**Parallel Flow Heat Exchange**

In a parallel flow heat exchanger, the hot fluid and the cold fluid flow in the same direction. This type of heat exchanger is commonly used in various industrial applications, including cooling systems for engines.

**Given Parameters:**

- Initial temperature of the engine oil (hot fluid), Th1 = 150°C
- Final temperature of the engine oil (hot fluid), Th2 = 80°C
- Initial temperature of the water (cold fluid), Tc1 = 25°C
- Final temperature of the water (cold fluid), Tc2 = 60°C

**Calculating the Temperature Difference**

To determine the effectiveness of the heat exchanger, we need to calculate the temperature difference between the hot and cold fluids.

ΔT1 = Th1 - Tc1 = 150°C - 25°C = 125°C
ΔT2 = Th2 - Tc2 = 80°C - 60°C = 20°C

**Calculating the Logarithmic Mean Temperature Difference (LMTD)**

The logarithmic mean temperature difference (LMTD) is an important parameter in heat exchanger analysis. It is calculated using the equation:

LMTD = ((ΔT1 - ΔT2) / ln(ΔT1 / ΔT2))

In this case, the LMTD can be calculated as follows:

LMTD = ((125°C - 20°C) / ln(125°C / 20°C)) = 84.64°C

**Calculating the Effectiveness**

The effectiveness (ε) of a heat exchanger is defined as the ratio of the actual heat transfer to the maximum possible heat transfer between the hot and cold fluids.

ε = (Q / Qmax)

To calculate the effectiveness, we need to determine the maximum possible heat transfer (Qmax) and the actual heat transfer (Q).

**Calculating Qmax**

Qmax can be calculated using the equation:

Qmax = mc × Cp × (Th1 - Tc1)

where mc is the mass flow rate of the cold fluid and Cp is its specific heat capacity.

**Calculating Q**

Q can be calculated using the equation:

Q = mh × Cp × (Th1 - Th2)

where mh is the mass flow rate of the hot fluid and Cp is its specific heat capacity.

**Substituting the Given Values**

Given that the specific heat capacity of water (Cp) is approximately 4.18 kJ/kg°C and the specific heat capacity of engine oil is unknown, we need to assume a value for Cp.

Let's assume the specific heat capacity of engine oil is 2.0 kJ/kg°C.

Substituting the values into the equations, we can calculate Qmax and Q.

Qmax = mc × Cp × (Th1 - Tc1)
Q = mh × Cp × (Th1 - Th2)

**Calculating the Effectiveness**

Finally, we can calculate the effectiveness using the equation:

ε = (Q / Qmax)

By substituting the values of Q and Qmax, we can determine the effectiveness of the heat exchanger.

Remember to always double-check your calculations and assumptions to ensure accurate results.

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