Counter flow heat exchanger

A counter flow heat exchanger is a type of heat exchanger in which the hot and cold fluids flow in opposite directions. This design allows for efficient heat transfer between the two fluids.

Specific heat and mass flow rate

The specific heat of a fluid is the amount of heat energy required to raise the temperature of a unit mass of the fluid by one degree Celsius. It is denoted by the symbol "C" and has units of J/(kg·°C). The mass flow rate is the amount of mass flowing through the heat exchanger per unit time and is denoted by the symbol "m_dot". It has units of kg/s.

Equation for effectiveness

The effectiveness of a heat exchanger is a measure of how well it transfers heat between the hot and cold fluids. It is denoted by the symbol "ε" and is defined as the ratio of the actual heat transfer to the maximum possible heat transfer.

The equation for effectiveness is given by:

ε = (Q_actual / Q_max)

Where:
Q_actual is the actual heat transfer between the hot and cold fluids
Q_max is the maximum possible heat transfer if the hot and cold fluids were completely mixed

NTU (Number of Transfer Units)

NTU is a dimensionless parameter used to characterize heat exchangers. It is defined as the product of the overall heat transfer coefficient (U), the heat transfer area (A), and the inverse of the minimum specific heat capacity rate (C_min) between the hot and cold fluids.

NTU = (U * A) / C_min

Relationship between NTU and effectiveness

The relationship between NTU and effectiveness for a counter flow heat exchanger is given by the following equation:

ε = 1 - exp(-NTU * (1 - C_min / C_max))

Where:
C_max is the maximum specific heat capacity rate between the hot and cold fluids

Given NTU = 0.5

In the given question, NTU is equal to 0.5. Plugging this value into the equation for effectiveness, we have:

ε = 1 - exp(-0.5 * (1 - C_min / C_max))

Since the product of specific heat and mass flow rate is the same for the hot and cold fluids, we can assume that C_min = C_max. Therefore, the equation simplifies to:

ε = 1 - exp(-0.5 * (1 - 1))

Simplifying further, we have:

ε = 1 - exp(0)

The exponential of 0 is equal to 1, so the equation simplifies to:

ε = 1 - 1 = 0

Therefore, the effectiveness of the heat exchanger is 0, which means that it is not able to transfer any heat between the hot and cold fluids.