Question

The radiator in an automobile is a cross-flow heat exchanger \(\left(U A_{s}=10 \mathrm{~kW} / \mathrm{K}\right)\) that uses air \(\left(c_{p}=1.00 \mathrm{~kJ} /\right.\) \(\mathrm{kg} \cdot \mathrm{K})\) to \(\mathrm{cool}\) the engine coolant fluid \(\left(c_{p}=4.00 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\). The engine fan draws \(30^{\circ} \mathrm{C}\) air through this radiator at a rate of \(10 \mathrm{~kg} / \mathrm{s}\) while the coolant pump circulates the engine coolant at a rate of \(5 \mathrm{~kg} / \mathrm{s}\). The coolant enters this radiator at \(80^{\circ} \mathrm{C}\). Under these conditions, what is the number of transfer units (NTU) of this radiator? (a) 1 (b) 2 (c) 3 (d) 4 (e) 5

Short answer

Answer: (a) 1

Step-by-step solution

Calculate the heat capacity for both streams

First, we'll calculate the heat capacity for both the hot (engine coolant) and cold (air) streams by multiplying the specific heat capacity (cp) with the mass flow rate (m_dot). For hot stream (engine coolant): \(C_h = m_{h} \times c_{p_h} = 5 \frac{kg}{s} \times 4 \frac{kW}{kg \cdot K} = 20 \frac{kW}{K}\) For the cold stream (air): \(C_c = m_{c} \times c_{p_c} = 10 \frac{kg}{s} \times 1 \frac{kW}{kg \cdot K} = 10 \frac{kW}{K}\)

Calculate the temperature difference

Next, we'll calculate the difference in temperature between the coolant at the entry of the radiator and the temperature of the incoming air: \(\Delta T = T_{h, in} - T_{c, in} = 80 ^\circ C - 30^\circ C = 50 K\)

Determine the minimum heat capacity rate (Cmin)

We need to find which stream has the lower heat capacity product to use it in the NTU formula: \(C_{min} = min(C_h, C_c) = min(20 \frac{kW}{K}, 10 \frac{kW}{K}) = 10 \frac{kW}{K}\)

Calculate the NTU

Now we can find the NTU using the overall heat transfer coefficient and surface area (UA), and the minimum heat capacity rate (Cmin). The formula for the NTU is: \(NTU = \frac{UA}{C_{min}}\) Substitute the values: \(NTU = \frac{10 \frac{kW}{K}}{10 \frac{kW}{K}} = 1\) So the answer is (a) 1.

Key concepts

Cross-Flow Heat Exchanger

A cross-flow heat exchanger is a system where two fluids at different temperatures pass perpendicularly to each other. Imagine a radiator in a car: hot coolant from the engine flows in one direction while cooler air is drawn through the radiator fins in a perpendicular direction. This arrangement allows for efficient heat transfer as the cooler air absorbs heat from the hot coolant.

In our exercise scenario, the automobile radiator effectively dissipates heat from the engine coolant, using the air as the secondary fluid. This type of heat exchanger is often preferred in cases where the two fluids are at vastly different temperatures and flow rates, as it allows for a compact design and effective heat transfer despite the mismatch.

Understanding the dynamics of cross-flow heat exchangers is crucial for optimizing their performance and ensuring efficient cooling, particularly in automotive applications. The effectiveness of such heat exchangers is often evaluated using the Number of Transfer Units (NTU) method, which is a dimensionless number representing the heat exchanger's capacity to alter a fluid's temperature.

Heat Capacity Rate

The heat capacity rate is a key concept in thermal engineering that represents the ability of a fluid to absorb or release heat. It is calculated by multiplying the fluid's mass flow rate by its specific heat capacity, a measure of how much energy is needed to raise the temperature of one kilogram of the substance by one degree Kelvin.

In the exercise, calculations for the heat capacity rate of both the air (cold stream) and the engine coolant (hot stream) were necessary to compare their respective abilities to retain or transfer heat. The lower heat capacity rate between the two, often denoted as \(C_{min}\), is crucial as it limits the maximum possible heat transfer in a heat exchanger. This value is then used in determining the NTU, which gives us insight into how efficiently the heat exchanger is operating.

The simplicity of this calculation makes it an essential step in solving various heat transfer problems and optimizing thermal systems, providing a straightforward way to gauge a fluid’s role in heat exchange processes.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient, often signified by the symbol \(U\), is an indicator of a heat exchanger's ability to conduct heat between two fluids separated by a solid barrier. It encompasses all the resistance to heat flow, including the conductive resistance of the exchanger material and the convective resistances on both the hot and cold sides.

In the given problem, the radiator’s overall heat transfer coefficient and its surface area are bundled into a single term \(UA\), which represents the product of the coefficient \(U\) and the heat exchanger's surface area \(A_s\). The larger the \(UA\) value, the more efficient the heat exchanger is at transferring heat.

The exercise demonstrates the use of \(UA\) in calculating the NTU, as it directly influences the heat exchanger’s effectiveness. Understanding how the overall heat transfer coefficient interacts with other elements, such as the heat capacity rates of the fluids involved, is critical for designing and evaluating the performance of heat exchangers across a variety of applications.