Simplified models of the symmetric single-pass parallel-plate counterflow heat exchanger: a tutorial

The heat exchanger is important in practical thermal processes, especially those of (i) the molten-salt storage schemes, (ii) compressed air energy storage schemes and (iii) other load-shifting thermal storage presumed to undergird a Smart Grid. Such devices, although central to the utilization of energy from sustainable (but intermittent) renewable sources, will be unfamiliar to many scientists, who nevertheless need a working knowledge of them. This tutorial paper provides a largely self-contained conceptual introduction for such persons. It begins by modelling a novel quantized exchanger,1 impractical as a device, but useful for comprehending the underlying thermophysics. It then reviews the one-dimensional steady-state idealization which demonstrates that effectiveness of heat transfer increases monotonically with (device length)/(device throughput). Next, it presents a two-dimensional steady-state idealization for plug flow and from it derives a novel formula for effectiveness of transfer; this formula is then shown to agree well with a finite-difference time-domain solution of the two-dimensional idealization under Hagen–Poiseuille flow. These results are consistent with a conclusion that effectiveness of heat exchange can approach unity, but may involve unwelcome trade-offs among device cost, size and throughput.


Introduction
A heat exchanger is a passive device through which two streams of liquid, separated by a partition, are passed for the purpose of transferring heat energy from one stream to the other. They are common in HVAC (heating, ventilating and air conditioning), petroleum refining and especially in the cooling of internal combustion engines where hot antifreeze from the engine block exchanges its heat with cooler ambient air at the radiator. Moreover, solar thermal electrical generating facilities are frequently constructed to include provision for the 2018 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. storage of sensible heat, so that generation can be extended into the evening hours; recovery of this stored thermal energy to make steam requires, of course, heat exchangers. For the more common applications, exchanger technology can be described as well developed [1][2][3], and exchanger design is commonly relegated to proprietary software [4]; moreover, the plate heat exchanger has been reviewed in some detail [5] (table 1).
Exchangers may, however, be due for a renaissance because, with the growing emphasis on energy efficiency and sustainability, a radically different energy infrastructure may have to be developed. For example, despite intense research since the dawn of the Atomic Era, neither controlled fusion [6] nor generally acceptable permanent storage of long-lived fission waste have been developed [7]. In turn, this has heightened perceptions that humankind may have to employ wind and/or direct insolation as a principal source of its supply of sustainable energy: both are intermittent. Perforce, metropolitan areas that desire a temporally trustworthy supply of energy must engage in storage of energy-storage which, metropolis by metropolis, is presumably so massive as to be best measured in terms of gigawatt-days. 2 In turn, storage of sustainable energy on that scale is reputed to mean storage either (i) by pumped hydro schemes or (ii) by advanced adiabatic compressed air energy storage (AA-CAES) [8]. The latter, however, depends in an essential fashion upon counterflow heat exchangers [9] if it is to operate with acceptable energy efficiency. And the exchangers needed for gigawatt-day storage schemes must be of a volumetric throughput and effectiveness of thermal exchange which may prove challenging to present technology. We feel that the unfamiliarity of such devices to virtually all policy-makers and to most scientists, make desirable a 'simple' introduction to counterflow heat exchangers. To this end, we introduce in §2, a quantized steady-state exchanger which real freshmen have indeed found to be intuitively simple. In §3, we reprise a one-dimensional exchanger which, in the steady state, permits both analytic solution and ready comprehension. In §4, we tackle two-dimensional exchangers. The quantitation of such devices, even when they are heavily idealized, turns out to be of discouraging complexity, including but not limited to unfamiliar eigenfunction expansions on a rectangle. Therefore, the mathematical arcana have been banished to three appendices. Nevertheless, the selected results we present do indicate clearly that scaling up exchanger-throughput to the high-effectiveness gigawatt-day levels prospectively needed by grid-sized long-term compressed air energy storage could be daunting in the extreme [10].
If a heat exchanger is adiabatic, heat is conserved between the two liquid streams and none is exchanged with the exchanger's ambient. If the flow directions of the two streams are predominantly antiparallel, the exchanger is described as 'counterflow' (or sometimes as 'countercurrent'). Consider the generic single-pass counterflow exchanger shown schematically in figure 1. A 'warm' influx stream at temperature Ş w,i enters at the left and runs past a predominantly antiparallel 'cool' influx stream at temperature Ş c,i which enters at the right: Ş w,i > Ş c,i . 3 The two streams are separated by a thin liquid-impermeable barrier of high thermal conductance so that the warm stream cools steadily during passage through the exchanger while the cool stream warms steadily as it accretes heat from the warm stream. Because heat energy cannot passively be transferred from regions of lower to regions of higher temperature, the efflux temperature of the warm stream must satisfy the inequality Ş w,i > Ş w,e > Ş c,i . (1.1) Similarly, In addition to this counterflow heat exchanger configuration, configurations known as crossflow (streams predominantly perpendicular) and coflow (streams predominantly parallel) exist; but they will not be discussed here.
In any era when useful energy is expensive and/or scarce, there are apt to be strong economic and/or regulatory pressures to capture the valuable heat energy, which may be of negligible worth to the system under study if it remains in the warm stream as it exits, and transfer it to some other stream where it will be useful. One measure of the quality of this heat exchange is the degree to which the warm stream can be stripped of its unneeded heat energy. Suppose that the number of moles per unit time of warm liquid transiting an adiabatic exchanger is M w and that its molar-specific heat is γ w ; then the actual rate of heat loss in transit will be just M w γ w [Ş w,i − Ş w,e ]. Since, by equation (1.1), the warm exit temperature 2 One gigawatt-day = 1.00 GWd = 86.4 TJ. This is approximately equal to energy of combustion of 14 000 bbl of petroleum, where one barrel of oil equivalent has been defined to be 5.8 × 10 6 Btu ∼ 6.12 GJ. 3 The nomenclature for these flows is admittedly complicated. The reader is encouraged to consult     can never be less than the cool-stream influx temperature, the maximum loss rate is

S
Hence one can define a figure of merit ε for the exchanger as (cf. figure 1) ε = actual heat transfer rate limiting heat transfer rate where, for any combination of subscripts, S =Ŝ/Ŝ 0 ; ε is commonly called the (thermal) 'effectiveness'. The goal, therefore, in heat exchanger design is to make ε as close to 1 as is economically and/or physically practical. In a real parallel-plate heat exchanger a common practice is to have a stack of many liquid layers separated by thin diaphragms and alternating as warm-cool-warm-cool-warm-cool, etc. This can be idealized as N of the unitary exchangers discussed above, but piped in parallel.
To someone educated in the physical sciences and drilled upon problems of passive heat flow in stationary (normally solid) media, the idea of effectively interchanging the heat contents of two flowing streams may well seem surprising. And, therefore, §2 is devoted to a tutorial thought-experiment which clearly shows that, in principle and for an adiabatic exchanger, such exchange is indeed possible and with thermal effectiveness approaching 1-if one has all the time in the world to wait plus material resources sufficient to lengthen the exchanger indefinitely. However, in a real rectilinear exchanger stream of uniform cross-section A, half-length L and mean (or nominal) liquid velocity U, the liquid has only the nominal dwell time 2L/U, where U = MV/A, M being the molar flux rate of the liquid and V being its molar volume. Moreover, the thermal conductivity κ of the membrane separating the two streams will be finite, and this will restrict the amount of heat which can be transferred during that dwell time. To estimate the effects of finite U and κ, §3 is given over to a brief review of the idealized one-dimensional convection-diffusion problem to show that the effectiveness is given by equation (3.4).
Moreover, if a liquid stream within the exchanger is free of turbulence, then heat transfer from it to the other stream must in some fashion depend upon a constant D of effective transverse thermal diffusivity. To estimate this effect, it is convenient to scrap the one-dimensional idealization of §3 and to replace it with a two-dimensional idealization in which κ does not appear. This is done in §4 where it is shown: (i) that the resulting problem reduces to a difficult boundary value problem, which requires a novel and non-obvious analytic and numerical treatment; (ii) that, under these circumstances, the effectiveness is given by equation (11.1); and (iii) that, remarkably, this simple form is qualitatively accurate whether the liquid velocity profile across a layer of moving liquid within the exchanger is uniform (plug flow) or Hagen-Poiseuille (laminar flow).
Our analyses will show clearly that high effectiveness, large throughput and low cost are unlikely to characterize the same exchanger: choose at most two.

A quantized counterflow heat exchanger
Heat exchangers as manufactured and used are continuous flow devices, and to our knowledge, a rigorously quantized device has not previously been proposed. Nevertheless, quantization is a useful tool for understanding the physics of exchanger operation.
Suppose, therefore, that two sequences of liquid-filled packets enter the exchanger of figure 1 by stepwise displacement; they are identical except for their liquid temperatures and their synchronized antisymmetric stepping. They do not exhibit steady flow but instead operate in stepping mode so that discrete boluses of liquid are periodically displaced. Within the exchanger, let there beÑ boluses (1, 2, . . . ,ñ, . . . ,Ñ) in each stream. Suppose that these isolated boluses are unstirred and thermally (iii) Right shift the warm stream by one bolus, with the bolus at the left end being replaced by a fresh bolus from the influx pipe and the one initially at the right end being discharged into the efflux pipe. Analogously, left shift the cool stream by one bolus. (iv) Allow the Ñ bolus warm/cool pairs to equilibrate thermally with each other.
In this fashion heat is exchanged between the warm and cool streams. Example:Ñ = 1. In this case, the equilibrium temperature of the only bolus-pair is always 0. Thus, the efflux pipes fill with liquid at temperature zero. Moreover, S w,e = 0 thereby setting Example:Ñ = 5. This case is shown in figure 2, which illustrates the operation of the first few cycles. Further calculation would show that, as the number of cycles becomes large, the equilibrated temperature distribution (from left to right) tends towards 2/3, 1/3, 0, −1/3, −2/3; and effectiveness during steady-state exchange becomes ε 5 = 5/6.
The upshot of this is that, at least in a thought experiment, it is possible to exchange the heat contents of two equivalent volumes of liquid which are distinguished only by their temperatures. This violates no thermodynamic law because (ideally) no work is done in the process.

A one-dimensional continuous heat exchanger
Under steady operation, a one-dimensional counterflow heat exchanger (with homogeneous composition in any transverse cross-section and constant inputs) reaches an equilibrium state within which the amount of heat in any cross-sectional slab is constant; the overall geometry of such an exchanger is shown in figure 3 for the case of antisymmetric warm and cool pathways. For a thin perfectly mixed transverse cross-section in the warm stream 4 [11], this heat balance is given by where M is the molar-specific weight, M is the molar flow rate of the influx, γ the molar-specific heat of the liquid influx,Ŝ 0 the symmetrized input temperature of the warm stream, κ the thermal conductivity of the membrane separating the two streams, 2a the thickness of the membrane and 2d the depth of the membrane in the x-direction. As z → 0, equation (3.1) becomes and, analogously, the heat balance equation for the cool stream becomes where the parameter Ω is given by  . Schematic of one functional warm-cool pair of a quantized counterflow heat exchanger. The warm-pipe inlet temperature is assumed to exceed the cool-pipe inlet temperature. There is no variation in the x-direction, which is assumed to extend from +d to -d. All flow within the exchanger is assumed to be parallel (or antiparallel) to the z-axis. Heat exchange with the surroundings, except that associated with the influx and efflux streams, is presumed minor because either (i) the unit is interior to an N-unit sandwich (N 1) or (ii) the outer surfaces of the unit are heavily insulated. Ideally, the fluid flow in each pipe will be steady and z-invariant; also the velocity profile in the y-direction will be symmetric about the line y = h.
The appropriate boundary conditions are and s c (+L) = −1. (3.2e) The solution of the system (3.2) is readily found by a variety of means. One which seems intuitively satisfying is to note from symmetry that s c (z) = -s w (-z), express s w (z) as the sum of odd and even functions, and deduce eventually from this that and From equations (3.3) it follows that, based upon this one-dimensional idealization, the prospective effectiveness of a counterflow heat exchanger is . (3.4) Suppose that, initially, an exchanger is designed to have 2ΩL = 1 and thus an effectiveness of only 50%: doubling the length gets the unit to 67%; tripling it gets 75% and making it longer still could well increase the cost prohibitively. 5 Raising effectiveness of the exchanger by increasing Ω may also be problematic: if anything, the exchanger's owner probably would like to increase the throughput M, which would decrease effectiveness; κ and γ are presumably constrained by the nature of the problem; greatly decreasing the membrane thickness 2a would doubtlessly weaken the device structurally; and increasing the depth 2d, like increasing the length 2L, will soon produce large cost increases for small effectiveness increases. In the very high flow rate limit as 2ΩL → 0, it is seen that ε 1D ∝ L/M.
In practice, counterflow exchangers are a well-developed and mature technology. In consequence, the recent literature on their one-dimensional idealization has been sparse in recent decades. However, the interested reader may wish to consult further references [12][13][14][15][16].

A two-dimensional continuous heat exchanger 4.1. General case
Suppose now that we take our steady-state idealization of the intra-exchanger liquid streams to a different limit. Let the thickness of the membrane separating the streams become vanishingly thin (a → 0) so that there will be continuity of both temperature and heat flux across the interface at y = 0. Assume that heat flux is (i) negligible in the x-direction, (ii) is overwhelmingly diffusive in the y-direction and (iii) is overwhelmingly convective in the z-direction, where the convection can be idealized as constant  Figure 4. Schematic of one symmetry unit of a counterflow heat exchanger. The fluid velocity will have a peak at each symmetry plane; but there will be no heat flux across such planes. Generic profiles of velocity across the warm and cool streams of the unit are shown as yellow dots. The profile in the warm stream will be Uf (υ) while that in the cool stream is taken to be its mirror image -Uf (-υ).
strictly axial flow velocity independent of transverse position. This is illustrated in figure 4, which will be recognized as a modification of figure 3. The principal differences between figures 3 and 4 are (i) that generic laminar flow profiles have been superposed upon the two channels and (ii) that symmetry planes rather than unit boundaries have been indicated. In a many-unit stack of warm-cool flow sheets, there will be N warm-cool flow sheet pairs (each 4 h thick) and approximately 2N functional pairs (each 2 h thick). Also, make the substitutions y = hυ and z = Lζ . Then, by repeating the formalism of any standard text on heat conduction ( [17], §1.7, esp. eqn 1.7(1)) and simplifying the x-directed, y-directed and z-directed fluxes, 6 where the boundary conditions are s c (υ, 1) = −1, −1 < υ < 0, ζ = 1, (4.1d) and where the constant Λ 2 is defined in the Symbol Table and where f (υ) is the liquid velocity profile over [0,1], assumed to be non-decreasing and of mean value 1. We note (i) that it seems reasonable to assume, from the antisymmetries of the physical problem, that and (ii) that this assumption seems to accord with equations (4.1). The information desired from the particular two-dimensional idealization of equations (4.1) and (4.2) is the effectiveness of the energy interchange as defined by (4.3a) where S w,e is the dimensionless warm-stream energy efflux at ζ = 1, which is given [18] by Technically, the question posed above is known as a 'steady-state conjugate-Graetz problem', a class which has been discussed infrequently in the archival literature [19]. 8 Generalized solutions for Graetz problems related to that of our highly idealized system (4.1) have been found [20][21][22][23]. More recently, Vera & Liñán [24] have investigated the system of equations (4.1) and (4.2) numerically. Also, Quintero & Vera [25] have investigated influence of wall and diaphragm conduction effects and found that they lower effectiveness. A fairly up-to-date review of conjugate problems in heat transfer is that of Dorfman & Renner [26]. We find these solutions distinctly recondite and point out that they seem not to have considered the simple variation of effectiveness with the dimensionless variable Λ.
To effect a theoretical solution, the nature of f (υ) must be specified. We define it so that the z-directed liquid velocity over half of a warm pipe is given by Uf(υ), where U is the mean velocity across the pipe. We then distinguish two special cases: . This is what one normally thinks in terms of for flow between stationary planar surfaces at Reynolds numbers below 50. and The plug flow problem might be analytically tractable, but would seem to require extreme turbulent flow to approximate its existence, thereby vitiating the convection diffusion model. On the other hand, recent computational fluid-dynamics modelling of counterflow exchangers suggests that the flow profile could in practice be laminar [27], thereby favouring a parabolic profile. The reasonableness of this finding can be seen by noting that Hagen-Poiseuille flow in a planar channel is expected to be stable when the Reynolds number R = 2hUρ/η ≤ 2500 [28] but to deteriorate into at least vortex shedding if there are obstacles or discontinuities [29]. For a thin water-filled channel with U ∼ 0.1 m s -1 , h ∼ 0.005 m, ρ = 1000 kg m -3 and η = 0.001 Pa s, R ∼ 1000, a situation in which laminarity should obtain. On the other hand, plumbing connections at input and output might induce modest tumbling that would scramble the temperature distributions.
Therefore, even though it may not be entirely relevant, our development will begin with plug flow for which one approximate solution will be presented. Then we shall tackle Hagen-Poiseuille flow for which only numerical data will be presented. Reassuringly, there is excellent qualitative agreement between the two. Both developments demonstrate that there is significant difference between the predictions of one-dimensional theory and those of the two-dimensional theory.

Plug flow
Because we could find no simple straightforward solution to (4.1), (4.2), (4.3) for plug flow, we opted for the approximate treatment of appendix A. It gives the effectiveness as a function of Λ 9 : To our knowledge, this remarkably simple approximation has not previously been published. Moreover, it reveals that effectiveness can be simply gauged using a single dimensionless constant. localized turbulent mixing and possible swirling in the exchanger's plumbing could serve to add the appearance of greater accuracy than the experimental reality might support.   1)) for the thermal effectiveness ε : equations ((4.6) ≡ (A.8)) for plug flow is plotted as a solid black line; overlying this is a sequence of blue dots generated for discrete values of Λ: these data, for Hagen-Poiseuille flow, were obtained using the wholly numerical FDTD method described in appendix B. For Λ ≤ 1, the two treatments yield similar results.

Hagen-Poiseuille flow
In our hands, simple analytic solutions to neither the parabolic Hagen-Poiseuille profile nor various approximations of it were found, although closed-form solutions to equation (A.5) do exist for a parabolic profile [24,30]. We concluded that a numerical model of the geometry of the previous section, only with a Hagen-Poiseuille flow profile, would be an interesting test of the approximate solution of appendix A. To this end, we obtained a plot of effectiveness versus Λ by means of an FDTD (finite-difference timedomain) approach; our finite difference algorithm is explained in appendix B, and our results are shown in figure 5. This figure shows that: (i) the plug flow and Hagen-Poiseuille (HP) flow models for effectiveness have qualitatively similar behaviours over the entire Λ-range [0. 1,10], only with HPflow showing qualitative superiority everywhere; (ii) that they agree acceptably well from Λ ∼ 0 out to Λ ∼ 1 and (iii) that, for higher values of Λ, the expected effectiveness is so low as to render irrelevant the precision of estimate effectiveness predicted by either model becomes so low as to discourage use of exchangers operating in that high Λ-range.

Discussion and conclusion
There are at least four things that one might ask of a counterflow heat exchanger. The first is that the passage of the exchanger liquids through it be swift so that much stored heat becomes available to be exchanged; this will be roughly indexed as [2hdU]. The second is that the cost the materials for the exchanger be small; this will be indexed as the plate area of a channel separator [4dL]. The third is that the diffusion velocity in the υ-direction be large so that heat exchange between the two streams is enhanced; this will be indexed as [D/2h]. This then yields And the last (fourth) thing that might be asked is that Λ be small so that the effectiveness approaches 1−. We have here a quadrilemma! And it seems impossible to get everything desired.
First, increasing the throughput increases Λ, thereby decreasing effectiveness. Second, decreasing the materials cost increases Λ, thereby decreasing the effectiveness. Third, increasing the diffusion velocity in the υ-direction decreases Λ, thereby increasing effectiveness.
Fourth, decreasing Λ to < ½ will raise the effectiveness to >0.94. And, by equation ( for example, to molten sodium would up the thermal diffusivity by a factor of roughly a thousandfold to approximately 65 × 10 −6 m 2 s −1 , but at the cost of a daunting increase in operating complexity. Alternatively, for a eutectic mixture of sodium and potassium nitrates, much used in concentrated solar power installations, the thermal diffusivity is a modest approximately 170 × 10 −9 m 2 s −1 [31].
The real-world work-around, which seems to have bypassed the theoretical limitations of static thermal diffusivity, is a deliberately induced local turbulence superposed upon to the bulk motion of the heat transfer liquids 10 : seemingly, this produces an effective diffusivity large enough to force the thermal impedance between the warm and cool streams towards the thermal resistance of the thin corrugated metal plates separating these streams. This strategy appears to have been followed the past 30 years [32,33]. And it has led to heat exchangers roughly the size of a pulpit Bible and costing less than 2 k$ apiece, which are able to sustain heat transfer rates on the order of 440 kW [11]. These are not expensive hand-crafted luxuries: they are a mass-produced commodity! Nevertheless, conversion of solar-thermal energy to and fro between stored heat and electricity still seems challenging. It has yet to been demonstrated cost-effectively at long-term utility-scale levels.
A comparison of the results of § §3 and 4 with a typical text on heat exchangers [4,34] will show that our notation is different, that our presentation is different and that our focus is different. This is because we are less interested in the hugely important practicalities of specifying, plumbing or using heat exchangers than we are in delivering to scientists from other fields an understanding how practical exchangers work and in what directions their theoretical limitations may lie.
Finally, remember that utility-scale storage and release of electric power presumably must take place at multi-megawatt levels. Endeavouring to store electrical energy as heat, which can then be converted to electricity, is therefore likely to require heat exchangers that operate at the megawatt level. Such exchangers now are commonplace and can be added as needed at a cost of around 5 dollars per kilowatt capacity (plumbing parts, labour and bulk liquid storage not included). But the small details of such generation, conversion and storage are as yet fluid because the structuring of America's smart resilient grid of the future (together with its component microgrids) is only now commencing. 11 In conclusion, this essay is intended as a tutorial paper, where originality is not necessarily a sine qua non. However, we believe the quantized heat exchanger to be new. Further, we are unaware of the occurrence elsewhere of the formula 'NTU = 2ΩL'; but the literature is vast, and we make no claim. Further, we believe the effectiveness formula of equation (4.6) to be original. These results could be interpreted as meaning that NTU, while highly useful for both intuitive and pedagogical purposes, could be and has proved difficult to predict accurately in practice. 14