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Selection of design parameters for closed-circuit forced-circulation solar heating systems
Solar Energy, Vol. 17, pp. 331-333. Pergamon Press 1975. Printed in Great Britain
SELECTION OF DESIGN PARAMETERS FOR CLOSED-CIRCUIT FORCED-CIRCULATION SOLAR HEATING SYSTEMS B. J. BRINKWORTH Solar Energy Unit, University College, Cardiff, Wales (Received 22 April 1975) Abstraet--A system effectiveness, determined only by the physical characteristics, is defined for a closed-circuit forced-circulation solar heating system. Conditions leading to high values of the effectiveness are identified and simple methods developed for the selection of design parameters giving a specified performance. INTRODUCTION
It seems inevitable that a closed-circuit solar heating system must be less efficient than a corresponding open-circuit one, because its collector is required to operate at a higher temperature. This is due to the imperfect effectiveness of the heat exchanger in the storage tank. The extent to which it might be offset by operating the collector more efficiently is not at first clear. For high collection efficiency, a high rate of fluid flow through the collector is required. On the other hand, a high effectiveness of the heat exchanger will be obtained only if the flow-rate through it is low. It appears that there might be an optimum flow-rate at which the proportion of the incident energy transferred to the storage tank would be a maximum. If it were found to be impracticable to use this flow-rate, perhaps because of high pumping costs, the designer would need to know what penalty would be paid in" loss of performance by using a smaller value. In addition, it is desirable to know what other system parameters have the most control in this situation so that the performance may be optimised at the design stage. It will be shown here that there exists a system effectiveness, determined by its physical characteristics only, to which the rate of energy retention by the system is proportional. At sufficiently high flow-rates this quantity reaches an asymptotic value determined largely by the characteristics of the heat exchanger. It will be seen that this asymptotic value and the flow-rate at which it is approached to any desired degree are easily determined.
Storage tank
7"2~_
--
~ e c t o r
0
Fig. 1. Basic closed-circuit forced-circulation solar heating systems. The temperature drop through the heat-exchanger may also be expressed in simple analytic form. If it is assumed that a reference temperature T3 can be assigned to the fluid in the storage tank and that there is an effective overall conductance Us between the two fluids, heatexchanger theory[3] predicts a relationship of the form (T~- T3)/(T2- T3) = exp (-U2A2/m G)
(3)
where the argument of the exponential is the number of heat transfer units or NTU of the exchanger. For brevity, this expression will be written in the form (T1- T3)/(T2- T3) = B
(4)
from which it follows that the temperature drop through the exchanger is given by
EFFICIENCY OF CLOSED-CIRCUIT SYSTEMS
A typical closed-circuit forced-circulation system is shown in Fig. I. If it may be assumed that a steady state exits and that losses from the connecting pipes may be neglected, it is clear that the temperature rise T2- T~ in the collector is equal to the temperature drop in the heat exchanger. The temperature rise in the collector is often represented[l, 2] by the expression T2- T, = ( q J U , - ( T ~ - To))(I - e x p (-UoAo/mG))
T : - T1 = ( T , - T3)(1- B)/B.
Equations (2) and (5) may be used to obtain T1 in terms of T3, yielding, after some manipulation, (T2 - T,) = (C - (T3 - T.))(1 - A)(1 - B)/(1 - AB). (6)
(1)
The rate of energy retention Q by the system is men (T2 - T1), and it follows from (1) and (2) that the product
which might be expressed briefly in the form T2- T, = (C - (T, - T.))(1 - A).
(5)
(2)
mcp
331
=
- UoAd In A = UoAo/In (l/A).
(7)
332
B. J.
BRINKWORTH
Hence the rate of energy retention may be cast in the form Q = Ao[qaUo/U~ - Uo(T3- T~)]D
(8)
where D
(1-A)(1-B) (1 -
AB) In (l/A)"
(9)
From eqn (9) we see that the energy retention is zero when qa = U~(T3-To). This shows that the effective average temperature of the collector plate is equal to T3, the temperature of the storage tank in the vicinity of the heat exchanger. The quantity D will be called the system effectiveness. It has two parameters, A = exp ( - UoAo/mG)
(10)
B = exp (-U2AJmcp),
(11)
and
with AB = exp (-( UoAo + U2A2)/mcp ).
(12)
Thus D is a function only of the physical characteristics of the collector and the heat exchanger together with the mass flow-rate and specific heat capacity of the fluid. It has values between 0 and 1, the latter value being reached only when A = 1, that is, when m ~ ~. The condition B = 0 represents a perfect heat exchanger, and for that case D reduces to the form (1 - A)/ln (I/A), which on expansion becomes Do = (1 - exp ( - UoAo/mCp))/(UoAo/mq,).
(13)
A always exceed 0.81, the value corresponding to the perfect heat exchanger case, B =0. For any real exchanger, it is apparent that A must be greater than this value if D > 0.9 is to be obtained. However, for any given value of A and lower limit for D there is a maximum permissible value of B. It is now possible to determine the effect of increasing the flow-rate. If all other quantities are constants, the maximum of D may be sought by equating to zero the derivative of eqn (9) with respect to m. It is found that no maximum exists below rn = ~. This means that any increase in m produces an increase in the collection efficiency which more than offsets the associated decrease in heat exchanger effectiveness. However, this is for constant values of all other quantities, and does not rule out the possibility of an optimum m occurring through the coupling between m and Uo or m and U2. In the collector the overall conductance is determined in practice largely by factors unconnected with the fluid flow, and it is expected that Uo would be only weakly dependent upon m. Heat exchangers of many kinds are possible, with widely differing characteristics. However, it has been found in tests with typical systems that eqn (3) represents the behaviour well and that a value may be assigned to U2 which is substantially independent of m. SELECTION OF PARAMETERS
As m increases, A and B approach their limiting values at different rates. By expanding the terms in eqn (9) as series, it is found that D tends to the asymptotic value Dl = U2Az/(UoAo + U2A2) = 1/(1 ÷ UoAo/U2A2).
(14)
This will be recognised as the flow factor [1, 2] for the collector alone,
It follows that the system must be designed to satisfy the criterion
DISCUSSION
UoAo/U2Az = (1 - D,)/D,.
Figure 2 shows part of the relationship between D and its parameters A and B, in carpet form for ease of interpolation. Evidently, it is desirable to operate the system in conditions corresponding with the upper part of the field, say D > 0.9. This particular objective would require that
Finally, it is necessary to determine the flow-rate at which the value of D approaches its asymptote within an agreed margin, so that the lowest effective value of m may be chosen. Expansion of eqn (9) to a further order and division by D1 leads to the expression
I'0
COllectparameter,ote;,~/~//A or / // / /
0-95
~" 0 9 0
8~8~/%./ ~
/
/"-./
0"85
If we choose a value of D which approaches the asymptote within a fraction E of its value, we must then use a value of m given by
Since, from eqn (14), any reasonable value of D requires that UoAo ~ U2A2, this criterion is, for practical purposes, about
0"80
075
(17)
)'1 0"2
E
~
D/D~ = 1-(UoAoU2Az/mcp(UoAo+ U2A2)). (16)
m >- UoAoU2A2/ecp(UoAo+ U2A2).
.>
(15)
Heat
exct~anger ~/
V^~ v06 m >- UoAo/eCp.
Fig.2. The systemeffectiveness.
(18)
For example, consider a water heating system with
Selection of design parameters for closed-circuit forced-circulation solar heating systems Ao = 5 m2, Uo = 5 W/m2Kand cp = 4200 J/kg K. If we wish to operate within 5% of the asymptotic value of the system effectiveness D1 (i.e. E = 0.05) then by eqn (18), the flow-rate must be greater than 0.119 kg/s. The desired operating value of D might be 0.9, so that the asymptotic value DI would be D/(1 - ~), that is, 0.947. By eqn (15), we are thus able to establish that the product U2A2 for the heat-exchanger must be 447 W/K. If on the other hand, the design had to be based on a known exchanger, with U2A2=2OOW/K, we then find from eqn (14) that D, = 0.889, leading to an operating value for D of 0.844. Equations (15) and (18) are simple but powerful design rules for closed-circuit systems. With their aid, the essential operating parameters may be obtained at the earliest stages of design.
NOMENCLATURE cp specificheat capacity of fluid m mass flow-rate of fluid in system q~ rate of absorption of radiation on collector plate abbreviation for 1 - D/D1 A abbreviation for exp (-UoAo/mcp) Ao reference surface area of collector plate A2 reference surface area of heat exchanger B abbreviation for exp (-U2AJmcp) C abbreviation for q./U1 D system effectiveness: abbreviation for (1 - B ) / ( 1 -
AB) In
333
(1 - A)
(l/A)
Do D, Q T, T, T~ T3 Uo
CONCLUSION
value of D with B =0 asymptotic value of D as m ~oc rate of retention of energy by system ambient temperature fluidtemperature at inlet to collector fluid temperature at outlet from collector tank fluid temperature in vicinity of heat exchanger overall conductance between fluid in collector and surroundings U, conductance between collector plate and surroundings U2 overall conductance between fluid in heat exchanger and tank fluid
It has been shown that a system effectiveness may be defined, determined only by the physical characteristics of the collector and the heat exchanger. Criteria have been established for obtaining high values of this factor. Simple rules have then been developed, which enable the system designer to select a suitable heat exchanger and to determine the flow-rate at which it should be operated for a given performance.
1. H. C. Hottel and A. Whillier, Evaluation of flat-plate solar collector performance. Trans. Conf. on Use of Solar Energy, Univ. Arizona, Tucson 2, 74-104 (1958). 2. R. W. Bliss, Jr., The derivation of several "plate efficiency factors" useful in the design of flat-plate solar heat collectors. Solar Energy 3, 55--64 (1959). 3. A. P. Fraas and M. N. Ozisik, Heat Exchanger Design. Wiley, New York (1%5).
REFERENCES
SELECTION OF DESIGN PARAMETERS FOR CLOSED-CIRCUIT FORCED-CIRCULATION SOLAR HEATING SYSTEMS B. J. BRINKWORTH Solar Energy Unit, University College, Cardiff, Wales (Received 22 April 1975) Abstraet--A system effectiveness, determined only by the physical characteristics, is defined for a closed-circuit forced-circulation solar heating system. Conditions leading to high values of the effectiveness are identified and simple methods developed for the selection of design parameters giving a specified performance. INTRODUCTION
It seems inevitable that a closed-circuit solar heating system must be less efficient than a corresponding open-circuit one, because its collector is required to operate at a higher temperature. This is due to the imperfect effectiveness of the heat exchanger in the storage tank. The extent to which it might be offset by operating the collector more efficiently is not at first clear. For high collection efficiency, a high rate of fluid flow through the collector is required. On the other hand, a high effectiveness of the heat exchanger will be obtained only if the flow-rate through it is low. It appears that there might be an optimum flow-rate at which the proportion of the incident energy transferred to the storage tank would be a maximum. If it were found to be impracticable to use this flow-rate, perhaps because of high pumping costs, the designer would need to know what penalty would be paid in" loss of performance by using a smaller value. In addition, it is desirable to know what other system parameters have the most control in this situation so that the performance may be optimised at the design stage. It will be shown here that there exists a system effectiveness, determined by its physical characteristics only, to which the rate of energy retention by the system is proportional. At sufficiently high flow-rates this quantity reaches an asymptotic value determined largely by the characteristics of the heat exchanger. It will be seen that this asymptotic value and the flow-rate at which it is approached to any desired degree are easily determined.
Storage tank
7"2~_
--
~ e c t o r
0
Fig. 1. Basic closed-circuit forced-circulation solar heating systems. The temperature drop through the heat-exchanger may also be expressed in simple analytic form. If it is assumed that a reference temperature T3 can be assigned to the fluid in the storage tank and that there is an effective overall conductance Us between the two fluids, heatexchanger theory[3] predicts a relationship of the form (T~- T3)/(T2- T3) = exp (-U2A2/m G)
(3)
where the argument of the exponential is the number of heat transfer units or NTU of the exchanger. For brevity, this expression will be written in the form (T1- T3)/(T2- T3) = B
(4)
from which it follows that the temperature drop through the exchanger is given by
EFFICIENCY OF CLOSED-CIRCUIT SYSTEMS
A typical closed-circuit forced-circulation system is shown in Fig. I. If it may be assumed that a steady state exits and that losses from the connecting pipes may be neglected, it is clear that the temperature rise T2- T~ in the collector is equal to the temperature drop in the heat exchanger. The temperature rise in the collector is often represented[l, 2] by the expression T2- T, = ( q J U , - ( T ~ - To))(I - e x p (-UoAo/mG))
T : - T1 = ( T , - T3)(1- B)/B.
Equations (2) and (5) may be used to obtain T1 in terms of T3, yielding, after some manipulation, (T2 - T,) = (C - (T3 - T.))(1 - A)(1 - B)/(1 - AB). (6)
(1)
The rate of energy retention Q by the system is men (T2 - T1), and it follows from (1) and (2) that the product
which might be expressed briefly in the form T2- T, = (C - (T, - T.))(1 - A).
(5)
(2)
mcp
331
=
- UoAd In A = UoAo/In (l/A).
(7)
332
B. J.
BRINKWORTH
Hence the rate of energy retention may be cast in the form Q = Ao[qaUo/U~ - Uo(T3- T~)]D
(8)
where D
(1-A)(1-B) (1 -
AB) In (l/A)"
(9)
From eqn (9) we see that the energy retention is zero when qa = U~(T3-To). This shows that the effective average temperature of the collector plate is equal to T3, the temperature of the storage tank in the vicinity of the heat exchanger. The quantity D will be called the system effectiveness. It has two parameters, A = exp ( - UoAo/mG)
(10)
B = exp (-U2AJmcp),
(11)
and
with AB = exp (-( UoAo + U2A2)/mcp ).
(12)
Thus D is a function only of the physical characteristics of the collector and the heat exchanger together with the mass flow-rate and specific heat capacity of the fluid. It has values between 0 and 1, the latter value being reached only when A = 1, that is, when m ~ ~. The condition B = 0 represents a perfect heat exchanger, and for that case D reduces to the form (1 - A)/ln (I/A), which on expansion becomes Do = (1 - exp ( - UoAo/mCp))/(UoAo/mq,).
(13)
A always exceed 0.81, the value corresponding to the perfect heat exchanger case, B =0. For any real exchanger, it is apparent that A must be greater than this value if D > 0.9 is to be obtained. However, for any given value of A and lower limit for D there is a maximum permissible value of B. It is now possible to determine the effect of increasing the flow-rate. If all other quantities are constants, the maximum of D may be sought by equating to zero the derivative of eqn (9) with respect to m. It is found that no maximum exists below rn = ~. This means that any increase in m produces an increase in the collection efficiency which more than offsets the associated decrease in heat exchanger effectiveness. However, this is for constant values of all other quantities, and does not rule out the possibility of an optimum m occurring through the coupling between m and Uo or m and U2. In the collector the overall conductance is determined in practice largely by factors unconnected with the fluid flow, and it is expected that Uo would be only weakly dependent upon m. Heat exchangers of many kinds are possible, with widely differing characteristics. However, it has been found in tests with typical systems that eqn (3) represents the behaviour well and that a value may be assigned to U2 which is substantially independent of m. SELECTION OF PARAMETERS
As m increases, A and B approach their limiting values at different rates. By expanding the terms in eqn (9) as series, it is found that D tends to the asymptotic value Dl = U2Az/(UoAo + U2A2) = 1/(1 ÷ UoAo/U2A2).
(14)
This will be recognised as the flow factor [1, 2] for the collector alone,
It follows that the system must be designed to satisfy the criterion
DISCUSSION
UoAo/U2Az = (1 - D,)/D,.
Figure 2 shows part of the relationship between D and its parameters A and B, in carpet form for ease of interpolation. Evidently, it is desirable to operate the system in conditions corresponding with the upper part of the field, say D > 0.9. This particular objective would require that
Finally, it is necessary to determine the flow-rate at which the value of D approaches its asymptote within an agreed margin, so that the lowest effective value of m may be chosen. Expansion of eqn (9) to a further order and division by D1 leads to the expression
I'0
COllectparameter,ote;,~/~//A or / // / /
0-95
~" 0 9 0
8~8~/%./ ~
/
/"-./
0"85
If we choose a value of D which approaches the asymptote within a fraction E of its value, we must then use a value of m given by
Since, from eqn (14), any reasonable value of D requires that UoAo ~ U2A2, this criterion is, for practical purposes, about
0"80
075
(17)
)'1 0"2
E
~
D/D~ = 1-(UoAoU2Az/mcp(UoAo+ U2A2)). (16)
m >- UoAoU2A2/ecp(UoAo+ U2A2).
.>
(15)
Heat
exct~anger ~/
V^~ v06 m >- UoAo/eCp.
Fig.2. The systemeffectiveness.
(18)
For example, consider a water heating system with
Selection of design parameters for closed-circuit forced-circulation solar heating systems Ao = 5 m2, Uo = 5 W/m2Kand cp = 4200 J/kg K. If we wish to operate within 5% of the asymptotic value of the system effectiveness D1 (i.e. E = 0.05) then by eqn (18), the flow-rate must be greater than 0.119 kg/s. The desired operating value of D might be 0.9, so that the asymptotic value DI would be D/(1 - ~), that is, 0.947. By eqn (15), we are thus able to establish that the product U2A2 for the heat-exchanger must be 447 W/K. If on the other hand, the design had to be based on a known exchanger, with U2A2=2OOW/K, we then find from eqn (14) that D, = 0.889, leading to an operating value for D of 0.844. Equations (15) and (18) are simple but powerful design rules for closed-circuit systems. With their aid, the essential operating parameters may be obtained at the earliest stages of design.
NOMENCLATURE cp specificheat capacity of fluid m mass flow-rate of fluid in system q~ rate of absorption of radiation on collector plate abbreviation for 1 - D/D1 A abbreviation for exp (-UoAo/mcp) Ao reference surface area of collector plate A2 reference surface area of heat exchanger B abbreviation for exp (-U2AJmcp) C abbreviation for q./U1 D system effectiveness: abbreviation for (1 - B ) / ( 1 -
AB) In
333
(1 - A)
(l/A)
Do D, Q T, T, T~ T3 Uo
CONCLUSION
value of D with B =0 asymptotic value of D as m ~oc rate of retention of energy by system ambient temperature fluidtemperature at inlet to collector fluid temperature at outlet from collector tank fluid temperature in vicinity of heat exchanger overall conductance between fluid in collector and surroundings U, conductance between collector plate and surroundings U2 overall conductance between fluid in heat exchanger and tank fluid
It has been shown that a system effectiveness may be defined, determined only by the physical characteristics of the collector and the heat exchanger. Criteria have been established for obtaining high values of this factor. Simple rules have then been developed, which enable the system designer to select a suitable heat exchanger and to determine the flow-rate at which it should be operated for a given performance.
1. H. C. Hottel and A. Whillier, Evaluation of flat-plate solar collector performance. Trans. Conf. on Use of Solar Energy, Univ. Arizona, Tucson 2, 74-104 (1958). 2. R. W. Bliss, Jr., The derivation of several "plate efficiency factors" useful in the design of flat-plate solar heat collectors. Solar Energy 3, 55--64 (1959). 3. A. P. Fraas and M. N. Ozisik, Heat Exchanger Design. Wiley, New York (1%5).
REFERENCES