Give step-by-step solution with explanation and final answer:
Consider you are an engineer working for Dan's Chemical Co. and it is your job to assess the viability of a batch process for a new type o

Question

Give step-by-step solution with explanation and final answer:
Consider you are an engineer working for Dan's Chemical Co. and it is your job to assess the viability of a batch process for a new type of sports drink. You want the product to be processed to a final state, in a timely fashion, and to be loaded to a shipment container at an appropriate temperature. You've determined you need to specify a new closed 4,000L reactor vessel that can deliver the necessary amount of thermal energy in a given time. The reactor vessel is supposed to come with its own heating system, both that can control the amount of heat delivered and the rate at which the heat is delivered. Consider the reactant fluid to be batch processed in the vessel is to have its desired phase change occur isothermally at 58.8°C, and exit the reactor at 72°C. The reactant and product fluids are known to have the same constant specific heat capacity of 1,536 J kg-1 K-1 and a density of 219kg m-3 up to 100°C, and has a latent heat of transformation of 8.103 kJ kg-1. Consider the plant to have an internal ambient temperature of (15°C) in the Winter. Part a) Determine the net heat energy required to have the desired produc

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# Step-by-Step Solution

Let's break down the problem into clear steps and solve for the **net heat energy required ($ Q_{duty} $)**.

---

## **Step 1: Gather Given Data**

- **Volume of reactor:** $ V = 4,000 $ L $ = 4. $ m³  (since $ 1,000 $ L = $ 1 $ m³)
- **Density of fluid:** $ \rho = 219 $ kg/m³
- **Specific heat capacity:** $ c_p = 1,536 $ J/(kg·K)
- **Latent heat of transformation:** $ L = 8.103 $ kJ/kg $ = 8,103 $ J/kg
- **Initial temperature:** $ T_{initial} = 15^\circ $C
- **Isothermal phase change temperature:** $ T_{phase} = 58.8^\circ $C
- **Exit temperature:** $ T_{final} = 72^\circ $C

---

## **Step 2: Calculate Mass of Fluid**

$$
\text{mass} = \rho \times V = 219~\text{kg/m}^3 \times 4.~\text{m}^3 = 876~\text{kg}
$$

---

## **Step 3: Identify the Heating Stages**

1. **Heat from $ 15^\circ $C to $ 58.8^\circ $C (pre-phase change)**
2. **Heat for the isothermal phase change at $ 58.8^\circ $C**
3. **Heat from $ 58.8^\circ $C to $ 72^\circ $C (post-phase change)**

---

## **Step 4: Calculate Heat for Each Stage**

### **1. Sensible Heat: $ 15^\circ $C to $ 58.8^\circ $C**

$$
Q_1 = m c_p (T_{phase} - T_{initial})
$$
$$
Q_1 = 876~\text{kg} \times 1,536~\text{J/kg·K} \times (58.8 - 15)~\text{K}
$$
$$
Q_1 = 876 \times 1,536 \times 43.8
$$
$$
Q_1 = 876 \times 67,276.8 = 58,946,956.8~\text{J}
$$

---

### **2. Latent Heat at $ 58.8^\circ $C**

$$
Q_2 = m L = 876~\text{kg} \times 8,103~\text{J/kg}
$$
$$
Q_2 = 876 \times 8,103 = 7,102,428~\text{J}
$$

---

### **3. Sensible Heat: $ 58.8^\circ $C to $ 72^\circ $C**

$$
Q_3 = m c_p (T_{final} - T_{phase}) = 876~\text{kg} \times 1,536~\text{J/kg·K} \times (72 - 58.8)~\text{K}
$$
$$
Q_3 = 876 \times 1,536 \times 13.2
$$
$$
Q_3 = 876 \times 20,275.2 = 17,757,235.2~\text{J}
$$

---

## **Step 5: Calculate Total Heat Required**

$$
Q_{duty} = Q_1 + Q_2 + Q_3
$$
$$
Q_{duty} = 58,946,956.8~\text{J} + 7,102,428~\text{J} + 17,757,235.2~\text{J}
$$
$$
Q_{duty} = 83,806,620~\text{J}
$$

---

## **Step 6: Convert to Megajoules (MJ)**

$$
Q_{duty} = \frac{83,806,620~\text{J}}{1,000,000}
$$
$$
Q_{duty} = 83.81~\text{MJ}
$$

---

# **Final Answer**

## **The net heat energy required is:**

$$
\boxed{Q_{duty} = 83.8~\text{MJ}}
$$

*(rounded to 1 decimal place)*

---

### **Summary Table**

| Stage                     | Calculation                          | Energy (J)     |
|---------------------------|--------------------------------------|----------------|
| 15°C → 58.8°C             | $ Q_1 = mc_p\Delta T_1 $           | 58,946,957     |
| Phase change @ 58.8°C     | $ Q_2 = mL $                       | 7,102,428      |
| 58.8°C → 72°C             | $ Q_3 = mc_p\Delta T_2 $           | 17,757,235     |
| **Total**                 |                                      | **83,806,620** |
| **Total (MJ)**            |                                      | **83.8**       |

---

**No heat losses are considered in this calculation.**

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