VIPSolutionsGive step-by-step solution with explanation and final answer:Recorded flow cates for a net (excess) rainfall of Zinch in 2 hours are shown in the table If the(Base flows 187.5 sec throughout the Storm shen determine the 2-he unit hydrogFapi. Also, estimate the 4-hr unit hydrograph. Time Flow-Q ( (hr) (sec) | s ! ORR Ts? si |” | | | [a | cor eg | [6 mm [ame = 10 325 | 12 | | ems [rr i | |
Question:
Give step-by-step solution with explanation and final answer:
Recorded flow cates for a net (excess) rainfall of Zinch in 2 hours are shown in the table
If the(Base flows 187.5 sec throughout the Storm shen determine the 2-he unit
hydrogFapi. Also, estimate the 4-hr unit hydrograph.
Time Flow-Q (
(hr) (sec) |
s !
ORR Ts? si |” |
| |
[a | cor eg |
[6 mm
[ame =
10 325 |
12 |
|
ems
[rr i
| |
This is AI generated Response.
Asked by: Test User 100248
Created at: 2025-05-17 23:25:50
Asked by: Test User 100248
Created at: 2025-05-17 23:25:50
Final Answer:
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Let's **solve the problem step by step**:
### **Step 1: Understand the Problem**
- **Given:** Flow rates (Q) due to a net rainfall of 2 inches in 2 hours.
- **Base flow:** 187.5 cfs (cubic feet per second).
- **Objective:**
1. Find the **2-hr unit hydrograph** (UH) (i.e., for 1-inch rainfall excess in 2 hours).
2. Estimate the **4-hr unit hydrograph**.
---
### **Step 2: Remove Base Flow**
Subtract the base flow (187.5 cfs) from the observed flow to get the **direct runoff hydrograph (DRH)**.
| Time (hr) | Flow-Q (cfs) | DRH = Q - Base flow (cfs) |
|-----------|--------------|---------------------------|
| 0 | 187.5 | 0 |
| 2 | 1100 | 912.5 |
| 4 | 1950 | 1762.5 |
| 6 | 1187.5 | 1000 |
| 8 | 625 | 437.5 |
| 10 | 325 | 137.5 |
| 12 | 200 | 12.5 |
| 14 | 187.5 | 0 |
| 16 | 187.5 | 0 |
---
### **Step 3: Find the 2-hr Unit Hydrograph**
The **2-inch rainfall excess** produced the DRH above. The **2-hr unit hydrograph** (UH) is the DRH **for 1 inch of rainfall excess in 2 hours**.
So, **divide each DRH ordinate by 2**:
| Time (hr) | DRH (cfs) | 2-hr Unit Hydrograph (cfs) |
|-----------|-----------|----------------------------|
| 0 | 0 | 0 |
| 2 | 912.5 | 456.25 |
| 4 | 1762.5 | 881.25 |
| 6 | 1000 | 500 |
| 8 | 437.5 | 218.75 |
| 10 | 137.5 | 68.75 |
| 12 | 12.5 | 6.25 |
| 14 | 0 | 0 |
| 16 | 0 | 0 |
---
### **Step 4: Estimate the 4-hr Unit Hydrograph**
The **4-hr unit hydrograph** can be estimated by summing two 2-hr UHs shifted by 2 hours each, and dividing by 2 (since it's for the same total rainfall excess, but over 4 hours).
Let’s write the 2-hr unit hydrograph ordinates as \( U \):
| Time (hr) | \( U \) (cfs) | \( U \) shifted by 2 hr |
|-----------|---------------|------------------------|
| 0 | 0 | 0 |
| 2 | 456.25 | 0 |
| 4 | 881.25 | 456.25 |
| 6 | 500 | 881.25 |
| 8 | 218.75 | 500 |
| 10 | 68.75 | 218.75 |
| 12 | 6.25 | 68.75 |
| 14 | 0 | 6.25 |
| 16 | 0 | 0 |
Sum the ordinates at each time and **divide by 2**:
| Time (hr) | \( U \) | \( U \) (shifted) | Sum | 4-hr UH (cfs) |
|-----------|---------|-------------------|-----|---------------|
| 0 | 0 | 0 | 0 | 0 |
| 2 | 456.25 | 0 |456.25| 228.13 |
| 4 | 881.25 | 456.25 |1337.5| 668.75 |
| 6 | 500 | 881.25 |1381.25| 690.63 |
| 8 | 218.75 | 500 |718.75| 359.38 |
| 10 | 68.75 | 218.75 |287.5 | 143.75 |
| 12 | 6.25 | 68.75 |75 | 37.5 |
| 14 | 0 | 6.25 |6.25 | 3.13 |
| 16 | 0 | 0 |0 | 0 |
---
### **Final Answers**
#### **2-hr Unit Hydrograph:**
| Time (hr) | 2-hr UH (cfs) |
|-----------|---------------|
| 0 | 0 |
| 2 | 456.25 |
| 4 | 881.25 |
| 6 | 500 |
| 8 | 218.75 |
| 10 | 68.75 |
| 12 | 6.25 |
| 14 | 0 |
| 16 | 0 |
#### **4-hr Unit Hydrograph (estimated):**
| Time (hr) | 4-hr UH (cfs) |
|-----------|---------------|
| 0 | 0 |
| 2 | 228.13 |
| 4 | 668.75 |
| 6 | 690.63 |
| 8 | 359.38 |
| 10 | 143.75 |
| 12 | 37.5 |
| 14 | 3.13 |
| 16 | 0 |
---
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Question:
i need detailed calculations along with the final Answer
This is AI generated Response.
Asked by: Test User 100248
Created at: 2025-05-17 23:26:51
Asked by: Test User 100248
Created at: 2025-05-17 23:26:51
Final Answer:
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Did you like this answer?
Let's work through the problem step-by-step with detailed calculations:
---
### **Given Data:**
| Time (hr) | Flow Q (cfs) |
|------------|--------------|
| 0 | 187.5 |
| 2 | 1100 |
| 4 | 1950 |
| 6 | 1187.5 |
| 8 | 625 |
| 10 | 325 |
| 12 | 200 |
| 14 | 187.5 |
| 16 | 187.5 |
- **Base flow:** 187.5 cfs (assumed constant throughout the storm)
- **Net rainfall excess (Z inch):** 2 inches in 2 hours
---
### **Step 1: Remove Base Flow to find the Direct Runoff Hydrograph (DRH)**
Subtract base flow from total flow at each time:
\[
\text{DRH} = Q - 187.5
\]
| Time (hr) | Q (cfs) | DRH (cfs) = Q - 187.5 |
|------------|---------|------------------------|
| 0 | 187.5 | 0 |
| 2 | 1100 | 912.5 |
| 4 | 1950 | 1762.5 |
| 6 | 1187.5 | 1000 |
| 8 | 625 | 437.5 |
| 10 | 325 | 137.5 |
| 12 | 200 | 12.5 |
| 14 | 187.5 | 0 |
| 16 | 187.5 | 0 |
---
### **Step 2: Determine the 2-hour unit hydrograph (UH)**
Since the net rainfall excess was **2 inches in 2 hours**, the **unit hydrograph (UH)** per inch of excess is obtained by dividing the DRH ordinates by 2:
\[
\text{2-hr UH at each time} = \frac{\text{DRH}}{2}
\]
| Time (hr) | DRH (cfs) | 2-hr UH (cfs) = DRH / 2 |
|------------|-----------|---------------------------|
| 0 | 0 | 0 |
| 2 | 912.5 | 456.25 |
| 4 | 1762.5 | 881.25 |
| 6 | 1000 | 500 |
| 8 | 437.5 | 218.75 |
| 10 | 137.5 | 68.75 |
| 12 | 12.5 | 6.25 |
| 14 | 0 | 0 |
| 16 | 0 | 0 |
This is the **2-hour unit hydrograph per inch of rainfall excess**.
---
### **Step 3: Estimating the 4-hour unit hydrograph**
The **4-hour UH** can be estimated by **convolution of the 2-hour UH with itself** or, more simply here, by summing two shifted 2-hour UH ordinates (each representing 1 inch of rainfall over 2 hours) and dividing by 2 for the total 4-hour period.
**Method:**
\[
\text{4-hr UH} (t) = \frac{1}{2} \left[ U_2(t) + U_2(t - 2) \right]
\]
where \( U_2(t) \) is the 2-hour UH. For each time, sum the 2-hour UH and the same UH shifted by 2 hours, then divide by 2.
Let's construct the 4-hour UH:
| Time (hr) | \( U_2(t) \) | \( U_2(t - 2) \) | Sum | 4-hr UH = Sum / 2 |
|------------|--------------|------------------|-------|---------------------|
| 0 | 0 | 0 | 0 | 0 |
| 2 | 456.25 | 0 | 456.25| 228.125 |
| 4 | 881.25 | 456.25 | 1337.5| 668.75 |
| 6 | 500 | 881.25 | 1381.25| 690.625 |
| 8 | 218.75 | 500 | 718.75| 359.375 |
| 10 | 68.75 | 218.75 | 287.5 | 143.75 |
| 12 | 6.25 | 68.75 | 75 | 37.5 |
| 14 | 0 | 6.25 | 6.25 | 3.125 |
| 16 | 0 | 0 | 0 | 0 |
**Thus, the estimated 4-hour unit hydrograph:**
| Time (hr) | 4-hr UH (cfs) |
|------------|--------------|
| 0 | 0 |
| 2 | 228.125 |
| 4 | 668.75 |
| 6 | 690.625 |
| 8 | 359.375 |
| 10 | 143.75 |
| 12 | 37.5 |
| 14 | 3.125 |
| 16 | 0 |
---
### **Final Summary:**
**(A) 2-hour unit hydrograph (per inch of rainfall excess):**
| Time (hr) | 2-hr UH (cfs) |
|------------|--------------|
| 0 | 0 |
| 2 | 456.25 |
| 4 | 881.25 |
| 6 | 500 |
| 8 | 218.75 |
| 10 | 68.75 |
| 12 | 6.25 |
| 14 | 0 |
| 16 | 0 |
**(B) 4-hour unit hydrograph (estimated):**
| Time (hr) | 4-hr UH (cfs) |
|------------|--------------|
| 0 | 0 |
| 2 | 228.125 |
| 4 | 668.75 |
| 6 | 690.625 |
| 8 | 359.375 |
| 10 | 143.75 |
| 12 | 37.5 |
| 14 | 3.125 |
| 16 | 0 |
---
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