3. Derive the operational Green-Ampt equation, t = and F(t)-F(tp)|yf|(þ−0; ) [F(tp)+|yf|(−0₁) + . In F(t)+|yf|(-0; ) K sat from the following two equations, - f(t)= K sat f(t)=-Ksat 1+ |(0-0₁) Kama [1410 / 1 (0-01)]. F(t) dF (t) dt P a 1stst +tp where to represents the time of ponding, t, represents the total time specified, and y is the effective tension at the wetting front. The meanings of the rest of symbols are the same as those defined in class. (Hint: Begin by inverting both sides of the equation, then x. dx x b separate variables and integrate. Also, ſ ax+b · In (a.x+b))
3. Derive the operational Green-Ampt equation, t = and F(t)-F(tp)|yf|(þ−0; ) [F(tp)+|yf|(−0₁) + . In F(t)+|yf|(-0; ) K sat from the following two equations, - f(t)= K sat f(t)=-Ksat 1+ |(0-0₁) Kama [1410 / 1 (0-01)]. F(t) dF (t) dt P a 1stst +tp where to represents the time of ponding, t, represents the total time specified, and y is the effective tension at the wetting front. The meanings of the rest of symbols are the same as those defined in class. (Hint: Begin by inverting both sides of the equation, then x. dx x b separate variables and integrate. Also, ſ ax+b · In (a.x+b))
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![**Derivation of the Operational Green-Ampt Equation**
To derive the operational Green-Ampt equation:
\[
t = \frac{F(t) - F(t_p)}{K_{sat}} + \frac{|\psi_f| (\phi - \theta_i)}{K_{sat}} \cdot \ln \left[ \frac{F(t_p) + |\psi_f| (\phi - \theta_i)}{F(t) + |\psi_f| (\phi - \theta_i)} \right] + t_p
\]
from the following two equations:
1. \[
f(t) = -K_{sat} \left[ 1 + \frac{|\psi_f| (\phi - \theta_i)}{F(t)} \right], \quad t_p \leq t \leq t_w
\]
2. \[
-f(t) = \frac{dF(t)}{dt}
\]
**Where:**
- \( t_p \) represents the time of ponding.
- \( t_w \) represents the total time specified.
- \( \psi_f \) is the effective tension at the wetting front.
- The meanings of the rest of symbols remain consistent with those defined in class.
**Hint:**
Begin by inverting both sides of the equation, then separate variables and integrate.
Also, consider the integral:
\[
\int \frac{x \cdot dx}{a \cdot x + b} = \frac{x}{a} - \frac{b}{a^2} \cdot \ln(a \cdot x + b)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ea503b5-9839-4a05-b482-50f8a035eb54%2F120bed1a-f4e7-47a5-b224-c70dc476d630%2Fpptj6y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Derivation of the Operational Green-Ampt Equation**
To derive the operational Green-Ampt equation:
\[
t = \frac{F(t) - F(t_p)}{K_{sat}} + \frac{|\psi_f| (\phi - \theta_i)}{K_{sat}} \cdot \ln \left[ \frac{F(t_p) + |\psi_f| (\phi - \theta_i)}{F(t) + |\psi_f| (\phi - \theta_i)} \right] + t_p
\]
from the following two equations:
1. \[
f(t) = -K_{sat} \left[ 1 + \frac{|\psi_f| (\phi - \theta_i)}{F(t)} \right], \quad t_p \leq t \leq t_w
\]
2. \[
-f(t) = \frac{dF(t)}{dt}
\]
**Where:**
- \( t_p \) represents the time of ponding.
- \( t_w \) represents the total time specified.
- \( \psi_f \) is the effective tension at the wetting front.
- The meanings of the rest of symbols remain consistent with those defined in class.
**Hint:**
Begin by inverting both sides of the equation, then separate variables and integrate.
Also, consider the integral:
\[
\int \frac{x \cdot dx}{a \cdot x + b} = \frac{x}{a} - \frac{b}{a^2} \cdot \ln(a \cdot x + b)
\]
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