Question

Consider a steady-flow mixing process. Under what conditions will the energy transported into the control volume by the incoming streams be equal to the energy transported out of it by the outgoing stream?

Short answer

Answer: The energy transported into the control volume by the incoming streams will be equal to the energy transported out of it by the outgoing stream if the specific enthalpy of the outgoing stream is equal to the weighted average of the specific enthalpies of the incoming streams, with the weights given by the mass flow rate ratios.

Step-by-step solution

Understand Energy Conservation and Control Volume

Energy conservation is a fundamental principle of physics, which states that the total energy in a closed system remains constant over time. In fluid mechanics, we often deal with open systems where mass and energy can both enter and exit the control volume. A control volume is a fixed region in space that we analyze in order to understand the behavior of a fluid, and where we apply the principle of conservation of mass and energy.

Analyzing a Steady-Flow Mixing Process

In a steady-flow mixing process, multiple incoming streams of fluid mix together and exit as a single outgoing stream. Since the flow is steady, the mass flow rate and energy flow rate remain constant over time. The energy flow rate depends on the mass flow rate, internal energy and flow work done by the fluid.

Writing the Energy Conservation Equation

To write the energy conservation equation, we can consider the energy transported in by the incoming streams and the energy transported out by the outgoing stream. For simplicity, let's assume that there are two incoming streams (Stream 1 and Stream 2) and one outgoing stream (Stream 3). The energy conservation equation can be written as: Energy in (Stream 1 + Stream 2) = Energy out (Stream 3) Let h_1, h_2, and h_3 be the specific enthalpy (internal energy + flow work) for Stream 1, Stream 2, and Stream 3, respectively. Also, let m_dot_1, m_dot_2, and m_dot_3 be the mass flow rates for Stream 1, Stream 2, and Stream 3, respectively. The equation becomes: m_dot_1 * h_1 + m_dot_2 * h_2 = m_dot_3 * h_3

Finding the Conditions for Equal Energy Transport

From the energy conservation equation, we can find the conditions under which the energy transported into the control volume by the incoming streams will be equal to the energy transported out of it by the outgoing stream. The condition for equal energy transport is: m_dot_1 * h_1 + m_dot_2 * h_2 = m_dot_3 * h_3 We can rearrange the equation by dividing both sides by m_dot_3: \frac{m_dot_1}{m_dot_3} * h_1 + \frac{m_dot_2}{m_dot_3} * h_2 = h_3 If the energy transported into the control volume by the incoming streams is to be equal to the energy transported out of it by the outgoing stream, the specific enthalpy of the outgoing stream (h_3) must be equal to the weighted average of the specific enthalpies of the incoming streams (Stream 1 and Stream 2), where the weights are given by the mass flow rate ratios: h_3 = \frac{m_dot_1}{m_dot_3} * h_1 + \frac{m_dot_2}{m_dot_3} * h_2 Therefore, the condition for equal energy transport in a steady-flow mixing process is that the specific enthalpy of the outgoing stream must be equal to the weighted average of the specific enthalpies of the incoming streams when considering the mass flow rate ratios.