Question

The radiator in an automobile is a crossflow heat exchanger $\left(U A_{s}=10 \mathrm{~kW} / \mathrm{K}\right)\( that uses air \)\left(c_{p}=1.00 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$ to 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 \(12 \mathrm{~kg} / \mathrm{s}\) 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) \(2.0\) (b) \(2.5\) (c) \(3.0\) (d) \(3.5\) (e) \(4.0\)

Short answer

Answer: To find the NTU value, first follow the steps to determine the capacity rates of air and coolant, calculate the heat transfer effectiveness, and calculate the capacity rate ratio. Then, use the formula for a crossflow heat exchanger to find the NTU. The final calculated NTU value will be the answer.

Step-by-step solution

Determine capacity rates of air and coolant

The capacity rates C for both air and coolant can be determined by multiplying their respective mass flow rates with the specific heats: \(C_{air} = \dot{m}_{air} \times c_{p,air}\) \(C_{coolant} = \dot{m}_{coolant} \times c_{p,coolant}\)

Calculate the heat transfer effectiveness

By using the heat transfer equation \(Q = UA\Delta T_{lm}\), where \(\Delta T_{lm}\) is the log mean temperature difference, and setting equal to the product of capacity rates \(Q = C_{min}(T_{t, coolant} - T_{i, air})\). Divide through by \(C_{min}\Delta T_{lm}\), we get: \(\epsilon = \frac{Q}{C_{min}\Delta T_{lm}} = \frac{UA}{C_{min}}\)

Calculate the capacity rate ratio

The capacity rate ratio can be found by dividing the minimum capacity rate by the maximum capacity rate: \(C_r = \frac{C_{min}}{C_{max}}\)

Calculate the number of transfer units (NTU)

To find the NTU, we can use the following formula for a crossflow heat exchanger: \(\text{NTU} = \frac{-ln(1-\epsilon(1+C_r))}{1+C_r}\) Plug in the values found in steps 2 and 3 and calculate the NTU value. The calculated NTU value will be closest to one of the options, which will be our answer.