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Analytical expression for transient heat transfer in concrete core activation.

INTRODUCTION

Concrete core activation (CCA, also called thermally activated building systems) is a heat and cold emission system where water tubes are embedded in the core of the concrete mass of a building. Although wall application is possible, in office buildings mostly the floor slab between two floors is thermally activated. Heated or cooled water flows through the tubes and heats up or cools down the concrete of the floor slab, which on its turn conditions the zones below and above. Due to the low heating and high cooling temperatures, CCA is well suited to combine with ground coupled heat pump systems and direct ground cooling. By using CCA, an important part of the thermal mass of a building is actively used as a thermal storage device and is integrated in the HVAC system. If controlled properly, this thermal storage can be used to reduce power peaks in the production system or, in combination with a heat pump, be deployed as a controllable electricity consumer in a smart grid application. However, the primary goal is guaranteeing thermal comfort in the building.

CCA as a thermal storage can be actively loaded or unloaded with heat by controlling water flow rate and water supply temperature in the tubes embedded in the concrete slab. However, heat transfer to the zones below and above is not controllable, but depends on the difference between CCA surface temperature and the zone air and radiant temperatures. Therefore, correctly assessing the transient thermal behavior of CCA is crucial, when the control strategy needs to be designed.

The controller needs to know how injection or extraction of heat will influence the zone temperature. For this purpose, dynamic thermal CCA models are required, where accuracy and applicability are important but often conflicting requirements. These models can be deployed in intelligent control strategies, such as model-based predictive control (MPC) (Braun 2003; Cho and Zaheer-uddin 2003; Gwerder and Todtli 2005; Gyalistras and Gwerder 2010; Privara et al. 2011; Ferkl 2013), to achieve thermal comfort against a low energy cost or in smart grid applications to actively control heat pumps as electricity users.

In this paper, an analytical thermal CCA model is presented which can be used in these applications. Three approaches to set up a dynamic thermal model can be distinguished, each of them having its own field of application. Very often a numerical approach, based on finite difference (FD), finite volume (FV), or finite element (FE) is used to calculate the 3D or 2D temperature distribution in a concrete slab (Table 1). However, these models lack flexibility and are time consuming to set up. Furthermore, integrating these models in a dynamic whole building simulation model slows down the simulation time.

A more simplified, more applicable, but less accurate approach is to describe the transient heat transfer in a material layer by using a lumped thermal resistance-capacitance (RC) model. For an RC-model having n thermal capacitances or nodes, it is assumed that the temperature inside the material layer is characterized by transient behavior of these nodal temperatures. Increasing the number of nodes improves the accuracy, but also the complexity of the model.

An analytical expression for the time- and space-dependent temperature profile in a CCA slab might be able to combine the conflicting requirements of accuracy and applicability. Analytical solutions offer important advantages: (1) transparency--because analytical solutions are merely mathematical expressions, they offer a clear view into how variables affect the result and their mutual interactions, and (2) applicability--algorithms and models expressed analytically are often more efficient than equivalent numeric implementations. Evidently, care should be taken to use the solution within the boundaries and assumptions of the analytical expression.

In heat transfer theory, lumped capacitance RC-solutions are typically used in situations where the conduction heat transfer coefficient [lambda]/d, with d the thickness of the material layer (m = 3.28 ft]) and [lambda] the heat conduction coefficient of that same layer [W/(m x K) = 6.935 1 Btu x in/(h x [ft.sup.2] x [degrees]F], is much larger than the convection coefficient h [W/([m.sup.2] x K) = 0.1761 Btu/(h x ft x [degrees]F)] at the elements surface. This is presented as the Biot number [Bi = (hd)/[lambda]] to be very small.

With the reinforced concrete as the main material, the CCA Bi-numbers are not very small: 0.1-1 is a typical range (Sourbron 2012). As a consequence, a certain temperature drop will appear inside the concrete slab. Therefore, describing the heat transfer from the water in the CCA to the room air by an RC-system will introduce errors. On the other hand, this approach leads to a simplified set of heat transfer equations, which are more easily solved and usable in dynamic building models and for controller design. Gluck (1999) presented first principle based models of various water-embedded systems. Ren and Wright (1998) used a second order RC-model for a ventilated hollow core CCA and integrated this in a second order building model. Schmidt (2004) used RC-models connected to a hydraulic model and both are combined into macro elements to describe a whole building element. The Opticontrol-project (Gyalistras and Gwerder 2010) deploys building RC-models to design advanced model based predictive controllers for CCA-buildings and in the GEOTABS-project (Ferkl 2013), such an MPC controller is implemented in a real building.

Pioneering work on simplified RC-modelling of CCA has been performed by EMPA, the Swiss Federal Laboratories for Materials Testing and Research (Koschenz and Lehmann 2000). This approach for model simplification is the starting point to set up the analytical model in this paper. The 3D heat flow pattern inside the concrete slab is trans ferred into a 1D model by means of an analytic steady-state solution for the temperature profile in a homogeneous concrete floor with embedded tubes and by means of a triangle-star transformation on the thermal resistances network. Deriving this expression introduces the equivalent thermal resistance [R.sub.x], while the temperature [T.sub.c] is the equivalent concrete core temperature. Figure 1 shows the different thermal resistances in the RC-network, which are presented in Table 2.

If the conditions

[[d.sub.i]/[d.sub.x]] > 0.3 and [[d.sub.t,o]/[d.sub.x]] < 0.2 (1)

are fulfilled (with [d.sub.i] the thickness of the concrete layer below or above the tubes, [d.sub.x] the distance between the tubes, and [d.sub.t,o] the outer tube diameter), the thermal resistances [R.sub.x], [R.sub.d1] and [R.sub.d2] are represented by the following:

[R.sub.x] [approximately equal to] [[d.sub.x]ln([d.sub.x]/[pi][d.sub.t,o])/2[pi][[lambda].sub.t]] = f([d.sub.t,o], [d.sub.x], [[lambda].sub.t]) (2)

[R.sub.di] = [d.sub.i]/[[lambda].sub.c] (3)

with i = 1 or 2, and [[lambda].sub.c] [W/(m x K), 1 Btu-in./(h x [ft.sup.2] x [degrees]F) = 0.1442 W/(m x K)] the thermal conductivity of concrete.

Mathematically speaking, meeting these conditions means that [R.sub.x] is only a function of the parameters [d.sub.t,o], [d.sub.x] and [[lambda].sub.c] and no longer a function of the total CCA thickness d (Koschenz and Lehmann 2000). Furthermore, [R.sub.d1] and [R.sub.d2] represent the heat resistances to the upper and lower room and are only a function of the related CCA thickness [d.sub.i] and no longer a function of tube parameters, nor of the opposite concrete layer. Physically speaking, meeting these conditions means that the temperature profile which exists at the tube level in the core of the tabs is flattened when reaching the upper or lower boundary of the slab and the assumption of 1D heat flux (uniform heat transfer coefficients) to the room below or above is justified.

The EMPA RC-model (Figure 1) is completed with thermal resistances [R.sub.t] for the tube wall; [R.sub.w] for the water-tube forced convection; and [R.sub.z], which takes into account the difference between the water supply temperature [T.sub.ws] and the mean water temperature in the slab [T.sub.wm] (Koschenz and Lehmann 2000). The positioning of the thermal capacitances is presented and analyzed by Weber and Johannesson (2005) and by Weber et al. (2005).

Weber and Johannesson (2005) demonstrated that it is possible to use this EMPA-RC-network to describe the two-dimensional heat flow occurring in an asymmetrical CCA floor. Only when the layer on top of the pipes becomes too thin, the results will deviate significantly from reality. Also, for cycling periods < 40 minutes of e.g., the water supply temperature, the heat flow from water to concrete deviates with more than 10% compared to finite elements results. Weber et al. (2005) report deviations < 0.2 K between a FE-model and an 11th order RC-star network model of a CCA with raised floor. Being relevant for whole building models, Weber et al. (2005) demonstrated that adding extra RC-links, one for the air between the concrete surface and the floor tiles and a second for the floor tiles, provides accurate temperatures inside the concrete floor.

Analytical solutions of the one-dimensional transient heat diffusion equation for slabs in a constant temperature environment are presented by Carslaw and Jaeger (1959). However, in a CCA-slab, water flows through the tubes at discrete points, which makes the one-dimensional heat transfer assumption invalid. On the other hand, the EMPA-model transforms the location dependent temperature profile at the CCA core into a fictive uniform core temperature [T.sub.c] (Koschenz and Lehmann 2000). Using this transformation, the existing analytical solution for transient heat conduction in slabs (Carslaw and Jaeger 1959) will be adapted to CCA-slabs in the next section in order to investigate the time dependent CCA behavior. The RC-approach to model the dynamic thermal behavior of the upper and lower slab part, as used in the EMPA-model by Koschenz and Lehmann (2000), is replaced by an equivalent analytical expression.

The acquired analytical dynamic model is extremely useful to analyze the effect of control strategies on the thermal performance of a CCA slab, as will be shown in the section Application of the Analytical Expression. A further simplification provides an explicit expression for the time required to either transfer a certain amount of heat to a zone or to store an amount of heat in a CCA element. Moreover, this is achieved by taking into account the initial state of the CCA element, the convective and radiative heat transfer rate at both sides of the slab and the water and zone temperatures. These expressions are valid for both the case with water flow, and for the case without water flow.

DERIVATION OFTHE ANALYTICAL EXPRESSION

Two Operational Modes

CCA operates in two modes: with and without water flow. Both modes can be derived from the same analytical expression, since they require similar boundary conditions. Figure 2 presents the nomenclature for the two operational modes. In the first mode, the so-called free running mode, without water flow, the slab is treated as a whole, with different combined convective and radiative heat transfer coefficients [h.sub.1] and [h.sub.2] above and below, and, in order to generalize the solution, with different temperatures [T.sub.1] and [T.sub.2] above and below the slab. In the second mode, the water flow mode, the upper and lower part of the slab are treated separately. At the water side, the equivalent concrete core temperature [T.sub.c] is the surface temperature. The inverse of the equivalent thermal resistance [R.sub.c] = [R.sub.z] + [R.sub.w] + [R.sub.t] + [R.sub.x] from the star network (see Introduction) determines the heat transfer coefficient [h.sub.1] from supply water temperature [T.sub.ws] to the concrete nodal temperature [T.sub.c]. It is important to notice that no thermal inertia is assumed between [T.sub.ws] and [T.sub.c]. At the room side, the temperature is, respectively, [T.sub.2,1] and [T.sub.2,2] for the upper and the lower part of the slab. The corresponding combined convective and radiative heat transfer coefficients are again [h.sub.2,1] and [h.sub.2,2].

Remark that the heat transfer coefficients ht from slab surface to room described the combined convective and radiative heat transfer from the CCA surface into the adjacent room (Awbi and Hatton 1999). The temperatures [T.sub.1] and [T.sub.2] are equivalent zone temperatures resulting from the air temperature and the mean radiant temperature, weighted by the convective and radiative heat transfer coefficient, respectively. The conventions used for the heat diffusion equation in the two operational modes are presented in Table 3.

Temperature Distribution T(x, t) in a Slab with Thickness L (0 < x < L)

Depending on the time scales of interest, boundary conditions need to be specified accordingly. In the following the boundary conditions for the CCA will be gradually built up, starting from temperature distribution solutions in a slab configuration. The solution is then extended for use in a CCA slab for the free running mode and for water flow mode. Carslaw and Jaeger (1959, paragraph 3.11) present a T(x, t)-equation for a slab (0 < x < L) subjected to different boundary conditions ([h.sub.i]) at upper and lower surface. The equation expresses the one-dimensional temperature distribution in a slab of general thickness L as a function of time and space for the same temperatures above and below the slab ([T.sub.1] = [T.sub.2] = 0).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

with

[alpha] = [lambda]/[rho]c the thermal diffusivity of concrete (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[Z.sub.n](x) = [lambda][[beta].sub.n] cos ([[beta].sub.n]x) + [h.sub.1] sin ([[beta].sub.n]x) (7)

and, with [T.sub.t] = 0 = f(x) the initial temperature profile of the CCA slab

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and [[beta].sub.n], n = 1, 2, ... the positive roots of

cot([beta]L) = [[[lambda].sup.2][[beta].sup.2] - [h.sub.1][h.sub.2]]/[beta][lambda]([h.sub.1] + [h.sub.2]) (9)

Solution for Different Upper and Lower Temperatures

Temperature distribution T(x, t). For the free running CCA, it is required that the temperature distribution can be determined for nonzero room temperatures, which differ above and below the slab. For the CCA with water flow, the upper and lower part of the slab are treated separately: x = 0 at the tube level, where the water supply temperature acts and x = L is, respectively the upper and lower slab surface, where the room temperatures act.

For both operational modes, the analytical expression (Equation 4) is extended in this paper to nonzero and different temperatures below and above the slab. This is implemented by separating the temperature variable T(x, t) into u(x) and w(x,t), as suggested by Carslaw and Jaeger (1959). The transient heat diffusion equation with corresponding boundary and initial conditions is written as follows. (For the water flow mode, [T.sub.2] must be replaced by [T.sub.2,1] and [T.sub.2,2] as shown in Figure 2[b]):

[partial derivative]T/[partial derivative]x = [alpha][[[partial derivative].sup.2]T/[partial derivative][x.sup.2]] (10)

[lambda][[partial derivative]T/[partial derivative]x] - [h.sub.1](T - [T.sub.1]) = 0, x = 0 (11)

[lambda][[partial derivative]T/[partial derivative]x] + [h.sub.2](T - [T.sub.2]) = 0, x = L (12)

T = f(x), t = 0 (13)

By substituting T(x, t) = u(x) + w(x, t) in these equations the following set of equations is obtained:

[[partial derivative].sup.2]u/[partial derivative][x.sup.2] = 0 or u = (Cx + D) (14)

[lambda][[partial derivative]u/[partial derivative]x] - [h.sub.1](u - [T.sub.1]) = 0, x = 0 (15)

[partial derivative][[partial derivative]u/[partial derivative]x] + [h.sub.2](u - [T.sub.2]) = 0, x = L (16)

and

[partial derivative]w/[partial derivative]t = [[partial derivative].sup.2]w/[partial derivative][x.sup.2] (17)

[lambda][[partial derivative]w/[partial derivative]x] - [h.sub.1] w = 0, x = 0 (18)

[lambda][[partial derivative]w/[partial derivative]x] + [h.sub.2] w = 0, x = L (19)

w = [bar.f](x) = f(x) - u, t = 0 (20)

Parameters C and D in Equation 14 can be found by using the appropriate boundary conditions (Equations 15 and 16):

[lambda]C - [h.sub.1] (Cx + D - [T.sub.1]) = 0, x = 0

[lambda]C + [h.sub.2](Cx + D - [T.sub.2]) = 0, x = L

resulting in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The solution for w(x, t) is equal to Equation 4, but with an adapted initial temperature distribution [bar.f](x) = f (x) - u.

Linearization of the

Initial Temperature Distribution f(x)

In order to avoid solving the integral in Equation 8 each time for different initial temperature distributionsf x), a general applicable approach is proposed: f(x) is approximated by discretizing the [T.sub.init]-profile into K parts and interpolating linearly between its values: [bar.f] [(x).sub.k ... k + 1] [approximately equal to] (M (x - O) + N) - (Cx + D) = M'x + N', where the values of M, N and O are determined by [T.sub.init](k) and [T.sub.init](k +1), k [member of] [0, K-1]. Combining this linearized initial temperature distribution with the integrand part of Equation 8 results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

This discretization approach offers an advantage for practical implementation of this analytical expression. If measurements in the CCA slab are available, these values can be used to define the initial state of the CCA slab.

The resulting temperature distribution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

In free running mode, for a situation with equal temperatures [T.sub.zone] below and above the slab, the C = 0 and D = [T.sub.zone]. In the case where [T.sub.zone] = 0 below and above the slab, the above expression for T reduces to the expression formulated by Carslaw and Jaeger (1959) (Equation 4).

Heat power [??] (x, t). With the known temperature distribution in the slab, the heat flux [W/[m.sup.2] = 0.317 Btu/(h x [ft.sup.2])] in the slab can be derived:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Equation 25 shows that the specific power reaches a steady-state value, determined by the parameter C, which is the steady-state heat flux [??] = [DELTA]T/R:

[[??].sub.t[much greater than]] = -[lambda]C = -[[[T.sub.2] - [T.sub.1]]/[[1/[h.sub.1]] + [L/[lambda]] + [1/[h.sub.2]]]] (26)

Cumulated specific heat q(x, t). The cumulated specific heat (J/[m.sup.2] = 8.8 x [10.sup.-5] Btu/[ft.sup.2]) is deduced from Equation 25:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Again, the specific heat reaches a steady-state value [lambda] Ct, which is linearly increasing with time, t. Moreover, since [[beta].sub.n] increases rapidly ([[beta].sub.1] [less than or equal to] [pi]/2, [[beta].sub.n > 2] = (n - 1 )[pi]), this relation can be simplified by using only the first term of the infinite sum.

[q.sub.t[greater than or equal to]] = -[lambda]{Ct + [[A.sub.1][[partial derivative][Z.sub.1](x)/[partial derivative]x][1/[alpha][[beta].sup.2.sub.1]]Z[k.sub.1, init]]} (28)

Combining the Solution for the Upper and Lower Slab Part for the CCA Slab in the Water Flow Mode

In the water flow mode, the approach to calculate the upper and lower slab part separately requires a correction when combining the two results. Since [T.sub.1] [not equal to] [T.sub.2] (different zone temperatures above and below the CCA slab), [h.sub.2,1] [not equal to] [h.sub.2,2] (different combined heat transfer coefficient at the slab surfaces) or [lambda]/[d.sub.1] [not equal to] [lambda]/[d.sub.2] (difference in slab thickness), the amount of heat transferred to the upper and lower part will also differ. If, as assumed up to now, the value of [h.sub.1] is equal for upper and lower slab part [([h.sub.1] = ([R.sub.z] + [R.sub.w] + [R.sub.t] + [R.sub.x]).sup.-1]/2), the T(x, t)-solution will result in a different revalue for upper and lower part. This is, considering the physical meaning of the equivalent core temperature [T.sub.c], impossible.

Figure 3 shows the difference between (a) the real heat flow pattern in the star network and (b) the resulting heat flow pattern by combining the solution for the upper and lower slab part. The correction factors [a.sub.1] and [a.sub.2] are introduced to make sure that [T.sub.c] = [T.sub.c1] = [T.sub.c2], and defined by

[q.sub.1]/q = [[1/[q.sub.1][R.sub.c]]([T.sub.ws] - [T.sub.c1])/[1/[R.sub.c]([T.sub.ws] - [T.sub.c])]] = [1/[a.sub.1]] (29)

[q.sub.1]/q = [[1/[q.sub.2][R.sub.c]]([T.sub.ws] - [T.sub.c2])/[1/[R.sub.c]([T.sub.ws] - [T.sub.c])]] = [1/[a.sub.2]] (30)

The correction factors [a.sub.1] and [a.sub.2] relate as 1/[a.sub.1] + 1/[a.sub.2] = 1. While solving the temperature distribution equation (Equation 24), an iteration is performed to find the values of [a.sub.1] and [a.sub.2] for which [T.sub.c] = [T.sub.c1] = [T.sub.c2]. The implemented analytical expression algorithm initiates with [a.sub.1] = [a.sub.2] = 1. With these [a.sub.1]- and [a.sub.2]-values, the temperature distribution in the upper slab part and in the lower slab part is calculated using Equation 24. At position x = 0 for each slab part, the fictive core temperatures [T.sub.c1] and [T.sub.c2] are determined. If [absolute value of ([T.sub.c1] - [T.sub.c2])] > [epsilon] (with [epsilon] are predefined threshold), then [a.sub.1] is increased or decreased (according to the sign of [[T.sub.c1] - [T.sub.c2]]) with a step [delta]. Again, the condition [absolute value of ([T.sub.c1] - [T.sub.c2])] > [epsilon] is checked and the iteration is repeated until the condition is met.

For reference, after circulating 30[degrees]C (86[degrees]F) water during 4 h through a CCA slab of 0.2m (0.66 ft) thickness with a room temperature of 20[degrees]C (68[degrees]F) above and below the slab, the correction values for the equivalent core thermal resistance [R.sub.c] are 1/[a.sub.1] = 1 - 1/[a.sub.2] = 0.51. Since a symmetrical slab is considered, with equal room temperatures above and below, this difference is induced by the different h-values above and below. The [a.sub.1]-value can be understood as follows:

For the uncovered slab in heating regime, the heat transfer coefficient at the ceiling is lower (thermal resistance larger) than at the floor surface ([h.sub.2,2] < [h.sub.2,1]). With equal room temperatures [T.sub.2,i] below and above the slab, [[??].sub.1] > [[??].sub.2]. If [a.sub.1] = [a.sub.2], then [T.sub.c1] < [T.sub.c2] (suppose heating regime, so [T.sub.ws] > [T.sub.2,i]), which is contradictory to the physical meaning of [T.sub.c] in the CCA-RC-model. In order to have [T.sub.c1] = [T.sub.c2] with [[??].sub.1] > [[??].sub.2], [a.sub.1] should be smaller than [a.sub.2]. An iteration is required to find the values of [a.sub.1] and [a.sub.2], because [a.sub.i][R.sub.c] is an integral part of the total resistance between T[.sub.ws] and [T.sub.2], which determines [[??].sub.i].

The approach of using the correction factors [a.sub.1] and [a.sub.2] can be seen as adapting the two competing parallel heat flows from the supply water temperature to the concrete node. In the above example, a larger [a.sub.2]-factor makes sure that more heat flows upwards while still having equal DT upwards and downwards.

Analytical Expression in Dimensionless Variables of Space and Time

In order to generalize the results, x and t are replaced by their dimensionless counterparts [xi] and Fo, together with the dimensionless Biot-number Bi. The temperature is not written in dimensionless form, since the initial temperature distribution f(x) is not a priori defined. With the dimensionless position, |[xi] = x/L, the Fourier number, Fo = t/([L.sup.2]/[alpha]), the Biot number at surface 1 and 2 [Bi.sub.i] = ([h.sub.1]L)/[lambda] (i = 1, 2),

[[beta]'.sub.n] = [beta]L,C' = CL, D' = D, [Z'.sub.n, init]([xi]) (1/h1)[Z.sub.n,init],

Equations 24-27 become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[Z'.sub.n]([xi]) = [[[beta]'.sub.n]/[Bi.sub.2]]cos([[beta]'.sub.n][xi]) + sin([[beta]'.sub.n][xi]) (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

and [[beta]'.sub.n], n = 1, 2, ... the positive roots of

cot[beta]' = [[[beta]'.sup.2] - [Bi.sub.1][Bi.sub.2]]/[beta]'([Bi.sub.1] + [Bi.sub.2]) (35)

The specific heat flow is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

with [lambda]/L the heat flux per unit temperature difference (W/[[m.sup.2] x K] = 0.1761 Btu/[h x ft x [degrees]F]).

The specific heat becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

with L[rho]c cumulated specific heat per unit temperature increase (J/[m.sup.2]K = 4.9 x [10.sup.-5]Btu/[ft.sup.2]).

Analysis of the CCAT([xi], Fo) Equation

The decay factors in the solution for T and [??] (Equations 24, 25), and in the solution for q (Equation 27) show that after a certain time, the influence of the initial condition extincts. For the higher order terms in the analytical expression (n [greater than or equal to] 2), this already occurs at a time which is only a fraction of the system's time constant. Remember that the Fourier number Fo relates time to the time constant of the CCA slab. Supposing the nth term is negligible when its value reaches 1% of it's initial value, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

E.g., for two uncovered CCA-slabs with a thickness of 10 cm (0.33 ft) and of 40 cm (1.31 ft), for n = 2, this 1%-time [Fo.sub.1%,2] is Fo = 0.4 (t =1.1 h) and Fo = 0.3 (t =13.9 h), respectively, so around 1/3 of the slab's time constant. For n = 3 and n = 4, this is approximately 10% and 5% of the time constant for both slabs.

The same reasoning applied to the first order term (n = 1) leads to the time at which the temperature has almost reached steady state. For the same CCA-slabs as above, this time is Fo = 6.8 (t = 19 h) to Fo = 2.0 (t = 90 h) for 10 cm (0.33 ft) to 40 cm (1.31 ft) respectively. Figure 4 shows the difference between a calculation of T([xi], Fo) using 100 terms in the analytical expression and using only the first term. As presented in Figure 4, for small values of Fo, it is crucial to include more than the first term of the temperature equation in order to obtain an accurate solution. Although the n = 2-decay term is very small for Fo > 0.4, the joint effect of the [T.sub.n-sum] (n = 1..100) affects the solution up to Fo [approximately equal to] 1. Further analysis of the solution (not shown in the figure) demonstrates that this holds more for larger Bi-numbers, e.g. thick CCA-slabs or for the free running CCA. For CCA with water flow, since upper and lower parts are treated separately, the Bi-numbers are lower, for which the (n [greater than or equal to] 2)-terms in the T(x, t)-solution decay more rapidly. However, as a general rule, using Fo = 1 as a threshold to neglect the n [greater than or equal to] 2-terms in the temperature equation seems appropriate.

APPLICATION OF THE ANALYTICAL EXPRESSION

Case 1: Heating Up CCA for Eight Hours

As an illustration, the solution of the analytical expression is presented for a CCA-slab being heated for 8 h with water at 30[degrees]C (86[degrees]F) (Table 4 shows the CCA-specifications). The initial temperature is 20[degrees]C (68[degrees]F) and the temperatures of the zones below and above are kept constant at 20[degrees]C (68[degrees]F). Heat transfer is considered in both directions, upwards and downwards, so into both zones.

The time is presented by the dimensionless Fourier number, with Fo = 1 [??] t = 11.2 h. The calculation is performed with a time step of 10 minutes, while Figure 5 shows the hourly results. The results are presented in 6 plots, (a)-(f), which are explained in Table 5.

From Figure 5, different aspects of the transient heat transfer occurring in the concrete slab can be analyzed. Initially, in Figure 5(a), the temperature increases more in the middle than at the edges, gradually evolving towards the steady-state situation given by C[xi] + D (see Equation 31). The specific heat power profile Figure 5(b)and 5(e) follows this trend with initially a large difference between the power transferred from the water to the CCA and the power transferred from the CCA to the room, while evolving to the steady-state heat power -[lambda]C (see Equation 25), which is 65 W/[m.sup.2] (20.6 Btu/h/[ft.sup.2]) in this case. If the heat pump is required to keep the 30[degrees]C (86[degrees]F) supply water temperature, its thermal power should be larger than the steady-state power for which the unit would conventionally be designed. A maximum of 153 W/[m.sup.2] (48.5 Btu/h/[ft.sup.2]) is attained (reached in the first hour of operation), which is a factor 2.4 higher compared to the steady-state design power. The centre of the CCA is loaded with heat prior to heat transfer starting up at the room side (Figure 5(c) and 5(f)). This results in a large amount of heat stored in the slab after 8 h runtime: 0.53 kWh/[m.sup.2] (0.17 Btu/[ft.sup.2]), as can be seen from the difference between the two curves in Figure 5(f).

Using a simplified coefficient of performance (COP) expression, the CCA operation is related to the electricity use of a heat pump (Table 6). Similar, the circulation pump electricity use is estimated. The electricity use [solid line (-) in Figure 5(d)] is directly related to the cumulated heat [Figure 5(f)] at [xi] = 0.5, the tube level of the slab. The electricity consumption of the circulation pump is only a minor fraction of the heat pump electricity use.

Pump Operation Time

Since the thermal resistance associated with heat conduction through the concrete slab is about double the one between the supply water and the concrete, it is possible to save on circulation pump operation time and therefore save on electricity cost. Shutting down the pump for a period of time allows the heat to diffuse through the concrete. To analyze this effect, five different cases are investigated, as shown in Figure 6.

For all cases, [T.sub.ws] = 30[degrees]C (86[degrees]F) and [T.sub.Room] = 20[degrees]C (68[degrees]F), which means the CCA is heated by the water flow. Figure 7 depicts the specific cumulated heat (a) from water to CCA, (b) from CCA to both rooms, and (c) stored in the CCA-slab. Each figure shows five lines, corresponding to one of the pump operation schedules.

This shows the following trends for the different pump operation modes. For pump-cases 2 to 5, the total pump runtime is equal over the 8 h period and half of the runtime of pump-case 1. After 8 h, the heat input from the water to the concrete slab appears to be almost equal for pump-cases 2-5 [Figure 7(a)]. However, the cumulated heat output to the rooms below and above differ from case to case [Figure 7(b)]. For a longer continuous operation time of the pump (e.g. pump-case 2), the surface temperatures of the CCAs reach higher values, resulting in a higher specific power output to the rooms. Numerical values supporting this observation are given in Table 7, which presents in the first row the mean specific heat power to the room over the 8 h period for the different pump-operation cases. Since the CCA has a large thermal capacity, operating the system implies that the CCA is loaded with heat.

Table 7 shows that operating the CCA with longer continuous pump operation times (pump-case 2) also leads to the lowest [q.sub.stored]. Pump-cases 3-5 are comparable, while pump-case 1 has the largest amount of energy stored in the concrete slab (which is mainly due to the larger amount of heat added by the water, which has been flowing for 8 h. Comparing pump-case 1 and pump-case 2 shows that, although pump-case 1 has 26% more heat output to the room, 71% more heat input was needed to reach this situation. The remainder of the heat is stored in the CCA. In pump-case 2 an amount of heat, almost equal to pump-case 3-5, is transferred to the CCA, but more heat is transferred to the room, due to the higher surface temperatures. As a consequence, less heat is stored.

These observations indicate that, in order to use the CCA as heating or cooling device, the pump operation time should be as continuous as possible for a certain operation time in order to reach high surface temperatures. On the other hand, if the CCA is used to store energy, rather than transferring heat to the office zones, it is beneficial to operate the circulation pump intermittently.

Simplified q(Fo) Expression

The previous conclusion suggests that a simplified expression that determines the time needed to reach a required amount of heat transferred or stored [q.sub.req] would provide essential control information towards star-up time and pump cycle time. In Figure 4 it was demonstrated that calculating the temperature profile with only the first term of the infinite sum, only yields acceptable results for Fo > 1. However, this appears not to be true for the cumulated heat q.

Figure 8 shows the detailed solution in solid line (-) (Equation 27) and the solution where only the first term of the sum is taken into account (markers x and *). The dashed line is the steady-state solution L [rho] cCFo indicating the slope to which both [q.sub.in] and [q.sub.out] evolve. The A markers present the stored energy [q.sub.sto], which is the difference between [q.sub.in] and [q.sub.out]. For small Fo numbers, there is a difference between the exact and the approximated solution. After the first 10 minutes, the relative error is 50% for [q.sub.in] and is very large (3.107%) for [q.sub.out]. The relative error drops below 5% for [q.sub.in] already at Fo > 0.1 and for [q.sub.out] at Fo > 0.6. [q.sub.out] is the most critical parameter. However, since [q.sub.out] is very small in the beginning (only 0.01 kWh/[m.sup.2] for Fo > 0.1), the absolute error remains small (see Figure 8). There fore, the q-solution with the first term can be used as an approximation of the cumulated heat. Starting from Equation 27, the cumulated heat transferred at the room side and at the water side can be approximated as presented in Equation 39 and Equation 40, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

Defining [K.sub.1] = ([A.sub.1]/[[beta].sub.1])[ cos ([[beta].sub.1]) - ([[beta].sub.1]/[Bi.sub.1]) sin ([[beta].sub.1])][Z.sub.1,init] and [K.sub.2] = ([A.sub.1]/[[beta].sub.1])[Z.sub.1,init], these expressions can be rearranged to Fo, leading to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

Furthermore, the heat stored in the slab is found by putting [q.sub.sto](Fo) = q(0, Fo) - q(1, Fo). The term C Fo is cancelled out, resulting in:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

Equations 41 and 42 are written in the form 1/[e.sup.x] = x, which is the exponential variant of the golden section equation 1/x = x - 1 (Hayes 2005). The solution of this exponential variant is the omega constant and is found by evaluating the Lambert W function in 1: W(1) = 0.567. This Lambert W function is used to find a solution for Fo for both heat transfer modes and the stored heat:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

Fo = -[1/[[beta].sup.2.sub.1]]ln[1 + [[q.sub.sto]/L[rho]c([K.sub.2] - [K.sub.1])]] (46)

The above expressions are extremely useful for control purposes. They provide an analytical solution to find the time required to either transfer a certain amount of heat to the room [q.sub.room] (Equation 44) or to put an amount of heat into the slab [q.sub.water] (Equation 45) or to store an amount of heat in the slab [q.sub.sto] (Equation 46). Moreover, this is achieved by taking into account the initial state of the slab (through [Z.sub.1,init]), the heat transfer rate at both sides of the slab (through [Bi.sub.1] and [Bi.sub.2]) and the water and room temperatures [T.sub.1] and [T.sub.2,i] (through the parameter C).

The Fo expressions are valid for both the case with water flow, where every slab part is treated separately, and for the case without water flow. Figure 9 shows the three different cumulated heat flows for a slab without water flow, initially at a temperature of 25[degrees]C (77[degrees]F) and cooling down towards the room temperature of 20[degrees]C (68[degrees]F).

CONCLUSIONS

An analytical solution is presented to calculate the transient heat transfer as a function of location in the slab x and of time t. This results in an (x, t)-expression for temperature, heat power and cumulated heat. The expressions can be used to analyze the transient behavior of CCA in a heating, cooling or free running situation.

A simplified expression can be derived from the detailed analytical expression relating time with the cumulated heat [q.sub.water] [right arrow] CCA, qCCA [right arrow] room and [q.sub.stored]. The simplified expression can be used to determine the time required to obtain a certain amount of heat transfer from CCA to room, from water to CCA or for thermal energy storage in the CCA-slab, while taking into account heat transfer parameters and initial slab conditions. The simplified expression is valid both for heating and cooling and is useful to determine start-up times and circulation pump cycle times of CCA, which are important control parameters in a CCA installation.

NOMENCLATURE

A              =  heat transfer surface area, [m.sup.2] = 10.76
                  [ft.sup.2]

[A.sub.n]      = heat transfer parameter in the analytical expression,
                 [m.sup.-1/2] [m.sup.2]K/W = 18.6 [ft.sup.-1/2] x
                 [ft.sup.2] x h x [degrees]F/Btu

a              = correction factor for difference in upward and
                  downward heat transfer,

c              = specific heat capacity, J/kg x K = 2.39 x [10.sup.-4]
                 Btu/(1b x [degrees]F)

C              = thermal capacitance, J/K = 5.26 x [10.sup.-4]
                 Btu/[degrees]F)

C              = variable in the analytical expression, K/m = 0.3048
                 K/ft

D              = variable in the analytical expression, K

d              = CCA thickness, m = 3.28 ft

d              = diameter, m = 3.28 ft

[d.sub.i]      = thickness of slab part i, m = 3.28 ft

[d.sub.x]      = tube spacing, m = 3.28 ft

[d.sub.t,o]    = tube outer diameter, m = 3.28 ft

M              = initial temperature profile of the concrete slab, K

h              = heat transfer coefficient, W/([m.sup.2] x K) =
                 0.176 Btu/h x ft x [degrees]F

[K.sub.i]      = (i = 1, 2), K

L              = length, m = 3.28 ft

[[??].sub.w]   = water flow rate, kg/s = 132.3 lb/min

q              = specific heat, J/[m.sup.2] = 8.8 x [10.sup.-5]
                 Btu/[ft.sup.2]

[??]           = specific heat power, W/[m.sup.2] = 0.317 Btu/
                 (h x [ft.sup.2])

[R.sub.cond]   = thermal conductive heat transfer resistance, K/W =
                 0.528 h x [degrees]F/Btu

[R.sub.concr]  = thermal resistance from water to concrete slab, K/W =
                 0.528 h x [degrees]F/Btu

[R.sub.conv]   = thermal convective heat transfer resistance, K/W =
                 0.528 h x [degrees]F/Btu

[R.sub.di]     = thermal conductive heat transfer resistance of slab
                 part i, K/W = 0.528 h x [degrees]F/Btu

[R.sub.i]      = thermal resistance, K/W = 0.528 h x [degrees]F/Btu

[R.sub.o]      = thermal resistance, K/W = 0.528 h x [degrees]F/Btu

[R.sub.a]      = thermal resistance from tube wall to upper
                 CCA surface, K/W = 0.528 h x [degrees]F/Btu

[R.sub.b]      = thermal resistance from tube wall to lower CCA
                 surface, K/W = 0.528 h x [degrees]F/Btu

[R.sub.d]      = Thermal resistance to heat transfer between upper
                 and lower CCA surface, K/W = 0.528 h x [degrees]F/Btu

[R.sub.x]      = equivalent core thermal resistance, K/W =
                 0.528 h x [degrees]F/Btu

[R.sub.t]      = thermal conductive resistance of tube wall, K/W =
                 0.528 h x [degrees]F/Btu

[R.sub.w]      = thermal resistance from water to tube wall
                 (forced convection), K/W = 0.528 h x [degrees]F/Btu

[R.sub.z]      = thermal resistance from water supply temperature to
                 mean water temperature, K/W = 0.528
                 h x [degrees]F/Btu

t              = time, s

T              = temperature, [degrees]C ([degrees]F)

T              = concrete core temperature, [degrees]C ([degrees]F)

[T.sub.s]      = surface temperature, [degrees]C ([degrees]F)

[T.sub.t]      = tube outer surface temperature, [degrees]C
                ([degrees]F)

[T.sub.wm]     = mean water temperature, [degrees]C ([degrees]F)

[T.sub.ws]     = water supply temperature, [degrees]C ([degrees]F)

u(x)           = intermediate temperature variable in the analytical
                 expression, K

w(x, t)        = intermediate temperature variable in the analytical
                 expression, K

x              = position variable, m = 3.28 ft

[Z.sub.n]      = position parameter in the analytical expression,
                 W/([m.sup.2] x K) = 0.176 Btu/(h x ft x [degrees]F)

Zn, init       = initial condition parameter in the analytical
                 expression, [m.sup.1/2] K= 1.811 [ft.sup.1/2] x K

Dimensionless Numbers

[xi]  = dimensionless distance
Fo    = dimensionless time
Bi    = Biot number

Greek Symbols

[alpha]   = k

[alpha]   = thermal diffusivity, [m.sup.2]/s = 10.76 [ft.sup.2]/s

[beta]n   = parameter in the analytic expression, 1/m = 0.3048 1/ft

[lambda]  = thermal conductivity, W/(m x K) = 6.935 Btu x in./h x
           [ft.sup.2] x [degrees]F

[rho]     = density, kg/[m.sup.3] = 0.0625 lb/[ft.sup.3]

[tau]     = time constant, s

Subscripts

c         = concrete core

c         = concrete

d1        = thickness of upper part of the concrete slab

d2        = thickness of lower part of the concrete slab

d         = slab thickness

init      = initial

s         = (concrete) surface

ss        = steady state

t         = tube

t, i      = tube inner

t, o      = tube outer

w         = water

ws        = water supply

w, mean   = water mean


REFERENCES

Antonopoulos, A., M. Vrachopoulos, and C. Tzivanidis. 1997. Experimental and theoretical studies of space cooling using ceiling-embedded piping. Applied Thermal Engineering, 17(4): 351-67.

Awbi, H., and A. Hatton. 1999. Natural convection from heated room surfaces. Energy and Buildings, 30(3): 233-44.

Babiak, J., M. Minarova, and B.W. Olesen. 2007. What is the effective thickness of a thermally activated concrete slab? In Clima 2007 WellBeing Indoors, page 5, Helsinki, Finland.

Barton, P., C.B. Beggs, and P.A. Sleigh. 2002. A theoretical study of the thermal performance of the TermoDeck hollow core slab system. Applied Thermal Engineering, 22: 1485-99.

Braun, J.E. 2003. Load control using building thermal mass. Journal of Solar Energy Engineering, 125(3): 292.

Carslaw, H., and J. Jaeger. 1959. Conduction of heat in solids. Oxford University Press, London, 2nd edition.

S. Cho and M. Zaheer-uddin. 2003. Predictive control of intermittently operated radiant floor heating systems. Energy Conversion and Management, 44(8): 1333-2.

European Committee for Standardization. 2007. EN 15377-3, Heating systems in buildings--Design of embedded water based surface heating and cooling systems--Part 3: Optimizing for use of renewable energy sources.

Ferkl, L. 2013. System operation and controller development. Technical report, Work Package 6 of ERASME-project GEOTABS: Towards optimal design and control of geothermal heat pumps combined with thermally activated building systems in offices, coordinator: KULeuven (www.geotabs.eu).

Fort, K. 2001. TRNSYS model Type 360: Floor heating and hypocaust. Available at www.transsolar.com/_software/download/de/ts_type_360_de.pdf, consulted.

Gluck, B. 1999. Thermisch Bauteilaktivierung (Bauteilheizung unf Bauteilk'uhlung). Rud. OttoMeyer-UmweltStiftung, Hamburg, Germany.

Gwerder, M., and J. Todtli. 2005. Predictive Control for Thermal Storage Management in Buildings. In 8th REHVA World Congress for Building Technologies CLIMA, pages 1-6, Lausanne.

Gyalistras, D., and M. Gwerder. 2010. Use of Weather and Occupancy Forecasts For Optimal Building Climate Control (OptiControl): Two Years Progress Report. Technical Report September, Terrestrial Systems Ecology ETH Zurich, Building Technologies Division, Siemens Switzerland Ltd., Zug, www.opticontrol.ethz.ch.

Hayes, B. 2005. Why W? On the Lambert W function, a candidate for a new 'elementary' function in mathematics. American Scientist, 93(2): 104-08.

Hoh, A., T. Tschirner, and D. Muller. 2005. A combined thermo-hydraulic approach to simulation of active building components applying Modelica Simulating basic building behavior. In 4th International Modelica Conference, Hamburg, Germany. March 7-8.

Koschenz, M., and B. Lehmann. 2000. Thermoaktive Bauteilsysteme tabs. EMPA Energiesysteme/Haustechnik, Duebendorf (Switzerland).

Meierhans, R. 1993. Slab cooling and earth coupling. ASHRAE Transactions, 99(2): 8.

Olesen, B.W., M.D. Carli, M. Scarpa, and M. Koschenz. 2006. Dynamic evaluation of the cooling capacity of thermo-active building systems. ASHRAE Transactions, 112(2).

Privara, S., J. Siroky, L. Ferkl, and J. Cigler. 2011. Model predictive control of a building heating system: The first experience. Energy and Buildings, 43(2-3): 564-72.

Ren, M.J., and J.A. Wright. 1998. Slab thermal storage system. Building and Environment, 1323(97): 43-52.

Russell, M.B., and P.N. 2001. Surendran. Influence of active heat sinks on fabric thermal storage in building mass. Applied Energy, 70: 17-33.

Schmidt, D. 2004. Methodology for the Modelling of Thermally Activated Building Components in Low Exergy Design (Doctoral Thesis). PhD thesis, Royal Institute of Technology.

Sourbron, M. 2012. Dynamic behaviour of buildings with Concrete Core Activation. PhD thesis. PhD thesis, KU Leuven, Leuven.

Weber, T., and G. Johannesson. 2005. An optimized RC-network for thermally activated building components. Building and Environment, 40(1): 1-14.

Weber, T., G. Johannesson, M. Koschenz, B. Lehmann, and T. Baumgartner. 2005. Validation of a FEM-program (frequency-domain) and a simplified RC-model (time-domain) for thermally activated building component systems (TABS) using measurement data. Energy and Buildings, 37(7): 707-24.

Weitzmann, P., and S. Svendsen. 2005. Detailed measurements and modelling of thermo active components using a room size test facility. In 7th Nordic Building Physics Symposium, Reykjavik, Iceland.

Maarten Sourbron is a post-doctoral researcher in the Faculty of Engineering Technology and Lieve Helsen and Martine Baelmans are professors in the Department of Mechanical Engineering at University of Leuven, Leuven, Belgium.

Table 1. Overview of Numerical CCA Models Used in Dynamic
Simulation Applications

(FD = Finite Difference, FV = Finite Volume, or
FE = Finite Element)

Reference               Type              Application

Meierhans (1993)         FD         Model of a full concrete
                                  CCA-slab comparison of a 3D-
                                 and 2D-method (neglecting the
                                 z-direction along the tubes),
                                    and experimental results

Antonopoulos et          FD         Comparison of a 3D- and
al. (1997)                         2D-method (neglecting the
                                 z-direction along the tubes),
                                    and experimental results

Russell and              FD         Analysis of the cooling
Surendran (2001)                 capacity and parametric study

Olesen et al. (2006)     FD        Model incorporated in the
                                 European standard EN15377-3 on
                                  water based embedded heating
                                 and cooling systems (European
                                 Committee for Standardization
                                             2007)

Fort (2001)              FD       Model for an embedded water
                                 based system integrated in the
                                  transient system simulation
                                         program TRNSYS

Barton et al. (2002)     FD      Analysis of a hollow core CCA
                                   for which conditioned air
                                    flows through the hollow
                                  cores. Bends are taken into
                                    account by an equivalent
                                         bend' length.

Hoh et al. (2005)        FV      Combination of a FV-model and
                                     hydraulic model in the
                                       Modelica-language

Weitzmann and            FV       Validation with measurements
Svendsen (2005)                  showed a good correlation for
                                  heat flows, but temperature
                                      deviations up to 2K

Babiak et al. (2007)     FE      Model used to observe a depth
                                 of on average 15 cm at which a
                                  sinusoidal room temperature
                                     with a 24 h period is
                                    transmitted through the
                                         concrete slab.

Table 2. Thermal Resistances in the One-Dimensional
Simplified CCA Model (Koschenz and Lehmann 2000)

Name, K/W               Thermal Resistance to Heat Transfer

[R.sub.a]       Between tube wall to upper CCA surface (conduction)

[R.sub.b]       Between tube wall to lower CCA surface (conduction)

[R.sub.d]         Between upper and lower CCA surface (conduction)

[R.sub.di]      Between upper (i = 1) or lower (i = 2) CCA slab and
                 the room at temperature [T.sub.zone,i] (conduction
                                  and convection)

[R.sub.x]        Between the tube wall and the Active concrete core
                  (Equivalent core thermal resistance--conduction)

[R.sub.t]          Between inner en outer tube wall (conduction)

[R.sub.w]           From water to tube wall (forced convection)

[R.sub.z]     From water supply temperature to mean water temperature

Table 3. Parameters of the CCA-Slab

Parameter                Slab in Free            Slab in Water
                         Running Mode              Flow Mode

                         Figure 2 (a)             Figure 2 (b)

                      Whole Slab Treated      Upper and Lower Part
                          in 1 Piece          of the Slab Treated
                                                   Separately

                         [T.sub.1] =              [T.sub.1] =
                      [T.sub.zone,below]       [T.sub.ws], water
                                               supply temperature
                                                     to CCA

Temperature              [T.sub.2] =             [T.sub.2,1] =
                      [T.sub.zone,above]      [T.sub.room, above]
                                                 [T.sub.2, 2] =
                                              [T.sub.room, below]

                            L = d                L = [d.sub.i]
                                                   (i = 1, 2)

                     x = 0 at the ceiling      x = 0 at the tube
                        surface (lower               level
                           surface)

Slab thickness L          T (0, t) =               T (0, t) =
                       [T.sub.ceiling-        [T.sub.c], the mean
                           surface]             core temperature

                      x = d at the floor      x = [d.sub.i] at the
                        surface (upper         upper or lower CCA
                           surface)                 surface

                          T (d, 0 =            T ([d.sub.i], t) =
                        [T.sub.floor-          [T.sub.surface i]
                           surface]

Heat transfer         [h.sub.1]: global           [h.sub.1] =
coefficient             heat transfer         ([R.sub.z] + Rw + Rt
                       coefficient for         + Rx)/' /ai is the
                        radiation and           equivalent heat
                      convection at the       transfer coefficient
                           ceiling              between Tws and
                                                 [T.sub.c], see
                                                 Introduction.

                      [h.sub.2]: global         [h.sub.2,1] and
                        heat transfer         [h.sub.2,2]: global
                       coefficient for           heat transfer
                        radiation and           coefficient for
                      convection at the          radiation and
                            floor              convection to the
                                                 adjacent zone

Table 4. Specifications of the CCA-Slab Used to Illustrate
the Analytical Expression (Same Layout as in Figure 2,
Uncovered CCA, Tubes in the Centre of the Slab)

                                 SI                  I-P
Thickness, d                    0.2 m              0.66 ft
Tubes spacing, [d.sub.x]       0.15 m              0.49 ft
Area, [A.sub.net]           10 [m.sup.2]        102 [ft.sup.2]
Tube length, [L.sub.t]         66.7 m               219 ft
Flow rate, [m.sub.w]         0.042 kg/s,         5.56 lb/min,
                              150 kg/h            20000 lb/h

Calculated [R.sub.Z]          0.006 K/W     0.001 h x [degrees]F/
                                                   Btu x ft

Table 5. Explanation of Figures 5 (a)- (c) Showing the
Transient Calculation Results

Figure Number          x-Axis Units         y-Axis
Showing                                    Linetype

(a)                     T (%, Fo)            [xi]

Temperature             [degrees]C            (-)
                       ([degrees]F)

(b)                    [??] (%, Fo)          [xi]

Specific heat      W/[m.sup.2] = 0.317        (-)
power             Btu/ (h x [ft.sup.2])

(c)                         q                [xi]

Specific          kWh/[m.sup.2] = 316.9       (-)
cumulated heat      Btu/ x [ft.sup.2]

Figure Number                        Explanation
Showing

(a)                The thick dashed line (- -) shows the upper and
                                  lower CCA surface

Temperature       Temperature distribution with a1h time step. E.g.,
                   when heating, T ([xi], Fo) is evolving from left
                                      to right.

(b)                 The sign is determined by the direction of the
                  heat flow; positive in the positive [xi]-direction
                  and negative in the negative [xi]-direction. E.g.,
                   in a heating situation, [??] is positive for the
                     upper part and negative for the lower part.

Specific heat     With water flow, the [??]-profile has the largest
power             difference between the centre and the surfaces at
                    small Fo, and evolves to a constant value (see
                   E[??]uation 26) for large values of Fo (after a
                   long time). Without water flow, the [??]-profile
                         has no discontinuity at [xi] = 0.5.

(c)                       Equal sign convention as for [??]

Specific           The [??]-profile starts from 0 at Fo = 0. In the
cumulated heat     center ([xi] = 0.5), [??] increases more rapidly
                   when the water flow (heating or cooling) is on.

Table 6. Explanation of Figures 5(d)-(f) Showing the
Transient Calculation Results (Continuation of Table 5)

Figure nr.        x-axis units     y-axis Linetype    Explanation
showing

(d)                    Fo          [E.sub.elec]       For an air/
                                                      water heat
                                                      pump with a
                                                      COP = 3.9, a
                                                      chiller with
                                                      an COP = 3.5
                                                      and a
                                                      circulation
                                                      pump consuming
                                                      0.84 W/
                                                      [m.sup.2]
                                                      (0.27 Btu/h x
                                                      [ft.sup.2]) or
                                                      252W/l/s (6.5
                                                      (Btu/h)/lb/
                                                      min)).

[E.sub.elec,    kWh/[m.sup.2] =    (-)                Cumulative
heatpump/          316.9 Btu/                         electricity
chiller]           [ft.sup.2]                         use for
                                                      production:
                                                      heat pump (in
                                                      heating mode)
                                                      or chiller (in
                                                      cooling mode).

[E.sub.elec,    kWh/[m.sup.2] =    ([+ or -])         Cumulative
total]             316.9 Btu/                         electricity
                   [ft.sup.2]                         use for
                                                      production and
                                                      circulation
                                                      pump.

(e)                    Fo          [??]               Specific heat
                                                      power

[[??].sub.       W/[m.sup.2] =     (x)                Sum of heat
water-CCA]       0.317 Btu/(h x                       power to upper
                  [ft.sup.2])                         and lower slab
                                                      part

[[??].sub.       W/[m.sup.2] =     (-)                Sum of heat
CCA-room]        0.317 Btu/(h x                       power to the
                  [ft.sup.2])                         room below and
                                                      above

(f)                    Fo          q                  Specific
                                                      cumulated heat

[q.sub.         kWh/[m.sup.2] =    ([degrees])        Sum of heat to
water-CCA]         316.9 Btu/                         upper and
                   [ft.sup.2]                         lower slab
                                                      part

[q.sub.         kWh/[m.sup.2] =    (-)                Sum of heat to
CCA-room]          316.9 Btu/                         the room below
                   [ft.sup.2]                         and above

Table 7. Characteristic Values over the 8 h Period
for Cases 1-5

Result                                       Case

                               1               2               3

[[??].sub.output,mean]    34.4 (10.9)     27.3 (8.7)      23.6 (7.5)
  x W/[m.sup.2] [Btu/
  (h x [ft.sup.2])]
[q.sub.input]             0.8 (253.5)    0.47 (148.9)    0.48 (152.1)
  (kWh/[m.sup.2])
  (Btu/[ft.sup.2])
[q.sub.output]            0.28 (88.7)     0.22 (69.7)     0.19 (60.2)
  (kWh/[m.sup.2])
  (Btu/[ft.sup.2])
[q.sub.stored]           0.53 (168.0)     0.25 (79.2)     0.29 (91.9)
  kWh/[m.sup.2]
  (Btu/[ft.sup.2])
[q.sub.stored]/               66              53              61
  [q.sub.input] (%)

Result                               Case

                               4               5

[[??].sub.output,mean]    21.4 (6.8)       20 (6.3)
  x W/[m.sup.2] [Btu/
  (h x [ft.sup.2])]
[q.sub.input]            0.48 (152.1)    0.47 (148.9)
  (kWh/[m.sup.2])
  (Btu/[ft.sup.2])
[q.sub.output]            0.17 (53.9)     0.16 (50.7)
  (kWh/[m.sup.2])
  (Btu/[ft.sup.2])
[q.sub.stored]            0.31 (98.2)     0.31 (98.2)
  kWh/[m.sup.2]
  (Btu/[ft.sup.2])
[q.sub.stored]/               64              66
  [q.sub.input] (%)
COPYRIGHT 2014 American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc. (ASHRAE)
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Author:Sourbron, Maarten; Helsen, Lieve; Baelmans, Martine
Publication:ASHRAE Transactions
Article Type:Report
Date:Jul 1, 2014
Words:9415
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