Question

Under what conditions is the heat transfer relation $$ \dot{Q}=\dot{m}_{c} c_{p c}\left(T_{c, \text { out }}-T_{c, \text { in }}\right)=\dot{m}_{h} c_{p h}\left(T_{h, \text { in }}-T_{h, \text { out }}\right) $$ valid for a heat exchanger?

Short answer

Answer: The heat transfer equation for a heat exchanger is valid under the conditions of steady-state operation, no heat losses to the surroundings, and satisfaction of the energy conservation principle.

Step-by-step solution

Describe the variables and terms

In the given equation, we have the following variables and terms: - \(\dot{Q}\): Heat transfer rate (W) - \(\dot{m}_c\): Mass flow rate of the cold fluid (kg/s) - \(c_{p c}\): Specific heat capacity of the cold fluid at constant pressure (J/kg K) - \(T_{c, \text { in }}\): Inlet temperature of the cold fluid (K) - \(T_{c, \text { out }}\): Outlet temperature of the cold fluid (K) - \(\dot{m}_h\): Mass flow rate of the hot fluid (kg/s) - \(c_{p h}\): Specific heat capacity of the hot fluid at constant pressure (J/kg K) - \(T_{h, \text { in }}\): Inlet temperature of the hot fluid (K) - \(T_{h, \text { out }}\): Outlet temperature of the hot fluid (K)

Identify energy balance and conservation laws

The given equation represents an energy balance in a heat exchanger. Since no work is involved in a heat exchanger process, the energy balance can be expressed as the heat transfer from the hot fluid equals the heat transfer to the cold fluid, as shown in the equation. The equation also has some implicit assumptions, such as steady-state operation (no time-varying process) and no heat losses to the surroundings. The energy conservation principle dictates that the energy input must equal the energy output. In the case of a heat exchanger, this means that the heat transfer from the hot fluid must equal the heat transfer to the cold fluid. $$ \dot{Q}_{\text{hot}} = \dot{Q}_{\text{cold}} $$

Deduce the necessary conditions

Now, we can deduce the conditions under which the given equation for heat transfer in a heat exchanger is valid: 1. Steady-state operation: The heat exchanger must be operating at a steady state, without any time-varying processes that would affect the heat transfer rate or temperatures. 2. No heat losses: There must be no heat loss to the surroundings. In other words, all the heat transfer from the hot fluid must be transferred to the cold fluid. 3. Energy conservation: The energy conservation principle, which states that the heat transfer from the hot fluid must equal the heat transfer to the cold fluid, must be satisfied. In summary, the given heat transfer equation for a heat exchanger is valid under the conditions of steady-state operation, no heat losses to the surroundings, and satisfaction of the energy conservation principle.

Key concepts

Steady-State Heat Transfer

Understanding the concept of steady-state heat transfer is essential when examining heat exchangers, as it is one of the primary conditions for the heat transfer relation \(\dot{Q}=\dot{m}_{c} c_{p c}\left(T_{c, \text { out }}-T_{c, \text { in }}\right)=\dot{m}_{h} c_{p h}\left(T_{h, \text { in }}-T_{h, \text { out }}\right)\) to be valid.

In a steady-state scenario, the temperatures, flow rates, and heat transfer rates remain constant over time. This means that there are no fluctuations in the heat exchange process, and the system has reached an equilibrium where the input and output are unchanging.

The importance of this condition cannot be overstated. Without it, calculations become complex, as time variations would have to be accounted for. For educational content that teaches this concept, visual aids such as diagrams showing a constant flow and temperature profile over time could be beneficial for students to visualize and comprehend steady-state operation.

Energy Balance

At the heart of every heat exchanger operation is the principle of energy balance. This concept is pivotal in ensuring the conservation of energy, a fundamental law of physics that dictates that energy cannot be created or destroyed within an isolated system.

When dealing with heat exchangers, the notion of energy balance is simplified to the idea that the amount of heat leaving one fluid must be equal to the amount of heat being absorbed by another fluid. Mathematically, this can be expressed as \(\dot{Q}_{\text{hot}} = \dot{Q}_{\text{cold}}\), where \(\dot{Q}_{\text{hot}}\) is the heat being lost by the hot fluid and \(\dot{Q}_{\text{cold}}\) is the heat being gained by the cold fluid.

In educational materials, it's advantageous to use real-world examples to illustrate energy balance, such as comparing it to a simple financial transaction where the amount given out is equal to the amount received. Such analogies make the concept more accessible and relatable to students.

Specific Heat Capacity

The specific heat capacity, denoted by \(c_p\), is a substance's heat capacity per unit mass. It represents the amount of energy required to raise the temperature of one kilogram of the substance by one Kelvin.

In the context of a heat exchanger, specific heat capacities of the fluids involved (hot and cold) are central to calculating the amount of heat transfer. The heat capacity dictates how much heat is needed to change the fluid's temperature, which directly affects the heat exchanger's efficiency. High specific heat capacities mean that the fluid can absorb or release a significant amount of heat without a large temperature change.

Educational content on this topic could benefit from incorporating interactive elements where students can simulate the effects of varying the specific heat capacities on the heat exchanger's performance. This hands-on approach encourages a deeper understanding of the role that specific heat capacity plays in thermal systems.