UPPSC AE Mechanical 2013 Official Paper II (Previous Year Paper)

Concept:

T_sp = Temperature of water sprayed = 42°C, DBT = Dry bulb temperature of air = 40°C

WBT = Dry bulb temperature of air = 20°C

Now,

Here we see that T_sp > DBT, Hence air gets heated. 

Also, water is added to it, so air gets humidified.

Hence in this process air gets heated and humidified.

The given specific enthalpies are

Exit of evaporator (h₁) = 232 kJ/kg, Inlet of condenser (h₂) = 283 kJ/kg,

Exit of condenser (h₃) = 116 kJ/kg

h₄ = 116 kJ/kg      --- From p-h curve we can see that h₃ = h₄.

Now,   

\(COP =\frac{{Refrigerationeffect}}{{Workdone}}\)

${\rm{COP}} = \frac{{{\rm{h}}1 - {\rm{h}}4}}{{{\rm{h}}2 - {\rm{h}}1}}$

Substitute the values, we get

${\rm{COP}} = \frac{{232 - 116}}{{283 - 232}}$

${\rm{COP}} = \frac{{116}}{{51}}$

∴ COP = 2.27

Concept:

If a mass of moist air in an airtight vessel heated to a higher temperature, then relative humidity of the air decreases.

Important Points

Heating and Humidifying: A Simultaneous increase in both the dry bulb temperature and humidity ratio of the air.

Cooling and Dehumidifying: It involves the removal of water from the air as the air temperature falls below the dewpoint temperature.

Sensible heating: During this process, the moisture content of air remains constant and its temperature increases as it flows over a heating coil.

Sensible cooling: During this process, the moisture content of air remains constant, but its temperature decreases as it flows over a cooling coil.

In sensible heating/cooling, specific humidity is constant while relative humidity varies. In heating/cooling with chemical humidification/ dehumidification, both specific humidity and relative humidity varies.

Explanation:

Dew point temperature (DPT): 

Water vapour in air exists in the superheated state (state 1) and the air is highly unsaturated. When it gets cool at constant pressure, it meets the saturation vapour curve where the moisture of condensation begins.

The temperature at this point is known as the dew point temperature.

Dry bulb temperature (DBT): 

It is the temperature of moist air measured by ordinary thermometer

Wet bulb temperature (WBT): 

It is the temperature of air shown by a thermometer whose bulb is covered with a wet cloth.

Additional Information

Normally for unsaturated air: DBT > WBT > DPT

But for saturated air: DBT = WBT = DPT

Concept:

The psychometric chart is

For air with Relative humidity 80%, the dew point will be less than wet bulb temperature.

General trend of temperatures is:

DPT ≤ WBT ≤ DBT

At saturation curve i.e. RH = 100%

DPT = WBT = DBT

Concept:

Function of Expansion Device: It reduces the pressure from the condenser to the evaporator and it supplies the flow of refrigerant to the evaporator as per the desired capacity.

Capillary Tube: It is a narrow tube of constant cross-section area. The pressure drop in the capillary tube is achieved due to frictional resistance and due to the acceleration of the fluid in the tube. The pressure drop is directly proportional to the length of the capillary tube and inversely proportional to the diameter of the capillary tube. It is used in the domestic refrigerators, water coolers, and window Air conditioners.

Concept:

• A ton of refrigeration (TR), also called a refrigeration ton (RT), is a unit of power used, especially to describe the heat-extraction capacity of refrigeration and air conditioning equipment.
• It is defined as the rate of heat transfer that results in the melting of 1 ton of pure ice at 0°C in 24 hours.
• The unit conversions is:
• 1 TR = 210 kJ/min = 3.5 kW
• 1 kCal = 4.18 kJ
• ∴ 1 kJ = 0.239 kCal
• ∴ 210 kJ = 50.19 kCal
• ∴ 1 TR = 50.19 kCal/min = 3011.52 kCal/hr

Concept:

Efficiency of Carnot cycle $\eta=\frac{W}{Q_H}=\frac{T_H-T_c}{T_H}=0.75$

By reversing the Carnot cycle we can have either heat pump or refrigerator.

Calculation:

COP of heat pump = $COP=\frac{Q_H}{W}=\frac{T_H}{T_H-T_C}=\frac{1}{\eta }=\frac{1}{0.75}=1.33$

Concept:

• The temperature at which motionless saturated air would induce, in a sedentary worker wearing ordinary indoor clothing, the same sensation of comfort as that induced by the actual conditions of temperature, humidity, and air movement.
• The effective temperature is used as a guide in air-conditioning practice, and, on the comfort chart (American Society of Heating and Air Conditioning Engineers), it appears as a family of curves that serves as one coordinate in defining comfort zones.

Every psychometric chart includes vertical lines that represent the dry bulb temperatures, diagonals line represents wet bulb temperature and curved lines represent relative humidity.

Dry Bulb Temperature: Actual temperature of gas or mixture of gases

Wet Bulb temperature: Temperature obtained by an accurate thermometer having a wick moistened with distilled water

Dew point temperature: Temperature at which the liquid droplets just appear when the moist air is cooled continuously.

Relative humidity along the saturation line is 100%.

From the Psychrometric Chart, we can conclude that at constant specific humidity, DPT remains constant and is independent of DBT.

Concept:

In a vapour compression cycle, a good refrigerator should have:

• The refrigerant should have a low boiling point and low freezing point.
• It must have low specific heat and high latent heat because high specific heat decreases the refrigerating effect per kg of refrigerant and high latent heat at low temperature increases the refrigerating effect per kg of refrigerant.
• It must have high critical pressure and temperature to avoid large power requirements.
• It should have a low specific volume to reduce the size of the compressor.
• It must have high thermal conductivity to reduce the area of heat transfer in the evaporator and condenser.
• It should be non-flammable, non-explosive, non-toxic and non-corrosive.
• It should not have any bad effects on the stored material or food when any leak develops in the system.

Concept:

R-12 or Freon 12 is used in domestic refrigerators and freezers, liquid chillers, dehumidifiers, ice makers, water coolers, water fountains and transport refrigeration. The wide range of applications of the refrigerant are due to its safe properties.

Properties of R12 and its Advantages

1. Safe properties: Refrigerant R12 is nontoxic, nonflammable, and non-explosive.

2. Suitable for a wide range of operating conditions: R12 has the boiling point of -29.8°C due to which it condenses at the moderate pressures at the atmospheric temperature. This means the discharge pressure of the compressor should be only moderate. This helps in using the compressor of a low compression ratio that has higher efficiency.

3. Miscibility with oil: Refrigerant R12 is miscible with the compressor oil under all the operating conditions. So, there is no problem of the oil return back to the compressor.

R12 has a low refrigerating effect.

Concept:

The thermo technical characteristics of low grade fuels:

• high moisture and ash content
• low calorific value
• sticking and congealing during transportation
• fragility
• small fractional composition

1. Turbo jet:

• The turbojet is an air breathing jet engine, typically used in aircraft. It consists of a gas turbine with a propelling nozzle.
• The gas turbine has an air inlet, a compressor, a combustion chamber, and a turbine (that drives the compressor).
• The compressed air from the compressor is heated by the fuel in the combustion chamber and then allowed to expand through the turbine.
• The turbine exhaust is then expanded in the propelling nozzle where it is accelerated to high speed to provide thrust.

2. Pulse jet:

• The pulse jet is a type of jet engine in which combustion occurs in pulses.
• A pulsejet engine can be made with few or no moving parts and is capable of running statically (i.e. it does not need to have air forced into its inlet typically by forward motion).
• Pulsejets are used today in target drone aircraft, flying control line model aircraft.

3. Turbo-prop:**

• In its simplest form a turboprop consists of an intake, compressor, combustor, turbine and a propelling nozzle. Air is drawn into the intake and compressed by the compressor. Fuel is then added to the compressed air in the combustor, where the fuel-air mixture then combusts.
• The hot combustion gases expand through the turbine. Some of the power generated by the turbine is used to drive the compressor.

4. Rocket:

• Rocket engines produce thrust by the expulsion of an exhaust fluid that has been accelerated to high speed through a propelling nozzle.
• The fluid is usually a gas created by high pressure (150-to-4,350-pound-per-square-inch (10 to 300 bar)) combustion of solid or liquid propellants, consisting of fuel and oxidiser components, within a combustion chamber.
• As the gases expand through the nozzle, they are accelerated to very high (supersonic) speed, and the reaction to this pushes the engine in the opposite direction.

Concept:

Humidity ratio/Specific humidity (ω ) (kg/kg) of a given moist air sample is defined as the ratio of the mass of water vapour (mw) to the mass of dry air (ma) contained in the sample.

$w = \frac{{{m_w}}}{{{m_a}}} = 0.622\frac{{{P_v}}}{{P - {P_v }}}$

Total pressure is given by:

P = P_v + P_a

where p = atmospheric pressure, p_a = partial pressure of dry-air and P_v = partial pressure of water-vapour.

Relative humidity (ϕ):

It is defined as the ratio of partial pressure of water vapour (pv)in a mixture to the saturation pressure (ps) of pure water at the same temperature of the mixture.

$\phi = \frac{{{p_v}}}{{{p_{vs}}}}$

Basic Processes in Conditioning of Air:

Sensible heating: Moisture content of air remains constant so specific humidity is constant, temperature increases as it flows over a heating coil

Sensible cooling: Moisture content of air remains constant so specific humidity is constant, but its temperature decreases as it flows over a cooling coil

Cooling and dehumidification: This process involves lowering of both the air temperature and the specific humidity. This process is commonly used in summer air conditioning in which air passes over a cooling coil. When moist air is cooled below its dew point, vapour is condensed from the air resulting in simultaneous cooling and dehumidification.

Heating and Humidification: During winter it is essential to heat and humidify the room air for comfort. It is done by first sensibly heating the air and then adding water vapour to the air stream through steam nozzles, as a result, both the temperature and humidity ratio of air increases.

Cooling & humidification: Air temperature drops and its humidity increases. 

Heating and de-humidification: This process can be achieved by using a hygroscopic material, which absorbs the water vapour from the moisture. If this process is thermally isolated, then the enthalpy of air remains constant, as a result, the temperature of air increases.

Concept:

• The Vapour compression cycle consists of phase change from gas to liquid in condenser and the compressor pumps liquid phased refrigerant.
• Making it more efficient as compression work per unit of mass flow is low. It is generally used in refrigeration purposes and air conditioning in buildings.
• The Air cycle uses air as a working fluid and doesn't have phase change in any part of the cycle. Meaning air always remains in the gas state.
• Hence the fluid circulation per KW of cooling is very high. Also, the compression work is more.
• One of the very few applications that satisfy its suitability is air conditioning in aircraft.

So (COP)_air < (COP)_vap.c

Concept:

In an aircraft, cooling systems are required to keep the cabin temperatures at a comfortable level. Even though the outside temperatures are very low at high altitudes, still cooling of cabin is required due to:

• Large internal heat generation due to occupants, equipment etc.
• Heat generation due to skin friction caused by the fast moving aircraft.
• At high altitudes, the outside pressure will be sub-atmospheric. When air at this low pressure is compressed and supplied to the cabin at pressures close to atmospheric, the temperature increases significantly.
• For example, when outside air at a pressure of 0.2 bar and temperature of 223 K (at 10000 m altitude) is compressed to 1 bar, its temperature increases to about 353 K. If the cabin is maintained at 0.8 bar, the
• temperature will be about 332 K. This effect is called as ram effect. This effect adds heat to the cabin, which needs to be taken out by the cooling system.

Sensible heating: It is the process of increasing the Dry bulb temperature at constant specific humidity.

Effects of sensible heating:

i) Dry bulb temperature increases

ii) Sp. Humidity or humidity ratio remains constant

iii) Dew point temperature is constant

iv) Relative humidity decreases

v) Enthalpy increases

vi) Specific Volume Increases

vii) Wet bulb temperature increases

Concept:

Kelvin-Plank gives the concept of heat engine hence, it is a conversion of heat into work. According to Kelvin-Planck statement of second law “It is impossible to construct a device (engine) operating in a cycle that will produce no effect other than extraction of heat from a single reservoir and convert all of it into work.”

Clausius statement: It is impossible to construct a system which will operate in a cycle, transfers heat from the low-temperature reservoir (or object) to the high-temperature reservoir (or object) without any external effect or work interaction with the surrounding.

Explanation:

Intensive Property: These are the properties of system which are independent of mass under consideration. For e.g. Pressure, Temperature, density

Extensive Properties: The properties which depend on the mass of system under consideration.

For e.g Internal Energy, Enthalpy, Volume, Entropy

Note: All specific properties are intensive properties. For e.g. specific volume, specific entropy etc.

Steady flow energy equation for a compressor from the above figure is given by:

h₁ + dQ = h₂ + dW ----(i)

Neglecting the changes of potential and kinetic energy. From the property relation

Tds = dh - νdp

For a reversible process,

Tds = dQ

So,                                          

dQ = dh - νdp    ----(ii)

If consider the process is reversible adiabatic then

dQ = 0

From equation (i) and (ii),      

h₁ - h₂ = dW  

dh = h₂ - h₁ = - dW    ----(iii)

And,                                        

dh = νdp    ----(iv)

From equation (iii) and (iv),   

-dW = νdp

W = -∫ νdp

Negative sign shows the work is done on the system (compression work) for initial and Final stage.

Concept:

For reversible heat engine efficiency

$\begin{array}{l} \eta = 1 - \frac{{{T_R}}}{{{T_S}}}\end{array}$

$\begin{array}{l} \eta = \frac{W}{{{Q_S}}} = 1 - \frac{{{Q_R}}}{{{Q_S}}}\ \end{array}$

Calculation:

$\begin{array}{l} \eta = 1 - \frac{{{T_R}}}{{{T_S}}} = 1 - \frac{{300}}{{500}} = 0.4\ \eta = \frac{W}{{{Q_S}}} = 1 - \frac{{{Q_R}}}{{{Q_S}}}\ {Q_R} = 150;kW \end{array}$

∴ Q = 150 kW or 150 kJ/s

Concept:

Thermodynamic cycle:

It can be defined as a series of state changes such that the final and initial state is identical.

Thermodynamic process:

It is a passage of a thermodynamic system from an initial state to the final state of the thermodynamic equilibrium. The initial and final states of the process are the defining elements. 

Thermodynamic property:

A thermodynamic property is any property that is measurable, and whose value describes a state of the system. Some e.g of the thermodynamic property are pressure, temperature, viscosity, density. Properties are point function i.e. these do not depend on the path followed.

Concept:

Throttling Calorimeter: A throttling calorimeter is based on the principle of throttling the wet steam so that it becomes superheated.

Separating Calorimeter: It is a vessel used initially to separate some of the moisture from the steam, to ensure superheat conditions after throttling.

Combined Separating and Throttling Calorimeter

A combined separating and throttling calorimeter is therefore found most suitable for accurate measurement of dryness of steam reducing the drawback of Separating Calorimeter (i.e. water particles from wet steam are not fully separated) and the drawback of Throttling Calorimeter (i.e. not suitable for very wet steam).

x₁ = dryness fraction of steam considering separating calorimeter

x₂ = dryness fraction of steam considering the throttling calorimeter

x = actual dryness fraction of steam in the sample

Calculation:

Actual dryness fraction of steam in the sample is:

X = x₁ × x₂

Given:             

x ₁ = 0.8, x₂ = 0.8   

Calculation:       

x = 0.8 × 0.8

∴ X = 0.64

Explanation:

The state of a system in which properties are definite as long as external conditions are unchanged is called an equilibrium state.

Thermodynamic equilibrium:

The system is said to be in thermodynamic equilibrium if the conditions for the following three equilibrium is satisfied:

1. Mechanical equilibrium: 

When there are no unbalanced forces within the system and between the system and the surrounding, the system is said to be under mechanical equilibrium.

This means that the pressure of the system is con**stant everywhere.

2. Chemical equilibrium: 

The system is said to be in chemical equilibrium when there are no chemical reactions going on within the system or there is no transfer of matter from one part of the system to other due to diffusion.

3. Thermal equilibrium: 

When the temperature of the system is uniform and not changing throughout the system and also in the surroundings, the system is said to be in thermal equilibrium.

This means that the temperature of the system is constant everywhere.

A system is said to be in thermodynamic equilibrium if it is simultaneously in mechanical, thermal, and chemical equilibrium.

Therefore, thermodynamic equilibrium is completely defined by the specification of internal energy (since it is a function of temperature only), enthalpy (enthalpy (H) = Internal energy (U) + PV), and generalized displacements (i.e. due to unbalanced forces).

Explanation:

If a process starts from the same initial state and goes to the same final volume then work done will be maximum for an isobaric process, i.e. constant pressure process.

The different processes given in the P-V diagram are:

1: Isochoric process

2: Adiabatic process

3: Isothermal process

4: Isobaric process

∴ It is evident from the P-V diagram that the area under the curve is maximum during the constant pressure process.

Concept:

• 1 kilowatt hour (kWh) = 3,600,000.00 newton meters (N-m)

• 1 newton meter (N-m) = 0.000238 kilocalories (kcal)

• 1 horsepower hour (Hp hr) = 0.75 kilowatt hours (kWh)

• 1 kilocalorie (kcal) = 4,186.80 newton meters (N-m)

Concept:

\({\eta _{th}}; = ;1 - \frac{1}{{r^{ \gamma -1}}}\left{ {\frac{{r_c^\gamma - 1}}{{\gamma \left( {{r_c} - 1} \right)}}} \right}\)

From the above equation, it is observed that the thermal efficiency of the diesel engine can be increased by increasing the compression ratio r, by decreasing the cut - off ratio rc, or by using gas with a large value of γ.

Hence we can say that from the given options, the second option is correct because the thermal efficiency decreases with an increase in heat addition cut-off ratio.

Concept:

Rankine cycle is a reversible cycle which have two constant pressure and two isentropic processes. These are four processes in the Rankine cycle:

Process 1 – 2: Isentropic compression

Process 2 – 3: Isobaric heat addition

Process 3 – 4: Isentropic expansion

Process 4 – 1: Isobaric heat rejection

Apply Steady Flow energy equation:

For boiler: Q_in = h₃ – h₂

For turbine: W_turb,out = h₃ – h₄ (change of enthalpy between inlet and outlet)

For Condenser: Q_out = h₄ – h₁

For Pump: W_pump,in = h₂ – h₁

Concept:

Polytropic Process is represented by

PVn = C

• n = 0 ⇒ P = C ⇒ Constant Pressure Process (Isobaric Process)
• n = 1 ⇒ PV = C ⇒ Constant Temperature Process (Isothermal process)
• n = γ ⇒ PVγ = C ⇒ Adiabatic Process
• n = ∞ ⇒ V = C ⇒ Constant Volume Process (Isochoric process)

From the above graph, we can see that in the expansion process, the work done increases as the value of n decreases.

Concept:

• The available energy of a system decreases as its temperature or pressure decreases and approaches to that of surroundings.
• When heat is transferred from a system, its temperature decreases, and hence the quality of the energy also decreases.
• The degradation is more for energy loss at a higher temperature.
• Quantity wise the losses may be the same but the quality-wise the losses are different.
• If the first law is the law of conservation of energy, then the second law is called the law of degradation of energy as it is the second law which tells us that energy always degrades quality-wise
• The Law of degradation of energy says that unavailable energy is gradually decreasing due to an increase in reversible processes.

Available Energy: Maximum amount of work extracted from a heat input in a cycle.

W_max (Available energy)= Q_s-T_od_s

Unavailable energy: Minimum amount of energy that is rejected to a sink.

Unavailable energy=T_od_s

 

As we can see in the above figure that when there is an increase in reversible processes (lesser will be irreversibility in the form of friction, heat dissipation) resulting in greater work output and decrease in heat rejection i.e. unavailable energy.

Concept:

Brayton Cycle:

The Brayton cycle (or Joule cycle) represents the operation of a gas turbine engine.

The cycle consists of four processes

• Process a - b: Reversible adiabatic compression in the inlet and compressor
• Process b - c: Isobaric heat addition (fuel combustion)
• Process c – d: Reversible adiabatic expansion in the turbine and exhaust nozzle
• Process d - a: Cool the air at constant pressure back to its initial condition.

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The efficiency of the Brayton cycle:

${\eta _{\rm{b}}} = 1 - \frac{1}{(r_p)^{\frac{\gamma -1}{\gamma}}}$

Where, r_p = (pressure ratio), compression ratio = (P_b/P_a) = (P_c/P_d )

Otto cycle:

Otto cycle consists of four processes. They are as follows:

Process 1-2: Reversible adiabatic or Isentropic compression

Process 2-3: Constant Volume Heat Addition

Process3-4: Isentropic (reversible adiabatic) expansion.

Process 4-1: Constant Volume Heat Rejection

The efficiency of the Otto cycle:

${\eta _{otto}} = 1 - \frac{1}{{{r^{\gamma - 1}}}}$

Where r = compression ratio $= ,,\frac{{V2}}{{V1}}$

When compression ratio is the same in both the Otto cycle and Brayton cycle i.e. r = r_p

Then, η_Otto = η_b.

∴ Option 2 is correct.

Concept:

The Brayton cycle is a theoretical cycle for simple gas turbine. Ideal Brayton cycle consists of two isentropic and two constant pressure processes.

As the plot depicts for a fixed turbine inlet temperature T₃, the net work output per cycle increases with the pressure ratio initially, it reaches a maximum value and then it starts to decrease.                                                        

For a fixed values of T_max and T_min pressure ratio for maximum work done by gas turbine:

(r_p )_{max work} = (T_max/T_min)^k/2(k-1)

where, k = specific heat ratio**, T_max **= Maximum temperature in gas turbine cycle

 T_min = Minimum temperature in the gas turbine cycle

Concept:

For a laminar fully developed flow through pipes, the friction factor, f is found from the exact solution of the Navier-Stokes equation is given by:

$f = \frac{{64}}{{Re}}$

Where ,  f = Friction factor, Re = Reynolds Number

Thus, the friction factor is inversely proportional to Reynolds Number. So, after plotting in the log plain it is like a straight line coming downwards. So, that is what you see for the laminar flow in the moody chart.

• When the pressure goes on increasing in the direction of flow, the pressure force acts against the direction of flow in the boundary layer and hence thickenings the boundary layer more rapidly.
• This and the boundary shear bring the fluid in the boundary layer to rest and cause backflow.
• Due to this the boundary layer no more sticks to the boundary but is shifted away from the boundary.

This phenomenon is called as “Boundary layer separation”.

A favorable pressure gradient is one in which the pressure decreases in the flow direction (i.e., dp/dx < 0)

On the other hand, an adverse pressure gradient is one in which pressure increases in the flow direction (i.e., dp/dx > 0)

At the verge of separation ${\left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}}$ is zero

∴ Shear stress $\tau = \mu {\left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}}$ is also zero

Concept:

Velocity Potential function: This function is defined as a function of space and time in flow such that the negative derivation of this function with respect to any direction gives the velocity of the fluid in that direction.

If velocity potential (ϕ) exist, there will be a flow.

$u = - \frac{{\partial ϕ }}{{\partial x}}$

$v = - \frac{{\partial ϕ }}{{\partial y}}$

$w = - \frac{{\partial ϕ }}{{\partial z}}$

Now, Angular velocity is given by:

${\omega _z} = \frac{1}{2}\left( {\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}}} \right)$

${\omega _z} = \frac{1}{2}\left[ {\frac{\partial }{{\partial x}}\left( { - \frac{{\partial ϕ }}{{\partial y}}} \right) - \frac{\partial }{{\partial y}}\left( { - \frac{{\partial ϕ }}{{\partial x}}} \right)} \right]$

${\omega _z} = \frac{1}{2}\left[ { - \frac{{{\partial ^2}ϕ }}{{\partial x\partial y}}; + \frac{{{\partial ^2}ϕ }}{{\partial y\partial x}};} \right]$

Since ϕ is a continuous function

$\frac{{{\partial ^2}ϕ }}{{\partial x\partial y}} = ;\frac{{{\partial ^2}ϕ }}{{\partial y\partial x}};$

Therefore, ωz  = 0

It is important to note that, for all irrotational flow, a velocity potential function ϕ exists whether the flow is compressible or incompressible.

Important points

• Stream function exists only when compressibility is negligible, irrespectively of whether the motion is rotational or irrotational.
• This is because the existence of stream function is merely a consequence of the assertion of the continuity of incompressible flow.

Concept:

• Mercury has very low vapor pressure and hence it is an excellent fluid to be used in a barometer.
• Another reason for using mercury is it doesn't wet the glass, good conductor of heat and it expands evenly with temperature.

Concept:

Cavitation is the phenomenon of formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below the vapour pressure of the fluid and sudden collapsing of these bubbles in the region of higher pressure.

Cavitation:

• It is the formation of gas bubbles of a flowing liquid in a region where the pressure of the liquid falls below its vapour pressure i.e. low pressure.
• It is the phenomenon of formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below the vapour pressure of the fluid and sudden collapsing of these bubbles in the region of higher pressure.
• In centrifugal pumps, the cavitation may occur at the inlet of the impeller of the pump or at the suction side of the pumps, where the pressure is considerably reduced.
• Hence if the pressure at the suction side of the pump drops below the vapour pressure of the liquid then the cavitation may occur.

Concept:

The stream function in a two-dimensional flow automatically satisfies the continuity equation.

Stream Function: It is the scalar function of space and time.

The partial derivative of stream function with respect to any direction gives the velocity component perpendicular to that direction. Hence it remains constant for a streamline

$u = - \frac{{\partial \psi }}{{\partial y}}$ & $v = \frac{{\partial \psi }}{{\partial x}}$

Stream function defines only for the two-dimensional flow which is steady and incompressible..

Properties of stream function:

1. If ψ exists, it follows continuity equation and the flow may be rotational or irrotational.
2. If ψ satisfies the Laplace equation, then the flow is irrotational.

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Continuity equation:

$\frac{{\partial {\rm{;u}}}}{{\partial {\rm{;x}}}} + \frac{{\partial {\rm{;v}}}}{{\partial {\rm{;y}}}} = 0$

$i.e.;\frac{{\partial {\rm{;}}}}{{\partial {\rm{;x}}}}\left( { - \frac{{\partial \psi }}{{\partial y}}} \right) + \frac{{\partial {\rm{;}}}}{{\partial {\rm{;y}}}}\left( {\frac{{\partial \psi }}{{\partial x}}} \right) = 0$

⇒  $- \frac{{{\partial ^2}\psi }}{{\partial x\partial y}} + \frac{{{\partial ^2}\psi }}{{\partial x\partial y}} = 0$

Concept:

For the flow to occur between two points in a pipeline the differential pressure between these points should be more than

• Surface friction
• Viscosity force
• Frictional force

So that the pressure force of fluid can overcome the surface friction, viscous effects and frictional resistances allowing the flow to take place in a pipeline between any two points in a pipeline.

Concept:

Free Vortex Flow: In free vortex flow the liquid particles describe circular paths about a fixed vertical axis without any external torque acting on them.

Example: The flow of water through the hole in the bottom of a washbasin.

Forced vortex flow: In this type of flow, the vessel containing the liquid is forced to rotate about the fixed vertical axis with the help of some external torque.

Explanation:

Reynold's number: 

• It is a dimensionless number that determines the nature of the flow of liquid through a pipe or flat plate.
• It is defined as the ratio of the inertial force to the viscous force for a flowing fluid.

Reynold's number is written as Re.

${Re} = \frac{{{\rm{Inertial;force}}}}{{{\rm{Viscous;force}}}}\Rightarrow\frac{ρ Vx}{μ}=\frac{Vx}{ν}$

where ρ = density of fluid, V = velocity of flowing fluid, μ = dynamic viscosity of fluid, ν = kinematic viscosity of fluid and x = characteristic diameter.

where $x=\frac{4A}{P}$

and A = area of cross-section and P = perimeter.

Explanation:

• In the boundary layer, viscous forces dominate over inertial forces.
• The viscous forces are those associated with transverse velocity gradients such as in the boundary layer while inertial forces dominate outside the boundary layer, as the flow outside boundary layer is the inviscid flow region.

Concept:

When a liquid flowing through a long pipe is suddenly brought to rest by closing the valve at the end of a pipe, then a pressure wave of high intensity is produced behind the valve. This pressure wave of high intensity has the effect of hammering action on the walls of the pipe. This phenomenon is known as water hammer or hammer blow.

The magnitude of water hammer depends upon

(a) The length of the pipeline,

(b) The elastic properties of the pipe material,

(c) The elastic properties of the liquid flowing through the pipe, and

(d) The time at which the valve is closed.

Concept:

${\rm{Mach;Number}} = \frac{{Inertia;force}}{{elastic;force}} = \frac{{Inertia;force}}{{Compressible;force}}$

• Incompressible flow
• M < 0.3
• From the above point, it is clear that the compressibility effect can be treated as negligible when the Mach number is up to 0.2
• Subsonic flow
• 0.3 < M < 1
• Sonic flow
• M = 1
• Supersonic flow
• 1 < M < 5
• Hypersonic flow
• M > 5

Explanation:

streamlined body:

It is defined as the body whose surface coincides with the streamline when the body is placed in the flow. Flow separation takes place only at the trailing edge in such cases.

• The boundary layer in such bodies must be attached to the surface of the body for a long time as possible and hence behind a streamlined body wake formation zone will be very small and consequently, pressure drag will be very small.
• Thus, the total drag on the streamlined body will be due to friction (shear) only.
• A streamlined body is a shape that lowers the friction drag between a fluid, like air and water, and an object moving through that fluid. Drag is a force that slows down motion. Friction drag is a special kind of drag.

Concept:

Mach number (M): It is defined as the ratio of inertia force to compressibility force.

$M = \frac{V}{c} = \frac{V}{{\sqrt {\frac{{dP}}{{d\rho }}} }}$

• Mach number is defined as the ratio of the speed of the fluid flow to the speed of sound in the same medium under similar conditions of temperature and pressure.
• If Mach Number is greater than 1, the body is called as Supersonic whereas it is said to be moving with Subsonic speed, if it is less than 1.
• If the Mach Number is greater than 5, the body is known as Hypersonic.

Reynolds number (Re): It is defined as the ratio of inertia force to viscous force

$Re = \frac{{\rho Vl}}{\mu }$

Euler number (Eu): It is defined as the ratio of inertia force.to pressure force.

$Eu = \frac{V}{\sqrt{\frac{P}{\rho}}}$

Froude number (Fr): It is defined as the ratio of inertia force to gravity force.         

$Fr = \frac{V}{{\sqrt {gL} }}$

Weber number (We): It is defined as the ratio of the inertia force to surface tension force.

$We = \frac{{\rho {V^2}l}}{\sigma }$

Concept:

Nominal boundary thickness as δ 

Displacement thickness is given as

\({δ ^*} = \mathop \smallint \limits_o^δ \left( {1 - \frac{u}{U}} \right)dy\)

Calculation:

\({δ ^*} = \mathop \smallint \limits_o^δ \left( {1 - \frac{y }{δ}} \right)dy\)

${δ ^*} =δ - \frac{δ }{2}$= $\frac{δ }{2}$

∴ The ratio of displacement thickness to nominal thickness will be ​ = $\frac{{\frac{δ }{2}}}{δ }$  

∴ After simplification, the answer will be 1/2.

Concept:

The governing equation is

$\frac{u}{{{u_0}}} = 1 - {\left( {\frac{r}{R}} \right)^2}$

Calculation:

The relation between the maximum velocity (u_m) and the average velocity (u₀) is

${u_m} = \frac{{{u_0}}}{2} = 0.5{u_0}$

Now,

Let α be the ratio of the total kinetic energy of fluid passing per second to the energy value obtained on the basis of average velocity.

∴ $α = \frac{{\smallint {u^3}dA}}{{Au_m^3}}$

∴ $α = \frac{{\smallint {u_0}{{\left[ {1 - {{\left( {\frac{r}{R}} \right)}^2}} \right]}^3}\left( {2\pi r} \right)dr}}{{\pi {R^2}{{\left( {0.5} \right)}^3}u_0^3}}$

Integrating the above equation

∴ α = 2

Note: The derivation of the above equation is a standard result and can be remembered directly as a relation.

Concept:

The acoustic velocity can alternatively be expressed with Hook's Law as

C = (K / ρ)^1/2

Where, C = speed of sound, K = Bulk modulus of elasticity, ρ = density of medium

C ∝ K^1/2

Now, Coefficient of compressibility is defined as inverse of Bulk modulus of elasticity,

${\rm{\beta }} = \frac{1}{K}$

Hence,                                    

${\rm{C}} \propto {{\rm{\beta }}^{ - \frac{1}{2}}}$

Therefore**,**

Sonic velocity in fluid is

• Directly proportional to square root of Bulk modulus of medium.
• Inversely proportional to square root of coefficient of compressibility of medium.

Concept:

• An adiabatic process is a process in which there is no heat interaction or heat transfer through the system boundary. The adiabatic process can be reversible or irreversible.
• However, an isentropic process is a process during which entropy of the system must remain constant, which means entropy does not change.
• In fluid dynamics, an isentropic flow is a fluid flow that is adiabatic as well as reversible i.e. no heat is added to the flow and no energy transformations take place due to friction or any dissipative effects.

Concept:

• A venturimeter is a device used for measuring the rate of flow of a fluid of a liquid flowing through a pipe

• The venturimeter always have a smaller convergent portion and larger divergent portion

• The size of the venturimeter is specified by its pipe diameter as well as throat diameter.

• This is done to ensures a rapid converging passage and a gradual diverging passage in the direction of flow to avoid the loss of energy due to separation

• In the course of flow through the converging part, the velocity increases in the direction of flow according to the principle of continuity, while the pressure decreases according to Bernoulli’s theorem

• The velocity reaches its maximum value and pressure reaches its minimum value at the throat

• Subsequently, a decrease in the velocity and an increase in the pressure take place in course of flow through the divergent part

• The angle of convergence ≈ 20°, Angle of divergence = 6° - 7° → It should be not greater than 7° to avoid flow separation

Concept:

• A stagnation point is a point in a flow field where the local velocity of the fluid is zero.
• The Bernoulli equation shows that the static pressure is highest when the velocity is zero and hence static pressure is at its maximum value at stagnation points. Because all the velocity head is converted into the pessure head at stagnation point.
• This static pressure is called the stagnation pressure.

Explanation:

In laminar flow between parallel plates, the two most important forces are viscous forces and pressure forces.

Viscous forces arise due to the friction between the fluid layers and act in the direction of decreasing velocity. They are responsible for the parabolic velocity profile observed in laminar flow, where the velocity is highest at the center of the channel and decreases towards the walls.

Pressure forces are the forces acting due to the difference in pressure between the two plates. They are responsible for driving the flow from the high-pressure region to the low-pressure region.

The other options are incorrect because:

Option 1: Inertial and viscous is not correct because inertial forces are only significant in turbulent flow, where the fluid motion is highly chaotic and irregular. In laminar flow, inertial forces are negligible.

Option 3: Gravity and pressure is not correct because gravity is only significant in flow situations where the fluid density is not uniform, such as in natural convection. In laminar flow between parallel plates, the fluid density is assumed to be uniform, and gravity does not play a significant role.

Option 4: Pressure and inertial are not correct because inertial forces are not significant in laminar flow, as explained earlier. The most important forces in laminar flow between parallel plates are viscous forces and pressure forces.

Concept:

Thermal diffusivity of material is given as, $\alpha = \frac{k}{{\rho c}}$. It is the property of a material.

Larger the value of α, faster heat will diffuse through the material. It means less heat will be stored and more conduction will occur.

A high value of α could result either from a high value of thermal conductivity or low value of thermal heat capacity ρc. Thermal diffusivity α has units of square meters per second (m2/s).

Concept:

Pipe with insulation can be assumed as cylinder. 

• The value of Router for which the heat transfer rate is maximum is called the critical radius of insulation.
• The thickness up to which heat flow increases and after which heat flow decreases is termed as Critical thickness.
• The insulation radius at which resistance to heat flow is minimum and consequently heat flow rate is maximum is called “critical radius”. And it starts decreasing after the critical thickness of insulation.

Note that the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.

The critical radius of insulation for a cylindrical body:

${r_{cr,cylinder}} = \frac{k}{h}$

Critical radius of insulation for a spherical shell:

${r_{cr,sphere}} = \frac{{2k}}{h}$

The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < rcr, reaches a maximum when r2 = rcr, and starts to decrease for r2 > rcr. Thus, insulating the pipe may increase the rate of heat transfer from the pipe instead of decreasing it when r2 < rcr.

Concept:

Prandtl Number (Pr) = $\frac{{kinematic;viscosity}}{{thermal;diffusivity}}$

Calculation:

$\begin{array}{l} \therefore Pr = \frac{\nu }{\alpha }\ \therefore Pr = ;\frac{{\mu C{\rm{p}}}}{k}\ \therefore {\rm{Pr}} = \frac{{0.001 \times 1000}}{1} \end{array}$

∴ Pr = 1

Where**, µ** = dynamic viscosity, C_p = Heat at constant pressure, k = Thermal Conductivity

Also, 

$\frac{\delta }{{\delta t}}$ = Pr^1/3      

∴ 𝞭 = 𝞭_t   (∵ Pr = 1) 

∴ Thermal boundary layer thickness: 𝞭_t = 1 mm

Explanation:

Effectiveness (ϵ)

Effectiveness is defined as the ratio of heat transfer rate with a fin to heat transfer rate without fin.

\(\epsilon = \frac{{{{\dot Q}{with;fin}};}}{{{{\dot Q}{without;fin}}}}\)

For a long fin

\(\epsilon = \frac{{{{\dot Q}{with;fin}};}}{{{{\dot Q}{without;fin}}}} = \sqrt {\frac{{KP}}{{h{A_c}}}} \)

where K = thermal conductivity of fin, P = perimeter of fin

h = heat transfer coefficient, A_c = cross-section area of fin

From the above formula, we can conclude:

Case 1: If $\frac{P}{{{A_c}}}$ increases, the effectiveness of fin increases.

• A_c must be small (thin fin) and fins must be closed (not too close).
• If fins are too closed it will abstract flow of air which will decrease the effectiveness of fin.

Case 2: if K increases, the effectiveness of fin increases.

• If K is large, the temperature drop along the length will be less.
• And the temperature drop between fin and surrounding will be more and heat transfer will be more.

Case 3: if h decreases, the effectiveness of fin increases.

• Fins are more effective under free convection ( h is less).

Concept:

Flow configuration in the condenser as shown below.

ΔT₁ = T_h,i - T_c,i,  ΔT₂ = T_h,o - T_c,o

$LMTD=\frac{{\Delta}T_1~-~{\Delta}T_2}{ln{\frac{\Delta{T_1}}{\Delta{T_2}}{}}}$

 

Calculation:

Given:

T_h,i = T_h,o = 60oC, T_c,i = 30oC, T_c,o = 45oC

 Therefore, ${\rm{Δ }}{T_1} = 30^\circ C,{\rm{Δ }}{T_2} = 15^\circ C,$

Now, we know that $LMTD = \frac{{{\rm{Δ }}{T_1} - {\rm{Δ }}{T_2}}}{{In\left( {\frac{{{\rm{Δ }}{T_1}}}{{{\rm{Δ }}{T_2}}}} \right)}} = \frac{{30 - 15}}{{In\left( {\frac{{30}}{{15}}} \right)}} = 21.6^\circ C$

Concept:

In a heat exchanger, the temperature of the hot fluid decreases while the temperature of the cold fluid increases. The increase and decrease follow an exponential law.

It can be parallel as well as counter flow.

Now,

Effectiveness of a parallel flow heat exchanger:

${\epsilon_{parallel}} = \frac{{1 - \exp \left[ { - NTU\left( {1 + R} \right)} \right]}}{{1 + R}}$

Effectiveness of a counter flow heat exchanger:

${\epsilon_{counter}} = \frac{{1 - \exp \left[ { - NTU\left( {1 - R} \right)} \right]}}{{1 - R\exp \left[ { - NTU\left( {1 - R} \right)} \right]}}$

From the above explanation, we can clearly see that it follows the exponential law.

Concept:

Among the following options given value of thermal conductivity of air is minimum.

${K_{Air}} \simeq 0.02$

The thermal conductivity of a material can be defined as the rate of heat transfer through a unit thickness of the material per unit area per unit temperature difference.

The thermal conductivity of a material is a measure of the ability of the material to conduct heat. A high value for thermal conductivity indicates that the material is a good heat conductor, and a low value indicates that the material is a poor heat conductor or insulator. T

The thermal conductivities of some common materials at room temperature are given as:

Concept:

The lumped heat capacity analysis is such an analysis in which temperature is the only function of time and not a function of space.

T ≠ f (space)

T = f (Time) only

For lumped parameter analysis

Biot Number < 0.1

${\rm{Biot;No}} = \frac{{{\rm{Internal;Conductive;Resistance;}}\left( {{\rm{ICR}}} \right)}}{{{\rm{External;Convective;Resistance;}}\left( {{\rm{ECR}}} \right)}}$

∴ for Biot No → 0, we should have Internal Conductive Resistance (ICR) as negligible.

Concept:

$Q = kA\frac{{dT}}{{dx}} \Rightarrow \frac{{dT}}{{dx}} = \frac{q}{k};$

For same q, is k is low (i.e. for an insulator), dT/dx will be large i.e. there will be a large temperature difference across the wall.

If k is high (i.e. for a conductor), dT/dx will be small i.e. there will be a small temperature difference across the wall.

Material with high thermal conductivities are good conductors of heat, whereas materials with low thermal conductivities are good thermal insulators.

Conduction of heat occurs most readily in pure metals, less so in alloys and much less readily in non-metals.

The very low thermal conductivities of certain thermal insulators e.g. cork is due to their porosity, the air trapped within the material acting as an insulator.

Concept:

In unsteady state heat transfer, the temperature varies with time as well as location within the system.

T = f(x, y, z, t) and $\frac{{\partial T}}{{\partial t}} \ne 0$

In steady state heat transfer, the temperature at any location within the system does not vary with time. The temperature in steady state is the function of space coordinates while it is independent of time.

T = f(x, y, z) and $\frac{{\partial T}}{{\partial t}} = 0$

Except in case of annealing in all other case temperature does not change with time at any location.

Hence annealing of castings is the correct answer

Concept:

To measure the high temperature inside a furnace, the options are:

The upper limit for thermistor devices is around 100°C (212°F) and RTDs are limited to around 750°C (1382°F). That leaves thermocouples and infrared pyrometers or imagers as the most suitable devices for measuring temperatures above 1000°C.

Hence Optical pyrometer is used to measure the temperature inside a furnace.

Concept:

A steam boiler is designed to absorb the maximum amount of heat released from the process of combustion.

Heat transfer within a steam boiler is accomplished by three methods: radiation, convection, and conduction.

The heating surface in the furnace area receives heat primarily by radiation. The remaining heating surface in the steam boiler receives heat by convection from the hot flue gases. Heat received by the heating surface travels through the metal by conduction, heat is then transferred from the metal to the water by convection.

Concept:

The maximum density of water is 103 kg m-3 at 4 oC temperature.

• The maximum water density occurs at 4°C, since two opposing effects are in equilibrium at this temperature.
• The water molecules in ice are in a crystal lattice that has plenty of empty space. When the ice melts into liquid water, the structure breaks and the liquid density increases.
• The molecules move more quickly and get further apart at temperatures well above zero. Density decreases as the temperature increases. The water also contains several ice-like clusters at temperatures close to 0°C.

Concept:

Values of some important bodies is given below

Explanation:

Concept of the critical thickness of insulation:

• In a plane wall thicker the insulation, lower the heat transfer rate. This is due to the fact the outer surface have always the same area.
• But in cylindrical and spherical coordinates, the addition of insulation also increases the outer surface area which decreases the convection resistance at the outer surface.
• At the same time conduction resistance increases due to increase in insulation thickness. So, to optimize the balance of conductive and convective resistance, a critical radius is decided. At this critical radius, the equivalent conduction and convection resistance will be minimum and heat transfer will be maximum.
• Moreover, in some cases, a decrease in the convection resistance due to the increase in surface area can be more significant than an increase in conduction resistance due to thicker insulation.

As a result, the total resistance may actually decrease resulting in increased heat flow.

Thickness up to which heat flow increases and beyond which heat flow decreases is termed as critical thickness. For cylinders and spheres, it is the critical radius (r_c ).

That can be derived from the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.

For cylinder

${{\bf{r}}_{\bf{c}}} = ;\frac{k}{h}$

For sphere

${{\bf{r}}_{\bf{c}}} = ;\frac{{2k}}{h}$

Concept:

It is defined as the ratio of the actual heat transfer that takes place from the fin to the heat that would be dissipated from the same surface area without f_in.

ε_fin = $\frac{{heat;transfer;with;fin}}{{heat;transfer;without;fin}}$

${\rm{\varepsilon }} = \frac{{\sqrt {hpk{A_c}} {\rm{\Delta }}T}}{{h{A_c}{\rm{\Delta T}}}}$

${\rm{\varepsilon }} = \sqrt {\frac{{{\rm{kP}}}}{{{\rm{hA}}}}}$

${{\rm{\varepsilon }}_{{\rm{fin}}}} \propto \frac{1}{{\sqrt h }}$

Where, h = Heat transfer coefficient, k = Thermal conductivity of fin, P = Perimeter of a fin, Ac = Cross-sectional area of a fin

Effectiveness is inversely proportional to the heat transfer coefficient. Hence if the heat transfer coefficient over the surface of a pin fin decreases, then its effectiveness will increase.

∴ Fins are never used with water or condensation heat transfer where the ‘h’ is relatively high and thus low effectiveness.

Significance:

The effectiveness of fin tells about the percentage increase in heat transfer rate while using a fin as compared to without fin. If the effectiveness of fin is low, then fins are not worth to be used because they will not be helpful in increasing the heat transfer rate.

Concept:

The thickness up to which heat flow increases and after which heat flow decreases is termed as Critical thickness.

The insulation radius at which resistance to heat flow is minimum and consequently heat flow rate is maximum is called “critical radius”.

Note that the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.

The critical radius of insulation for a cylindrical body:

${r_{cr,cylinder}} = \frac{k}{h}$

The critical radius of insulation for a spherical shell:

${r_{cr,sphere}} = \frac{{2k}}{h}$

The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < rcr, reaches a maximum when r2 = rcr, and starts to decrease for r2 > rcr. Thus, insulating the pipe may increase the rate of heat transfer from the pipe instead of decreasing it when r2 < rcr.

Concept:

For a cylinder under steady state conduction with uniform heat generation:

Heat generated in the rod = Heat conducted radially at the surface

$\dot q \times \pi {r^2}{\rm{l}} = - {\rm{k;}}\left( {2{\rm{\pi rl}}} \right)\left( {\frac{{{\rm{dT}}}}{{{\rm{dr}}}}} \right)$

$\frac{{dT}}{{dr}} = - \frac{{\dot qr}}{{2k}}$

Temperature gradient is proportional to radius.

∴ The temperature gradient at half the radius location (i.e. at r = R/2) will be one half of that at surface (i.e. at r = R)

Concept:

The response of a thermocouple is defined as the time required for the thermocouple to reach the source temperature when it is exposed to it.

Referring to the lumped-parameter solution for transient heat conduction

$\ln \left(\frac{{T_0 - T_\infty }}{{{T_i} - T_\infty }}\right) = - \frac{{h{A_s}}}{{\rho V{C_p}}}t$

where,

T_i = Initial temperature of body

T =  temperature of source temperature

h = convective heat transfer coefficient

A_s = surface area of body

ρ = density of body

V = volume of body

C_p = specific heat of body

It is evident that larger the parameter hA/ρVC_p, the faster the exponential term will reach zero or more rapid will be the response of the thermocouple. A larger value of hA/ρVC_p can be obtained either by increasing the value of convective coefficient, or by decreasing the wire diameter, density and specific heat.