Thermal calculation of heat exchangers with simplified consideration of axial wall heat conduction

. The usual thermal design and rating methods [1, 2] for heat exchangers neglect axial wall heat conduction in the separating walls and external shells of recuperators and in the solid matrix of regenerators. This may lead to undesirable undersizing. In this paper a simplified model is developed for the fast estimation of axial wall conduction effects in counterflow, parallel flow and mixed-mixed cross-flow recuperators. The dispersion model [3] is used to describe the performance deterioration of the exchanger with an effective fluid dispersion Peclet number for the correction of the heat transfer coefficients or mean temperature difference. The method is tested against analytical and numerical calculations for counterflow and parallel flow with good results. It is also shown how the method can be adapted to thermal regenerators and the related thermal calculation methods [1, 2]. An alternative approach is suggested for the consideration of lateral heat conduction resistance in the solid matrix.


Introduction
The usual simple heat exchanger design and rating methods [1,2] neglect the negative effect of axial heat conduction in the separating wall and the outer shells as well as in the solid matrix of regenerators. This may lead to undesirable underdesign. Kroeger [4] was the first to present an analytical closed form solution for balanced and unbalanced counterflow with axial conduction in the separating wall with adiabatic ends. Recently Aminuddin and Zubair [5] published several analytical solutions and calculated results on the performance of counterflow heat exchangers with consideration of axial heat conduction in the separating wall and heat losses from the shell to the vicinity. They assumed various realistic boundary conditions at the wall ends which are useful for the application of the solutions to the cell method or to series connections of exchangers.
In this paper a simple approximation is developed for the fast estimation of the performance deterioration due to axial heat conduction not only in the separating wall but also in the outer shells. Heat losses to the vicinity are neglected. Since axial wall heat conduction has a similar effect as axial fluid dispersion, the dispersion model [3] is used in the approximation method. First counterflow and parallel flow recuperators are considered.

System of governing differential equations
The steady state heat transfer process in a counterflow or parallel flow heat exchanger with adiabatic outside surface and axial wall heat conduction can be described with the following system of ordinary differential equations: , 0 For counterflow and parallel flow L 1 = L 2 = L and λ w1 A qw1 = λ w2 A qw2 = λ w A qw . In eq (2) the positive sign is for counterflow and the negative sign for parallel flow.
The boundary conditions and further conditions in brackets are: The system of five ordinary differential equations of first and second order together with the required eight boundary conditions are implemented into the numeric computing environment of the mathematical software Maple for counterflow and parallel flow, respectively. The resulting real-valued two-point (x = 0 and x = 1) boundary value problems are solved by a finite difference method based on the trapezoidal scheme with Richardson extrapolation enhancing the accuracy [6]. The numerical solutions and analytical solutions of Kroeger [4] for special cases are used for the test of the approximation method developed in this paper.

Infinite thermal conductivities of the walls
First the most disadvantageous case of infinite thermal conductivities of the walls is considered. The wall Peclet numbers become zero and the system of differential equations, eqs (1) -(4), reduces to eqs (1) and (2). The wall temperatures t w , t wa1 and t wa2 take constant values. The following equation has been derived for the calculation of P 1 = P 1,0 with index 0 for zero wall Peclet numbers.

Weak wall heat conduction effects
The limiting case of very low thermal wall conductivities (λ w →0, λ wa1 →0, λ wa2 →0) is considered in which fluid and wall temperatures do not change remarkably. The wall temperatures of the outside walls assume the temperature of the adjacent fluid i = 1,2, T wa,i = T i , as the outside surface is adiabatic. So the local axial conductive heat flows in the outer walls a,i = a,1 and a,2 are With the local temperature of the separating wall The substitutive effective axial dispersive heat flows in fluids i = 1,2 are The basic idea of this model is to shift the conductive axial heat flows from the walls into the adjacent fluids. This yields effective Peclet numbers for the dispersion model. Since the separating wall is in contact with both fluids, its heat flow has to be distributed according to the (unknown) fractions ϕ 1 for fluid 1 and ϕ 2 = 1 -ϕ 1 for fluid 2. The substitutive dispersive heat flows are accordingly expressed as The eqs (16) and (17)   The positive sign is for counterflow and the negative sign for parallel flow. The negative sign occurs as for parallel flow dT 2 /dT 1 ≤ 0. The Peclet numbers Pe 1 and Pe 2 could be used if the fractions ϕ 1 and ϕ 2 were known.
They are determined in the following.
The overall conductive effect can be expressed with the dimensionless dispersive mean temperature difference [3] 2 , 1 This equation defines also Pe = Pe 1,2 which is equal for both the fluids. The deteriorative conduction effect should yield positive values Pe = Pe 1,2 > 0 for all cases of counterflow and parallel flow.
Multiplying eqs (18) and (19) by P 1 and P 2 , respectively, summation of both equations and dividing by P 1 yields with P 2 /P 1 The eq (23) remains valid if the indices 1 and 2 are exchanged. The positive sign is for counterflow and the negative sign is for parallel flow. The dispersive Peclet number Pe has to be applied to both fluids.
Substituting eq (22) into eqs (18) and (19) yields formulas for Pe 1 and Pe 2 which have to be applied to fluid 1 and fluid 2, respectively.
With counterflow Pe 1 > 0 and Pe 2 > 0, which means that, owing to wall heat conduction, for fixed inlet and outlet temperatures the mean temperature difference θ m = T 1m -T 2m decreases, the mean temperature T 1m decreases and T 2m increases.
With parallel flow one of the Peclet numbers may be negative. If e.g. Pe wa,1 = Pe wa,2 = ∞ (no outer wall effect) and N 1 > N 2 , then Pe 1 > 0 and Pe 2 < 0. This means that in this case θ m = T 1m -T 2m decreases, T 1m decreases and T 2m decreases as well, but less than T 1m . This is in qualitative accordance with the real process.
The application of Pe 1 and Pe 2 from eqs (18), (19) and (22) gives the same results as Pe from eq (23). The two Peclet numbers Pe 1 and Pe 2 yield more insight in the process, applying only one Peclet number Pe according to eq (23) is more appropriate for the approximation method developed in the following.

Approximation equation for Pe
The relative error of Pe from eq (23) In eq (25) Pe 1,0 is determined from eq (11) and Θ lg is the dimensionless logarithmic mean temperature difference for counterflow or parallel flow, respectively, calculated with P 1,0 and R 1 .
The eqs (20) and (25) are also exactly valid for the pure cross-flow and are approximations for other onepass flow arrangements, if Θ lg is replaced by the correct mean temperature difference for the flow arrangement under consideration.
With the limiting Peclet number Pe 0 from eq (25) and the Peclet number from eq (23) which is now denoted with Pe ∞ the approximation equation is formed which yields exact values of Pe for Pe ∞ = ∞ and for Pe ∞ = 0. The comparison of eq (26) with analytical and numerical results from eqs (1) -(10) provides a value of m = 0.87. In the range 5 ≤ Pe ≤ 23 for counterflow and 9 ≤ Pe ≤ 140 for parallel flow the mean relative error of Pe is about 5 % and the mean relative error of P 1 or P 2 , calculated with Pe, is about 0.5 %.
The temperature change P 1 can be calculated by correcting the overall heat transfer coefficient (kA) according to and applying the corrected value (kA) λ to the known formulas for counterflow and parallel flow. Other ways are possible as shown in [3].

Application of the model to mixedmixed cross-flow
The eqs (11) and (24) -(26) can directly be applied to mixed-mixed cross-flow. Only the eq (23) has to be replaced by a new equation and in eq (25) Θ lg has to be replaced by the mean temperature difference for mixedmixed cross-flow. Now in the separating wall axial heat conduction takes place in the two flow directions and both the flow lengths L i may be different. So in the wall Peclet numbers usually L 1 ≠ L 2 and λ w1 A qw1 ≠ λ w2 A qw2 (see eq (7)).
Applying again the principle of shifting the conductive heat flows into the fluids leads to the following equation of Pe = Pe ∞ : Using eq (28) and (29) the eq (23) for counterflow (+) and parallel flow (-) can be rearranged to a similar form: In the limiting cases of one constant fluid temperature, ψ 1 = ∞ or ψ 2 = ∞, the eqs (30) and (31) become identical. Since both equations have been derived according to the same model they should reach approximately the same accuracy, also in the general case ψ 1 < ∞ and ψ 2 < ∞.
Using the Peclet numbers Pe (30) from eq (30) and Pe (31,±) from eq (31) enables the general presentation: Other one-pass symmetrical flow arrangements may be described with individual values of 0 ≤ a ≤ 1.

Application of the model to regenerators under steady periodic operation
The method derived for counterflow and parallel flow recuperators can also be adapted and applied to fixedbed and rotary regenerators. The outer walls are considered as part of the solid matrix and need not be considered separately.

Infinite axial thermal conductivity of the solid matrix
This limiting case corresponds to the steady state heat transfer process in two mixed-unmixed cross-flow heat exchangers coupled by a circulating transversely mixed flow stream [7,8]. The temperature change P 1 The capacity C s,i in eq (34) is the capacity of one matrix "i" and C s,t in eq (35) is the total capacity of the rotating matrix. The time τ i is the duration of period "i" and the sum (τ 1 + τ 2 ) in eq (35)  ( They take the lateral heat conduction resistance inside the solid matrix into account. The estimation of the additional resistance 1/a s = φ δ/λ s is subject to Hausen's theory of regenerators [9, 1, 2] and will be discussed later in this paper. Once the temperature change P 1 is determined from eq (33), the Peclet number for λ s = ∞ can be calculated.
For regenerators the previous eq (25) has to be replaced by For fixed-bed regenerators: The correction factor k/k 0 , which is in reality a correction of the mean temperature difference, takes the deviations of the time averaged longitudinal temperature profiles in the regenerator from those in the equivalent counterflow recuperator into account. This correction has been and still is discussed and calculated by many researchers [9,10]. It is calculated for a solid matrix with infinite lateral and zero axial thermal conductivity. Results are given in the relevant literature [9,10].

Low axial thermal conductivity of the solid matrix
The previous eqs (23) or (31), (28) and (29) can directly be applied using the following definitions of the wall Peclet numbers.
Fixed-bed regenerator: ( ) The cross-section for heat conduction A qs,i in eq (40) belongs to the one solid matrix "i" under consideration. The cross-section A qs,t in eq (41) is the total axial crosssection of the rotating solid matrix.
The products a i A i in eqs (23), (28) and (29) have to be substituted according to Fixed-bed regenerator: Rotary regenerator:
For the rotary regenerator and eq (27) yields the corrected value (kA) λ . In eq (44) A 1 and A 2 are the actual heat transfer surfaces in the hot and cold section. For the fixed-bed regenerator eq (27) has to be adapted to the definitions in Hausen's theory [9,1,2] leading to The surface A in eq (45) is the heat transfer surface of one solid matrix. If inlet and mean outlet temperatures are given, eq (20) can be used for the correction of the mean temperature difference instead of eq (27) [3].

Equivalent wall thickness
The function φ in eq (36) can be calculated according to Hausen [9,1,2] for the plane wall of thickness δ, for the cylinder of diameter δ and the sphere of diameter δ. For other geometries Hausen introduced the equivalent wall thickness to be used in the equations for the plane wall.
A new formula for the equivalent wall thickness is proposed which gives better agreement with exact calculations for spherical elements than eq (46) s eq V A 6 1 1 3 For the cylinder eqs (46) and (47) yield the same results.

Alternative calculation model
In the theory on thermal regenerators [9,10]  from eq (36) is added. According to previous investigations on temperature oscillation heat transfer processes [11], not only the heat transfer coefficient has to be corrected (eq (36)) but also the wall thickness or the capacity of the elements. (Index s in this paper is index w in ref [11] can be calculated according to the temperature oscillation model [11]. The relevant equations are given in the appendix. The resulting values of s a are equal to or a few percent lower than those of Hausen's approach. The required capacity reduction yields another minor or negligible correction on the safe side. The limiting case of infinite storage capacity does no longer occur.

Conclusions
1. The effect of axial heat conduction in the walls of recuperators and in the solid matrix of regenerators can be described and estimated with the axial dispersion model and an effective dispersive Peclet number.
2. The derived approximation model can be applied to counterflow, parallel flow and mixed-mixed cross-flow recuperators as well as counterflow and parallel flow regenerators.