Final answer:
The heat transfer rate in a counter-flow heat exchanger cannot be precisely determined without additional information on the exit temperatures of the water streams. Calculating the rate requires the overall heat transfer coefficient, surface area, mass flow rates, specific heat capacities, and the log mean temperature difference (LMTD), which is dependent on the unknown exit temperatures.
Step-by-step explanation:
To calculate the heat transfer rate for the counter-flow heat exchanger, we use the formula Q = m * c * ΔT, where m is the mass flow rate, c is the specific heat capacity, and ΔT is the temperature change. However, since this is a heat exchanger, we have to consider the energy balance across the system, which depends on the properties of both the hot and cold water streams, their respective mass flow rates, and the overall heat transfer coefficient (U).
The temperature change for the hot water (ΔTh) is the difference between its entry temperature (Th1 = 100°C) and its exit temperature (Th2), which can be presumed to be closer to the temperature of the incoming cold water. Similarly, the temperature change for the cold water (ΔTc) is the difference between its exit temperature (Tc2) and its entry temperature (Tc1 = 20°C).
Given the data, hot water enters at 100°C and the cold water enters at 20°C. To determine the heat transfer rate (Q), we need to use the overall heat transfer coefficient (U = 1000 W/m²·K) and the surface area (A = 27 m²) as well in the equation Q = U * A * ΔTm, where ΔTm is the log mean temperature difference (LMTD) for the heat exchanger.
The formula for LMTD in a counter-flow heat exchanger is given by:
ΔTm = ( (Th1 - Tc2) - (Th2 - Tc1) ) / ln( (Th1 - Tc2) / (Th2 - Tc1) )
But without the exit temperatures, we can't directly calculate the LMTD. Therefore, we cannot provide the exact heat transfer rate without more information.