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Thermal, Electrical and Viscous (TEV) Characterization of Biomodified Geomaterials

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𝑑𝑥. 𝑑𝑦. 𝑑𝑧

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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

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∆θ

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o o

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λ

dQ

dt = λAdTdx

dQ dt = λ A

dT dx

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(𝜌1)

(𝜌)

1 𝜌 = 1

𝑅×𝐿 𝐴 = (𝐼

𝑉) × 𝑎

𝜏 = 𝜂𝑑𝑣 𝑑𝑦

𝜏 𝜂

𝑑𝑣 𝑑𝑦

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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

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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

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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

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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

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(𝜆)

λ = Q/A × ∆x/∆t

𝑄 𝐴

∆𝑥

∆𝑡

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∂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

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𝑆 𝑉 𝐼 𝑅

𝜆𝑤𝑎𝑡𝑒𝑟 = 0.607𝑚.𝐾𝑊 𝜆𝑔𝑙𝑦𝑐𝑒𝑟𝑖𝑛𝑒 = 0.292𝑚.𝐾𝑊

Calibration constant(Cn) = λmaterial λmeasured

λmaterial λmeasured

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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)

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0.1 1000 1E7 1E11 30

45 60 75 90 105 120 135

Temperature (O C)

Time (Seconds)

Temperature at Hostrock-buffer interface

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References

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