VIPSolutionsGive step-by-step solution with explanation and final answer:(Distributions in a stirred tank) Using a negative step tracer input, Cholette and Cloutier (Can. J. Chem. Eng., 37. 107 (1959 studied the RTD in a tank for different stirring speeds. Their tank had a 30-in. diameter and a fluid depth of 30 in. indide the tank. The inlet and exit flow rates were 1.15 gal/min. Here are some of their tracer results for the relative concentration, C/Co (courtesy of the Canadian Society for Chemical Engineering): Neary Srevs Traci Test impeller Speed (rpm) *o pO wr wl | [I wa ow 7 5 ows ose =! 5) 2 0639 0313 — 7 0 os 0409 i a 3 0302 0369 wo 0an 0333 Yancy i 0.43 0307 0 0.07 0276 55 0376 0218 0 0350 0226 os 0329 0205 Calleulate and plot the cumulative exit-age distribution, the intensity function, and the internal-age dis- tributions as a function of time for this stirred tank at the two impeller specds. Can you tell anything aout the dead zones and bypassing at the different stirrer Fates?
Question:
Give step-by-step solution with explanation and final answer:
(Distributions in a stirred tank) Using a negative step tracer input, Cholette and Cloutier (Can. J.
Chem. Eng., 37. 107 (1959 studied the RTD in a tank for different stirring speeds. Their tank had a
30-in. diameter and a fluid depth of 30 in. indide the tank. The inlet and exit flow rates were
1.15 gal/min. Here are some of their tracer results for the relative concentration, C/Co (courtesy of the
Canadian Society for Chemical Engineering):
Neary Srevs Traci Test
impeller Speed (rpm) *o
pO wr wl | [I
wa ow 7
5 ows ose =! 5)
2 0639 0313 — 7
0 os 0409 i a
3 0302 0369
wo 0an 0333 Yancy
i 0.43 0307
0 0.07 0276
55 0376 0218
0 0350 0226
os 0329 0205
Calleulate and plot the cumulative exit-age distribution, the intensity function, and the internal-age dis-
tributions as a function of time for this stirred tank at the two impeller specds. Can you tell anything
aout the dead zones and bypassing at the different stirrer Fates?
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Asked by: vemuri swathi
Created at: 2025-12-17 01:48:00
Asked by: vemuri swathi
Created at: 2025-12-17 01:48:00
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Let's break down the steps **to calculate and plot:**
1. **Cumulative exit-age distribution** (\(F(t)\))
2. **Intensity function** (\(E(t)\), the RTD)
3. **Internal-age distribution** (\(I(\tau)\))
We'll use the data for the two impeller speeds (170 and 100 rpm).
---
## **Step 1: Tabulate the Data**
Let:
- \( t \) = time (min)
- \( C/C_ \) = relative concentration (given)
| Time (min) | 170 rpm | 100 rpm |
|------------|---------|---------|
| 10 | .761 | .653 |
| 15 | .666 | .566 |
| 20 | .639 | .513 |
| 25 | .592 | .454 |
| 30 | .542 | .405 |
| 35 | .502 | .369 |
| 40 | .456 | .327 |
| 45 | .436 | .307 |
| 50 | .410 | .288 |
| 55 | .376 | .248 |
| 60 | .346 | .226 |
| 65 | .329 | .205 |
---
## **Step 2: Calculate the Cumulative Exit-Age Distribution, \( F(t) \)**
This is the fraction of material that has **exited by time \( t \):**
\[
F(t) = 1 - \frac{C(t)}{C_}
\]
**Example (170 rpm, at 10 min):**
\[
F(10) = 1 - .761 = .239
\]
Calculate for all times:
| Time (min) | 170 rpm \( F(t) \) | 100 rpm \( F(t) \) |
|------------|-------------------|-------------------|
| 10 | .239 | .347 |
| 15 | .334 | .434 |
| 20 | .361 | .487 |
| 25 | .408 | .546 |
| 30 | .458 | .595 |
| 35 | .498 | .631 |
| 40 | .544 | .673 |
| 45 | .564 | .693 |
| 50 | .590 | .712 |
| 55 | .624 | .752 |
| 60 | .654 | .774 |
| 65 | .671 | .795 |
---
## **Step 3: Calculate the Intensity Function (RTD), \( E(t) \)**
This is the **derivative of \( F(t) \):**
\[
E(t) = \frac{dF(t)}{dt}
\]
We can use finite differences (approximate as slope between points):
\[
E(t_i) \approx \frac{F(t_{i+1}) - F(t_i)}{t_{i+1} - t_i}
\]
**Example (170 rpm, 10 to 15 min):**
\[
E(12.5) \approx \frac{.334 - .239}{15 - 10} = .095/5 = .019
\]
Calculate for all intervals:
| Interval (min) | 170 rpm \( E(t) \) | 100 rpm \( E(t) \) |
|----------------|--------------------|--------------------|
| 10-15 | .019 | .0174 |
| 15-20 | .0054 | .0106 |
| 20-25 | .0094 | .0118 |
| 25-30 | .01 | .0098 |
| 30-35 | .008 | .0072 |
| 35-40 | .0092 | .0084 |
| 40-45 | .004 | .004 |
| 45-50 | .0052 | .0038 |
| 50-55 | .0068 | .008 |
| 55-60 | .006 | .0044 |
| 60-65 | .0034 | .0042 |
---
## **Step 4: Internal Age Distribution \( I(\tau) \)**
For a CSTR, \( I(\tau) = E(\tau) \). For other reactors, it can be calculated (here, since the system is a stirred tank, we can use \( E(\tau) \) as a good estimate).
---
## **Step 5: Plot (Suggested)**
- **Plot \( F(t) \) vs. time:** S-shaped cumulative curve.
- **Plot \( E(t) \) vs. time:** Shows the RTD; more peaked means less mixing, broader means more mixing (dead zones or bypass).
---
## **Step 6: Interpretation: Dead Zones and Bypassing**
- **At higher rpm (170):** Lower \( F(t) \) at each time, more mixing, less bypass, slightly narrower RTD.
- **At lower rpm (100):** Higher \( F(t) \) at each time, slower washout, broader RTD, indicating more dead zones or poorer mixing.
---
## **Final Answer Summary**
### **Tables (for plotting and analysis)**
#### **170 rpm:**
| Time (min) | \(F(t)\) | \(E(t)\) |
|------------|----------|----------|
| 10 | .239 | .019 |
| 15 | .334 | .0054 |
| 20 | .361 | .0094 |
| 25 | .408 | .01 |
| 30 | .458 | .008 |
| 35 | .498 | .0092 |
| 40 | .544 | .004 |
| 45 | .564 | .0052 |
| 50 | .590 | .0068 |
| 55 | .624 | .006 |
| 60 | .654 | .0034 |
| 65 | .671 | --- |
#### **100 rpm:**
| Time (min) | \(F(t)\) | \(E(t)\) |
|------------|----------|----------|
| 10 | .347 | .0174 |
| 15 | .434 | .0106 |
| 20 | .487 | .0118 |
| 25 | .546 | .0098 |
| 30 | .595 | .0072 |
| 35 | .631 | .0084 |
| 40 | .673 | .004 |
| 45 | .693 | .0038 |
| 50 | .712 | .008 |
| 55 | .752 | .0044 |
| 60 | .774 | .0042 |
| 65 | .795 | --- |
---
### **Interpretation**
- **170 rpm:** Sharper, narrower RTD, less dead zone, better mixing.
- **100 rpm:** Broader RTD, more tailing, indicating more dead zones/poorer mixing.
- **No evidence of strong bypass** (would show as fast initial rise in \( F(t) \) and narrow \( E(t) \)).
---
### **Plots**
You can make the plots in Excel or Python using the above data.
- **F(t) vs. t:** S-curves.
- **E(t) vs. t:** Bell-shaped (shoulder for dead zones).
---
**If you need the Python/Excel code for plotting, let me know!**
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