I just need I, J, K answered! ### Heat Transfer and Heat Exchanger EfficiencyThis section covers advanced concepts in thermodynamics related to heat exchangers and their performance optimization.#### Questions and Conceptsf. **Heat Transfer Coefficient (U):** - Calculate the overall heat transfer coefficient (U) for two different conditions. - Analyze whether the two U values are the same and discuss.g. **Influence on Efficiency:** - Using your calculated U and the log mean temperature difference (ΔTₘ), determine which parameter affects the heat exchanger’s efficiency the most.h. **Heating Milk from 4 to 60 °C:** - Explore modifications required in the heat exchanger, such as changing its length or operational conditions (flow rate, temperatures), to achieve this heating goal.i. **Effect of Tube Length on U:** - Consider how increasing the tube's length affects the overall heat transfer coefficient (U) and explain why this change occurs.j. **Effect of Flow Rate on U:** - Examine if and how increasing the flow rate of milk will alter the heat transfer coefficient (U), providing a rationale for your answer.k. **Con-current vs. Counter-current Flow Options:** - Discuss situations where a con-current flow may be preferred over a counter-current flow in a heat exchanger.l. **Impact of Fouling:** - Evaluate how fouling inside pipes affects milk pasteurization. - Propose actions to maintain pasteurization goals, considering heat transfer and fluid flow dynamics.These questions are designed to deepen understanding of heat exchanger efficiency and the operational factors that influence it. ### Tubular Heat Exchanger Analysis#### BackgroundA tubular heat exchanger is described with the following specifications:- Outer tube radius: 4 cm- Inner tube radius: 2 cm- Length: 15 m- Milk processing rate: 3 kg/s- Inlet temperatures: Milk at 4 °C, Hot water at 90 °C- Specific heat capacity of water: 4.19 kJ/(kg °C)- Specific heat capacity of milk: 4.0 kJ/(kg °C)#### Tasks**a. Con-current Flow Analysis:**- **Objective:** Determine the outlet temperature of the milk.- **Given:** - Water flow rate: 6 kg/s - Outlet temperature of hot water: 70.13 °C- **Instructions:** Perform calculations for the outlet temperature of the milk and compare your results with a virtual experiment.**b. Counter-current Flow Analysis:**- **Objective:** Determine the flow rate of water.- **Given:** - Outlet temperatures: Hot water at 68.78 °C, Milk at 48.46 °C- **Instructions:** Calculate the water flow rate and compare your findings with a virtual experiment.**c. Energy Used:**- **Question:** Is the energy of water used during pasteurization sensible heat or latent heat?**d. Heat Transfer Efficiency:**- **Task:** Calculate the heat transfer rates for con-current and counter-current flows.- **Objective:** Compare these rates to determine which flow direction is more efficient.**e. Log Mean Temperature Difference (LMTD):**- **Task:** Calculate the LMTD for both flow conditions.- **Objective:** Compare results to identify which flow direction has a higher LMTD.This problem set involves a combination of thermodynamics and heat transfer principles, focusing on the functionality and efficiency of a tubular heat exchanger.

Answered: i. If you increase the length of the…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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I just need I, J, K answered!

### Heat Transfer and Heat Exchanger Efficiency

This section covers advanced concepts in thermodynamics related to heat exchangers and their performance optimization.

#### Questions and Concepts

f. **Heat Transfer Coefficient (U):**
- Calculate the overall heat transfer coefficient (U) for two different conditions.
- Analyze whether the two U values are the same and discuss.

g. **Influence on Efficiency:**
- Using your calculated U and the log mean temperature difference (ΔTₘ), determine which parameter affects the heat exchanger’s efficiency the most.

h. **Heating Milk from 4 to 60 °C:**
- Explore modifications required in the heat exchanger, such as changing its length or operational conditions (flow rate, temperatures), to achieve this heating goal.

i. **Effect of Tube Length on U:**
- Consider how increasing the tube's length affects the overall heat transfer coefficient (U) and explain why this change occurs.

j. **Effect of Flow Rate on U:**
- Examine if and how increasing the flow rate of milk will alter the heat transfer coefficient (U), providing a rationale for your answer.

k. **Con-current vs. Counter-current Flow Options:**
- Discuss situations where a con-current flow may be preferred over a counter-current flow in a heat exchanger.

l. **Impact of Fouling:**
- Evaluate how fouling inside pipes affects milk pasteurization.
- Propose actions to maintain pasteurization goals, considering heat transfer and fluid flow dynamics.

These questions are designed to deepen understanding of heat exchanger efficiency and the operational factors that influence it.

### Tubular Heat Exchanger Analysis

#### Background
A tubular heat exchanger is described with the following specifications:
- Outer tube radius: 4 cm
- Inner tube radius: 2 cm
- Length: 15 m
- Milk processing rate: 3 kg/s
- Inlet temperatures: Milk at 4 °C, Hot water at 90 °C
- Specific heat capacity of water: 4.19 kJ/(kg °C)
- Specific heat capacity of milk: 4.0 kJ/(kg °C)

#### Tasks
**a. Con-current Flow Analysis:**
- **Objective:** Determine the outlet temperature of the milk.
- **Given:**
- Water flow rate: 6 kg/s
- Outlet temperature of hot water: 70.13 °C
- **Instructions:** Perform calculations for the outlet temperature of the milk and compare your results with a virtual experiment.

**b. Counter-current Flow Analysis:**
- **Objective:** Determine the flow rate of water.
- **Given:**
- Outlet temperatures: Hot water at 68.78 °C, Milk at 48.46 °C
- **Instructions:** Calculate the water flow rate and compare your findings with a virtual experiment.

**c. Energy Used:**
- **Question:** Is the energy of water used during pasteurization sensible heat or latent heat?

**d. Heat Transfer Efficiency:**
- **Task:** Calculate the heat transfer rates for con-current and counter-current flows.
- **Objective:** Compare these rates to determine which flow direction is more efficient.

**e. Log Mean Temperature Difference (LMTD):**
- **Task:** Calculate the LMTD for both flow conditions.
- **Objective:** Compare results to identify which flow direction has a higher LMTD.

This problem set involves a combination of thermodynamics and heat transfer principles, focusing on the functionality and efficiency of a tubular heat exchanger.

Expert Solution

Step 1: Introduction

A heat exchanger typically involves two fluids separated by a solid wall. Heat is first transferred from the hot fluid to the wall by convection, through the wall by conduction, and from the wall to the cold fluid again by convection. Any radiation effects are usually included in the convection heat transfer coefficients. The thermal resistance network associated with this heat transfer process involves two convection and one conduction resistances.

The relation for the overall heat transfer coefficient can be written as follows:

$\frac{1}{U A_{s}} = \frac{1}{U_{i} A_{i}} = \frac{1}{U_{o} A_{o}} = R = \frac{1}{h_{i} A_{i}} + R_{w a l l} + \frac{1}{h_{o} A_{o}}$

Perhaps you are wondering that why we have two overall heat transfer coefficients, $U_{i}$ and $U_{o}$ , for a heat exchanger. The reason is that every heat exchanger has two heat transfer surface areas $A_{i}$ and $A_{o}$ , which in general, are not equal to each other.

Note that, $U_{i} A_{i} = U_{o} A_{o}$ , but $U_{i} does not equal U_{o}$ unless, $A_{i} = A_{o}$ .Therefore, the overall heat transfer coefficient U of a heat exchanger is meaningless unless the area on which it is based is specified.

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i. If you increase the length of the tube, will the overall heat transfer coefficient U change? Explain why? j. If you increase the flow rate of milk, will U change? Explain why?