A Domestic Waste Heat Recovery System Mathematical Model of a Green Kitchen Module

Abstract

Keywords: appliances; energy saving; clean energy

1. Introduction

The present environmental concerns, and the fact that the conventional energy sources are bound to be exhausted, make the search of alternative energy sources more urgent than ever. Energy saving is a form of alternative energy; in this context, waste heat recovery becomes particularly important. To appreciate the importance of this form of energy saving, one can consider the fact that in the UK alone 40 TWh per year could come from waste heat recovery in the industrial sector [MOXOFF, 2012a]. In this paper, the results obtained in a design study of heat storage/ recovery systems (aimed at green kitchen applications) are described, together with the model employed to achieve them. In the green kitchen approach [MOXOFF, 2013a Mukherjee, 2011], the thermal energy dissipated by various kitchen appliances is recovered by various means, typically for the production of hot water for domestic use. There are various forms of waste heat recovery [Hussam et al., 2018]: regenerative and recuperative burners [Institute for Industrial Productivity, 2017 - BDF Industries, 2017]

capture the waste heat from the combustion of hot flue gases, economizers [Thermtech, 2014] recover waste heat to be used for heating liquids, waste heat boilers [Ganapathy, 2015] are suitable to recover heat from exhaust gases and mainly used to produce steam for power generation, air preheaters [Yodrak et al., 2010] are employed for exhaust heat recovery. Heat exchangers [IPIECA, 2022] and heat pipes [JHCSS, 2017] are employed in such devices. The recovered heat can be employed to preheat the gases entering burners and/or furnaces, to produce electricity by steam or thermoelectric generators. In green kitchen applications the recovered waste heat comes from domestic appliances. To have an idea of the potentialities of such technologies, one can consider the fact that savings of the order of 5-10% can be achieved by preheating gases in burners [Spirax Sarco, 2011], producing steam via a Rankine cycle can reach an efficiency of 22% [Stefanou, 2017], thermoelectric generators have a 2-5% efficiency but through the use of nanotechnologies efficiencies greater than 15% can be achieved [Caillat et al., 1999].

*Corresponding author. *E-mail address: scimodsim@gmail.com, stefanspag@gmail.com Published by Yıldız Technical University Press, İstanbul, Türkiye This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

In regenerative burners, the heat coming from the hot flue gases is first stored in a heat exchange medium such as aluminum oxide, then recovered by heating the cold gas through contact with the medium: this is an example of the utility of sensible heat materials in waste heat recovery. Sensible heat materials [Sarbu et al., 2018] store heat by increasing their temperature; in phase change material [Sarbu et al., 2018 - Ahmed et al., 2018], energy is mainly stored in the form of latent heat, although there can also be a temperature increase. Among sensible heat materials one can cite water, rock, sand and steel. Phase change materials can be classified as organic (paraffins and non-paraffins), inorganic (salt hydrates, low melting point metals), eutectic mixtures. At present, the cheapest and more performant phase change materials are paraffins [Sarbu et al., 2018]. In this paper, the lumped-parameter model employed is first described. To the best of the author's knowledge, this model substantially innovates the existing literature. On the base of the developed model, a number of solutions are proposed. The focus is on a thermal bus, with a refrigerator, an induction cook-top and a domestic oven connected in series. Whirlpool Europe presented a patent application [Mukherjee, 2011] concerning a green kitchen heat storage/ recovery system, based on a water tank as the storage medium. The use of cylindrical modules, with cylindrical or spherical storage modules, was only proposed as an ancillary system, improving the water tank performance. The system described here, whose feasibility study was commissioned by Whirlpool Europe, substantially improves the solution proposed in the patent. In the main body of the paper, the formulae needed for practical use are only given; a full treatment is in Annexes A1-A7.

2. Problem Setting

In Table 1, some typical sensible heat materials (SHMs) and phase change materials (PCMs) are illustrated. The table shows that the energy storage density for PCMs is up to ~10 times larger than for SHMs. For appliance temperatures <50°C, the module charging process has to stop before the onset of fusion. In this case, PCMs only count for their specific heat. In this temperature range, SHMs may be more suitable than PCMs. The first proposed approach is based on sensible heat materials [Dincer et al., 1997 - Sharma et al., 2005]. In this approach, the dissipated heat, carried by a fluid, heats the material. In the heat recovery phase, a cold fluid is put in touch with the material and heats up. If ρm is the material density, C its specific heat, ΔT the difference between the fluid and solid temperatures, then the material energy density is: ρ =ρ . C . ΔT(1)

form of latent heat of fusion. During the phase transition, the material does not change its temperature. If ρm is the material density, ΔH the transition (fusion) enthalpy, the material energy density is: (2) ρ =ρ . ΔH e

In Table 3, the thermophysical properties of common PCMs are shown. In heat storage/recovery systems, the storage material is part of a heat exchanger so that, in the module charging phase, a hot thermo-vector fluid (heated by the thermal energy to recover) is able to transfer heat to the material. In the module discharging phase, a cold thermo-vector fluid gets heated, removing heat from the material. In [MOXOFF, 2013a - Mukherjee, 2011], a possible design solution, based on a fixed bed (Fig. 1), was proposed. A fixed bed [Holdich, 2002] is a metal cylinder, filled with spheres of a suitable material and crossed by a thermovector fluid that exchanges heat with them. In the fixed bed regime, the viscous drag force does not prevail over gravity and the spheres are stuck in their position. Beyond the minimal fluidization velocity, the viscous drag force overcomes gravity and the spheres acquire a turbulent motion. One is then in the fluidized bed regime; the latter is unstable for spheres of diameter ≥1 cm [Holdich, 2002]. The diameter that had to be chosen falls in this range [MOXOFF, 2013b]; consequently, a fluidized bed solution was not considered. In [MOXOFF, 2013a - Mukherjee, 2011] a possible design solution, based on a shell-and-tube heat exchanger [Mukherjee, 1998] (Fig. 2), was also proposed. Here it is only pointed out that the shell-and-tube heat exchanger has to be equipped with baffles, whose aim is described later. In [MOXOFF, 2013a], appliances are classified, according to their time behaviour, as discrete or continuous. As an example, one may consider a domestic oven and a refrigerator. Heat is recovered from a domestic oven when the appliance is turned off and its temperature is ~180250°C [Zavattoni, 2012]. One then has a discrete process. A refrigerator, instead, dissipates energy continuously. A typical refrigerator is reported to dissipate 84 W at 40°C [Zavattoni, 2012]. Appliances can either dissipate at a nearly constant temperature (as in the case of a refrigerator or cook-top induction plane) or a variable temperature (as in the case of a domestic oven). The following module classification is then proposed: •

In Table 2, the thermophysical properties of typical SHMs are shown.

Phase change materials [Sharma et al., 2009] work similarly, except that most of the thermal energy is accumulated in the

Table 1. Heat storage characteristics of typical materials [Dincer, 1997 - Sharma et al., 2009]. ΔT = 15°C is assumed for sensible heat materials Property Rock Water Organic PCM Density (kg/m3) Specific heat (kJ/kg.°K) Latent heat (kJ/kg) Mass per 106 J stored (kg) Volume per 106 J stored (m3) Relative mass Relative volume

Table 2. Heat storage characteristics of typical sensible heat materials [Dincer, 1997 - Mardiana-Idayua et al., 2012] Material

Density (kg/m3) Specific heat (kJ/kg.°K) Thermal conductivity (W/m.°K)

Table 3. Thermophysical properties of various phase change materials [Sharma et al., 2005 - Sharma et al., 2009]. The literature data are inconsistent and fragmentary. Where a parameter is available for only one phase, the value for the remaining phase was taken to coincide with the available value. Where a piece of data is missing for both phases, when possible, it was approximated with the average value for materials of the same category, when not possible, it was approximated with the typical values of Table 1. In the case of inconsistent data from different sources, the most pessimistic value was chosen. Materials with a transition temperature in the 20-100°C range were only considered. (l) = liquid state, (s) = solid state Material

Fusion Fusion Thermal Thermal Density (l) Density (s) Specific Specific temperature heat conductivity conductivity (kg/m3) (kg/m3) heat (l) heat (s) (°C) (kJ/kg) (l) (W/m.°K) (s) (W/m.°K) (kJ/kg.°K) (kJ/kg.°K)

4. Modules

The charging process merely allows a limited charge efficiency. On the other hand, the choice of a factor fd>1 increases efficiency. In fact, the energy accumulated into a module is given by ηc∙E0. If one increases the energy size by making the module taller by a factor fd, one can pretend to have the same energy size with an increased charge efficiency. As noticed above, in this paper it is assumed that topt≈τ. Therefore, see the comments to Eq. (24), one has θ=1/fd+θf where, as shown in Appendix A5, θf is the non-dimensional time taken by fusion to occur. The charging efficiency for a non-upscaled module must then be multiplied by fd, to give the effective charging efficiency. Based on these facts, it is then possible to give expressions for the module efficiency. In particular, see Annex A4, an expression valid for all the previously defined module categories is:

~ T∞=T∞−k1∙(1−e−1)∙(T∞−T0)(37) ~ ~ T=T∞+(T0−T∞)∙exp(−θ)(38) 1 (39) ηcopt=fd ∙(1−k2 ∙exp(− )) fd  1 ηeopt=fd ∙(1−exp(− )) (40) fd  In the previous equations, k1 is 0, unless for VITSHMs and VITPCMs, in which case it is 1. k2 is 1, unless for CITPCMs and VITPCMs, in which case it is 1/2.

at will (within technological limits) by choosing a suitable air extraction pump. In the case of a refrigerator, the charge time depends on how much energy must be accumulated. Envisaging possible synchronization among the appliances, a common charge time of 18 min was then chosen [MOXOFF, 2012a - MOXOFF, 2013b]. The appliances considered were: •

A refrigerator, dissipation temperature: 40°C, dissipated power: 84 W, charging time: 18 min, stored energy: 0.1 MJ, air flow rate: 4.0 l/s.

An induction cook plane, dissipation temperature: 60°C, dissipated power: 100 W, charging time: 18 min, stored energy: 0.1 MJ, air flow rate: 2.4 l/s.

An oven, equivalent dissipation temperature: 80°C, real dissipation temperature range (as time proceeds): 20100°C, charging time: 18 min, stored energy: 1.1 MJ, dissipated power: 365 W, air flow rate: 5.9 l/s.

5. Approximations

In Annex A4, it is shown that the previous equations are approximations to the exact formulae. Using these approximate formulae, however, leads to a conservative design choice (to slightly overestimating module height, which can only increase thermal exchange efficiency, given the previously mentioned NTU conditions). Based on the results in Annex A3, the model is expected to reproduce the experimental results within ~3%. In Annex A4, it is equally shown that, for VITSHMs and VITPCMs, the efficiency expressions should be multiplied by correction factors ~1, ignoring which leads to slightly overestimating module height, with beneficial effects on thermal exchange efficiency. The expression for the PCM module charging time, in principle, should be modified by adding the fusion time to the set 18 min. On the other hand, Table 3 shows that the fusion heats of interest are <514 MJ/m3. Furthermore, Tables 4-7 show that transferred power is larger than 84·0.8=67 W. The storage units have 5 mm diameter and height <20 cm. A fusion process duration <30 sec, therefore negligible, is then calculated.

6. Results

The mathematical model was simulated by an Excel spreadsheet. Using the model developed here, optimal configurations for both fixed beds and shell-and-tube heat exchangers, loaded with spherical or cylindrical heat storage units (made up of suitable heat storage materials), were calculated. The storage modules have an energy size of 1.4 MJ, approximately corresponding to the energy needed for heating water in a dish-washer cycle [MOXOFF, 2013a]. As for the thermo-vector fluid, air, with a maximum flow rate of 6 l/s per module, was chosen, as determined by structural and acoustical considerations (water would require lower flows, which would not pass through the heat exchanger [Zavattoni et al., 2014]. A domestic kitchen heat storage module was considered, coupled to a thermal bus that connects a refrigerator, an induction cook-top plane and a domestic oven. For the refrigerator, one must have two parallel heat storage modules [Mukherjee et al., 2011], since while one module is charging, the other is in the discharge phase. For discrete behaviour appliances, instead, a single module suffices. The cook-top induction plane charge time is ~18 min [Zavattoni et al., 2014]. The domestic oven charge time can be determined

In Tables 4-7, results for cylindrical modules with 50 cm diameter are shown. As for efficiencies and volume, the smallest factor needed for satisfying model approximations was chosen and efficiencies were calculated accordingly (and turned out to be ~80-150%, since the effective efficiency, due to the way it is defined, can be >100%). In order to rank the various solutions, a figure of merit, the power density (up to ~200 W/kg), was devised: E∙ηc (41) ρp= ρm∙Vm∙topt  As for shell-and-tube heat exchangers, the following solutions were found: •

On the other hand, fixed beds turned out to be ~30% shorter and ~10% lighter. Since module height is not critical, and fixed beds are harder and more expensive to build, the choice of election is PCM-based shell-and-tube heat exchangers.

7. Discussion

As shown in Annex A3, a maximum error of ~3% is predicted, with respect to the real temperature behaviour of the system. A priori, one might worry about pressure drop in the heat exchangers. However (Annex A7), the maximum pressure drop is <0.5% of the atmospheric pressure and can then be neglected. Let us finally notice that both the fixed bed and the shell-and-tube exchanger have to be equipped with baffles. Baffles have different functions [Mukherjee, 1998]: they contribute to the structural stability, drive the flow by avoiding blind corners, make the flow orthogonal to the axis, improving the heat exchange rate. Simulation results imposed to devise a mechanism for increasing the mean area velocity, so as to increase the system NTUs. Baffles turned out to be the answer. The model described here was conceived in order to fulfill the request, by Whirlpool Europe, to dimension a green kitchen system for which, to the best of the author's knowledge, no model existed in the literature. Experimental tests were

performed by Whirlpool Europe [Zavattoni et al., 2014] but no details were released in the open literature. Our exercise was meant to obtain a proof-of-concept, so that Whirlpool might decide whether to build and experiment a prototype. To the best of our knowledge, however, no prototype was built, therefore our model, although its approximations were demonstrated to be reasonable, was not validated experimentally.

8. Conclusion

The model is based on a lumped parameter approach, which makes it sufficiently simple and manageable. As previously stated, and shown in Appendix A3, a maximum error of ~3% is predicted, with respect to the real behaviour of the system. The model, however, could not be validated experimentally. Unfortunately, not all the possible heat storage materials could be considered, due to inconsistencies in literature data. However, ~20 SHMs and PCMs were considered, representing all the available material typologies. We acknowledge the existence of thermoplastic materials [Oguzhan et al., 2019], for which, however, we could not find useful data. Our simulations allowed us to find a PCM-based and a SHM-based configuration, satisfying our specifications with the largest exergetic efficiency, namely ~50x50x25 cm, ~30 kg (PCM) and ~50x50x40 cm, ~110 kg (SHM). The PCM-based module is more compact and less heavy, so we tend to prefer it. Starting from the results achieved, future work might involve numerical optimization of the best configurations, using CFD software. Needless to say, the model would need experimental validation, which at the moment is lacking. In any case, the efficiencies calculated with a lumped parameter model are a lower estimate of those calculated with a 2D model [Taylor et al., 1991a Taylor et al., 1991b]; the model, consequently, produces a conservative design. An interesting development might be an extension of the model to restaurant or industrial kitchens, especially in connection with a financial analysis.

Nomenclature

: Material density (kg.m-3) : Material specific heat (J.K-1.kg-1) : Fluid minus solid temperature (K) : Material energy density (J.m-3) : Material fusion enthalpy (J) : Material volume (m3) : Module energy size (J) : Module volume factor (n.d.) : 1-ε3 is 3D porosity (n.d.) : 1-ε2 is 2D porosity (n.d.) : Module volume (m3) : Module height (m) : Module shape factor (n.d.) : Module unit shape factor (n.d.)

: Optimal module charging temperature (K) Topt : Optimal module charging time (s) topt W : Power transported by the fluid (W) Φ : Fluid volume throughput (m3.s-1) E : Energy deposed in the material (J) Nu : Nusselt number (n.d.) Pr : Prandtl number (n.d.) Re : Reynolds number (n.d.) : Fluid thermal conductivity (W.m-1.K-1) kf : Material thermal conductivity (W.m-1.K-1) km : Fluid viscosity (N.s.m-2) μf : Fluid mean velocity (m/s) vf : Baffle pitch over heat module diameter (n.d.) fp : Module charging efficiency (n.d.) ηc : Module energetic efficiency (n.d.) ηe opt : Optimum module charging efficiency (n.d.) ηc opt : Optimum module energetic efficiency (n.d.) ηe : Module power density (J.m-3) ρp t : Time coordinate (s) : Fourier number (n.d.) Fo : Module energetic efficiency correction factor (n.d.) fe : Module charging efficiency correction factor (n.d.) fc : Power exchanged between material and fluid (W) Wex LMTD : Log Mean Temperature Difference (K) S : Heat exchange area (m2) : Fluid outlet temperature (K) Tf : Solid material temperature (K) Ts : Heat exchanged between material and fluid (J) Wf : Power exchanged between material and fluid (J) Ws P : Heat exchanger wetted perimeter (m) x : Distance coordinate (m) λ : Spatial temperature decay constant (m) ~ : Effective oven dissipation temperature (K) T∞

: Module charging time (s) : Infinitesimal mass entering the oven (kg) : Oven thermal disturbance standard deviation (m) : Air thermal diffusivity (m2.s-1) : Oven characteristic length (m) : Oven thermal disturbance characteristic time (s) : Biot number (n.d.) : Material thermal diffusivity (m2.s-1) : Module unit heat diffusion time scale (s) : Module unit thermal equilibrium time scale (s) : Ratio between material and fluid thermal conductivities (n.d.) εfit : Error due to temperature fitted by an exponential (K) : Error due to non-constant temperature (K) εnct : Error due to material lagging behind fluid (K) εdyn ε : Total temperature error (K) : Temperature drop across the material (K) ΔTm : Material fusion temperature (K) Tf : Material fusion duration (τ depending on Cl) (s) θf : Material fusion ending time (τ depending on Cl) (s) θ(l)f (s) : Material fusion ending time (τ depending on Cs) (s) θ f : Material solid specific heat (J.K-1.kg-1) Cs : Material liquid specific heat (J.K-1.kg-1) Cl : Material average specific heat (J.K-1.kg-1) Ca : Material fusion specific heat (J.K-1.kg-1) Cf : Material total specific heat (J.K-1.kg-1) Ct : Material solid non-dimensional specific heat (n.d.) ξs : Material liquid non-dimensional specific heat (n.d.) ξl : Material average non-dimensional specific heat (n.d.) ξa ξf material fusion non-dimensional specific heat (n.d.) : Material total non-dimensional specific heat (n.d.) ξt : Optimum module charging time (τ depending on Cl) (s) θopt E : System exergy (J) U : System internal energy (J) : Environmental pressure (Pa) p0 V : System volume (m3) S system entropy (J.K-1) W : Work performed by the system (J) : System plus environment entropy variation (J.K-1) ΔStot : Dissipated work (J) Wdis : Exergetic efficiency (n.d.) ηex Wmax : Maximum work performed by the system (J) : A non-dimensional temperature (n.d.) τ∞ <θopt> : Average value of θopt (s) α : A function of <θopt> (n.d.) : Temperature at which the modulus charges (K) Tc : Non-dimensional Tc (τ depending on Cl) (n.d.) θc : Fluidized bed pressure drop (Pa) Δpfb

: Module unit diameter (m) : Heat exchanger pressure drop (Pa) : Number of heat exchanger baffles (n.d.) : Effective number of heat exchanger tube rows (n.d.) : Friction factor (n.d.) : Module diameter (m) : Minimal fluidization velocity (m.s-1)

Acknowledgements

I express my gratitude to dr. Paolo Ferrandi, dr. Chiara Riccobene and dr. Matteo Longoni (MOXOFF) for their invaluable help. My gratitude, also, to prof. Alfio Quarteroni (Politecnico di Milano) and dr. Ottavio Crivaro (MOXOFF) for their constant encouragement and for allowing and supporting the publication of this paper. I also gratefully acknowledge the contribution of dr. John Doyle (Whirlpool Europe).

Data Availability Statement

The published publication includes all graphics and data collected or developed during the study.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Financial Disclosure

The authors declared that this study has received no financial support.

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