A new method for the measurement of solar collector time constant

Renewable Energy 30 (2005) 855–865 www.elsevier.com/locate/renene

A new method for the measurement of solar collector time constant H.J. Houa,b, Z.F. Wangc, R.Z. Wanga,b,*, P.M. Wangc a

Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, Shanghai 200030, China b Engineering Center for Solar Power and Refrigeration, MOE China, Shanghai 200030, China c Solar Energy Lab, Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100080, China Received 24 April 2004; accepted 13 August 2004 Available online 30 November 2004

Abstract A new test method about the time constant of the solar collector has been presented in this paper. It is simple and has been validated through experiments. With the new method it is not necessary to adjust the inlet temperature of the transfer fluid as closely as possible to the ambient air temperature. Also, it is not necessary to know the characteristic parameters of the collector in advance. The model used in the paper is a first order system model, as in most cases. The experimental data obtained from the test of solar collector time constant shows that the solar collector is not a strictly first order system. A criterion is proposed to decide whether the system is a first order system or not and the resemblance of the system to the first order system. q 2004 Elsevier Ltd. All rights reserved. Keywords: Solar collector; Time constant; First order system

1. Introduction As one kind of solar energy absorbers, solar collectors usually work outdoors, so they are subjected to unsteady weather conditions. For example, when a cloud shades the sun abruptly, the incident solar energy on the collectors abruptly reduces to zero, thus a * Corresponding author. Address: Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, Shanghai 200030, China. Tel.: C86 21 6293 3838; fax: C86 21 6293 3250. E-mail address: [email protected] (R.Z. Wang). 0960-1481/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2004.08.005

856

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

Nomenclature A system amplification coefficient (–) Ac collector aperture (m2) Cp specific heat of the heat transfer fluid (J/(kg 8C)) FR solar collector heat removal factor (–) G solar irradiance (W/m2) m_ mass flow rate of heat transfer fluid (kg/s) (mc)e effective heat capacity of the solar collector (J/(m2 8C)) T time constant (s) Ta ambient air temperature (8C) Tf,i collector inlet temperature (8C) Tf,o collector outlet temperature (8C) UL solar collector heat loss coefficient (W/(m2 8C)) Tf,o,initial collector outlet temperature at the beginning of time constant test period (8C) Tf,o,N collector outlet temperature at time tZN(8C) Greek (ta) effective transmittance absorptance product (–) t time (s)

transient driving force is generated on the solar collectors. The collectors then begin to operate under a transient condition. Thus besides to steady state or quasi-steady state, transient process also needs to be studied. Time constant is one of the most important parameters of solar collectors thermal performance under transient condition. It indicates the response characteristics of solar collector under transient forces. Therefore the solar collector time constant test is one important part of the thermal performance tests. Although the time constant of the solar collector can be obtained through the two established standards: ISO 9806-1 and ASHRAE 93-86 standard, many limitations are imposed by them. In the two standards, we need to adjust the temperature of the transfer fluid at the inlet Tf,i as closely as possible (preferably within G1 8C) to the ambient air temperature Ta otherwise ISO 9806-1standard is invalid or another parameter of the solar collector FRUL is needed to meet the requirements of ASHRAE 93-86 standard. 2. Test methods about solar collector time constant Relevant solar collector time constant tests procedures are different in different standards [1,2], which deal with thermal performance test of solar collectors. In ISO9806-1 standard [1] the test procedure for collector time constant is described by: 1. Testing shall be performed either outdoors or in a solar irradiance simulator. 2. During the test the solar irradiance on the plane of the collector aperture shall be greater than 800 W mK2.

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

857

3. The heat transfer fluid shall be circulated through the collector at the same flow rate as that used during collector thermal efficiency tests. 4. The temperature Tf,i shall be set approximately equal to the ambient air temperature Ta. 5. Initially, the aperture of the collector shall be shielded from the solar radiation by means of a solar-reflecting cover. When a steady state has been reached, the cover shall be removed and measurements continued until steady state conditions have been achieved again. The time constant of collector T is defined as the time taken for the collector outlet temperature to rise by 0.632 of the total increase from Tf;o;t K Ta jtZ0 to Tf;o;t K Ta jtZN: In ASHRAE 93-86 [2], the first three steps are the same as that ISO9806-1 standard. While step 4 for ISO9806-1 is not necessarily for ASHRAE 93-86. Then the step 5 states: The collector is opposed to an incident solar flux of greater than 800 W mK2. When a steady state has been reached, the incident solar energy is then abruptly reduced to zero by either shielding the collector from the sun or shutting off the solar simulator. Simultaneously, the temperature Tf,i and the temperature Tf,o(t) are continuously monitored. If the temperature Tf,i is set approximately equal to the ambient air temperature Ta during the test period the time constant of collector T is the time required for the quantity ðTf;o ðtÞK Tf;i Þ=ðTf;o;initial K Tf;i Þ to change from 1.0 to 0.368. Otherwise, Tfi sTa during the test period, the time constant T will be the time required for the quantity _ p ½Tf;o ðtÞ K Tf;i  Ac FR UL ðTf;i K Ta Þ C mc _ p ðTf;o;initial K Tf;i Þ Ac FR UL ðTf;i K Ta Þ C mc to change from 1.0 to 0.368. Here as we can see the parameter FRUL is introduced and needs to be known in advance.

3. Theoretical basis When the collectors work under the condition in which the heat transfer fluid moves through the collector and carries the useful energy gained away. The energy balance equation is: ðmcÞe

dTf;m _ p ðTf;o K Tf;i Þ Z Ac F 0 ½S K UL ðTf;m K Ta Þ K mc dt _ p ðTf;o K Tf;i Þ Z Ac FR ½S K UL ðTf;i K Ta Þ K mc

(1)

where SZ GðtaÞ: The model is based on the Hottel Whillier Bliss equation [3] with correction term of thermal capacitance. When a steady state for the collector has been reached, if the solar irradiation G is suddenly changed and then held constant, the solar collector behavior will change from one steady state to another. During the transient process, assuming that the temperature Tf,i, Ta, wind speed and mass flow rate of the heat transfer fluid m_ are kept constant, then the rate of change of the transfer fluid exit temperature with time is related to the rate of

858

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

change of transfer fluid average temperature with time by [4]: dTf;m dT Z K f;o dt dt Thus, the energy balance equation of transient process is changed as 8 dT < _ p ðTf;o K Tf;i Þ ðmcÞe K f;o Z Ac FR ½S K UL ðTf;i K Ta Þ K mc dt :T ZT tZ0 f;o

(2)

(3)

f;o;initial

Eq. (3) can be rearranged into the following: 8 AF < ðmcÞe K dðTf;o K Tf;i Þ C ðTf;o K Tf;i Þ Z c R ½S K UL ðTf;i K Ta Þ _ p _ p mc dt mc : Tf;o Z Tf;o;initial t Z 0

(4)

The solution of Eq. (4) is:   _ p ½Tf;o ðtÞ K Tf;i  _ pt Ac FR ½S K UL ðTf;i K Ta Þ K mc mc Z exp K _ p ðTf;o;initial K Tf;i Þ Ac FR ½S K UL ðTf;i K Ta Þ K mc KðmcÞe

(5)

In ISO9806-1 standard, by controlling the temperature Tf,i to be almost equal to Ta, thus the Eq. (5) becomes   _ p ½Tf;o ðtÞ K Ta  _ pt Ac FR S K mc mc Z exp K (6) _ p ðTf;o;initial K Ta Þ Ac FR S K mc KðmcÞe Under the former steady state condition, the solar irradiance on the plane of the collector aperture was adjusted to be equal to zero. Thus, the equation Tf;o Z Tf;i Z Ta is established during the period. Accordingly the initial condition of the transient process between the two steady state conditions is: Tf;o;t jtZ0 Z Tf;o;initial Z Tf;i Z Ta : Then Eq. (6) becomes    _ pt mc Ac FR S Tf;o ðtÞ K Ta Z 1 K exp K (7) _ p KðmcÞe mc _ p Þ; thus When t/N then Tf;o;NK Ta Z ðAc FR SÞ=ðmc    _ pt mc Tf;o ðtÞ K Ta Z 1 K exp K KðmcÞe Tf;o;N K Ta

(8)

The time constant of the collector is the time required for the quantity ðTf;o ðtÞK Ta Þ=ðTf;o;NK Ta Þ to change from 0 to 0.632. Whereas Tf;o sTa during the former steady state condition that means the solar irradiance on the collector is not zero at this time. The time constant of the collector should be the time required for the quantity ðTf ;o K Ta Þjt K ðTf ;o K Ta Þjt¼0 ðTf ;o K Ta Þjt¼N K ðTf ;o K Ta Þjt¼0 to change from 0 to 0.632. This is the test method of collector time constant described in ISO9806-1 standard.

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

859

In ASHRAE 93-86 standard, under the former steady state condition, the incident solar energy is controlled greater than 790 W mK2. Then it is abruptly reduced to zero and kept constant during the following period of the transient process. Thus the Eq. (5) becomes   _ p ½Tf;o ðtÞ K Tf;i  _ pt Ac FR ½UL ðTf;i K Ta Þ C mc mc Z exp K _ p ðTf;o;initial K Tf;i Þ Ac FR ½UL ðTf;i K Ta Þ C mc KðmcÞe If Tf;i Z Ta is valid, Eq. (9) becomes   _ pt mc Tf;o;t K Tf;i Z exp K KðmcÞe Tf;o;initial K Tf;i

(9)

(10)

The two Eqs. (10) and (9) are the test methods of collector time constant described in ASHRAE 93-86. The time constant of the collector is the time required for the left hand terms of the Eqs. (10) and (9) to change from 1 to 0.368. Under the two conditions the temperature Tf,i is either adjusted to be close to the ambient air temperature Ta or not respectively. By observing Eq. (4), we find it has the same form as that of a first-order system has. T

dy C y Z Ax dt

(11)

Eq. (11) is the typical differential equation of a first-order system [5]. Where y is the system output, x is the system input, T is the time constant of the object, A is the system amplification coefficient. They are expressed as y Z Tf;o K Tf;i

x Z S K UL ðTf;i K Ta Þ

TZ

ðmcÞe K _ p mc

AZ

Ac FR _ p mc

(12)

where T is the time constant of the collector, which is defined in the two standards mentioned above.

4. New test method In many cases, it is difficult to adjust the temperature Tf,i to be almost equal to the ambient air temperature Ta. For example during winter in Beijing, the water temperature is always higher than Ta. In order to obtain the time constant under this condition, besides the method described by the ASHRAE 93-86 standard, a new method is presented. The method is simpler than the one described by ASHRAE 93-86 standard. For the new method, it is not required to have Tf,i equal to Ta. The detailed formula derivation is given below. _ p Þ to Eq. (5) and making rearrangement of it, it is then By adding T Z ððmcÞe KÞ=ðmc changed as h ti _ p ½ðTf;o ðtÞ K Tf;i Þ K ðTf;o;initial K Tf;i Þ mc Z 1 K exp K (13) _ p ðTf;o;initial K Tf;i Þ Ac FR ½S K UL ðTf;i K Ta Þ K mc T

860

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

Thus, Tf;o ðtÞ K Tf;o;initial Z

t i _ p ðTf;o;initial K Tf;i Þ h Ac FR ½S K UL ðTf;i K Ta Þ K mc 1 K exp K _ p mc T

(14)

If tZN; Tf;o;N K Tf;o;initial Z

_ p ðTf;o;initial K Tf;i Þ Ac FR ½S K UL ðTf;i K Ta Þ K mc _ p mc

(15)

Then

t i Tf;o ðtÞ K Tf;o;initial h Z 1 K exp K T Tf;o;N K Tf;o;initial

(16)

Here as we can see, the time constant of the collector is the time required for the quantity ðTf;o ðtÞK Tf;o;initial Þ=ðTf;o;NK Tf;o;initial Þ to change from 0 to 0.632. In this method, we used the Eq. (5) directly so we need to meet the requirement that described by the paragraph before Eq. (2). According to the requirement, during the test period, the solar irradiation G can be changed into the two modes described in the two established standards and the temperature Tf,i, Ta and the mass flow rate m_ keep constant. It is not necessary to adjust the temperature Tf,i to be equal to Ta as it is needed in ISO9806-1. Also it is not necessary to know the FRUL in advance while it is necessary in ASHRAE 93-86.

5. Experimental verification The tests were conducted outdoors in Beijing, China. During the experiment the weather was clear. The measurements were made on a glazed flat plate collector and the heat transfer fluid was water. The aperture area of the flat plate collector was 1.2 m2. _ G, Tf,i, Tf,o and Ta are connected to a data acquisition Various transducers about m; instrument interfaced to a computer. The sampling rate is set as 4 s for each data. During the test period the ambient air temperature was around 30 8C. The flow rate of heat transfer fluid was adjusted to 1 l/min. In order to validate the new method, three experiments were conducted. During Experiment 1 and 2, the temperature Tfi was regulated at 38 8C, the solar irradiation G was changed into the two modes described by ISO9806-1 and ASHRAE 93-86, respectively. As Tfi sTa and FRUL is unknown, the time constant of the solar collector cannot be obtained by the two standards mentioned above. But it can be obtained by the new method. Experiment 1. When the solar irradiance G was changed as was required in ISO9806-1 _ the ambient air temperature Ta, the temperature Tfi and Tf,o of the standard, the flow rate m; collector were continuously monitored at the same time until the next steady state established. Here, the equation Tf;o Z Tf;o;N is established. For the purpose of this test, a steady state condition is assumed to exist when the temperature Tf;o varies by less than

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

861

Fig. 1. R1 as a function of time for Experiment 1.

0.05 8C per minute. By plotting R1 Z

Tf;o ðtÞ K Tf;o;initial Tf;o;N K Tf;o;initial

as a function of time in Fig. 1, the time constant of the collector is 76 s. The temperature Tf,o as a function of time is plotted in Fig. 2. Experiment 2. The solar irradiance G was changed as required by ASHRAE 93-86 _ the ambient air temperature Ta, the temperature Tfi and Tf,o of standard and the flow rate m; the collector were monitored continuously. By plotting R1 Z

Tf;o ðtÞ K Tf;o;initial Tf;o;N K Tf;o;initial

as a function of time in Fig. 3. The time constant of the collector is 73 s. The temperature Tf,o as a function of time is plotted in Fig. 4. Experiment 3. The time constant of the collector was tested according to the method described by ISO9806-1. During the test period the temperature Tfi was adjusted

Fig. 2. Tf,o as a function of time.

862

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

Fig. 3. R1 as a function of time for Experiment 2.

approximately equal to the ambient air temperature Ta. The time constant of the collector is 78 s. The temperature Tf,o as a function of time is plotted in Fig. 5.

6. Results and discussion The experiment results showed that the three time constants obtained by the new method are approximately equal to each other. By observing Eqs. (8) and (16), taking natural logarithm for both sides, the two equations become Eqs. (17) and (18), respectively   _ pt mc T ðtÞ K Ta ln 1 K f;o (17) ZK KðmcÞe Tf;o;N K Ta   _ pt mc Tf;o ðtÞ K Tf;o;initial ln 1 K ZK KðmcÞe Tf;o;N K Tf;o;initial

Fig. 4. Tf,o as a function of time.

(18)

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

863

Fig. 5. Tf,o as a function of time.

If TZ

KðmcÞe ; _ p mc

  Tf;o ðtÞ K Tf;o;initial Y Z ln 1 K ; Tf;o;N K Tf;o;initial and   T ðtÞ K Ta Z Z ln 1 K f;o Tf;o;N K Ta are defined, then the Eqs. (17) and (18) becomes: t (19) Y ZK T t Z ZK (20) T It indicates that the variables Y, Z and the time t are of linear relationship. By using the experiment data of Experiments 1, 2, and 3, Figs. 6–8 could be plotted.

Fig. 6. Y–t relationship corresponding to Experiment 1.

864

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

Fig. 7. Y–t relationship corresponding to Experiment 2.

Fig. 8. Z–t relationship corresponding to Experiment 3.

Figs. 6–8 have shown the variables Y, Z and the time t are not of linear relationship apparently. The reason is that the solar collector model used here is a first-order model as in most cases [1,2,6–8]. But in fact it is not a strictly first order system, a solar collector includes many energy storage elements, such as working fluid in absorber tubes, absorber tubes, etc. it is a multi-capacity object. An n-order model is better than a first order model. A first order system model is just a simplified model. The deviation of the experiment data to its linear regression curve reversely indicates the resemblance of the system to the linear system. In other words, the less deviation, the more likely to be a first order system.

7. Conclusions 1. A new method for testing the time constant of the solar collector was validated. During the tests it was not necessary to adjust the temperature Tfi to be equal to Ta as it was needed in ISO9806-1 and it was also not necessary to know the FRUL in advance like in ASHRAE 93-86.

H.J. Hou et al. / Renewable Energy 30 (2005) 855–865

865

2. In fact a solar collector is an n-order system. To consider it as a first order system is just a simplified model. 3. A criterion is presented to decide whether a system is a first order system or not and the resemblance of the system to the first order system.

References [1] ISO 9806-1. Test methods for solar collectors. Part 1: thermal performance of glazed liquid heating collectors including pressure drop, 1994. [2] ANSI/ASHRAE 93-method of testing to determine the thermal performance of solar collectors. New York: ASHRAE Inc.; 1986. [3] Duffie J, Beckman W. Solar engineering of thermal processes, 2nd ed. New York: Wiley; 1991. [4] Simon F. Flat-plate solar-collector performance evaluation with a solar simulator as a basis for collector selection and performance prediction. Solar Energy 1976;18:451–66. [5] Zhang FW, et al. Thermal system process control, analysis, design and debugging. Nanjing: Southeast University Press; 1999 [in Chinese]. [6] Muschaweck J, Spirkl W. Dynamic solar collector performance testing. Solar Energy Mater Solar Cells 1993; 30:95–105. [7] Amer EH, Nayak JK, Sharma GK. Transient method for testing flat-plate solar collectors. Energy Convers Manage 1998;39(7):549–58. [8] Amer EH, Nayak JK, Sharma GK. A new dynamic method for testing solar flat-plate collectors under variable weather. Energy Convers Manage 1999;40:803–23.