Control strategy and thermal inertia influence on evacuated tube solar water heating systems

Abstract

The main objective of this article is to investigate the effect of a control scheme on the behavior of a solar domestic hot water system, consisting of two evacuated tubes with compound-parabolic concentrator (CPC) collectors and a storage tank. In this regard, a transient thermal analysis of the system was performed using an original mathematical model. The developed model was validated by comparing the results with the data provided by the manufacturer. Simulations were conducted during a 48-h period in the meteorological conditions of Bucharest, Romania (latitude 44°24′N, longitude 26°05′E). The second objective of the present work relates to the working fluid thermal inertia. An analysis was made to determine whether the solar collectors’ fluid thermal inertia has an impact on the storage tank water temperature. Three different working fluids have been considered. The results showed that for the entire 2-day period in which simulations have been conducted, differences between the useful energy delivered to the storage tank water in the case with mass flow rate adjusted depending on the incident solar radiation and the cases with constant mass flow rate were approximately 1 kWh in summer and 0.72 kWh in winter. In addition, it has been concluded that the time required to heat the storage tank water from 30 to 60 degrees centigrade is approximately an hour less when the system is working with water compared to the propylene glycol case.

Appendices

Appendix 1: Solar collector equations

The heat fluxes from Eqs. (1–6) are calculated as follows:

$$Q_{\text{og}} = \alpha_{\text{og}} I_{\text{S}}$$

(24)

$$Q_{\text{ig}} = \alpha_{\text{ig}} \tau_{\text{og}} I_{\text{S}}$$

(25)

$$Q_{\text{co} - \text{a}} = h_{\text{c,co} - \text{a}} S_{\text{co} - \text{a}} \left( {T_{\text{co}} - T_{\text{a}} } \right)$$

(26)

$$Q_{\text{co} - \text{w}} = h_{\text{c,co} - \text{w}} S_{\text{co} - \text{w}} \left( {T_{\text{co}} - \bar{T}_{\text{w}} } \right)$$

(27)

$$Q_{\text{al} - \text{a}} = h_{\text{c,al }- \text{a}} S_{\text{al} - \text{a}} \left( {T_{\text{al}} - T_{\text{a}} } \right)$$

(28)

$$Q_{al - co} = \frac{{S_{al - co} \left( {T_{al} - T_{co} } \right)}}{{\frac{{\delta_{al} }}{{\lambda_{al} }} + \frac{{\delta_{co} }}{{\lambda_{co} }}}}$$

(27)

$$Q_{ig - al} = \frac{{2\pi L_{c} \left( {T_{ig} - T_{al} } \right)}}{{\frac{1}{{\lambda_{ig} }}\ln \left( {\frac{{d_{ig,e} }}{{d_{ig,i} }}} \right) + \frac{1}{{\lambda_{al} }}\ln \left( {\frac{{d_{al,e} }}{{d_{al,i} }}} \right)}}$$

(28)

$$Q_{ig - og} = \left( {h_{c,ig - og} + h_{r,ig - og} } \right)S_{ig - og} \left( {T_{ig} - T_{og} } \right)$$

(29)

$$Q_{og - o} = \left[ {h_{c,og - 0} \left( {T_{og} - T_{0} } \right) + h_{r,og - 0} \left( {T_{og} - T_{sky} } \right)} \right] \cdot S_{og - 0}$$

(30)

where h _c,co-a is convection heat transfer coefficient between the copper pipe and the inner air, W/(m²K); h _c,co-w convection heat transfer coefficient between the copper pipe and the working fluid, W/(m²K); h _c,al-a convection heat transfer coefficient between the aluminium fin and the inner air, W/(m²K); h _c,ig-og convection heat transfer coefficient between the vacuum environment [86], between the two glass tubes, W/(m²K); h _r,ig-og radiation heat transfer coefficient between the inner and outer glass, W/(m²K); h _c,og-0 convection heat transfer coefficient from the outer glass to the ambient, W/(m²K); h _r,og-0 radiation heat transfer coefficient from the outer glass to the ambient, W/(m²K); λ _al aluminium fin conductivity, W/(mK); λ _co copper pipe conductivity, W/(mK); λ _ig inner glass conductivity, W/(mK).

1.1 Heat transfer from the copper pipe to the inner air

The convection heat transfer coefficient from the copper pipe to the inner air was calculated with the following formula [19]:

$$h_{c,co - a} = \frac{{\lambda_{\text{equivalent}} }}{{\delta_{m} }}$$

(31)

where δ _m molecular diameter, cm; λ _equivalent equivalent thermal conductivity, W/(mK) [19]:

$$\lambda_{\text{equivalent}} = \varepsilon_{a} \lambda_{\text{med}}$$

(32)

where λ _med is the thermal conductivity at the air medium temperature, ε _a coefficient related to convection influence on the heat transfer [19]:

$$\varepsilon_{a} = A\sqrt[4]{{\delta_{m}^{3} \Delta T_{co - a} }}$$

(33)

$$A = 0.18\frac{{\left( {\beta_{v} \cdot g \cdot \Pr_{a} } \right)^{0.25} }}{{\nu_{a}^{0.5} }}$$

(34)

$$\Pr\nolimits_{a} = \frac{{\nu_{a} \cdot \rho_{a} \cdot c_{a} }}{{\lambda_{med} }}$$

(35)

where β _v is the volumetric thermal expansion coefficient (1/K); g gravitational acceleration (g = 9.81 m/s²); Pr _a Prandtl number; ρ _a air density, kg/m³; c _a air specific heat capacity, J/(kgK); ν _a kinematic viscosity, m²/s.

1.2 Heat transfer from the copper pipe to the working fluid

Convection heat transfer coefficient between the copper pipe and the working fluid [20]:

$$h_{c,co - w} = \frac{{Nu_{co - w} \lambda_{w} }}{{d_{co,i} }}$$

(36)

For laminar flow (Re _w ≤ 2300), Nusselt number is calculated with the formula [20]:

$$Nu_{co - w} = \left\{ {Nu_{1}^{3} + 0.7^{3} + \left[ {Nu_{2} - 0.7} \right]^{3} + Nu_{3}^{3} } \right\}^{1/3}$$

(37)

where

$$Nu_{1} = 3.66$$

(38)

$$Nu_{2} = 1.615 \cdot \sqrt[3]{{\text{Re}_{w} \Pr\nolimits_{w} d_{coi} /L_{co} }}$$

(39)

$$\text{Re}_{w} = \frac{{w_{w} d_{coi} \rho_{w} }}{{\mu_{w} }}$$

(40)

$$Nu_{3} = \left( {\frac{2}{{1 + 22\Pr\nolimits_{w} }}} \right)^{1/6} \left( {\text{Re}_{w} \Pr\nolimits_{w} d_{coi} /L_{co} } \right)^{1/2}$$

(41)

For turbulent flow (Re _w ≥ 10⁴ and 0.1 ≤ Pr _w ≤ 1000) the following relationship has been used [20]:

$$Nu_{co - w} = \frac{{\left( {\xi /8} \right)\text{Re}_{w} \Pr_{w} }}{{1 + 12.7\sqrt {\xi /8} \left( {\Pr_{w}^{2/3} - 1} \right)}}\left[ {1 + \left( {d_{co,i} /L_{co} } \right)^{2/3} } \right]$$

(42)

where

$$\xi = \left( {1.8 \cdot \log_{10} \text{Re}_{w} - 1.5} \right)^{ - 2}$$

(43)

For transition flow (2300 < Re _w < 10⁴) we used the expression [20]:

$$Nu_{co - w} = \left( {1 - \gamma } \right) \cdot Nu_{l,2300} + \gamma \cdot Nu_{{t,10^{4} }}$$

(44)

where γ is given by:

$$\gamma = \frac{{\text{Re}_{w} - 2300}}{{10^{4} - 2300}}$$

(45)

Here $Nu_{l,2300}$ is the Nusselt number at $\text{Re}_{w}$ = 2300 calculated from Eq. (37) and $Nu_{{t,10^{4} }}$ is the Nusselt number from Eq. (42) at $\text{Re}_{w}$ = 10⁴.

1.3 Heat transfer in the vacuum environment between the two glass tubes

The pressure between the two glass tubes is approximately 10⁻⁵ mbar (7.5 × 10⁻⁶ Torr). When the space between the two glass tubes is vacuumed (pressure less than 1 Torr), heat transfer takes place by free molecular convection [21]. The convection heat transfer coefficient in the vacuum environment has been calculated with the following formula:

$$h_{c,ig - og} = \frac{{k_{std} }}{{\left( {{{d_{ig,e} } \mathord{\left/ {\vphantom {{d_{ig,e} } {2\ln \left( {{{d_{og,i} } \mathord{\left/ {\vphantom {{d_{og,i} } {d_{ig,e} }}} \right. \kern-0pt} {d_{ig,e} }}} \right) + b\lambda_{m} \left( {{{d_{ig,e} } \mathord{\left/ {\vphantom {{d_{ig,e} } {d_{og,i} + 1}}} \right. \kern-0pt} {d_{og,i} + 1}}} \right)}}} \right. \kern-0pt} {2\ln \left( {{{d_{og,i} } \mathord{\left/ {\vphantom {{d_{og,i} } {d_{ig,e} }}} \right. \kern-0pt} {d_{ig,e} }}} \right) + b\lambda_{m} \left( {{{d_{ig,e} } \mathord{\left/ {\vphantom {{d_{ig,e} } {d_{og,i} + 1}}} \right. \kern-0pt} {d_{og,i} + 1}}} \right)}}} \right)}}$$

(46)

$$b = \frac{{\left( {2 - a_{c} } \right)\left( {9\gamma_{c} - 5} \right)}}{{2a_{c} \left( {\gamma_{c} + 1} \right)}}$$

(47)

$$\lambda_{m} = \frac{{2.331 \cdot 10^{ - 20} \left( {T_{med} + 273.15} \right)}}{{P_{a} \delta_{m}^{2} }}$$

(48)

where k _std is air thermal conductivity in standard conditions (20 °C și 101.325 kPa), W/(mK); b interaction coefficient; λ _m mean free path between two molecules, cm; γ _c specific heat report; a _c accommodation coefficient; T _med medium temperature between the glass tubes (T _med = (T _ig  + T _og )/2); P _a pressure, mmHg; δ _m molecular diameter, cm.

The characteristics of air (at T = 300 °C and I _s = 940 W/m²) are given in Table 5 [21].

Table 5 Air characteristics [21]

Radiation heat transfer coefficient between the two glass tubes [20]:

$$h_{r,ig - og} = \frac{{\sum \left( {T_{ig} + T_{og} } \right)\left( {T_{ig}^{2} + T_{og}^{2} } \right)}}{{\left( {1/\varepsilon_{og} } \right) + \left( {1/\varepsilon_{ig} } \right) - 1}}$$

(49)

1.4 Heat transfer from the outer glass to the ambient

Convection heat transfer coefficient from the outer glass to the ambient [20]:

$$h_{c,og - 0} = \frac{{Nu_{0} \lambda_{og - 0} }}{{d_{og,e} }}$$

(50)

For wind speed greater than 0.1 m/s, forced convection heat transfer will occur. Nusselt number in this case is estimated using the correlation proposed by Zukauskas [22]:

$$Nu_{0} = C\text{Re}_{og}^{p} \Pr\nolimits_{0}^{n} \left( {\frac{{\Pr_{0} }}{{\Pr_{og} }}} \right)^{1/4}$$

(51)

The values of p and C are given in Table 6, and n is equal to 0.37 (for Pr₀ ≤ 10 : n = 0.37 ; for Pr₀ > 10 : n = 0.36).

Table 6 Values used for p and C depending on the Reynolds number [22]

Equation (A.30) is available for 0.7 < Pr _og < 500 and 1 < Re _og < 10⁶. All the air properties have been evaluated at ambient temperature, except Pr _og , which is evaluated at outer glass temperature.

Radiation heat transfer coefficient from the outer glass to the ambient:

$$h_{r,og - 0} = \varepsilon_{og} \cdot \sigma \left( {T_{og} + T_{sky} } \right)\left( {T_{og}^{2} + T_{sky}^{2} } \right) ,$$

(52)

where T _sky is the sky temperature, $T_{sky} = T_{0} - 6$, K.

Appendix 2: Storage tank equations

1.1 Heat transfer from the working fluid to the serpentine wall

The convection heat transfer coefficient in helically coiled tubes has been calculated with the correlation proposed by Gnielinski [20]:

$$h_{c,ws - sl} = \frac{{Nu_{ws} \lambda_{ws} }}{{d_{sl,i} }}$$

(53)

Critical Reynolds number [20]:

$$\text{Re}_{crit} = 2300 \cdot \left[ {1 + 8.6 \cdot \left( {\frac{{d_{sl,i} }}{D}} \right)^{0.45} } \right]$$

(54)

For laminar flow, Re < Re _crit , the Nusselt number is obtained from [20]:

$$Nu_{ws} = 3.66 + 0.08\left[ {1 + 0.8\left( {\frac{{d_{sl,i} }}{D}} \right)^{0.9} } \right]\text{Re}_{ws}^{p} \Pr\nolimits_{ws}^{{^{1/3} }} \left( {\frac{{\Pr_{ws} }}{{\Pr_{sl} }}} \right)^{0.14}$$

(55)

$$p = 0.5 + 0.2903\left( {\frac{{d_{sl,i} }}{D}} \right)^{0.194}$$

(56)

$$D = D_{C} \left[ {1 + \left( {\frac{{b_{s} }}{{\pi D_{C} }}} \right)} \right]$$

(57)

$$D_{C} = \sqrt {D_{S}^{2} - \left( {\frac{{b_{s} }}{\pi }} \right)^{2} }$$

(58)

$$D_{S} = \frac{{L_{sl} }}{{n_{c} \pi }}$$

(59)

where D average diameter of the coil curvature, m; D _C projected diameter of a winding, m; b _s pitch, m; D _S average diameter of a spiral with n_c turns and a pitch bs, formed from the tube of length L_sl, m; L _ol coil length, m; n _c number of turns. Pr _ws Prandtl number at working fluid in the serpentine temperature, Pr _sl Prandtl number at the serpentine wall temperature, Prandtl number is evaluated at medium fluid temperature between inlet and outlet from the serpentine. For turbulent flow regime, Re _ws > 2.2 × 10⁴, the following correlation has been used [20]:

$$Nu_{ws} = \frac{{\left( {f/8} \right)\text{Re}_{ws} \Pr_{ws} }}{{1 + 12.7\sqrt {f/8} \left( {\Pr_{ws}^{{^{2/3} }} - 1} \right)}}\left( {\frac{{\Pr_{ws} }}{{\Pr_{sl} }}} \right)^{0.14}$$

(60)

$$f = \left[ {\frac{0.3164}{{\text{Re}_{ws}^{0.25} }} + 0.03\left( {\frac{{d_{sl,i} }}{D}} \right)^{0.5} } \right]\left( {\frac{{\mu_{sl} }}{{\mu_{ws} }}} \right)^{0.27}$$

(61)

where μ _sl dynamic viscosity evaluated at the serpentine wall temperature, Pa∙s; μ _ws dynamic viscosity evaluated at the mean fluid temperature, Pa∙s.

In the case of Re _crit < Re _ws > 2.2 × 10⁴, the Nusselt number corresponding to transition regime is [20]:

$$Nu_{ws} = \gamma_{t} Nu_{l} \left( {\text{Re}_{crit} } \right) + \left( {1 - \gamma_{t} } \right)Nu_{t} \left( {\text{Re} = 2.2 \times 10^{4} } \right)$$

(62)

where Nu _l Nusselt number for the laminar flow, calculated with Eq. (55), for Re_ws = Re_crit, Nu _t Nusselt number for the turbulent flow, calculated with Eq. (60), for Re_ws = 2.2 × 10⁴.

1.2 Heat transfer from the serpentine wall to the water in the storage tank

If there is no water consumption from the tank, the mass flow rate is zero, which means that the heat transfer occurs by free convection. The corresponding Nusselt number has the expression [20]:

$$Nu_{wb} = \left\{ {0.6 + 0.387\left[ {Ra_{L} f_{2} \left( {\Pr\nolimits_{wb} } \right)} \right]^{1/6} } \right\}^{2}$$

(64)

$$f_{2} \left( {\Pr\nolimits_{wb} } \right) = \left[ {1 + \left( {\frac{0.559}{{\Pr_{wb} }}} \right)^{9/16} } \right]^{ - 16/9}$$

(65)

$$h_{c,sl - wb} = \frac{{Nu_{wb} \lambda_{wb} }}{{d_{sl,e} }}$$

(66)

When there is consumption from the storage tank, the water in the first layer is replaced by cold water from the network, which is inserted into the boiler through the bottom. The Nusselt number for forced convection is calculated using the formula [20]:

$$Nu_{wb} = 0.3 + \sqrt {Nu_{lam}^{2} + Nu_{turb}^{2} }$$

(67)

$$Nu_{lam} = 0.664\sqrt {\text{Re}_{L} } \sqrt[3]{{\Pr\nolimits_{wb} }}$$

(68)

$$Nu_{turb} = \frac{{0.037\text{Re}_{L}^{0.8} \Pr_{wb} }}{{1 + 2.443\text{Re}_{L}^{ - 0.1} \left( {\Pr_{wb}^{2/3} - 1} \right)}}$$

(69)

$$\text{Re}_{L} = \frac{{w_{wb} l_{sl} }}{{v_{w} }}$$

(70)

$$h_{c,sl - wb} = \frac{{Nu_{wb} \lambda_{wb} }}{{l_{sl} }}$$

(71)

$$l_{sl} = \frac{{\pi d_{sl,e} }}{2}$$

(72)