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Optimization of a hybrid solar forced-convection water heating system
Energy Convers. Mgmt Vol. 27, No. 4, pp. 367-377, 1987 Printed in Great Britain. All rights reserved
0196-8904/87 $3.00 + 0.00 Copyright © 1987 Pergamon Journals Ltd
OPTIMIZATION OF A HYBRID SOLAR FORCED-CONVECTION WATER HEATING SYSTEM A. K. G O Y A L , A S H V I N I K U M A R ? and M . S. S O D H A Centre of Energy Studies, Indian Institute o f Technology, Hauz Khas, New Delhi 110 016, India (Received 23 June 1986) Abstract--In this paper, a techno-economic model has been developed for a hybrid solar forcedconvection water heating system. Two options of auxiliary energy use, viz. (A) an instant electric heater and (B) use of diesel as the auxiliary energy fuel, have been considered. Numerical calculations have been made for the climate of Delhi, India, corresponding to the two representative demand patterns, viz. (i) hot-water demand of big residential buildings and (ii) industrial hot-water demand. Taking into account the life, capital cost and the maintenance cost o f the solar and auxiliary systems, the cost o f useful energy has been calculated for different values of collector area and tank capacity. This exercise, thereby, yields the optimum values of collector area and tank capacity corresponding to the minimum cost of useful energy. The effect of government subsidy on the optimized values of collector area, tank capacity and cost of useful energy has also been investigated. Optimization
Solar water heating
NOMENCLATURE Ac = Area of solar collector (m 2) A~ = Ac/M (m2/kg) A t = Surface area o f tank T l (m 2) A ~ = Surface area of tank T 2 (m 2) Bf = Annual bill of conventional fuel per unit of hot water demand M (Rs/yr/kg) Cb = Specific heat o f fluid heated by boiler and circulated through heat exchanger in tank T 2 (J/kg °C) C~ = Cost of collector per unit area (Rs/m 2) C = Cost of useful energy, independent of demand (Rs/kWh) C'/M = Cost of useful energy, depending upon demand (Rs/kWh) C, = Cost of installation (Rs) C w = Specific heat o f water in tank T 1 (J/kg °C) C T = Cost o f tank per unit volume (Rs/m 3) Cwc = Specific heat o f collector fluid (J/kg °C) f l , f 2 = A n n u i t y factors, defined by equation (13) F' = Plate efficiency factor of solar collector hf~ = Convective heat transfer coefficient (per unit length) between fluid in heat exchanger and water in tank T z (W/m °C) hf~ = Convective heat transfer coefficient (per unit length) between fluid in heat exchanger and water in tank T 2 (W/m °C) I0 = Initial investment o f conventional system (Rs) I, = Initial investment of solar system (Rs) L = Lifetime o f solar system (yr) L 0 = Lifetime of auxiliary energy system (yr) L~ = Length o f heat exchanger in tank Tt (m) L 2 = Length of heat exchanger in tank T 2 (m) rh,, = Rate of hot water withdrawal (kg/s) rnb = Rate o f hot fluid flow in heat exchanger in tank T 2 (kg/s) rhe = Rate of fluid flow in collector (kg/s) M = Daily hot water demand (kg) M , = Maintenance cost for conventional system (Rs/yr)
INTRODUCTION S o l a r h y b r i d h o t - w a t e r s y s t e m s are t h o s e in w h i c h the s o l a r h o t - w a t e r s y s t e m is a u g m e n t e d w i t h a n auxiliary source o f energy. T h e s e s y s t e m s h a v e t w o m a i n
1"Present address: Solar Energy Centre, Block 134 (Ninth Floor), C.G.O. Complex, Lodi Road, New Delhi 110003, India. EC.M.27/~4:
Mwt = Storage capacity of tank T~ (kg) Mw = g w J M Ms = Maintenance cost for solar system (Rs/yr) P = Price of auxiliary energy (Rs/kWh) Q, = Auxiliary power (W) Qu, = Total annual useful energy (kWh) Q~ = Solar contribution to total useful power (W) S = Solar radiation incident over collectors (W/m 2) Sc = Annual cost of water heating system (Rs/yr) t = Time (s) Ta = Ambient air temperature (°C) Tb = Temperature of boiler which is maintained at constant temperature (°C) Tb~= Fluid temperature when it enters boiler from tank T 2 (°C) Td = Collector fluid temperature when it is fed to collector for recirculation (°C) T~o = Collector fluid temperature when it enters tank T t (°C) T. = Desired water temperature (°C) Tf = Fluid temperature (°C) Ti = Water temperature at mains (°C) Tw = Water temperature in tank T l (°C) T~, = Water temperature in tank I"2 (°C) U L = Heat loss coefficient o f solar collector (W/m 2 °C) Ut = Heat loss coefficient of tank T l (W/m 2 °C) U~ = Heat loss coefficient of tank T 2 ("~V/m 2 °C) x = Position coordinate measured vertically downward (m) Z = Rate of interest (fraction) ~t = Absorptivity o f collector plate fl = Fraction which is defined as ratio o f storage capacity o f tank T2 and daily hot water demand = Transmittivity of cover glass r/E = Utilization efficiency of electricity r/D = Utilization efficiency of diesel
367
368
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM
advantages over the purely solar systems which are as follows: (i) solar hot-water systems cannot meet the hot-water demand on the days of low/no sunshine, and (ii) solar systems designed for minimum insolation conditions will be too large to be cost-effective. Extensive investigations have been made for studying various aspects of solar hot-water systems, which include operation under thermosyphonic mode [1-7], forced-convection mode[8-10], use of heat exchangers in the storage tank[10-12] and use of antifreeze solutions [12]. Experimental [13-17] and theoretical [18-21] studies have also been carried out to predict the thermal performance of these systems. Many of the mathematical models developed, however, have limited application for the evaluation of the techno-economic performance of the solar hybrid water heating system because (i) there is no provision of auxiliary energy, (ii) a very small time step (of the order of seconds) makes the model impractical for evaluation of long-term performance, (iii) a provision for the time-depend.ent hot-water withdrawal is usually not made, and (iv) economic aspects are not included. Amongst others, Gordon and Rabl[22] and Collares-Pareira et al. [23] have presented a method for the design and optimization of single-pass water heating systems used in industrial process heat plants with and without storage; the annual collected energy was estimated by polynomial fitting of the yearly averaged threshold energy. Gutierrez et al. [24] have studied the effects of auxiliary energy supply, load type and storage capacity on the thermal performance of a forced-convection hybrid water heating system. Recently, Sodha et al. [25] and Goyal et al. [26] have presented techno-economic models for a forced-convection hybrid water heating system which uses diesel and electricity as the auxiliary fuel, respectively. These models, however, considered only a typical domestic demand pattern. At present, two approaches are commonly used for predicting the system performance. One involves the use of large scale computer simulations [27], which require a large computer and are complicated in use. The second approach is the parameterization of a large number of simulation runs in a form that serves as a tool for easy calculations; the f-chart [28] and ~b-f chart [29] methods are typical examples of this approach. This, however, produces results which are dependent upon the collector specifications, type of demand and climatic conditions. In case any of these specifications change, the whole exercise has to be repeated. In this paper, a simple and realistic technoeconomic model has been developed for a hybrid solar forced-convection hot-water system. The system consists of a solar water-heating system and an arrangement for providing auxiliary energy if and when required. Two options of auxiliary energy use have been considered, viz. (A) an instant electric heater fitted in the tap from which the hot water is withdrawn, and (B) use of diesel as the auxiliary
energy fuel for heating a second water tank which is always maintained at a constant desired temperature. In order to study the relative performance of the two cases (defined by A and B), numerical calculations have also been made for the typical climate of Delhi, India. For this purpose, 12 months of the year are represented by 12 days, averaged over the months. The solar contribution to the total useful energy is first calculated for the 12 days and then summed, after multiplying each by the number of days in the respective months for calculating the annual solar contribution. Two forms of hot-water demand have been considered: (i) hot-water demand of big residential buildings, represented by a suitable demand pattern, and (ii) industrial hot-water demand, assumed to be constant throughout the days and months. The cost of useful energy for various values of the collector area and tank capacity has been calculated taking into account the life, capital cost and maintenance cost of the solar and auxiliary systems. From these results, the optimum collector area (for a given tank capacity), which corresponds to the minimum cost of useful energy, is obtained. The calculations are repeated for other tank capacities, and thus, an optimum value of tank capacity and a corresponding optimum value of collector area are obtained. The calculations show that, on account of the low cost of diesel, it provides the more economical design of the solar hybrid water heating system. The effect of government subsidy on the optimum values of the collector area and tank capacity has also been investigated. SYSTEMS A solar hybrid water heating system, required to meet the hot-water demands of an industry or the residents of a large apartment building, consists of a solar water-heating system and an auxiliary source of energy. The solar water-heating system consists of a bank of solar collectors operating under a forced convection mode of water flow, a storage tank, controls and pipings. In connection with the use of auxiliary energy, two options have been considered which are as follows: (A) using an instant electric heater in the tap from which the hot water is withdrawn; (B) using a second tank which is always kept at a desired temperature by burning diesel. Schematics of the above two options have been shown in Figs l(a) and (b), respectively. Their operation is as follows: Case A (Fig. la)
(i) The pump P operates only if the difference of temperature between the collector outlet and the top of the storage tank is greater than 5°C; (ii) the instant electric heater is switched on only if the water temperature in the storage tank (Tw) is less than the desired temperature (Ta); (iii) If Tw > Td, the electric heater is switched off.
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM SoLar
369
D
(o)
7
~
7"
7 Thermostat
Instant electric heater
Hot water
Pump Water mains
Fig. 1 (a). Schematics of a typical hybrid (solar + electricity) hot-water system.
(b) SoLar
Thermos t
Tank I TI ~ - - 1 - ~
~
T
2
~
. Hot water
H - Heat exchanger
os
Pure
Water mains F u e L ~ air
~
BoiLer C
Fig. I (b). Schematics of a typical hybrid (solar + diesel) hot-water system.
Case B (Fig. lb) (i) The pump P1 operates when the difference of temperature between the collector outlet and the top of the storage tank T~ is greater than 5°C; (ii) the burner, boiler and pump P2 operate only when the temperature of the water in tank T2 is less than the desired temperature; (iii) if the temperature of the water in tank T~ is greater than the desired temperature, water from tank T 2 bypasses tank T 2 and goes directly to the hot-water line; (iv) if the temperature of the water in tank T] is less than the desired temperature, water from tank T~ goes to tank T2 and then to the hot-water line.
Use of valves V~ and V2 enables the change of direction of water flow whenever desired. THERMAL ANALYSIS In developing the thermal models, the following assumptions have been made: (i) solar collectors are joined in parallel; (ii) water in the storage tanks (tank T in Case A and tanks T l and T 2 in Case B) is at uniform temperature; (iii) solar collectors face south and are inclined from the horizontal at an angle equal to the latitude of the place; (iv) heat losses from a storage tank are proportional to its volume; and (v) pipe losses are negligible.
370
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM
The thermal analysis has been discussed in the following two parts which correspond to the aforementioned cases of auxiliary energy use.
The amount of heat required per unit time to raise the water temperature (Tw) to a desired level (Td) by the electric heater may be evaluated by the expression Q. = thwCw(Td - Tw)+,
(6)
Case A
Following Goyal et al. [26], the energy-balance conditions for collector-outlet temperature, tank TI, and heat-exchanger in tank T2 may, respectively, be written as T~o -- T, - ( a z ) S / U L = exp
~--~ - ,
Tci_ Ta __ (o~)S/UL
/'HcCwc ]
where the superscript plus shows that only the positive values of the expression have to be considered. The solar contribution to the total useful power may be calculated as Q,s=m~Cw(T~-~),
(l)
= thwC,(Td - TO,
for
Tw
for
Tw > Td.
(7)
rh¢(Lo - Li)Cw¢ = M~C wdT~ ~ - + U t A t ( T w - Ta) Case B + thw(t)Cw(Tw - T~) (2)
and
rhcfwc ~ x 1 AX = -hf,(Tf-- Tw)~kX.
(3)
Solving equations (1) and (3) for Too and Tci, respectively, and substituting the obtained values in equation (2), the following equation is obtained:
dT~
(4)
- - + g(t)Tw = f ( t ) , dt
This part of the analysis closely follows the mathematical model of Sodha et al. [25]. Energy-balance conditions for collector-outlet temperature, tank T1 and the heat exchanger in tank T 2 will be the same as given for Case A. The energy balance condition for the water temperature in tank T2 and the associated heat exchanger may be written as F(t)rhw(t)Cw(T~, - Tw) + U~A~(T~, - Ta)
= mbCb(Tb--
where
(8)
and
1 [rh¢ Cwc(1 - F~)(l - F 2 ) g(t)=M~-~wt_ (-1 F IFz)
l =
rnbCb~xfAx =
mwCwT~],
x { T ~ + (°tz)S'~
:(')
Tbi)
-hf2(T f - Tw)AX,
where
[moC~c(1 - FI)(I - F~) _
F(t) = 0,
for
Tw > Tw;
=1,
for
Tw
Solving equation (9) for Tb~, the fluid temperature at the boiler inlet, using the condition that
"4- UtA t q'- tlvtwCw/ and
T f = T b at
( \
(9)
hr, L , ~ ,~oc~J
and substituting the obtained value of Tbi in equation (8), one obtains
F 1= e x p | - - - - | ,
F2= e x p ( F ' U L A ~ . ) rh~Cwc }"
rhb(t)Cb[1
Since the solar insolation S and the ambient air temperature Ta are periodic functions of time with a period of 24 h, the steady state solution for T~ (t) will also be periodic in nature. Thus, T~(t), g(t) and f ( t ) can be each represented by a Fourier series in time. Substitution of these values of T~ and g(t) in equation (4) yields the following matrix equation for T~:
A13 × 13Tw13× 1 --'--Y13 × 1.
x=0
(5)
It may be mentioned here that, as a consequence of earlier studies, only the first six terms of the Fourier series have been retained in obtaining equation (5). This equation can easily be solved for Tw by operating with the inverse of matrix A on both sides.
=
f
hf~L2"~-I
-exp l - m-~-~jJ [F(t)mw(t)C.(T" - T.) + U~A~(T;. - 7".)]. (10)
Equation (10) is a transcendental equation for rhb(t ) and has been solved by an iterative method. The various other parameters occurring in the equation have been assumed to be constant over a period of lh. The amount of heat required from the auxiliary energy source may be calculated by the following expression: Q. = mbCb(Tb -- Tbi).
(1 l)
The solar contribution to the total useful power may, however, be calculated using the expression given by equation (7).
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM ECONOMIC ANALYSIS
Corresponding to a given initial investment, the annual cost of the total system (solar + auxiliary) may be expressed as follows [26]: S c = I J I + / j 2 + M ~ I ~ + M ~ I c + P ~01 yr Qadt,
lo 0.20 c o
.~ o 12
"o
~ o.oa
whereft and f2 are known as the annuity factors and may be defined as
~_ o.o,
y_o
F Z ( 1 - Z) L ]
U' ]
[ z(1 +
f2 =
(o)
E o 16
(12)
f' =
371
I
I
2
4
iI 6
8
I
10 12 14
I 16 18 20 22 24
Time ( h ) (b)
L(T~-z-~-~_-~j,
(13)
~ ,.o
where Z is in fractions. From the prices given by Indian manufacturers, it is seen that the cost of the solar hot-water system, including pump, controls, piping etc., is given by
~" o.s ~ o6 ~" 0.4
Is=(A~Cc+MwCT)M+C I.
(14)
Similarly, the cost of the auxiliary energy system may be expressed as
lc = flM.CTM
=
taF
+ C'p
We have also neglected the cost of pumping energy. Substituting for I~ from equation (14) in (12), one gets Sc
.¢_o O. 2
M
A
M
d
A
S
0
N
Months
Fig. 2 (a). A typical profile for daily hot-water demand; (b) a typical pattern for monthly hot-water demand.
(A'cC¢ + M~CT)(f + M~)M + C=(f + M~) + I~(f + Ma) + MBf,
(15)
G(f, + M,) + C~(A + Ma)
where
MQ,t B - Ca ~/ lyr
f----Mjo
C' =C+-M"
Qadt.
The annual amount of useful energy (in kWh) is given by Q~t = -3.6 - - - x- ~10
earM C.(Ta - Ti) dt = MQut.
The overall cost of the useful energy (Rs/kWb) can be calculated by the following expression:
sc c~ = MQ=' or
co=
(A;Co + M ; G ) ( A +
MJ
+ sf
Qut
+ Cl(fl + Ms) + It(f2 + Ma) MQut C' M
= C +--.
(16)
In the case of the auxiliary energy system using diesel as the fuel, the expression for C~ gets modified to
c~=
[(A~Cc + M~ CT) (fi + Ms) + Bf + [JM~Cx(f2 + Ma)]
Qu,
NUMERICAL RESULTS AND DISCUSSION
For a quantitative appreciation of the relative performance of the two options of auxiliary energy use (defined by Cases A and B), numerical calculations have beenlmade for the climatic data of Delhi, India. For this purpose, the whole year is divided into 12 days which represent the monthly average days. Corresponding to these days, the solar radiation data for a horizontal surface is converted to that for an inclined collector surface by using the monthly averaged hourly values of beam and diffuse radiation [30] in the Liu and Jordan [31] formula. Monthly averaged hourly values of the ambient air temperature and the solar intensity over a collector surface have been given by Goyal et al. and Mani [26, 30]. For the calculations, the following two demand patterns have been chosen: (i) Hot-water demand of big residential apartment buildings: the hot-water demand varies with the hour of the day. In this case, two cases of monthly demand, viz. (a) constant monthly demand and, (b) variable monthly demand, have been considered. Typical profiles for daily and monthly demands are shown in Figs 2(a) and (b) respectively.
372
GOYAL et al.: 0.64
A HYBRID SOLAR WATER HEATING SYSTEM Table 1. List of parameters used in numerical calculations
Case A Constant monthly demand Z Without subsidy Tr With subsidy
0.62 0.60
Thermal analysis parameters F ' = 0.84 U L = 6 W/m 2 °C me = 0.01 kg/sm 2 (during sunshine hours) 0 (during off-sunshine hours) =0.9 r = 0.95 Td ~ 6 0 ° C
0.50 0.56 ~
0.54
"~ m
0.52 -
~
0- ~
Cost analysis parameters Cc = 1560 Rs/m 2 CT = 5 Rs/kg C~ ~ Rs 3000 C r = Rs 2000 (for diesel) Ic = Rs 300 (for instant heater) M~ = 5% of the capital cost of solar system
M,=0.0 P = r/E = ~/D= Z = L = Lo =
0.75 Rs/kWh 1.0 0.62 10% 10yr 10yr
0.4,
~
j
0.42 0,40
I
I
I
I 0.375
I
I
I
I
I
I
I
o.o,
~
0.011
'~ E ~
0.009 0.007 0.005 [ 0.125
I I 0.625
085
I 1.125 1.250
Mw / M F i g . 3. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case A .
(ii) Industrial hot-water demand: in this case, a constant value of daily and monthly demands is considered. " In order to evaluate the integral appearing in equation (12), the auxiliary energy is first calculated for each of the 12 days and then summed, after multiplying it by the number of days in the respective months. Various heat-transfer coefficients have been
calculated using standard heat-transfer relationships [30, 31]. A set of typical values for the various parameters is given in Table 1. It is to be noted here that the inlet water temperature is taken to be equal to the daily average of the ambient air temperature. The calculations show that the annually averaged solar fraction, which represents the total useful energy provided by the solar collectors as a fraction of the total energy required for meeting a certain hot-water demand at a specified temperature, keeps on increasing with an increase in collector area and/or tank capacity. In this connection, however, the following inferences are easy to make: (i) the percentage increase in the solar fraction diminishes non-linearly with increasing collector area and/or tank capacity; and (ii) the cost of the solar system increases linearly with collector area and tank capacity. In view of the above=mentioned facts, the cost of useful energy can easily be chosen as a key parameter for the optimization of collector area and tank capacity for a specific demand pattern. Table 2 presents the cost of useful energy for Case A as a function of collector area and tank capacity, assuming monthly demand to be a constant throughout the year. From this table, it can be noted that the
Table 2. Cost of useful energy for Case A as a function of collector area and tank capacity With constant monthly demand M. 0.25 0.375 0.50 0.625 0.750 0.875 1.0 1.125
1.25
With variable monthly demand
Ac
0.005
0.010
0.015
0.02
0.005
0.010
0.015
0.020
I II I II I II I II I II I II I II I II I II
0.609 0.570 0.603 0.561 0.606 0.561 0.612 0.565 0.620 0.569 0.628 0.575 0.636 0.580 0.644 0.586 0.653 0.592
0.612 0.538 0.572 0.496 0.553 0.475 0.547 0.465 0.547 0.463 0.550 0.463 0.555 0.466 0.562 0.469 0.569 0.470
0.699 0.562 0.613 0.504 0,581 0.468 0.563 0.447 0.554 0.436 0.550 0.430 0.550 0.427 0.553 0.426 0.557 0.428
0.745 0.604 0.682 0.538 0.642 0.496 0.618 0.469 0.604 0.452 0.597 0.443 0.594 0.437 0.595 0.434 0.597 0.434
0.603 0.551 0.597 0.541 0.602 0.542 0.610 0.547 0.620 0.554 0.631 0.561 0.642 0.569 0.654 0.576 0.665 0.564
0.631 0.534 0.592 0,491 0.577 0.473 0.575 0.467 0.578 0.466 0.585 0.469 0.593 0.474 0.602 0.480 0.612 0.486
0.722 0.580 0.670 0.524 0.644 0.495 0.633 0.480 0.630 0.473 0.632 0.472 0.637 0.473 0.644 0.476 0.652 0.481
0.833 0.646 0.776 0.585 0.745 0.550 0.730 0,531 0,724 0.522 0.724 0.518 0.727 0.518 0.733 0.520 0.741 0.524
GOYAL et al.:
0.64
A HYBRID SOLAR WATER HEATING SYSTEM
Case A VariabLe monthLy demand ]~ Without subsidy 11" With subsidy Z
0.62 0.60 0.58
A
0.56 0.54
~" re to
0.52 0.50 0.48
0.46
J
n
J
0.44 0.42 0.40 0.15
#
L..
11
o.o13 0.011 E 0"009
iz-f
---
~
o.007 0
0.005 0.125
0375
J 0.625
0.875
i 1.125
M~, /M
Fig. 4. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case A.
variation of collector area for a fixed tank capacity yields an optimum collector area corresponding to the minimum cost of useful energy. This optimum, however, can be seen to change for other values of tank capacity. This necessitates a need for double optimization. It can be mentioned here that, although Table 2 summarizes the results for one of the several cases studied for optimization purposes, a similar behaviour is observed in the other cases also. The results for these cases have been given in the subsequent figures and tables. Figures 3-6 illustrate the variation of the optimum collector area (obtained for a specific tank capacity) with the cost of useful energy corresponding to the hot water demands of the big residential apartment buildings for Cases A and B, respectively. From these figures, the grand-optimum of the collector area and tank capacity which corresponds to the minimum cost of useful energy is obtained in terms of M, which represents the maximum of the daily hot-water demand in the whole year (Table 3). It can be seen from the figures that, owing to the low cost of diesel, it provides a more economical option of auxiliary energy use as compared to electricity. In this case, the auxiliary system which is designed to meet the water demand independently, however, becomes underutilized and hence indicates the need of its redesigning in accordance with the available useful energy from the solar system. The effect of reduction in the size of the storage tank (T2) on the minimum cost of useful energy and the optimum values of collector
0.555 .r
Case B Constant monthly demand I Without subsidy Tr With subsidy
O. 49
044
O. 39
~
rr 0.010
0.005
E a. 0
0
0.125
373
0.250 0.375 0.500 0.625 0 7 5 0
0.875
1.0
M,,/M Fig. 5. Variation o f optimum collector area and cost o f useful energy with capacity o f storage tank for
Case B.
G O Y A L et al.:
374
A H Y B R I D SOLAR W A T E R H E A T I N G SYSTEM
0.582
0.556
~
/ Case B Variable monthlydemand "r Without subsidy TT With subsidy
.z:
0.486
g.
G 0436 ,
A
0.386 I
~E 0.010 -
0.005
E_
I
I
I
I
I
I
II
F
0
I
I
0125
[ 0250
I I ] I 0.375 0.500 0.625 0.750 0.875
0
I 1.0
M,,/M
Fig. 6. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case B.
Table 3. Optimum value of design parameters in the case of hot-water demand of big residential buildings S. No.
Condition
Parameters
Case A
Case B
1
Constant monthly demand
Ac/M (m2/kg) Mw/M
0.0128 0.8125 0.537
0.0045 0.3125 0.502
2
Variable monthly demand
AJM (m2/kg) M./M
3
(1) + (subsidy)
Ac/M (m2/k8) Mw/M
0.0079 0.4250 0.561 0.0164 1.106 0.419
0.0035 0.1875 0.523 0.0095 0.625 0.439
4
(2) + (subsidy)
0.0135 0.875 0.457
0.007 0.500 0.462
C (Rs/kWh)
C (Rs/kWh) C (Rs/kWh)
AJM (m2/kg) M,,,/M C (Rs/kWh)
Table 4. Optimum value of various design parameters as a function of the capacity of storage tank T2 Tank capacity (kg) S. No.
Conditions
I
Constant monthly demand
2
Variable monthly demand
Parameters
0.25 M
0.5 M
0.75 M
1.0 M
AJM (m2/kg) M./M
0.0052 0.3436 0.459
0.0055 0.375 0.468
0.005 0.3436 0.489
0.0045 0.3125 0.502
0.00375 0.2814 0.467
0.0035 0,250 0.468
0.0035 0.250 0.504
0.0035 0.1875 0.523
0.01 0.556 0,395
0.00925 0.625 0.419
0.00925 0.6875 0.424
0.00925 0.625 0.439
0.0075 0.500 0.405
0.007 0.50 0.424
0.007 0,500 0.444
0.007 0.500 0.462
C (Rs/kWh)
Ac/M (m2/k8) M, IM C (Rs/kWh)
3
(I) + (subsidy)
4
(2) + (subsidy)
Ac/M (m2/kg) M,/M C (Rs/kWh)
AJM (m2/kg) M,,,/M C (Rs/kWh)
GOYAL et al.:
A HYBRID SOLAR WATER HEATING SYSTEM
0.75 i
Case A Industrial water heating [constant demand throughout the day and months)
0.70
¢-
0.65
~
O60
G
0.55
I Without subsidy With subsidy
0.50 0.45
~
0.40
I
I '
I
11
I
I
I
I
oo15 ~
-
0.010
E 0 005
E_
0~"
I
0
I
0125 0.2500375 0.500 0625 0750 0875
L
1.0
1125
12
M,,/M Fig. 7. Variation of optimum collector area and cost Of useful energy with capacity of storage tank for Case A.
0.5(
0.4!
0.40
O.35
0
~
~------i
.
1
5
~
o,o ~
0.005
E ~.
0
__i._ 0.125 0.250 0.375 0.500 0.625 0.7'50 0.8'75 1.0
0 M w /M
Fig. 8. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case B.
375
376
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM Table 5. Optimum value of design parameters for industrial hot-water demand S. No. Conditions Parameters Case A Case B 1 Constant monthly demand Ac/M (m2/kg) 0.012 0.0053 Mw/M 0.625 0.250 C (Rs/kWh)t 0.537 0.4825 2 (1) + (subsidy) AJM (m2/kg) 0.0165 0.0105 Mw/M 1.0 0.5063 C (Rs/kWh)t 0.430 0,3925 eDefined by equation (16).
Table 6. Value of factor C' in the expression (C'/M) Auxiliary energy source Diesel Electricity (a) Hot-water demand of big residential buildings 1. Constant monthly demand 61.59 42.56 2. Variable month demand 80.88 55.98 3. (1) + (subsidy) 48.13 28.94 4. (2) + (subsidy) 63.20 38.06 (b) Industrial hot-water demand I. Without subsidy 64.61 46.07 2. With subsidy 50.49 31.94
cost o f the useful energy for various values o f M a n d c o r r e s p o n d i n g to the Cases A a n d B are given in Table 7. T h e effect o f g o v e r n m e n t subsidy o n the o p t i m u m values o f the collector area, t a n k capacity a n d the m i n i m u m cost o f the useful energy has also been s h o w n in Figs 3-8. The a m o u n t o f subsidy has been t a k e n to be 33% o f the total cost o f the solar system. As expected, this m a k e s the system more economical. REFERENCES
area a n d capacity of storage tank T~ is s h o w n in T a b l e 4. Figures 7 a n d 8 show the v a r i a t i o n of the o p t i m u m collector area o b t a i n e d for a specific value o f t a n k capacity as a function o f the cost o f the useful energy c o r r e s p o n d i n g to the industrial h o t - w a t e r d e m a n d for Cases A a n d B, respectively. T h e g r a n d - o p t i m u m o f collector area a n d t a n k capacity c o r r e s p o n d i n g to the m i n i m u m cost o f useful energy is given in T a b l e 5. It m a y be n o t e d here t h a t the cost o f the useful energy in the preceding discussion does not include the factor C ' / M which c o r r e s p o n d s to the fixed costs o f the system. The values o f this factor for various cases are given in T a b l e 6. It is obvious t h a t this factor has only a little effect on the total cost o f the useful energy for higher values of M. In order to appreciate the effect o f C'/M, the values o f the total
1. D. J. Close, Sol. Energy 6, 33-40 (1962). 2. C. L. Gupta and H. P. Garg, Sol. Energy 12, 163-182 (1968). 3. Y. Zvirin, A. Shitzer and G. Grossman, Int. J. Heat Mass Transfer 20, 997-999 (1977). 4. A. Shitzer, D. Kalmanoviz, Y. Zvirin and G. Grossman, Sol. Energy 22, 27-35 (1979). 5. G. L. Morrison and D. B. J. Ranatunga, Sol. Energy 24, 55-61 (1980). 6. G. L. Morrison and D. B. J. Ranatunga, Sol. Energy 24, 191-198 (1980). 7. A. Mertol, W. Place, Webster and R. Greif, Sol. Energy, 27(5), 367-386 (1981). 8. G. Gutierrez, F. Hincapie, A. Dutfie and W. A. Beckman, Sol. Energy 15, 287-296 (1974). 9. D. Wolf, A. I. Kudish and A. N. Simbira, Energy 6, 333-349 (1981). 10. B. J. Brinkworth, Sol. Energy 17, 331 (1975). 11. F. de Winter, Sol. Energy 17, 335 (1975). 12. J. M. Bradley, 2nd Miami Int. Conf. Alternative Energy Sources, Miami Beach, 10-13 December (1979).
Table 7. Total cost of useful energy for various values of M Cu(Rs/kWh) for case of
Auxiliary M energy (kg) source CMD VMD S. No. (a) Hot-water demand of big residential buildings 1 1000 Diesel 0.563 0.603 2000 0.532 0.563 3000 0.522 0.549 4000 0.517 0.543 2 1000 Electricity 0.579 0.617 2000 0.558 0.589 3000 0.551 0.579 4000 0.541 0.575 (b) Industrial hot-water demand 1 1000 Diesel 0.546 2000 0.514
2
CMD + subsidy
VMD + subsidy
0.487 0.463 0.455 0.451 0.448 0.434 0.429 0.426
0.525 0.493 0.483 0.478 0.495 0.476 0.470 0.467
0.442 0.417
3000
0.503
0.408
4000 1000
0.498 0.583
0.404 0.461
0.560 0.552 0.548
0.446 0.440 0.437
2000 3000 4000
Electricity
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM 13. J. T. Czarnacki, Sol. Energy 2, 2 (1958). 14. R. Farrington, D. Norcen and L. M. Murphy, Proc. System Simulation and Economic Analysis, San Diego, Calif., 23-25 January, pp. 131-136 (1980). 15. A. H. Fanney, S. T. Liu and J. E. Hill, Proc. 3rd A. Solar Heating and Cooling, Research and Development Branch Contractors Meeting, Washington, D.C., 24-27 September, pp. 440-444 (1978). 16. A. H. Fanney and S. T. Liu, Proc. ISES, Silver Jubilee Congr., Atlanta, Ga, May, Vol. 2, pp. 972-976 (1979). 17. M. S. Sodha, S. N. Shukla and G. N. Tiwari, J. Energy 7(2), 107 (1983). 18. A. Whillier and G. Saluja, Sol. Energy 9, 821 (1965). 19. D. J. Close, Sol. Energy 11, 112 (1967). 20. M. S. Sodha, G. N. Tiwari and S. N. Shukla, In Review of Renewable Energy Resources, Vol. 1, pp. 139-229. Wiley Eastern, New Delhi (1983). 21. W. E. Buckles and S. A. Klein, (1980), Sol. Energy 25, 417-424 (1980). 22. J. M. Gordon and A. Rabl, Sol. Energy 28, 519 (1982).
377
23. M. Collares-Pareira, J. M. Gordon, A. Rabl and Y. Zarmi, Sol. Energy 39, 121 (1984). 24. G. Gutierrez, F. Hincapie, A. Duffle and W. A. Beckman, Sol. Energy 15, 287-296 (1976). 25. M. S. Sodha, A. Kumar, N. K. Bansal and I. C. Goyal, Int. J. Energy Res. In press. 26. A. Goyal, A. Kumar and M. S. Sodha, Technoeconomics of Solar Forced Flow Hybrid Hot Water Systems. Reidel, Dordrecht (1986). 27. TRANSYS, Report 38, Solar Energy Laboratory, The University of Wisconsin, Madison, Wis. (1976). 28. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design by the f-Chart Method. Wiley, New York (1977). 29. J. E. Braun, S. A. Klein and K. A. Pearson, Sol. Energy 31, 597 (1983). 30. A. Mani, Handbook of Solar Radiation for India. Allied Publishers, New Delhi (1981). 31. J. A. Duffle and W. A. Beckman, Solar Engineering oJ Thermal Processes. Wiley, New York (1978).
0196-8904/87 $3.00 + 0.00 Copyright © 1987 Pergamon Journals Ltd
OPTIMIZATION OF A HYBRID SOLAR FORCED-CONVECTION WATER HEATING SYSTEM A. K. G O Y A L , A S H V I N I K U M A R ? and M . S. S O D H A Centre of Energy Studies, Indian Institute o f Technology, Hauz Khas, New Delhi 110 016, India (Received 23 June 1986) Abstract--In this paper, a techno-economic model has been developed for a hybrid solar forcedconvection water heating system. Two options of auxiliary energy use, viz. (A) an instant electric heater and (B) use of diesel as the auxiliary energy fuel, have been considered. Numerical calculations have been made for the climate of Delhi, India, corresponding to the two representative demand patterns, viz. (i) hot-water demand of big residential buildings and (ii) industrial hot-water demand. Taking into account the life, capital cost and the maintenance cost o f the solar and auxiliary systems, the cost o f useful energy has been calculated for different values of collector area and tank capacity. This exercise, thereby, yields the optimum values of collector area and tank capacity corresponding to the minimum cost of useful energy. The effect of government subsidy on the optimized values of collector area, tank capacity and cost of useful energy has also been investigated. Optimization
Solar water heating
NOMENCLATURE Ac = Area of solar collector (m 2) A~ = Ac/M (m2/kg) A t = Surface area o f tank T l (m 2) A ~ = Surface area of tank T 2 (m 2) Bf = Annual bill of conventional fuel per unit of hot water demand M (Rs/yr/kg) Cb = Specific heat o f fluid heated by boiler and circulated through heat exchanger in tank T 2 (J/kg °C) C~ = Cost of collector per unit area (Rs/m 2) C = Cost of useful energy, independent of demand (Rs/kWh) C'/M = Cost of useful energy, depending upon demand (Rs/kWh) C, = Cost of installation (Rs) C w = Specific heat o f water in tank T 1 (J/kg °C) C T = Cost o f tank per unit volume (Rs/m 3) Cwc = Specific heat o f collector fluid (J/kg °C) f l , f 2 = A n n u i t y factors, defined by equation (13) F' = Plate efficiency factor of solar collector hf~ = Convective heat transfer coefficient (per unit length) between fluid in heat exchanger and water in tank T z (W/m °C) hf~ = Convective heat transfer coefficient (per unit length) between fluid in heat exchanger and water in tank T 2 (W/m °C) I0 = Initial investment o f conventional system (Rs) I, = Initial investment of solar system (Rs) L = Lifetime o f solar system (yr) L 0 = Lifetime of auxiliary energy system (yr) L~ = Length o f heat exchanger in tank Tt (m) L 2 = Length of heat exchanger in tank T 2 (m) rh,, = Rate of hot water withdrawal (kg/s) rnb = Rate o f hot fluid flow in heat exchanger in tank T 2 (kg/s) rhe = Rate of fluid flow in collector (kg/s) M = Daily hot water demand (kg) M , = Maintenance cost for conventional system (Rs/yr)
INTRODUCTION S o l a r h y b r i d h o t - w a t e r s y s t e m s are t h o s e in w h i c h the s o l a r h o t - w a t e r s y s t e m is a u g m e n t e d w i t h a n auxiliary source o f energy. T h e s e s y s t e m s h a v e t w o m a i n
1"Present address: Solar Energy Centre, Block 134 (Ninth Floor), C.G.O. Complex, Lodi Road, New Delhi 110003, India. EC.M.27/~4:
Mwt = Storage capacity of tank T~ (kg) Mw = g w J M Ms = Maintenance cost for solar system (Rs/yr) P = Price of auxiliary energy (Rs/kWh) Q, = Auxiliary power (W) Qu, = Total annual useful energy (kWh) Q~ = Solar contribution to total useful power (W) S = Solar radiation incident over collectors (W/m 2) Sc = Annual cost of water heating system (Rs/yr) t = Time (s) Ta = Ambient air temperature (°C) Tb = Temperature of boiler which is maintained at constant temperature (°C) Tb~= Fluid temperature when it enters boiler from tank T 2 (°C) Td = Collector fluid temperature when it is fed to collector for recirculation (°C) T~o = Collector fluid temperature when it enters tank T t (°C) T. = Desired water temperature (°C) Tf = Fluid temperature (°C) Ti = Water temperature at mains (°C) Tw = Water temperature in tank T l (°C) T~, = Water temperature in tank I"2 (°C) U L = Heat loss coefficient o f solar collector (W/m 2 °C) Ut = Heat loss coefficient of tank T l (W/m 2 °C) U~ = Heat loss coefficient of tank T 2 ("~V/m 2 °C) x = Position coordinate measured vertically downward (m) Z = Rate of interest (fraction) ~t = Absorptivity o f collector plate fl = Fraction which is defined as ratio o f storage capacity o f tank T2 and daily hot water demand = Transmittivity of cover glass r/E = Utilization efficiency of electricity r/D = Utilization efficiency of diesel
367
368
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM
advantages over the purely solar systems which are as follows: (i) solar hot-water systems cannot meet the hot-water demand on the days of low/no sunshine, and (ii) solar systems designed for minimum insolation conditions will be too large to be cost-effective. Extensive investigations have been made for studying various aspects of solar hot-water systems, which include operation under thermosyphonic mode [1-7], forced-convection mode[8-10], use of heat exchangers in the storage tank[10-12] and use of antifreeze solutions [12]. Experimental [13-17] and theoretical [18-21] studies have also been carried out to predict the thermal performance of these systems. Many of the mathematical models developed, however, have limited application for the evaluation of the techno-economic performance of the solar hybrid water heating system because (i) there is no provision of auxiliary energy, (ii) a very small time step (of the order of seconds) makes the model impractical for evaluation of long-term performance, (iii) a provision for the time-depend.ent hot-water withdrawal is usually not made, and (iv) economic aspects are not included. Amongst others, Gordon and Rabl[22] and Collares-Pareira et al. [23] have presented a method for the design and optimization of single-pass water heating systems used in industrial process heat plants with and without storage; the annual collected energy was estimated by polynomial fitting of the yearly averaged threshold energy. Gutierrez et al. [24] have studied the effects of auxiliary energy supply, load type and storage capacity on the thermal performance of a forced-convection hybrid water heating system. Recently, Sodha et al. [25] and Goyal et al. [26] have presented techno-economic models for a forced-convection hybrid water heating system which uses diesel and electricity as the auxiliary fuel, respectively. These models, however, considered only a typical domestic demand pattern. At present, two approaches are commonly used for predicting the system performance. One involves the use of large scale computer simulations [27], which require a large computer and are complicated in use. The second approach is the parameterization of a large number of simulation runs in a form that serves as a tool for easy calculations; the f-chart [28] and ~b-f chart [29] methods are typical examples of this approach. This, however, produces results which are dependent upon the collector specifications, type of demand and climatic conditions. In case any of these specifications change, the whole exercise has to be repeated. In this paper, a simple and realistic technoeconomic model has been developed for a hybrid solar forced-convection hot-water system. The system consists of a solar water-heating system and an arrangement for providing auxiliary energy if and when required. Two options of auxiliary energy use have been considered, viz. (A) an instant electric heater fitted in the tap from which the hot water is withdrawn, and (B) use of diesel as the auxiliary
energy fuel for heating a second water tank which is always maintained at a constant desired temperature. In order to study the relative performance of the two cases (defined by A and B), numerical calculations have also been made for the typical climate of Delhi, India. For this purpose, 12 months of the year are represented by 12 days, averaged over the months. The solar contribution to the total useful energy is first calculated for the 12 days and then summed, after multiplying each by the number of days in the respective months for calculating the annual solar contribution. Two forms of hot-water demand have been considered: (i) hot-water demand of big residential buildings, represented by a suitable demand pattern, and (ii) industrial hot-water demand, assumed to be constant throughout the days and months. The cost of useful energy for various values of the collector area and tank capacity has been calculated taking into account the life, capital cost and maintenance cost of the solar and auxiliary systems. From these results, the optimum collector area (for a given tank capacity), which corresponds to the minimum cost of useful energy, is obtained. The calculations are repeated for other tank capacities, and thus, an optimum value of tank capacity and a corresponding optimum value of collector area are obtained. The calculations show that, on account of the low cost of diesel, it provides the more economical design of the solar hybrid water heating system. The effect of government subsidy on the optimum values of the collector area and tank capacity has also been investigated. SYSTEMS A solar hybrid water heating system, required to meet the hot-water demands of an industry or the residents of a large apartment building, consists of a solar water-heating system and an auxiliary source of energy. The solar water-heating system consists of a bank of solar collectors operating under a forced convection mode of water flow, a storage tank, controls and pipings. In connection with the use of auxiliary energy, two options have been considered which are as follows: (A) using an instant electric heater in the tap from which the hot water is withdrawn; (B) using a second tank which is always kept at a desired temperature by burning diesel. Schematics of the above two options have been shown in Figs l(a) and (b), respectively. Their operation is as follows: Case A (Fig. la)
(i) The pump P operates only if the difference of temperature between the collector outlet and the top of the storage tank is greater than 5°C; (ii) the instant electric heater is switched on only if the water temperature in the storage tank (Tw) is less than the desired temperature (Ta); (iii) If Tw > Td, the electric heater is switched off.
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM SoLar
369
D
(o)
7
~
7"
7 Thermostat
Instant electric heater
Hot water
Pump Water mains
Fig. 1 (a). Schematics of a typical hybrid (solar + electricity) hot-water system.
(b) SoLar
Thermos t
Tank I TI ~ - - 1 - ~
~
T
2
~
. Hot water
H - Heat exchanger
os
Pure
Water mains F u e L ~ air
~
BoiLer C
Fig. I (b). Schematics of a typical hybrid (solar + diesel) hot-water system.
Case B (Fig. lb) (i) The pump P1 operates when the difference of temperature between the collector outlet and the top of the storage tank T~ is greater than 5°C; (ii) the burner, boiler and pump P2 operate only when the temperature of the water in tank T2 is less than the desired temperature; (iii) if the temperature of the water in tank T~ is greater than the desired temperature, water from tank T 2 bypasses tank T 2 and goes directly to the hot-water line; (iv) if the temperature of the water in tank T] is less than the desired temperature, water from tank T~ goes to tank T2 and then to the hot-water line.
Use of valves V~ and V2 enables the change of direction of water flow whenever desired. THERMAL ANALYSIS In developing the thermal models, the following assumptions have been made: (i) solar collectors are joined in parallel; (ii) water in the storage tanks (tank T in Case A and tanks T l and T 2 in Case B) is at uniform temperature; (iii) solar collectors face south and are inclined from the horizontal at an angle equal to the latitude of the place; (iv) heat losses from a storage tank are proportional to its volume; and (v) pipe losses are negligible.
370
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM
The thermal analysis has been discussed in the following two parts which correspond to the aforementioned cases of auxiliary energy use.
The amount of heat required per unit time to raise the water temperature (Tw) to a desired level (Td) by the electric heater may be evaluated by the expression Q. = thwCw(Td - Tw)+,
(6)
Case A
Following Goyal et al. [26], the energy-balance conditions for collector-outlet temperature, tank TI, and heat-exchanger in tank T2 may, respectively, be written as T~o -- T, - ( a z ) S / U L = exp
~--~ - ,
Tci_ Ta __ (o~)S/UL
/'HcCwc ]
where the superscript plus shows that only the positive values of the expression have to be considered. The solar contribution to the total useful power may be calculated as Q,s=m~Cw(T~-~),
(l)
= thwC,(Td - TO,
for
Tw
for
Tw > Td.
(7)
rh¢(Lo - Li)Cw¢ = M~C wdT~ ~ - + U t A t ( T w - Ta) Case B + thw(t)Cw(Tw - T~) (2)
and
rhcfwc ~ x 1 AX = -hf,(Tf-- Tw)~kX.
(3)
Solving equations (1) and (3) for Too and Tci, respectively, and substituting the obtained values in equation (2), the following equation is obtained:
dT~
(4)
- - + g(t)Tw = f ( t ) , dt
This part of the analysis closely follows the mathematical model of Sodha et al. [25]. Energy-balance conditions for collector-outlet temperature, tank T1 and the heat exchanger in tank T 2 will be the same as given for Case A. The energy balance condition for the water temperature in tank T2 and the associated heat exchanger may be written as F(t)rhw(t)Cw(T~, - Tw) + U~A~(T~, - Ta)
= mbCb(Tb--
where
(8)
and
1 [rh¢ Cwc(1 - F~)(l - F 2 ) g(t)=M~-~wt_ (-1 F IFz)
l =
rnbCb~xfAx =
mwCwT~],
x { T ~ + (°tz)S'~
:(')
Tbi)
-hf2(T f - Tw)AX,
where
[moC~c(1 - FI)(I - F~) _
F(t) = 0,
for
Tw > Tw;
=1,
for
Tw
Solving equation (9) for Tb~, the fluid temperature at the boiler inlet, using the condition that
"4- UtA t q'- tlvtwCw/ and
T f = T b at
( \
(9)
hr, L , ~ ,~oc~J
and substituting the obtained value of Tbi in equation (8), one obtains
F 1= e x p | - - - - | ,
F2= e x p ( F ' U L A ~ . ) rh~Cwc }"
rhb(t)Cb[1
Since the solar insolation S and the ambient air temperature Ta are periodic functions of time with a period of 24 h, the steady state solution for T~ (t) will also be periodic in nature. Thus, T~(t), g(t) and f ( t ) can be each represented by a Fourier series in time. Substitution of these values of T~ and g(t) in equation (4) yields the following matrix equation for T~:
A13 × 13Tw13× 1 --'--Y13 × 1.
x=0
(5)
It may be mentioned here that, as a consequence of earlier studies, only the first six terms of the Fourier series have been retained in obtaining equation (5). This equation can easily be solved for Tw by operating with the inverse of matrix A on both sides.
=
f
hf~L2"~-I
-exp l - m-~-~jJ [F(t)mw(t)C.(T" - T.) + U~A~(T;. - 7".)]. (10)
Equation (10) is a transcendental equation for rhb(t ) and has been solved by an iterative method. The various other parameters occurring in the equation have been assumed to be constant over a period of lh. The amount of heat required from the auxiliary energy source may be calculated by the following expression: Q. = mbCb(Tb -- Tbi).
(1 l)
The solar contribution to the total useful power may, however, be calculated using the expression given by equation (7).
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM ECONOMIC ANALYSIS
Corresponding to a given initial investment, the annual cost of the total system (solar + auxiliary) may be expressed as follows [26]: S c = I J I + / j 2 + M ~ I ~ + M ~ I c + P ~01 yr Qadt,
lo 0.20 c o
.~ o 12
"o
~ o.oa
whereft and f2 are known as the annuity factors and may be defined as
~_ o.o,
y_o
F Z ( 1 - Z) L ]
U' ]
[ z(1 +
f2 =
(o)
E o 16
(12)
f' =
371
I
I
2
4
iI 6
8
I
10 12 14
I 16 18 20 22 24
Time ( h ) (b)
L(T~-z-~-~_-~j,
(13)
~ ,.o
where Z is in fractions. From the prices given by Indian manufacturers, it is seen that the cost of the solar hot-water system, including pump, controls, piping etc., is given by
~" o.s ~ o6 ~" 0.4
Is=(A~Cc+MwCT)M+C I.
(14)
Similarly, the cost of the auxiliary energy system may be expressed as
lc = flM.CTM
=
taF
+ C'p
We have also neglected the cost of pumping energy. Substituting for I~ from equation (14) in (12), one gets Sc
.¢_o O. 2
M
A
M
d
A
S
0
N
Months
Fig. 2 (a). A typical profile for daily hot-water demand; (b) a typical pattern for monthly hot-water demand.
(A'cC¢ + M~CT)(f + M~)M + C=(f + M~) + I~(f + Ma) + MBf,
(15)
G(f, + M,) + C~(A + Ma)
where
MQ,t B - Ca ~/ lyr
f----Mjo
C' =C+-M"
Qadt.
The annual amount of useful energy (in kWh) is given by Q~t = -3.6 - - - x- ~10
earM C.(Ta - Ti) dt = MQut.
The overall cost of the useful energy (Rs/kWb) can be calculated by the following expression:
sc c~ = MQ=' or
co=
(A;Co + M ; G ) ( A +
MJ
+ sf
Qut
+ Cl(fl + Ms) + It(f2 + Ma) MQut C' M
= C +--.
(16)
In the case of the auxiliary energy system using diesel as the fuel, the expression for C~ gets modified to
c~=
[(A~Cc + M~ CT) (fi + Ms) + Bf + [JM~Cx(f2 + Ma)]
Qu,
NUMERICAL RESULTS AND DISCUSSION
For a quantitative appreciation of the relative performance of the two options of auxiliary energy use (defined by Cases A and B), numerical calculations have beenlmade for the climatic data of Delhi, India. For this purpose, the whole year is divided into 12 days which represent the monthly average days. Corresponding to these days, the solar radiation data for a horizontal surface is converted to that for an inclined collector surface by using the monthly averaged hourly values of beam and diffuse radiation [30] in the Liu and Jordan [31] formula. Monthly averaged hourly values of the ambient air temperature and the solar intensity over a collector surface have been given by Goyal et al. and Mani [26, 30]. For the calculations, the following two demand patterns have been chosen: (i) Hot-water demand of big residential apartment buildings: the hot-water demand varies with the hour of the day. In this case, two cases of monthly demand, viz. (a) constant monthly demand and, (b) variable monthly demand, have been considered. Typical profiles for daily and monthly demands are shown in Figs 2(a) and (b) respectively.
372
GOYAL et al.: 0.64
A HYBRID SOLAR WATER HEATING SYSTEM Table 1. List of parameters used in numerical calculations
Case A Constant monthly demand Z Without subsidy Tr With subsidy
0.62 0.60
Thermal analysis parameters F ' = 0.84 U L = 6 W/m 2 °C me = 0.01 kg/sm 2 (during sunshine hours) 0 (during off-sunshine hours) =0.9 r = 0.95 Td ~ 6 0 ° C
0.50 0.56 ~
0.54
"~ m
0.52 -
~
0- ~
Cost analysis parameters Cc = 1560 Rs/m 2 CT = 5 Rs/kg C~ ~ Rs 3000 C r = Rs 2000 (for diesel) Ic = Rs 300 (for instant heater) M~ = 5% of the capital cost of solar system
M,=0.0 P = r/E = ~/D= Z = L = Lo =
0.75 Rs/kWh 1.0 0.62 10% 10yr 10yr
0.4,
~
j
0.42 0,40
I
I
I
I 0.375
I
I
I
I
I
I
I
o.o,
~
0.011
'~ E ~
0.009 0.007 0.005 [ 0.125
I I 0.625
085
I 1.125 1.250
Mw / M F i g . 3. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case A .
(ii) Industrial hot-water demand: in this case, a constant value of daily and monthly demands is considered. " In order to evaluate the integral appearing in equation (12), the auxiliary energy is first calculated for each of the 12 days and then summed, after multiplying it by the number of days in the respective months. Various heat-transfer coefficients have been
calculated using standard heat-transfer relationships [30, 31]. A set of typical values for the various parameters is given in Table 1. It is to be noted here that the inlet water temperature is taken to be equal to the daily average of the ambient air temperature. The calculations show that the annually averaged solar fraction, which represents the total useful energy provided by the solar collectors as a fraction of the total energy required for meeting a certain hot-water demand at a specified temperature, keeps on increasing with an increase in collector area and/or tank capacity. In this connection, however, the following inferences are easy to make: (i) the percentage increase in the solar fraction diminishes non-linearly with increasing collector area and/or tank capacity; and (ii) the cost of the solar system increases linearly with collector area and tank capacity. In view of the above=mentioned facts, the cost of useful energy can easily be chosen as a key parameter for the optimization of collector area and tank capacity for a specific demand pattern. Table 2 presents the cost of useful energy for Case A as a function of collector area and tank capacity, assuming monthly demand to be a constant throughout the year. From this table, it can be noted that the
Table 2. Cost of useful energy for Case A as a function of collector area and tank capacity With constant monthly demand M. 0.25 0.375 0.50 0.625 0.750 0.875 1.0 1.125
1.25
With variable monthly demand
Ac
0.005
0.010
0.015
0.02
0.005
0.010
0.015
0.020
I II I II I II I II I II I II I II I II I II
0.609 0.570 0.603 0.561 0.606 0.561 0.612 0.565 0.620 0.569 0.628 0.575 0.636 0.580 0.644 0.586 0.653 0.592
0.612 0.538 0.572 0.496 0.553 0.475 0.547 0.465 0.547 0.463 0.550 0.463 0.555 0.466 0.562 0.469 0.569 0.470
0.699 0.562 0.613 0.504 0,581 0.468 0.563 0.447 0.554 0.436 0.550 0.430 0.550 0.427 0.553 0.426 0.557 0.428
0.745 0.604 0.682 0.538 0.642 0.496 0.618 0.469 0.604 0.452 0.597 0.443 0.594 0.437 0.595 0.434 0.597 0.434
0.603 0.551 0.597 0.541 0.602 0.542 0.610 0.547 0.620 0.554 0.631 0.561 0.642 0.569 0.654 0.576 0.665 0.564
0.631 0.534 0.592 0,491 0.577 0.473 0.575 0.467 0.578 0.466 0.585 0.469 0.593 0.474 0.602 0.480 0.612 0.486
0.722 0.580 0.670 0.524 0.644 0.495 0.633 0.480 0.630 0.473 0.632 0.472 0.637 0.473 0.644 0.476 0.652 0.481
0.833 0.646 0.776 0.585 0.745 0.550 0.730 0,531 0,724 0.522 0.724 0.518 0.727 0.518 0.733 0.520 0.741 0.524
GOYAL et al.:
0.64
A HYBRID SOLAR WATER HEATING SYSTEM
Case A VariabLe monthLy demand ]~ Without subsidy 11" With subsidy Z
0.62 0.60 0.58
A
0.56 0.54
~" re to
0.52 0.50 0.48
0.46
J
n
J
0.44 0.42 0.40 0.15
#
L..
11
o.o13 0.011 E 0"009
iz-f
---
~
o.007 0
0.005 0.125
0375
J 0.625
0.875
i 1.125
M~, /M
Fig. 4. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case A.
variation of collector area for a fixed tank capacity yields an optimum collector area corresponding to the minimum cost of useful energy. This optimum, however, can be seen to change for other values of tank capacity. This necessitates a need for double optimization. It can be mentioned here that, although Table 2 summarizes the results for one of the several cases studied for optimization purposes, a similar behaviour is observed in the other cases also. The results for these cases have been given in the subsequent figures and tables. Figures 3-6 illustrate the variation of the optimum collector area (obtained for a specific tank capacity) with the cost of useful energy corresponding to the hot water demands of the big residential apartment buildings for Cases A and B, respectively. From these figures, the grand-optimum of the collector area and tank capacity which corresponds to the minimum cost of useful energy is obtained in terms of M, which represents the maximum of the daily hot-water demand in the whole year (Table 3). It can be seen from the figures that, owing to the low cost of diesel, it provides a more economical option of auxiliary energy use as compared to electricity. In this case, the auxiliary system which is designed to meet the water demand independently, however, becomes underutilized and hence indicates the need of its redesigning in accordance with the available useful energy from the solar system. The effect of reduction in the size of the storage tank (T2) on the minimum cost of useful energy and the optimum values of collector
0.555 .r
Case B Constant monthly demand I Without subsidy Tr With subsidy
O. 49
044
O. 39
~
rr 0.010
0.005
E a. 0
0
0.125
373
0.250 0.375 0.500 0.625 0 7 5 0
0.875
1.0
M,,/M Fig. 5. Variation o f optimum collector area and cost o f useful energy with capacity o f storage tank for
Case B.
G O Y A L et al.:
374
A H Y B R I D SOLAR W A T E R H E A T I N G SYSTEM
0.582
0.556
~
/ Case B Variable monthlydemand "r Without subsidy TT With subsidy
.z:
0.486
g.
G 0436 ,
A
0.386 I
~E 0.010 -
0.005
E_
I
I
I
I
I
I
II
F
0
I
I
0125
[ 0250
I I ] I 0.375 0.500 0.625 0.750 0.875
0
I 1.0
M,,/M
Fig. 6. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case B.
Table 3. Optimum value of design parameters in the case of hot-water demand of big residential buildings S. No.
Condition
Parameters
Case A
Case B
1
Constant monthly demand
Ac/M (m2/kg) Mw/M
0.0128 0.8125 0.537
0.0045 0.3125 0.502
2
Variable monthly demand
AJM (m2/kg) M./M
3
(1) + (subsidy)
Ac/M (m2/k8) Mw/M
0.0079 0.4250 0.561 0.0164 1.106 0.419
0.0035 0.1875 0.523 0.0095 0.625 0.439
4
(2) + (subsidy)
0.0135 0.875 0.457
0.007 0.500 0.462
C (Rs/kWh)
C (Rs/kWh) C (Rs/kWh)
AJM (m2/kg) M,,,/M C (Rs/kWh)
Table 4. Optimum value of various design parameters as a function of the capacity of storage tank T2 Tank capacity (kg) S. No.
Conditions
I
Constant monthly demand
2
Variable monthly demand
Parameters
0.25 M
0.5 M
0.75 M
1.0 M
AJM (m2/kg) M./M
0.0052 0.3436 0.459
0.0055 0.375 0.468
0.005 0.3436 0.489
0.0045 0.3125 0.502
0.00375 0.2814 0.467
0.0035 0,250 0.468
0.0035 0.250 0.504
0.0035 0.1875 0.523
0.01 0.556 0,395
0.00925 0.625 0.419
0.00925 0.6875 0.424
0.00925 0.625 0.439
0.0075 0.500 0.405
0.007 0.50 0.424
0.007 0,500 0.444
0.007 0.500 0.462
C (Rs/kWh)
Ac/M (m2/k8) M, IM C (Rs/kWh)
3
(I) + (subsidy)
4
(2) + (subsidy)
Ac/M (m2/kg) M,/M C (Rs/kWh)
AJM (m2/kg) M,,,/M C (Rs/kWh)
GOYAL et al.:
A HYBRID SOLAR WATER HEATING SYSTEM
0.75 i
Case A Industrial water heating [constant demand throughout the day and months)
0.70
¢-
0.65
~
O60
G
0.55
I Without subsidy With subsidy
0.50 0.45
~
0.40
I
I '
I
11
I
I
I
I
oo15 ~
-
0.010
E 0 005
E_
0~"
I
0
I
0125 0.2500375 0.500 0625 0750 0875
L
1.0
1125
12
M,,/M Fig. 7. Variation of optimum collector area and cost Of useful energy with capacity of storage tank for Case A.
0.5(
0.4!
0.40
O.35
0
~
~------i
.
1
5
~
o,o ~
0.005
E ~.
0
__i._ 0.125 0.250 0.375 0.500 0.625 0.7'50 0.8'75 1.0
0 M w /M
Fig. 8. Variation of optimum collector area and cost of useful energy with capacity of storage tank for Case B.
375
376
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM Table 5. Optimum value of design parameters for industrial hot-water demand S. No. Conditions Parameters Case A Case B 1 Constant monthly demand Ac/M (m2/kg) 0.012 0.0053 Mw/M 0.625 0.250 C (Rs/kWh)t 0.537 0.4825 2 (1) + (subsidy) AJM (m2/kg) 0.0165 0.0105 Mw/M 1.0 0.5063 C (Rs/kWh)t 0.430 0,3925 eDefined by equation (16).
Table 6. Value of factor C' in the expression (C'/M) Auxiliary energy source Diesel Electricity (a) Hot-water demand of big residential buildings 1. Constant monthly demand 61.59 42.56 2. Variable month demand 80.88 55.98 3. (1) + (subsidy) 48.13 28.94 4. (2) + (subsidy) 63.20 38.06 (b) Industrial hot-water demand I. Without subsidy 64.61 46.07 2. With subsidy 50.49 31.94
cost o f the useful energy for various values o f M a n d c o r r e s p o n d i n g to the Cases A a n d B are given in Table 7. T h e effect o f g o v e r n m e n t subsidy o n the o p t i m u m values o f the collector area, t a n k capacity a n d the m i n i m u m cost o f the useful energy has also been s h o w n in Figs 3-8. The a m o u n t o f subsidy has been t a k e n to be 33% o f the total cost o f the solar system. As expected, this m a k e s the system more economical. REFERENCES
area a n d capacity of storage tank T~ is s h o w n in T a b l e 4. Figures 7 a n d 8 show the v a r i a t i o n of the o p t i m u m collector area o b t a i n e d for a specific value o f t a n k capacity as a function o f the cost o f the useful energy c o r r e s p o n d i n g to the industrial h o t - w a t e r d e m a n d for Cases A a n d B, respectively. T h e g r a n d - o p t i m u m o f collector area a n d t a n k capacity c o r r e s p o n d i n g to the m i n i m u m cost o f useful energy is given in T a b l e 5. It m a y be n o t e d here t h a t the cost o f the useful energy in the preceding discussion does not include the factor C ' / M which c o r r e s p o n d s to the fixed costs o f the system. The values o f this factor for various cases are given in T a b l e 6. It is obvious t h a t this factor has only a little effect on the total cost o f the useful energy for higher values of M. In order to appreciate the effect o f C'/M, the values o f the total
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Table 7. Total cost of useful energy for various values of M Cu(Rs/kWh) for case of
Auxiliary M energy (kg) source CMD VMD S. No. (a) Hot-water demand of big residential buildings 1 1000 Diesel 0.563 0.603 2000 0.532 0.563 3000 0.522 0.549 4000 0.517 0.543 2 1000 Electricity 0.579 0.617 2000 0.558 0.589 3000 0.551 0.579 4000 0.541 0.575 (b) Industrial hot-water demand 1 1000 Diesel 0.546 2000 0.514
2
CMD + subsidy
VMD + subsidy
0.487 0.463 0.455 0.451 0.448 0.434 0.429 0.426
0.525 0.493 0.483 0.478 0.495 0.476 0.470 0.467
0.442 0.417
3000
0.503
0.408
4000 1000
0.498 0.583
0.404 0.461
0.560 0.552 0.548
0.446 0.440 0.437
2000 3000 4000
Electricity
GOYAL et al.: A HYBRID SOLAR WATER HEATING SYSTEM 13. J. T. Czarnacki, Sol. Energy 2, 2 (1958). 14. R. Farrington, D. Norcen and L. M. Murphy, Proc. System Simulation and Economic Analysis, San Diego, Calif., 23-25 January, pp. 131-136 (1980). 15. A. H. Fanney, S. T. Liu and J. E. Hill, Proc. 3rd A. Solar Heating and Cooling, Research and Development Branch Contractors Meeting, Washington, D.C., 24-27 September, pp. 440-444 (1978). 16. A. H. Fanney and S. T. Liu, Proc. ISES, Silver Jubilee Congr., Atlanta, Ga, May, Vol. 2, pp. 972-976 (1979). 17. M. S. Sodha, S. N. Shukla and G. N. Tiwari, J. Energy 7(2), 107 (1983). 18. A. Whillier and G. Saluja, Sol. Energy 9, 821 (1965). 19. D. J. Close, Sol. Energy 11, 112 (1967). 20. M. S. Sodha, G. N. Tiwari and S. N. Shukla, In Review of Renewable Energy Resources, Vol. 1, pp. 139-229. Wiley Eastern, New Delhi (1983). 21. W. E. Buckles and S. A. Klein, (1980), Sol. Energy 25, 417-424 (1980). 22. J. M. Gordon and A. Rabl, Sol. Energy 28, 519 (1982).
377
23. M. Collares-Pareira, J. M. Gordon, A. Rabl and Y. Zarmi, Sol. Energy 39, 121 (1984). 24. G. Gutierrez, F. Hincapie, A. Duffle and W. A. Beckman, Sol. Energy 15, 287-296 (1976). 25. M. S. Sodha, A. Kumar, N. K. Bansal and I. C. Goyal, Int. J. Energy Res. In press. 26. A. Goyal, A. Kumar and M. S. Sodha, Technoeconomics of Solar Forced Flow Hybrid Hot Water Systems. Reidel, Dordrecht (1986). 27. TRANSYS, Report 38, Solar Energy Laboratory, The University of Wisconsin, Madison, Wis. (1976). 28. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design by the f-Chart Method. Wiley, New York (1977). 29. J. E. Braun, S. A. Klein and K. A. Pearson, Sol. Energy 31, 597 (1983). 30. A. Mani, Handbook of Solar Radiation for India. Allied Publishers, New Delhi (1981). 31. J. A. Duffle and W. A. Beckman, Solar Engineering oJ Thermal Processes. Wiley, New York (1978).