LESSON - 5 WORK & HEAT AND THEIR NUMERICAL PROBLEMS

It is necessary to understand clearly the definition of work and heat before we discuss Laws of Thermodynamics. 

5.1. WORK

Work is usually defined as force F acting through a displacement x, the displacement being in the direction of the force. Mathematically it can be expressed as

                                 (SI units: J or kJ)                                

 But in thermodynamics, we define work as:

Work is said to be done by a system if the sole effect on the surroundings could be the raising of a weight. Here raising of a weight is equivalent to a force acting through a distance.

In real life there may not be any raising of a weight but still there is work done by system on surrounding. To make it clear, the following illustration is discussed:

Consider an arrangement in which the battery rotates a fan with the help of an electric motor as shown in Fig. 5.1(a). Battery and motor constitute the system. Since no weight is being raised so it seems that no work is being done by the system on surroundings. In fact it is false. Let us now replace the fan with a pulley and weight arrangement as shown in Fig. 5.1(b). Now here the battery can raise the weight with the help of an electric motor, and the sole effect external to the system is the raising of a weight. Thus, for system with a fan in Fig. 5.1(a), we can conclude that work is being done by the system on the surroundings even there may not be any raising of a weight.

So rotation of shaft across the system boundary constitutes transfer of energy in the form of work.

 Fig. 5.1. Transfer of energy in the form of work because of a rotation of shaft across   the system boundary

Now in the above example if battery alone is considered to constitute the system as shown in Fig. 5.2 then electric current crossing the system boundary is able to raise a weight with the help of a motor and pulley.

So passing of electric current across the system boundary also constitutes transfer of energy in the form of work.

 Fig. 5.2. Transfer of energy in the form of work because of a flow of an electric current across the system boundary

5.1.1. Different forms of work done during a process

During a process, there are many ways in which work can be done on or by a system. These include work done at the moving boundary of a system (displacement work or ‘pdV’ work), paddle-wheel work or stirrer work, shaft work, electric work, work done in moving an electric charge through an electric field, work of polarization and magnetization, work of chemical cell, work in stretching a liquid film, charging a capacitor, work done in stretching a wire, etc.      

5.1.1.1. Work done at the moving boundary of a system (displacement work or ‘pdV’ work)

Consider an example of piston and cylinder arrangement (Fig. 5.3) to find work done at the moving boundary of a system. Say, the air in the cylinder undergoes change in state from 1 to 2 which is represented by a quasi-static process 1-2 on the p-V diagram.

First find the work done for small ‘dL’ movement of the piston. During this small movement the pressure ‘p’inside the cylinder is considered constant.

Then work done for small ‘dL’ is given by

       δW = F.dL = p.A .dL                     ....................................................  (5.1)

            where, F is the force acting on the piston

                        A is the area of the piston face

As   A.dL = change in volume of the gas due to dL movement of piston, it is taken as dV.

Therefore from equation (5.1),

       δW =  p.dV

To find work done during process 1-2, integrate the above equation.

               

 

 Fig. 5.3. Use of pressure-volume diagram to show work done at the moving boundry of a system.

  • In the above piston cylinder arrangement, two types of works are possible.

   One is due to compression process the work done by the surrounding on the system (process 1-2)

   Other is work done by the system on surrounding due to expansion process (process 2-1).

  • It is clear from the P-V diagram that the   work done during the process ‘1-2’is the area under curve ‘1-2’ i.e. area ‘a-1-2-b-a’ shown by section line in Fig. 5.3.

  • Now, consider a system in which it is possible to change the state of a system from state ‘1’ to state ‘2’ through different paths ‘A’, ‘B’ and ‘C’  as shown on PV diagram in Fig. 5.4. Since the area underneath each curve represents the work, so the amount of work done for different paths between states ‘1’ and ‘2’ will be different i.e. WC> WB> WA. For this reason work is called a path function, or δW is an inexact differential.

 

Fig. 5.4. Various quasistatic process between two given states, indicating that work is a path function.

5.1.1.2. Paddle-wheel work, stirrer work, or shaft work

Refer Fig. 5.5. The system consists of a gas in the container. Here, rotation of shaft across the system boundary by the falling weight constitutes work called paddle-wheel work, stirrer work, or shaft work.Though there is no change in volume of the system still work has been done on the system.

Fig. 5.5. Example of paddle-wheel work, stirring work, or shaft work
5.1.2. Total or net work done by a system

In many engineering situations, during a process executed by a system, different forms of work transfer by/on the system occur simultaneously. Under such circumstances, the total or net work done of the system would be

Wnet = Wdisplcament + Wpaddle + Welectrical + other forms of work done

5.1.3.  Cases where work done is zero

Following are some cases where work done is zero

5.1.3.1.

Refer Fig. 5.6. In this shaft and weight are also included in the system along with gas and container. As rotation of shaft does not cross the system boundary, no work is being done by the falling of weight.

Fig. 5.6. Example of a system so selected that the lowering of a weight   does not involve work.

 5.1.3.2.

Consider the arrangement in which gas and vacuum in a container are separated by a membrane as shown in Fig. 5.7 (a). Gas in the container constitutes a system and it is at an initial state ‘1’. Now, let the membrane be ruptured and gas fills the entire volume of the container and it is at final state ‘2’ as shown in Fig. 5.7 (b).Neglect any work associated with the rupturing of the membrane.

Even there is moving boundary of a system still work done by the system is zero because there is no resistance (no force) at the system boundary as the volume increases. 

 

 

                 (a)

                  (b)

           Fig. 5.7. Example of a process involving a change of volume for which the work is zero.
5.1.4. Important points for work:

  • Work done by the system on the surroundingas shown in Fig. 5.1 is taken as +ve.

  • Work done by the surrounding on a system as shown in Fig. 5.5 is taken as –ve.

  • The symbol W designates the work done.

  • As work is a path function, work has meaning when a process occurs. It is always written as

                               

  • We never speak the work at state ‘1’ or at state ‘2’ as we speak for properties,   therefore the amount of work done during a change of state from state ‘1’ to ‘2’can  never be written as  W2 – W1.

Problem 5.1:   A non flow reversible process 1-2 can be written by the following equation:    

P = (V2 + 8/V) bar.  Determine the work done if volume changes from 1 to 3 m3.

Solution:

Given:  V1 = 1 m3;    V2 = 3 m3 ;  P = (V2 + 8/V) bar;  

Determine the work done, 1W2 ;

         Formula:          1W2 =

                             or           =

                                           = 105[V3/3 + 8ln(V)]

                                           = 105[V23/3 + 8ln(V2)] - [V13/3 + 8ln(V1)]

Answer:   Work done,  1W2   = 105[V23/3 + 8ln(V2)] - [V13/3 + 8ln(V1)]

                                                   = 105  x { [33/3 + 8ln(3)] - [13/3 + 8ln(1)] }

      = 1.745 x 105 J

Problem 5.2: A gas in the cylinder and piston arrangement comprises the system. It expands from 1.5 m3 to 2 m3 while receiving 200 kJ of work from a paddle wheel. The pressure on the gas remains constant at 600 kPa. Determine the network done by the system.

Solution:

Given: V1 = 1.5 m3;    V2 = 2 m3 ;   Wpaddle= −200 kJ = 2 x 105 J;   

             P = 600 kPa = 6 x 105 N/m2

  To determine network done by the system, Wnet :

     Formula: Wnet = Wdisplcament + Wpaddle

               Finding unknown, Wdisplacement :

               Wdisplacementis the displacement work done by the system = ∫pdV

                                                         =         (because p is constant)

                                                         = P(V2 − V1)

                                                         = 6 x 105 (2 − 1.5) = 3 x 105  Nm  or J

Answer: Net work by the system is,

Wnet = Wdisplcament + Wpaddle

          = 3 x 105  - 2 x 105   = 1 x 105  Nm  or 100kJ

 5.2.  HEAT

Transfer of energy because of temperature difference across the system boundaries is called heat interaction.

Consider an arrangement of gas contained in a vessel in Fig. 5.8 (a) as system. Since the heating element on the outer wall of the container is outside the system, the transfer of energy is in the form of heat due to the temperature difference between the surrounding (heating element) and the system. Now consider another arrangement in which heating element is part of the system as shown in Fig. 5.8 (b). In this arrangement, as the heat is not crossing the system boundary but current is flowing across the system boundary, therefore transfer of energy in the system from the surrounding is not in the form of heat but in the form of work.

 

(a)                                                                                (b)

Fig. 5.8. An example showing heat and work

5.2.1. Important points for heat:

  • Heat given to the system is taken as +ve.

  • Heat given out of the system is taken as –ve.

  • The symbol Q designates the heat transfer.

  • Like work, heat has meaning only when a process occurs as heat is also a path function. It is also recognized as an inexact differential.

So, It is always written as

                                                (SI units: J or kJ)                                

where, 1Q2 is the heat transferred during the given process between state 1 and state 2.

Comparison of heat and work

  1. Both heat and work are path functions and are recognized as inexact differentials. So instead of dW and dQ these are expressed as δW and δQ, respectively

  2. Heat and work are not a property of the system.

  3. Both are boundary phenomena as both heat and work are observed only at the system boundary and not in the system. So, both heat and work have meaning only when these enter or leave the system in the form of energy during a process.

  4. Both heat and work are energy in transit and they have meaning only when they flow.

  5. Both heat and work transfer are energy interactions.

Last modified: Saturday, 7 September 2013, 4:41 AM