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Effective heat capacity of exterior Planar Thermal Mass (ePTM) subject to periodic heating and cooling
Energy and Buildings 47 (2012) 394–401
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Effective heat capacity of exterior Planar Thermal Mass (ePTM) subject to periodic heating and cooling Peizheng Ma ∗ , Lin-Shu Wang Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300, United States
a r t i c l e
i n f o
Article history: Received 13 June 2011 Received in revised form 3 October 2011 Accepted 12 December 2011 Keywords: Dynamic heat transfer Planar Thermal Mass (PTM) Heat capacity Thermal storage Periodic heating and cooling
a b s t r a c t Thermal mass can be effectively used for controlling temperature in buildings. Based on its location and function, building thermal mass can be classified as interior thermal mass and exterior (envelope) thermal mass. In this paper, the dynamic heat transfer performance of exterior Planar Thermal Mass (ePTM) subject to sinusoidal heating and cooling is investigated. When the mean value of outdoor air temperature is equal to indoor air temperature, solutions of temperature distribution and heat flux in ePTM are deduced analytically. Using the analytical solutions, heat exchange and thermal storage of ePTM are attained. Correspondingly, the time lag and the decrement factor are obtained; they show similarity characteristics—independent of environment temperatures. When the two temperatures are not equal, an approximated temperature distribution solution is developed based on the principle of superposition, which has been validated by numerical solutions. The approximated solution is then used to investigate the heat exchange between the ePTM and the environment. In contrast to the common belief that a large thermal mass has an apparent insulating effect, no “insulating effect of mass” under sinusoidal heating and cooling is found in this investigation. Another two rules of thumb based on common belief are corrected: a wood wall is better than a concrete wall of same thickness as exterior walls in terms of the time-lag effect and the decrement factor. The effect of the time lag is not the longer the time lag is the better its contribution to thermal comfort becomes. © 2011 Elsevier B.V. All rights reserved.
1. Introduction and problem description Based on its location and function, building thermal mass can be classified as interior thermal mass (such as furniture, appliances and interior walls, ceiling and floors) and exterior (envelope) thermal mass (such as exterior walls, roof/ceiling, floors and windows). In Ref. [1], analytical method was used to solve the dynamic heat transfer problem of interior Planar Thermal Mass (iPTM) subject to sinusoidal heating and cooling, and the effective thermal storage capacity of the iPTM was investigated. For exterior Planar Thermal Mass (ePTM), unlike iPTM, we are concerned more about the heat exchange between the ePTM and the indoor air, the time-lag effect and the decrement factor (which will be explained later), rather than the thermal storage capacity. In this paper, the dynamic heat transfer problem of ePTM under external sinusoidal thermal wave and internal constant thermal environment will be investigated.
Consider the ePTM shown in Fig. 1. The ePTM surface temperature Ts is a sinusoidal function with a mean value Tm [K or ◦ C]. The peak amplitude and the period of Ts are (T)s and P, respectively. Suppose that all the thermo-physical properties of the ePTM (i.e., density [kg/m3 ], mass specific heat cp [kJ/kg K], volumetric specific heat cvol [kJ/m3 K], conductivity k [W/m K], and thermal diffusivity ˛ [m2 /s]) are constant. This problem can be dealt with as a one-dimensional (x direction) heat transfer problem. The temperature changes in the ePTM will repeat themselves periodically if the surface temperature variation has occurred for a sufficiently long period of time. In the absence of internal energy sources/sinks, the heat diffusion equation should be used to calculate the temperature distribution in the ePTM: G.D.E. :
∂2 T 1 ∂T = ˛ ∂t ∂x2
(1)
and the Boundary Conditions (B.C.s) of the Governing Differential Equation (G.D.E.) are: ∗ Corresponding author. Tel.: +1 631 745 8937. E-mail addresses: [email protected] (P. Ma), [email protected] (L.-S. Wang). 0378-7788/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2011.12.015
B.C.s :
T (0, t) = Ts = (T )s sin
2 P
t
+ Tm
(2)
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
395
Ts Temperature, T
k, ρ, cvol, cp, α
Ts
TsL
Outdoor air, Tout
L
Tm (ΔT)s
Tin
Indoor air, Tin
Time, t Note: Tin may be higher, lower or equal to Tm
0
x Fig. 1. EPTM and its surface temperatures.
−k
∂T ∂x
= h[T (L, t) − Tin ] = h[TsL − Tin ]
(3)
L
where Tin is the indoor air temperature, TsL is the inside surface temperature, and h [W/m2 K] is the heat transfer coefficient. In this paper, h = 8.10 W/m2 K is chosen for the indoor air film [2]. 2. Analytical solution when Tin = Tm Generally, the indoor air temperature is not constant and is not equal to Tm . However, if the amount of the inside air is large enough, Tin can be considered as constant. In this paper, we will first treat Tin as constant and equal to Tm . Let (x, t) = T (x, t) − Tm
(4)
and when Tin = Tm , the G.D.E. and B.C.s can be rewritten as G.D.E. :
∂ ∂x
∂x2
=
1 ∂ ˛ ∂t
(5)
(0, t) = Ts − Tm = (T )s sin
B.C.s :
∂2
L
2 P
t
h = − (L, t) k
B.C.s :
∂c ∂x
L
(13)
h = − c (L, t) k
(14)
The general solution of the D.E. above can be gotten as √ √ c (x, t) = (C1 ex 2i/(˛P) + C2 e−x 2i/(˛P) )e2it/P
C1 = C2 =
(1 − ε + i)(T )s
(16)
(1 + ε + i)e2(1+i)ı + (1 − ε + i) (1 + ε + i)e2(1+i)ı (1 + ε + i)e2(1+i)ı + (1 − ε + i)
(T )s
hLeff ε≡ √ 2k
and Leff ≡
(18)
˛P
2˛P
˜ (0, t) = (T )s cos
B.C.s :
∂˜ ∂x
L
(8)
2 P
t
h˜ = − (L, t) k
(9)
(10)
Define a (complex) temperature function as: ˜ c (x, t) = (x, t) + i(x, t) (11) √ where i = −1. The complex function c (x, t) satisfies the following differential equation (D.E.) and B.C.s: D.E. :
∂2 c 1 ∂c = ˛ ∂t ∂x2
(19)
(20)
which are called dynamic Biot number, dimensionless thickness and effective thickness, respectively. Then
Using the method described in Ref. [1], introduce an auxiliary problem with the G.D.E. and B.C.s: ∂2 ˜ 1 ∂˜ = ˛ ∂t ∂x2
(17)
where
(6)
2.1. Temperature distribution in the ePTM
G.D.E. :
(15)
Subject the two B.C.s to the equation above, C1 and C2 can be obtained as
ı≡L
(7)
c (0, t) = (T )s e2it/P
(12)
Fig. 2. Inner effective heat exchange coefficient for certain ε and ı.
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P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
c (x, t) = (T )s
√ 2x/Leff
(1 − ε + i)e(1+i)
(1 + ε +
⇒ c (x, t) = (T )s
√ 2x/Leff
+ (1 + ε + i)e2(1+i)ı e−(1+i)
i)e2(1+i)ı
+ (1 − ε + i)
e2it/P
A1 + iA2 (A1 B1 + A2 B2 ) + i(A2 B1 − A1 B2 ) = (T )s B1 + iB2 B12 + B22
(21)
where
√ ⎧ ⎫ √ 2x/Leff ) cos(2ı− 2x/Leff +2t/P) e(2ı− ⎪ ⎪ ⎪ ⎪ √ √ ⎨ε ⎬ eff −e 2x/L √cos( 2x/Leff + 2t/P)
√ A1 = (22) eff √ e(2ı− 2x/L ) cos 2ı− 2x/Leff +2t/P+/4 ⎪ ⎪ ⎪ ⎪ √ √ ⎩+ 2 ⎭ eff 2x/L eff +e
cos( 2x/L
+ 2t/P + /4)
+e
sin( 2x/L
+ 2t/P + /4)
⎧ ⎫
√ √ 2x/Leff ) sin(2ı − 2x/Leff + 2t/P) e(2ı− ⎪ ⎪ ⎪ ⎪ √ √ ⎨ε ⎬ eff −e 2x/L √sin( 2x/Leff + 2t/P)
A2 = (23) √ eff 2x/L ) sin(2ı− 2x/Leff +2t/P+/4) ⎪ √ ⎪ e(2ı− ⎪ ⎪ √ √ ⎩+ 2 ⎭ eff 2x/L eff B1 = ε[e2ı cos(2ı) − 1] + [e2ı cos(2ı) − e2ı sin(2ı) + 1]
(24)
and
Fig. 3. Effective thermal storage coefficient.
B2 = ε[e2ı sin(2ı)] + [e2ı cos(2ı) + e2ı sin(2ı) + 1]
(25) where
At last, the temperature distribution in the ePTM can be obtained as (x, t) = (T )s
A2 B1 − A1 B2 B12
(26)
+ B22
√
L
ε,ı
2 = 2
F12 + F22
(37)
B12 + B22
and
2.2. EPTM heat exchange and thermal storage
eff
carea ≡ cp Leff
Since A1 and A2 are functions of x and B1 and B2 are not, ∂T ∂ (T )0 = −k = −k 2 ∂x ∂x B + B2
q = −k
1
B1
2
∂A2 ∂A1 − B2 ∂x ∂x
(27)
Let x = 0 and the heat flux at the outer surface can be gotten as
q
=−
0
√ 2k(T )0
D12 + D22 B12
Leff
+ B22
sin
2t
− ϕ0
P
(28)
which can be called inner effective heat exchange coefficient for certain ε and ı, as shown in Fig. 2, and effective area specific heat, respectively. As shown in Fig. 2, the heat exchange between the surface and the indoor air decreases as ı increases when ε is constant; and it increases as ε increases when ı is constant. The amount of heat stored in the ePTM in a period from t1 to t2 can be gotten as
where 2ı
D1 = ε(e
sin 2ı + e
2ı
2ı
cos 2ı + 1) + 2(e
cos 2ı − 1)
(29)
D2 = ε(e2ı sin 2ı − e2ı cos 2ı − 1) + 2e2ı sin(2ı)
B1 D1 + B2 D2 B1 D2 − B2 D1
(31)
Let x = L and the heat flux at the inner surface can be gotten as √ 2k(T )0
q L = −
Leff
where
F12 B12
+ F22 sin + B22
2t P
− ϕL
(32)
F1 = 2εe (sin ı + cos ı)
(34)
and B1 F1 + B2 F2 B1 F2 − B2 F1
(35)
At the inner surface, the amount of heat exchanged in the duration from t1 to t2 can be gotten as QL = A
t2
t1
√
2 q L dt = 4
F12 + F22
cp Leff (T )0 cos 2
B12 + B2
QL 1 L eff carea (T )0 cos = ε,ı A 2
2t − ϕstor P
2t P
− ϕL
t2 t1
2t P
− ϕL
t2 t1
(36)
2 = 4
2
(D1 − F1 ) + (D2 − F2 ) B1 2 + B2 2
2 eff
carea (T )0
t2 t1
1 stor eff Qstor = ε,ı carea (T )0 cos ⇒ 2 A
2t − ϕstor P
t2
(39)
t1
where tan ϕstor =
(33)
F2 = 2εeı (sin ı − cos ı)
−
qL )dt
t1
and
ı
tan ϕL =
√
t2
(q0
× cos
tan ϕ0 =
⇒
Qstor = A
(30)
and
(38)
√
stor
ε,ı
2 = 2
B1 D1 + B2 D2 − B1 F1 − B2 F2 B1 D2 − B2 D1 − B1 F2 + B2 F1
(40)
(D1 − F1 )2 + (D2 − F2 )2 B1 2 + B2 2
(41)
which is called effective thermal storage coefficient for certain ε and ı in this paper, as shown in Fig. 3. stor increases almost linearly when ı is small (ı ≤ 1 for In Fig. 3, ε,ı small ε ePTMs to ı ≤ 2 for large ε ePTMs), which means that the amount of stored heat energy increases almost linearly with the thickness of the ePTM when ı is not large. Generally, it is natural to try to thicken an ePTM to make it be a better thermal mass. However, from Fig. 3, we realize that if we continue to thicken the ePTM after a certain thickness, the heat stored in the ePTM actually decreases a little. That is because the previously stored energy flows out of the ePTM and interfaces the influent heat.
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397
3. The time-lag effect and the decrement factor when Tin = Tm For ePTMs, the time-lag effect and the decrement factor are much more important than that for iPTMs. The definitions [3] of the time lag ϕ and the decrement factor f are as follows:
ϕ=
tTi,min − tTe,min tTi,max − tTe,max
(42)
and Ti,max − Ti,min Te,max − Te,min
f =
(43)
where t means time, T means temperature, i means interior surface, e means exterior surface, max means maximum and min means minimum. When x = 0, the exterior surface temperature is Ts . Therefore, Te,min = Tm − (T)s when tTe,min = 3P/4 = 18 h, and Te,max = Tm + (T)s when tTe,max = P/4 = 6 h. When x = L, √ 2t 2 2eı (L, t) = (T )s (44) sin + +ı− L P 4 B2 + B2 1
Fig. 4. The time lag.
2
where = arctan
L
B2 ± n B1
(n = 0, 1, 2, 3, . . .)
(45)
For certain ε and ı, B1 and B2 are constant. Thus, L has maximum value: √ 2 2eı L,max = (T )s (46) B12 + B22 when tL,max =
P 2
arctan
B2 nP + −ı ± B1 4 2
(47)
and has minimum value: √ 2 2eı L,min = −(T )s B12 + B22 when tL,min
P = 2
(48)
B2 5 nP −ı ± arctan + B1 4 2
(49)
Therefore,
⎧
⎨ tTi,min − tTe,min = tL,min − 18 h = 1 arctan B2 + 5 − ı ± n − 18 h ϕ P B1 P P 2 4 2 = P ⎩ tTi,max − tTe,max = tL,max − 6 h = 1 arctan B2 + − ı ± n − 6 h P P B1 P 2 4 2 (50) and f =
Fig. 5. The decrement factor.
Ti,max − Ti,min L,max − L,min = = Te,max − Te,min (T )s − [−(T )s ]
√ 2 2eı
B12 + B22
(51)
The two equations above show the similarity nature of dimensionless ϕ/P and f – independent of environment temperatures. Thus, let us change the value of ε and ı to see what will happen to ϕ and f. Some curves of ϕ and f are plotted in Figs. 4 and 5, respectively. From Fig. 4, we can find that: when ı is smaller than about 1 and ε is the same, ϕ increases non-linearly as the increase of ı; when ı is the same, as ε increases from zero to positive infinity, ϕ decreases about 2.5 h, except when ı is smaller than about 1; when ε is the same, as ı increases from about 1 to about 6 or 7,
Fig. 6. The time lag of wood/concrete ePTMs (against ı).
ϕ increases almost linearly to 24 h (one period); at about 6 or 7, ϕ decreases from 24 h to 0 suddenly, which means that the time lags more than one period; and the shift point decreases as ε increases; the change is very small when ε is larger than about 5. In Fig. 5, the decrement factor is obviously non-linear. When ı is the same, as ε increases, f decreases. When ε trends to positive infinity, f is zero for all ı since both Ti,max and Ti,min are zero when the inside surface temperature keeps constant. When ε is the same, as ı increases, f decreases from nearly 1 to 0 and the decrease is very sharp when ı is small. Again taking wood and concrete for examples, the time lag and the decrement factor are shown in Figs. 6–11. In Fig. 6, when ı is the same, the time lag of a concrete wall is larger about 80 min than that of a wood wall, except when ı is smaller than about 1.
Fig. 7. The decrement factor of wood/concrete ePTMs (against ı).
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P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
4. Approximated solution when Tin = / Tm 4.1. Temperature distribution and heat flux / Tm , by using the principle of superposition, a linWhen Tin = ear part should be added because of the temperature difference of the two surfaces of the ePTM. From Eq. (26), the temperature distribution becomes Fig. 8. The time lag of wood/concrete ePTMs (against L).
II (x, t) = (T )s
A2 B1 − A1 B2 B12 + B22
− (Tm − TmL )
x L
(52)
Here, TmL , which is the mean value of the temperature at the ePTM interior surface, is an unknown to be determined. Consider the limit of a steady-state heat transfer problem by taking (T)s → 0, i.e., Ts = Tm . Then
Fig. 9. The decrement factor of wood/concrete ePTMs (against L).
Tm − TmL L/k 1 1 1 = = = = √ Tm − Tin L/k + 1/h 1 + k/hL 1 + 1/ıε 1 + 2k/ıhLeff (53) Let T = Tm − Tin
(54)
Eq. (52) can be changed into √ T 2x 1 + 1/ıε ıLeff B12 + B22 √ A2 B1 − A1 B2 2x = (T )s − 2 2 eff ı + 1/ε L B +B
II (x, t) = (T )s
Fig. 10. The time lag of wood/concrete ePTMs (against L/k).
A2 B1 − A1 B2
1
−
(55)
2
where In Fig. 7, the decrement factor of a wood wall is much smaller than that of a concrete wall, when ı is the same (especially when ı is smaller than about 3). In Fig. 8, when the real thickness L is the same, the time lag of a wood wall is much larger than that of a concrete wall. In Fig. 9, the decrement factor of a wood wall is much smaller than that of a concrete wall, when L is the same (especially when L is smaller than about 0.6 m). In Fig. 10, when the thermal resistance L/k is the same, the time lag of a concrete wall is much larger than that of a wood wall. However, the L/k value of a concrete wall cannot be too large, since when L/k ≈ 0.5 m2 K/W, the real thickness of the concrete wall is already about 1 m, which is too thick to be used in common buildings. Contrarily, for a typical wood wall, the corresponding L/k value can be large, since its thermal conductivity is much smaller than a concrete wall (for the same L/k = 0.5 m2 K/W, the real thickness of the wood wall is just about 0.06 m or 2.36 in.). In Fig. 11, the decrement factor of a concrete wall is smaller than that of a wood wall, when the thermal resistance is the same (the result is not surprising since the concrete wall is about 15 times thicker than the wood wall under this condition). Both Figs. 10 and 11 are for reference purpose only. They serve no useful purpose for the direct comparison of realistic wood walls and concrete walls.
= T /(T )s
(56)
Numerical method was used in Ref. [2] to validate the accuracy of II . It was demonstrated [2] that II is a sufficiently accurate approximation useful for the determination of heat flux. From the temperature distribution function, the heat flux can be gotten as √ ∂A ∂A B1 ∂x2 − B2 ∂x1 ∂ 2 q II = −k − (57) = −k(T )0 ∂x Leff (ı + 1/ε) B2 + B2 1
When x = 0, q II0
=−
√ 2k(T )0
=−
√ 2k(T )0
D12 + D22 B12 + B22
Leff
and when x = L, q IIL
F12 + F22 B12 + B22
Leff
2
sin
sin
2t P
2t P
− ϕ0
− ϕL
− ı + 1/ε
(58)
− ı + 1/ε
(59)
From the two equations above, we know that, for certain ε and ı, the effective thermal storage coefficient is the same, no matter / Tm , since the linear parts ( /(ı + 1/ε)) of q II0 and Tin = Tm or Tin = q IIL are cancelled out. 4.2. Heat exchange coefficients Let us define some coefficients as follows:
0in
≡
Q0in
0out ≡ Fig. 11. The decrement factor of wood/concrete ePTMs (against L/k).
max
eff
(60)
Acarea (T )s Q0out eff
max
Acarea (T )s
(61)
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
Fig. 12. Coefficients of wood walls when ≤ 0.
Fig. 13. Coefficients of wood walls when ≥ 0.
Fig. 14. Coefficients of concrete walls when ≤ 0.
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P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
Fig. 15. Coefficients of concrete walls when ≥ 0.
Lin ≡
QLin
max
(62)
eff
Acarea (T )s
Lout ≡
QLout
max
(63)
eff
Acarea (T )s
and
steady ≡
Qsteady
max
(64)
eff
Acarea (T )s
where Q0in max and Q0out max are the total heat flowed into or out of the ePTM outside surface in one period, respectively; QLin max and QLout max are the total heat flowed into or out of inside surface in one period, respectively; and Qsteady max is the total heat flowed through the ePTM in one period if treat Ts = Tm . From the knowledge of one-dimension heat transfer, it is easy to get Qsteady
max
A
=
(T )s P Tm − Tin P (T )s P = k= ˛cp L/k + 1/h 2 L + k/h 2 L + k/h 2
=
(T )s cp Leff Leff (T )s Leff eff = c 2 L + k/h 2 L + k/h 4 area
Therefore,
steady =
Qsteady
max
eff Acarea (T )s
=
Leff = 4 L + k/h
√
2 √ √ eff 4 2L/L + 2k/hLeff
√ =
2 4 ı + 1/ε
(65)
Take wood/concrete ePTM for example, some curves of the coefficients against ı are plotted in Figs. 12–15. Note: < 0 means that Tm < Tin ; < (− 1) means that Tin is higher than the highest value of the outside surface temperature; > 0 means that Tm > Tin ; > 1 means that Tin is lower than lowest the value of the outside surface temperature.
0in : As increases, 0in increases when ı is the same; when ≤ (− 1), for certain , 0in increases from zero to near 1 (a little smaller than 1) as ı increases; when −1 < ≤ 0, 0in decreases first and increases a little later, then trends to a constant value near 1.
0out : As increases, 0out increases when ı is the same; for certain , 0out increases as ı increases and the increase velocity becomes smaller when ı is larger; when ı is large, 0out trends to a constant value near −1 (a little smaller than −1, except when = 0); however, when is close to zero and is ı small, 0out increases first
and decreases a little later, which phenomena was discussed before when = 0.
Lin : As increases, Lin increases when ı is the same; for certain , Lin increases as ı increases and the increase velocity becomes smaller when ı is larger; when ı is large, Lin trends to a constant value near zero (a little smaller than zero).
Lout : when ≤ (− 1), Lout = 0; when −1 < ≤ 0, for certain ,
Lout decreases as ı increases and finally becomes zero when ı is large enough. For = 0, 0in = 0out and Lin = Lout when ı is the same. For certain and ı, 0in + Lin = 0out + Lout , which means that in one period, the net heat stored is zero. Comparing Fig. 13 with Fig. 12, when ı is the same and is contrary, 0in exchanges with 0out , and Lin exchanges with Lout . Comparing Figs. 14 and 15 with Figs. 12 and 13, all absolute values of the coefficients of concrete walls are smaller the that of wood walls, correspondingly. However, this does not mean the amount of heat flowed through concrete walls is smaller than the amount of heat flowed through wood walls. That is because that when ı is the same, the thickness of a concrete wall is much larger than that of a wood wall. The amount of heat flowed into the inside air is exactly equal to the steady value when ı is the same ( Lout = steady ); that is to say, in these two cases, if we just concern the total heat flowed into the inside air, Ts can be treated as Tm , although Ts is a sinusoidal function. In Ref. [4] page 51, it states the “insulating effect of mass” as follows: “If the temperature difference across a massive material fluctuates in certain specific ways, then the massive material will act as if it had high thermal resistance.” That is, there is an apparent “thermal resistance” in a massive wall. In fact, the “thermal resistance” insulating effect of mass subject to sinusoidal heating and cooling never occurs. When > 1, whether the material is massive or not, Ts can be treated as Tm and the amount of heat flowed into or out of the inside air is exactly equal to the steady state value. When < 1 and ı is small, walls seem to act as if it had low thermal resistance because of the outdoor temperature fluctuation ( Lin is bigger than steady , which means that more heat flowed into the inside wall surface from the indoor air; this greater part of heat flows back later, which is presented by Lout ; combining Lin and Lout , the net heat flowed through the inside surface is again exactly equal to the steady state value). Therefore, based on the investigation in this paper, there is no such “thermal resistance” insulating effect of mass under sinusoidal heating and cooling.
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
4.3. Correction of some rules of thumb Some rules of thumb need to be corrected based on the investigation above. Rule of thumb 1: “In hot and dry climates, one usually finds massive walls used for their time-lag effect.” (Ref. [4] page 3) Compared to wood walls, concrete walls are massive. However, when the thickness is the same, the time-lag effect of a wood wall is larger than that of a concrete wall as shown in Fig. 8. Moreover, the decrement factor of a wood wall is smaller than that of a concrete wall as shown in Fig. 9. That is to say, for an exterior wall, a wood wall is much better than a samethickness concrete wall, if just considering the time-lag effect and the decrement factor. Therefore, the massive walls usually used in hot and dry climates may not be due to their time-lag effect, but other reasons. For instance, wood is rarer in desert regions. Rule of thumb 2: “Walls with high time lags and small decrement factors, give comfortable inside temperatures even if the outside is very hot”. [5,6] In fact, this statement is not very exact. For the decrement factor, the smaller, of course, the better; but for the time lag, it is not the higher, the better. For example, a wall with a one-period time lag seems no time lag at all. A wall with a half-period time lag and a very small decrement factor is considered the best. 5. Conclusions In this paper, the dynamic heat transfer of ePTM subject to sinusoidal heating and cooling has been investigated. An approximated
401
analytical solution is developed and is used to obtain the coefficients of exterior wood/concrete walls, the time-lag effect, and the decrement factor. Some main conclusions are summarized as follows: (1) For a certain ePTM under sinusoidal heating and cooling, the thermal storage is constant and there is no “insulating effect of mass”. (2) The time lag and the decrement factor show similarity characteristics—independent of environment temperatures. (3) For exterior walls, a wood wall is better than a concrete wall of same thickness in terms of the time-lag effect and the decrement factor. References [1] P. Ma, L.-S. Wang, Effective heat capacity of interior planar thermal mass (iPTM) subject to periodic heating and cooling, Energy Buildings 47 (2012) 44–52. [2] P. Ma, Dynamic heat transfer: effective heat capacity of planar thermal mass subject to periodic heating and cooling, Master of Science thesis, Stony Brook University, May, 2010. [3] K.J. Kontoleon, D.K. Bikas, Thermal mass vs. thermal response factors: determining optimal geometrical properties and envelope assemblies of building materials, in: International Conference “Passive and Low Energy Cooling for the Built Environment”, Santorini, Greece, May, 2005. [4] N. Lechner, Heating, Cooling, Lighting: Sustainable Design Methods for Architects, 3rd edition, John Wiley & Sons, 2008. [5] H. Asan, Y.S. Sancaktar, Effects of Wall’s thermophysical properties on time lag and decrement factor, Energy and Buildings 28 (1998) 159–166. [6] T.R. Knowles, Proportioning composites for efficient thermal storage walls, Solar Energy 31 (3) (1983) 319–326.
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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild
Effective heat capacity of exterior Planar Thermal Mass (ePTM) subject to periodic heating and cooling Peizheng Ma ∗ , Lin-Shu Wang Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300, United States
a r t i c l e
i n f o
Article history: Received 13 June 2011 Received in revised form 3 October 2011 Accepted 12 December 2011 Keywords: Dynamic heat transfer Planar Thermal Mass (PTM) Heat capacity Thermal storage Periodic heating and cooling
a b s t r a c t Thermal mass can be effectively used for controlling temperature in buildings. Based on its location and function, building thermal mass can be classified as interior thermal mass and exterior (envelope) thermal mass. In this paper, the dynamic heat transfer performance of exterior Planar Thermal Mass (ePTM) subject to sinusoidal heating and cooling is investigated. When the mean value of outdoor air temperature is equal to indoor air temperature, solutions of temperature distribution and heat flux in ePTM are deduced analytically. Using the analytical solutions, heat exchange and thermal storage of ePTM are attained. Correspondingly, the time lag and the decrement factor are obtained; they show similarity characteristics—independent of environment temperatures. When the two temperatures are not equal, an approximated temperature distribution solution is developed based on the principle of superposition, which has been validated by numerical solutions. The approximated solution is then used to investigate the heat exchange between the ePTM and the environment. In contrast to the common belief that a large thermal mass has an apparent insulating effect, no “insulating effect of mass” under sinusoidal heating and cooling is found in this investigation. Another two rules of thumb based on common belief are corrected: a wood wall is better than a concrete wall of same thickness as exterior walls in terms of the time-lag effect and the decrement factor. The effect of the time lag is not the longer the time lag is the better its contribution to thermal comfort becomes. © 2011 Elsevier B.V. All rights reserved.
1. Introduction and problem description Based on its location and function, building thermal mass can be classified as interior thermal mass (such as furniture, appliances and interior walls, ceiling and floors) and exterior (envelope) thermal mass (such as exterior walls, roof/ceiling, floors and windows). In Ref. [1], analytical method was used to solve the dynamic heat transfer problem of interior Planar Thermal Mass (iPTM) subject to sinusoidal heating and cooling, and the effective thermal storage capacity of the iPTM was investigated. For exterior Planar Thermal Mass (ePTM), unlike iPTM, we are concerned more about the heat exchange between the ePTM and the indoor air, the time-lag effect and the decrement factor (which will be explained later), rather than the thermal storage capacity. In this paper, the dynamic heat transfer problem of ePTM under external sinusoidal thermal wave and internal constant thermal environment will be investigated.
Consider the ePTM shown in Fig. 1. The ePTM surface temperature Ts is a sinusoidal function with a mean value Tm [K or ◦ C]. The peak amplitude and the period of Ts are (T)s and P, respectively. Suppose that all the thermo-physical properties of the ePTM (i.e., density [kg/m3 ], mass specific heat cp [kJ/kg K], volumetric specific heat cvol [kJ/m3 K], conductivity k [W/m K], and thermal diffusivity ˛ [m2 /s]) are constant. This problem can be dealt with as a one-dimensional (x direction) heat transfer problem. The temperature changes in the ePTM will repeat themselves periodically if the surface temperature variation has occurred for a sufficiently long period of time. In the absence of internal energy sources/sinks, the heat diffusion equation should be used to calculate the temperature distribution in the ePTM: G.D.E. :
∂2 T 1 ∂T = ˛ ∂t ∂x2
(1)
and the Boundary Conditions (B.C.s) of the Governing Differential Equation (G.D.E.) are: ∗ Corresponding author. Tel.: +1 631 745 8937. E-mail addresses: [email protected] (P. Ma), [email protected] (L.-S. Wang). 0378-7788/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2011.12.015
B.C.s :
T (0, t) = Ts = (T )s sin
2 P
t
+ Tm
(2)
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
395
Ts Temperature, T
k, ρ, cvol, cp, α
Ts
TsL
Outdoor air, Tout
L
Tm (ΔT)s
Tin
Indoor air, Tin
Time, t Note: Tin may be higher, lower or equal to Tm
0
x Fig. 1. EPTM and its surface temperatures.
−k
∂T ∂x
= h[T (L, t) − Tin ] = h[TsL − Tin ]
(3)
L
where Tin is the indoor air temperature, TsL is the inside surface temperature, and h [W/m2 K] is the heat transfer coefficient. In this paper, h = 8.10 W/m2 K is chosen for the indoor air film [2]. 2. Analytical solution when Tin = Tm Generally, the indoor air temperature is not constant and is not equal to Tm . However, if the amount of the inside air is large enough, Tin can be considered as constant. In this paper, we will first treat Tin as constant and equal to Tm . Let (x, t) = T (x, t) − Tm
(4)
and when Tin = Tm , the G.D.E. and B.C.s can be rewritten as G.D.E. :
∂ ∂x
∂x2
=
1 ∂ ˛ ∂t
(5)
(0, t) = Ts − Tm = (T )s sin
B.C.s :
∂2
L
2 P
t
h = − (L, t) k
B.C.s :
∂c ∂x
L
(13)
h = − c (L, t) k
(14)
The general solution of the D.E. above can be gotten as √ √ c (x, t) = (C1 ex 2i/(˛P) + C2 e−x 2i/(˛P) )e2it/P
C1 = C2 =
(1 − ε + i)(T )s
(16)
(1 + ε + i)e2(1+i)ı + (1 − ε + i) (1 + ε + i)e2(1+i)ı (1 + ε + i)e2(1+i)ı + (1 − ε + i)
(T )s
hLeff ε≡ √ 2k
and Leff ≡
(18)
˛P
2˛P
˜ (0, t) = (T )s cos
B.C.s :
∂˜ ∂x
L
(8)
2 P
t
h˜ = − (L, t) k
(9)
(10)
Define a (complex) temperature function as: ˜ c (x, t) = (x, t) + i(x, t) (11) √ where i = −1. The complex function c (x, t) satisfies the following differential equation (D.E.) and B.C.s: D.E. :
∂2 c 1 ∂c = ˛ ∂t ∂x2
(19)
(20)
which are called dynamic Biot number, dimensionless thickness and effective thickness, respectively. Then
Using the method described in Ref. [1], introduce an auxiliary problem with the G.D.E. and B.C.s: ∂2 ˜ 1 ∂˜ = ˛ ∂t ∂x2
(17)
where
(6)
2.1. Temperature distribution in the ePTM
G.D.E. :
(15)
Subject the two B.C.s to the equation above, C1 and C2 can be obtained as
ı≡L
(7)
c (0, t) = (T )s e2it/P
(12)
Fig. 2. Inner effective heat exchange coefficient for certain ε and ı.
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P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
c (x, t) = (T )s
√ 2x/Leff
(1 − ε + i)e(1+i)
(1 + ε +
⇒ c (x, t) = (T )s
√ 2x/Leff
+ (1 + ε + i)e2(1+i)ı e−(1+i)
i)e2(1+i)ı
+ (1 − ε + i)
e2it/P
A1 + iA2 (A1 B1 + A2 B2 ) + i(A2 B1 − A1 B2 ) = (T )s B1 + iB2 B12 + B22
(21)
where
√ ⎧ ⎫ √ 2x/Leff ) cos(2ı− 2x/Leff +2t/P) e(2ı− ⎪ ⎪ ⎪ ⎪ √ √ ⎨ε ⎬ eff −e 2x/L √cos( 2x/Leff + 2t/P)
√ A1 = (22) eff √ e(2ı− 2x/L ) cos 2ı− 2x/Leff +2t/P+/4 ⎪ ⎪ ⎪ ⎪ √ √ ⎩+ 2 ⎭ eff 2x/L eff +e
cos( 2x/L
+ 2t/P + /4)
+e
sin( 2x/L
+ 2t/P + /4)
⎧ ⎫
√ √ 2x/Leff ) sin(2ı − 2x/Leff + 2t/P) e(2ı− ⎪ ⎪ ⎪ ⎪ √ √ ⎨ε ⎬ eff −e 2x/L √sin( 2x/Leff + 2t/P)
A2 = (23) √ eff 2x/L ) sin(2ı− 2x/Leff +2t/P+/4) ⎪ √ ⎪ e(2ı− ⎪ ⎪ √ √ ⎩+ 2 ⎭ eff 2x/L eff B1 = ε[e2ı cos(2ı) − 1] + [e2ı cos(2ı) − e2ı sin(2ı) + 1]
(24)
and
Fig. 3. Effective thermal storage coefficient.
B2 = ε[e2ı sin(2ı)] + [e2ı cos(2ı) + e2ı sin(2ı) + 1]
(25) where
At last, the temperature distribution in the ePTM can be obtained as (x, t) = (T )s
A2 B1 − A1 B2 B12
(26)
+ B22
√
L
ε,ı
2 = 2
F12 + F22
(37)
B12 + B22
and
2.2. EPTM heat exchange and thermal storage
eff
carea ≡ cp Leff
Since A1 and A2 are functions of x and B1 and B2 are not, ∂T ∂ (T )0 = −k = −k 2 ∂x ∂x B + B2
q = −k
1
B1
2
∂A2 ∂A1 − B2 ∂x ∂x
(27)
Let x = 0 and the heat flux at the outer surface can be gotten as
q
=−
0
√ 2k(T )0
D12 + D22 B12
Leff
+ B22
sin
2t
− ϕ0
P
(28)
which can be called inner effective heat exchange coefficient for certain ε and ı, as shown in Fig. 2, and effective area specific heat, respectively. As shown in Fig. 2, the heat exchange between the surface and the indoor air decreases as ı increases when ε is constant; and it increases as ε increases when ı is constant. The amount of heat stored in the ePTM in a period from t1 to t2 can be gotten as
where 2ı
D1 = ε(e
sin 2ı + e
2ı
2ı
cos 2ı + 1) + 2(e
cos 2ı − 1)
(29)
D2 = ε(e2ı sin 2ı − e2ı cos 2ı − 1) + 2e2ı sin(2ı)
B1 D1 + B2 D2 B1 D2 − B2 D1
(31)
Let x = L and the heat flux at the inner surface can be gotten as √ 2k(T )0
q L = −
Leff
where
F12 B12
+ F22 sin + B22
2t P
− ϕL
(32)
F1 = 2εe (sin ı + cos ı)
(34)
and B1 F1 + B2 F2 B1 F2 − B2 F1
(35)
At the inner surface, the amount of heat exchanged in the duration from t1 to t2 can be gotten as QL = A
t2
t1
√
2 q L dt = 4
F12 + F22
cp Leff (T )0 cos 2
B12 + B2
QL 1 L eff carea (T )0 cos = ε,ı A 2
2t − ϕstor P
2t P
− ϕL
t2 t1
2t P
− ϕL
t2 t1
(36)
2 = 4
2
(D1 − F1 ) + (D2 − F2 ) B1 2 + B2 2
2 eff
carea (T )0
t2 t1
1 stor eff Qstor = ε,ı carea (T )0 cos ⇒ 2 A
2t − ϕstor P
t2
(39)
t1
where tan ϕstor =
(33)
F2 = 2εeı (sin ı − cos ı)
−
qL )dt
t1
and
ı
tan ϕL =
√
t2
(q0
× cos
tan ϕ0 =
⇒
Qstor = A
(30)
and
(38)
√
stor
ε,ı
2 = 2
B1 D1 + B2 D2 − B1 F1 − B2 F2 B1 D2 − B2 D1 − B1 F2 + B2 F1
(40)
(D1 − F1 )2 + (D2 − F2 )2 B1 2 + B2 2
(41)
which is called effective thermal storage coefficient for certain ε and ı in this paper, as shown in Fig. 3. stor increases almost linearly when ı is small (ı ≤ 1 for In Fig. 3, ε,ı small ε ePTMs to ı ≤ 2 for large ε ePTMs), which means that the amount of stored heat energy increases almost linearly with the thickness of the ePTM when ı is not large. Generally, it is natural to try to thicken an ePTM to make it be a better thermal mass. However, from Fig. 3, we realize that if we continue to thicken the ePTM after a certain thickness, the heat stored in the ePTM actually decreases a little. That is because the previously stored energy flows out of the ePTM and interfaces the influent heat.
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
397
3. The time-lag effect and the decrement factor when Tin = Tm For ePTMs, the time-lag effect and the decrement factor are much more important than that for iPTMs. The definitions [3] of the time lag ϕ and the decrement factor f are as follows:
ϕ=
tTi,min − tTe,min tTi,max − tTe,max
(42)
and Ti,max − Ti,min Te,max − Te,min
f =
(43)
where t means time, T means temperature, i means interior surface, e means exterior surface, max means maximum and min means minimum. When x = 0, the exterior surface temperature is Ts . Therefore, Te,min = Tm − (T)s when tTe,min = 3P/4 = 18 h, and Te,max = Tm + (T)s when tTe,max = P/4 = 6 h. When x = L, √ 2t 2 2eı (L, t) = (T )s (44) sin + +ı− L P 4 B2 + B2 1
Fig. 4. The time lag.
2
where = arctan
L
B2 ± n B1
(n = 0, 1, 2, 3, . . .)
(45)
For certain ε and ı, B1 and B2 are constant. Thus, L has maximum value: √ 2 2eı L,max = (T )s (46) B12 + B22 when tL,max =
P 2
arctan
B2 nP + −ı ± B1 4 2
(47)
and has minimum value: √ 2 2eı L,min = −(T )s B12 + B22 when tL,min
P = 2
(48)
B2 5 nP −ı ± arctan + B1 4 2
(49)
Therefore,
⎧
⎨ tTi,min − tTe,min = tL,min − 18 h = 1 arctan B2 + 5 − ı ± n − 18 h ϕ P B1 P P 2 4 2 = P ⎩ tTi,max − tTe,max = tL,max − 6 h = 1 arctan B2 + − ı ± n − 6 h P P B1 P 2 4 2 (50) and f =
Fig. 5. The decrement factor.
Ti,max − Ti,min L,max − L,min = = Te,max − Te,min (T )s − [−(T )s ]
√ 2 2eı
B12 + B22
(51)
The two equations above show the similarity nature of dimensionless ϕ/P and f – independent of environment temperatures. Thus, let us change the value of ε and ı to see what will happen to ϕ and f. Some curves of ϕ and f are plotted in Figs. 4 and 5, respectively. From Fig. 4, we can find that: when ı is smaller than about 1 and ε is the same, ϕ increases non-linearly as the increase of ı; when ı is the same, as ε increases from zero to positive infinity, ϕ decreases about 2.5 h, except when ı is smaller than about 1; when ε is the same, as ı increases from about 1 to about 6 or 7,
Fig. 6. The time lag of wood/concrete ePTMs (against ı).
ϕ increases almost linearly to 24 h (one period); at about 6 or 7, ϕ decreases from 24 h to 0 suddenly, which means that the time lags more than one period; and the shift point decreases as ε increases; the change is very small when ε is larger than about 5. In Fig. 5, the decrement factor is obviously non-linear. When ı is the same, as ε increases, f decreases. When ε trends to positive infinity, f is zero for all ı since both Ti,max and Ti,min are zero when the inside surface temperature keeps constant. When ε is the same, as ı increases, f decreases from nearly 1 to 0 and the decrease is very sharp when ı is small. Again taking wood and concrete for examples, the time lag and the decrement factor are shown in Figs. 6–11. In Fig. 6, when ı is the same, the time lag of a concrete wall is larger about 80 min than that of a wood wall, except when ı is smaller than about 1.
Fig. 7. The decrement factor of wood/concrete ePTMs (against ı).
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P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
4. Approximated solution when Tin = / Tm 4.1. Temperature distribution and heat flux / Tm , by using the principle of superposition, a linWhen Tin = ear part should be added because of the temperature difference of the two surfaces of the ePTM. From Eq. (26), the temperature distribution becomes Fig. 8. The time lag of wood/concrete ePTMs (against L).
II (x, t) = (T )s
A2 B1 − A1 B2 B12 + B22
− (Tm − TmL )
x L
(52)
Here, TmL , which is the mean value of the temperature at the ePTM interior surface, is an unknown to be determined. Consider the limit of a steady-state heat transfer problem by taking (T)s → 0, i.e., Ts = Tm . Then
Fig. 9. The decrement factor of wood/concrete ePTMs (against L).
Tm − TmL L/k 1 1 1 = = = = √ Tm − Tin L/k + 1/h 1 + k/hL 1 + 1/ıε 1 + 2k/ıhLeff (53) Let T = Tm − Tin
(54)
Eq. (52) can be changed into √ T 2x 1 + 1/ıε ıLeff B12 + B22 √ A2 B1 − A1 B2 2x = (T )s − 2 2 eff ı + 1/ε L B +B
II (x, t) = (T )s
Fig. 10. The time lag of wood/concrete ePTMs (against L/k).
A2 B1 − A1 B2
1
−
(55)
2
where In Fig. 7, the decrement factor of a wood wall is much smaller than that of a concrete wall, when ı is the same (especially when ı is smaller than about 3). In Fig. 8, when the real thickness L is the same, the time lag of a wood wall is much larger than that of a concrete wall. In Fig. 9, the decrement factor of a wood wall is much smaller than that of a concrete wall, when L is the same (especially when L is smaller than about 0.6 m). In Fig. 10, when the thermal resistance L/k is the same, the time lag of a concrete wall is much larger than that of a wood wall. However, the L/k value of a concrete wall cannot be too large, since when L/k ≈ 0.5 m2 K/W, the real thickness of the concrete wall is already about 1 m, which is too thick to be used in common buildings. Contrarily, for a typical wood wall, the corresponding L/k value can be large, since its thermal conductivity is much smaller than a concrete wall (for the same L/k = 0.5 m2 K/W, the real thickness of the wood wall is just about 0.06 m or 2.36 in.). In Fig. 11, the decrement factor of a concrete wall is smaller than that of a wood wall, when the thermal resistance is the same (the result is not surprising since the concrete wall is about 15 times thicker than the wood wall under this condition). Both Figs. 10 and 11 are for reference purpose only. They serve no useful purpose for the direct comparison of realistic wood walls and concrete walls.
= T /(T )s
(56)
Numerical method was used in Ref. [2] to validate the accuracy of II . It was demonstrated [2] that II is a sufficiently accurate approximation useful for the determination of heat flux. From the temperature distribution function, the heat flux can be gotten as √ ∂A ∂A B1 ∂x2 − B2 ∂x1 ∂ 2 q II = −k − (57) = −k(T )0 ∂x Leff (ı + 1/ε) B2 + B2 1
When x = 0, q II0
=−
√ 2k(T )0
=−
√ 2k(T )0
D12 + D22 B12 + B22
Leff
and when x = L, q IIL
F12 + F22 B12 + B22
Leff
2
sin
sin
2t P
2t P
− ϕ0
− ϕL
− ı + 1/ε
(58)
− ı + 1/ε
(59)
From the two equations above, we know that, for certain ε and ı, the effective thermal storage coefficient is the same, no matter / Tm , since the linear parts ( /(ı + 1/ε)) of q II0 and Tin = Tm or Tin = q IIL are cancelled out. 4.2. Heat exchange coefficients Let us define some coefficients as follows:
0in
≡
Q0in
0out ≡ Fig. 11. The decrement factor of wood/concrete ePTMs (against L/k).
max
eff
(60)
Acarea (T )s Q0out eff
max
Acarea (T )s
(61)
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
Fig. 12. Coefficients of wood walls when ≤ 0.
Fig. 13. Coefficients of wood walls when ≥ 0.
Fig. 14. Coefficients of concrete walls when ≤ 0.
399
400
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
Fig. 15. Coefficients of concrete walls when ≥ 0.
Lin ≡
QLin
max
(62)
eff
Acarea (T )s
Lout ≡
QLout
max
(63)
eff
Acarea (T )s
and
steady ≡
Qsteady
max
(64)
eff
Acarea (T )s
where Q0in max and Q0out max are the total heat flowed into or out of the ePTM outside surface in one period, respectively; QLin max and QLout max are the total heat flowed into or out of inside surface in one period, respectively; and Qsteady max is the total heat flowed through the ePTM in one period if treat Ts = Tm . From the knowledge of one-dimension heat transfer, it is easy to get Qsteady
max
A
=
(T )s P Tm − Tin P (T )s P = k= ˛cp L/k + 1/h 2 L + k/h 2 L + k/h 2
=
(T )s cp Leff Leff (T )s Leff eff = c 2 L + k/h 2 L + k/h 4 area
Therefore,
steady =
Qsteady
max
eff Acarea (T )s
=
Leff = 4 L + k/h
√
2 √ √ eff 4 2L/L + 2k/hLeff
√ =
2 4 ı + 1/ε
(65)
Take wood/concrete ePTM for example, some curves of the coefficients against ı are plotted in Figs. 12–15. Note: < 0 means that Tm < Tin ; < (− 1) means that Tin is higher than the highest value of the outside surface temperature; > 0 means that Tm > Tin ; > 1 means that Tin is lower than lowest the value of the outside surface temperature.
0in : As increases, 0in increases when ı is the same; when ≤ (− 1), for certain , 0in increases from zero to near 1 (a little smaller than 1) as ı increases; when −1 < ≤ 0, 0in decreases first and increases a little later, then trends to a constant value near 1.
0out : As increases, 0out increases when ı is the same; for certain , 0out increases as ı increases and the increase velocity becomes smaller when ı is larger; when ı is large, 0out trends to a constant value near −1 (a little smaller than −1, except when = 0); however, when is close to zero and is ı small, 0out increases first
and decreases a little later, which phenomena was discussed before when = 0.
Lin : As increases, Lin increases when ı is the same; for certain , Lin increases as ı increases and the increase velocity becomes smaller when ı is larger; when ı is large, Lin trends to a constant value near zero (a little smaller than zero).
Lout : when ≤ (− 1), Lout = 0; when −1 < ≤ 0, for certain ,
Lout decreases as ı increases and finally becomes zero when ı is large enough. For = 0, 0in = 0out and Lin = Lout when ı is the same. For certain and ı, 0in + Lin = 0out + Lout , which means that in one period, the net heat stored is zero. Comparing Fig. 13 with Fig. 12, when ı is the same and is contrary, 0in exchanges with 0out , and Lin exchanges with Lout . Comparing Figs. 14 and 15 with Figs. 12 and 13, all absolute values of the coefficients of concrete walls are smaller the that of wood walls, correspondingly. However, this does not mean the amount of heat flowed through concrete walls is smaller than the amount of heat flowed through wood walls. That is because that when ı is the same, the thickness of a concrete wall is much larger than that of a wood wall. The amount of heat flowed into the inside air is exactly equal to the steady value when ı is the same ( Lout = steady ); that is to say, in these two cases, if we just concern the total heat flowed into the inside air, Ts can be treated as Tm , although Ts is a sinusoidal function. In Ref. [4] page 51, it states the “insulating effect of mass” as follows: “If the temperature difference across a massive material fluctuates in certain specific ways, then the massive material will act as if it had high thermal resistance.” That is, there is an apparent “thermal resistance” in a massive wall. In fact, the “thermal resistance” insulating effect of mass subject to sinusoidal heating and cooling never occurs. When > 1, whether the material is massive or not, Ts can be treated as Tm and the amount of heat flowed into or out of the inside air is exactly equal to the steady state value. When < 1 and ı is small, walls seem to act as if it had low thermal resistance because of the outdoor temperature fluctuation ( Lin is bigger than steady , which means that more heat flowed into the inside wall surface from the indoor air; this greater part of heat flows back later, which is presented by Lout ; combining Lin and Lout , the net heat flowed through the inside surface is again exactly equal to the steady state value). Therefore, based on the investigation in this paper, there is no such “thermal resistance” insulating effect of mass under sinusoidal heating and cooling.
P. Ma, L.-S. Wang / Energy and Buildings 47 (2012) 394–401
4.3. Correction of some rules of thumb Some rules of thumb need to be corrected based on the investigation above. Rule of thumb 1: “In hot and dry climates, one usually finds massive walls used for their time-lag effect.” (Ref. [4] page 3) Compared to wood walls, concrete walls are massive. However, when the thickness is the same, the time-lag effect of a wood wall is larger than that of a concrete wall as shown in Fig. 8. Moreover, the decrement factor of a wood wall is smaller than that of a concrete wall as shown in Fig. 9. That is to say, for an exterior wall, a wood wall is much better than a samethickness concrete wall, if just considering the time-lag effect and the decrement factor. Therefore, the massive walls usually used in hot and dry climates may not be due to their time-lag effect, but other reasons. For instance, wood is rarer in desert regions. Rule of thumb 2: “Walls with high time lags and small decrement factors, give comfortable inside temperatures even if the outside is very hot”. [5,6] In fact, this statement is not very exact. For the decrement factor, the smaller, of course, the better; but for the time lag, it is not the higher, the better. For example, a wall with a one-period time lag seems no time lag at all. A wall with a half-period time lag and a very small decrement factor is considered the best. 5. Conclusions In this paper, the dynamic heat transfer of ePTM subject to sinusoidal heating and cooling has been investigated. An approximated
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analytical solution is developed and is used to obtain the coefficients of exterior wood/concrete walls, the time-lag effect, and the decrement factor. Some main conclusions are summarized as follows: (1) For a certain ePTM under sinusoidal heating and cooling, the thermal storage is constant and there is no “insulating effect of mass”. (2) The time lag and the decrement factor show similarity characteristics—independent of environment temperatures. (3) For exterior walls, a wood wall is better than a concrete wall of same thickness in terms of the time-lag effect and the decrement factor. References [1] P. Ma, L.-S. Wang, Effective heat capacity of interior planar thermal mass (iPTM) subject to periodic heating and cooling, Energy Buildings 47 (2012) 44–52. [2] P. Ma, Dynamic heat transfer: effective heat capacity of planar thermal mass subject to periodic heating and cooling, Master of Science thesis, Stony Brook University, May, 2010. [3] K.J. Kontoleon, D.K. Bikas, Thermal mass vs. thermal response factors: determining optimal geometrical properties and envelope assemblies of building materials, in: International Conference “Passive and Low Energy Cooling for the Built Environment”, Santorini, Greece, May, 2005. [4] N. Lechner, Heating, Cooling, Lighting: Sustainable Design Methods for Architects, 3rd edition, John Wiley & Sons, 2008. [5] H. Asan, Y.S. Sancaktar, Effects of Wall’s thermophysical properties on time lag and decrement factor, Energy and Buildings 28 (1998) 159–166. [6] T.R. Knowles, Proportioning composites for efficient thermal storage walls, Solar Energy 31 (3) (1983) 319–326.