Introduction

This project aims to answer the question how much energy can be saved if (part of) the thermal energy carried by the water going into the drain of a shower is recovered? Consider the scheme shown in where a person is taking a shower. This person configures the tap such that a temperature T_shower that he likes comes out from the shower. This is achieved by mixing hot and cold water at temperatures T_hot and T_cold respectively. The water is used by the individual taking the shower and is afterwards disposed through the drain at a temperature T_drain, as shown in the figure.

A scheme of a person in a shower.

Due to the nature of the showering process, the water is actually used for a very brief time and usually T_drain is still substantially higher than T_cold. This poses a motivation to recover part of the thermal energy carried by the water flowing into the drain to be reused. One possible way of recovering this thermal energy is with the addition of a heat exchanger as shown in . Now, a heat exchanger is added to recover the thermal energy flowing into the drain and bring it back into the shower using the cold water source.

A scheme of a person in a shower using a heat exchanger to recover part of the thermal energy carried by the water through the drain.

In the following sections this approach will be studied and an estimation of the energy savings that can be achieved will be presented.

Analysis of the problem

In this section the problem will be analyzed from a physical point of view.

Regular shower scheme

Let's start with the regular shower scheme illustrated in . The temperature T_shower is achieved by mixing some amount of hot and cold water. When multiple fluids at different temperatures are mixed the final temperature is given by  $$T=\frac{\sum_{i}T_{i}c_{i}m_{i}}{\sum_{i}m_{i}c_{i}}$$ where $T_i$ is the temperature of each fluid, $m_i$ the mass of each fluid and $c_i$ the specific heat. Differentiating with respect to time and assuming a steady state flow in which the temperatures $T_i$ are constant the previous equation converts into $$T=\frac{\sum_{i}T_{i}\dot{m}_{i}c_{i}}{\sum_{i}\dot{m}_{i}c_{i}}$$ where $\dot{m}_{i}=\frac{dm_{i}}{dt}$ is the mass flow rate of each fluid. For the simple case of the hot and cold water shown in the equation becomes simply $$T_{\text{shower}}=\frac{T_{\text{hot}}\dot{m}_{\text{hot}}+T_{\text{cold}}\dot{m}_{\text{cold}}}{\dot{m}_{\text{hot}}+\dot{m}_{\text{cold}}}$$ When $T_\text{hot}$ and $T_\text{cold}$ are fixed, as is usually the case in these scenarios, then the way to change $T_\text{shower}$ is to change the mass flow rates of hot and cold water, as we all know from experience.

Defining $\dot{M}=\dot{m}_{\text{hot}}+\dot{m}_{\text{cold}}$ as the total flow rate, i.e. the total amount of water coming out from the shower, then from we can obtain the required amount of hot water as a function of the temperatures and the total mass flow rate in the shower: $$\dot{m}_{\text{hot}}=\dot{M}\frac{T_{\text{shower}}-T_{\text{cold}}}{T_{\text{hot}}-T_{\text{cold}}}$$ It is important to note that this equation is valid only for $T_{\text{cold}}\leq T_{\text{shower}}\leq T_{\text{hot}}$. As expected, we see that the closer $T_\text{shower}$ is to $T_\text{hot}$ the higher the amount of hot water.

Shower with heat exchanger

Let's now move on to the modified scheme with a heat exchanger shown in . Before this, a brief formulation of how a generic heat exchanger works will be presented.

Heat exchanger

Consider the heat exchanger shown in . In this heat exchanger there are two lines, one through which the hot fluid enters at a rate $\dot{m}_{\text{hot}}$ and temperature $T_{\text{hot}}$, the other through which the cold fluid enters at a rate $\dot{m}_{\text{cold}}$ and temperature $T_{\text{cold}}$. During their passage through this heat exchanger these fluids exchange heat (obviously) and they come out at temperatures $T_\text{hot final}$ and $T_\text{cold final}$, as shown in the scheme.

A generic heat exchanger.

In the steady state, i.e. when the flow rate of the fluids is constant and all the transients are already stabilized, the thermal energy rate transferred from the hot fluid into the cold fluid can be expressed as $$\dot{Q}=\dot{m}_\text{cold}\intop_{T_\text{cold}}^{T_\text{cold final}}c_\text{cold}\,dT$$ where $c_\text{cold}$ is the specific heat of the cold fluid. The maximum transferred heat will be obtained when $T_{\text{cold final}}=T_{\text{hot}}$, which could be possible for example if the two pipes are in thermal contact for a long enough distance and all heat loses can be neglected. This would be the exchanged heat in an ideal heat exchanger, $$\dot{Q}_{\text{max}}=\dot{m}_{\text{cold}}\intop_{T_{\text{cold}}}^{T_{\text{hot}}}c_{\text{cold}}\,dT$$ We can then express the transferred heat as a function of $\dot{Q}_{\text{max}}$ and some efficiency $\eta$ of the heat exchanger which will depend upon its implementation details: $$\dot{Q}=\eta\dot{Q}_{\text{max}}$$

Combining equations , and we can calculate $T_\text{final cold}$ from $$\intop_{T_{\text{cold}}}^{T_{\text{cold final}}}c_{\text{cold}}\,dT=\eta\intop_{T_{\text{cold}}}^{T_{\text{hot}}}c_{\text{cold}}\,dT$$

Since the specific heat of water at atmospheric pressure is roughly constant, the previous equation simplifies to $$T_{\text{cold final}}=T_{\text{cold}}+\eta\left(T_{\text{hot}}-T_{\text{cold}}\right)$$ This simple equation allows to express the final temperature of the cold fluid as a function of the other known temperatures and the efficiency of the exchanger.

Introducing a heat exchanger in the shower

Now let us study how the amount of required hot water changes when a heat exchanger is introduced as shown in . We start from but this time the temperature of the cold water is given by the temperature in the output of the heat exchanger, i.e. $T_{\text{cold}}\leftarrow T_{1}$, so $$\dot{m}_{\text{hot}}=\dot{M}\frac{T_{\text{shower}}-T_{1}}{T_{\text{hot}}-T_{1}}$$ Now, the temperature at the output of the heat exchanger can be expressed using replacing the appropriate temperatures according to , i.e. $$T_{1}=T_{\text{cold}}+\eta\left(T_{\text{drain}}-T_{\text{cold}}\right)$$ and so replacing into we obtain $$\dot{m}_{\text{hot}}=\dot{M}\frac{T_{\text{shower}}-\eta T_{\text{drain}}-\left(1-\eta\right)T_{\text{cold}}}{T_{\text{hot}}-\eta T_{\text{drain}}-\left(1-\eta\right)T_{\text{cold}}}$$ This equation expresses the required amount of hot water as a function of all the parameters.

Let's assume that $T_{\text{drain}}\approx T_{\text{shower}}$, which in the steady state is reasonable since the water is only used for a brief time and rapidly flows into the drain. In this case reads $$\dot{m}_{\text{hot}}\approx\dot{M}\frac{\left(1-\eta\right)\left(T_{\text{shower}}-T_{\text{cold}}\right)}{T_{\text{hot}}-T_{\text{cold}}-\eta\left(T_{\text{shower}}-T_{\text{cold}}\right)}$$

Savings in energy when using a heat exchanger

In this section it will be calculated how much energy is saved when using a heat exchanger as compared with the regular case.

The required power to heat water for a shower can be calculated as $$P=\dot{m}_{\text{hot}}\intop_{T_{\text{cold}}}^{T_{\text{hot}}}c\,dT$$ where it was assumed that the water is heated from the same source as the cold water, i.e. the starting temperature is $T_\text{cold}$. Note that since $\dot{m}_{\text{hot}}$ is a flow rate then so $P$ is an energy rate, i.e. a power. From this equation we see that $$\frac{P_{\text{with heat exchanger}}}{P_{\text{regular shower}}}=\frac{\dot{m}_{\text{hot with heat exchanger}}}{\dot{m}_{\text{hot regular shower}}}$$ i.e. the fraction of saved power is the same as the fraction of saved hot water.

The fraction of saved hot water can be computed by dividing by . This gives $$\frac{\dot{m}_{\text{hot with heat exchanger}}}{\dot{m}_{\text{hot regular shower}}}=\frac{\left(1-\eta\right)\Delta_{\text{source}}}{\Delta_{\text{source}}-\eta\Delta_{\text{shower}}}$$ where $\Delta_{\text{source}}=T_{\text{hot}}-T_{\text{cold}}$ and $\Delta_{\text{shower}}=T_{\text{shower}}-T_{\text{cold}}$ were defined. It can be seen that when $\eta=0$, which is equivalent to having no heat exchanger, the ratio is 1 which means that there is no saving at all, as expected. Likewise, as $\eta\to 1$ the ratio goes to 0 meaning that an ideal heat exchanger will recover all the heat flowing into the drain and so a tiny amount of hot water (or equivalently heating power) would be required. Of course in reality an efficiency of 1 is never achieved.

In a plot of is shown. As seen, the lower the temperature setting for the shower the higher the benefit. Typical efficiencies of heat exchangers lie between 70-90 %  which implies savings in the range of 50-80 %.

Impact for Zürich

In this section the impact of adding a heat recovery system in the shower of residential homes will be estimated.

Impact in a single household

We start by calculating the required energy to heat up the water of one shower using the regular scheme from , which is given by $$\text{shower energy}=\intop_{t_{\text{start}}}^{t_{\text{stop}}}\dot{m}_{\text{hot}}\,dt\,\intop_{T_{\text{cold}}}^{T_{\text{hot}}}c\,dT $$ with $t$ being the time. $\dot{m}_{\text{hot}}$ can be obtained from . In the steady state this reduces to $$\text{shower energy}\approx\dot{M}\Delta_{\text{shower}}t_{\text{shower}}c_{\text{water}}$$ where $t_\text{shower}$ is the time it takes to take the shower.

In order to continue with the calculation, the different parameters such as temperatures and water flow rates have to be determined. These were measured by the authors at their residences, which are regular rented apartment in the greater Zürich area. The measured parameters are

• T_cold = 15 °C
• T_hot = 54 °C
• T_shower = 38 °C
• T_drain = 36 °C
• Shower flow rate = 0.09 l s^-1
• Shower time = 14 min

The volumetric shower flow rate can be converted into the shower mass flow rate $\dot{M}$ with the density of the water which is approximately 1 g ml^-1 . This gives a shower mass flow rate of 0.09 kg s^-1. The specific heat $c_{\text{water}}$ is, under the conditions considered here, approximately constant with a value of 4184 J kg⁻¹ K⁻¹ .

Inserting all the numbers into we obtain $$\text{shower energy}\approx7.3\times10^{6}\text{ J}=2\text{ kWh}$$ which is in agreement with other sources. As seen in this depends on the temperature of the cold water so it could vary in different regions and will probalby be higher during the winter.

From we see that the energy is proportional to the hot water mass flow rate. Thus $$\frac{\text{shower energy with heat exchanger}}{\text{shower energy}}=\frac{\dot{m}_{\text{hot with heat exchanger}}}{\dot{m}_{\text{hot regular shower}}}$$ so we can obtain the savings in energy using , i.e. $$\text{shower energy with HE}=\text{shower energy without HE}\frac{\left(1-\eta\right)\Delta_{\text{source}}}{\Delta_{\text{source}}-\eta\Delta_{\text{shower}}}$$ shows a plot of this equation using the parameters values previously determined. We see that for high efficiencies of the heat exchanger the savings are drastic.

Impact for the city

In order to estimate the impact for the city of Zürich we can proportionally scale the results from . The city of Zürich has approximately 4.3×10⁵ inhabitants . Let's assume that each of them takes a shower (similar to the one studied in ) once every two days. This amounts to approximately 7.8×10⁷ showers per year which in terms of energy is approximately 1.5 GWh per year. By installing a heat exchanger in each household with an efficiency of around 70 % this consumption can be cut by 50 %, i.e. 750 MWh per year.

Considering the 20 kWH energy consumption for a single shower (from ), and assuming that an individual takes a shower once every two days, then the averaged power used for showering by this individual is about 420 W. This is almost a quarter of the 2000 W per capita quota that Zürich is aiming to achieve as part of its 2000 W Society initiative . This further motivates the introduction of an efficient energy recovery system for household showers.

Conclusions

It was demonstrated that the introduction of a heat recovery system in domestic showers can lead to drastic reductions in the consumption of hot water. The translation into a reduction of energy for heating up the water was quantified and compared against the 2000 W limit initiative of the city of Zürich. According to our estimation almost a fourth part of this limit is used only for showering when no heat recovery systems are in use. The introduction of a heat recovery system could reduce this component by 40-60 %.