Questions 7.4. Using a nominal annual interest rate of 15%, find the effective annual interest rate if compounding is (1) annual, (2) quarterly, (3) monthly, and (4) continuous. For uniform annual payments of $10000 for 10 years, compare the compound amount accumulated at the end of 10 years under (1) through (4).
11 -1 (1) annual p=1 i=0,15
1 0,15 , 11 0,15 0,15 %15 1 (2) quarterly p=4 i=0,15
1 0,159 , 11 0,15 0,159 %15,9 %15,9 4 (3) monthly p=12 i=0,15
1 0,161 , 11 0,15 0,161 %16,1 %16,1 12 (4) continious p=∞
1 , 1 0,162 %16,2 A=10000 $
− ( ) + − +, (1) 10000 , $ 203037,182 +,− $ 212173,123 (2) 10000 , +, − $ 214261,649 (3) 10000 , +, − $ 215314,021 (4) 10000 , 7.12. Another boiler must be added to t o the steam-generating system s ystem of a small plant. Two boiler manufacturers have submitted bids from which the following information is obtained. Boiler A
Boiler B
First Cost ($)
50000
120000
Salvage Value ($)
10000
20000
Useful Life (years)
20
40
Annual Operating Cost ($)
9000
6000
If the minimum acceptabie rate of return is 10%, which boiler should be purchased?
When two options’ useful lives are different, the planning horizons (or planning period) are
transformed to a common multiple of the lives. In this question, the useful lives become 40 years for each boilers.
Boiler A 10000
1
2
3
4
5
10000
20
40
…… 9000
……………..
9000 9000 9000 9000
9000 9000
9000
9000 50000
50000
Boiler B 20000
1
2
3
4
5
20
40
…… 6000
6000 6000 6000 6000
…………….. 6000 6000 6000
6000
120000
The present value teorem is used for the calculation. The following equations are also used;
1 1 1 1 1 50000 50000 ,0.10,209000 ,0.10,40 [10000 ,0.10,2010000 ,0.10,40] 1
120000 6000 ,0.10,4020000 ,0.10,40 1 10,10 1 50000 50000 10,10 9000 0,1010,10 1 10000 1 ] $143736,252 [10000 10,10 10,10 1 10,10 120000 6000 0,1010,10 1 $178232,406 20000 10,10
< so Boiler A should be purchased.
7.5. For the cogeneration system case study, calculate the leveliz ed annual costs associated with carrying charges, operation and maintenance, and fuel using the following simplified model: (1) The purchased-equipment costs should be calculated from the functions and constants given in Appendix B. These values are expressed in mid-1994 dollars. (2) The average annual inflation rate is 4.0%; the average nominal escalation rate of all (except fuel) costs is 4.0%, and the average nominal escalation rate of the natural gas (CH4) cost is 4.2%. The economic life is 20 years, the average annual cost of mone y is 12.0%, and the average capacity factor is estimated to be 90%. The unit cost of fuel is 3.0 mid-1994 dollars per GJ lower heating value. (3) The effects of taxes, insurance, and financing of capital expenditures are neglected. (4) Overnight construction is assumed (i.e., the allowance for funds used during construction is zero). (5) Startup costs, working capital, and costs of licensing, research, and development are assumed to be zero. (6) The total capital investment (TCI) is four times higher than the sum of purchased-equipment costs expressed in mid- 1997 dollars.
4.0∑ =
(7) Commercial operation of the plant starts on January 1, 1998. The purchased-equipment costs will be escalated with the inflation rate. (8) The annual operating and maintenance expenses (expressed in mid-1997 dollars) are estimated to be 20% of the total purchasedequipment costs expressed in mid- 1997 dollars: 2
0.20∑ =
(9) The annual levelized carrying charges are obtained by multiplying the total capital investment by the capital-recovery factor. (10) The annual levelized operating and maintenance costs are obtainedmby multiplying the O&M costs in mid-1997 dollars by themconstant escalation levelization factor. Calculate in current dollars for a 20-year period the annualized levelized (a) carrying charges, (b) operating and maintenance costs, and (c) fuel costs.
Parameter Average general inflation rate (%) Average nominal escalation rate of all (except fuel) costs (%) Average nominal escalation of natural gas costs (%) Beginning of the design and construction period Date of commercial operation Plant economic life (years) Average capacity factor (%) Unit cost of fuel ($/GJ-LHV) Lower Heating Value of Fuel LHV (MJ/kg)
4.0 4.0 4.2 Jan 1. 1998 Jan 1. 1998 20 90.0 3.0 50,01
In order to calculate the levelized annual costs associated with carrying charges, operation and maintenance, and fuel; the following equations shoul be used.
(1 ) (1 ) 1 1 1 11 , , , ,
ℎ 365 24 ℎ 0,9036524 ℎ 3600 0,003 $ 50,01 1,6419 $ 6,991 10 1994 3
$ 6,991 10 1 0,042 8,241 10 1998 levelized annual fuel costs in current dollars
0,042 0,93036 11 110,12 0,12 10,12 10,12 1 0,13388 1 , 1 0, 9 30361 0, 9 3036 10,93036 0,13388 8,241 10 1 0,042− 10,805 10 1997 0,04 0,92857 110,12 0,13388 0,9285710,92857 , 10,92857 0,13388
4
