Sage Journals: Discover world-class research

Abstract

Radiant floor cooling systems are increasingly used in practice. The temperature distribution on the floor surface and inside the floor structure, especially the minimum and average temperature of floor surface, determines the thermal performance of radiant floor systems. A good temperature distribution of the floor structure is very important to prevent occupant discomfort and avoid possible condensation in summer cooling. In this study, based on the heat transfer model of the single-layer homogeneous floor structure when there is no internal heat radiation in the room, this paper proposes a heat transfer model of single-layer floor radiant cooling systems when the room has internal heat radiation. Using separation variable methods, an analytical solution was developed to estimate temperature distribution of typical radiant floor cooling systems with internal heat radiation, which can be used to calculate the minimum temperature and the average temperature of typical composite floor structure. The analytical solution was validated by experiments. The values of the measured experiments are in a good agreement with the calculations. The absolute error between the calculated and the measured floor surface temperatures was within 0.45°C. The maximum relative error was within 2.31%. Prove that this model can be accepted. The proposed method can be utilized to calculate the cooling capacity of a typical multi-layer composite floor and will be developed in the future study for design of a typical radiant floor cooling system.

Keywords

Radiant floor cooling temperature distribution internal heat radiation surface temperature

Introduction

In recent years, radiant floor cooling systems are increasingly used in many eastern Asian and European countries, which have the advantages of energy saving, sustainable development and green health (Hu and Niu, 2012; Rhee and Kim, 2015; Srivastava et al., 2018). Radiant floor cooling systems are also widely used in large space buildings, such as airports and greenhouses (Zhao et al., 2016). These applications can effectively use natural cold sources. Nowadays, more and more office buildings in northern China try to use the floor radiant heating systems in winter as cooling systems in summer (Werner-Juszczuk, 2019).

The most key parameter of a radiant floor cooling system is the temperature distribution of floor surface, which can represent the thermal performance of the floor, and also determines the cooling capacity of a radiant floor. In order to estimate the floor surface temperature, many relevant researches have been carried out. Li et al. (2014, 2021) propose the concept of equivalent thermal resistance, and get the average and minimum temperature expressions of the floor surface, then propose a new method to estimate thermal performance of multilayer radiant floor. Wu et al. (2015) propose a simplified method named the shape coefficient method to calculate the floor surface mean temperature. Zhang et al. (2012) provide analytical solutions for the floor surface temperature distribution under heating and cooling conditions respectively and simplify the equations.

Due to the complexity of the coupling of radiation and convection, many experts and scholars use numerical simulations and experimental methods to obtain the temperature distribution law of the floor surface. Numerical simulation can be used to obtain the discrete temperature distribution of floor surface (Larsen et al., 2010; Li et al., 2017; Liu et al., 2011; Wang et al., 2014). Fernández-Gutiérrez et al. (2015) combine experimental and numerical methods to study the radiant floor cooling systems. Pantelic et al. (2018) conduct full scale laboratory experiment to research the cooling capacity of a radiant floor system.

The researches of the thermal performance of radiant floors with other disturbances like solar radiation has been performed in recent years. Feng et al. (2016) take the solar radiation into account and propose a new design method for radiant floor cooling systems. Tang et al. (2020) propose a new approximate method for the dynamic cooling capacity prediction of radiant floors with time variable solar radiation. Zhao et al. (2015) take the airport as a typical research case, in which the cooling capacity of a radiant floor of large spaces with longwave radiant is calculated.

Scholars do excellent works on the radiant floor systems, but researches are mostly concentrated on heating in winter, and mostly use CFD and TRYSNS simulation methods to study radiant floor systems. Few studies focus on the use of radiant floor systems for cooling in summer. Few studies focus on the temperature distribution on the floor surface and inside. Few studies focus on the analytical methods to study the heat transfer problems of a typical multi-layer composite floor. Based on heat transfer analysis of coupling of convection and radiation, this study put forward an analytical method adapted to a typical two-layer composite floor structure that can provide accurate temperature distribution models inside the floor structure. By the analytical solutions, internal heat source terms such as sunlight and light are considered, and the analytical solutions are corrected. Using the modified equations, it can quickly estimate the whole surface temperature of floor, and get the thermal performance of radiant floor cooling systems. This method provides new ideas and methods to optimize design parameters of floor structure in a radiant floor cooling system.

Heat transfer process with internal heat radiation on floor surface

The heat transfer process between floor surface and indoor environment

The floor surface exchanges heat with the indoor environment through radiation and convection. The total heat flux $q_{tot}$ through the floor surface is the sum of convective heat flux $q_{c}$ , radiant heat flux $q_{r}$ and internal heat radiation taken away by the floor surface $q_{n f}$ .

q_{tot} = q_{r} + q_{c} + q_{n f}

(1)

As shown in Figure 1, the floor surface exchanges heat with the indoor air by convection and exchanges heat with the inner surfaces of envelopes by radiation. The floor surface also accepts heat radiation from internal heat sources such as lights and solar.

Figure 1.

Heat exchange between floor surface and indoor environment.

The floor surface make radiant heat exchange with the inner surfaces of envelopes. The inner surfaces of room are assumed to be a closed system consisting of two gray surfaces: the floor surface and an imaginary surface composed of other inner surfaces of envelopes. The radiation between the two surfaces can be calculated as

q_{r} = \frac{σ [{(AUST + 273.15)}^{4} - {(t_{s} + 273.15)}^{4}]}{(\frac{1 - ϵ_{1}}{ϵ_{1}}) + (\frac{1}{ϕ_{12}}) + (\frac{(1 - ϵ_{2}) A_{1}}{ϵ_{2} A_{2}})}

(2)

where

ϕ_{12}

is the view factor from the floor surface to the imaginary inner surface;

ϵ_{1}

and

ϵ_{2}

are the emissivity of the floor surface and imaginary inner surface, respectively, which are almost equal, with the value of 0.95 (Li et al., 2014);

A_{1}

and

A_{2}

are the respective areas of the floor surface and imaginary inner surface;

t_{s}

is the average temperature of the floor surface;

AUST

is the average uncooled surface temperature. For convenience,

AUST

can be calculated as

AUST = \frac{\sum_{i}^{n} A_{i} t_{i}}{\sum_{i}^{n} A_{i}}

(3)

We define

h_{r} = \frac{σ [{(AUST + 273.15)}^{2} + {(t_{s} + 273.15)}^{2}] * [(AUST + 273.15) + (t_{s} + 273.15)]}{(\frac{1 - ϵ_{1}}{ϵ_{1}}) + (\frac{1}{ϕ_{12}}) + (\frac{(1 - ϵ_{1}) A_{1}}{ϵ_{1} A_{2}})}

(4)

Radiant heat exchange with inner surfaces of envelopes can be calculated as

q_{r} = h_{r} (AUST - t_{s})

(5)

The value of $h_{r}$ is related to $AUST$ and varies only slightly with temperature. A constant value of $h_{r} = 5.5 W / (m^{2} \cdot K)$ has less than 4% error (Olesen et al., 2000).

Convective heat transfer occurs between the floor surface and indoor air. The heat flux $q_{c}$ can be expressed as

q_{c} = h_{c} (t_{a} - t_{s})

(6)

where

t_{a}

is the indoor air temperature and

h_{c}

is the convective heat transfer coefficient between the floor surface and the indoor air, whose value is usually

1.5 W / (m^{2} \cdot K)

(Olesen et al., 2000).

Based on the radiant heat transfer coefficient and convective heat transfer coefficient, we fit the average radiant temperature and the indoor air temperature at operative temperature $t_{o p}$ for simplification

t_{o p} = \frac{h_{c} t_{a} + h_{r} A UST}{h_{c} {+ h}_{r}}

(7)

Therefore, choose the operating temperature $t_{o p}$ as the reference. The total heat flux between the floor surface and indoor environment is considered to be

q_{tot} = h_{tot} (t_{o p} - t_{s}) + q_{n f}

(8)

where

h_{tot}

is the total heat transfer coefficient, which is set to 7

W / (m^{2} \cdot K)

(Olesen et al., 2000).

The internal heat radiation taken away by per unit floor surface area is $q_{n f}$ .

The heat transfer process below floor surface

The homogeneous floor structure used in this section is shown in Figure 2. The filling layer consists of fine stone concrete with thickness $δ$ ; The pipes spacing is $L$ ; The outer diameter of pipes is $D$ ; The distance from the center of the pipes to the upper surface is $l_{1}$ , and the distance to the thermal insulation layer is $l_{2}$ .

Figure 2.

Heat transfer process under the floor surface.

The heat transfer process below the floor surface is shown in Figure 2. First, convective heat transfer occurs from the chilled water to the inner pipes’ wall. Second, heat is transferred in the pipe wall and layer of floor by conduction. Finally, heat transfer to the floor surface. To simplify the model, some resistances can be ignored (Jin et al., 2010), such as convective heat transfer resistance between water and the inner pipe wall, the thermal resistance of the pipes’ wall and the contact thermal resistance between the outer pipes’ wall and the filling layer. So, the temperature of outer pipes’ wall equals the average temperature $t_{w m}$ of the supply and return water. Through the center of the pipes, a virtual layer is assumed with temperature $t_{m}$ . The floor heat transfer process is simplified that the outer wall temperature first exchanges heat with the virtual layer with thermal resistance $R_{m}$ . Then the virtual layer exchanges heat with the floor surface, and the thermal resistance is $R_{t}$ .

R_{f} = R_{t} + R_{m}

(9)

Thermal resistance analysis

The overall heat transfer process is shown in Figure 3.

Figure 3.

Total heat transfer process.

The heat transfer above the floor surface and the heat transfer below the floor surface are connected in series. According to the principle of thermal resistance in series, the total thermal resistance $R_{all}$ can be expressed as Figure 4.

R_{all} = R_{t o t} + R_{f}

(10)

Figure 4.

Thermal resistances and heat flux analysis of the overall heat transfer process.

Models

The model without internal heat radiation on floor surface

In this section, to facilitate analysis, we assume that there is no internal heat radiation in room, which is

q_{n f} = 0

(11)

Based on this assumption, the two-dimensional heat transfer model is shown in Figure 5. The center of pipe is taken as the zero point of the coordinate system.

Figure 5.

A homogeneous floor heat transfer model without internal heat radiation.

The governing equation of this heat transfer process is

\frac{\partial}{\partial x} (λ \frac{\partial t}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial t}{\partial y}) = 0

(12)

The boundary conditions are summarized as follows:

In $x$ direction

\frac{\partial t}{\partial x} |_{x = 0} = 0

(13)

\frac{\partial t}{\partial x} |_{x = L} = 0

(14)

In $y$ direction

- λ \frac{\partial t}{\partial y} |_{y = l_{1}} = h_{tot} (t - t_{o p})

(15)

\frac{\partial t}{\partial y} |_{y = - l_{2}} = 0

(16)

At the pipes’ wall

t_{pipe} = t_{p m}

(17)

According to Koschenz and Dorer (1999), the analytical solution of temperature distribution of single-layer homogeneous floor structure can be obtained, as follows

t (x, y) = t_{o p} - Γ (t_{p m} - t_{o p}) [\frac{π}{L} (y - \frac{2 λ}{U} + |y|) - \sum_{i}^{\infty} \frac{1}{i} (e^{- (\frac{2 π i}{L}) |y|} + G (i) e^{- (\frac{2 π i}{L}) y}) \cos (\frac{2 π i}{L} x)

(18)

where

Γ = \frac{1}{\ln (\frac{L}{π D}) + (\frac{2 π λ}{L U}) + \sum_{i = 1}^{\infty} \frac{G (i)}{i}}

(19)

U = \frac{1}{(\frac{1}{h_{tot}}) + (\frac{l_{1}}{λ})}

(20)

G (i) = \frac{\frac{(B i + 2 π i)}{(B i - 2 π i)} e^{- (\frac{4 π i}{L} l_{2})} - 2 e^{- \frac{4 π i}{L} ({l_{1} + l}_{2})} - e^{- (\frac{4 π i}{L} l_{1})}}{e^{- \frac{4 π i}{L} ({l_{1} + l}_{2})} + \frac{B i + 2 π i}{B i - 2 π i}}

(21)

B i = \frac{h_{tot} L}{λ}

(22)

The model with internal heat radiation on floor surface

When there is internal heat radiation such as lights and sunlight on the floor, we assume that the solar radiation is evenly distributed on the floor surface. So $q_{n f}$ is a positive constant. At the same time, we assume that there is no lateral heat transfer at the edge of irradiation.

Based on all above assumptions, the heat transfer diagram is shown in Figure 6.

Figure 6.

A homogeneous floor heat transfer model with internal heat radiation.

The mathematical description of the two-dimensional heat transfer model is as follows:

The governing equation is

\frac{\partial}{\partial x} (λ \frac{\partial t}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial t}{\partial y}) = 0

(23)

The boundary conditions are:

In $x$ direction

\frac{\partial t}{\partial x} |_{x = 0} = 0

(24)

\frac{\partial t}{\partial x} |_{x = L} = 0

(25)

In $y$ direction

- λ_{2} \frac{\partial t}{\partial y} |_{y = l_{1}} = h_{tot} (t - t_{o p}) - q_{n f}

(26)

\frac{\partial t}{\partial y} |_{y = {- l}_{2}} = 0

(27)

At the pipes’ wall

t_{pipe} = t_{p m}

(28)

For the boundary condition of $y$ direction, we rewrite the boundary condition at $y = - l_{2}$ as

- λ \frac{\partial t}{\partial y} = h_{tot} (t - t_{o p} - \frac{q_{n f}}{h_{tot}})

(29)

Then, we make

t_{z} = t_{o p} + \frac{q_{n f}}{h_{tot}}

(30)

The boundary condition at $y = - l_{2}$ eventually becomes

- λ \frac{\partial t}{\partial y} = h_{tot} (t - t_{z})

(31)

According to Koschenz and Dorer (1999), combined with the solution process of Equation (18), the analytical solution of the above mathematical description is

t (x, y) = t_{z} - Γ (t_{p m} - t_{z}) [\frac{π}{L} (y - \frac{2 λ}{U} + |y|) - \sum_{i = 1}^{\infty} \frac{1}{i} (e^{- \frac{2 π i}{L} |y|} + G (i) e^{- \frac{2 π i}{L} y}) \cos (\frac{2 π i}{L} x)]

= t_{o p} + \frac{q_{n f}}{h_{tot}} - Γ (t_{p m} - t_{o p} - \frac{q_{n f}}{h_{tot}}) [\frac{π}{L} (y - \frac{2 λ}{U} + |y|) - \sum_{i = 1}^{\infty} \frac{1}{i} (\begin{matrix} e^{- \frac{2 π i}{L} |y|} + \\ G (i) e^{- \frac{2 π i}{L} y} \end{matrix}) \cos (\frac{2 π i}{L} x)]

(32)

Temperature distribution in a typical floor structure with internal heat radiation

Description of a typical multi-layer composite floor structure

In practice, a typical radiant floor system is mostly a multi-layer composite floor structure. In China, a typical radiant floor structure is shown in Figure 7.

Figure 7.

A typical multi-layer composite floor structure used in China.

In most applications, the surface layer is granite. The toweling layer is cement mortar which thermal conductivity is close to the surface layer. For calculation convenience, we consider these two layers as one homogeneous layer. In this way, the typical multi-layer composite floor structure is simplified into a two-layer composite floor structure, as shown in the Figure 8.

Figure 8.

Two-layer composite floor structure.

The above layer and the filling layer are homogeneous materials respectively. The thickness of above layer is $H_{1}$ and the thermal conductivity is $λ_{1}$ . The thermal conductivity of filling layer is $λ_{2}$ . The diameter of the round pipes is $D$ and the pipes spacing is $L$ .

There is continuous flowing chilled water in the buried pipes of filling layer. The temperature of the pipes wall is $t_{p m}$ , which has the relationship with cold water temperature:

t_{p m} = t_{w m} + Δ t

(33)

According to Jin et al. (2010), the thermal resistance between the chilled water and the outer pipes’ wall can be ignored, which is

t_{p m} = t_{w m}

(34)

Calculation of temperature distribution in the two-layer floor structure with internal heat radiation

Mathematical description of the two-layer floor structure

Because of the periodicity of buried pipes installation, the temperature distribution of the floor structure also has periodicity. The area between any two pipes only to be studied to get the temperature distribution law.

Variable separation methods are often used to solve the heat conduction problems in the regions with regular geometric shapes. Because the cut face of floor is usually a horizontal regular rectangle as shown in Figure 9, this paper selects rectangular coordinate system and rectangular calculation regions for research easily. Therefore, the region enclosed by a–b–c–d–a in Figure 9 is selected as research regions. The rectangular coordinate system is established as shown in Figure 10. The $x$ axis is between the two layers.

Figure 9.

Schematic of heat exchange.

Figure 10.

Calculation region, coordinate system and boundary conditions.

Governing equations and boundary conditions of the two calculation regions

Calculation regions are shown in Figure 10. We set the function $t$ to represent the temperature distribution in the calculation area.

We make

θ = t - t_{z}

(35)

then Region 1 and Region 2 are

θ_{1}

and

θ_{2}

respectively.

The governing equations for regions 1 and 2 are the following forms

\frac{\partial^{2} θ_{1}}{\partial x^{2}} + \frac{\partial^{2} θ_{1}}{\partial y^{2}} = 0

(36)

\frac{\partial^{2} θ_{2}}{\partial x^{2}} + \frac{\partial^{2} θ_{2}}{\partial y^{2}} = 0

(37)

Based on the principle of symmetry, the boundary conditions in the $x$ direction are as follows

x = 0, \frac{\partial θ_{1}}{\partial x} = 0 \frac{\partial θ_{2}}{\partial x} = 0

(38)

x = L, \frac{\partial θ_{1}}{\partial x} = 0 \frac{\partial θ_{2}}{\partial x} = 0

(39)

For the $y$ direction

When $y = H_{1}$ , from equation (31), we can get the boundary condition as follows.

y = H_{1}, - λ_{1} \frac{\partial θ_{1}}{\partial y} = h_{tot} θ_{1}

(40)

There is no contact thermal resistance between Region 1 and Region 2. So, the boundary condition at $y = 0$ is

{y = 0, λ}_{1} \frac{\partial θ_{1}}{\partial y} = λ_{2} \frac{\partial θ_{2}}{\partial y}

(41)

{y = 0, θ}_{1} = θ_{2}

(42)

At $y = - H_{2}$ , the boundary condition is

θ_{2} + t_{z} = f (x)

(43)

In previous section, we obtained the expression of the temperature distribution of the single-layer homogeneous floor structure when there is solar radiation on the floor surface. According to the equivalent thermal resistance method (Li et al., 2014), we can convert the two-layer floor structure into a single-layer homogeneous floor structure, and then we can obtain the temperature distribution expression $f (x)$ on the tangent line at the upper end of pipes.

According to equation (30) and (32), the temperature distribution of the tangent line at the upper end of pipes along the $x$ direction is expressed as

f (x) = t (x, \frac{D}{2}) = t_{z} - Γ (t_{p m} - t_{z}) [\frac{π}{L} (D - \frac{2 λ_{2}}{U}) - \sum_{i = 1}^{\infty} \frac{1}{i} (e^{- \frac{π D i}{L}} + G (i) e^{\frac{π D i}{L} -}) \cos (\frac{2 π i}{L} x)]

(44)

Where (Li et al., 2014)

l_{1} = H_{2} + \frac{D}{2} + H_{1} \frac{λ_{2}}{λ_{1}}

(45)

l_{2} = D / 2

(46)

Solutions

Separation of variables

With the above two governing equations and closed boundary conditions, we can solve the temperature distribution expression of the two regions by the methods of separation variables.

In Regions 1 and 2 respectively, it is assumed that $θ$ is the product of two functions $X (x)$ and $Y (y)$ . The governing equations of Regions 1 and 2 can be written as follows

θ_{1} (x, y) = X_{1} (x) Y_{1} (y)

(47)

θ_{2} (x, y) = X_{2} (x) Y_{2} (y)

(48)

Substituting equations (47)–(48) to equations (36)–(37), we can get

\frac{{X_{1}}^{''}}{X_{1}} + \frac{{Y_{1}}^{''}}{Y_{1}} = 0

(49)

\frac{{X_{2}}^{''}}{X_{2}} + \frac{{Y_{2}}^{''}}{Y_{2}} = 0

(50)

The boundary conditions can be written as

x = 0 X_{1}^{'} = 0 X_{2}^{'} = 0

(51)

x = L X_{1}^{'} = 0 X_{2}^{'} = 0

(52)

y = H_{1} - λ_{1} Y_{1}^{'} = h_{tot} Y_{1}

(53)

y = 0 θ_{1} = θ_{2}

(54)

When $y = - H_{2}$ . Both sides of the equation go to $t_{z}$

θ_{2} = - Γ (t_{p m} - t_{z}) [\frac{π}{L} (D - \frac{2 λ_{2}}{U}) - \sum_{i = 1}^{\infty} \frac{1}{i} (e^{- \frac{π D i}{L}} + G (i) e^{\frac{π D i}{L} -}) \cos (\frac{2 π i}{L} x)]

(55)

The general solution of function $X (x)$ and $Y (y)$

Because the two function values at $y = 0$ in Region 1 and Region 2 are exactly the same, the general solution of function $X (x)$ and $Y (y)$ , in $x$ direction, is

\frac{X_{1}^{″}}{X_{1}} = \frac{X_{2}^{″}}{X_{2}} = - ε^{2}

(56)

whose solutions are

X_{1} (x) = X_{2} (x) = \sin (ε x) + \cos (ε x)

(57)

According to the boundary conditions at $x = 0$ and $x = L$

We can get

ε = \frac{n π}{L}, (when n = 0, 1, 2, 3 \dots .)

(58)

For the function $Y (y)$ , because

\frac{{Y_{1}}^{''}}{Y_{1}} = ε^{2}

(59)

The general solutions in the $y$ direction of the two regions are

Y_{1 n} (y) = A_{0} + B_{0} y, (when n = 0)

(60)

Y_{1 n} (y) = A_{n} e^{ε y} + B_{n} e^{- ε y}, (when n = 1, 2, 3 . . . . . .)

(61)

Y_{2 n} (y) = C_{0} + D_{0} y, (when n = 0)

(62)

Y_{2 n} (y) = C_{n} e^{ε y} + D_{n} e^{- ε y}, (when n = 1, 2, 3 . . . . . .)

(63)

$A_{n}$ , $B_{n}$ , $C_{n}$ , $D_{n}$ are the coefficients to be determined.

General solution

According to equations (57) (58) (60) (61) (62) (63)

We get the general solutions of regions 1 and 2 as

θ_{1} (x, y) = A_{0} + B_{0} y + \sum_{n = 1}^{\infty} (A_{n} e^{ε y} + B_{n} e^{- ε y}) \cos (ε x)

(64)

θ_{2} (x, y) = C_{0} + D_{0} y + \sum_{n = 1}^{\infty} (C_{n} e^{ε y} + D_{n} e^{- ε y}) \cos (ε x)

(65)

Calculation of coefficients

In the analysis of the boundary conditions, we choose the order from top to bottom.

At $y = H_{1}$

B_{0} = A_{0} \frac{- B i_{1}}{L + H_{1} B i_{1}}, (n = 0)

(66)

B_{n} = A_{n} e^{2 ε H_{1}} \frac{n π + B i_{1}}{n π - B i_{1}}, (n = 1, 2, 3 . . . . . .)

(67)

where

B i_{1} = \frac{h_{tot} L}{λ_{1}}

(68)

We use the boundary conditions at $y = 0$ that is equations (41) and (42), then we get the following relationships

A_{0} = C_{0}

(69)

A_{n} = \frac{C_{n} + D_{n}}{2} + \frac{λ_{2}}{2 λ_{1}} (C_{n} - D_{n})

(70)

B_{0} = \frac{λ_{2}}{λ_{1}} D_{0}

(71)

B_{0} = \frac{C_{n} + D_{n}}{2} - \frac{λ_{2}}{2 λ_{1}} (C_{n} - D_{n})

(72)

From these, we get the following relationships

C_{n} = \frac{1}{2} [(1 + \frac{λ_{1}}{λ_{2}}) A_{n} + (1 - \frac{λ_{1}}{λ_{2}}) B_{n}]

(73)

D_{n} = \frac{1}{2} [(1 - \frac{λ_{1}}{λ_{2}}) A_{n} + (1 + \frac{λ_{1}}{λ_{2}}) B_{n}]

(74)

According to the boundary conditions at $y = - H_{2}$ . We get the following formula

\begin{array}{l} θ_{2} (x, - H_{2}) = C_{0} - D_{0} H_{2} + \sum_{n = 1}^{\infty} (C_{n} \cdot e^{- ε H_{2}} + D_{n} e^{ε H_{2}}) \cdot cos (\frac{n π}{L} \cdot x) \\ = - Γ (t_{p m} - t_{z}) [\frac{π}{L} (D - \frac{2 λ_{2}}{U}) - \sum_{n = 1}^{\infty} \frac{1}{i} (e^{- \frac{π D i}{L}} + G (i) e^{- \frac{π D i}{L}})] \cos \frac{2 π i}{L} \\ \times (n = 2i,n = 2,4,6 \dots) \end{array}

(75)

From the above relationship, we can get the relationships between $C$ and $D$ as follows

C_{0} - D_{0} H_{2} = - Γ \cdot (t_{p} - t_{z}) \frac{π}{L} (D - \frac{2 λ_{2}}{U})

(76)

(C_{n} \cdot e^{- ε H_{2}} + D_{n} e^{ε H_{2}}) = Γ (t_{p} - t_{z}) [\frac{2}{n} (e^{- \frac{π D n}{2 L}} + G (n) e^{- \frac{π D n}{2 L}})] (n = 2, 4, 6 \dots \dots)

(77)

According to all the above relational expressions, we can solve and get the coefficients as follows

A_{0} = 2 π Γ (t_{p} - t_{z}) \frac{L + {B i}_{1} H_{1}}{L {B i}_{2}}

(78)

A_{n} = χ_{n} Γ (t_{p} - t_{z}), (n = 2, 4, 6 \dots)

(79)

B_{0} = - 2 π Γ (t_{p m} - t_{z}) \frac{{B i}_{1}}{L {B i}_{2}}

(80)

B_{n} = ϕ_{n} χ_{n} Γ (t_{p m} - t_{z}), (n = 2, 4, 6 \dots)

(81)

{C_{0} = A}_{0} = 2 π Γ (t_{p m} - t_{z}) \frac{L + {B i}_{1} H_{1}}{L {B i}_{2}}

(82)

D_{0} = \frac{λ_{1}}{λ_{2}} B_{0} = - \frac{{2 π λ}_{1}}{λ_{2}} Γ (t_{p m} - t_{z}) \frac{{B i}_{1}}{L {B i}_{2}}

(83)

C_{n} = \frac{1}{2} [\frac{λ_{1}}{λ_{2}} (1 - ϕ_{n}) + 1 + ϕ_{n}] χ_{n} Γ (t_{p m} - t_{z})

(84)

D_{n} = \frac{1}{2} [- \frac{λ_{1}}{λ_{2}} (1 - ϕ_{n}) + 1 + ϕ_{n}] χ_{n} Γ (t_{p m} - t_{z}), (n = 2,4, 6 \dots)

(85)

where

ϕ_{n} = \frac{n π + {B i}_{1}}{n π - {B i}_{1}} e^{2 ε H_{1}}

(86)

χ_{n} = \frac{[\frac{2}{n} (e^{- \frac{π D n}{2 L}} + G (\frac{n}{2}) e^{- \frac{π D n}{2 L}})]}{\frac{1}{2} [\frac{λ_{1}}{λ_{2}} (1 - ϕ_{n}) + 1 + ϕ_{n}] e^{- ε H_{2}} - \frac{1}{2} [\frac{λ_{1}}{λ_{2}} (1 - ϕ_{n}) - 1 - ϕ_{n}] e^{ε H_{2}}}

(87)

The temperature distribution models in region 1 and 2

The expressions of temperature distribution in Region 1 as follows

t_{1} = t_{z} + 2 π Γ (t_{pm} - t_{z}) \frac{L + {B i}_{1} H_{1}}{L {B i}_{2}} - 2 π Γ (t_{pm} - t_{z}) \frac{{B i}_{1}}{L {B i}_{2}} y + \sum_{n = 2}^{\infty} [χ_{n} Γ (t_{pm} - t_{z}) e^{ε y} + ϕ_{n} χ_{n} Γ (t_{pm} - t_{z}) e^{- {εy}_{}}] \cos (ε x)

(88)

The temperature distribution of the floor surface is that $y = H_{1}$ , as follows.

t_{1} (x, H_{1}) = t_{z} + 2 π Γ (t_{p m} - t_{z}) \frac{1}{{B i}_{2}} + \sum_{n = 2}^{\infty} [χ_{n} Γ (t_{p m} - t_{z}) e^{ε H_{1}} + ϕ_{n} χ_{n} Γ (t_{p m} - t_{z}) e^{- ε H_{1}})] \cos (ε x) = 2 π Γ t_{p m} \frac{1}{{B i}_{2}} + (1 - 2 π Γ \frac{1}{{B i}_{2}}) (t_{o p} + \frac{q_{n f}}{h_{tot}}) + \sum_{n = 2}^{\infty} [χ_{n} Γ (t_{p m} - t_{o p} - \frac{q_{n f}}{h_{tot}}) {(e}^{ε H_{1}} - ϕ_{n} e^{- ε H_{1}})] \cos (ε x)

(89)

The expression of temperature distribution in Region 2 as follows

t_{2} (x, y) = t_{z} + 2 π Γ (t_{p m} - t_{z}) \cdot (\frac{L + B i_{1} H_{1}}{L B i_{2}} - \frac{λ_{1} B i_{1}}{λ_{2} L B i_{2}} \cdot y) + \sum_{n = 2}^{\infty} \frac{1}{2} χ_{n} Γ (t_{p m} - t_{z}) \cdot \{[1 + ϕ_{n} + \frac{λ_{1}}{λ_{2}} (1 - ϕ_{n})] \cdot e^{ε y} + [1 + ϕ_{n} - \frac{λ_{1}}{λ_{2}} (1 - ϕ_{n})] e^{- ε y}\} \cdot \cos (ε x)

(90)

After the analytical solutions of temperature distribution in Regions 1 and 2 being obtained respectively. We use Paraview software to visualize the temperature distribution of a real case, as shown in Figure 11.

Figure 11.

An example of temperature distribution visualization.

It can be seen that the temperature near the two buried pipes is the lowest, and the isotherms are in circular distribution. At the junction of Region 1 and Region 2, the temperature values of the two regions are the same. The maximum temperature appears at the symmetrical center of the two pipes. All these are in accordance with the basic law of heat transfer (Zhang et al., 2007), and basically verify the correctness of the model.

For a radiant floor cooling system, the most important parameters are mean floor surface temperature and the lowest temperature in floor surface. With the equation (89), we can obtain the mean floor surface temperature as follows

t_{s} = t_{o p} + \frac{q_{n f}}{h_{tot}} + Γ \cdot \frac{2 π}{B i_{2}} (t_{p m} - t_{o p} - \frac{q_{n f}}{h_{tot}})

(91)

As can be seen from Figure 11, the lowest surface temperature is located at $x = 0$ . The formula for estimating the lowest temperature in floor surface is derived and shown as

t_{\min} = t_{o p} + \frac{q_{n f}}{h_{tot}} + Γ \cdot \frac{2 π}{B i_{2}} (t_{p m} - t_{o p} - \frac{q_{n f}}{h_{tot}}) + \sum_{n = 2}^{\infty} χ_{n} Γ (t_{p m} - t_{o p} - \frac{q_{n f}}{h_{tot}}) \cdot (e^{ε H_{1}} + ϕ_{n} e^{- ε H_{1}})

(92)

Validation of the models

In this section, certain experiments are conducted to verify the above results.

Experimental conditions

In the cold regions of China, many office buildings use radiant floor heating systems for heating in winter. We select two office rooms with radiant floor heating system as the experimental platform. The Room 1 is located on the basement one of buildings. The test Room 2 is located on the top floor of buildings. The dimensions of Room 1 are 4 $m$ (length)×4 $m$ (width)×3.3 $m$ (height). The dimensions of Room 2 are 3.1 $m$ (length)×3.6 $m$ (width)×2.6 $m$ (height). The building envelopes of Room 1 except the floor and ceiling are made of 150-mm-thick. The ceiling of Room 1 is the floor of upper level. So, the ceiling structure is the same as the floor. The building envelopes of Room 2 except the floor are made of iron board. The materials of floor structure from top to bottom is tile, cement mortar, filling layer and polystyrene insulation. The tile layer and cement mortar layer can be regarded as one upper layer. The floor structure parameters of Room 1–2 are shown in Table 1. The buried pipes of Room 1 are PVC pipes and the buried pipes of Room 2 are copper pipes. The outer pipe diameter of all the buried pipes are 20 mm. The heat transfer coefficient of insulation can provide good insulation performance.

Table 1.

Parameters of the experimental floor structures.

Room	Spacing	Above layer		Filling layer		Insulation layer
Room	$L (m m)$	Thickness $H_{1} (m m)$	Heat conductivity $λ_{1} (W / (m \cdot K))$	Thickness $H_{2} (m m)$	Heat conductivity $λ_{2} (W / (m \cdot K))$	Thickness $H_{3} (m m)$	Heat conductivity $λ_{3} (W / (m \cdot K))$
1	150	30	3.45	30	1.49	40	0.025
2	200	19	2.1	20	1.51	25	0.029

In this paper, the fold-back type layout and in-line type of buried pipes are selected to experiment respectively in order to eliminate the influence of different layout types on the experimental results. The layout types of buried pipes in Room 1–2 are shown in Figure 12.

Figure 12.

Layout of buried pipes in Room 1 and Room 2.

An infrared radiation board is used as indoor heat sources to simulate internal heat radiation. The infrared radiation board uses 220 V power supply and has 3 power levels. Heat flux densitometers are used to measure the heat flux density of the radiant board to the floor surface, which is $q_{n f}$ . In order to observe the experimental results of the internal heat radiation under different value ranges, this paper selects four types of working conditions. That is, $q_{n f}$ is a high value, $q_{n f}$ is a median value, $q_{n f}$ is a low value and $q_{n f}$ is equal to 0. These conditions can show how different internal heat radiation affect the temperature distribution of floor surface.

During the experiments in summer, floor and fresh air can achieve combined cooling. Chilled water is passed into the buried pipes to real floor cooling. 13 temperature measuring points are set to measure the temperature of floor surface in each room as shown in Figure 13. Each envelope has 4 temperature measuring points on the inner surface. Sticking thermometer are used to record the temperature of solid surfaces. In the vertical direction of the indoor space, 4 temperature measuring points are set from top to bottom which can measure the temperature of indoor air.

Figure 13.

Temperature measuring points of floor surface in Room 1 and Room 2.

Comparison of results

Both the experimental measurement results and calculation results are show in Table 2.

Table 2.

Comparison of calculations with experimental results.

Room	Case	$t_{p m}$ (°C)	$t_{o p}$ (°C)	$q_{n f} (W)$	$t_{s}$ (°C)			$t_{m}$ (°C)
Room	Case	$t_{p m}$ (°C)	$t_{o p}$ (°C)	$q_{n f} (W)$	Measured	Calculated	Error	Measured	Calculated	Error
1	1	14.19	25.01	76.66	19.54	19.40	−0.14	19.40	19.27	−0.13
	2	15.50	26.38	78.12	20.71	20.47	−0.24	20.61	20.34	−0.27
	3	16.11	27.45	48.74	20.12	19.73	−0.39	20.09	19.64	−0.45
	4	14.55	28.27	47.12	19.06	18.62	−0.44	18.81	18.51	−0.30
	5	14.50	28.89	16.97	17.41	17.37	−0.04	17.47	17.30	−0.17
	6	17.03	27.81	18.10	19.30	19.11	−0.19	19.15	19.06	−0.09
	7	18.21	28.10	0.00	19.57	19.24	−0.33	19.42	19.21	−0.21
	8	18.01	27.89	0.00	19.52	19.08	−0.44	19.30	19.06	−0.24
2	1	13.78	25.71	75.33	19.95	19.89	−0.06	18.87	18.82	−0.05
	2	15.13	26.38	77.46	21.03	20.90	−0.13	20.09	19.88	−0.21
	3	16.94	27.68	48.57	21.01	20.81	−0.20	20.43	20.12	−0.31
	4	13.43	26.02	48.22	18.41	18.34	−0.07	17.46	17.47	0.01
	5	14.85	26.23	18.81	18.05	17.95	−0.10	17.74	17.41	−0.33
	6	16.46	27.55	17.60	19.13	19.05	−0.08	18.58	18.60	0.02
	7	17.99	27.77	0.00	19.46	19.22	−0.24	19.25	19.00	−0.25
	8	17.65	28.31	0.00	19.21	19.03	−0.18	18.96	18.79	−0.17

We can find that the calculated values agree well with the corresponding experimental values. The maximum absolute error is 0.45°C. The maximum relative error is 2.31%. As can be seen from the Table 2, almost all the calculated values are smaller than the corresponding measured values. Few calculated values are greater than measured data. This is because the measured value of solar radiation is too small or the thermal inertia of the floor structure is too large. The verified experiments reveal that the analytical models are in excellent agreement with the measured experiments. Analytical models of temperature distribution can be accepted.

Conclusions

This paper studies the floor radiation cooling systems. Using the equivalent thermal resistance methods and separate variable methods, this paper proposes a novel analytical calculation methods to solve the problem of temperature distribution inside floor structure and on the floor surface. The main conclusions of this paper are as follows:

Based on the floor surface temperature distribution model without solar radiation on floor surface. This paper proposes a calculation model that can calculate temperature distribution of floor structure when any solar radiation on the floor surface. The distribution of surface temperature for a homogeneous floor structure can be conveniently calculated, when internal heat radiation hits on the floor.

Based on the model for the homogeneous floor structure with internal heat radiation, this paper also proposes a model to calculate temperature distribution for typical multi-layer floor structure. Using this model, temperature distribution of the two-layer floor structure can be calculated easily, when there is internal heat radiation on floor surface. This model also can be used to calculate the minimum temperature and the average temperature of typical composite floor structure.

This paper selects two actual cases for experimental researches and compares the calculated values with the measured values, and find that the calculated values agree well with the corresponding experimental values. The maximum absolute error is 0.45°C. The maximum relative error is 2.31%.

This paper supplies the calculation method of floor surface temperature for radiant floor cooling systems. The research results can be applied to the floor cooling system of large space buildings with glass curtain walls and ceilings such as Beijing Daxing International Airport. The results in this paper also can be used in agriculture to estimate soil surface temperature in greenhouses and calculate the temperature distribution inside the soil. The calculation models can also effectively prevent condensation inside the floor structure of the radiant floor cooling systems and guide the design.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the Plan of Guidance and Cultivation for Young Innovative Talents of Shandong Provincial Colleges and Universities, and was funded by both the Natural Science Foundation of China (Grant Number 51908333) and Doctoral Scientific Fund Project of Shandong Jianzhu University (Grant Number XNBS1605).

ORCID iD

Xiaolong Wang

References

Feng

Schiavon

Bauman

(2016) New method for the design of radiant floor cooling systems with solar radiation. Energy and Buildings 125: 9–18.

Fernández-Gutiérrez

González-Prieto

Parras

, et al. (2015) Experimental and numerical study of a small-scale and low-velocity indoor diffuser coupled with radiant floor cooling. International Journal of Heat and Mass Transfer 87: 71–78.

Niu

(2012) A review of the application of radiant cooling & heating systems in mainland China. Energy and Buildings 52: 11–19.

Jin

Zhang

Luo

, et al. (2010) Numerical simulation of radiant floor cooling system: The effects of thermal resistance of pipe and water velocity on the performance. Building and Environment 45(11): 2545–2552.

Koschenz

Dorer

(1999) Interaction of an air system with concrete core conditioning. Energy and Buildings 30(2): 139–145.

Larsen

Filippín

Lesino

(2010) Transient simulation of a storage floor with a heating/cooling parallel pipe system. Building Simulation 3(2): 105–115.

Sun

(2017) Development of a simplified resistance and capacitance (RC)-network model for pipe-embedded concrete radiant floors. Energy and Buildings 150: 353–375.

Chen

Zhang

, et al. (2014) Simplified thermal calculation method for floor structure in radiant floor cooling system. Energy and Buildings 74(2): 182–190.

Zhang

Guo

, et al. (2021) Development of a new method to estimate thermal performance of multilayer radiant floor. Journal of Building Engineering 33: 101562.

10.

Liu

Tian

Zhang

, et al. (2011) Establishment and validation of modified star-type RC-network model for concrete core cooling slab. Energy and Buildings 43(9): 2378–2384.

11.

Olesen

Michel

Bonnefoi

, et al. (2000) Heat exchange coefficient between floor surface and space by floor cooling – Theory or a question of definition. ASHRAE Transactions 6(1): 684–694.

12.

Pantelic

Schiavon

Ning

, et al. (2018) Full scale laboratory experiment on the cooling capacity of a radiant floor system. Energy and Buildings 170: 134–144.

13.

Rhee

Kim

(2015) A 50 year review of basic and applied research in radiant heating and cooling systems for the built environment. Building and Environment 91: 166–190.

14.

Srivastava

Khan

Bhandari

, et al. (2018) Calibrated simulation analysis for integration of evaporative cooling and radiant cooling system for different Indian climatic zones. Journal of Building Engineering 19: 561–572.

15.

Tang

Zhang

Liu

, et al. (2020) A novel approximate harmonic method for the dynamic cooling capacity prediction of radiant slab floors with time variable solar radiation. Energy and Buildings 223: 110117.

16.

Wang

Liu

Wang

, et al. (2014) Numerical and experimental analysis of floor heat storage and release during an intermittent in-slab floor heating process. Applied Thermal Engineering 62(2): 398–406.

17.

Werner-Juszczuk

(2019) Analysis of the use of radiant floor heating as a cooling system. Proceedings 16(1): 23.

18.

Zhao

Olesen

, et al. (2015) A new simplified model to calculate surface temperature and heat transfer of radiant floor heating and cooling systems. Energy and Buildings 105: 285–293.

19.

Zhang

Liu

Jiang

(2012) Simplified calculation for cooling/heating capacity, surface temperature distribution of radiant floor. Energy and Buildings 55: 397–404.

20.

Zhao

Liu

Jiang

(2015) Cooling capacity prediction of radiant floors in large spaces of an airport. Solar Energy 113: 221–235.

21.

Zhao

Liu

Jiang

(2016) Application of radiant floor cooling in large space buildings – A review. Renewable and Sustainable Energy Reviews 55: 1083–1096.

22.

Zhang

Ren

Mei

(2007) Heat Transfer. Beijing: China Construction Industry Press.

An analytical method to estimate temperature distribution of typical radiant floor cooling systems with internal heat radiation

Abstract

Keywords

Introduction

Heat transfer process with internal heat radiation on floor surface

The heat transfer process between floor surface and indoor environment

The heat transfer process below floor surface

Thermal resistance analysis

Models

The model without internal heat radiation on floor surface

The model with internal heat radiation on floor surface

Temperature distribution in a typical floor structure with internal heat radiation

Description of a typical multi-layer composite floor structure

Calculation of temperature distribution in the two-layer floor structure with internal heat radiation

Mathematical description of the two-layer floor structure

Governing equations and boundary conditions of the two calculation regions

Solutions

Separation of variables

The general solution of function X ( x ) and Y ( y )

General solution

Calculation of coefficients

The temperature distribution models in region 1 and 2

Validation of the models

Experimental conditions

Comparison of results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

The general solution of function $X (x)$ and $Y (y)$