Question

Under what conditions will the temperature rise of the cold fluid in a heat exchanger be equal to the temperature drop of the hot fluid?

Short answer

Answer: The temperature rise of the cold fluid is equal to the temperature drop of the hot fluid when the product of the mass flow rate and specific heat capacity of the cold fluid is equal to the product of the mass flow rate and specific heat capacity of the hot fluid, i.e., (m_cold * c_cold) / (m_hot * c_hot) = 1.

Step-by-step solution

Understand the heat exchange process

In a heat exchanger, the hot fluid transfers heat to the cold fluid, which causes the temperature of the hot fluid to drop and the temperature of the cold fluid to rise. The amount of heat transferred between the two fluids can be calculated using the formula: Q = mcΔT where Q is the heat transferred, m is the mass flow rate of the fluid (kg/s), c is the specific heat capacity of the fluid (J/kg*K), and ΔT is the change in temperature (K).

Set the heat gained by the cold fluid equal to the heat lost by the hot fluid

In order for the temperature rise of the cold fluid to be equal to the temperature drop of the hot fluid, the heat gained by the cold fluid (Q_cold) must be equal to the heat lost by the hot fluid (Q_hot). We can set up an equation to represent this condition: Q_cold = Q_hot Using the heat transfer formula for each fluid, we can write: m_cold * c_cold * ΔT_cold = m_hot * c_hot * ΔT_hot

Solve for the required conditions

Our goal is to find the conditions under which ΔT_cold = ΔT_hot. In other words, we want the change in temperature of the cold fluid to be equal to the change in temperature of the hot fluid. To accomplish this, let's rearrange our equation from step 2 and divide both sides by ΔT_cold: m_cold * c_cold = m_hot * c_hot * (ΔT_hot / ΔT_cold) Since we want ΔT_cold = ΔT_hot, we can replace (ΔT_hot / ΔT_cold) with 1: m_cold * c_cold = m_hot * c_hot

Determine the final condition

The condition under which the temperature rise of the cold fluid in a heat exchanger is equal to the temperature drop of the hot fluid is: m_cold * c_cold = m_hot * c_hot or equivalently, (m_cold * c_cold) / (m_hot * c_hot) = 1 This means that the product of the mass flow rate and specific heat capacity of the cold fluid must be equal to the product of the mass flow rate and specific heat capacity of the hot fluid.