Question

Consider a shell and tube heat exchanger in a milk be heated from ${20}^{\circ }\mathrm{C}$ by hot water initially at ${140}^{\circ }\mathrm{C}$ and flowing at a rate of $5\mathrm{kg}/\mathrm{s}$. The milk flows through 30 thin-walled tubes with an inside diameter of $20\mathrm{mm}$ with each tube making 10 passes through the shell. The average convective heat transfer coefficients on the milk and water side are $450\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$ and $1100\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$, respectively. In order to complete the pasteurizing process and hence restrict the microbial growth in the milk, it is required to have the exit temperature of milk attain at least ${70}^{\circ }\mathrm{C}$. As a design engineer, your job is to decide upon the shell width (tube length in each pass) so that the milk exit temperature of ${70}^{\circ }\mathrm{C}$ can be achieved. One of the design requirements is that the exit temperature of hot water should be at least ${10}^{\circ }\mathrm{C}$ higher than the exit temperature of milk.

Short answer

Solving for $Q_{v,milk}$, we have: $Q_{v,milk}=\frac{Q}{{\rho }_{milk}\times C_{p,milk}\times (\mathrm{\Delta }T{)}_{milk}}$ We are given the hot water flow rate of 5 kg/s and its initial temperature of 140°C. Considering a hot water exit temperature at least 10°C higher than the milk exit temperature, that would mean the hot water exit temperature is at least 70°C + 10°C = 80°C. So, the available temperature difference for the hot water is 140°C - 80°C = 60°C. Therefore, we can also calculate the heat transfer rate (Q) using the hot water side as follows: $Q=m_{water}\times C_{p,water}\times (\mathrm{\Delta }T{)}_{water}$ where: $m_{water}$ = mass flow rate of hot water (given, 5 kg/s) $C_{p,water}$ = specific heat capacity of water ($4180\mathrm{W}/(\mathrm{kg}\cdot \mathrm{K})$) $(\mathrm{\Delta }T{)}_{water}$ = hot water temperature difference, which is ${60}^{\circ }\mathrm{C}$ Thus, $Q=5\times 4180\times 60=1,254,000\mathrm{W}$ Now, using the formula for $Q_{v,milk}$, we get: $Q_{v,milk}=\frac{1,254,000}{1000\times 4180\times 50}=0.006{\mathrm{m}}^3/\mathrm{s}$ #tag_title# Step 2: Calculate the overall heat transfer coefficient#tag_content# Now we have the overall heat transfer coefficient: $U=\frac{1}{\frac{1}{h_{milk}\times A_m}+\frac{1}{h_{water}\times A_w}}$ where: $h_{milk}$ = heat transfer coefficient on the milk side (given, 450 W/m²·K) $h_{water}$ = heat transfer coefficient on the water side (given, 1100 W/m²·K) $A_m$ = heat transfer area on the milk side (to be determined) $A_w$ = heat transfer area on the water side (to be determined) Since the heat transfer area is the same on both sides, we can rewrite this formula as: $U=\frac{1}{\frac{1}{h_{milk}}+\frac{1}{h_{water}}}=\frac{1}{\frac{1}{450}+\frac{1}{1100}}=299.2\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$ #tag_title# Step 3: Use the LMTD method to determine the required area#tag_content# Using the logarithmic mean temperature difference (LMTD) method, we have: $Q=U\times A\times \mathrm{\Delta }{T}_{lm}$ where: $\mathrm{\Delta }{T}_{lm}$ = logarithmic mean temperature difference, which is given by the formula $\frac{(\mathrm{\Delta }{T}_1-\mathrm{\Delta }{T}_2)}{\ln (\mathrm{\Delta }{T}_1/\mathrm{\Delta }{T}_2)}$ In this case, $\mathrm{\Delta }{T}_1$ is the temperature difference at the inlet (140°C - 20°C = 120°C) and $\mathrm{\Delta }{T}_2$ is the temperature difference at the outlet (80°C - 70°C = 10°C). $\mathrm{\Delta }{T}_{lm}=\frac{(120-10)}{\ln (120/10)}={40.7}^{\circ }\mathrm{C}$ Now we can solve for the required heat transfer area (A): $A=\frac{Q}{U\times \mathrm{\Delta }{T}_{lm}}=\frac{1,254,000}{299.2\times 40.7}=102.29{\mathrm{m}}^2$ #tag_title# Step 4: Determine the shell width (tube length in each pass)#tag_content# The heat transfer area on the milk side can be calculated as follows: $A=n\times L\times (D\times \pi )$ where: $n$ = number of tubes (given, 30) $L$ = length of each pass $D$ = inside diameter of each tube (given, 20mm or 0.02m) We know that the milk makes 10 passes through the shell, so the total length of the tubes is $L_{total}=10\times L$. Hence, $A=30\times (10\times L)\times (0.02\times \pi )$ Now, solving for L: $L=\frac{A}{30\times 10\times (0.02\times \pi )}=\frac{102.29}{30\times 10\times (0.02\times 3.14)}=1.71\mathrm{m}$ So, the required shell width, or tube length in each pass, is approximately 1.71 meters.

Step-by-step solution

Calculate the required heat transfer rate

To calculate the required heat transfer rate (Q), we can use the equation: $Q=m_{milk}\times C_{p,milk}\times (\mathrm{\Delta }T{)}_{milk}$ where: $m_{milk}$ = mass flow rate of milk (to be determined) $C_{p,milk}$ = specific heat capacity of milk (approximately equal to water, $4.18\mathrm{kJ}/(\mathrm{kg}\cdot \mathrm{K})$ or $4180\mathrm{W}/(\mathrm{kg}\cdot \mathrm{K})$) $(\mathrm{\Delta }T{)}_{milk}$ = milk temperature difference, which is ${70}^{\circ }\mathrm{C}-{20}^{\circ }\mathrm{C}={50}^{\circ }\mathrm{C}$ First, we need to determine the mass flow rate of milk ($m_{milk}$). Assuming incompressible flow, we can use the continuity equation: $Q=a\cdot v$, where: $Q$ = volumetric flow rate (m³/s) $a$ = cross-sectional area of a single tube (m²) $v$ = average velocity of fluid within the tube (m/s) We are given the diameter of each tube ($20\mathrm{mm}$), so the cross-sectional area of a single tube can be calculated as: $a=\frac{1}{4}\times \pi \times (0.02\mathrm{m}{)}^2=3.14\times {10}^{-4}{\mathrm{m}}^2$ For a total of 30 tubes and since the milk makes 10 passes through the shell, the actual volume flow rate of the milk is: $Q_{v,milk}=30\times a\times \frac{v}{10}$ We know that: $m_{milk}={\rho }_{milk}\times Q_{v,milk}$ where ${\rho }_{milk}$ is the density of milk (approximately equal to water, $1000\mathrm{kg}/{\mathrm{m}}^3$). Now we can substitute this equation into the equation for the required heat transfer rate: $Q={\rho }_{milk}\times Q_{v,milk}\times C_{p,milk}\times (\mathrm{\Delta }T{)}_{milk}$

Key concepts

Convective Heat Transfer Coefficient

In any heat exchanger system, understanding the convective heat transfer coefficient is crucial. It's a measure of a heat exchanger's ability to transfer heat between a fluid and a solid surface. Higher coefficients signify more efficient heat transfer, which means a heat exchanger can achieve the desired temperature change more quickly.

The coefficient is influenced by several factors, such as the type of fluid, its velocity, temperature, viscosity, and the nature of the surface it's in contact with. In the given exercise, the convective heat transfer coefficients are provided for both the milk and water sides, enabling us to calculate the amount of heat that can be exchanged based on the milk’s flow characteristics and tube dimensions. When designing a shell and tube heat exchanger, achieving a balance between size, cost, and efficiency often involves optimizing these coefficients.

Pasteurization Temperature

Pasteurization is a process used to kill harmful bacteria and pathogens in food products, such as milk, without altering their taste significantly. It involves heating the product to a specific temperature, known as the pasteurization temperature, for a given period.

In our exercise, milk needs to reach at least 70°C to be pasteurized effectively. This target temperature ensures that any potentially harmful microorganisms are eliminated. Designing a heat exchanger for pasteurization must account for this requirement, making sure that the milk exits the system at or above the pasteurization temperature. Consequently, design calculations must include the flow rate and the heat transfer required to elevate the temperature of the milk from the inlet to the pasteurization temperature.

Shell and Tube Heat Exchanger

A shell and tube heat exchanger is a type of heat exchanger that is commonly used in industrial applications for heating or cooling fluids. It consists of a series of tubes (the tube bundle) through which one fluid flows, while another fluid flows over the tubes within a shell to facilitate the heat exchange.

The exercise describes a specific shell and tube exchanger where milk flows through the tubes, and hot water flows around them in the shell. The design engineer must ensure that the size of the shell—including parameters like the shell width—is sufficient for the hot water to transfer enough heat to the milk to reach the necessary pasteurization temperature of 70°C. The length of the tube in each pass affects the residence time of fluid and is critical to achieving the required heat transfer, considering also that the hot water must retain a higher exit temperature than the milk. Such considerations are pivotal in tailoring the dimensions and flow characteristics of the exchanger to meet the pasteurization needs.