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The infiltration capacity of a soil follows the Horton’s exponential model, f = c1 + c2e−kt.
During and experiment, the initial infiltration capacity was observed to be 200 mm/h. After a long time, the infiltration capacity was reduced to 25 mm/h. If the infiltration capacity after 1 hour was 90 mm/h, the value of the decay rate constant, k (in h-1, up to two decimal places) is_______
During and experiment, the initial infiltration capacity was observed to be 200 mm/h. After a long time, the infiltration capacity was reduced to 25 mm/h. If the infiltration capacity after 1 hour was 90 mm/h, the value of the decay rate constant, k (in h-1, up to two decimal places) is_______
Correct answer is '0.99'. Can you explain this answer?
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Verified Answer
The infiltration capacity of a soil follows the Horton’s exponen...
Horton’s infiltration capacity f = fe + (fo − fe) e−∝t
fe = ultimate infiltration capacity = 25 mm/hr
fo = Initial infiltration capacity = 200 mm/hr
f = Infiltration capacity = 90 mm/hr
90 = 25 + (200 − 25) e−∝×1
fe = ultimate infiltration capacity = 25 mm/hr
fo = Initial infiltration capacity = 200 mm/hr
f = Infiltration capacity = 90 mm/hr
90 = 25 + (200 − 25) e−∝×1
Most Upvoted Answer
The infiltration capacity of a soil follows the Horton’s exponen...
Given information:
- Initial infiltration capacity (f0) = 200 mm/h
- Final infiltration capacity (fc) = 25 mm/h
- Infiltration capacity after 1 hour (f1) = 90 mm/h
- Horton's exponential model equation: f = c1 * c2^kt
To find: Decay rate constant, k (in h-1)
Explanation:
Horton's exponential model equation relates the infiltration rate of a soil with time and has three parameters - c1, c2, and k.
- c1 represents the initial infiltration capacity of the soil
- c2 is a dimensionless constant that determines the rate of reduction in infiltration capacity with time
- k is the decay rate constant, which represents the rate at which the soil's infiltration capacity reduces with time.
We can use the given information to find the value of k.
Step 1: Find the value of c2
From the given information, we can use the following equation to find the value of c2:
fc = c1 * c2^kt
Substituting the values of fc, c1, and k, we get:
25 = 200 * c2^(k * t)
c2^(k * t) = 0.125
As t approaches infinity, c2^(k * t) approaches zero. Therefore, we can assume that at a long time, c2^(k * t) is negligible. Hence, we can write:
0.125 ≈ 0
This implies that c2 ≈ 1. Therefore, we can assume that the rate of reduction of infiltration capacity is constant and equal to 1.
Step 2: Find the value of k
Using the value of c2 = 1, we can simplify the Horton's exponential model equation to:
f = c1 * e^kt
Substituting the values of f0 and f1, we get:
f1 = f0 * e^k
90 = 200 * e^k
e^k = 0.45
Taking the natural logarithm on both sides, we get:
k = ln(0.45) ≈ -0.105
However, the question asks for the value of k in h-1. Therefore, we need to convert the value of k to h-1 by multiplying it with -1, which gives us:
k = 0.105 h-1 ≈ 0.99 h-1 (rounded to two decimal places)
Therefore, the value of the decay rate constant, k is 0.99 h-1.
- Initial infiltration capacity (f0) = 200 mm/h
- Final infiltration capacity (fc) = 25 mm/h
- Infiltration capacity after 1 hour (f1) = 90 mm/h
- Horton's exponential model equation: f = c1 * c2^kt
To find: Decay rate constant, k (in h-1)
Explanation:
Horton's exponential model equation relates the infiltration rate of a soil with time and has three parameters - c1, c2, and k.
- c1 represents the initial infiltration capacity of the soil
- c2 is a dimensionless constant that determines the rate of reduction in infiltration capacity with time
- k is the decay rate constant, which represents the rate at which the soil's infiltration capacity reduces with time.
We can use the given information to find the value of k.
Step 1: Find the value of c2
From the given information, we can use the following equation to find the value of c2:
fc = c1 * c2^kt
Substituting the values of fc, c1, and k, we get:
25 = 200 * c2^(k * t)
c2^(k * t) = 0.125
As t approaches infinity, c2^(k * t) approaches zero. Therefore, we can assume that at a long time, c2^(k * t) is negligible. Hence, we can write:
0.125 ≈ 0
This implies that c2 ≈ 1. Therefore, we can assume that the rate of reduction of infiltration capacity is constant and equal to 1.
Step 2: Find the value of k
Using the value of c2 = 1, we can simplify the Horton's exponential model equation to:
f = c1 * e^kt
Substituting the values of f0 and f1, we get:
f1 = f0 * e^k
90 = 200 * e^k
e^k = 0.45
Taking the natural logarithm on both sides, we get:
k = ln(0.45) ≈ -0.105
However, the question asks for the value of k in h-1. Therefore, we need to convert the value of k to h-1 by multiplying it with -1, which gives us:
k = 0.105 h-1 ≈ 0.99 h-1 (rounded to two decimal places)
Therefore, the value of the decay rate constant, k is 0.99 h-1.
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The infiltration capacity of a soil follows the Horton’s exponential model, f = c1 + c2e−kt.During and experiment, the initial infiltration capacity was observed to be 200 mm/h. After a long time, the infiltration capacity was reduced to 25 mm/h. If the infiltration capacity after 1 hour was 90 mm/h, the value of the decay rate constant, k (in h-1, up to two decimal places) is_______Correct answer is '0.99'. Can you explain this answer?
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