1. Energy Balance Equation
Objectives
Steady Flow Process
Mass Balance Equation
Energy Balance Equation

2. Control Volume
For any system the first law can be written as,
Q = ∆E + W
Where E represents all forms of energy stored in the system .
For pure substance ,
E = Ek + Ep + U
Where Ek is the kinetic energy , Ep is the potential energy and U is the residual energy which is stored in the molecular structure of the substance.
so, Q = ∆Ek +∆Ep +∆U + W
When there is a mass transfer across the system boundary , the system is called an open system.
Most of the engineering devices are open systems involving the flow of fluids through them.

3. The steam leaves the turbine at low pressure through the exhaust pipe
Control Volumes
Consider the steam turbine in which steam enters at a high pressure and does work upon the turbine rotor.
The steam leaves the turbine at low pressure through the exhaust pipe
If a certain mass of steam is considered as the thermodynamic system then the energy equation becomes,
Q = ∆Ek +∆Ep +∆U + W
Where Ek is the kinetic energy in Joules
Ep is the potential energy and in Joules
U is the residual energy which is stored in the molecular structure of the substance in Joules
Flow Process Involving Work and Heat Interactions

4. Control Volumes
The expansion process is not analyzed in this process.
The moving system is followed through the turbine, taking in to account the concept of heat and work all the way through.
Instead of concentrating the attention upon the certain quantity of the fluid ,attention is focused upon the certain fixed region in the space called a control volume through which the moving substance flows.
The broken line in the diagram represents the surface of the control volume which is known as control surface.

5. Mass Balance in a Simple steady Flow Process
As a fluid flows through the certain control volume , its thermodynamic properties may vary along the space co-ordinates as well as with time.
If the rates of flow of mass and energy through the control surface change with time , the mass and energy within the control volume also would change with time.
Steady flow means the that the rates of flow of mass and energy across the control surface are constant.

6. Mass Balance in a Simple Steady Flow Process
In most of the engineering devices there is a constant rate of flow of mass and energy through the control surface, and the control volume in course of time attains a steady state.
At the steady state of a system ,any thermodynamic property will have a fixed value at a particular location , and will not alter with time.
Thermodynamic properties may vary along space co-ordinates , but do not vary with time.

7. Mass Balance in a Simple Steady Flow Process
A steady flow system is shown in the diagram.
In that one stream of fluid enters and an another stream of fluid leaves the control volume.
There is no accumulation of mass and energy in the control volume and also the properties at any location in the control volume are same and steady.
Here section 1.1 and 2.2 indicates the entrance and exit of the fluid across the control surface respectively.
Steady Flow Process

8. Mass Balance in a Simple Steady Flow process
A1 , A Cross section of inlet and outlet stream , m2
w1, w Mass flow rate , kg/s
p1, p pressure,absolute,N/m2
u1,u Specific internal energy , J/kg
v1 , v Specific volume , m3/kg
V1, V velocity , m/s
Z1,Z elevation above an arbitrary datum ,m
đQ/đτ net rate of heat transfer through the control surface , J/s
đWx/đτ - net rate of work transfer through the control surface , J/s
Exclusive of work done at sections 1 and 2 in transferring the fluid through the control surface
τ time ,s
Subscripts 1 and 2 refer to the inlet and exit sections

9. Mass Balance in a Simple Steady Flow Process
By the conservation of mass , if there is no accumulation of mass with in the control volume , the mass flow rate entering must equal the mass flow rate leaving . It is given by,
w1 = w2
A1V1/v1 = A2V2/v2
This equation is the equation of continuity.
Energy Balance :
In flow process the work transfer may be of two types they are,
The external work and
The flow work

10. Mass Balance in a Simple Steady Flow Process
External work:
The external work refers to all the work transfer across the control surface other than due to normal fluid forces .
In thermodynamics the kinds of external work importance are ,
Shear Work ( shaft or stirring ) and
Electrical Work
Here we considered as the external work done in the form of shear work Wx
Flow Work:
Here flow work is the displacement work done by the fluid of mass dm1 at the inlet section 1 and that of the mass dm2 at the exit section 2, which are
( - p1v1dm1) and (p2v2dm2) respectively.

11. Mass Balance in a Simple Steady Flow Process
The total work transfer is given by,
W= Wx - p1v1dm1 + p2v2dm2
In the rate form,
đ W/ đ τ = đ Wx / đ τ - p1v1 dm1 / đ τ + p2v2 dm2 / đ τ
đ W/ đ τ = đ Wx / đ τ - p1v1 w1 + p2v2 w2
Since there is no accumulation of energy , by the conservation of energy , the total rate of flow of energy entering the control volume must equal to the rate of flow of all energy streams leaving the control volume. This is expressed in the form of,
w1e đ Q/ đ τ = w2e2 + dW / đ τ

12. Mass Balance in a Simple Steady flow Process
Substituting the value of đ W/ đ τ we can get,
w1e1 + w1 p1v1+ đ Q/ đ τ = w2e2 + p2v2 w2 + đ Wx / đ τ
where e1 and e2 refer to the energy carried in to or out of the control volume with unit mass of fluid.
The specific energy e is given by,
e = ek + ep + u
= V2 / 2 + Zg + u
Substituting the value of e , we can get,
w1 ( V1 2 / 2 + Z1g + u1 ) + w1 p1v1 + đ Q/ đ τ
= w2 ( V22 / 2 + Z2g + u2 ) + w2 p2v2 + đ Wx / đ τ

13. Mass Balance in a Simple Steady Flow Process
Or w1 (h1 + V1 2 / 2 + Z1g) + đ Q/ đ τ
= w2 ( h2 + V22 / 2 + Z2g) + đ Wx / đ τ
Where h = u + pv
And since w1 = w2 let w = w1 = w2 = dm/ d τ
Dividing the above equation by dm/ d τ, we can get,
h1 + V1 2 / 2 + Z1g + đ Q/ dm
= h2 + V22 / 2 + Z2g + đ Wx / dm
The above equation is energy flow per unit mass of fluid , (J/kg)
The above two energy flow equations are known as steady flow energy equations for a single stream of fluid entering and single stream of fluid leaving.

14. Mass Balance in a Simple Steady Flow Process
When only a single stream of fluid enters and leaves the control volume the basis of energy flow per unit mass is convenient.
When more than one fluid stream is involved the basis of energy flow per unit time is suitable.
Energy flow per unit mass of fluid can be written as,
Q - Wx = ( h2 - h1 ) + V22 - V1 2 / 2 + g (Z2 - Z1)
Where Q and Wx refer to energy transfer per unit mass . In the differential form the steady flow energy equation becomes,
đ Q - đWx = dh + V dV + gdZ

15. Mass Balance in a Simple Steady Flow Process
When more than one stream of fluid enters or leaves the control volume as shown in the diagram, the mass balance and energy balance equation for steady is given as follows,
Mass Balance:
w1 + w2 = w3 + w4
A1V1/v1 + A2V2/v2 = A3V3/v3 + A4V4/v4
Energy Balance :
w1 (h1 + V1 2 / 2 + Z1g)+w2( h2 + V22 / 2 + Z2g)+đ Q/ đ τ
= w3 ( h3 + V32 / 2 + Z3g)
+ w4 (h4 + V4 2 / 2 + Z4g) + đ Wx / đ τ
This energy flow equation is used for wide variety of process like pipe line flows, heat transfer process.
Mass Balance in Steady Flow Process

16. Summary
Steady flow means the that the rates of flow of mass and energy across the control surface are constant.
Mass Balance is defined by the conservation of mass , if there is no accumulation of mass with in the control volume , the mass flow rate entering must equal the mass flow rate leaving . It is given by,
w1 = w2
The equation for energy flow per unit time (J/s) is represented as
w1 (h1 + V1 2 / 2 + Z1g) + đ Q/ đ τ = w2 ( h2 + V22 / 2 + Z2g) + đ Wx / đ τ
Energy flow per unit mass of fluid can be written as,
Q - Wx = ( h2 - h1 ) + V22 - V1 2 / 2 + g (Z2 - Z1)