One kilogram carbon dioxide at 100 kPa, 400 K is mixed with 2 kg carbon dioxide at 200 kPa, 20

step 1 In this question it is given that we have a rigid insulated tank. 1 kilogram of carbon dioxide i

One kilogram carbon dioxide at 100 kPa, 400 K is mixed with...

Key Concepts

Mixing Process Analysis

In the analysis of mixing, especially for gases, the final state is determined by using both energy and mass balances. The mixing of portions at different temperatures and pressures leads to thermodynamic interactions that establish a new equilibrium state. Understanding how to calculate the final pressure and temperature involves integrating concepts from energy conservation, adiabatic processes, and the specific thermodynamic properties of the substance, such as its constant heat capacity.

Entropy Generation

Entropy generation quantifies the irreversibility of a process. In mixing processes, especially when components start at different temperatures and pressures, irreversibility is inherent and results in an increase in the total entropy of the system. By comparing the entropy change calculated based on the final equilibrium state with the sum of the entropy changes of the individual parts, the amount of entropy generated due to the irreversibility of the mixing process can be determined.

Adiabatic Process

An adiabatic process is one in which no heat is transferred into or out of the system. In the context of a rigid, insulated tank, the mixing process is adiabatic, meaning that the only changes in the state are due to internal energy redistribution rather than any external heat exchange. This helps simplify the analysis since the energy balance does not include any heat transfer terms.

Constant Heat Capacity

Using a constant heat capacity assumption means that the specific heat of the fluid remains constant over the temperature range under consideration. This simplification allows the internal energy change to be expressed linearly with temperature change, and is particularly useful in problems where the fluid properties do not vary significantly. The assumption is crucial in calculating the final temperature from the change in internal energy during mixing.

Conservation of Energy

In an isolated or closed system (such as a rigid, insulated tank) the total energy remains constant during the process. This principle, also known as the first law of thermodynamics, is used to determine the final state by equating the initial total internal energy of the system to the final internal energy. In mixing problems, the energy stored in the different fluid portions is summed to find the overall energy content, which then allows for calculation of the final temperature using the appropriate energy balance equations.

Transcript

00:01 In this question it is given that we have a rigid insulated tank.

00:06 1 kilogram of carbon dioxide in this tank is mixed with 2 kilogram of carbon dioxide.

00:13 The pressure and temperature of this 1 kilogram carbon dioxide is 100 kioskal and 400 kelvin respectively.

00:22 And the pressure and temperature of this 2 kilogram carbon dioxide is 200 kilo -pascal and 2 ,000.

00:32 Kelvin.

00:35 We are required to evaluate the final pressure and temperature and entropy generation in the process using constant heat capacity.

00:43 So let's see how to solve this question.

00:46 So first of all, let's name these variables.

00:50 So we can say ma is equal to 1 kilogram, mb is equal to 2 kilogram, pa is equals to 100 kyl pascal, pb is equal to 200 kyl pascal, pb is equal to 400 kyl, p .a is equal to 400 and p b is equal to 2000 kelvin.

01:11 Now let's find the total mass or we can say m2 is equal to ma plus mb.

01:21 Now substitute all the values so we will have total mass of carbon dioxide and this will be equal to 1 kilogram plus 2 kilogram is equal to 3 kilogram.

01:32 Now, let's write the expression to calculate final temperature of the carbon dioxide so we can write t is equal to ma, ta, t2, mb, divided by m.

01:50 Now substitute all the values, so we will have final temperature t is equal to 1 kilogram multiplied by 400 plus 2 kilogram multiplied by 2000 divided by m that means 3 kilogram so it was m2 total mass so when we further calculate we get final temperature t is equals to 1 ,466 .67 kelvin so this is the final answer for the final temperature now let's write the expression to calculate total volume of the carbon dioxide so we can write total volume v is equal to v a plus vb now substitute all the values so we will have v is equal to va that means m a r p a divided by pa plus mb r pb r pb divided by pb now substitute all the values we will have v is equals to 1 into r and from the table a5 the value of r is equals to 0 .189 kilojoule per kilogram kelvin so now substitute that value we will have 0 .1889 multiplied by pa that means 400 divided by pa that means 100 plus 2 into 0 .18892 into 2000 divided by 200.

03:50 When we further calculate we get total volume of carbon dioxide is equals to 4 .5336 meter cube.

04:01 Now let's write the expression to calculate final pressure of carbon dioxide...

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