Water Age Calculation in a Distribution Network

Context: Water AgeThe average time water takes to travel from its source (like a treatment plant) to a specific point in the network..

Water age is a critical metric in hydraulic engineering and water quality management. As water travels through a distribution network, the disinfectant residual (typically chlorine) decays. If water becomes too "old," the residual can fall below effective levels, increasing the risk of bacterial regrowth. The EPAEnvironmental Protection Agency. The US government agency that sets standards for drinking water quality. recommends maintaining a detectable residual throughout the system. This exercise will guide you through calculating water age in a simple branched network using US customary units.

Pedagogical Note: This exercise demonstrates the two fundamental concepts of water age analysis: advection (travel time in pipes) and complete mixing at junctions.

Learning Objectives

Understand the concept of water age and its importance for water quality.
Calculate water travel time (advection) in a pipe using flow rate and dimensions.
Apply the complete mixing model to calculate water age at a pipe junction.
Master unit conversions between inches, feet, gallons, and minutes (US Customary Units).
Combine advection and mixing to determine the final water age at a delivery point.

Study Data

We will analyze a simple water network supplied by two sources: a main reservoir (Source A) and a local well (Source B). These sources merge at Junction J1, which then supplies a customer at Node C1. All calculations must be performed using US Customary Units.

System Specifications

Characteristic	Value
Source A (Reservoir)	Water Age at source = 0 hours
Source B (Well)	Water Age at source = 0 hours
Unit Conversion 1	1 cubic foot (ft³) = 7.481 Gallons
Unit Conversion 2	1 foot (ft) = 12 inches (in)

Network Schematic

\(L_1\) \(D_1\) \(Q_1\) \(L_2\) \(D_2\) \(Q_2\) \(L_3\) \(D_3\)

Parameter	Symbol	Value
Pipe 1 Length	2000	ft
Pipe 1 Diameter	8	in
Pipe 1 Flow Rate	350	GPM
Pipe 2 Length	500	ft
Pipe 2 Diameter	6	in
Pipe 2 Flow Rate	150	GPM
Pipe 3 Length	1000	ft
Pipe 3 Diameter	10	in

Questions to Address

Calculate the travel time (in hours) for water in Pipe 1 (Reservoir A to Junction J1).
Calculate the travel time (in hours) for water in Pipe 2 (Well B to Junction J1).
Calculate the age of the water at Junction J1, assuming complete mixing.
Calculate the travel time (in hours) for water in Pipe 3 (Junction J1 to Customer C1).
What is the final age of the water delivered to Customer C1?

Fundamentals of Water Age Analysis

To solve this exercise, we need two key concepts. All calculations must use consistent units.

1. Advection (Travel Time) in Pipes
This assumes "plug flow," where water travels as a single block without mixing. The time it takes is simply the pipe's volume divided by the flow rate. \[ \text{Time} = \frac{\text{Volume}}{\text{Flow Rate}} \] To use this, all units must match. We will convert pipe dimensions (feet, inches) to a volume in Gallons and the flow rate to Gallons per Minute (GPM). \[ \text{Volume (gal)} = \text{Area (ft}^2\text{)} \times \text{Length (ft)} \times \left(\frac{7.481 \text{ gal}}{1 \text{ ft}^3}\right) \] \[ \text{Area (ft}^2\text{)} = \frac{\pi \times (\text{Diameter (in)} / 12)^2}{4} \]

2. Complete Mixing at Junctions
When two or more pipes meet, we assume the water mixes completely and instantaneously. The age of the water leaving the junction is the flow-weighted average of the ages of all incoming flows. \[ \text{Age}_{\text{out}} = \frac{\sum (Q_{\text{in}} \times \text{Age}_{\text{in}})}{\sum (Q_{\text{in}})} \] For our case with two inputs: \[ \text{Age}_{J1} = \frac{(Q_1 \times \text{Age}_1) + (Q_2 \times \text{Age}_2)}{Q_1 + Q_2} \]

Solution: Water Age Calculation in a Distribution Network

Question 1: Calculate the travel time (in hours) for water in Pipe 1 (Reservoir A to Junction J1).

Principle

We need to find the advection time in Pipe 1. This requires calculating the pipe's volume in gallons and dividing it by the flow rate in gallons per minute (GPM). The final result must be converted from minutes to hours.

Mini-Lesson

The key is to meticulously track units. We will convert the 8-inch diameter to feet to find the area, then use the 2000-foot length to find the volume in cubic feet. We then convert cubic feet to gallons.

Pedagogical Note

This first step is the foundation for all water age calculations. We are calculating the "travel time," which is the age the water *gains* while traveling from the source (Age 0) to the junction.

Norms

We will use the provided conversion factors: \(1 \text{ ft} = 12 \text{ in}\) and \(1 \text{ ft}^3 = 7.481 \text{ gal}\). These are standard US customary unit conversions.

Formula(s)

The formulas below will be applied in sequence.

Pipe Area

First, the formula for the area of a circle, converting diameter (D) from inches to feet.

\text{Area (ft}^2\text{)} = \frac{\pi \times (D \text{ (in)} / 12)^2}{4}

This gives us the cross-sectional area in square feet.

Pipe Volume

Next, the formula for the volume of the pipe in gallons. This is the area (ft²) times the length (ft) multiplied by the conversion factor for gallons per cubic foot.

\text{Volume (gal)} = \text{Area (ft}^2\text{)} \times L \text{ (ft)} \times 7.481

This gives us the total volume of water in the pipe.

Time

Finally, the formula for travel time. We divide the total volume by the flow rate (GPM) to get minutes, then divide by 60 to get hours.

\text{Time (hr)} = \frac{\text{Volume (gal)}}{\text{Flow Rate (GPM)}} \times \left(\frac{1 \text{ hr}}{60 \text{ min}}\right)

This is our final formula for advection time in hours.

Hypotheses

[List the assumptions made for this calculation.]

We assume "plug flow," meaning the water moves as a single block with no mixing inside the pipe.
We assume the pipe is flowing full.
We assume the age at Source A is 0, as given.

Data

Data for Pipe 1:

Parameter	Symbol	Value	Unit
Pipe 1 LengthLength of the pipe from Source A to Junction J1.	\(L_1\)	2000	ft
Pipe 1 Diameter	\(D_1\)	8	in
Pipe 1 Flow	\(Q_1\)	350	GPM

Tips

Unit Conversion Tip: The most common error is forgetting to convert diameter from inches to feet. Always divide the inch value by 12 *before* squaring it.

Schematic (Before Calculations)

This calculation focuses on the path from Source A to J1.

Schematic for Q1

Calculation(s)

We follow a step-by-step process, plugging in the values for Pipe 1.

Step 1: Calculate Pipe 1 Area (ft²)

We start with the formula for the area of a circle. We take the diameter of Pipe 1, \(D_1 = 8 \text{ in}\), and divide by 12 to convert it to feet.

\begin{aligned} \text{Area}_1 &= \frac{\pi \times (D_1 / 12)^2}{4} \\ \text{Area}_1 &= \frac{\pi \times (8 \text{ in} / 12)^2}{4} \\ \text{Area}_1 &= \frac{\pi \times (0.667 \text{ ft})^2}{4} \\ \text{Area}_1 &= 0.349 \text{ ft}^2 \end{aligned}

This area (0.349 ft²) is the basis for our volume calculation.

Step 2: Calculate Pipe 1 Volume (Gallons)

Now, we find the total volume. We use the area from Step 1 (\(0.349 \text{ ft}^2\)) and multiply by the length of Pipe 1, \(L_1 = 2000 \text{ ft}\), to get cubic feet. Then, we convert to gallons using the given factor of 7.481.

\begin{aligned} \text{Volume}_1 (\text{ft}^3) &= \text{Area}_1 \times L_1 \\ \text{Volume}_1 (\text{ft}^3) &= 0.349 \text{ ft}^2 \times 2000 \text{ ft} = 698.1 \text{ ft}^3 \\ \text{Volume}_1 (\text{gal}) &= \text{Volume}_1 (\text{ft}^3) \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_1 (\text{gal}) &= 698.1 \text{ ft}^3 \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_1 (\text{gal}) &= 5222.6 \text{ gal} \end{aligned}

The total volume to be moved is 5222.6 gallons.

Step 3: Calculate Travel Time (Hours)

Finally, we calculate the travel time. We take the volume from Step 2 (\(5222.6 \text{ gal}\)) and divide by the flow rate of Pipe 1, \(Q_1 = 350 \text{ GPM}\). This gives us the time in minutes. We then divide by 60 to get hours.

\begin{aligned} \text{Time}_1 (\text{min}) &= \frac{\text{Volume}_1 (\text{gal})}{Q_1 (\text{GPM})} \\ \text{Time}_1 (\text{min}) &= \frac{5222.6 \text{ gal}}{350 \text{ GPM}} = 14.92 \text{ min} \\ \text{Time}_1 (\text{hr}) &= \frac{14.92 \text{ min}}{60 \text{ min/hr}} \\ \text{Time}_1 (\text{hr}) &= 0.2487 \text{ hr} \end{aligned}

This gives our final travel time for Pipe 1, 0.2487 hours.

Schematic (After Calculations)

Not applicable for this step. The result is a scalar value (time), not a new diagram.

Analysis

The travel time in Pipe 1 is 14.92 minutes, which we will use as 0.25 hours (or 0.2487 for precision) in subsequent calculations. This is the "age" of the water arriving at J1 from Source A.

Cautionary Points

Always use the *internal* diameter of the pipe. Here "Diameter" is assumed to be internal. In real-world problems, you might be given a "nominal" diameter, and you'd need to look up the true internal diameter based on the pipe material (e.g., PVC, Ductile Iron).

Key Takeaways

Water age is fundamentally a \(\text{Volume} / \text{Flow Rate}\) calculation.
Unit consistency (feet, inches, gallons) is the most critical part of the calculation.

Did You Know?

The water doesn't truly move in a perfect "plug." Water near the pipe walls moves slower due to friction than water in the center. This is called "dispersion," but for network-scale models, plug flow is a very effective and standard assumption.

FAQ

[List common follow-up questions students might have.]

Final Result

The travel time in Pipe 1 is 0.25 hours (or 14.92 minutes).

Your Turn

If the flow in Pipe 1 *decreased* to 250 GPM due to higher demand elsewhere, what would the new travel time be in hours?

Memo Card

Question 1 Summary:

Key Concept: Advection (Travel Time)
Essential Formula: \(\text{Time} = \text{Volume} / \text{Flow Rate}\)
Key Data: L=2000ft, D=8in, Q=350 GPM

Question 2: Calculate the travel time (in hours) for water in Pipe 2 (Well B to Junction J1).

Principle

This calculation is identical to Question 1, but uses the dimensions and flow rate for Pipe 2.

Mini-Lesson

The same principles of unit conversion (inches to feet, ft³ to gallons) apply. The velocity in this pipe will likely be different from Pipe 1 due to the different diameter and flow rate. Each pipe in the network must be calculated individually.

Pedagogical Note

Pay attention to how the different parameters interact. Pipe 2 is much shorter than Pipe 1 (which *decreases* time), but also has a smaller diameter (which *decreases* volume) and a lower flow rate (which *increases* time). You must calculate to see the net effect.

Norms

We will use the provided conversion factors: \(1 \text{ ft} = 12 \text{ in}\) and \(1 \text{ ft}^3 = 7.481 \text{ gal}\).

Formula(s)

The formulas are the same as in Question 1, but applied to Pipe 2's data.

Pipe Area

First, the formula for the area of a circle, converting diameter (D) from inches to feet.

\text{Area (ft}^2\text{)} = \frac{\pi \times (D \text{ (in)} / 12)^2}{4}

This gives us the cross-sectional area in square feet.

Pipe Volume

Next, the formula for the volume of the pipe in gallons. This is the area (ft²) times the length (ft) multiplied by the conversion factor for gallons per cubic foot.

\text{Volume (gal)} = \text{Area (ft}^2\text{)} \times L \text{ (ft)} \times 7.481

This gives us the total volume of water in the pipe.

Time

Finally, the formula for travel time. We divide the total volume by the flow rate (GPM) to get minutes, then divide by 60 to get hours.

\text{Time (hr)} = \frac{\text{Volume (gal)}}{\text{Flow Rate (GPM)}} \times \left(\frac{1 \text{ hr}}{60 \text{ min}}\right)

This is our final formula for advection time in hours.

Hypotheses

We assume "plug flow" in Pipe 2. We also assume the age of the water at the source (Source B) is 0 hours, as stated in the problem.

Data

Data for Pipe 2:

Parameter	Symbol	Value	Unit
Pipe 2 Length	\(L_2\)	500	ft
Pipe 2 Diameter	\(D_2\)	6	in
Pipe 2 Flow	\(Q_2\)	150	GPM

Tips

A 6-inch diameter is 0.5 feet. A 12-inch diameter is 1 foot. Using fractions (like 6/12) or decimals (0.5) for the diameter in feet is fine, but be consistent. Using 0.5 is often easier for calculators.

Schematic (Before Calculations)

This step focuses only on the advection in Pipe 2.

Schematic for Q2

Calculation(s)

We follow the same steps as Q1, but using the values for Pipe 2.

Step 1: Calculate Pipe 2 Area (ft²)

We take the diameter of Pipe 2, \(D_2 = 6 \text{ in}\), and divide by 12 to get 0.5 ft.

\begin{aligned} \text{Area}_2 &= \frac{\pi \times (D_2 / 12)^2}{4} \\ \text{Area}_2 &= \frac{\pi \times (6 \text{ in} / 12)^2}{4} \\ \text{Area}_2 &= \frac{\pi \times (0.5 \text{ ft})^2}{4} \\ \text{Area}_2 &= 0.1963 \text{ ft}^2 \end{aligned}

This area (0.1963 ft²) is the basis for our volume calculation.

Step 2: Calculate Pipe 2 Volume (Gallons)

We use the area from Step 1 (\(0.1963 \text{ ft}^2\)) and multiply by the length of Pipe 2, \(L_2 = 500 \text{ ft}\). Then, we convert to gallons.

\begin{aligned} \text{Volume}_2 (\text{ft}^3) &= \text{Area}_2 \times L_2 \\ \text{Volume}_2 (\text{ft}^3) &= 0.1963 \text{ ft}^2 \times 500 \text{ ft} = 98.17 \text{ ft}^3 \\ \text{Volume}_2 (\text{gal}) &= \text{Volume}_2 (\text{ft}^3) \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_2 (\text{gal}) &= 98.17 \text{ ft}^3 \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_2 (\text{gal}) &= 734.5 \text{ gal} \end{aligned}

The total volume to be moved is 734.5 gallons.

Step 3: Calculate Travel Time (Hours)

We take the volume from Step 2 (\(734.5 \text{ gal}\)) and divide by the flow rate of Pipe 2, \(Q_2 = 150 \text{ GPM}\). Finally, we convert minutes to hours.

\begin{aligned} \text{Time}_2 (\text{min}) &= \frac{\text{Volume}_2 (\text{gal})}{Q_2 (\text{GPM})} \\ \text{Time}_2 (\text{min}) &= \frac{734.5 \text{ gal}}{150 \text{ GPM}} = 4.897 \text{ min} \\ \text{Time}_2 (\text{hr}) &= \frac{4.897 \text{ min}}{60 \text{ min/hr}} \\ \text{Time}_2 (\text{hr}) &= 0.0816 \text{ hr} \end{aligned}

This gives our final travel time for Pipe 2, 0.0816 hours.

Schematic (After Calculations)

Not applicable for this intermediate calculation step. The result is a time value, not a new diagram.

Analysis

The travel time of 0.08 hours (about 4.9 minutes) is very fast. This is due to the combination of a short pipe length (500 ft) and a relatively high flow for a 6-inch pipe.

Cautionary Points

Note that this pipe is shorter (500 ft) but also smaller (6 in). Both length and diameter affect the volume.

Key Takeaways

Pipe 2 contributes "fresh" water very quickly to the junction.
Pipe dimensions (L and D) and flow (Q) all play a role in travel time.

Did You Know?

In real systems, well water (like Source B) often has a different chemical signature (e.g., higher minerals) and a different (often zero) disinfectant residual compared to reservoir water. This makes mixing calculations even more important.

FAQ

Common questions about this step.

Final Result

The travel time in Pipe 2 is 0.08 hours (or 4.90 minutes).

Your Turn

If the flow in Pipe 2 *increased* to 200 GPM, what would the new travel time be in hours?

Memo Card

Question 2 Summary:

Key Concept: Advection (Travel Time)
Essential Formula: \(\text{Time} = \text{Volume} / \text{Flow Rate}\)
Key Data: L=500ft, D=6in, Q=150 GPM

Question 3: Calculate the age of the water at Junction J1, assuming complete mixing.

Principle

The age at Junction J1 is the flow-weighted average of the two water streams entering it. First, we must find the age of each stream *as it arrives* at J1. Then, we can apply the mixing formula.

Mini-Lesson

Age at Arrival = Age at Source + Travel Time. Since both sources (A and B) have an age of 0 hours, the age of water arriving at J1 from each pipe is simply the travel time we just calculated.

Pedagogical Note

Think of this like mixing two cups of coffee. One is hot (old age) and one is cold (young age). If you pour more from the hot cup (higher flow), the final mixture will be warmer (older).

Norms

This calculation is based on the principle of Conservation of Mass. We are conserving a "substance" called "age," which is a common practice in water quality models.

Formula(s)

The complete mixing formula for two inflows:

The formula for a flow-weighted average. The numerator is the sum of each inflow's "age load" (Flow × Age). The denominator is the sum of the inflows (Total Flow).

\text{Age}_{J1} = \frac{(Q_1 \times \text{Age}_{\text{in},1}) + (Q_2 \times \text{Age}_{\text{in},2})}{Q_1 + Q_2}

The flow (Q) units cancel, leaving the final age in hours.

Hypotheses

We assume complete and instantaneous mixing at Junction J1. This means the two streams are fully blended the moment they meet, and the water leaving J1 has a single, uniform age. We also rely on our previous hypothesis that the age at both sources is 0.

Data

We combine data from the problem statement and our previous answers.

Parameter	Symbol	Value	Unit
Pipe 1 Flow	\(Q_1\)	350	GPM
Pipe 1 Arriving Age	\(\text{Age}_{\text{in},1}\)	0.2487	hours
Pipe 2 Flow	\(Q_2\)	150	GPM
Pipe 2 Arriving Age	\(\text{Age}_{\text{in},2}\)	0.0816	hours

Tips

Notice the units: (GPM * hours) / (GPM). The GPM units cancel out, leaving the final answer in hours, which is what we want. This is a good way to check your formula.

Schematic (Before Calculations)

This step models the mixing *at* the junction.

Schematic for Q3

Calculation(s)

Now we plug the values into the mixing formula. We use the flow rates (Q1, Q2) and the arrival ages (Time1, Time2) we just calculated.

Step 1: Calculate Total Flow (Denominator)

The total flow leaving the junction, \(Q_{\text{total}}\), is the sum of the inflows \(Q_1 = 350 \text{ GPM}\) and \(Q_2 = 150 \text{ GPM}\).

\begin{aligned} Q_{\text{total}} &= Q_1 + Q_2 \\ Q_{\text{total}} &= 350 \text{ GPM} + 150 \text{ GPM} \\ Q_{\text{total}} &= 500 \text{ GPM} \end{aligned}

This total flow (500 GPM) is the denominator for our mixing formula.

Step 2: Calculate Weighted Age Sum (Numerator)

We multiply each pipe's flow by its corresponding arrival age. We use the precise values from Q1 (\(0.2487 \text{ hr}\)) and Q2 (\(0.0816 \text{ hr}\)).

\begin{aligned} \text{Sum} &= (Q_1 \times \text{Age}_{\text{in},1}) + (Q_2 \times \text{Age}_{\text{in},2}) \\ \text{Sum} &= (350 \text{ GPM} \times 0.2487 \text{ hr}) + (150 \text{ GPM} \times 0.0816 \text{ hr}) \\ \text{Sum} &= 87.045 \text{ (GPM} \cdot \text{hr)} + 12.24 \text{ (GPM} \cdot \text{hr)} \\ \text{Sum} &= 99.285 \text{ (GPM} \cdot \text{hr)} \end{aligned}

This sum (99.285 GPM·hr) is the numerator, representing the total "age load" arriving at the junction.

Step 3: Calculate Mixed Age at J1

We divide the "Weighted Age Sum" from Step 2 by the "Total Flow" from Step 1.

\begin{aligned} \text{Age}_{J1} &= \frac{\text{Sum}}{Q_{\text{total}}} \\ \text{Age}_{J1} &= \frac{99.285 \text{ (GPM} \cdot \text{hr)}}{500 \text{ GPM}} \\ \text{Age}_{J1} &= 0.1986 \text{ hr} \end{aligned}

This is the final mixed age of the water at Junction J1.

Schematic (After Calculations)

The result is the age of water *at* node J1, which becomes the starting age for Pipe 3.

Result at J1

Analysis

The resulting age (0.20 hours) is between the two incoming ages (0.25 hr and 0.08 hr). Because the flow from Pipe 1 (350 GPM) is larger than from Pipe 2 (150 GPM), the final age is pulled closer to Pipe 1's age.

Cautionary Points

This model does *not* account for the physical volume of the junction itself, which is usually negligible. If this junction were a large storage tank, we would need a different, more complex calculation (a "mass balance over time").

Key Takeaways

The age of mixed water is a flow-weighted average, not a simple arithmetic average.
The stream with the higher flow rate (Q) has more influence on the final mixed age.

Did You Know?

Hydraulic models like EPANET use this exact mixing algorithm (or more complex ones for tanks) at every single junction and tank in the network, thousands of times, to simulate water quality over many days.

FAQ

Common questions about this step.

Final Result

The age of the water at Junction J1 is 0.20 hours (or 11.91 minutes).

Your Turn

Recalculate the age at J1, but this time, assume the water from Source 1 (Reservoir) was already 0.5 hours old *at the source*. (Use T1=0.25, T2=0.08)

Memo Card

Question 3 Summary:

Key Concept: Complete Mixing
Essential Formula: \(\text{Age}_{\text{out}} = \frac{\sum (Q_{\text{in}} \times \text{Age}_{\text{in}})}{\sum Q_{\text{in}}}\)
Key Data: Ages and Flows from Q1 & Q2.

Question 4: Calculate the travel time (in hours) for water in Pipe 3 (Junction J1 to Customer C1).

Principle

This is another advection (travel time) calculation, just like in Q1 and Q2. The key missing piece of data is the flow rate in Pipe 3, \(Q_3\).

Mini-Lesson

Conservation of Mass: At a junction, the total flow *in* must equal the total flow *out* (assuming no storage). \[ Q_{\text{in}} = Q_{\text{out}} \] \[ Q_1 + Q_2 = Q_3 \]

Pedagogical Note

The age of the water *entering* Pipe 3 (the 0.20 hr we just calculated) has no effect on the *travel time* in Pipe 3. Travel time is a purely hydraulic calculation (Volume / Flow Rate). The age will be used in the next step.

Norms

The principle of Conservation of Mass at Junction J1 (\(\sum Q_{in} = \sum Q_{out}\)) is the governing principle. We also use the same unit conversion factors as before.

Formula(s)

We use the advection time formulas again, but first must find \(Q_3\).

Flow Rate

First, the formula for conservation of mass. The flow *out* of the junction (Q3) must equal the total flow *in* (Q1 + Q2).

\[ Q_3 = Q_1 + Q_2 \]

We must calculate this flow rate *before* we can find the travel time.

Time

Next, the standard advection time formula, using the volume of Pipe 3 and the flow rate (Q3) we just determined.

\text{Time}_3 (\text{hr}) = \frac{\text{Volume}_3 (\text{gal})}{Q_3 \text{ (GPM)}} \times \left(\frac{1 \text{ hr}}{60 \text{ min}}\right)

This formula will give us the *additional* time the water spends traveling in Pipe 3.

Hypotheses

We assume plug flow in Pipe 3. We also assume no other demands or inflows/outflows between Junction J1 and Customer C1. The flow \(Q_3\) is constant along the entire length of Pipe 3.

Data

Data for Pipe 3:

Parameter	Symbol	Value	Unit
Pipe 3 Length	\(L_3\)	1000	ft
Pipe 3 Diameter	\(D_3\)	10	in
Pipe 3 Flow	\(Q_3\)	?	GPM

Tips

The flow rate in Pipe 3 is the *sum* of the inflows (Q1 + Q2). This is the first step. The age of the water *entering* Pipe 3 (from Q3) does not affect the *travel time* in Pipe 3.

Schematic (Before Calculations)

This step focuses on the advection in Pipe 3.

Schematic for Q4

Calculation(s)

First, we must find the flow in Pipe 3, \(Q_3\), by summing the inflows. Then, we repeat the travel time calculation for Pipe 3.

Step 1: Calculate Flow Rate \(Q_3\)

Using conservation of mass at Junction J1:

\begin{aligned} Q_3 &= Q_1 + Q_2 \\ Q_3 &= 350 \text{ GPM} + 150 \text{ GPM} \\ Q_3 & = 500 \text{ GPM} \end{aligned}

This flow rate of 500 GPM will be used to calculate the advection time in Pipe 3.

Step 2: Calculate Pipe 3 Area (ft²)

We take the diameter of Pipe 3, \(D_3 = 10 \text{ in}\), and divide by 12.

\begin{aligned} \text{Area}_3 &= \frac{\pi \times (D_3 / 12)^2}{4} \\ \text{Area}_3 &= \frac{\pi \times (10 \text{ in} / 12)^2}{4} \\ \text{Area}_3 &= \frac{\pi \times (0.833 \text{ ft})^2}{4} \\ \text{Area}_3 &= 0.5454 \text{ ft}^2 \end{aligned}

This is the cross-sectional area for Pipe 3.

Step 3: Calculate Pipe 3 Volume (Gallons)

We use the area from Step 2 (\(0.5454 \text{ ft}^2\)) and multiply by the length of Pipe 3, \(L_3 = 1000 \text{ ft}\).

\begin{aligned} \text{Volume}_3 (\text{ft}^3) &= \text{Area}_3 \times L_3 \\ \text{Volume}_3 (\text{ft}^3) &= 0.5454 \text{ ft}^2 \times 1000 \text{ ft} = 545.4 \text{ ft}^3 \\ \text{Volume}_3 (\text{gal}) &= \text{Volume}_3 (\text{ft}^3) \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_3 (\text{gal}) &= 545.4 \text{ ft}^3 \times 7.481 \text{ gal/ft}^3 \\ \text{Volume}_3 (\text{gal}) &= 4079.9 \text{ gal} \end{aligned}

The total volume of Pipe 3 is 4079.9 gallons.

Step 4: Calculate Travel Time (Hours)

We take the volume from Step 3 (\(4079.9 \text{ gal}\)) and divide by the flow rate from Step 1, \(Q_3 = 500 \text{ GPM}\).

\begin{aligned} \text{Time}_3 (\text{min}) &= \frac{\text{Volume}_3 (\text{gal})}{Q_3 (\text{GPM})} \\ \text{Time}_3 (\text{min}) &= \frac{4079.9 \text{ gal}}{500 \text{ GPM}} = 8.16 \text{ min} \\ \text{Time}_3 (\text{hr}) &= \frac{8.16 \text{ min}}{60 \text{ min/hr}} \\ \text{Time}_3 (\text{hr}) &= 0.136 \text{ hr} \end{aligned}

This is the travel time *within* Pipe 3, 0.136 hours.

Schematic (After Calculations)

Not applicable. The result is a time value.

Analysis

This pipe is 10-inch, which is larger than Pipe 1, but the flow rate (500 GPM) is also higher. The resulting travel time is 0.14 hours.

Cautionary Points

In a real network, C1 would be a "demand node." The flow \(Q_3\) is determined by the customer's demand, and the system pressures adjust to deliver it. Here, we've simplified it by *giving* the inflows \(Q_1\) and \(Q_2\), which in turn *set* the outflow \(Q_3\).

Key Takeaways

Flow rates are additive at junctions (for inflows) and determine the travel time in the downstream pipes.
A larger pipe (10-in) holds more volume, but a higher flow rate can still result in a short travel time.

Did You Know?

Larger diameter pipes like this 10-inch one are often 'transmission mains,' designed to move large volumes of water between key points, while smaller pipes (like Pipe 2) might be 'distribution mains' for local supply.

FAQ

Common questions about this step.

Final Result

The travel time in Pipe 3 is 0.14 hours (or 8.16 minutes).

Your Turn

If Pipe 3 was shorter, only 750 ft, what would the new travel time be in hours? (Assume \(Q_3\) is still 500 GPM).

Memo Card

Question 4 Summary:

Key Concept: Advection & Conservation of Mass
Essential Formula: \(Q_3 = Q_1 + Q_2\) and \(\text{Time}_3 = \text{Vol}_3 / Q_3\)
Key Data: L=1000ft, D=10in, Qs from Q1 & Q2

Question 5: What is the final age of the water delivered to Customer C1?

Principle

The final age at Customer C1 is a simple summation. It's the age of the water *as it enters* Pipe 3 (which is the mixed age at \(\text{Age}_{J1}\)) *plus* the time it spent traveling *through* Pipe 3 (\(\text{Time}_3\)).

Mini-Lesson

This is the core concept of water age modeling. Age is cumulative. The age at a downstream node is always the age at the upstream node *plus* the travel time in the pipe connecting them. \[ \text{Age}_{\text{downstream}} = \text{Age}_{\text{upstream}} + \text{Travel Time} \]

Pedagogical Note

This is the final step where all our previous work comes together. We see how the "history" of the water (the 0.20 hr age from mixing at J1) is carried forward and added to its new travel time (0.14 hr) to get a final result.

Norms

This summation assumes a 'First-In, First-Out' (FIFO) advection model within the pipe, which is the basis of the 'plug flow' assumption. The age 'packet' that enters Pipe 3 simply adds the travel time by the time it exits.

Formula(s)

The final age calculation.

The final age formula is a simple summation. The age at the customer (C1) is the age when the water *entered* the pipe (at J1) *plus* the *travel time* it spent in the pipe (Time_3).

\text{Age}_{C1} = \text{Age}_{J1} + \text{Time}_{3}

This shows how water age is cumulative as it travels through the network.

Hypotheses

All our previous assumptions (plug flow, complete mixing, 0 source age, conservation of mass) hold true and are combined for this final answer.

Data

We use our results from Q3 and Q4.

Parameter	Symbol	Value	Unit
Mixed Age at J1	\(\text{Age}_{J1}\)	0.1986	hours
Travel Time in Pipe 3	\(\text{Time}_3\)	0.1360	hours

Tips

This is a simple addition. The 'hard work' was finding the mixed age at J1 and the travel time in Pipe 3. This final step combines them. Make sure both values are in the same units (hours) before adding.

Schematic (Before Calculations)

This step summarizes the entire path.

Schematic for Q5

Calculation(s)

We simply add the age of the water *when it entered* Pipe 3 (which is \(\text{Age}_{J1}\)) to the time it *spent traveling* in Pipe 3 (which is \(\text{Time}_3\)).

We use the precise result from Q3 (\(\text{Age}_{J1} = 0.1986 \text{ hr}\)) and the precise result from Q4 (\(\text{Time}_3 = 0.1360 \text{ hr}\)).

\begin{aligned} \text{Age}_{C1} &= \text{Age}_{J1} + \text{Time}_{3} \\ \text{Age}_{C1} &= 0.1986 \text{ hr} + 0.1360 \text{ hr} \\ \text{Age}_{C1} &= 0.3346 \text{ hr} \end{aligned}

The final age at the customer is 0.3346 hours, or about 20 minutes.

To convert this to minutes for a better understanding: \(0.3346 \text{ hr} \times 60 \text{ min/hr} \approx 20.08 \text{ minutes}\).

Schematic (After Calculations)

The final result at the customer node.

Final Result at C1

Analysis

The final age of water delivered to the customer is 0.3346 hours, which is approximately 20 minutes (\(0.3346 \times 60 \approx 20.08\)).

This is a very "fresh" water, as expected in a small, simple system. In a large municipal network with storage tanks and complex loops, water ages can easily range from several hours to several days. An age of 2-3 days is often a target maximum to ensure disinfectant residuals remain effective.

Cautionary Points

This calculation does not account for any storage (e.g., a customer's own water tank) at C1, which would further age the water before consumption. This is the age *at the meter*.

Key Takeaways

Final water age is cumulative.
It is the sum of the mixed age from upstream junctions and the advection (travel) time from those junctions.

Did You Know?

Water utilities use flushing programs—intentionally opening hydrants in "dead-end" areas—to force fresh, low-age water into parts of the network where high water age is a problem.

FAQ

Common questions about this step.

Final Result

The final age of water at Customer C1 is 0.33 hours (or 20.08 minutes).

Your Turn

Using your "Your Turn" answer from Q3 (\(\text{Age}_{J1} \approx 0.58 \text{ hr}\)), what would the final age at C1 be? (Use the original \(\text{Time}_3 = 0.14 \text{ hr}\)).

Memo Card

Question 5 Summary:

Key Concept: Cumulative Age
Essential Formula: \(\text{Age}_{\text{Final}} = \text{Age}_{\text{Start}} + \text{Travel Time}\)
Key Data: Result from Q3 (Age_J1) + Result from Q4 (Time_3)

Interactive Tool: Junction Mixing Calculator

This tool simulates the mixing at Junction J1 (Question 3). Use the sliders to see how changing the flow rates and incoming ages of Source 1 and Source 2 affects the final mixed water age.

Input Parameters

Source 1 Flow (Q1) (GPM) 350 GPM

Source 1 Age (Age1) (Hours) 0.25 Hours

Source 2 Flow (Q2) (GPM) 150 GPM

Source 2 Age (Age2) (Hours) 0.08 Hours

Key Requirements

Total Outflow (Q1 + Q2) - GPM

Resulting Mixed Age - Hours

Final Quiz: Test Your Knowledge

1. What is the *primary* reason water age is a key concern for water quality?

Disinfectant (chlorine) residual decays over time.
The water gets too warm in the pipes.
The pressure drops as water gets older.

2. If the flow rate (GPM) in a pipe *doubles*, what happens to the water's travel time?

It doubles.
It stays the same.
It is halved.

3. The calculation for water age at a mixing junction is a...

Simple average of the incoming ages.
Flow-weighted average of the incoming ages.
Geometric mean of the incoming flows.

[Optional: Correct! The flow rate (Q) of each stream determines its "vote" in the final mixed age.]

Glossary

Advection: The transport of a substance (like water or a contaminant) by the bulk motion of the fluid. In this exercise, it represents the travel time in a pipe.
EPA (Environmental Protection Agency): The US federal government agency responsible for setting and enforcing national standards for drinking water quality.
GPM (Gallons Per Minute): A standard US customary unit for measuring volumetric flow rate, commonly used in water distribution and plumbing.
Water Age: The average time water takes to travel from its source (e.g., a treatment plant) to a specific point in the network. A key indicator of water quality.