_
𝑑𝑥. 𝑑𝑦. 𝑑𝑧
λ dQ
dtA
dT dx
1 ρ
Τ η dv dy
qi i (i = x or y or z)
λu
λf W ρd
λr λdry
λs
Q
∆x
∆t Α R S
∆θ
o o
λ
dQ
dt = λAdTdx
dQ dt = λ A
dT dx
(𝜌1)
(𝜌)
1 𝜌 = 1
𝑅×𝐿 𝐴 = (𝐼
𝑉) × 𝑎
𝜏 = 𝜂𝑑𝑣 𝑑𝑦
𝜏 𝜂
𝑑𝑣 𝑑𝑦
𝑑𝑥. 𝑑𝑦. 𝑑𝑧
𝑑𝑥. 𝑑𝑦 𝑑𝑧 𝑞𝑖
𝑖 (𝑖 = 𝑥 𝑜𝑟 𝑦 𝑜𝑟 𝑧)
(𝑞𝑥+𝜕𝑞𝑥
𝜕𝑥 𝑑𝑥) − 𝑞𝑥+ (𝑞𝑦+𝜕𝑞𝑦
𝜕𝑦 𝑑𝑦) − 𝑞𝑦+ (𝑞𝑧+𝜕𝑞𝑧
𝜕𝑧 𝑑𝑧) − 𝑞𝑧 = 0
𝜕𝑞𝑥
𝜕𝑥 +𝜕𝑞𝑦
𝜕𝑦 +𝜕𝑞𝑧
𝜕𝑧 = 0
𝜕(−𝜆 𝜕𝑇 𝜕𝑥⁄ )
𝜕𝑥 +𝜕 (−𝜆 𝜕𝑇 𝜕𝑦⁄ )
𝜕𝑦 +𝜕(−𝜆 𝜕𝑇 𝜕𝑧⁄ )
𝜕𝑧 = 0
𝜆𝜕2𝑇
𝜕𝑥2 +𝜕𝜆
𝜕𝑥
𝜕𝑇
𝜕𝑥+ 𝜆𝜕2𝑇
𝜕𝑦2+𝜕𝜆
𝜕𝑦
𝜕𝑇
𝜕𝑦+ 𝜆𝜕2𝑇
𝜕𝑧2 +𝜕𝜆
𝜕𝑧
𝜕𝑇
𝜕𝑧 = 0
∂qx
∂x +∂qy
∂y +∂qz
∂z = 𝜁𝑑𝑇 𝑑𝑡
𝜕(−𝜆 𝜕𝑇 𝜕𝑥⁄ )
𝜕𝑥 +𝜕 (−𝜆 𝜕𝑇 𝜕𝑦⁄ )
𝜕𝑦 +𝜕(−𝜆 𝜕𝑇 𝜕𝑧⁄ )
𝜕𝑧 = 𝜁𝑑𝑇 𝑑𝑡
𝜆𝜕2𝑇
𝜕𝑥2+𝜕𝜆
𝜕𝑥
𝜕𝑇
𝜕𝑥+ 𝜆𝜕2𝑇
𝜕𝑦2+𝜕𝜆
𝜕𝑦
𝜕𝑇
𝜕𝑦+ 𝜆𝜕2𝑇
𝜕𝑧2 +𝜕𝜆
𝜕𝑧
𝜕𝑇
𝜕𝑧 = 𝜁𝑑𝑇 𝑑𝑡
𝜁
𝜆𝑢
𝜆𝑓 𝑤
𝜌𝑑
𝜆𝑢 = 0.1442[0.7𝑙𝑜𝑔(𝑤) − 0.4] × 100.6243𝜌𝑑
𝜆𝑓 = 0.001096 × 100.8116𝜌𝑑+ 0.00461𝑤 × 100.9115𝜌𝑑
𝜆𝑢 = 0.1442[0.9𝑙𝑜𝑔(𝑤) − 0.2] × 100.6243𝜌𝑑
𝜆𝑓 = 0.001442 × 101.373𝜌𝑑+ 0.01226𝑤 × 100.4994𝜌𝑑
𝜆𝑟𝑢 𝜆𝑟𝑓
𝑆𝑢
𝜆𝑟𝑢 = 0.7𝑙𝑜𝑔(𝑆𝑢) + 1 𝜆𝑟𝑓 = 𝑆𝑢
𝜆𝑟𝑢= 𝑙𝑜𝑔(𝑆𝑢) + 1
𝜆𝑟𝑓 = 𝑆𝑢
(𝜆) (𝜆𝑟)
𝜆𝑑𝑟𝑦 𝜆𝑠
𝜆𝑟 = 𝜆 − 𝜆𝑑𝑟𝑦 𝜆𝑠− 𝜆𝑑𝑟𝑦
𝜆 = 𝜆𝑟(𝜆𝑠− 𝜆𝑑𝑟𝑦) + 𝜆𝑑𝑟𝑦
𝐼𝑓 𝑆 = 0, 𝜆 = 𝜆𝑑𝑟𝑦 𝑎𝑛𝑑 𝜆𝑟 = 0 𝑎𝑛𝑑 𝑖𝑓 𝑆 = 1, 𝜆 = 𝜆𝑠 𝑎𝑛𝑑 𝜆𝑟 = 1 𝜆𝑠𝑢 𝜆𝑠𝑓
𝜆𝑝 𝜆𝑤 𝜆𝑖 𝑛
𝜆𝑠𝑢 = 𝜆1−𝑛𝑝 𝜆𝑛𝑤 𝜆𝑠𝑓 = 𝜆1−𝑛𝑝 𝜆𝑛𝑖
(𝑞)
𝜆𝑝 = {2.01−𝑞 × 7.7𝑞, 𝑤ℎ𝑒𝑛 𝑞 > 0.2 3.01−𝑞 × 7.7𝑞, 𝑤ℎ𝑒𝑛 𝑞 ≤ 0.2
(𝜆)
λ = Q/A × ∆x/∆t
𝑄 𝐴
∆𝑥
∆𝑡
∂T
∂t = α (∂2T
∂r2 +1 r
∂T
∂r)
𝛼 𝑟 𝑇
∆𝜃
∆θ = Q
4πλln (t2 t1)
λ =2.303 × CnQ
4πS =2.303 × CnVI
4πS = 2.303 × CnI2R 4πS
𝑆 𝑉 𝐼 𝑅
𝜆𝑤𝑎𝑡𝑒𝑟 = 0.607𝑚.𝐾𝑊 𝜆𝑔𝑙𝑦𝑐𝑒𝑟𝑖𝑛𝑒 = 0.292𝑚.𝐾𝑊
Calibration constant(Cn) = λmaterial λmeasured
λmaterial λmeasured
100000 1000000 1E7 14.7
14.8 14.9 15.0 15.1
Temperature at the sheath and backfill interface
Minimum Temperature (Deg C)
Time (Seconds)
0.1 1000 1E7 1E11 30
45 60 75 90 105 120 135
Temperature (O C)
Time (Seconds)
Temperature at Hostrock-buffer interface