## Appendix A

The conservation of energy during the energy efficient DHW generation operation mode (c)—cf.

Figure 1c—in the process water cycle is given by:

where

${\dot{Q}}_{\mathrm{con}}$ is the heat transferred from the refrigerant to the water by the condenser,

${\dot{Q}}_{\mathrm{RPW},\mathrm{P}}$ is the heat transferred by discharging the PCM in the RPW-HEX to the water,

${\dot{Q}}_{\mathrm{RPW},\mathrm{R}}$ is the heat transferred from the refrigerant in the RPW-HEX directly to the water and

${\dot{Q}}_{\mathrm{DHW}}$ is the heat transferred to the DHW storages. The latter can be calculated by the in- and outlet temperature of the water:

and if the HP is running, the contribution of the condenser can be described with:

where

ϑ_{sp} is the set-point temperature for heating which is also used for DHW generation in operating mode (c) to preheat the water at a high

COP. The sensible energy of the hot gas transferred through the RPW-HEX can be calculated from the efficiency of the RPW-HEX

where

ε_{RPW} is the fraction of the total heat at the hot side of the HP that can be transferred to the RPW-HEX and has to be taken from experiments or simulations (cf. [

15]). The heat transfer rate needed to be transferred from the PCM to the DHW Δ

Q_{RPW,P} is the heat that cannot be provided by the refrigerant of the HP. It can be calculated from (A1) and the energy extracted from the RPW-HEX after the time

τ is:

To calculate the energies (A2–A5) the change in time of the outlet temperature of the DHW storage (${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{out}\right)}$) is crucial, because this temperature strongly influences the contribution of the HP to the heat transferred to the DHW storages. If ${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{out}\right)}$ is significantly smaller than the set point temperature ϑ_{sp}, the HP can operate at a high part load (since the volume flow rate is fixed) and therefore can provide a high share of thermal energy to the DHW charging process. If ${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{out}\right)}$ is close to the condensing temperature or even higher, the HP will turn off, and all the energy must be provided by the PCM. Hence, ${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{out}\right)}$ has a high impact to the RPW-HEX storage capacity needed for providing a certain amount of thermal energy for the DHW storages.

In the present work, we use a moving boundary approach to describe the thermocline behavior of the DHW storages to find an analytic solution for A5.

Figure A1a shows the principle of the approach.

**Figure A1.**
(

**a**) Scheme of the moving boundary approach. (

**b**) Fitting (A12) to experimental data of a charging process of a 140 L enerboxx

^{®®} storage (storage used in [

28]) for different temperatures and volume flow rates. The solid lines in (

**b**) represent the moving boundary approach calculated from (A12), the dotted line represent a lumped parameter approach (not described) and the markers denote measured temperatures from the experiment.

**Figure A1.**
(

**a**) Scheme of the moving boundary approach. (

**b**) Fitting (A12) to experimental data of a charging process of a 140 L enerboxx

^{®®} storage (storage used in [

28]) for different temperatures and volume flow rates. The solid lines in (

**b**) represent the moving boundary approach calculated from (A12), the dotted line represent a lumped parameter approach (not described) and the markers denote measured temperatures from the experiment.

The storage consists of an upper volume at hot temperature (

ϑ_{h}), which is always heated to a temperature corresponding to the constant inlet temperature and a lower volume at cold temperature (

ϑ_{c}), whose temperature corresponds to the fresh water temperature. Both volumes are separated by a moving boundary (

$\ell $) which depends directly on the stored energy:

where

A_{DHW} is the ground area of the DHW storage,

L is the height of the DHW storage,

ρ_{w} is the density and

c_{p,w} is the specific heat capacity of water. Note that with the aid of the three-way valve (component M in

Figure 1), the inlet temperature to the DHW storage

${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{in}\right)}$ is held constant at a fixed temperature by mixing the process water leaving the RPW-HEX with the water leaving the condenser in the real machine. Hence it can be assumed as constant.

The maximum energy can be stored if the entire storage is charged to

ϑ_{h} (

$\ell $ = 0) and the storage is empty if the entire storage has a temperature of

ϑ_{c} (

$\ell $ =

L). Heating of the DHW storage only takes place in the cold region of the storage. Using an NTU description for a constant wall temperature (

ϑ_{c}) one finds for the heat transfer to the DHW storage:

where

w is the width of the storage (i.e., the heat transfer cross-section between the cold fresh water and the process water coming from the HP is

w $\ell $), α is the heat transfer coefficient between the cold fresh water cell and the process water, and

${\dot{m}}_{\mathrm{w}}$ is the mass flow rate of the process water. A turbulent flow in the storage is considered, and therefore, the heat transfer coefficient α is assumed to change with

${\dot{m}}_{\mathrm{w}}$ according to:

where

${\dot{m}}_{\mathrm{w},\mathrm{nom}}$ is the nominal mass flow rate calculated at a volume flow fate of 300 l/h and the factor 0.8 follows from the

Colburn equation.

Inserting (A6) in (A8), one finds, after carrying out algebraic transformations, a differential equation that can be solved analytically to:

where

a, b and

d are constants that have been introduced for better handling of the equations:

and

k_{1} can be determined by solving the initial value problem at

t = 0:

Using (A6) and (A9) in (A2) gives an equation for the outlet temperature of the DHW storage:

Figure A1b shows (A12) fitted with

α_{nom} = 484 Wm

^{−2} K

^{−1} to measured data from a charging experiment with an enerboxx

^{®®} storage. Furthermore, the presented approach is compared to a lumped parameter approach, where a perfectly mixed water volume was assumed. The behavior of the outlet temperature in the region of interest (low temperatures indicated by the strong solid line of the moving boundary approach in

Figure A1b) can be reproduced significantly better with the presented approach. Nevertheless, the moving boundary method will always underestimate the real temperature whereas the lumped parameter approach will always overestimate the real temperature.

Inserting (A12) in (A2) gives an equation for the heat transfer rate to the DHW storage dependent on time:

The heat transfer at the condenser is limited by two constraints. Firstly, by the set point temperature

ϑ_{sp} (outlet temperature of the water after the condenser) and secondly by the minimum heat transfer that is provided at the minimal rotational speed of the compressor

${\dot{Q}}_{\mathrm{con},\mathrm{msc}}$. The compressor turns off, once

${\vartheta}_{\mathrm{DHW},\mathrm{msc}}^{\left(\mathrm{out}\right)}$ at the minimum speed of the compressor is reached:

Inserting (A14) in (A12) and numerical rearrangement gives an equation for the time when the minimum rotational speed is reached:

With (A2), (A3) and the considerations discussed before, one finds an equation for the energy transferred over the condenser during operation mode (c) depending on the time

t:

If τ_{msc} from (A15) is negative as in the first case of (A16), the HP is always off because the outlet temperature of the DHW storage is already above ${\vartheta}_{\mathrm{DHW},\mathrm{msc}}^{\left(\mathrm{out}\right)}$ at the beginning (t = 0). Therefore no energy is transferred directly from the refrigerant to the process water (Q_{con} = 0, Q_{RPW,R} = 0) and all the energy for the storage has to be provided by the RPW-HEX. If the outlet temperature of the DHW storage is always below ${\vartheta}_{\mathrm{DHW},\mathrm{msc}}^{\left(\mathrm{out}\right)}$, 0 ≤ t < τ_{msc}, as in the second case of (A16) the HP is turned on all the time and the compressor is controlled to provide always as much energy as needed to reach the set-point ϑ_{sp} at the condenser water outlet. If the HP reaches its minimum power within the time of the operation 0 ≤ τ_{msc} ≤ t, the HP will be turned off at τ_{msc} and the energy is then provided by the RPW-HEX, only.

The energy transferred from the refrigerant to the hot water over the RPW-HEX follows by integrating (A4) and finally the energy extracted from the stored energy in the RPW-HEX follows from integrating (A5) to:

From (A9) one can calculate the time

τ_{DHW} needed to charge the DHW storages to a certain energy set-point

${Q}_{\mathrm{DHW}}^{\mathrm{sp}}$ (e.g., the upper limit) analytically:

Inserting τ_{DHW} from (A18) as t in (A17) gives finally the energy Q_{RPW,P} that has to be extracted from the RPW-HEX to charge the DHW-storage to this certain set-point ${Q}_{\mathrm{DHW}}^{\mathrm{sp}}$.

Figure A2 shows the contribution of the refrigerant and the PCM to the hot water generation in operating mode (c) over the time at a typical operation condition (

ϑ = 0 °C,

ϑ_{sp} = 43 °C,

ϑ_{c} = 12 °C,

${\vartheta}_{\mathrm{DHW}}^{\left(\mathrm{in}\right)}$ = 60 °C, 210 L storage, standard-mode). Due to the increasing temperature of the water leaving the storage

${\vartheta}_{\mathrm{DHW},\mathrm{msc}}^{\left(\mathrm{out}\right)}$ (decreasing cf.

Figure A1a), the heat transfer rate decreases with time (cf.

Figure A2a). Additionally, due to the restriction of the fixed set-point for the HP (

ϑ_{sp} = 43 °C), the heat provided by the condenser has to decrease. Hence, the HP reaches the minimum operation speed of the compressor at

τ_{msc} and the compressor turns off. Until the DHW storage is fully charged at

τ_{DHW}, the remaining energy must be provided by the PCM. At the end of the charging process, about one third of the energy for providing DHW at 60 °C was taken from the HP operating with a heating

COP and about two third of energy were taken from the energy stored in the RPW-HEX (which was also stored when the HP was operated in heating mode with a high COP (cf.

Figure A2b).

**Figure A2.**
(**a**) Calculated heat transfer rate to the decentralized DHW storages during the energy efficient DHW charging operating mode (c) for a storage at the lower charging limit in standard operating mode at t = 0. The heat transfer rate consists of contributions from the refrigerant via the condenser (green) and the RPW-HEX (blue) and of the contribution from the PCM and the aluminum in the RPW-HEX (orange). (**b**) shows the calculated total amount of energy transferred to the DHW storages during the charging process.

**Figure A2.**
(**a**) Calculated heat transfer rate to the decentralized DHW storages during the energy efficient DHW charging operating mode (c) for a storage at the lower charging limit in standard operating mode at t = 0. The heat transfer rate consists of contributions from the refrigerant via the condenser (green) and the RPW-HEX (blue) and of the contribution from the PCM and the aluminum in the RPW-HEX (orange). (**b**) shows the calculated total amount of energy transferred to the DHW storages during the charging process.

The time τ_{RPW} to extract a certain amount of energy from the RPW-HEX can be calculated numerically, e.g., with a Newton-solver, by inserting (A9) in (A17). Note that this is only possible if the energy stored in the RPW-HEX is lower than the maximum transferable energy from the RPW-HEX to the DHW storage which is limited by the DHW storage size or the defined maximum charging level of the DHW storage, respectively. If this is the case, the charging time can be calculated with (A18) and the remaining difference of energy remains in the RPW-HEX after the DHW-charging operation process.