Question

Two mass streams of the same ideal gas are mixed in a steady-flow chamber while receiving energy by heat transfer from the surroundings. The mixing process takes place at constant pressure with no work and negligible changes in kinetic and potential energies. Assume the gas has constant specific heats. (a) Determine the expression for the final temperature of the mixture in terms of the rate of heat transfer to the mixing chamber and the inlet and exit mass flow rates. (b) Obtain an expression for the volume flow rate at the exit of the mixing chamber in terms of the volume flow rates of the two inlet streams and the rate of heat transfer to the mixing chamber. (c) For the special case of adiabetic mixing, show that the exit volume flow rate is the sum of the two inlet volume flow rates.

Short answer

Question: Determine expressions for the final temperature, volume flow rate at the exit, and the exit volume flow rate for an adiabatic mixing process of a steady-flow mixing chamber with two mass streams of an ideal gas. Assume constant specific heats, constant pressure, and negligible changes in kinetic and potential energies. a) Derive the expression for the final temperature of the mixture. b) Obtain the expression for the volume flow rate at the exit. c) Determine the exit volume flow rate for the special case of adiabatic mixing.

Step-by-step solution

(a) Derive the final temperature expression

First, let's apply the conservation of mass. The mass flow rate of the two inlet streams is given by: \(m_1\) and \(m_2\), and the mass flow rate at the exit is \(m_e\). According to the conservation of mass, the total mass flow rate should be constant. Therefore: 1) \(m_e = m_1 + m_2\) Next, let's apply the conservation of energy. Since there is no work done, volume changes are negligible, and potential and kinetic energy changes are negligible as well, the internal energy change is due to heat transfer. Therefore, for a steady-flow device: 2) \(Q = m_1 c_p T_1 + m_2 c_p T_2 - m_e c_p T_e\) From 1) 3) \( T_e = (T_1 \frac{m_1}{m_1 + m_2} + T_2 \frac{m_2}{m_1 + m_2}) + \frac{Q}{(m_1 + m_2)c_p} \) This is the expression for the final temperature of the mixture in terms of the rate of heat transfer to the mixing chamber and the inlet and exit mass flow rates.

(b) Obtain the expression for the volume flow rate at the exit

To find the volume flow rate at the exit of the mixing chamber, we need the relation between mass flow rate and volume flow rate: 4) \(m_i = \rho_i V_i\) Since it is an ideal gas, we can use the relation: 5) \(PV = mRT\) So: 6) \(\rho_i = \frac{m_i}{V_i} = \frac{P}{RT_i}\) From 4) and 6), we can write the volume flow rates as: 7) \(V_i = \frac{m_i RT_i}{P}\) Putting these expressions in the expression for the final temperature derived in (a), we have: 8) \( T_e = (T_1 \frac{V_1}{V_1 + V_2} + T_2 \frac{V_2}{V_1 + V_2}) + \frac{Q}{(V_1 + V_2)\frac{P}{R} c_p} \) Using the relationship between specific heat and the gas constant (\(c_p = c_v + R\)), the final temperature is: 9) \( T_e = (T_1 \frac{V_1}{V_1 + V_2} + T_2 \frac{V_2}{V_1 + V_2}) + \frac{Q}{(V_1 + V_2)c_v} \) Now, as we are interested in the volume flow rate at the exit, let's use equations 7) and insert the final temperature at the exit (9): 10) \(V_e = \frac{m_e RT_e}{P} = \frac{(m_1+m_2)-Q/c_p}{(V_1+V_2)\frac{R}{P}c_v}\) This equation gives the expression for the volume flow rate at the exit of the mixing chamber in terms of the volume flow rates of the two inlet streams and the rate of heat transfer to the mixing chamber.

(c) Exit volume flow rate for an adiabatic mixing

For an adiabatic mixing process, there is no heat transfer, i.e., \(Q = 0\). From the expression (10), we can rewrite the volume flow rate at the exit for an adiabatic process: 11) \(V_e = \frac{(m_1+m_2)}{(V_1+V_2)\frac{R}{P}c_v}\) From 7), we can write: \(m_1 =\frac{V_1}{RT_1}\) and \(m_2 = \frac{V_2}{RT_2}\) Adding these equations, we get: 12) \(m_1+m_2 = \frac{V_1+V_2}{R}\) Now, putting 12) back into the expression for the volume flow rate at the exit (11), we get: 13) \(V_e = V_1+V_2\) For an adiabatic mixing process, the exit volume flow rate is the sum of the two inlet volume flow rates.