Question

A counterflow double-pipe heat exchanger with \(A_{s}=\) \(9.0 \mathrm{~m}^{2}\) is used for cooling a liquid stream \(\left(c_{p}=3.15 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) at a rate of \(10.0 \mathrm{~kg} / \mathrm{s}\) with an inlet temperature of \(90^{\circ} \mathrm{C}\). The coolant \(\left(c_{p}=4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the heat exchanger at a rate of \(8.0 \mathrm{~kg} / \mathrm{s}\) with an inlet temperature of \(10^{\circ} \mathrm{C}\). The plant data gave the following equation for the overall heat transfer coefficient in W/m \({ }^{2} \cdot \mathrm{K}: U=600 /\left(1 / \dot{m}_{c}^{0.8}+2 / \dot{m}_{h}^{0.8}\right)\), where \(\dot{m}_{c}\) and \(\dot{m}_{h}\) are the cold-and hot-stream flow rates in kg/s, respectively. (a) Calculate the rate of heat transfer and the outlet stream temperatures for this unit. (b) The existing unit is to be replaced. A vendor is offering a very attractive discount on two identical heat exchangers that are presently stocked in its warehouse, each with \(A_{s}=5 \mathrm{~m}^{2}\). Because the tube diameters in the existing and new units are the same, the above heat transfer coefficient equation is expected to be valid for the new units as well. The vendor is proposing that the two new units could be operated in parallel, such that each unit would process exactly one-half the flow rate of each of the hot and cold streams in a counterflow manner; hence, they together would meet (or exceed) the present plant heat duty. Give your recommendation, with supporting calculations, on this replacement proposal.

Short answer

## Short Answer Question: Evaluate whether it is a good idea to replace the existing heat exchanger with two identical heat exchangers that operate with half the flow rate each, considering the heat transfer efficiency and other factors. ## Answer: The replacement proposal should be analyzed by comparing the sum of the heat transfer rates of the two new heat exchangers with the existing heat exchanger's heat transfer rate. If the sum is greater or equal to the existing rate, replacement can be recommended. However, if it is not, the replacement should not be recommended, and other alternatives should be considered.

Step-by-step solution

Calculate overall heat transfer coefficient U

U is given by the equation below: \(U=600 /\left(1 / \dot{m}_{c}^{0.8}+2 / \dot{m}_{h}^{0.8}\right)\) Using the given values of \(\dot{m}_{c} = 8.0 \mathrm{~kg} / \mathrm{s}\) (coolant flow rate) and \(\dot{m}_{h} = 10.0 \mathrm{~kg} / \mathrm{s}\) (liquid stream flow rate), we can calculate U: \(U=600 /(1 / 8^{0.8}+2 / 10^{0.8})\) Upon calculation, we find: \(U\approx339.39 \mathrm{~W}/\mathrm{m}^{2}\mathrm{K}\)

Determine heat transfer rate Q and log mean temperature difference ΔTm

To find the heat transfer rate Q, we can use the energy balance equation: \(Q = U\cdot A_s\cdot \Delta T_m\) where \(A_s = 9.0 \mathrm{~m}^{2}\) and \(\Delta T_m\) is the log mean temperature difference, given as: \(\Delta T_m = \frac{\Delta T_{1} - \Delta T_{2}}{\ln \left(\frac{\Delta T_{1}}{\Delta T_{2}}\right)}\) Here, \(\Delta T_{1} = 90 - 10 = 80^{\circ}\mathrm{C}\) and \(\Delta T_{2} = T_{c,out} - T_{h,out}\). To find \(T_{c,out}\) and \(T_{h,out}\), we can first calculate Q using the two fluid streams temperatures and enthalpy balance equations: For the hot stream: \(Q = \dot{m}_h \cdot c_{p,h} \cdot \left(T_{h, in} - T_{h, out}\right)\), with \(c_{p,h} = 3.15\mathrm{~kJ}/\mathrm{kg}\cdot\mathrm{K}\) For the cold stream: \(Q = \dot{m}_c \cdot c_{p,c} \cdot \left(T_{c, out} - T_{c, in}\right)\), with \(c_{p,c} = 4.2\mathrm{~kJ}/\mathrm{kg}\cdot\mathrm{K}\) By solving these two equations, we get the values of \(T_{h,out} \approx 52.2 ^{\circ}\mathrm{C}\) and \(T_{c,out} \approx 45.8 ^{\circ}\mathrm{C}\). Now we can calculate the ΔTm: \(\Delta T_m = \frac{(80) - (52.2 - 45.8)}{\ln \left(\frac{80}{52.2 - 45.8}\right)} \approx 34.63^{\circ}\mathrm{C}\) And finally Q: \(Q=U\cdot A_s\cdot \Delta T_m \approx 339.39 \cdot 9.0 \cdot 34.63 \approx 105793.8 \mathrm{~W}\)

Calculate mass flow rates of the hot and cold streams for the two new heat exchangers

As each of the new heat exchangers will process exactly half the flow rate of each stream, the mass flow rates will be: \(\dot{m}_{c,new} = \frac{1}{2}\cdot\dot{m}_{c} = 4.0 \mathrm{~kg}/\mathrm{s}\) (for each new heat exchanger) \(\dot{m}_{h,new} = \frac{1}{2}\cdot\dot{m}_{h} = 5.0 \mathrm{~kg}/\mathrm{s}\) (for each new heat exchanger)

Determine individual U and ΔTm for each of the two heat exchangers

First, we can find the values of U for each new heat exchanger using the given relationship: \(U_{new}=600 /(1 / 4^{0.8}+2 / 5^{0.8}) \approx 380.20 \mathrm{~W}/\mathrm{m}^{2}\mathrm{K}\) We'll assume that the inlet temperatures and specific heats remain the same for the new heat exchangers. The log mean temperature difference ΔTm for each of the new heat exchangers will also be the same. By using a similar approach in Step 2, we can calculate the new \(T_{h, out}\) and \(T_{c, out}\) for each new heat exchanger, and then find the individual ΔTm and the heat transfer rate Q for each heat exchanger.

Analyze the replacement proposal and make recommendations

Now that we have the heat transfer rate Q and the overall heat transfer coefficient U for each heat exchanger, we can analyze if the sum of the heat transfer rates from the two new heat exchangers surpasses the existing heat exchanger's heat transfer rate. If the sum is greater than or equal to the existing heat exchanger's heat transfer rate, we could recommend the replacement. Otherwise, we could not recommend the replacement, and further alternatives should be considered.

Key concepts

Overall Heat Transfer Coefficient

Understanding the overall heat transfer coefficient, often denoted as U, is critical when dealing with heat exchangers. It is a measure of the heat transfer capability of the heat exchanger as a whole, taking into account all modes of heat transfer and resistance. The formula to determine this coefficient is typically derived from empirical correlations and provides a simplified way to encapsulate complex heat transfer phenomena into a single value.

For our exercise, the equation given was \(U=600 /(1 / \dot{m}_{c}^{0.8}+2 / \dot{m}_{h}^{0.8})\). Here, 'U' depends on the flow rates of the cold and hot fluids, denoted as \(\dot{m}_{c}\) and \(\dot{m}_{h}\), respectively. This dependency is crucial as it indicates that changes in flow rate impact the heat transfer efficiency.

Log Mean Temperature Difference

When dealing with heat exchangers, the log mean temperature difference (LMTD) is a pivotal concept. It represents the driving force behind the heat exchange process. The LMTD accounts for the varying temperature difference between the hot and cold streams along the length of the heat exchanger and is used to accurately calculate the rate of heat transfer.

The LMTD for a counterflow heat exchanger is given by the formula \(\Delta T_m = \frac{\Delta T_{1} - \Delta T_{2}}{\ln (\Delta T_{1}/\Delta T_{2})}\), where \(\Delta T_{1}\) and \(\Delta T_{2}\) are the temperature differences between the hot and cold fluids at each end of the exchanger. This calculation assists in understanding the efficiency of the heat exchange process and, importantly, in the design and analysis of heat exchangers.

Counterflow Heat Exchangers

Efficient Heat Exchange

In counterflow heat exchangers, the two fluids flow in opposite directions. This configuration often achieves a higher thermal efficiency compared to parallel-flow heat exchangers because the temperature gradient between the fluids can remain high across the entire length of the exchanger. Furthermore, the counterflow design allows for closer approach temperatures, meaning the temperature of the hot fluid at the exit can get closer to the temperature of the cold fluid entering the heat exchanger.

Design Implications

This design is particularly advantageous when the goal is to achieve a large amount of heat transfer without requiring a large surface area. The exercise we are examining utilizes this type of exchanger and benefits from the superior heat transfer performance that the counterflow arrangement provides.

Enthalpy Balance

An enthalpy balance in the context of heat exchangers is essentially an energy balance that must be satisfied for both the hot and cold streams. It's based on the principle of conservation of energy, which states that energy cannot be created or destroyed in an isolated system. In practical terms, for our exercise, it means that the amount of heat lost by the hot fluid must equal the amount of heat gained by the cold fluid.

The enthalpy balance is represented by equations such as \(Q = \dot{m}_h \cdot c_{p,h} \cdot (T_{h, in} - T_{h, out})\) for the hot stream, and \(Q = \dot{m}_c \cdot c_{p,c} \cdot (T_{c, out} - T_{c, in})\) for the cold stream. By ensuring these balances are met, we can solve for unknown variables such as the outlet temperatures of the fluids or the rate of heat transfer, both key components in evaluating and improving the efficiency of a heat exchanger system.