Turunkan persamaan difusi panas, Persamaan 2.26, untuk koordinat silindris dengan memulai dari volume kendali diferensial yang ditunjukkan pada Gambar 2.12.
$$\frac{1}{r} \frac{\partial}{\partial r} \left(kr \frac{\partial T}{\partial r} \right) + \frac{1}{r^2} \frac{\partial}{\partial \phi} \left(k \frac{\partial T}{\partial \phi} \right) + \frac{\partial}{\partial z} \left(k \frac{\partial T}{\partial z} \right) + \dot{q} = \rho c_p \frac{\partial T}{\partial t}$$
Definisi Geometri Volume Kendali Diferensial
Tinjau volume kendali diferensial dalam koordinat silindris dengan dimensi:
- Arah radial ($r$): ketebalan $dr$
- Arah azimutal ($\phi$): panjang busur $r \, d\phi$
- Arah aksial ($z$): tinggi $dz$
Maka volume kendali ($dV$) adalah:
$$dV = (dr)(r \, d \phi)(dz)$$
$$dV = r \ dr \ d \phi \ dz$$
Luas penampang diferensial untuk masing-masing permukaan adalah:
- $A_r = r \ d \phi \ dz$ (luas permukaan tegak lurus sumbu-$r$)
- $A_\phi = dr \ dz$ (luas permukaan tegak lurus sumbu-$\phi$)
- $A_z = r \ dr \ d \phi$ (luas permukaan tegak lurus sumbu-$z$)
Prinsip Konservasi Energi
Berdasarkan Hukum Pertama Termodinamika untuk volume kendali:
$$\dot{E}_{in} – \dot{E}_{out} + \dot{E}_{gen} = \dot{E}_{st}$$
- Laju Energi Masuk ($\dot{E}_{in}$):
$$\dot{E}_{in} = q_r + q_\phi + q_z$$
- Laju Energi Keluar ($\dot{E}_{out}$):
$$\dot{E}_{out} = q_{r+dr} + q_{\phi+d\phi} + q_{z+dz}$$
- Laju Pembangkitan Energi ($\dot{E}_{gen}$):
$$\dot{E}_{gen} = \dot{q} \cdot dV$$
$$\dot{E}_{gen} = \dot{q} (r \ dr \ d\phi \ dz)$$
- Laju Penyimpanan Energi ($\dot{E}_{st}$):
$$\dot{E}_{st} = \rho \cdot dV \cdot c_p \frac{\partial T}{\partial t}$$
$$\dot{E}_{st} = \rho c_p (r \, dr \, d\phi \, dz) \frac{\partial T}{\partial t}$$
Ekspansi Deret Taylor untuk Laju Perpindahan Panas
Menggunakan ekspansi deret Taylor (dengan mengabaikan orde yang lebih tinggi), laju perpindahan panas yang keluar dari permukaan dapat dinyatakan sebagai:
- Arah Radial ($r$):
$$q_{r+dr} = q_r + \frac{\partial q_r}{\partial r} dr$$
$$q_r – q_{r+dr} = -\frac{\partial q_r}{\partial r} dr$$
- Arah Azimutal ($\phi$):
$$q_{\phi+d\phi} = q_\phi + \frac{\partial q_\phi}{\partial \phi} d\phi$$
$$q_\phi – q_{\phi+d\phi} = -\frac{\partial q_\phi}{\partial \phi} d\phi$$
- Arah Aksial ($z$):
$$q_{z+dz} = q_z + \frac{\partial q_z}{\partial z} dz$$
$$q_z – q_{z+dz} = -\frac{\partial q_z}{\partial z} dz$$
Penerapan Hukum Fourier
Berdasarkan Hukum Fourier, laju konduksi panas pada masing-masing arah adalah:
- Pada arah $r$:
$$q_r = -k A_r \frac{\partial T}{\partial r} = -k (r \, d\phi \, dz) \frac{\partial T}{\partial r}$$
- Sehingga:
$$q_r – q_{r+dr} = -\frac{\partial}{\partial r} \left[ -k (r \, d\phi \, dz) \frac{\partial T}{\partial r} \right] dr = \frac{\partial}{\partial r} \left( k r \frac{\partial T}{\partial r} \right) dr \, d\phi \, dz$$
- Pada arah $\phi$:
$$q_\phi = -k A_\phi \frac{\partial T}{r \, \partial \phi} = -k (dr \, dz) \frac{1}{r} \frac{\partial T}{\partial \phi}$$
- Sehingga:
$$q_\phi – q_{\phi+d\phi} = -\frac{\partial}{\partial \phi} \left[ -k (dr \, dz) \frac{1}{r} \frac{\partial T}{\partial \phi} \right] d\phi = \frac{1}{r} \frac{\partial}{\partial \phi} \left( k \frac{\partial T}{\partial \phi} \right) dr \, d\phi \, dz$$
- Pada arah $z$:
$$q_z = -k A_z \frac{\partial T}{\partial z} = -k (r \, dr \, d\phi) \frac{\partial T}{\partial z}$$
- Sehingga:
$$q_z – q_{z+dz} = -\frac{\partial}{\partial z} \left[ -k (r \, dr \, d\phi) \frac{\partial T}{\partial z} \right] dz = \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) r \, dr \, d\phi \, dz$$
Substitusi ke Neraca Energi Total
Substitusikan seluruh komponen ke dalam persamaan neraca energi $(\dot{E}_{in} – \dot{E}_{out}) + \dot{E}_{gen} = \dot{E}_{st}$:
$$\frac{\partial}{\partial r} \left( k r \frac{\partial T}{\partial r} \right) dr \, d\phi \, dz + \frac{1}{r} \frac{\partial}{\partial \phi} \left( k \frac{\partial T}{\partial \phi} \right) dr \, d\phi \, dz + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) r \, dr \, d\phi \, dz + \dot{q} (r \, dr \, d\phi \, dz) = \rho c_p (r \, dr \, d\phi \, dz) \frac{\partial T}{\partial t}$$
Bagi seluruh ruas dengan volume diferensial $dV = r \, dr \, d\phi \, dz$:
$$\frac{1}{r} \frac{\partial}{\partial r} \left( k r \frac{\partial T}{\partial r} \right) + \frac{1}{r^2} \frac{\partial}{\partial \phi} \left( k \frac{\partial T}{\partial \phi} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) + \dot{q} = \rho c_p \frac{\partial T}{\partial t}$$
