Jika kanal pada Soal-1 terisi penuh, berapa debit maksimum yang bisa dibawa oleh kanal (sebelum air meluap ke tepian sungai)?
Identifikasi Data
- Lebar dasar ($b$): $6\text{ m}$
- Lebar permukaan atas ($T$): $12\text{ m}$
- Kedalaman aliran ($y$): $2,2\text{ m}$
- Koefisien kekasaran Manning ($n$): $0,016$ (untuk lapisan aspal)
- Debit aliran ($Q$): $120\text{ m}^3\text{/s}$
- Panjang saluran ($L$): $1\text{ km} = 1000\text{ m}$
Parameter Geometri Penampang Pertama
Kita hitung luas penampang basah
$$A_1 = \frac{b + T}{2} \times y$$
$$A_1 = \frac{6 + 12}{2} \times 2,2$$
$$A_1 = 18 \times 1,1$$
$$\mathbf{A_1 = \mathbf{19,8 \text{ m}^2}}$$
Kita hitung proyeksi horizontal sisi miring
$$z_1 = \frac{12 – 6}{2} = \frac{6}{2} = 3 \text{ m}$$
Kita hitung sisi miring
$$s_1 = \sqrt{3^2 + 2,2^2}$$
$$s_1 = \sqrt{9 + 4,84}$$
$$s_1 = \sqrt{13,84}$$
$$\mathbf{s_1 = 3,72 \text{ m}}$$
Kita hitung keliling basah
$$P_1 = b + 2s$$
$$P_1 = 6 + 2 \cdot 3,72$$
$$P_1 = 6 + 7,44$$
$$\mathbf{P_1 = 13,44 \text{ m}}$$
Kita hitung radius hidrolik
$$R_{h1} = \frac{A}{P}$$
$$R_{h1} = \frac{19,8}{13,44}$$
$$\mathbf{R_{h1} = 1,4732 \text{ m}}$$
Parameter Geometri Penampang Kedua
Kita hitung luas penampang basah
$$A_2 = \frac{b + T}{2} \times y$$
$$A_2 = \frac{6 + 14,73}{2} \times 3,2$$
$$A_2 = 10,365 \times 3,2$$
$$\mathbf{A_2 = \mathbf{33,168 \text{ m}^2}}$$
Kita hitung proyeksi horizontal sisi miring
$$z_2 = \frac{14,73 – 6}{2} = \frac{8,73}{2} = 4,365 \text{ m}$$
Kita hitung sisi miring
$$s_2 = \sqrt{4,365^2 + 3,2^2}$$
$$s_2 = \sqrt{19,053 + 10,24}$$
$$s_2 = \sqrt{29,293}$$
$$\mathbf{s_2 = 5,412 \text{ m}}$$
Kita hitung keliling basah
$$P_2 = b + 2s$$
$$P_2 = 6 + 2 \cdot 5,412$$
$$P_2 = 6 + 10,824$$
$$\mathbf{P_2 = 16,824 \text{ m}}$$
Kita hitung radius hidrolik
$$R_{h2} = \frac{A}{P}$$
$$R_{h2} = \frac{33,168}{16,824}$$
$$\mathbf{R_{h2} = 1,9714 \text{ m}}$$
Perbandingan Debit
Persamaan debit Manning adalah:
- $Q = \frac{1}{n} \times A \times R_h^{2/3} \times S_0^{1/2}$
- $Q_1 = \frac{1}{n} \times A_1 \times R_{h1}^{2/3} \times S_0^{1/2}$
- $Q_2 = \frac{1}{n} \times A_2 \times R_{h2}^{2/3} \times S_0^{1/2}$
Kita bandingkan debit kedua dengan debit pertama
$$\frac{Q_2}{Q_1} = \frac{\frac{1}{n} \times A_2 \times R_{h2}^{2/3} \times S_0^{1/2}}{\frac{1}{n} \times A_1 \times R_{h1}^{2/3} \times S_0^{1/2}}$$
$$\frac{Q_2}{Q_1} = \frac{ A_2 \times R_{h2}^{2/3}}{A_1 \times R_{h1}^{2/3}}$$
$${Q_2} = \frac{A_2 \times R_{h2}^{2/3}}{A_1 \times R_{h1}^{2/3}} \cdot Q_1$$
Kita cari nilai debit kedua
$${Q_2} = \frac{33,168 \times 1,9714^{2/3}}{19,8 \times 1,4732^{2/3}} \cdot 120$$
$${Q_2} = \frac{33,168 \times 1,5722}{19,8 \times 1,2947} \cdot 120$$
$${Q_2} = \frac{52,146}{25,635} \cdot 120$$
$${Q_2} = 2,034 \cdot 120$$
$$\mathbf{{Q_2} = 244,08 \text{ m}^3/s}$$
