A counter-flow heat exchanger is used to cool oil \(\left(c_{p}=\right.\) \(2.20 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K})\) from \(110^{\circ} \mathrm{C}\) to \(85^{\circ} \mathrm{C}\) at a rate of \(0.75 \mathrm{~kg} / \mathrm{s}\) by cold water \(\left(c_{p}=4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) that enters the heat exchanger at \(20^{\circ} \mathrm{C}\) at a rate of \(0.6 \mathrm{~kg} / \mathrm{s}\). If the overall heat transfer coefficient is \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the heat transfer area of the heat exchanger is (a) \(0.745 \mathrm{~m}^{2}\) (b) \(0.760 \mathrm{~m}^{2}\) (c) \(0.775 \mathrm{~m}^{2}\) (d) \(0.790 \mathrm{~m}^{2}\) (e) \(0.805 \mathrm{~m}^{2}\)

In the realm of heat exchangers, the heat transfer rate is a fundamental concept that gauges the amount of heat transferred per unit of time. It is commonly denoted by the symbol 'q'. To calculate this rate, we utilize the formula:
\(q = m \cdot c_p \cdot \Delta T\),
where 'm' represents the mass flow rate, 'c_p' is the specific heat capacity of the fluid, and \(\Delta T\) is the difference in temperature between the inlet and outlet of the fluid in the heat exchanger. It's crucial to remember that in systems like the one in our exercise, we must ensure energy balance, meaning the heat lost by the hot fluid (oil, in our case) should be equal to the heat gained by the cold fluid (water here), barring any system losses.
Understanding the heat transfer rate not only helps in designing the heat exchanger but also in process control, ensuring efficient energy usage and the proper functioning of thermal systems.

The Logarithmic Mean Temperature Difference, or LMTD, is a vital stepping stone in quantifying the driving force for heat exchange in a heat exchanger. Particularly for scenarios where the temperature difference between the hot and cold fluids changes along the length of the exchanger, as in counterflow or parallel flow systems, the LMTD provides an average temperature difference that accurately reflects the gradients involved. The LMTD is calculated using the formula:
\[LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln (\Delta T_1/\Delta T_2)}\],
where \(\Delta T_1\) is the temperature difference at one end of the heat exchanger and \(\Delta T_2\) is the temperature difference at the other end. Employing LMTD is particularly helpful when determining the size of a heat exchanger, as it directly relates to the heat transfer area required for a given heat transfer rate.

The overall heat transfer coefficient, symbolized by 'U', plays a crucial role in heat exchanger design. It encapsulates the total resistance to heat transfer, which includes resistances due to conduction within the materials and convection on both the fluid sides. Mathematically, 'U' is defined by:
\[U = \frac{1}{\frac{1}{h_1} + \frac{d}{k} + \frac{1}{h_2}}\],
where 'h' represents the individual convection heat transfer coefficients and 'k' is the thermal conductivity of the wall material, and 'd' is the thickness of the wall. This coefficient is measured in units of \(W/(m^2\cdot K)\) and indicates how well a heat exchanger can transfer heat; the higher the 'U', the more efficient the heat exchanger. This parameter is essential when calculating the required heat transfer area for a specific heat transfer rate.

Specific heat capacity, often referred to simply as 'specific heat', is denoted by \(c_p\) and is one of the most pivotal properties in thermodynamics. Specific heat is the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). Expressed in units of \(kJ/(kg\cdot K)\), this intrinsic characteristic depends on the material substance and its phase. For instance, in our exercise, the specific heat capacity plays a significant role in determining how much heat the oil and water can carry. Essentially, it impacts the heat transfer rate directly: substances with a high specific heat can absorb or release large quantities of heat with minimal temperature changes, thus influencing the design and operation of heat exchangers.