# First Law of Thermodynamics for an Open System

In this video we will learn first law of thermodynamics in a practical way.

Detailed description of above video lecture is given below.

## Applications of First Law of Thermodynamics

First law of thermodynamics when it is applied to an open system has got tremendous applications all across industries. Using this law you can predict how much is the pressure drop across the nozzle, or how much is the energy required by the pump to pump the fluid out, or what is the heat transfer in heat exchanger, or what is the amount of work produced by the turbine.

 Fig.1 Industrial applications of first law of thermodynamics for an open system
In a nutshell first law simply means conservation of energy, or it states that energy is getting transformed from one form to another form.

## First Law for Closed System

We will understand how first law is applied for a thermodynamic system by analyzing a simple example, an example of piston cylinder arrangement. Here the cylinder has got some gas inside it. Assume there is no air leakage to the surrounding. So this is an example of closed system where mass does not change. Assume the gas is absorbing some heat Q from the surrounding; also assume that this gas is able to push the piston upwards due to high pressure of gas. So the gas is doing some work on the piston with quantity W.

 Fig.2 Piston-cylinder arrangement to demonstrate first law for a closed system
So there are 2 energy interactions to the gas, it will increase by a quantity Q, because it is absorbing energy. And it will decrease by a quantity W since it is losing energy by doing some work. So you can write change in energy (E) of gas as follows.
This is first law of thermodynamics for a closed system. Same equation you can write in differential form as follows. It is in form of rate of change of quantities per unit time.

## First Law for an Open System

Now we are going to open the system, or open the cylinder as shown below.

 Fig.3 A piston-cylinder open system made by making the cylinder open
The system is no more closed now, it’s an open system. The mass is continuously varying. It can have an inlet mass flow rate at particular pressure and particular velocity. Similarly there will be exit flow rate of particular pressure and velocity. Here also our objective is the same. We want to find out energy change of the gas or the system. But here it is not possible to pin point a particular quantity of gas. The gas is continuously flowing. So before proceeding to the energy change calculation, we have to define a system first, a control volume where you will do energy balance.
 Fig.4 Defining a control volume for energy balance
Here the dotted line represents the control volume, or the space at which we will do energy balance. Here you can see there are 4 energy interactions to the system. 2 energy interactions which are coming to the system and another 2 energy interactions which leave the system. So if you want to find out energy change in system you should add energy transfer due to heat flow and inlet mass flow and subtract energy transfer due to work done and exit mass flow. So for an open system change in energy will be as follows.
Note that the flow stream has got 3 components of energy. Internal energy, kinetic energy and potential energy. Z represents the altitude of flow stream. This equation is the first law of thermodynamics for an open system.

## Concept of Flow work & Enthalpy – More Useful Form of First Law

But for an open system the term W, work done by the gas should be carefully examined. Here the gas is doing work to push the cylinder up, plus it is doing work to suck the fluid in or eject the fluid out. Or to maintain the flow gas has to do some work. This kind of work, the work which is required to maintain the flow is known as flow work. So the total work done by the system will be summation of visible work and flow work.

Wcv represents the visible work, in this case the work done by the gas on the piston. And we know flow work is the work required to eject the fluid out or suck the fluid out. The work required to eject the fluid out will be force at exit portion multiplied by velocity of this stream. Force is same as pressure at that portion times area. So we can represent flow work like this.
If you do some rearrangement to the equation by substituting volumetric flow rate as mass flow rate into specific volume, by representing u+Pv as a new property enthalpy,
the above equation will be simplified like this.
This is the final and most useful form of first law of thermodynamics for an open system.

## One Application of First Law

We will work out one interesting example using firs law equation in this section. A pump problem, where fluid is getting pumped from point 1 to point 2.

 Fig.5 Work required by the pump for pumping fluid from state 1 to 2 is to be calculated
We want to find out what’s the energy required by the pump to perform this action. To find out that we will use equation derived for first law of thermodynamics for an open system.We can assume the pump operation is in steady state. So energy of the pump does not change with time. So you can put first term in equation as zero. And usually there will not be any heat transfer to the pump. So you can put that term also as zero.
If cross sectional areas of point 1 and point 2 are equal, then velocities will be equal, so from this equation velocity part also get cancelled out. You can also assume height difference between inlet and outlets are negligible. So the altitude term also gets cancelled out. And finally what remains is this.
Work done by the control volume is mass flow rate times change in enthalpy. If you want work done on control volume or energy required by the pump, you have to just reverse the sign.
Using the same approach you can solve lot of other energy transfer problems in industries.

# Fatigue Failure Analysis

Even if you design mechanical components satisfying mechanical strength criteria it may fail due to a phenomenon called fatigue. Historically many design disasters have happened by neglecting effect of Fatigue.In this video lecture we will learn how to predict and quantify fatigue effect.

Detailed description of above video lecture is given below

## A Wire Breaking problem

To understand what is fatigue let’s consider this metal wire. You have to break it. So how will you break it? Will you pull it from both ends or will you bend the wire upward and downward repetitively.

 Fig.1 Two methods to break metal wire, Either bend it upward and downward repetitively or pull it
Your answer is obviously the second option. Because this method requires less effort compared to the first case. This is a well known example of fatigue failure. So how does material fail due to fatigue? To get answer for this question let us have a close look at stress variation in wire cross section.

## Reason Behind Fatigue Failure - Crack Propagation

When you bend it downwards bending stress induced is in the wire cross section. There will be tension at top area and compression at bottom area. When wire is at equilibrium there will not be any stress on wire cross section. When wire is bending upwards there will be compression at top and tension at bottom.

 Fig.2 Stress variation in wire cross-section, as wire is bent downward and upward
So if you trace stress induced at a point with respect to time it will vary like this. As a fluctuating stress with time.
 Fig.3 Stress variation at a point is plttod on stress vs time graph
Initially the point will have positive stress, after that zero, then negative stress. The same cycle repeats again and again. Such fluctuating stress is root cause of fatigue failure. When such fluctuating load act on a material it will initiate something called micro crack. This crack will begin to grow with fluctuating load and over time it will cause an abrupt failure. Unlike failure due to static load failure due to fatigue happens without any warning, it does not make necking. And the failure is unpredictable.

## Fatigue Failure in Real Life Engineering Problems

 Fig.4 Some practical cases which could result in fatigue failure, if not designed properly

The same phenomenon can happen for axle of this motor where it is undergoing fluctuating stress due to gravity effect of this mass. A rail wheel when it is in contact with with the track produces a high contact stress, but when the wheel rotates stress gets relieved. When it comes back to original position again contact stress arises. So this also is a case of fluctuating stress case. Again will lead to fatigue failure if we do not design it carefully. Same is the case with a gear pair. Here contact stress arised at contact point fluctuates with time.

## Effect of Stress Amplitude on Number of Cycles - S N Curve

This is the most important part in fatigue analysis. Relationship between stress amplitude and number of cycles it can execute before it fails. As you can guess as stress amplitude increases number of cycles for failure decreases. We will draw number of cycles in x axis, Stress amplitude in y axis. Both in logarithmic scale. Let’s start with the maximum stress a material can withstand, its ultimate stress. So this will happen, as you increase the stress even before completing one cycle the material will get broken. If you decrease the stress amplitude it will execute more number of cycles before it fails. Decreasing stress further even more number of cycles.

 Fig.5 Number of cylces for fatigue failure increases with decrease in stress amplitude
So this will follow a trend like this, but not forever. You can see after particular stress amplitude, even with slight decrease in stress number of cycles required to make it fail increases drastically.
 Fig.6 Stress amplitude Vs number of cycles, green region represents safe design area
Or in short if you have stress amplitude below this limit number of cycles to make to fail jumps ton infinity. Or material never fails after this limit. the material never fails. This limit is known as endurance limit; below endurance limit it is safe to operate the material. Engineers always try to design their components by keeping stress amplitude below endurance limit. You can see that endurance limit is way below ultimate stress value.

## Fatigue Failure, when there is no Complete Stress Reversal

The case we discussed had complete stress reversal. What will be maximum stress limit for this case ?. When stress reversal does not happen. It has got a mean value and amplitude.

 Fig.7 Fluctuating stress case which is not fully reversed
For this purpose we have to use something called Goodman diagram. Where mean value of stress is drawn on x axis. Amplitude of stress is drawn on y axis. When mean value of zero, we know safe stress limit is same as endurance limit. When amplitude of stress is zero, it is same as a static loading condition. So safe limit for tension is ultimate tensile stress at tension and safe limit for compression is ultimate tensile stress for compression. According to Goodman analysis safe stress amplitude limits for other cases lie on straight lines connecting this points. So for a particular stress mean value, we can find what’s the maximum allowable safe stress limit from this diagram. It will be here.
 Fig.8 Use of Goodman diagram to find safe stress amplitude when stresmm mean value is not zero
Similar analysis can be done considering, safe limit of amplitude zero condition as yield strength of material. This is known as Soberberg diagram. Generally Goodman analysis is the most preferred one.

# Centrifugal Pumps | Design Aspects

In this lecture, we will learn design aspects of centrifugal pumps. More precisely we will learn how to select a centrifugal pump and motor for pumping fluid at a specified rate, for a given system.

Summary of above lecture is given below

Before going to design part we will extent the theoretical knowledge gained in first video to more practical sense.

## Energy Loss in Pump – Head Reduction

Energy head developed by backward curved pump decreases linearly with flow. But this is theoretically maximum energy head possible. Obtained assuming whole shaft power input got transformed to fluid energy. This is true, only for ideal cases. In practice, there will be lots of energy losses associated with pump flow.

### Frictional Loss

One of the main energy loss is due to effect of friction in the flow. This loss increases quadratically with velocity. A similar loss occurs when there is sudden expansion or contraction.

 Fig.1 Vortices generated due to sudden contractio or sudden expansion of flow
Magnitude of this is also proportional to velocity square. So head curve will come down as shown in figure.
 Fig.2 Energy loss due to friction and flow area change and corresponding drop in pump head curve
This is the reason why we always try to transform dynamic part of fluid energy to static part in a centrifugal pump.

### Recirculation Loss

Next is due to recirculation effect in the flow. When flow is below the designed flow rate, recirculation losses become predominant as shown in figure. When pump operates at its designed flow rate recirculation loss is almost zero.

 Fig.3 Phenomenon of flow circulation and corresponding head drop

### Incidence Loss

If there is a difference in blade angle and flow angle, it will cause further loss. Here energy loss happens due to flow impingement and recirculation effect. This is again prominent in off design flow conditions. So it tends to have higher losses as we move away from designed flow rate point.

 Fig.4 Flow incidence and corresponding head loss
Energy losses we have discussed so far, which reduce head of the flow is known as hydraulic losses.

## Pump Performance Curve

The effective head verses flow rate curve is shown in Fig. 5(a).

 Fig.5 Typical pump performance curves
The shape could be as in Fig. 5(b) depending upon pump parameters. Such curves are known as pump performance curves. Please note that it is quite difficult to determine pump performance curve theoretically, rather they are determined experimentally.

## Pressure Rise across the Pump

Using pump performance curve one can easily predict what is the pressure rise across the pump, by applying energy equation across it.

 Fig.6 Pressure rise across the pump due to energy addition to it
Where value of h is determined from pump performance curve for corresponding flow rate.

## Power Gained by Fluid

Power gained by fluid will be lower than the power supplied.

 Fig.7 Power input to pump and power gained by the fluid
One main factor is hydraulic loss as we discussed. Other factors are volumetric loss and mechanical loss. So efficiency of a pump can be defined as power gained by fluid by power supplied to the pump.
For a typical centrifugal pump, efficiency will vary as shown in figure.
 Fig.8 Change in efficiency and pump shaft power input with flow rate
Corresponding shaft power variation is also shown. You can note that, there is an operating point in pump, where efficiency is maximum. It is known as best efficiency point. Corresponding point is marked on head and shaft power input curves.

## Impeller Selection

For a particular casing we could fit different sized impellers in it. Performance curves of different sized impellers are shown on same graph. Best efficiency points are also marked.

 Fig.9 Different pump performance curves as we chnge size of impeller
So back to the basic question, how to select a centrifugal pump for this application. Main condition is that fluid should be pumped at a particular flow rate to a specified height.
 Fig.10 The fluid pumping problem, where we have to pump fluid at a specified flow rate for a given system
Performance characteristic of the system is given in a system curve. That means how pressure drop varies in system with flow rate. Depending upon minor losses, major losses and altitude of network it would vary as shown in figure below. Please note that system curve will change drastically depending upon valve opening. Assume following is the one system curve at a particular valve opening. Required flow rate is also marked.
 Fig.11 System curve of the piping network
The operating point of pump will be intersection point of system curve and pump performance curve.
 Fig.12 Different pump operating points possible depending upon selectio of impeller
So depending upon selection of impeller the pump could operate anywhere at dotted points. But we have requirement, a requirement of specified flow rate. Out of these operating points the blue one is most near to the required flow rate. So we will select corresponding impeller. In the same graph we can represent iso-efficiency curves.
 Fig.13 From iso-efficieny curves we can determine efficiency of pump at operating condition
So efficiency at the operating condition also can be determined. The required shaft power can be calculated, using following equation.
Knowledge of power input requirement will lead to proper electric motor selection.

## Problem of Cavitation

This pump will operate well if it can overcome one more problem, problem of cavitation. We will learn how to design against cavitation in a separate article.

# Centrifugal Pump

Centrifugal pumps are the most commonly used turbo machinery devices. They are used to raise the pressure or induce flow in a control volume. Centrifugal pumps are radial flow devices. Various kinds of centrifugal pumps are available in the market with different construction details. But working principle behind all of them remain same. In this video we will analyze, working principle of a centrifugal pump with single suction, semi open impeller.

Following article gives detailed description of the video lecture.

## Working of Centrifugal Pumps

One of such pump (single suction, semi-open) is shown in figure below, with one part of its casing removed for ease of understanding.

 Fig.1 Single suction, semi open centrifugal pump with one portion of casing removed
Working of centrifugal pump is simple; as the impeller rotates it creates very low pressure at inlet of the impeller, called as eye of impeller. This low pressure helps in sucking fluid surrounding in. The fluid is pushed radially along the impeller to the casing. Casing collects the fluid , and it is pumped out through discharge nozzle.These processes are shown schematically in following figure. We will go through main components of a centrifugal pump in a detailed way.
 Fig.2 Fluid flow in a centrifugal pump

## Impeller

Impeller is the device which rotates, and transfer energy to fluid. It has got collection of vanes fitted to a hub plate. Shape and geometry of impeller blades are critical in pump performance.

 Fig.3 Details of impeller

## Casing

Casing collects fluid from impeller in an efficient way. The casing has got a special shape, with area of cross section increases from inlet to outlet. As the impeller ejects fluid throughout casing, along length of casing mass flow rate increases. But, increasing area of casing helps in maintaining almost same velocity. Thus volute shaped casing helps in converting dynamic part of fluid energy to static part.

## Construction Details of Casing

Casing is made on 2 volute curves, which are at offset. A three dimensional volute is made from this curves. A portion is removed from volute shape, in order to accommodate the impeller in it. A discharge nozzle is fit at exit portion of the casing, most of the time discharge nozzle is diverging in shape. The steps followed are shown in following figure.

 Fig.4 Construction details of volute casing

For centrifugal pumps of small capacity as we discussed, impeller and casing are its main components. But for larger centrifugal pumps, there will be additional diffuser blades also present, in order to reduce velocity further. Or they aid in dynamic to static energy conversion.

 Fig.5 Use of diffuser blades in large capacity centrifugal pumps

Blade and fluid velocities at inlet and outlet are shown in the figure below.

 Fig.6 Flow and Blade velocities at inlet and outlet of impeller
Here you can see fluid velocity increases from inlet to outlet due to energy addition to fluid. The work required for changing inlet velocity condition to outlet is given by following equation.

Details of such turbomachinery analysis will be discussed in a separate article. Here Q is the flow rate and Vtheta represents tangential velocity component of flow.From here we can find what’s the head rise in meters of fluid. Please note that this is energy head rise. It comprises of both pressure head and velocity head.
For a centrifugal pump, inlet velocity will be parallel to radius. So tangential component of velocity at inlet is zero.
Outlet blade angle beta can be derived in terms of velocities.
Also flow rate through impeller is given as flow area times radial velocity.
So head rise in a centrifugal pump, can be derived in terms of flow rate.
Using this equation we can predict what’s the head rise, as we change the flow rate for particular pump geometry and for a particular impeller angular velocity. Most important parameter in this equation is, blade outlet angle, beta. There can be 3 different pump characteristics depending upon value of this angle.

First case, if beta is less than 90 degree. Since second term in LHS of head vs flow equation is positive in this case, pressure head decreases with increase in flow. These kinds of impellers are called backward curved.

 Fig.7 Head vs Flow rate curve for a bacward curved blade impeller

If beta is 90 degree, with flow rate there is no change in pressure rise. Because second term in LHS of head vs flow equation is zero here. They are called Radial type.

If beta is more than 90 degree, pressure increases with increase in flow rate. Such blades are called forward curved blades.

 Fig.9 Head vs Flow rate curve for a forward curved blade impeller

## Most Suited Blade for Industrial Use

The big question is that out of these blade profiles, which one is the most suited for industrial use ?. To get answer for this question let’s see how power consumption varies with discharge for each of these cases. For backward curved blades as energy head decreases with discharge power consumption stabilizes with flow. In radial blades since head does not have any connection with flow rate, power consumption increases linearly. In forward curved blades since energy head increases with flow power consumption increases exponentially.This will make the operation unstable and will eventually lead to burnout of motor.

 Fig.9 Power consumption in different blade geometries
So backward curved blades which has got self stabilizing characteristics in power consumption is the most preferred one in industry.

You can find an article explaining the working of Centrifugal pumps in a practical way here .