Question

An ideal gas undergoes a reversible, steady-flow process in which pressure and volume are related by the polytropic equation \(P v^{n}=\) constant. Neglecting the changes in kinetic and potential energies of the flow and assuming constant specific heats, \((a)\) obtain the expression for the heat transfer per unit mass flow for the process and ( \(b\) ) evaluate this expression for the special case where \(n=k=c_{p} / c_{v}\).

Short answer

Question: Derive the expression for the heat transfer per unit mass flow for a reversible, steady-flow process of an ideal gas following a polytropic equation. Also, evaluate the expression for the special case where n equals the ratio of specific heats. Answer: The expression for the heat transfer per unit mass flow for the given process is: \(\displaystyle \frac{Q}{m} = c_{p}\left(T_{1}\left[ \displaystyle \frac{P_{1}^{n-1}}{P_{2}^{n-1}}\right]^{\displaystyle\frac{1}{n-1}} - T_{1}\right)\) For the special case where \(n = k = \displaystyle \frac{c_{p}}{c_{v}}\), the expression is: \(\displaystyle \frac{Q}{m} = c_{p}\left(T_{1}\left[ \displaystyle \frac{P_{1}^{k-1}}{P_{2}^{k-1}}\right]^{\displaystyle\frac{1}{k-1}} - T_{1}\right)\)

Step-by-step solution

Write down the general energy balance equation

First, we need to write the general energy balance equation for a steady-flow process. It can be represented as: \(\displaystyle \frac{Q}{m} = \Delta h + \Delta KE + \Delta PE\) However, we are told to neglect the changes in kinetic and potential energies. Therefore, the equation simplifies to: \(\displaystyle \frac{Q}{m} = \Delta h\)

Implement the polytropic process relation

Now, we need to implement the polytropic process relation \(Pv^{n} =\) constant. This can be represented as: \(P_{1} v_{1}^{n} = P_{2} v_{2}^{n}\) By inserting the ideal gas equation \(P_{1} v_{1} = mR_{u} T_{1}\) and \(P_{2} v_{2} = mR_{u} T_{2}\), we get: \((P_{1} v_{1})^{n-1} = (P_{2} v_{2})^{n-1}\) which can further be written as: \((\displaystyle \frac{mR_{u} T_{1}}{mR_{u} T_{2}})^{n-1} = \frac{P_{1}^{n-1}}{P_{2}^{n-1}}\) Now we can write the change in enthalpy: \(\Delta h = h_{2} - h_{1} = c_{p} (T_{2} - T_{1})\)

Simplify the energy balance equation

Combine the energy balance equation with the change in enthalpy expression: \(\displaystyle \frac{Q}{m} = c_{p}(T_{2} - T_{1})\) Now, we can solve for \(T_{2}\): \(T_{2} = T_{1} + \displaystyle \frac{Q}{mc_{p}}\) Insert the polytropic process relation: \(T_{2} = T_{1}\left[ \displaystyle \frac{P_{1}^{n-1}}{P_{2}^{n-1}}\right]^{\displaystyle\frac{1}{n-1}}\) Now, insert this expression for \(T_{2}\) into the energy balance equation: \(\displaystyle \frac{Q}{m} = c_{p}\left(T_{1}\left[ \displaystyle \frac{P_{1}^{n-1}}{P_{2}^{n-1}}\right]^{\displaystyle\frac{1}{n-1}} - T_{1}\right)\)

Evaluate the expression for the special case where n = k

In the special case where \(n = k = \displaystyle \frac{c_{p}}{c_{v}}\), evaluate the expression for the heat transfer per unit mass flow: \(\displaystyle \frac{Q}{m} = c_{p}\left(T_{1}\left[ \displaystyle \frac{P_{1}^{k-1}}{P_{2}^{k-1}}\right]^{\displaystyle\frac{1}{k-1}} - T_{1}\right)\) Therefore, the expression for the heat transfer per unit mass flow for the given process is: \(\displaystyle \frac{Q}{m} = c_{p}\left(T_{1}\left[ \displaystyle \frac{P_{1}^{k-1}}{P_{2}^{k-1}}\right]^{\displaystyle\frac{1}{k-1}} - T_{1}\right)\)