ) N(Ts - Tf) ∂τ ∂x
(A.1)
∂Ts ) N(Tf - Ts) ∂τ
(A.2)
B
Case I (1/ε ) N/B . 1). Let us expand the unknown variables as Ts ) Tso + Ts1ε, Tf ) Tfo + Tf1ε and substitute them in the above system of equations. The result for eq A.2 is ε
∂Tso ) (Tfo - Tso) + ε(Tf1 - Ts1) + higher order terms ∂τ (A.3)
So, at zeroth-order with respect to ε: Tfo ) Tso. And at first order, Tf1 - Ts1 ) ∂Tso/∂τ.
A
∂Tfo ∂Tfo + F(τ) ) B(Ts1 - Tf1) ∂τ ∂x
(A.5)
Now replacing the relations resulting from the expansion of (A.2) in the above equation gives the final result (A + B)
∂Tso ∂Tso + F(τ) )0 ∂τ ∂x
(A.6)
Case II (1/ε ) N . 1). Again a first-order expansion of the unknown temperature with respect to ε is attempted. Substitution in eq A.1 leads to εA
∂Tfo ∂Tfo + εF(τ) ) (Tso - Tfo) + ε(Ts1 - Tf1) + ∂τ ∂x higher order terms (A.7)
The zeroth- and first-order expansions with respect to ε of the above equation are (under the restriction that A takes a finite value): Tso ) Tfo Ts1 - Tf1 ) A
∂Tfo ∂Tfo + F(τ) ∂τ ∂x
(A.8) (A.9)
Now substituting the expansions for the two temperatures in (A.2), the following zeroth order expansion is obtained: B
∂Tso ) (Tf1 - Ts1) ∂τ
(A.10)
Using eqs A.9 and A.10, the following equation is derived for the temperature in the present case of thermal equilibrium: (A + B)
∂Tso ∂Tso + F(τ) )0 ∂τ ∂x
(A.11)
As it was shown in both cases of thermal equilibrium, the governing equation for the temperature evolution is identical. The physical meaning of the two cases is different. In the first case, the solid thermal capacity is very small so the solid temperature is obliged to follow the local fluid temperature. In the second case, the fluid thermal capacity is very small so fluid temperature is obliged to follow the local solid temperature. The distinction between the second region of thermal equilibrium and the first one becomes evident from the fact that this second thermal equilibrium exists even for N/B finite (but larger than 1 to preclude that 1/B f 0 also) as 1/N f 0 (whereas N/B . 1 for the first case). As an example to understand the new region of thermal equilibrium, the pair N ) 1000, B ) 300 leads to a thermal equilibrium, not the classical one (since N/B ) 3.33 is not much larger than 1) but the new one (1/N f 0).
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Appedix B: Asymptotic Expansions for the Case 1/B ) 0
Nomenclature
The governing equations in this case is the system of eqs 23 and 24 which consists of partial differential equations despite the fact that only one partial derivative appears. Small N. Let us say that ε ) N , 1. Assuming the expansions Tf ) Tfo + εTf1 and Ts ) Tso + εTs1 and substituting in (23) and (24) results in
cp ) isobaric heat capacity (J kg-1 K-1) F(t) ) oscillation pattern of the flow (dimensionless) fi ) Fourier coefficients defined in eq 12 G ) mass flow rate (kg s-1 m-2) Go ) maximum mass flow rate (kg s-1 m-2) h ) interstitial heat transfer coefficient (W m-2 K-1) k ) thermal conductivity (W m-1 K-1) L ) length of porous media in flow direction N, A, B ) dimensionless parameters defined in eq 6 P ) flow oscillation period (s) Pe ) Peclet number T ) temperature (dimensionless except in the Problem Formulation section). t ) time (s) T1 ) cold side temperature T2 ) hot side temperature Tfo, Tf1 ) zeroth- and first-order terms of asymptotic expansions for the fluid temperature Tso, Ts1 ) zeroth- and first-order terms of asymptotic expansions for the solid temperature x ) longitudinal coordinate (dimensionless except in the Problem Formulation section). X(τ) ) transformed time defined in eq 20a Xo ) transformed half-period time τ ) dimensionless time Greek Symbols R ) interfacial area per unit volume of porous media (m-1) γ ) slope of the spatial temperature profile θ(τ) ) dimensionless temperature defined in eqs 13 F ) density (kg/m3) Subscripts and Others s ) solid f ) fluid overbar ) dimensionless variable
∂Tfo )0 ∂x Tsi )
∫
1
0
F(τ)
(B.1)
for i ) 0, 1
Tfi dτ
∂Tf1 ) (Tso - Tfo) ∂x
(B.2) (B.3)
The zeroth-order solution is coming directly from the application of the boundary conditions i.e.: Tfo ) 0
0 < τ < 1/2
(B.4)
Tfo ) 1
1/2 < τ < 1
(B.5)
Tso ) 0.5
(B.6)
The first-order terms can be found as Tf1 ) Tf1 )
1 x 2 F(τ)
0 < τ < 1/2
1 (1 - x) 2 F(τ)
Ts1 ) (x - 1/2)
∫
(B.7)
1/2 < τ < 1
(B.8)
1 dτ F(τ)
(B.9)
1/2
0
For the case of the square wave F(τ), it can be shown that the above solution is equivalent to the expansion of the exact solution (eqs 25 and 26) with respect to N. The linearity of T with respect to F(τ) shown in (26) is a simple coincidence since for the particular form of F it holds that F ) 1/F (when F is nonzero). In every other case of F(τ), the slope of the gas temperature profile is timedependent (see eqs B.7 and B.8) invalidating the approximate solution. It is noted that for F(τ) taking very small values for a large τ period (e.g., sinusoidal) the quantity ε/F(τ) is not small and the above expansion is not valid. Large N. Let us say that ε ) 1/N , 1. Assuming the expansions Tf ) Tfo + εTf1 and Ts ) Tso + εTs1 and substituting in eqs 23 and 24 results eventually to Tf ) x +
1 (Z(x) + F(τ)) N
(B.10)
Where Z(x) is an undetermined spatial profile. For the case of a square wave F(τ) direct application of the boundary conditions leads to the first order expansion of the exact solution (eqs 25 and 26) with respect to 1/N. But for arbitrary F(τ), matching of the relation to the boundary conditions is not possible. In this case, the expansion (B.10) is an outer one and there are two boundary layers with widths depending on N where inner expansions are required. Nevertheless far from these layers Z(x) is linear so the approximating solution is valid. This is confirmed by the fact that the above solution is compatible with the asymptotic case II in ref 1 (corresponding to B . 1, N . 1).
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ReceiVed for reView September 17, 2009 ReVised manuscript receiVed March 18, 2010 Accepted April 9, 2010 IE9014638
