OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
The objective function given in Eq. (32) can be plotted with respect to the coefficient of performance of the heat
pump given in Eq. (27) for various k values, as shown in Fig-2. It is apparent from the Fig 2 that there exists a
certain value of COP hp at which the objective function bF hp is maximum for a given k value. Thus Eq. (32) can
be maximized with respect to y and x.
Therefore, ∂bF hp / ∂y = 0 and ∂bF hp / ∂x = 0 gives:
y * = √(kk L T L1 ) and (34)
x * = T H1 / [√{(T L1 – y) / (kk H )(1-((kk L )/y))} - 1] (35)
The optimal values of x * and y * can be submitted in Eq. (32) to obtain the value of bF max
bF hp = 1 / [k{(k H / x * ) + (k L (T L1 – y * ) / y * (T H1 + x * ))}+ b {1 – ((T L1 – y * ) / (T L1 + x * ))}]
(36)
and the optimum coefficient of performance is
COP * hp = Q H / W = (T H1 + x * ) / [(T H1 + x * ) - (T L1 – y * )] (37)
2.2 Optimization for Endoreversible Refrigerator system
The model and T-S diagram as given in Fig 1 for an endoreversible heat pump hold schematically for
endoreversible refrigerator also. However, in this case (T H2 – T H1 ) is the ambient temperature and (T L1 – T L2 ) is
the temperature of the cooled space. Therefore, Eqs. (1) – (26) can be used for endoreversible refrigerator. The
coefficient of performance (COP) for endoreversible refrigerator is:
COP ref = Q L / W = (T L1 - y) / [(T H1 + x) - (T L1 - y)] (38)
The objective function of thermoeconomic optimization as proposed by earlier researchers [9, 12-13] is given by
F ref = Q L / (C i + C e + C m ) (39)
C i = a(A H + A L ) + b 1 W
C e = b 2 W = b 2 (Q H - Q L )
And C m = b p (Q H - Q L )
Here b p is equal to equivalent annual operation hours per unit power input.
Substituting the value of C i , C e and C m into Eq. (39), gives
F ref = Q L / {a (A H + A L ) + b (Q H - Q L )} (40)
Where b = b 1 + b 2 + b p
Using Eqs. (20), (24), and (26) gives
F ref = Q L / Q L [a{(k H / x)((T H1 + x) / (T L1 - y)) + (k L / y) }+ b {((T H1 + x) / (T L1 - y)) - 1}]
Or b F ref = 1 / [k{(k H / x)((T H1 + x) / (T L1 - y)) + (k L / y) }+ {((T H1 + x) / (T L1 - y)) - 1}]
(41)
Where k = a/b; Economical parameter
The objective function given in Eq. (41) can be plotted with respect to the coefficient of performance of the heat
pump given in Eq. (38) for various k values, as shown in Fig-3. It is apparent from the Fig 3 that there exists a
certain value of COP hp at which the objective function bF hp is maximum for a given k value. Thus Eq. (41) can
be maximized with respect to y and x.
Therefore, ∂bF hp / ∂y = 0 and ∂bF hp / ∂x = 0 gives:
x * = √(kk H T H1 ) and` (42)
y * = T L1 / [√{(T H1 + x) / (kk L )(1+ ((kk H )/x))} + 1] (43)
The optimal values of x * and y * can be submitted in Eq. (32) to obtain the value of bF max
b F ref = 1 / [k{(k H / x * )((T H1 + x * ) / (T L1 – y * )) + (k L / y * ) }+ {((T H1 + x * ) / (T L1 – y * )) - 1}]
(44)
and the optimum coefficient of performance is
COP ref = Q L / W = (T L1 – y * ) / [(T H1 + x * ) - (T L1 – y * )] (45)
3.PERFORMANCE ANALYSIS
The variation of objective function for refrigerators and heat pumps with respect to the coefficient of
performance (COP) for various values of economic parameter (k), corresponding to both fixed and variable
temperature heat reservoir are shown in Fig 2 and Fig 3, respectively. In each figure, the dotted line represents
curves for a system with fixed temperature heat reservoir, while the curves with solid line show the variation, for
the same system but with variable temperature heat reservoir. In both the cases, for each system, a similar pattern
of variation is observed; however, the value of COP that maximizes the objective function for the given value of
31