gate mechanical engineering notes on heat transfer
TRANSCRIPT
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HEAT TRANSFER Steady state conduction Steady state conduction process means, that the temperature does not vary with respect to time. Therefore, the heat flux remains unchanged with time during steady state heat transfer, through a medium at any location. For example, heat transfer through the walls of a house will be steady when the conditions inside the house and the outdoors remain constant for several hours. Transient heat transfer, on the other hand is one in which conditions change w.r.t. time. During such process temperature and heat flux normally vary with time as well as position. In special cases, the variation is w.r.t. time and not with position, such heat transfer systems are called lumped systems. Steady state conduction through a plane wall Consider a thin element of thickness L in a large plane wall, k as the thermal conductivity, T1 and T2 as the wall face temperatures (T1 > T2).
When the thermal conductivity is a constant value, Fourier law yields,
π = βππ΄
πΏ(π2 β π1)
If the thermal conductivity varies w.r.t temperature as,
π = π0(1 + π½π)
The resultant heat flow equation becomes,
π = βπ0π΄
πΏ[(π2 β π1) +
π½
2(π2
2 β π12)]
Thermal Resistance concept Rewriting the conduction equation for a plane wall,
π =π1 β π2
π π‘β
Where,
π π‘β =πΏ
ππ΄
Is known as the thermal resistance of the wall material against heat conduction.
Thermal resistance depends upon the geometry and thermal properties of the material
The heat conduction equation can be considered to be analogous to flow of electric current in a circuit. As per Ohmβs law,
πΌ =π
π
Where I = electric current (the flow quantity) V = potential difference (driving force for the flow quantity) R = electrical resistance Similarly, putting up the same concept for Fourier Law,
βπππ‘ ππππ€ πππ‘π =βπ(πππ‘πππ‘πππ ππ ππππ£πππ πππππ)
π‘βπππππ πππ ππ π‘ππππ
The figures given below, illustrate the above mentioned concept,
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Consider a case of convection heat transfer across a surface of area AS, and temperature TS to a fluid with temperature πβ, convective heat transfer coefficient h. Newtonβs law of cooling gives us,
π = βπ΄π(ππ β πβ)
π =ππ β πβ
π π‘β
Where,
π π‘β =1
βπ΄π
Is the thermal resistance against convection heat transfer.
When the value of convection heat transfer coefficient becomes too large, i.e. π β β the convection resistance becomes almost zero
For a multilayered wall, as shown in the figure below, the heat flow through each material is assumed to be the same,
π = βπ1π΄π2 β π1
π₯1= βπ2π΄
π3 β π2
π₯2= βπ3π΄
π4 β π3
π₯3
Upon solving all sections, the heat flow through the complete system is written as,
π =π1 β π4
(π₯1 π1π΄β ) + (π₯2 π2π΄β ) + (π₯3 π3π΄)β=
βπππ£πππππ
β π π‘βπππππ
In the above illustration, the multi layers are joined in series with each other, so to find out the overall thermal resistance, apply the series combination law of electrical resistances
Heat conduction flow through a cylindrical wall Consider a long cylinder of inside radius r1, outside radius r2 and length L. The cylinder is exposed to a temperature difference T1 β T2. The heat flow occurs in the radial direction. The following boundary conditions exist in the cylinder,
π = π1 ππ‘ π = π1
π = π2 ππ‘ π = π2
Invoking the heat conduction equation we get the following expression,
π =2πππΏ(π1 β π2)
ln(π2 π1β )=
βπππ£πππππ
β π π‘βπππππ
Where π π‘βπππππ =ππ (π2 π1β )
2πππΏ for a cylinder
Heat conduction for a spherical body The above mentioned analysis can be run for a spherical body subjected to same temperature differences.
π =π1 β π2
β π π‘βπππππ(π πβπππ)
Where,
π π‘βπππππ(π πβπππ) =π2 β π1
4ππ1π2π
Consider a 1-dimensional heat flow through a cylindrical or spherical layer that is exposed to convection on both sides due to fluids at temperatures πβ1 and πβ2 with heat transfer coefficients h1 and h2, as shown in the figure below,
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The thermal resistance network in this case is a combination of one conduction and two convection resistances in series. The heat flow through this system can be expressed as,
π =πβ1 β πβ2
π π‘ππ‘ππ
Thermal contact resistance When we analyse heat conduction through multi layered solids, we assume perfect contact at the interface, which results in NO temperature drop at the interface. This is possible only when the surfaces in contact are perfectly smooth and free from any sort of irregularity. In reality, even high smooth surfaces are microscopically rough. This is illustrated with the help of the figure below,
Due to the gaps visible in the actual thermal contact, there will be air entrapped in them which acts as an insulator, thereby increasing the thermal resistance at the interface. The thermal resistance at the interface is called thermal contact resistance (RC). Critical thickness of insulation By adding more insulation to a wall decreases the heat transfer across it. The thicker the insulation, the lower is the heat transfer rate. This happens because the heat transfer area A remains constant, and by adding more insulation will result in increase in the thermal resistance of the wall without increasing the convection resistance.
But, adding more insulation to a cylindrical pipe does not always result in decrease in heat transfer. Additional insulation increases the thermal resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection. Consider a cylindrical pipe of outer radius r1 and outer surface temperature T1. The pipe is now insulated with an insulating material of thermal conductivity βkβ and outer radius r2. Heat is lost to the surroundings maintained at temperature πβ and convection heat transfer coefficient βhβ. the heat flow through the pipe is expressed as,
π =π1 β πβ
π ππππ + π ππππ£=
π1 β πβ
ππ(π2 π1β )2ππΏπ
+1
β(2ππ2πΏ)
The variation of βqβ with the outer radius is plotted, and the value of r2 at which the βqβ reaches a maximum value is called as critical thickness of insulation.
ππ (ππ¦π) =π
β
The value of critical radius will be largest when
βkβ is large and βhβ is small.
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The lowest value of βhβ encountered in practice is about 5 W/m2-0C (in the case of convection of gases)
Thermal conductivity of commonly used insulating materials is about 0.05 W/m-0C
The largest value of critical radius is about 1 cm For the spherical body, the value of critical value is given as,
ππ (π πβ) =2π
β
Heat source systems In a number of engineering applications heat is generated within a system. Plane wall with heat source (internal heat generation) Consider a plane wall with uniformly distributed heat sources. The thickness of the wall is β2Lβ, and the heat conduction is assumed to be in one dimension only. Let the heat generated per unit volume is qg, and the thermal conductivity does not vary with temperature. The differential equation for such a condition is,
π2π
ππ₯2+
ππ
π= 0
The boundary conditions can be specified as follows,
π = ππ€ ππ‘ π₯ = Β±πΏ
The general solution of the differential equation then becomes,
π = βππ
2ππ₯2 + πΆ1π₯ + πΆ2
On applying the boundary conditions, the temperature distribution can be found to be a parabolic expression as follows,
π β ππ
ππ€ β π0= (
π₯
πΏ)
2
At steady state conditions, the total heat generated must equal the heat lost at the faces.
Relation between, Tw and To is,
π0 =πππΏ2
2π+ ππ€
The maximum temperature in a symmetrical solid with uniform heat generation occurs at its centre, as illustrated in the figure,
Heat transfer through finned surfaces The heat conducted through a body must be frequently removed by some convection process. For this purpose, we make use of finned surfaces. The heat transfer from internally flowing fluid to fin wall is via convection. Heat is conducted through the finned surface and finally dissipated to the surroundings via convection. To increase the heat transfer via convection:-
Increase the convective heat transfer coefficient βhβ, or
Increase the surface area βASβ Increasing βhβ is not always a practical approach, so we increase the surface area by using finned surfaces. Fins are extended surfaces, made of highly conductive materials like aluminium, and are manufactured with the help of extrusion, welding etc. The finned surfaces are commonly used to enhance the heat transfer rate, and they often increase the heat transfer from a surface severalfold. In analysis of fins, we assume steady state operation with no heat generation in the fin, constant thermal conductivity βkβ of the fin material. The value of convective heat transfer coefficient varies along the fin length as well as its circumference, and its value at a point is a function of the fluid motion at that point. The value of βhβ is much lower at the fin base that at its tip. This is because that fluid is surrounded by solid surfaces near the fin base, this disrupts its motion, while fluid near the fin tip has little contact with the solid surface and thus encounters less resistance to flow.
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Keeping in mind the above point, adding too many fins on a surface may actually decrease the overall heat transfer when the decrease in βhβ offsets any gain resulting from increase in surface area
Fin equation For the volume element shown in the figure, illustrating the fin, under steady state conditions, the energy balance is,
(πππ‘π ππ βπππ‘
πππππ’ππ‘πππ πππ‘ππ‘βπ πππππππ‘
)
= (πππ‘π ππ βπππ‘
πππππ’ππ‘πππ ππππ π‘βππππππππ‘
)
+ (πππ‘π ππ βπππ‘
ππππ£πππ‘πππ ππππ π‘βππππππππ‘
)
The differential equation governing the heat flow from a fin is given as,
π2π
ππ₯2 β π2π = 0
Where,
π= πβ πβ ππ ππππ€π ππ π‘βπ π‘πππππππ‘π’ππ ππ₯πππ π , πππ π2
=βπ
ππ΄π
At the fin base,
ππ = ππ β πβ
There are four different boundary conditions at the fin base and the fin tip,
1. Infinitely long fin (π»πππ πππ = π»β)
The boundary condition at the tip is π(πΏ) = 0 ππ πΏ β β
The expression for the heat transfer for an infinitely long fin is given as,
πππππ πππ = ββπππ΄π(ππ β πβ)
Where p = perimeter AC = cross section area of the fin The temperature along the fin in this case decreases exponentially from ππ to πβ.
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2. Fin tip with negligible heat loss (ππππ πππ = π)
The fins should not be so long, so as to make their temperature approach the surrounding temperature at the tip. A more realistic situation can be realised in this case where the heat transfer from the fin tip is negligible, because the heat transfer from the fin is proportional to its surface area, and the surface area of the fin tip is taken to be negligible as compared to the total area of the fin. The mathematical expression for heat transfer from an insulated fin tip is,
ππππ π’πππ‘ππ π‘ππ πππ = ββπππ΄π(ππ
β πβ) π‘ππβ (ππΏ)
The temperature distribution for an insulated tip fin is expressed as,
π(π₯) β πβ
ππ β πβ=
πππ β π(πΏ β π₯)
πππ β (ππΏ)
3. Fin of finite length and loses heat by convection from its end The fin tips, in actual, are exposed to surroundings, which include convection and also effects of radiation. In this case, the temperature distribution can be expressed via an expression,
π(π₯) β πβ
ππ β πβ
=πππ β π(πΏ β π₯) + (β ππ)β π ππβ π(πΏ β π₯)
πππ β ππΏ + (β ππ)β π ππβ (ππΏ)
And the heat flow from such a fin is given as,
ππππ
= ββπππ΄π(ππ
β πβ)π ππβ (ππΏ) + (β ππ)β πππ β (ππΏ)
πππ β (ππΏ) + (β/ππ) π ππβ (ππΏ)
Corrected fin length It is a better way to determine the heat loss from the fin tip. We replace the fin length βLβ, in case of an insulated tip, by a corrected length βLCβ, defined as
πΏπΆ = πΏ +π΄πΆ
π
The corrected length approximation gives us good results when the variation of temperature near the fin tip is small, and the heat transfer coefficient at the fin tip is almost same as that at the lateral surface of the fin.
Fins subjected to convection at the tip are equivalent to fins with insulated tips by replacing the actual length by corrected length
πΏπΆ,ππππ‘ππππ’πππ = πΏ +π‘
2, π€βπππ π‘
= π‘βππππππ π ππ ππππ‘ππππ’πππ πππ
πΏπΆ,ππ¦πππππππππ = πΏ +π·
4, π€βπππ π·
= ππππππ‘ππ ππ ππ¦πππππππππ πππ
Fin efficiency Fin efficiency is mathematically defined as, ππππ
=πππ‘π’ππ βπππ‘ π‘ππππ πππ πππ‘π ππππ π‘βπ πππ
πππππ βπππ‘ π‘ππππ πππ πππ‘π ππππ π‘βπ πππ ππ π‘βπ πππ‘πππ πππ π€πππ ππ‘ πππ π π‘πππ.
For insulated fin tips,
ππππ =ββπππ΄πΆ (ππ β πβ) π‘ππβ (ππΏ)
βππΏ(π0 β πβ)=
π‘ππβ (ππΏ)
ππΏ
Fins with triangular and parabolic profiles
contain less material and are more efficient than ones with rectangular profiles, thus they are more suitable for weight sensitive applications
An important design consideration for fins is the fin length L. usually, a longer fin means larger heat transfer area and higher heat transfer rate from the fin. But a longer fin means, more material, and more mass
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Fin effectiveness Fins are used to enhance heat transfer from a surface, and the use of fins on a surface cannot be justified, if the enhanced heat transfer from the surface justifies the added cost and design complexity associated with fins. The performance of the fin is judged on the basis of enhancement in heat transfer w.r.t. no-fin case. Therefore, the fin performance is expressed in terms of fin effectiveness,
ππππ =ππππ
πππ πππ
ππππ = π then the addition of fins has no effect
on heat transfer
ππππ < 1 then the fins acts as an insulation, i.e.
slowing down the heat transfer
ππππ > 1 then the fins are enhancing the heat
transfer The use of fin is only justified, if the effectiveness is way more than 1, as finned surfaces are designed on the basis of maximizing effectiveness for a specified cost or minimizing cost for a specified effectiveness.
Relation between fin efficiency, and fin effectiveness
ππππ =π΄πππ
π΄πππππ
Where Afin = pL Ab = surface area with no fin
The effectiveness for a long fin can be determined by using the relation,
πππππ πππ = βππ
ππ¨π
Some important conclusions regarding the design and selection of fins:
The thermal conductivity of the fin material should be as high as possible. More commonly used materials for fins are copper, aluminium and iron. The most widely used is aluminium as it is of low cost and has high corrosion resistance
The ratio of perimeter to the cross section area of the fin should be as high as possible. This can be satisfied by thin plate fins and slender pin fins
The fins are most effective in use involving low convective heat transfer coefficient, i.e. when we use a gas instead of a liquid and heat transfer occurs via free convection and not forced convection. It is due to the above reason, why, in liquid-to-gas heat exchangers, e.g. car radiators, fins are placed on the gas side
1. Unsteady State Conduction Lumped system analysis In many heat transfer applications, some bodies are said to behave like a lump, i.e. their interior temperatures remain essentially uniform at all times, during a given heat transfer process. Thus, for such bodies, the temperature can be said to be a part of time only, i.e. T = f(t). In lumped system analysis we utilise the simplification provided to analyse such cases with considerable amount of accuracy. Consider a copper ball heated to a certain temperature. Measurements indicate that the temperature of the copper ball changes with time, but it doesnβt change with position. This way the temperature of the ball remains uniform at all times, and it behaves as a lumped system.
Consider an arbitrary shaped body with the following parameters,
During a small time interval βdtβ, the temperature of the body rises by a small amount βdTβ.
βπππ‘ π‘ππππ πππ πππ‘π π‘βπ ππππ¦ ππ’ππππ ππ‘= πππππππ π ππ ππππππ¦ ππ ππππ¦ ππ’ππππ ππ‘
or,
βπ΄π(πβ β π)ππ‘ = ππΆπππ (1)
where,
π = ππ, πππ ππ = π(π β πβ)
Therefore, rearranging equation (1)
π(π β πβ)
(π β πβ)= β
βπ΄π
πππΆπππ‘ (2)
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Integrating equation (2) from t=0, at which T = Ti
ππ π(π‘) β πβ
ππ β πβ= β
βπ΄π
πππΆππ‘ (3)
Taking exponent on both sides of equation (3)
π(π‘) β πβ
ππ β πβ= πβππ‘
Where
π =βπ΄π
πππΆπ
The term βbβ is called time constant, a positive quantity, whose units are (time)-1. The figure below, shows a plot of equation (3) for different values of βbβ,
The temperature of a body reaches the ambient temperature πβ exponentially. A larger value of βbβ indicates that the body will reach the ambient temperature in a short time, i.e. higher temperature decay rate.
Time constant is proportional to surface area and inversely proportional to the mass and the specific heat of the body.
This indicates that it takes longer to cool or heat a larger mass
The lumped system analysis provides convenient heat transfer analysis, but it is important to know when to apply it. To establish a criterion to apply the lumped system analysis, let us define a parameter, characteristic length as,
πΏπΆ =π
π΄π
And another parameter, Biot number (Bi) as,
π΅π = βπΏπΆ/π Biot number can also expressed as,
π΅π =ππππ£πππ‘πππ ππ‘ π π’πππππ ππ π ππππ¦
πππππ’ππ‘πππ π€ππ‘βππ π‘βπ ππππ¦
=πππππ’ππ‘πππ πππ ππ π‘ππππ
ππππ£πππ‘πππ πππ ππ π‘ππππ=
πΏπΆ/π
1/β
As per the above expression, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body. Lumped system analysis assumes uniform temperature distribution within the body, and it is possible only, when conduction resistance is zero.
Lumped system analysis is exact when Bi = 0, and it is approximate when Bi > 0
For a highly accurate lumped system analysis, the value of Bi should be as low as possible
Keeping in mind the uncertainties in the convection process, the generally acceptable value of Bi, to make lumped system analysis applicable is
π΅π β€ 0.1
The first step when applying lumped system analysis, is to calculate Biot number and to check the criteria for its applicability
Small bodies, with high thermal conductivity are best suited for application of lumped system analysis, when they are in a medium which is not a good conductor of heat
The hot small copper ball shown in the diagram, satisfies the criterion for the application of lumped system analysis,
Fourier Number (Fo) Mathematically, it is expressed as,
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πΉπ =πΌπ‘
πΏ2
Fourier number is used in calculations, when heat conduction and heat energy storage concurrently.
2. Convection Introduction The phenomenon of conduction and convection are similar to each other in the context, that both require a material medium for their occurrence. But convection requires the presence of fluid motion. In a solid, the main mode of heat transfer is conduction, but in liquids or gases, the mode can be either conduction or convection, depending upon the presence of bulk fluid motion. The fluid motion enhances the heat transfer, as it brings together hot and cool streams of fluid in contact with each other. The heat transfer rate, via convection is higher than the heat transfer rate via conduction, and with a higher fluid velocity, the heat transfer rate is also high. Convection depends upon fluid properties like dynamic viscosity (Β΅), thermal conductivity (k), density (Ο), specific heat (CP) and fluid velocity (v). In addition to these properties, it also depends upon geometry, roughness of the solid surface, over which the fluid is flowing and the type of fluid flow (i.e. laminar or turbulent). The governing mathematical expression for convection heat transfer process is called Newtonβs law of cooling and is expressed as,
πππππ£πππ‘πππ = βπ΄π(ππ β πβ)
Where, h = convection heat transfer coefficient (W/m2 β 0C) AS = surface area for heat transfer (m2) TS = surface temperature πβ = temperature of the fluid sufficiently far from the surface Convection process is of two types:
a) Forced convection b) Natural/ Free convection
The diagram below illustrates the two types of convection processes named above,
When a fluid is made to flow over a nonporous solid surface, it is observed that the fluid
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velocity at the solid surface is zero, i.e. the fluid layer in immediate contact with the sold surface is stagnant. This condition is known as no slip condition, and is mainly due to the viscosity of the fluid. The no slip condition is responsible for the development of the velocity profile of the flow, which is a parabolic profile, due to friction between adjacent layers of the fluid and viscosity.
A phenomenon, similar to no slip condition occurs, when a fluid and solid surface will have the same temperature at the point of contact. This is known as the no-temperature-jump condition. An implication of no-slip and no-temperature-jump condition is that the heat transfer from the solid surface is via pure conduction. This heat is then carried away by the fluid motion, via convection. Nusselt Number (Nu) Nusselt number is defined as,
ππ’ =βπΏπΆ
π=
πππππ£πππ‘πππ
ππππππ’ππ‘πππ
Nusselt number represents the enhancement of heat transfer as a result of convection relative to conduction across a layer of fluid. For effective convection, Nu should be as large as possible. If Nu =1, then it represents a condition of heat transfer by pure conduction. Velocity Boundary Layer Consider a parallel flow of a fluid over a flat plate, as shown in the figure. The fluid approaches the plate in the x-direction with a uniform upstream velocity βVβ, and can be put equal to free stream velocity π’β over the plate, away from the surface. The no slip condition exists in the vicinity of the plate surface. Due to this, the adjacent layers of fluid slow down. Therefore, the presence of the plate is experienced upto a height πΏ from the plate surface. Beyond this heightπΏ, the effect of the plate, slowing down the adjacent layers is no longer there, and the free stream velocity is achieved. The variation of fluid velocity w.r.t. y is as follows,
π’ = 0 ππ‘ π¦ = 0
π’ = π’β ππ‘ π¦ = πΏ
The region of the flow above the plate bounded in πΏ in which the effects of the viscous shearing effects are felt is called the velocity boundary layer. The boundary layer thickness, πΏ, is usually defined as the distance βyβ from the surface at which π’ = 0.99 π’β.
Thermal Boundary Layer A velocity boundary layer develops when a fluid flows over a surface as a result of fluid adjacent to the surface exhibiting no-slip condition. Likewise, a thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature, as shown in the figure,
The thickness of the thermal boundary layer increases in the flow direction. The convection heat transfer rate, along the surface is directly proportional to the temperature gradient. Thus,
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the temperature profile shape in the thermal boundary layer dictates the convective heat transfer between solid surface and the fluid flowing over it. Prandtl Number (Pr) Prandtl number is defined as,
ππ =ππππππ’πππ πππππ’π ππ£ππ‘π¦ ππ ππππππ‘π’π
ππππππ’πππ πππππ’π ππ£ππ‘π¦ ππ βπππ‘=
ππΆπ
π
The value of Prandtl numbers for fluids range from less than 0.01 for liquid metals and more than 10,000 for heavy oils. Prandtl number for gases is about 1, which indicates that momentum and heat dissipate at the same rate through it. Reynolds Number (Re) Reynolds number is defined as,
π π =πππππ‘ππ ππππππ
π£ππ πππ’π ππππππ =
πππΏπΆ
π
At large values of Re, the inertia forces dominate the viscous forces, and the viscous forces cannot prevent the large and random fluctuations of the fluid, thus the flow is termed as turbulent. At low values of Re, the viscous forces dominate over the inertia forces, which help in keeping the fluid flow in line, thus the flow in this case is termed as laminar. The value of Re, at which the transition occurs from laminar flow to turbulent flow takes place is called critical Reynolds number.
The transition from laminar to turbulent flow depends upon the surface geometry, surface roughness, free stream velocity, and the type of fluid
Physical significance of Dimensionless parameters Consider the Reynoldβs number (Re), which is defined as the ratio of inertia to viscous forces in a region of characteristic dimension L. Inertia forces are associated with an increase in the momentum of a moving fluid. The inertia forces dominate for large values of Re and the viscous forces dominate for small values of Re. Reynolds number is also used to determine the existence of laminar or turbulent flow. The small disturbances present in any kind of flow can be amplified to produce turbulent flow conditions. For small values of Re, the viscous forces are very high as compared to the inertia forces to produce this kind of amplification, hence maintaining laminar flow conditions. As the value of Re increases the impact of viscous forces become less and the dominance of inertia forces set in, thereby making small disturbances amplified to create a transition to turbulent flow. Prandtl number (Pr) is the ratio of momentum diffusivity to the thermal diffusivity, and this number provides the measure of the relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers. The value of Pr for gases is almost unity, so the momentum and energy transfer by diffusion are comparable. For liquid metals, Pr<<1, which indicate that
the energy diffusion rate exceeds the momentum diffusion rate. For oils Pr>>1, which indicates the opposite as compared to liquid metals. Differential equations for Convection equations Consider a flat plate with parallel flow of a fluid over its surface. The flow directions along the surface is x and in a direction normal to the surface is y. The fluid flow has uniform free stream velocity π’β.
There are three differential equations for laminar flow in boundary layers, namely:
1. Conservation of mass equation 2. Conservation of momentum equation 3. Conservation of energy equation
Conservation of Mass equation According to this principle, the mass cannot be destroyed or created. In a steady flow the amount of mass flowing through the control volume remains constant, (πππ‘π ππ πππ π ππππ€ ππ) = (πππ‘π ππ πππ π ππππ€ ππ’π‘)
Mathematically the conservation of mass equation is given as,
ππ’
ππ₯+
ππ£
ππ¦= 0
Where u = flow velocity in x-direction and v =flow velocity in y-direction Conservation of Momentum equation For this we make use of Newtonβs second law, which states, the net force acting on the control volume is equal to the mass times the acceleration of the fluid element within the control volume, which is also equal to the net rate of momentum outflow from the control volume.
(πππ π )(πππππππππππ ππ π π ππππππππ ππππππ‘πππ)= (πππ‘ πππππ πππ‘πππ ππ π‘βππ‘ ππππππ‘πππ)
The momentum conservation in x-direction is mathematically expressed as,
π (π’ππ’
ππ₯+ π£
ππ’
ππ¦) = π
π2π’
ππ¦2
βππ
ππ₯, π€βπππ π ππ π‘βπ ππππ π π’ππ
Conservation of Energy equation
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The energy equation for a steady two-dimensional flow of a fluid with constant properties and negligible shear stresses is expressed as,
ππΆπ (π’ππ
ππ₯+ π£
ππ
ππ¦) = π (
π2π
ππ₯2 +π2π
ππ¦2)
The above equation states that the net energy convected by the fluid out of the control volume is equal to the net energy transferred into the control volume by heat conduction. Thermal Boundary Layer Thermal boundary layer may be defined as that region where temperature gradients are present in the flow. These temperature gradients result from a heat exchange process between the fluid and the stationary surface. Let the temperature of the surface is TS and the temperature of the fluid outside the thermal boundary layer is πβ. The thickness of the thermal boundary layer is designated as πΏπ‘. At the wall, the velocity of the fluid is zero, and the heat transfer into the fluid takes place via conduction,
π = βππ΄ππ
ππ¦|
π π’πππππ
From Newtonβs law of cooling,
πππππ£πππ‘πππ = βπ΄π(ππ β πβ)
Equating the above heat transfer rates, we get,
β =
(βπππππ¦
|π π’πππππ
)
ππ β πβ
The following boundary conditions apply,
π = ππ ππ‘ π¦ = 0
ππ
ππ¦= 0 ππ‘ π¦ = πΏπ‘
π = πβ ππ‘ π¦ = πΏπ‘
After mathematical manipulation and substitution, we get the following relation for the temperature profile, which indicates a cubic curve,
π β ππ
ππ β πβ=
π
πβ=
3
2(
π¦
πΏπ‘) β
1
2(
π¦
πΏπ‘)
3
To find out the value of thermal boundary layer thickness, use the following formula,
πΏπ‘
πΏ=
1
1.026ππβ1/3
Where πΏ = velocity boundary layer thickness Pr = Prandtl Number
The above analysis is based on the assumption, that the fluid properties are constant throughout the flow
When there exists, an appreciable variation between wall and free stream conditions, we use a term film temperature to evaluate the properties. The film temperature is defined as the arithmetic mean of surface temperature and free stream temperature,
ππ =ππ + πβ
2
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3. Radiation Introduction Consider a hot object that is suspended in a vacuum chamber, with its walls at room temperature.
The hot object will cool down and attain thermal equilibrium with its surroundings, i.e. the object will lose heat until its temperature becomes equal to the temperature of the temperature of the walls of the chamber. In a vacuum, conduction and convection cannot bring about heat transfer, so the process which comes into action in a vacuum, is radiation. As it is clear from the above example, that radiation does not require any material medium for its occurrence. Radiation takes place across all three states of matter.
Radiation heat transfer can occur between two bodies separated by a medium colder than both bodies, e.g. solar radiations reach the surface of the earth after passing through cold layers at high altitiudes
We associate thermal radiation with the rate at which energy is emitted by the matter as a result of its finite temperature. The energy emission mechanism is related to the energy released as a result of oscillations of infinite number of electrons, comprised in a matter. These oscillations then in turn increase the internal energy of the matter, which means a rise in its temperature.
All forms of matter emit radiation
For gases it is a volume phenomenon
For opaque solids, like metals, woods etc. radiation is considered to be a surface phenomena, since the radiation emitted only by the molecules at the surface can escape the solid
Thermal radiation is viewed as electromagnetic waves propagation. For thermal radiation propagation through a medium, the wavelength (Ξ») and frequency (Ξ½) are related as,
π =π
π
Where c = speed of light in the given medium For vacuum, speed of light is given as cO = 2.998 X 108 m/s Electromagnetic radiation is the propagation of a collection of discrete packets of energy called photons or quanta. Each photon of frequency Ξ½ is considered to have an energy of,
π = βπ =βπ
π
Where h = 6.6256 X 10-34 J.sec is the Planckβs constant Although all EM waves have the same general features, waves with different wavelength differ significantly in their behaviour. The electromagnetic spectrum lists all the EM radiation encountered in daily practice, i.e gamma rays, X-rays, UV radiations and radio waves.
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Blackbody Radiations A body at a temperature above the absolute zero emits radiation in all directions over a wide range of wavelengths, and the amount of radiation energy emitted from a surface at a particular wavelength depends upon the material of the body and its surface temperature. A blackbody is an object that emits maximum amount of radiation, and is defined as a perfect emitter and absorber of radiation. At a given value of temperature and wavelength, no surface can emit more energy than a blackbody. It also absorbs all incident energy on it, regardless of t he wavelength and direction.
A blackbody is a diffuse emitter, which means emissions from a blackbody are independent of direction
The energy emitted by a blackbody per unit time and per unit surface area is given by Stefan-Boltzmann Law,
πΈπ(π) = ππ4
Where Ο = 5.67 X 10-8 W/m2 K4 is known as Stefan-Boltzmann constant. T = absolute temperature of the surface in K Eb = blackbody emissive power and is a function of temperature
Any surface that absorbs the visible portion of the spectrum would appear black to the eye
Any surface that reflects the visible portion of the spectrum would appear white to the eye
As per Stefan-Boltzmannβs law, blackbody emissive power is ONLY a function of absolute temperature of the surface, but as per Planckβs law, the blackbody emissive power is a function of wavelength and temperature, as follows,
πΈππ(π, π) =πΆ1
π5 [exp (πΆ2ππ
) β 1] π/π2 . ππ
Where,
πΆ1 = 2πβππ2 = 3.742 π 108π. ππ4/π2
πΆ2 =βππ
π= 1.439 π 108ππ. πΎ
Where T = absolute temperature of the surface Ξ» = wavelength of the radiation emitted k = Boltzmannβs constant = 1.38065 X 10-23 J/K The relation discussed is valid for a surface in a vacuum or a gas, for other mediums it needs to be changed by replacing,
πΆ1 ππ¦πΆ1
π2
Where n = index of refraction of the medium The variation of the spectral emissive power of blackbody with wavelength is illustrated in the figure below,
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Some important points to note regarding the above figure are as follows:
The emitted radiation is a continuous function of wavelength, and it increases with temperature, reaches a maximum value and then starts to decrease with an increasing wavelength
At any wavelength, the quantity of emitted radiation increases with increasing temperature
A larger fraction of the radiation is emitted at shorter wavelengths at higher temperature, and this is indicated on the curve with a shift of the curve towards shorter wavelength region as temperature increases. This shift can be specified by Weinβs displacement law as,
(ππ)max πππ€ππ = 2897.8 π π. πΎ
Surfaces at Tβ€ 800 K emit radiations in Infrared region which is not visible to our eye, unless it is reflected from some surface
The colour of an object is not dependent upon its emission, which is primarily in the infrared zone, unless the temperature is above 1000 K value. Actually the colour of the object depends upon the absorption and reflection characteristics of the surface, i.e. selective absorption and selective reflection. In the figure given below, an incident light with red, yellow, green and blue colours is incident upon a surface, which absorbs all the colours except the red one. Red colour is reflected by the surface, thus the surface appears red to the human eye.
The total radiation energy emitted by a blackbody at a particular temperature is given by the area under EbΞ» and Ξ» chart as shown below in the figure,
Radiation Intensity Radiation is emitted by parts of a plane surface in all directions into the hemisphere above the surface, and the directional distribution of emitted radiation is usually not uniform. So, we need a quantity that describes the magnitude of radiation emitted in a particular direction in space. This quantity is known as radiation intensity (I). The direction of radiation passing through a point is best described in spherical coordinates in terms of the zenith angle ΞΈ and azimuth angle Ο, ass illustrated in the figure below
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If all surfaces emitted radiation uniformly in all directions, the emissive power would be sufficient to quantify radiation
Radiosity Surfaces emit and reflect radiation, so the radiation leaving the surface consists of emitted and reflected components, as illustrated in the figure below,
Therefore, radiosity J, is defined as the total radiation leaving the surface, without any dependence of origin. Its units are W/m2.
For a black body, radiosity is equivalent to the emissive power Eb
Emissivity The emissivity of a surface represents the ratio of the radiation emitted by the surface at a given temperature to the radiation emitted by a black body at the same temperature. It is denoted by βΞ΅β and its value varies between 0 and 1. It is a measure of how closely a surface approximates a blackbody.
For a blackbody, Ξ΅ = 1
Incident radiation on a surface is called IRRADIATION and is denoted by βGβ
Absorptivity, Reflectivity and Transmissivity When radiation strikes a surface a part of it is absorbed, a part is reflected and the remaining part is transmitted. The fraction of incident radiation which is absorbed by the surface is called absorptivity (Ξ±),
πΌ =πππ πππππ ππππππ‘πππ
π‘ππ‘ππ ππππππππ‘ ππππππ‘πππ, 0 β€ πΌ β€ 1
The fraction of incident radiation which is reflected is called reflectivity (Ο),
π =πππππππ‘ππ ππππππ‘πππ
π‘ππ‘ππ ππππππππ‘ ππππππ‘πππ, 0 β€ π β€ 1
The fraction of incident radiation which is transmitted is called transmissivity (Ο),
π =π‘ππππ πππ‘π‘ππ ππππππ‘πππ
π‘ππ‘ππ ππππππππ‘ ππππππ‘πππ, 0 β€ π β€ 1
πΌ + π + π = 1
For opaque surface, π = 0
View factor Radiation heat transfer between surfaces depends upon the orientation of the surfaces w.r.t. each other as well as their radiation properties and temperatures, as illustrated by the figure below,
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This dependence on orientation is accounted for by the view factor, also called shape factor and configuration factor. This is a purely geometric quantity and is independent of the surface properties and temperature. The view factor from a surface i to j is denoted by πΉπβπ or
just πΉππ and is defined as the fraction of the radiation
leaving surface i that strikes surface j directly.
The radiation that strikes a surface does not need to be absorbed by that surface, and the radiation that strikes a surface after being reflected by other surfaces is not considered in the evaluation of view factors
The figure below shows the view factors for different surface configurations,
4. Heat Exchangers Introduction Heat exchangers are devices that help in exchange of heat energy between two fluids that are at different temperatures, which are kept from mixing with each other. These devices use convection in each fluid and conduction through the wall separating the two fluids. While analysing heat exchangers we consider overall heat transfer coefficient, U which encompasses all the effects in a heat exchanger. Types of heat exchangers The simplest type of heat exchanger consists of two pipes of different diameters, called the double pipe heat exchanger. One fluid flows through the inner pipe while the other fluid flows through the outer pipe, and depending upon the direction of flow we classify the double pipe heat exchanger as:
Parallel flow: In this type, both the fluids flow in the same direction
Counter flow: In this type, both the fluids flow in opposite directions
The figures below illustrate the two types of heat exchangers described above,
The ratio of heat transfer surface area of a heat exchanger to its volume is called the area density (Ξ²)
When two fluids move perpendicular to each other, the heat exchanger is termed as cross flow heat exchanger. This type of heat exchanger is further classified as:-
Unmixed
Mixed flow
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The most common type of heat exchanger used in industrial applications is a shell & tube heat exchanger, as illustrated in the figure below,
The figure shown above is a one-pass shell and tube heat exchanger. Baffles are commonly placed in the shell to force the shell-side fluid to flow across the shell to enhance heat transfer and to maintain uniform spacing between the tubes. These types of heat exchangers are classified according to the number of shell and tube passes involved:
One shell pass and two tube passes
Two shell passes and four tube passes
Overall Heat Transfer Coefficient Heat in a heat exchanger is first transferred from hot fluid to the wall by convection, through the wall by conduction and from the wall to the cold fluid via convection.
Any radiation effects are usually taken care off in the convective heat transfer coefficient
The figure below illustrates the thermal resistance network associated with heat transfer in a double pipe heat exchanger,
The total thermal resistance of the network is given as,
π π‘βπππππ = π π + π π€πππ + π π =1
βππ΄π+
[ln (π·ππ·π
)]
2πππΏ+
1
βππ΄π
Where Ai = inner surface area of the wall = ππ·ππΏ Ao = outer surface area of the wall = ππ·ππΏ L = length of the tube The heat transfer rate between two fluids is expressed as,
οΏ½ΜοΏ½ =βπ
π π‘βπππππ= ππ΄βπ
π π‘βπππππ =1
ππ΄
Where U = overall heat transfer coefficient in W/m2
When the wall thickness is small and the thermal conductivity of the tube material is high, this means that the thermal resistance of the wall, Rwall = 0 and inner and outer surface
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areas are almost equal, Ai =Ao, then the overall heat transfer coefficient is,
1
πβ
1
βπ+
1
βπ
The value of U is dominated by the smaller convection coefficient
Fouling factor The performance of heat exchangers deteriorates with time due to accumulation of deposits on heat transfer surfaces. This additional layer of deposits creates extra thermal resistance, and decreases the heat transfer rate in the heat exchanger. The total effect of this accumulation is represented by Rf, which is basically a measure of the thermal resistance introduced by fouling.
The most common type of fouling is the precipitation of solid deposits in a fluid on the heat transfer surfaces
The other common type of fouling is chemical fouling
The fouling factor depends upon the operating temperature and the velocity of the fluids, and the length of the service. The value of fouling factor increases with increasing temperature and decreasing velocity. For an unfinned shell-and-tube heat exchanger, it can be expressed as,
1
ππ΄= π π‘βπππππ =
1
βππ΄π+
π π,π
π΄π+
[ln (π·ππ·π
)]
2πππΏ+
π π,π
π΄π
+1
βππ΄π
Where Rf,i = fouling factor at inner surfaces Rf,o = fouling factor at outer surfaces Log Mean Temperature Difference (LMTD) As per this method, heat transfer rate in a heat exchanger is given as,
οΏ½ΜοΏ½ = ππ΄βπππ
Where, ΞTln = LMTD and it is expressed as,
βπππ =βπ1 β βπ2
[ln (βπ1βπ2
)]
The values of βπ1 and βπ2 depends upon the flow type in the double pipe heat exchanger, i.e. whether it is parallel flow or counter flow heat exchanger. The figure illustrates the above mentioned point,
The arithmetic mean temperature is given as,
βπππππ =βπ1 + βπ2
2
It should be noted that LMTD is always less then arithmetic mean temperature
Effectiveness-NTU The LMTD method discussed above is easy to use in a heat exchanger analysis where the inlet and the outlet temperatures of the hot and cold fluids are known or can be determined. This means, that the LMTD method is useful for determining the size of a heat exchanger when the mass flow rates and the inlet and outlet temperatures of the hot and cold fluids are given. Another type of problem encountered in heat exchanger analysis is that of determination of the βheat transfer rateβ and the βoutlet temperaturesβ of the hot and cold fluids for the given fluid mass flow rates and inlet temperatures when the type and size of the heat exchange are given. In this situation the heat transfer surface area A of the heat exchanger is known but not the outlet temperatures. To solve the heat exchanger analysis in the above condition, we will use the effectiveness-NTU method. In
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this method, we define a dimensionless parameter, heat transfer effectiveness as,
ν =πππ‘π’ππ βπππ‘ π‘ππππ πππ πππ‘π
πππ₯πππ’π πππ π ππππ βπππ‘ π‘ππππ πππ πππ‘π=
οΏ½ΜοΏ½
ππππ₯Μ
οΏ½ΜοΏ½ = πΆπΆ(ππ,ππ’π‘ β ππ,ππ) = πΆβ(πβ,ππ β πβ,ππ’π‘)
πΆπΆ = ππΆΜ πππ πππ πΆβ = πβΜ ππβ
To calculate the maximum possible heat transfer rate, we have to consider the minimum value out of CC and Ch, and substitute as Cmin in the relation,
ππππ₯Μ = πΆπππ(πβ,ππ β ππ,ππ)
Going by the above given relations, we can determine the actual heat transfer rate, if the heat exchanger effectiveness is given, as,
οΏ½ΜοΏ½ = ν ππππ₯Μ
This way, the effectiveness of a heat exchanger enables us to determine the heat transfer rate without knowing the outlet temperatures of the fluids handled.
The effectiveness of a heat exchanger depends upon the geometry of the exchanger and the flow arrangement
For a parallel flow heat exchanger,
νππππππππ ππππ€ =
1 β exp [βππ΄π
πΆπππ(1 + (
πΆπππ
πΆπππ₯))]
1 + (πΆπππ
πΆπππ₯)
The quantity βππ΄π
πΆπππ is a dimensionless parameter called the
number of transfer units (NTU),
πππ =βππ΄π
πΆπππ
Where, U = overall heat transfer coefficient AS = heat transfer surface area Cmin = minimum heat capacity
NTU is proportional to the surface area
The larger the value of NTU, the larger will be the heat exchanger
Another dimensionless parameter is defined, capacity ratio (c),
π =πΆπππ
πΆπππ₯
Effectiveness is a function of the NTU and the
capacity ratio (c) as,
ν = π (ππ΄π
πΆπππ,
πΆπππ
πΆπππ₯) = π(πππ, π)
The value of effectiveness ranges from 0 to 1, and it increases rapidly with NTU for small values but slowly for large values, so the use of a heat exchanger with a large value of NTU cannot be justified economically
For a given value of NTU and capacity ratio, the counter-flow heat exchanger has the highest effectiveness
The effectiveness of a heat exchanger is independent of the capacity ratio for NTU values less than 0.3
The value of capacity ratio ranges between 0 and 1. For a given NTU, the effectiveness becomes maximum for c = 0 and a minimum for c = 1. If c = 0, it indicates a phase change process, as in a condenser or a boiler.
Questions: 1. In descending order of magnitude, the thermal
conductivity of (a) pure iron, (b) liquid water, (c) saturated water vapour and (d) aluminum can be arranged as
(A) abcd (B) bcad (C) dabc (D) dcba 2. For the circular tube of equal length and diameter
shown below, the view factor πΉ13 is 0.17. The view factor πΉ12 in this case will be
ππ·
(A) 0.17 (B) 0.21 (C) 0.79 (D) 0.83 3. For the same inlet and outlet temperatures of hot
and cold fluids, the Log mean Temperature Difference (LMTD) is (A) greater for parallel flow heat exchanger than for counter flow heat exchanger (B) greater for counter flow heat exchanger than for parallel flow heat exchanger (C) same for both parallel and counter flow heat exchangers (D) dependent on the properties of the fluids.
Common Data For π.4 and π.5 Heat is being transferred by convection ππ: ππ water at 48ππΆ to a glass plate whose surface that is exposed to the water is at 40π C. The thermal conductivity of water is 0.6 π/ππΎ and the thermal conductivity of glass is 1.2 π/ππΎ. The spatial gradient of temperature in the water at the waterβ glass interface is πππππ¦ = 1 Γ 104πΎ/π.
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4. The value of the temperature gradient in the glass at
the waterβglass interface in πΎ/π is (A) β2 Γ 104 (B) 0.0 Γ 104
(C) 0. 5 Γ 104 (π·) 2 Γ 104
5. The heat transfer coefficient β in π/π2πΎ is (A) 0.0 (B) 4.8 (C) 6 (D) 750 6. Consider a laminar boundary layer over a heated flat
plate. The free stream velocity is πβ. At some distance π₯ from the leading edge the velocity boundary layer thickness is πΏπ£ and the thermal boundary layer thickness is πΏπ― . If the Prandtl number is greater than 1, then (A) πΏπ£ > πΏπ (B) πΏπ > πΏπ£
(C) πΏπ£ β πΏπ βΌ (πβπ₯)β1/2 (D) πΏπ£ β πΏπ βΌ π₯β1/2
7. In a counter flow heat exchanger, for the hot fluid the heat capacity = 2ππ½/πππΎ, mass flow rate =5ππ/π , inlet temperature = 150ππΆ, outlet temperature = 100π C. For the cold fluid, heat capacity = 4ππ½/πππΎ, mass flow rate = 10ππ/π , inlet temperature = 20π C. Neglecting heat transfer to the surroundings, the outlet temperature of the cold fluid in 0πΆ is
(A) 7.5 (B) 32.5 (C) 45.5 (D) 70.0 8. A plate having 10 ππ2 area each side is hanging in
the middle of a room of 100 π2π‘ ot al surface area. π he plat et emp erature and emissivity are respectively 800 πΎ and 0.6. The temperature and emissivity values for the surfaces of the room are 300 πΎ and 0.3 respectively. Boltzmannβs constant π = 5.67 Γ 10β8π/π2πΎ4 The total heat loss ffom the two surfaces of the plate is
(A) 13.66 π (B) 27.32 π (C) 27.87 π (D) 13.66 MW 9. In a condenser, water enters at 30ππΆ and flows at
the rate 1500 ππ/βπ. The condensing steam is at a temperature of 120ππΆ and cooling water leaves the condenser at 80π C. Specific heat of water is 4. 187 ππ½/πππΎ. If the overall heat transfer coefficient is 2000 π/π2πΎ, then heat transfer area is
(A) 0.707 π2 (B) 7.07 π2 (C) 70.7 π2 (D) 141.4 π2 10. A spherical thermocouple junction of diameter
0.706 mm is to be used for the measurement of
temperature of a gas stream. The convective heat transfer coβefficient on the π ead surface is 400 π/π2πΎ. π hermoβphysical properties of thermocouple material are π = 20π/ππΎ, π =400π½/ kg πΎ and π = 8500ππ/π3 If the thermocouple initially at 30ππΆ is placed in a hot stream of 300ππΆ, then time taken by the bead to reach 298ππΆ, is
(A) 2.35 π (B) 4.9 π (C) 14.7 π (D) 29.4 π 11. A stainless steel tube (ππ = 19π/ππΎ) of 2 cm πΌπ·
and 5 cm ππ· is insulated with 3 cm thick asbestos (ππ = 0.2π/ππΎ) . If the temperature difference between the innermost and outermost surfaces is 600ππΆ, the heat transfer rate per unit length is
(A) 0.94 π
π (B) 9.44 π/π
(C) 944.72 π
π (D) 9447.21 π/π
12. One dimensional unsteady state heat transfer
equation for a sphere with heat generation at the rate of πβ can be written as
(A) 1
π
π
ππ(π
ππ
ππ) +
π
π=
1ππ€
πΌππ‘
(B) 1π
π2ππ(π2 ππ
ππ) +
π
π=
1ππ
πΌππ‘
(C) π2π
ππ2 +π
π=
1ππ
πΌππ‘
(D) π2
ππ2 (ππ) +π
π=
1ππ
πΌππ‘
Common Data For π.13 and π.14 An uninsulated air conditioning duct of rectangular cross section 1 Γ 0.5π , carrying air at 20ππΆ with a velocity of 10 π/π , is exposed to an ambient of 30π C. Neglect the effect of duct construction material. For air in the range of 20 β 30ππΆ, data are as follows; thermal conductivity = 0.025π/ππΎ ; viscosity = 18ππ as, π randtl number = 0.73; density = 1.2ππ/π3 π he laminar flow Nusselt number is 3.4 for constant wall temperature conditions and for turbulent flow, Nu = 0.023Re08ππ033 13. The Reynolds number for the flow is (A) 444 (B) 890 (C) 4. 44 Γ 105 (π·) 5. 33 Γ 105
14. The heat transfer per meter length of the duct, in watts is
(A) 3.8 (B) 5.3 (C) 89 (D) 769 15. Hot oil is cooled ππ: ππ80 to 50ππΆ in an oil cooler
which uses air as the coolant. The air temperature rises ffom 30 to 40π C. The designer uses a LMTD value of 26π C. The type of heat exchange is
(A) parallel flow (B) double pipe (C) counter flow (D) cross flow 16. A solid cylinder (surface 2) is located at the centre of
a hollow sphere (surface 1). The diameter of the sphere is 1 π, while the cylinder has a diameter and length of 0.5 π each. The radiation configuration factor πΉ11 is
(A) 0.375 (B) 0.625
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(C) 0.75 (D) 1 17. A small copper ball of 5 mm diameter at 500 πΎ is
dropped into an oil bath whose temperature is 300 K. The thermal conductivity of copper is 400 π/ππΎ, its density 9000 ππ/π3 and its specific heat 385 π½/πππΎ. If the heat transfer coefficient is 250 π/π2πΎ and lumped analysis is assumed to be valid, the rate of fall of the temperature of the ball at the beginning of cooling will be, in πΎ/π ,
(A) 8.7 (B) 13.9 (C) 17.3 (D) 27.7 18. Heat flows through a composite slab, as shown
below. The depth ofthe slab is 1 π. The π values are in π/ππΎ. The overall thermal resistance in πΎ/π is
(A) 17.2 (B) 21.9 (C) 28.6 (D) 39.2 19. The following figure was generated ffom
experimental data relating spectral black body emissive power to wavelength at three temperature π1, π2 and π3(π1 > π2 > π3) .
The conclusion is that the measurements are (A) correct because the maxima in πΈππ show the correct trend (B) correct because Planckβs law is satisfied (C) wrong because the Stefan Boltzmann law is not satisfied (D) wrong because Wienβs displacement law is not satisfied 20. In a case of one dimensional heat conduction in a
medium with constant properties, π is the
temperature at position π₯, at time π‘. Then ππ
ππ‘ is
proportional to
(A) π
π₯ (B)
ππ
ππ₯
(C) π2π
ππ₯ππ‘ (D)
π2π
ππ₯2
21. With an increase in the thickness of insulation around a circular pipe, heat loss to surrounding due to (A) convection increase, while that the due to
conduction decreases (B) convection decrease, while that due to conduction increases (C) convection and conduction decreases (D)convection and conduction increases
22. A thin layer of water in a field is formed after a
farmer has watered it. The ambient air conditions are: temperature 20ππΆ and relative humidity 5%. An extract of steam tables is given below.
Temp (βπΆ)
β15 β10 β5 0.01 5 10 15 20
Saturation Pressure (πππ)
0.10 0.26 0.40 0.61 0.87 1.23 1.71 2.34
Neglecting the heat transfer between the water and the ground, the water temperature in the field after phase equilibrium is reached equals
(A) 10. 3ππΆ
(B) β10. 3ππΆ
(C) β14. 5ππΆ
(D) 14. 5ππΆ
23. π΄ 100 π electric bulb was switched on in a 2.5 π Γ3π Γ 3π size thermally insulated room having a temperature of 20π C. The room temperature at the end of 24 hours will be
(A) 321ππΆ
(B) 341ππΆ
(C) 450ππΆ
(D) 470ππΆ
24. In a composite slab, the temperature at the interface (ππππ‘ππ) between two material is equalto the average ofthe temperature at the two ends. Assuming steady oneβdimensional heat conduction, which of the following statements is true about the respective thermal conductivities?
(A) 2π1 = π2 (B) π1 = π2
(C) 2π1 = 3π2 (D) π1 = 2π2
0.5 m
1 m
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Common Data For π.25 and π.26 Consider steady oneβdimensional heat flow in a plate of 20 mm thickness with a uniform heat generation of 80 ππ/π3 The lefl and right faces are kept at constant temperatures of 160ππΆ and 120ππΆ respectively. The plate has a constant thermal conductivity of 200 π/ππΎ.
25. The location of maximum temperature within the plate ππππ its left face is
(A) 15 mm (B) 10 mm (C) 5 mm (D) 0 mm
26. The maximum temperature within the plate in 0πΆ is (A) 160 (B) 165 (C) 200 (D) 250
27. The average heat transfer coβefficient on a thin hot vertical plate suspended in still air can be determined ππ: ππ observations of the change in plate temperature with time as it cools. Assume the plate temperature to be uniform at any instant of time and radiation heat exchange with the surroundings negligible. The ambient temperature is 25ππΆ, the plat has a total surface area of 0.1 π2 and a mass of 4 ππ. The specific heat ofthe plate material is 2.5 ππ½/πππΎ. The convective heat transfer coβefficient in π/π2πΎ, at the instant when the plate temperature is 225ππΆ and the change in plate temperature with time πππππ‘ = β0.02πΎ/π , is
(A) 200 (B) 20 (C) 15 (D) 10 28. In a counter flow heat exchanger, hot fluid enters at
60ππΆ and cold fluid leaves at 30π C. Mass flow rate of the fluid is 1 ππ/π and that of the cold fluid is 2 ππ/π . Specific heat of the hot fluid is 10 ππ½/πππΎ and that of the cold fluid is 5 ππ½/πππΎ. The Log Mean Temperature Difference (LMTD) for the heat exchanger in 0πΆ is
(A) 15 (B) 30 (C) 35 (D) 45 29. The temperature distribution within the thermal
boundary layer over a heated isothermal flat plate is given by
π β ππ€
πβ β ππ€=
3
2(
π¦
πΏπ‘) β
1
2(
π¦
πΏπ‘)3,
where ππ€ and πβ are the temperature ofplate and ππ: ππ stream respectively, and π¦ is the normal distance measured ffom the plate. The local Nusselt number based on the thermal boundary layer thickness πΏπ‘ is given by (A) 1.33 (B) 1.50 (C) 2.0 (D) 4.64 30. Steady twoβdimensional heat conduction takes
place in the body shown in the figure below. The normal temperature gradients over surfaces π and π can be considered to be uniform. The temperature gradient πππππ₯ at surface π is equal to 10 πΎ/π. Surfaces π and π are maintained at constant
temperature as shown in the figure, while the remaining part ofthe boundary is insulated. The body has a constant thermal conductivity of 0.1
π/ππΎ. The values of ππ
ππ₯ and
ππ
ππ¦ at surface π are
π¦
(A)
ππ
ππ₯= 20πΎ/π,
ππ
ππ¦= 0πΎ/π
(B) ππ
ππ₯= 0πΎ/π,
ππ
ππ¦= 10πΎ/π
(C) ππ
ππ₯= 10πΎ/π,
ππ
ππ¦= 10πΎ/π
(D) ππ
ππ₯= 0πΎ/π, π = 20πΎ/π
31. A hollow enclosure is formed between two infinitely long concentric cylinders of radii 1 π and 2 π, respectively. Radiative heat exchange takes place between the inner surface of the larger cylinder (surfaceβ2) and the outer surface ofthe smaller cylinder (surfaceβl). The radiating surfaces are difffise and the medium in the enclosure is nonβparticipating. The fiaction of the thermal radiation leaving the larger surface and striking itselfis
(A) 0.25 (B) 0.5 (C) 0.75 (D) 1 32. For the threeβdimensional object shown in the figure
below, five faces are insulated. The sixth face (πππ π), which is not insulated, interacts thermally with the ambient, with a convective heat transfer coefficient of 10 π/π2πΎ The ambient temperature is 30π C. Heat is uniformly generated inside the object at the rate of 100 π/π3 Assuming the face πππ π to be at uniform temperature, its steady state temperature is
Surface β 2
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π||
|1π|π»
(A) 10ππΆ (B) 20ππΆ
(C) 30ππΆ (D) 40ππΆ
33. The logarithmic mean temperature difference
(LMTD) of a counter flow heat exchanger is 20π C. The cold fluid enters at 20ππΆ and the hot fluid enters at 100π C. Mass flow rate of the cold fluid is twice that of the hot fluid. Specific heat at constant pressure of the hot fluid is twice that of the cold fluid. The exit temperature of the cold fluid
(A) is 40ππΆ
(B) is 60ππΆ
(C) is 80ππΆ (D) cannot be determined 34. For flow of fluid over a heated plate, the following
fluid properties are known Viscosity = 0.001 ππ β π ;
Specific heat at constant pressure = 1ππ½/ kg.πΎ; Thermal conductivity = lπ/mβπΎ The hydrodynamic boundary layer thickness at a specified location on the plate is 1 mm. The thermal boundary layer thickness at the same location is
(A) 0.001 mm (B) 0.01 mm (C) 1 mm (D) 1000 mm Common Data For π.35 and π.36
Radiative heat transfer is intended between the inner surfaces of two very large isothermal parallel metal plates. While the upper plate (designated as plate 1) is a black surface and is the warmer one being maintained at 727ππΆ , the lower plate (plate 2) is a diffise and gray surface with an emissivity of 0.7 and is kept at 227ππΆ. Assume that the surfaces are sufficiently large to form a twoβsurface enclosure and steadyβstate conditions to exits. StefanβBoltzmann constant is given as 5.67 Γ 10β8π/π2πΎ4
35. The irradiation (ππ ππ/π2) for the plate (plate 1) is (A) 2.5 (B) 3.6 (C) 17.0 (D) 19.5 36. If plate 1 is also diffuse and grey surface with an
emissivity value of 0.8, the net radiation heat exchange (ππ ππ/π2) between plate 1 and plate 2 is
(A) 17.0 (B) 19.5 (C) 23.0 (D) 31.7 37. Consider steadyβstate conduction across the
thickness in a plane composite wall (as shown in the figure) exposed to convection conditions on both sides.
βπ , πβπ
Given: βπ = 20π/π2πΎ, βπ = 50π/π2πΎ;, πβπ =
20ππΆ; πβ,0 = β2ππΆ, π1 = 20π/ππΎ; π2 = 50π/ππΎ; πΏ1 = 0.30π and πΏ2 = 0.15π. Assuming negligible contact resistance between the wall surfaces, the interface temperature, π (in 0πΆ), of the two
walls will be (A) β0.50 (B) 2.75 (C) 3.75 (D) 4.50 38. In a parallel flow heat exchanger operating under
steady state, the heat capacity rates (product of specific heat at constant pressure and mass flow rate) of the hot and cold fluid are equal. The hot fluid, flowing at 1 ππ/π with ππ = 4ππ½/ kg πΎ, enters
the heat exchanger at 102ππΆ while the cold fluid has an inlet temperature of 15π C. The overall heat transfer coefficient for the heat exchanger is estimated to be 1 ππ/π2πΎ and the corresponding heat transfer surface area is 5 π2 Neglect heat transfer between the heat exchanger and the ambient. The heat exchanger is characterized by the following relations:
2ν = β exp ( β2 NTU) The exit temperature (in 0πΆ) for the cold fluid is
(A) 45 (B) 55 (C) 65 (D) 75 39. A coolant fluid at 30ππΆ flows over a heated flat plate
maintained at constant temperature of 100π C. The boundary layer temperature distribution at a given location on the plate may be approximated as π =30 + 70 exp (βπ¦) where π¦ (in m) is the distance normal to the plate and π is in β C. Ifthermal conductivity of the fluid is 1. 0π/ππΎ, the local convective heat transfer coefficient (ππ π/π2πΎ) at that location will be
(A) 0.2 (B) 1 (C) 5 (D) 10 40. A fin has 5 mm diameter and 100 mm length. The
thermal conductivity of fin material is 400 Wm β1πΎ β 1 One end ofthe fm is maintained at 130ππΆ and its remaining surface is exposed to ambient air at 30π C. If the convective heat transfer coefficient is 40 Wm β2πΎ β 1, the heat loss (in π) from the fm is
(A) 0.08 (B) 5.0 (C) 7.0 (D) 7.8
1 2
L2 L1
βπ, πβ,π βπ, πβ,π
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41. The ratios of the laminar hydrodynamic boundary layer thickness to thermal boundary layer thickness of flows of two fluids π and π on a flat plate are 1/2 and 2 respectively. The Reynolds number based on the plate length for both the flows is 104 The Prandtl and Nusselt numbers for π are 1/8 and 35 respectively. The Prandtl and Nusselt numbers for π are respectively
(A) 8 and 140 (B) 8 and 70 (C) 4 and 40 (D) 4 and 35
42. A spherical steel ball of 12 mm diameter is initially at 1000 K. It is slowly cooled in surrounding of 300 K. The heat transfer coefficient between the steel ball and the surrounding is 5 π/π2 K. The thermal conductivity of steel is 20 π/ππΎ. The temperature difference between the centre and the surface of the steel ball is (A) large because conduction resistance is far higher than the convective resistance. (B) large because conduction resistance is far less than the convective resistance. (C) small because conduction resistance is far higher than the convective resistance. (D) small because conduction resistance is far less than the convective resistance.
43. A pipe of 25 mm outer diameter carries steam. The
heat transfer coefficient between the cylinder and surroundings is 5 π/π2 K. It is proposed to reduce the heat loss ffom the pipe by adding insulation having a thermal conductivity of 0.05 π/π K. Which one of the following statements is TRUE? (A) The outer radius of the pipe is equal to the critical radius. (B) The outer radius of the pipe is less than the critical radius. (C) Adding the insulation will reduce the heat loss. (D) Adding the insulation will increases the heat loss.
44. In a condenser of a power plant, the steam
condenses at a temperatures of 60π C. The cooling water enters at 30ππΆ and leaves at 45π C. The logarithmic mean temperature difference (LMTD) of the condenser is
(A) 16. 2ππΆ (B) 21. 6ππΆ (C) 30ππΆ (D) 37. 5ππΆ 45. Water (ππ = 4.18ππ½/πππΎ) at 80ππΆ enters a
counter flow heat exchanger with a mass flow rate of 0.5 ππ/π . Air (ππ = 1ππ½/πππΎ) enters at 30ππΆ
with a mass flow rate of 2.09 ππ/π . If the effectiveness of the heat exchanger is 0.8, the LMTD (inπΆ) is
(A) 40 (C) 10 (B) 20 (D) 5
46. Consider two infinitely long thin concentric tubes of circular cross section as shown in the figure. If π·1 and π·2 are the diameters of the inner and outer tubes respectively, then the view factor πΉ22 is give by
(A) (π·2
π·1) β 1 (B) zero
(C) (π·1
π·2) (π·) 1 β (
π·1
π·2)
47. Which one of the following configurations has the highest fm effectiveness?
(A) Thin, closely spaced fins (B) Thin, widely spaced fins
(C) Thick, widely spaced fins (D) Thick, closely spaced fins 48. For an opaque surface, the absorptivity (πΌ) ,
transmissivity (π) and reflectivity (π) are related by the equation:
(A) πΌ + π = π
(B) π + πΌ + π = 0
(C) πΌ + π = 1
(D) πΌ + π = 0
Answers:
1 C 31 B
2 D 32 D
3 C 33 C
4 C 34 C
5 D 35 D
6 A 36 D
7 B 37 C
8 B 38 B
9 A 39 B
10 B 40 B
11 C 41 A
12 B 42 D
13 C 43 C
14 D 44 B
15 D 45 C
16 C 46 D
17 C 47 A
18 C 48 C
19 D
20 D
21 B
22 C
23 D
24 D
25 C
26 B
27 D
28 B
29 B
30 D