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This paper presents a new air-heating system concept for energy-efficient dwellings. It is a system designed to heat a low-energy building by coupling a heat-recovery ventilation system with a three-fluid heat exchanger located on the chimney of a wood-pellet stove. The proposed work focuses on the heat transfer that occurs between flue gases, the ventilation air and the combustion air within a triple concentric tube heat exchanger with no insulation at its outer surface. The main objective is to predict outlet temperature for the specific geometry of the heat exchanger studied here. Thus, the governing differential equations are derived for a counter-co-current flow arrangement of the three fluids. Then analytical solutions for the steady-state temperature distribution are obtained as well as the amount of heat transferred to the outside. An expression for the effectiveness of the heat exchanger is also proposed. Based on these results, calculations are performed on a case study to predict the fluid temperature distribution along the heat exchanger. Finally, a parametric study is carried out on this case study to assess the influence of the relevant parameters on the effectiveness of the heat exchanger. In addition, computation of heat losses to the outside justifies whether insulation is needed.

Low energy buildings are characterized by a suitable orientation, a high thermal insulation and a very low air leakage [

However, wood burning appliances present the disadvantage of concentrating heat only in the room where they are installed. Therefore, systems mechanically extracting hot air around the device are often used to redistribute heat to other rooms in the house. Nevertheless, the Consumer Safety Commission [

Instead of taking hot air above the heating appliance, as do conventional systems, the combined system presented in this study recovers heat by blowing fresh air into a specific heat exchanger, which is both integrated into the chimney of a wood pellet stove and connected to the ventilation air supply network. Hence, while ensuring occupant safety, this combined system aims to meet the requirements of both heating and indoor air quality in energy efficient dwellings. A detailed description of this combined system comprising of a room-sealed wood pellet stove (RSWPS), a mechanical ventilation heat recovery (MVHR) and a triple concentric tube heat exchanger (TCTHE), is given in [

Considering that ideal configuration should combine all the advantages of the RSWPS and the MVHR while also introducing the TCTHE to better distribute heat in the house, the authors recommend coupling the three main components of the combined system as it is shown in

The main objective of this work is to develop a theoretical model of a triple concentric tube heat exchanger in order to compute steady-state temperature distribution of the three fluids namely flue gases, ventilation air and combustion air. This will provide heat fluxes and effectiveness of the heat exchanger.

Scheme of the combined system configuration.

As described on the diagram in

Diagram of the TCTHE.

After the first analysis performed by Morley [

As depicted in

The theoretical analysis of the TCTHE-NI in a specific combination of counter-co-current flow arrangement, as defined in [

The system operates under steady-state conditions;

Phase change does not take place;

Bulk mean temperatures only depend on axial (

The thermo-physical properties of the fluid streams are constant and uniform;

Thermal conduction is not assumed in fluids or walls parallel to the fluid flow direction;

Heat transfer coefficients are independent of time and axial position;

Temperature of the outside remains constant.

Physical model for the TCTHE-NI.

Under the above assumptions, the energy balance on a control volume of length _{1} = _{12(a)} _{2} = _{12(a)} + _{32(b)} _{3} = − _{32(b)} − _{30(c)}

In Equations (1) and (3), the differential heat flow rates _{1} and _{3} represent the heat lost by hot fluids 1 and 3 between locations _{2} denotes the heat retrieved by the cold fluid 2 between locations _{p}

In the meantime and on the same differential control volume, the differential heat flow rates _{ij(k)}_{i}_{j}_{k}_{k}_{ij(k)}_{k}dA_{k}_{i}_{j}_{k}_{k}dx_{k}dx_{k}dx

According to an electrical analogy, the thermal circuit presented in _{w}_{r}

Electrical analogy of the combined modes of heat transfer in the TCTHE-NI.

Concerning radiation, it should be noticed that flue gases, ventilation air and combustion air are non-participating fluids. Here, while maintaining the same radiant thermal resistance, the expression of the heat transfer by radiation in the annuli is assumed to be split in two parts involving the bulk temperature of the fluid between tube walls. In this way, the thermal resistance _{k}_{k}_{k}_{int} = 2 π _{k}_{int}

Taking the example of this study where fins are added on the external wall of the inner tube _{o}_{2a} should be replaced in Equation (7) by _{2a f}:

where nb_{f}_{f}_{tot}_{f}_{f}

Then, the following expressions are used to assess the various coefficients that describe heat transfers by conduction, convection and radiation.

Conductive heat transfer coefficient

Through a cylindrical wall _{wk}_{wk}_{k int}_{k}_{k}_{ext}/_{k}_{int} and heat exchanger length

Convective heat transfer coefficient

The convective heat transfer coefficient _{i}_{i}_{h}_{i}_{i} / D_{h}_{h}_{outer} − _{inner} for an annular space between two (inner and outer) cylindrical walls. Then the average convection coefficients are determined thanks to correlations from the literature.

For forced convection laminar flow (_{h}^{1/3}

For forced convection transition and turbulent flows (

^{4/5}

^{1/3}Annulus:

^{4/5}

^{1/3}(

_{outer}/

_{inner})

^{0,53}for the inner wall

^{4/5}

^{1/3}for the outer wall

For free convection, the following correlations [^{4} ≤ ^{9}
^{1/4}
Turbulent flow:
10^{9} ≤ ^{13}
^{1/3}

According to the three flow rates and the three tubes diameters studied here and presented in

Radiant heat transfer coefficient

The radiant heat transfer coefficient _{r kl}_{co}_{c}

and:

and where ∀

Then, considering the differential surface areas and the overall heat transfer coefficients given by Equations (6)–(9), the set of governing Equations (1)–(3) can thus be written in the following form:
_{1} dT_{1} = _{a} dA_{a}_{1} − _{2}) _{2} dT_{2} = _{a} dA_{a}_{1} − _{2}) + _{b} dA_{b}_{3} − _{2}) _{3} dT_{3} = −_{b} dA_{b}_{3} − _{2}) + _{c} dA_{c}_{3} − _{0})

To provide a formulation with non-dimensional parameters and for brevity, following definitions are used:

Hence, the set of governing equations can be written in the form:
_{1}(1) = _{1 in} _{2}(0) = _{2 in} _{3}(0) = _{3 in}

By combining Equations (17)–(19), the following set of third order ordinary differential equations is obtained:
_{0} + _{1}(_{r}_{1} − 1) + _{3}(_{r}_{3} + 1), _{1}(_{0} + _{3})(_{r}_{1} − 1) + _{3}_{r}_{3}(_{0} − _{1}), _{0}_{1}_{3}_{r}_{3}

General solutions of the above system of 3 linear homogeneous third order ordinary differential Equation (21) are in the form of a sum of exponential functions in _{n}_{n}_{n}_{1}, λ_{2} and λ_{3} are solutions of the characteristic Equation (23):
^{3} + ^{2} +

Cardano’s method is used to solve the cubic Equation (23). Thus, the quadratic term is eliminated and the so-called ^{3} + ^{2} +

Considering the range of variation of the parameters used in this study, the discriminant Δ of the reduced form is always strictly negative:

Hence, Equation (24) has three real roots:

Then, using Equations (17)–(19), the boundary conditions given by Equation (20) and the form of the solutions given by Equation (22) yields a set of three equations for each fluid:
_{1} = _{1} + _{1} + _{1} _{1}(1) = _{1}^{λ1} + _{1}^{λ2} + _{1}^{λ3} _{2}(0) = _{2} + _{2} + _{2} _{3}(0) = _{3} + _{3} + _{3}

After some manipulations combining equations, coefficients α_{n}_{n}_{n}_{n}_{1}(0):
_{n}_{n}^{'}_{n}^{"}_{n}_{n}^{'}_{n}^{"}_{0} , _{1} , _{3} , _{r}_{1} , _{r}_{3} , λ_{1} , λ_{2} , λ_{3} , _{1}(1), _{2}(0) and _{3}(0) (cf.

Applying _{1}(_{2}(_{1}(_{1}, _{2}, _{3} , _{4}, _{5} and _{6} are coefficients which can be expressed in terms of λ_{1}, λ_{2}, λ_{3}, _{1}, _{n}_{n}^{'}_{n}^{"}_{n}_{n}^{'}_{n}^{"}

Equations (40) and (41) yield the following expression for the unknown _{1}(0):

Thus, for _{n}_{n}_{n}_{n}_{n}_{n}_{0} + Δ_{in} _{n}

Outlet temperatures of the three fluids are computed from:
_{1 out} = _{1}(0) = _{0} + Δ_{in} _{1}(0) _{2 out} = _{2}(1) = _{0} + Δ_{in} _{2}(1) _{3 out} = _{3}(1) = _{0} + Δ_{in} _{3}(1)

The heat rate _{30(c)} transferred across the non-adiabatic outside surface can be now determined through the integration of _{3}(

For _{n}_{n}_{n}_{out} − _{in})_{n}

Considering an analogy with a two-fluid heat exchanger, the overall effectiveness _{actual} to the maximum possible heat transfer rate _{max}:

As recommended by Sekulic and Kmecko [_{actual}, is recovered by the fluid 2:
_{actual} = _{2} = _{2}(_{2 out} − _{2 in})

Then, the maximum possible heat transfer rate, _{max}, that could be recovered by the fluid 2, from both fluids 1 and 3, can be expressed as:
_{max} = _{min (1;2)} Δ_{max (1;2)} + _{min (3;2)} Δ_{max (3;2)} ^{2} / _{min (i;j)} = min(_{i}_{j}_{max(i;j)} = max(_{iin};_{jin})−min(_{iin};_{jin}).

It is worth mentioning that owing to the fluids properties considered in this study [8], in any case _{max} is given by:
_{max} = _{1} (_{1 in} − _{2 in}) + _{3} (_{3 in} − _{2 in})

Thus, heat exchanger effectiveness

Using previous notations, the heat exchanger effectiveness

Since in any case _{2} > _{1} + _{3}, the non-dimensional parameter _{min} / _{max} =(_{1} + _{3})/_{2} = _{r1} + _{r3}

Finally, the heat exchanger effectiveness _{2out} − _{2in})/Δ_{in}

The input data used in the parametric study are given in

Input data and outlet temperatures for both TCTHE-NI and TCTHE-PI models (

_{0} = 25 °C |
Fluid 1 | Fluid 2 | Fluid 3 | ||
---|---|---|---|---|---|

FG | VA | CA | |||

Input data | _{a, b, c} |
(mm) | 80 | 180 | 230 |

_{in 1, 2, 3} |
(°C) | 180 | 15 | 60 | |

_{1, 2, 3} |
(W/K) | 10 | 20 | 5 | |

Computed data | _{a, b, c} |
(W/(m^{2} K)) |
5.0 | 3.2 | 2.0 |

TCTHE-NI | _{out 1, 2, 3} |
(°C) | 153.09 | 31.6 | 38.2 |

TCTHE-PI | _{out 1, 2, 3} |
(°C) | 153.13 | 32.1 | 45.2 |

The overall heat transfer coefficients _{c} is equal to zero. The last two rows of

The heat capacity rate values, noted C, correspond to flow rates of about 35, 130 and 15 m^{3}/h for the flue gases, the ventilation air and the combustion air respectively. These values were measured during laboratory tests and ventilation air flow rate meets the regulatory requirements of the ventilation of dwellings in France. Numerical calculations have been carried out for both TCTHE-NI and PI models to predict the temperature distributions of the three fluids along the heat exchanger (

Temperature distribution along the TCTHE-NI and PI (

Then, the three graphs in _{c} ≠ 0), TCTHE-NI (_{c} = 0) and TCTHE-PI models. Each graph totalizes on its left part the heat flux assigned by the flue gases and by the combustion air, and on its right part the heat flux retrieved by the ventilation air and by the outside in case of no insulation. As expected, the algebraic sum of _{OUT}, _{FG}, _{VA} and _{CA} is always equal to zero. Thus, the heat balance is verified for all models studied. Moreover, if the overall heat transfer coefficient of the external wall tube is taken very close to zero, like here with _{c} = 1.10^{−10} W/(m^{2} K) for failing to divide by zero and lead to errors in the code, it must be emphasized that results of the TCTHE-NI (_{c} = 0) and TCTHE-PI models are in perfect agreements.

Heat balance for the three TCTHE models (

Although heat losses to the surroundings represent about 12% of the total heat transferred in the case of the TCTHE-NI, it is worth mentioning that the heat recovered by the ventilation air is only about 3% lower than in the case of the TCTHE-PI. Indeed and as expected, it is the amount of heat assigned by the combustion air to the surroundings which is mainly affected depending on whether the heat exchanger is insulated from the outside or not. Thus, as this study is conducted with an industrial approach, adding a fourth tube containing thermal insulation is not a relevant solution to increase the heat recovered by the ventilation air. In addition, efficiency of the pellet stove is not degraded because the combustion air temperature increases highly between the outlet of the heat exchanger and the inlet of the stove. Indeed, just behind the appliance, the combustion air recovers heat directly from flue gases in the concentric tube used for connecting the chimney with the stove.

As the amount of heat recovered by the ventilation air is quite similar for both models, as well as there are identical values for _{max} = 1875 W and

It should also be emphasized that the temperature distribution appears to be quasi-linear along the exchanger on the graph in

Temperature distribution along the TCTHE-NI and PI (

On the graph in

Of course, such a 10m length heat exchanger is not really appropriate for the combined system studied here. However, it should be stressed that in this case again, heat losses to the surroundings represent about 12% of the total heat transferred by both the flue gases and the combustion air. Here, as the maximum heat transfer rate is still the same, _{max} = 1875 W, the effectiveness of the triple concentric tube heat exchanger reaches 53.0% if there is no insulation from the surroundings and 56.6% when perfectly insulated from the outside.

Outlet temperatures for TCTHE-NI and TCTHE-PI models (

_{0} = 25 °C |
Fluid 1 | Fluid 2 | Fluid 3 | ||
---|---|---|---|---|---|

FG | VA | CA | |||

Computed data | _{a, b, c} |
(W/(m^{2} K)) |
5.0 | 2.5 | 1.5 |

TCTHE-NI | _{out 1, 2, 3} |
(°C) | 76.5 | 64.7 | 42.0 |

TCTHE-PI | _{out 1, 2, 3} |
(°C) | 77.5 | 68.0 | 53.0 |

In this study, fins could be added on the external wall of the inner tube to increase the amount of heat recovered by the ventilation air on the flue gas side. Considering the original set of input data for the TCTHE-NI model, the thickness, width and length of fins are 0.4 mm, 45 mm and 1160 mm, respectively. The fins are located in the longitudinal direction of the flow and their volume represent less than 2% of the total ventilation air flow volume. So, we consider initially the flow is not affected significantly by adding fins.

Several values of _{2} are considered to reflect the various flow rates of the ventilation air that may be encountered in the heat exchanger,

Effectiveness as a function of the number of fins for the TCTHE-NI model with the original set of input data and with ventilation air flow variations.

Considering the original set of input data presented in _{2}, _{b}, _{2 in} and _{0}. Thus, the four graphs in

For each graph, several values of NTU are obtained by varying the length of the heat exchanger between 1 and 30 m. However, it is worth noting that the NTU value is always lower than 0.5 when calculations are carried out with the original set of input data defined in this study.

First, _{2}. Indeed, when the mass flow rate of the fluid2increases,itcausesbotha decrease of the non-dimensional parameter _{2 int} and _{2 ext} in the first annulus. As in the case of two-fluid heat exchangers, the effectiveness value is minimum for _{2} has a fairly significant influence on the effectiveness, even for low values of NTU.

Effectiveness charts for the TCTHE-NI model.

Then, as shown in _{b} decreases. In simple words, reducing the flow area for the fluid 2 results in a significant increase of the convective heat transfer coefficients _{2 int} and _{2 ext} in the first annulus and this effect overrides the negative effect of the decrease in heat transfer surface area with fluid 3. Here again, the choice of the size of _{b} has a significant influence on the effectiveness, even for low values of NTU and this observation is consistent with the findings of Ünal [

_{2 in} decreases, _{in} increases. Considering the specific combination of the counter-current co-current flow arrangement studied here, the actual heat recovered by the fluid 2 increases at a greater rate than the maximum heat recoverable. Nevertheless, it is of some interest to note that the variation of _{2 in} has a relatively limited influence on the effectiveness, especially for low values of NTU.

When the external temperature _{0} increases, _{0} has little influence on the effectiveness for low NTU values, but its impact becomes more significant for higher NTU values.

The analysis of the four graphs in _{2} or _{b} than by parameters _{2 in} or _{0}.

The heat transferred to the outside, _{OUT}, may be compared to the total heat assigned or retrieved, _{TOT} = _{VA} + _{OUT} = _{FG} + _{CA}, in the TCTHE-NI. _{OUT}/_{TOT} with increasing NTU values. Of course, this ratio is always equal to zero in case of perfect insulation (PI). But depending on the surroundings temperature, _{0}, heat losses to the outside may represent a significant part of heat transfer that occurs in the exchanger.

Considering the original set of input data, _{0} > 10 °C, reflecting the majority of cases that may be encountered in our application. Indeed, in the case of the combined system studied here, the TCTHE-NI is commonly located just behind the heating appliance in a closet in which the temperature remains between 15 °C and 25 °C.

Percentage of heat transferred to the surroundings for the TCTHE-NI model with the original set of input data (

In addition, as shown in

Analytical solutions for the steady-state temperature of three heat exchanging fluids along the length of a triple concentric tube heat exchanger with no insulation at the outer surface are obtained for a specific combination of counter-current and co-current arrangement. The amount of heat transferred to the outside through the external tube is also calculated. For a case study, the temperature distribution of the three fluids is graphically represented and its exponential variation is stressed by increasing the length of the heat exchanger. Heat balance calculations and comparison with the perfectly insulated model have provided validation of the mathematical model developed in this paper. With the input data values of the parametric study conducted here, the temperature of the ventilation air increases by about 15 °C in the heat exchanger while the temperature of flue gases and combustion air decreases about 30 °C and 20 °C respectively. Then the importance of adding fins to the external surface of the inner tube of the TCTHE-NI is pointed out, with a potential increase of about 6% for eight fins, and the influence of several parameters, especially _{2} and _{b}, on the heat exchanger effectiveness is also assessed through a parametric study conducted in the continuity of the case study. Heat losses to the outside represent around 10% of the total heat transferred in the TCTHE-NI for the set of input data used in the case study, so insulation of the heat exchanger is not really required if it is placed in the heated volume of the dwelling.

Finally, this theoretical study could be extended to the various flow arrangements encountered in a triple concentric tube heat exchanger and the equations derived here could also be used to perform sizing calculations. In addition, we are aware that this model developed in this study is quite simple but comparison between experimental and numerical results showed that heat transfer rates are predicted with a relative difference lower than 5% [

heat transfer area (m^{2})

_{p}

specific heat capacity at constant pressure [J/(kg·K)]

heat capacity rate (J/K)

_{r}

dimensionless heat capacity ratio (-)

differential heat transfer area (m^{2})

differential heat flow rate (W)

axial discretization step (m)

diameter (m)

_{h}

hydraulic diameter (m)

thickness (m)

effectiveness (%)

convective heat transfer coefficient [W/(m^{2}·K)]

_{r}

radiant heat transfer coefficient (W/(m^{2} K))

_{w}

conductive heat transfer coefficient [W/(m^{2}·K)]

length (m)

mass flow rate (kg/s)

dimensionless ratio of overall conductance

Nusselt number (-)

number of transfer units (-)

perimeter (m)

Prandtl number (-)

radius (m)

Rayleigh number (-)

Reynolds number (-)

temperature (K)

overall heat transfer coefficient [W/(m^{2}·K)]

dimensionless length coordinate (-)

axial coordinate (m)

heat capacity rate ratio (min/max) (-)

emissivity (-)

thermal conductivity [W/(m·K)]

Stephan-Boltzmann constant [5.67 × 10^{−8} W/(m^{2}·K^{4})]

heat flow rate (W)

_{f}

fin efficiency (-)

_{o}

overall surface efficiency (-)

dimensionless temperature (-)

temperature difference (K)

_{in}

temperature difference depending on inlet temperatures and heat capacity rates (K)

ambient air (surroundings)

first fluid (flue gases)

second fluid (ventilation air)

third fluid (combustion air)

internal tube

intermediate tube

external tube

outside (surroundings)

fin(s)

inlet

outlet

external

internal

inner

outer

actual

mean

maximum

minimum

total

flue gases

ventilation air

combustion air

outside

total

room-sealed wood pellet stove

mechanical ventilation heat recovery

triple concentric tube heat exchanger

triple concentric tube heat exchanger with perfect insulation

triple concentric tube heat exchanger with no insulation

double concentric tube heat exchanger with no insulation

This work was supported in part by the Poitou-Charentes Regional Council and by the Calyxis cluster of expertise in risk.

Coefficients _{n} , _{n} and _{n} are given by Equations (36)–(38):
_{n}_{n}^{'}_{n}^{"}_{n}_{n}^{'}_{n}^{"}_{0}, _{1}, _{3}, _{r}_{1}, _{r}_{3}, λ_{1}, λ_{2} , λ_{3}, _{1}(1), _{2}(0) and _{3}(0).

As presented in Equations (22) and (39), general solutions of the system of 3 linear homogeneous third order ordinary differential equations are in the form of a sum of exponential functions in _{1}, _{2}, _{3}, _{4} , _{5} and _{6} are coefficients which can be expressed in terms of _{1}, _{2}, _{3}, _{1}, _{n}_{n}^{'}_{n}^{"}_{n}_{n}^{'}_{n}^{"}