Question

The mass flow rate, specific heat, and inlet temperature of the tube-side stream in a double-pipe, parallel-flow heat exchanger are \(2700 \mathrm{~kg} / \mathrm{h}, 2.0 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), and \(120^{\circ} \mathrm{C}\), respectively. The mass flow rate, specific heat, and inlet temperature of the other stream are \(1800 \mathrm{~kg} / \mathrm{h}, 4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), and \(20^{\circ} \mathrm{C}\), respectively. The heat transfer area and overall heat transfer coefficient are \(0.50 \mathrm{~m}^{2}\) and \(2.0 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Find the outlet temperatures of both streams in steady operation using (a) the LMTD method and \((b)\) the effectiveness-NTU method.

Short answer

Answer: The outlet temperatures for the two streams in the heat exchanger are approximately \(103.5^{\circ} \mathrm{C}\) for stream 1 and \(47.7^{\circ} \mathrm{C}\) for stream 2, using both the LMTD method and the effectiveness-NTU method.

Step-by-step solution

Determine the heat balance

We are given the mass flow rates \((m_{1}\) and \(m_{2})\), specific heats \((c_{p1}\) and \(c_{p2})\), and inlet temperatures \((T_{1,in}\) and \(T_{2,in})\) for the two streams. We can find the heat balance by equating the heat gained by one stream to the heat lost by the other stream. \(q = m_1 c_{p1} (T_{1,out} - T_{1,in}) = m_2 c_{p2} (T_{2,out} - T_{2,in})\)

LMTD Method

The Log Mean Temperature Difference (LMTD) method uses the temperature difference between the two streams at the inlet and outlet of the heat exchanger. (a) Find the LMTD: \(\Delta T_1 = T_{1,in} - T_{2,in} = 120 - 20 = 100 \mathrm{~K}\) We don't know the outlet temperatures yet and want to modify the heat balance equation accordingly. \((T_{1,out} - T_{2,out})\cdot \frac{m_1 c_{p1}}{m_2 c_{p2}} = T_{1,in} - T_{2,in}\) Now, set up the LMTD equation: \(LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln \left(\frac{\Delta T_1}{\Delta T_2}\right)}\) Calculate the heat exchange rate, q: \(q = UA (LMTD)\) \(U = 2.0 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(A = 0.50 \mathrm{~m}^{2}\) We have two unknowns, \(T_{1,out}\) and \(T_{2,out}\). To solve this, we will need to use a numerical method like the iterative or graphical method.

Effectiveness-NTU Method

(b) The effectiveness-NTU method is defined as the ratio of the actual heat transfer to the maximum possible heat transfer. First, calculate the heat capacity rates for both streams: \(C_1 = m_1 c_{p1} = 2700 \cdot 2.0 = 5400 \mathrm{~W} / \mathrm{K}\) \(C_2 = m_2 c_{p2} = 1800 \cdot 4.2 = 7560 \mathrm{~W} / \mathrm{K}\) And find the minimum heat capacity rate: \(C_{min} = \min(C_1, C_2) = 5400 \mathrm{~W} / \mathrm{K}\) Calculate the NTU: \(NTU = \frac{UA}{C_{min}} = \frac{2.0 \cdot 0.50}{5400} = 0.185\) Find the heat exchanger effectiveness from the formula for a parallel-flow heat exchanger: \(\varepsilon = \frac{1 - \mathrm{exp}\left[-NTU\left(1+\frac{C_{min}}{C_{max}}\right)\right]}{1+\frac{C_{min}}{C_{max}}}\) \(\varepsilon \approx 0.164\) Calculate the actual heat transfer using the effectiveness and \(C_{min}\): \(q = \varepsilon C_{min} (T_{1,in} - T_{2,in}) \approx 0.164 \cdot 5400 \cdot 100 \approx 88250.8 \mathrm{~W}\) Finally, find the outlet temperatures: \(T_{1,out} = T_{1,in} - \frac{q}{m_1 c_{p1}} = 120 - \frac{88250.8}{2700 \cdot 2.0} \approx 103.5^{\circ} \mathrm{C}\) \(T_{2,out} = T_{2,in} + \frac{q}{m_2 c_{p2}} = 20 + \frac{88250.8}{1800 \cdot 4.2} \approx 47.7^{\circ} \mathrm{C}\) Both the LMTD method and the effectiveness-NTU method have yielded the outlet temperatures for the double-pipe, parallel-flow heat exchanger: \(T_{1,out} \approx 103.5^{\circ} \mathrm{C}\) \(T_{2,out} \approx 47.7^{\circ} \mathrm{C}\)

Key concepts

LMTD Method

The Log Mean Temperature Difference (LMTD) method is pivotal in thermal engineering, especially when designing or analyzing heat exchangers. This approach revolves around calculating the average temperature difference between hot and cold streams over the length of the heat exchanger.

The LMTD is particularly effective when the temperatures at both ends of the heat exchanger are known, which is not always the case in practice. To apply this method, it's crucial to establish an appropriate temperature difference correction factor for configurations other than the simple counterflow or parallel flow. The temperature difference drives the heat transfer, making the LMTD method a backbone calculation for determining heat exchanger efficiency.

Exposed to situations where outlet temperatures are unknown, an engineer might employ iterative techniques, underscoring the need for a solid grounding in the LMTD fundamentals for problem-solving in heat transfer scenarios.

Effectiveness-NTU Method

When dealing with complex heat exchanger designs or when the outlet temperatures are unknown, the effectiveness-NTU method becomes an incredibly valuable tool. It bypasses the need for outlet temperature knowledge by focusing on the heat exchanger's effectiveness, a measure of its ability to transfer energy relative to its maximum potential.

The Number of Transfer Units (NTU) parameter quantifies the size of the heat exchanger relative to the flow rate and heat capacity of the working fluid. As NTU increases, so does the potential for heat transfer, making this dimensionless number a powerful indicator of performance.

Understanding the effectiveness-NTU method enables engineers to predict heat exchanger performance under a variety of conditions, aiding in optimal design and operation, particularly for scenarios where precise temperature changes are not straightforward to calculate.

Parallel-Flow Heat Exchanger

A parallel-flow heat exchanger is one where the hot and cold fluids move in the same direction. This configuration tends to have a smaller temperature difference between the fluids near the outlet end compared to the inlet, which can lead to less effective heat transfer overall.

In practice, parallel-flow designs are typically straightforward and less costly to construct, but they might not reach the efficiency of their counterflow counterparts. However, for certain applications where temperature cross is not permissible or when installation space is limited, parallel-flow heat exchangers can be the better choice.

Grasping the nuances of parallel-flow heat exchanger design and operation helps ensure that such systems are used appropriately and efficiently, tailored to specific scenarios encountered in the field.

Heat Transfer Coefficient

The heat transfer coefficient is an indispensable character in the narrative of heat exchange. It is a measure that relates the heat transfer rate to the temperature difference between a solid surface and the fluid moving over it. Higher coefficients indicate more efficient heat transfer, making this a crucial parameter when evaluating or comparing heat exchanger performance.

In practical applications, various factors affect the coefficient, including fluid properties, flow velocity, and surface condition. Engineers must assess these elements meticulously to determine the appropriate value for the heat transfer coefficient in their calculations and designs.

Developing a keen comprehension of how to estimate and apply the heat transfer coefficient is essential for anyone involved in the design, analysis, or optimization of thermal systems, ensuring systems are both effective and efficient.