Kunci Jawaban AOSTK Tugas III

1. Alat penukar kalor shell and tube, harga total dinyatakan dengan fungsi sbb :

D dan L dalam ft.

Batasan :

D dan L pada C minimum dengan cara kalkulus dan nilai C minimum !

Jawab :

Agar nilai C minimum, maka :

2. Proses reaksi : T hari

Fraksi massa : x kg produk/kg umpan → x = f(T)

Biaya pemisahan : Cs per batch → Cs = f(x)

Hubungan x dan T :

x (kg produk/kg umpan) 0 0,25 0,50 0,75 1,00

T (hari) 0 0,5 1,5 3 6

Hubungan x dan Cs :

x (kg produk/kg umpan) 0 0,25 0,50 0,75 1,00

Cs (Rp) - 4.000.000 2.000.000 800.000 0

Biaya operasi : Rp 1.000.000 / hari

Harga produk : Rp 40.000 / kg produk

Keuntungan kotor :

Waktu batch (T) sehingga keuntungan kotor (R) maksimum dengan cara Golden Section !

Jawab :

x = f(T) → x = -0,0291 T2 + 0,3329 T + 0,044

x = -0,0291T2 + 0,3329T + 0,044 R² = 0,991

0

0,2

0,4

0,6

0,8

1

1,2

0 1 2 3 4 5 6 7

x (k

g p

rod

uk/

kg u

mp

an)

T (hari)

Cs = f(x) → x = f(T) → Cs = f(T)

Cs = f(T) → Cs = -1.634.852,8628 ln (T) + 2.763.753,6708

Persamaan menjadi :

Cs = -2.893.825,5261ln(x) - 12.510,6845 R² = 0,9999

-500000

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

0 0,2 0,4 0,6 0,8 1 1,2

Cs

(Rp

)

x (kg produk/kg umpan)

Cs = -1.634.852,8628ln(T) + 2.763.753,6708 R² = 0,9916

-500000

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

0 1 2 3 4 5 6 7

Cs

(Rp

)

T (hari)

Dengan cara kalkulus, agar nilai R maksimum :

Dengan cara Golden Section :

Tujuan : Maksimasi

(TQ – TP)I = (TB – TQ)II

(r – (1 – r)) (TB – TA)I = (1 – r) (TB –

TA)II

(r – (1 – r)) (TB – TA)I = (1 – r) r (TB –

TA)I

2r – 1 = r – r2

r2 + r – 1 = 0

r = 0,6180

Jika RP > RQ, maka :

II I

TA TA

TP

TQ TP

TB TQ

TP = TB – r (TB – TA)

Jika RP < RQ, maka :

II I

TA TP

TP TQ

TQ

TB TB

TQ = TA + r (TB – TA)

Ta Tp Tq Tb Ra Rp Rq Rb

1 1,0000 2,5280 3,4720 5,0000 Rp 3.192.246 Rp 4.041.315 Rp 3.680.844 Rp 2.897.488

2 1,0000 1,9443 2,5280 3,4720 Rp 3.192.246 Rp 4.116.637 Rp 4.041.315 Rp 3.680.844

3 1,9443 2,5280 2,8884 3,4720 Rp 4.116.637 Rp 4.041.315 Rp 3.925.129 Rp 3.680.844

4 1,9443 2,3050 2,5280 2,8884 Rp 4.116.637 Rp 4.091.543 Rp 4.041.315 Rp 3.925.129

5 1,9443 2,1673 2,3050 2,5280 Rp 4.116.637 Rp 4.110.922 Rp 4.091.543 Rp 4.041.315

6 1,9443 2,0821 2,1673 2,3050 Rp 4.116.637 Rp 4.117.325 Rp 4.110.922 Rp 4.091.543

7 1,9443 2,0295 2,0821 2,1673 Rp 4.116.637 Rp 4.118.801 Rp 4.117.325 Rp 4.110.922

8 1,9443 1,9969 2,0295 2,0821 Rp 4.116.637 Rp 4.118.672 Rp 4.118.801 Rp 4.117.325

9 1,9969 2,0295 2,0495 2,0821 Rp 4.118.672 Rp 4.118.801 Rp 4.118.476 Rp 4.117.325

10 1,9969 2,0170 2,0295 2,0495 Rp 4.118.672 Rp 4.118.849 Rp 4.118.801 Rp 4.118.476

11 1,9969 2,0094 2,0170 2,0295 Rp 4.118.672 Rp 4.118.819 Rp 4.118.849 Rp 4.118.801

12 2,0094 2,0170 2,0218 2,0295 Rp 4.118.819 Rp 4.118.849 Rp 4.118.845 Rp 4.118.801

13 2,0094 2,0141 2,0170 2,0218 Rp 4.118.819 Rp 4.118.843 Rp 4.118.849 Rp 4.118.845

14 2,0141 2,0170 2,0189 2,0218 Rp 4.118.843 Rp 4.118.849 Rp 4.118.850 Rp 4.118.845

15 2,0170 2,0189 2,0200 2,0218 Rp 4.118.849 Rp 4.118.850 Rp 4.118.849 Rp 4.118.845

T = (TA + TB)/2 = 2,0194

R = Rp 4.118.849