Question

What is the effect of heat gain and heat loss on the entropy of the fluid during Rayleigh flow?

Short answer

Answer: During Rayleigh flow, heat gain causes an increase in entropy due to the increased disorder in the fluid molecules, while heat loss results in a decrease in entropy due to the decreased disorder in the fluid molecules.

Step-by-step solution

Define Rayleigh flow

Rayleigh flow is the flow of a compressible fluid that undergoes changes in pressure, temperature, and volume due to heat transfer, while the mass flow rate and cross-sectional area of the flow passage remain constant.

Establish governing equations

For Rayleigh flow, we will need to consider thermodynamic and energy equations. Key equations include: 1. The continuity equation, which states that the mass flow rate remains constant: \(ρAv= \text{constant}\), where \(ρ\) is density, \(A\) is the cross-sectional area, and \(v\) is the fluid velocity. 2. Conservation of energy, expressed by the energy equation: \(h_1+v_1^2/2 = h_2+v_2^2/2+q\), where \(h\) is the enthalpy, \(v\) is the fluid velocity, and \(q\) is the heat transfer per unit mass. 3. The second law of thermodynamics, to evaluate the entropy change: \(ds = \frac{dh−vdp}{T}\), where \(ds\) is the change in entropy, \(dh\) is the change in enthalpy, \(dp\) is the change in pressure, and \(T\) is the temperature.

Heat gain

When heat is added to the fluid during Rayleigh flow (heat gain), the temperature and enthalpy increase while the fluid expands, causing a decrease in the pressure. Since \(ds > 0\), during heat gain, the entropy increases (\(ds = c_p \ln\frac{T_2}{T_1}-R\ln\frac{P_2}{P_1}>0\)). This is due to the fact that more disorder is generated in the fluid molecules when energy (in the form of heat) is supplied.

Heat loss

When heat is removed from the fluid (heat loss), the temperature and enthalpy decrease while the fluid volume contracts, causing an increase in pressure. Since \(ds < 0\), during heat loss, the entropy decreases (\(ds= c_p\ln\frac{T_2}{T_1}-R\ln\frac{P_2}{P_1}<0\)). This is due to the fact that the removal of energy decreases disorder in the fluid molecules.

Summary of effects on entropy

In conclusion, the effect of heat gain during Rayleigh flow is an increase in entropy, while heat loss results in a decrease in entropy. The main reason for these changes in entropy is the thermodynamic behavior of the fluid molecules when energy (in the form of heat) is added or removed, affecting disorder within the fluid system.