Consider a counter-flow heat exchanger with the inlet temperatures of two fluids (1 and 2) being T₁, in = 300 K and T₂, _in = 350 K. The heat capacity rates of the two fluids are C₁= 1000 W/K and C₂ = 400 W/K, and the effectiveness of the heat exchanger is 0.5. The actual heat transfer rate is _____ kW. (Answer in integer)

The correct answer is 10.

Based on the heat exchanger diagram and given data:
- Inlet temperature of fluid 1 (cold fluid), $T_{1,\text{in}}=300\text{K}$ (indicated in the diagram as T_c1 = 300K)
- Inlet temperature of fluid 2 (hot fluid), $T_{2,\text{in}}=350\text{K}$ (indicated in the diagram as T_h1 = 350K)
- Heat capacity rate of fluid 1, $C_1=1000\text{W/K}$
- Heat capacity rate of fluid 2, $C_2=400\text{W/K}$
- Effectiveness of the heat exchanger, $\epsilon =0.5$

To calculate the actual heat transfer rate, we first determine the minimum heat capacity rate, $C_{\text{min}}$:
$C_{\text{min}}=\text{min}(C_1,C_2)=\text{min}(1000,400)=400\text{W/K}$

The maximum possible heat transfer rate ($q_{\text{max}}$) occurs when the temperature difference is at its maximum (the difference between the hot and cold fluid inlet temperatures) and is governed by the minimum heat capacity rate:
$q_{\text{max}}=C_{\text{min}}(T_{2,\text{in}}-T_{1,\text{in}})$

Substituting the values into the formula:
$q_{\text{max}}=400\times (350-300)=400\times 50=20000\text{W}=20\text{kW}$

The effectiveness ($\epsilon$) is defined as the ratio of the actual heat transfer rate ($q_{\text{actual}}$) to the maximum possible heat transfer rate:
$\epsilon =\frac{q_{\text{actual}}}{q_{\text{max}}}$

Rearranging the formula to solve for the actual heat transfer rate:
$q_{\text{actual}}=\epsilon \times q_{\text{max}}$
$q_{\text{actual}}=0.5\times 20\text{kW}=10\text{kW}$