Question

A control volume has one inlet and one exit. The mass flow rates in and out are, respectively, \(\dot{m}_{i}=1.5\) and \(\dot{m}_{\mathrm{e}}=\) \(1.5\left(1-e^{-0.002}\right.\) ), where \(t\) is in seconds and \(\dot{m}\) is in \(\mathrm{kg} / \mathrm{s}\). Plot the time rate of change of mass, in \(\mathrm{kg} / \mathrm{s}\), and the net change in the amount of mass, in \(\mathrm{kg}\), in the control volume versus time, in s, ranging from 0 to \(3600 \mathrm{~s}\).

Short answer

Integrate and plot \(1.5e^{-0.002t}\) for mass rate change and \(1.5 \left(1 - e^{-0.002 t}\right)\) for net mass change over time.

Step-by-step solution

Write down the given information

Given information: - Inlet mass flow rate, \(\dot{m}_{i}=1.5 \, \text{kg/s}\)- Exit mass flow rate, \(\dot{m}_{e}=1.5\left(1-e^{-0.002 t}\right) \, \text{kg/s}\)- Time range is from 0 to 3600 seconds

Write the mass balance equation

Using the mass balance equation for a control volume, we have: \[ \frac{d m_\text{cv}}{d t} = \dot{m}_{i} - \dot{m}_{e}\]where \(m_\text{cv}\) is the mass in the control volume.Substitute the given mass flow rates:\[ \frac{d m_\text{cv}}{d t} = 1.5 - 1.5\left(1 - e^{-0.002 t}\right)\]

Simplify the expression

Simplify the expression for the rate of change of mass:\[ \frac{d m_\text{cv}}{d t} = 1.5 - 1.5 + 1.5 e^{-0.002 t} = 1.5 e^{-0.002 t}\]This is the time rate of change of mass in the control volume.

Calculate the net change in the amount of mass

Integrate the rate of change of mass to find the net change in mass, \(\Delta m\), over time:\[ \Delta m = \int_0^t \frac{d m_\text{cv}}{d t} \, dt = \int_0^t 1.5 e^{-0.002 t} \, dt\]\(u = -0.002 t\), then \(\frac{du}{dt} = -0.002\)So we need to change the limits accordingly and integrate:\[ \Delta m = 1.5 \int_0^t e^{-0.002 t} \, dt = -750 \left[e^{-0.002 t}\right]_0^t = 1.5 \left[1 - e^{-0.002 t}\right]\]

Plot the results

Using a computational tool like MATLAB or Python, plot the following:1. Time rate of change of mass, \(\frac{d m_\text{cv}}{d t} = 1.5 e^{-0.002 t}\), vs time (0 to 3600 seconds).2. Net change in the amount of mass, \(\Delta m = 1.5 \left[1 - e^{-0.002 t}\right]\), vs time (0 to 3600 seconds).

Key concepts

Mass Flow Rate

The mass balance equation is essential to understand the changes within a control volume. The general form is:
\[ \frac{d m_{cv}}{d t} = \dot{m}_{i} - \dot{m}_{e} \]
In this exercise, it translates to:
\[ \dot{m}_{cv} = 1.5 - 1.5(1 - e^{-0.002 t}) \]
Mass balance ensures the difference between inlet and outlet flow rates equals the rate of mass change within the control volume. Simplifying, we have:
\[ \frac{d m_{cv}}{d t} = 1.5e^{-0.002 t} \]

Rate of Change of Mass

In thermodynamics, analyzing the rate of change of mass helps us understand how mass within the control volume adjusts over time. Using the simplified equation:
\[ \frac{d m_{cv}}{d t} = 1.5e^{-0.002 t} \]
This function shows that as time increases, the rate of mass change decays exponentially. Initially, the rate is at its peak, then gradually diminishes. When implementing this in a graph, expect a downward trend starting from a maximum value of 1.5kg/s, tapering off as time progresses.

Integration in Thermodynamics

Integration is key to finding the total change or 'net change' in the amount of mass over a period. From the rate of change expression, we integrate:
\[ \Delta m = \int_0^t1.5e^{-0.002 t}dt \]
Using substitution and integration process:
\[ \Delta m = 1.5\big[ 1 - e^{-0.002 t}\big] \]
This integral encompasses the sum of all infinitesimal changes over time. It provides the cumulative mass added or removed from the control volume.

Time-Dependent Analysis

Time-dependent analysis helps us visualize how a system evolves. In this problem, both the exit mass flow rate and the rate of mass change are time-dependent. They rely on the exponential term (e^{-0.002 t}). Such time-based fluctuations are common in real-world systems, where inputs and outputs rarely remain constant.
Tools like MATLAB or Python are ideal for plotting these variations. The plots illustrate the system dynamics:
- Rate of change versus time: Shows the exponential decay
- Net change versus time: Displays how cumulative mass shifts over the period of 3600 seconds. These visualizations provide powerful insights into the long-term behavior of the control volume.