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Batch processes are often used in chemical and pharmaceutical operations to achicve a desired chemical composition for the final product and typically involve a transient heating operation to take the product from room temperature to the desired process temperature. Consider a situation for which a chemical of density \(\rho=\) \(1200 \mathrm{~kg} / \mathrm{m}^{3}\) and specific heat \(c=2200 \mathrm{~J} / \mathrm{kg}-\mathrm{K}\) occupies a volume of \(V=2.25 \mathrm{~m}^{3}\) in an insulated vessel. The chemical is to be heated from room temperature, \(T_{f}=\) \(300 \mathrm{~K}\), to a process temperature of \(T=450 \mathrm{~K}\) by passing saturated steam at \(T_{h}=500 \mathrm{~K}\) through a coiled, thinwalled, 20 -mm-diameter tube in the vessel. Steam condensation within the tube maintains an interior convection cocfficient of \(h_{y}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the highly agitated liquid in the stirned vessel maintains an outside convection coefficient of \(h_{e}=2000 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\). If the chemical is to be heated from 300 to \(450 \mathrm{~K}\) in 60 minutes, what is the required length \(L\) of the submerged tubing?

Step by step solution

01

Calculate Heat Required

First, calculate the total amount of heat required to raise the temperature of the chemical from room temperature, 300 K to the desired process temperature, 450 K. Use the formula: \[ Q = m imes c imes (T - T_f) \]where \( m = \rho \times V = 1200 \times 2.25 \). So, \( Q = 1200 \times 2.25 \times 2200 \times (450 - 300) \) Joules.

02

Calculate the Heat Transfer Rate

Since the heating process takes 60 minutes (or 3600 seconds), we need the heat transfer rate. The heat transfer rate (\( \dot{Q} \)) is given by dividing total heat \( Q \) by time: \[ \dot{Q} = \frac{Q}{3600} \] Plug in the value of \( Q \) calculated in Step 1 to find \( \dot{Q} \).

03

Determine Tube Surface Area Required

Use the heat transfer equation: \[ \dot{Q} = U \times A \times \Delta T_{lm} \] where \( U \) is the overall heat transfer coefficient, \( A \) is the surface area of the tube, and \( \Delta T_{lm} \) is the log mean temperature difference. First, we need \( U \), which can be calculated by \[ U = \frac{1}{\frac{1}{h_i} + \frac{1}{h_o}} \] Fill in \( h_i = 10000 \) and \( h_o = 2000 \) to find \( U \).

04

Log Mean Temperature Difference

Calculate the log mean temperature difference using: \[ \Delta T_{lm} = \frac{(T_h - T)(T_h - T_f)}{\ln{\left( \frac{T_h - T}{T_h - T_f} \right)}} \] Substitute \( T_h = 500 \), \( T = 450 \), and \( T_f = 300 \) to find \( \Delta T_{lm} \).

05

Solve for Tube Length

Now, solve for the tube length \( L \). The tube is cylindrical, so \( A = \pi \times d \times L \). From our previous equations: \[ \dot{Q} = U \times (\pi \times d \times L) \times \Delta T_{lm} \].Rearrange to solve for \( L \): \[ L = \frac{\dot{Q}}{U \times \pi \times d \times \Delta T_{lm}} \]Substitute \( \dot{Q} \), \( U \), \( \Delta T_{lm} \), and \( d = 0.02 \) m to find \( L \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Batch Heating Operations

Batch heating operations are critical in chemical and pharmaceutical industries for producing products with specific chemical compositions. In such operations, materials are heated in batches rather than continuously. This allows for precise control over the heating process. Batch heating involves heating insulated vessels that contain the materials. It targets achieving a desired process temperature at an ensured uniform temperature distribution within the vessel.

Batch heating operations are commonly used to convert raw ingredients into final products. They provide flexibility and control over the chemical reactions. The batch process ends when the desired temperature is reached and the necessary chemical transformations have occurred. This method is effective when dealing with small to moderate production scales where product consistency is paramount.

Transient Heating Analysis

Transient heating analysis is used to understand how the temperature of materials in batch operations changes over time. Unlike steady-state heating, transient heating involves changes in temperature up until the material reaches thermal equilibrium.

To accurately evaluate transient heating, it requires analyzing the heat transfer during the heating process. This involves understanding the rate at which heat enters the process, the thermal properties of the material, and how quickly the material approaches the target temperature. Transient heating is significant in understanding the time-dependent temperature profiles of batch processed materials. It helps in designing systems that achieve the desired temperature within a specified time frame, ensuring efficiency and safety in operations.

Specific Heat Calculation

Specific heat calculation is vital in batch heating operations to determine the amount of thermal energy needed to raise the temperature of a material. Specific heat, denoted by the symbol \( c \), is the amount of heat per unit mass required to increase the temperature by one degree Kelvin.

The formula for calculating the total heat required \( Q \) is:

• \( Q = m \times c \times (T - T_f) \)
• where \( m \) is the mass, \( c \) is the specific heat, and \((T - T_f)\) is the temperature change.

By using specific heat values, engineers can design systems to supply adequate energy and ensure efficient temperature increases. This calculation is crucial for conserving energy and optimizing the time required to reach the process temperature.

Log Mean Temperature Difference

The Log Mean Temperature Difference (LMTD) is used in calculating the heat transfer rate across a heat exchanger. It is essential since it accounts for the varying temperature difference along the length of the heat exchanger.

The formula for LMTD is:

• \[ \Delta T_{lm} = \frac{(T_h - T)(T_h - T_f)}{\ln{\left( \frac{T_h - T}{T_h - T_f} \right)}} \]
• where \( T_h \), \( T \), and \( T_f \) are the inlet steam, process, and initial temperatures respectively.

LMTD provides a "mean" effective temperature difference, crucial for accurately determining the required heat exchanger surface area. It simplifies the design and analysis by providing a single temperature difference value that represents the entire heat exchange process. This makes it easier for engineers to ensure the system is sufficient to transfer the desired amount of heat efficiently.