1 | www.mindvis.in

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,

𝑞 = −𝑘𝐴

𝐿(𝑇2 − 𝑇1)

If the thermal conductivity varies w.r.t temperature as,

𝑘 = 𝑘0(1 + 𝛽𝑇)

The resultant heat flow equation becomes,

𝑞 = −𝑘0𝐴

𝐿[(𝑇2 − 𝑇1) +

𝛽

2(𝑇2

2 − 𝑇12)]

Thermal Resistance concept Rewriting the conduction equation for a plane wall,

𝑞 =𝑇1 − 𝑇2

𝑅𝑡ℎ

Where,

𝑅𝑡ℎ =𝐿

𝑘𝐴

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 Ohm’s law,

𝐼 =𝑉

𝑅

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,

ℎ𝑒𝑎𝑡 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 =∆𝑇(𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑟 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒)

𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒

The figures given below, illustrate the above mentioned concept,

2 | www.mindvis.in

Consider a case of convection heat transfer across a surface of area AS, and temperature TS to a fluid with temperature 𝑇∞, convective heat transfer coefficient h. Newton’s law of cooling gives us,

𝑞 = ℎ𝐴𝑆(𝑇𝑆 − 𝑇∞)

𝑞 =𝑇𝑆 − 𝑇∞

𝑅𝑡ℎ

Where,

𝑅𝑡ℎ =1

ℎ𝐴𝑆

Is the thermal resistance against convection heat transfer.

When the value of convection heat transfer coefficient becomes too large, i.e. 𝒉 → ∞ 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,

𝑞 = −𝑘1𝐴𝑇2 − 𝑇1

𝑥1= −𝑘2𝐴

𝑇3 − 𝑇2

𝑥2= −𝑘3𝐴

𝑇4 − 𝑇3

𝑥3

Upon solving all sections, the heat flow through the complete system is written as,

𝑞 =𝑇1 − 𝑇4

(𝑥1 𝑘1𝐴⁄ ) + (𝑥2 𝑘2𝐴⁄ ) + (𝑥3 𝑘3𝐴)⁄=

∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙

∑ 𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙

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 – T2. The heat flow occurs in the radial direction. The following boundary conditions exist in the cylinder,

𝑇 = 𝑇1 𝑎𝑡 𝑟 = 𝑟1

𝑇 = 𝑇2 𝑎𝑡 𝑟 = 𝑟2

Invoking the heat conduction equation we get the following expression,

𝑞 =2𝜋𝑘𝐿(𝑇1 − 𝑇2)

ln(𝑟2 𝑟1⁄ )=

∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙

∑ 𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙

Where 𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙 =𝑙𝑛 (𝑟2 𝑟1⁄ )

2𝜋𝑘𝐿 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.

𝑞 =𝑇1 − 𝑇2

∑ 𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙(𝑠𝑝ℎ𝑒𝑟𝑒)

Where,

𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙(𝑠𝑝ℎ𝑒𝑟𝑒) =𝑟2 − 𝑟1

4𝜋𝑟1𝑟2𝑘

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 𝑇∞1 and 𝑇∞2 with heat transfer coefficients h1 and h2, as shown in the figure below,

3 | www.mindvis.in

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,

𝑞 =𝑇∞1 − 𝑇∞2

𝑅𝑡𝑜𝑡𝑎𝑙

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 ‘k’ and outer radius r2. Heat is lost to the surroundings maintained at temperature 𝑇∞ and convection heat transfer coefficient ‘h’. the heat flow through the pipe is expressed as,

𝑞 =𝑇1 − 𝑇∞

𝑅𝑐𝑜𝑛𝑑 + 𝑅𝑐𝑜𝑛𝑣=

𝑇1 − 𝑇∞

𝑙𝑛(𝑟2 𝑟1⁄ )2𝜋𝐿𝑘

+1

ℎ(2𝜋𝑟2𝐿)

The variation of ‘q’ with the outer radius is plotted, and the value of r2 at which the ‘q’ reaches a maximum value is called as critical thickness of insulation.

𝑟𝑐 (𝑐𝑦𝑙) =𝑘

ℎ

The value of critical radius will be largest when

‘k’ is large and ‘h’ is small.

4 | www.mindvis.in

The lowest value of ‘h’ 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,

𝑟𝑐 (𝑠𝑝ℎ) =2𝑘

ℎ

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 ‘2L’, 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,

𝑑2𝑇

𝑑𝑥2+

𝑞𝑔

𝑘= 0

The boundary conditions can be specified as follows,

𝑇 = 𝑇𝑤 𝑎𝑡 𝑥 = ±𝐿

The general solution of the differential equation then becomes,

𝑇 = −𝑞𝑔

2𝑘𝑥2 + 𝐶1𝑥 + 𝐶2

On applying the boundary conditions, the temperature distribution can be found to be a parabolic expression as follows,

𝑇 − 𝑇𝑜

𝑇𝑤 − 𝑇0= (

𝑥

𝐿)

2

At steady state conditions, the total heat generated must equal the heat lost at the faces.

Relation between, Tw and To is,

𝑇0 =𝑞𝑔𝐿2

2𝑘+ 𝑇𝑤

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 ‘h’, or

Increase the surface area ‘AS’ Increasing ‘h’ 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 ‘k’ 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 ‘h’ 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.

5 | www.mindvis.in

Keeping in mind the above point, adding too many fins on a surface may actually decrease the overall heat transfer when the decrease in ‘h’ 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,

(𝑟𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡

𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑜𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡

)

= (𝑟𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡

𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒𝑒𝑙𝑒𝑚𝑒𝑛𝑡

)

+ (𝑟𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡

𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒𝑒𝑙𝑒𝑚𝑒𝑛𝑡

)

The differential equation governing the heat flow from a fin is given as,

𝑑2𝜃

𝑑𝑥2 − 𝑎2𝜃 = 0

Where,

𝜃= 𝑇− 𝑇∞ 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠 𝑡ℎ𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑒𝑥𝑐𝑒𝑠𝑠, 𝑎𝑛𝑑 𝑎2

=ℎ𝑝

𝑘𝐴𝑐

At the fin base,

𝜃𝑏 = 𝑇𝑏 − 𝑇∞

There are four different boundary conditions at the fin base and the fin tip,

1. Infinitely long fin (𝑻𝒇𝒊𝒏 𝒕𝒊𝒑 = 𝑻∞)

The boundary condition at the tip is 𝜃(𝐿) = 0 𝑎𝑠 𝐿 → ∞

The expression for the heat transfer for an infinitely long fin is given as,

𝑞𝑙𝑜𝑛𝑔 𝑓𝑖𝑛 = √ℎ𝑝𝑘𝐴𝑐(𝑇𝑏 − 𝑇∞)

Where p = perimeter AC = cross section area of the fin The temperature along the fin in this case decreases exponentially from 𝑇𝑏 to 𝑇∞.

6 | www.mindvis.in

2. Fin tip with negligible heat loss (𝒒𝒇𝒊𝒏 𝒕𝒊𝒑 = 𝟎)

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,

𝑞𝑖𝑛𝑠𝑢𝑙𝑎𝑡𝑒𝑑 𝑡𝑖𝑝 𝑓𝑖𝑛 = √ℎ𝑝𝑘𝐴𝑐(𝑇𝑏

− 𝑇∞) 𝑡𝑎𝑛ℎ (𝑎𝐿)

The temperature distribution for an insulated tip fin is expressed as,

𝑇(𝑥) − 𝑇∞

𝑇𝑏 − 𝑇∞=

𝑐𝑜𝑠ℎ 𝑎(𝐿 − 𝑥)

𝑐𝑜𝑠ℎ (𝑎𝐿)

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,

𝑇(𝑥) − 𝑇∞

𝑇𝑏 − 𝑇∞

=𝑐𝑜𝑠ℎ 𝑎(𝐿 − 𝑥) + (ℎ 𝑎𝑘)⁄ 𝑠𝑖𝑛ℎ 𝑎(𝐿 − 𝑥)

𝑐𝑜𝑠ℎ 𝑎𝐿 + (ℎ 𝑎𝑘)⁄ 𝑠𝑖𝑛ℎ (𝑎𝐿)

And the heat flow from such a fin is given as,

𝑞𝑓𝑖𝑛

= √ℎ𝑝𝑘𝐴𝑐(𝑇𝑏

− 𝑇∞)𝑠𝑖𝑛ℎ (𝑎𝐿) + (ℎ 𝑎𝑘)⁄ 𝑐𝑜𝑠ℎ (𝑎𝐿)

𝑐𝑜𝑠ℎ (𝑎𝐿) + (ℎ/𝑎𝑘) 𝑠𝑖𝑛ℎ (𝑎𝐿)

Corrected fin length It is a better way to determine the heat loss from the fin tip. We replace the fin length ‘L’, in case of an insulated tip, by a corrected length ‘LC’, defined as

𝐿𝐶 = 𝐿 +𝐴𝐶

𝑝

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

𝐿𝐶,𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 = 𝐿 +𝑡

2, 𝑤ℎ𝑒𝑟𝑒 𝑡

= 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑖𝑛

𝐿𝐶,𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 = 𝐿 +𝐷

4, 𝑤ℎ𝑒𝑟𝑒 𝐷

= 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑓𝑖𝑛

Fin efficiency Fin efficiency is mathematically defined as, 𝜂𝑓𝑖𝑛

=𝑎𝑐𝑡𝑢𝑎𝑙 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑟𝑎𝑡𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑓𝑖𝑛

𝑖𝑑𝑒𝑎𝑙 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑟𝑎𝑡𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑓𝑖𝑛 𝑖𝑓 𝑡ℎ𝑒 𝑒𝑛𝑡𝑖𝑟𝑒 𝑓𝑖𝑛 𝑤𝑒𝑟𝑒 𝑎𝑡 𝑏𝑎𝑠𝑒 𝑡𝑒𝑚𝑝.

For insulated fin tips,

𝜂𝑓𝑖𝑛 =√ℎ𝑝𝑘𝐴𝐶 (𝑇𝑏 − 𝑇∞) 𝑡𝑎𝑛ℎ (𝑎𝐿)

ℎ𝑝𝐿(𝑇0 − 𝑇∞)=

𝑡𝑎𝑛ℎ (𝑎𝐿)

𝑎𝐿

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

7 | www.mindvis.in

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,

𝜖𝑓𝑖𝑛 =𝑞𝑓𝑖𝑛

𝑞𝑛𝑜 𝑓𝑖𝑛

𝝐𝒇𝒊𝒏 = 𝟏 then the addition of fins has no effect

on heat transfer

𝝐𝒇𝒊𝒏 < 1 then the fins acts as an insulation, i.e.

slowing down the heat transfer

𝝐𝒇𝒊𝒏 > 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

𝜖𝑓𝑖𝑛 =𝐴𝑓𝑖𝑛

𝐴𝑏𝜂𝑓𝑖𝑛

Where Afin = pL Ab = surface area with no fin

The effectiveness for a long fin can be determined by using the relation,

𝝐𝒍𝒐𝒏𝒈 𝒇𝒊𝒏 = √𝒌𝒑

𝒉𝑨𝒄

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 doesn’t 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 ‘dt’, the temperature of the body rises by a small amount ‘dT’.

ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑖𝑛𝑡𝑜 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 𝑑𝑢𝑟𝑖𝑛𝑔 𝑑𝑡= 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑏𝑜𝑑𝑦 𝑑𝑢𝑟𝑖𝑛𝑔 𝑑𝑡

or,

ℎ𝐴𝑆(𝑇∞ − 𝑇)𝑑𝑡 = 𝑚𝐶𝑝𝑑𝑇 (1)

where,

𝑚 = 𝜌𝑉, 𝑎𝑛𝑑 𝑑𝑇 = 𝑑(𝑇 − 𝑇∞)

Therefore, rearranging equation (1)

𝑑(𝑇 − 𝑇∞)

(𝑇 − 𝑇∞)= −

ℎ𝐴𝑆

𝜌𝑉𝐶𝑝𝑑𝑡 (2)

8 | www.mindvis.in

Integrating equation (2) from t=0, at which T = Ti

𝑙𝑛 𝑇(𝑡) − 𝑇∞

𝑇𝑖 − 𝑇∞= −

ℎ𝐴𝑆

𝜌𝑉𝐶𝑝𝑡 (3)

Taking exponent on both sides of equation (3)

𝑇(𝑡) − 𝑇∞

𝑇𝑖 − 𝑇∞= 𝑒−𝑏𝑡

Where

𝑏 =ℎ𝐴𝑆

𝜌𝑉𝐶𝑝

The term ‘b’ 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 ‘b’,

The temperature of a body reaches the ambient temperature 𝑇∞ exponentially. A larger value of ‘b’ 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,

𝐿𝐶 =𝑉

𝐴𝑆

And another parameter, Biot number (Bi) as,

𝐵𝑖 = ℎ𝐿𝐶/𝑘 Biot number can also expressed as,

𝐵𝑖 =𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑎 𝑏𝑜𝑑𝑦

𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦

=𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒=

𝐿𝐶/𝑘

1/ℎ

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

𝐵𝑖 ≤ 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,

9 | www.mindvis.in

𝐹𝑜 =𝛼𝑡

𝐿2

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 (µ), thermal conductivity (k), density (ρ), 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 Newton’s law of cooling and is expressed as,

𝑞𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 = ℎ𝐴𝑆(𝑇𝑆 − 𝑇∞)

Where, h = convection heat transfer coefficient (W/m2 – 0C) AS = surface area for heat transfer (m2) TS = surface temperature 𝑇∞ = 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

10 | www.mindvis.in

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,

𝑁𝑢 =ℎ𝐿𝐶

𝑘=

𝑞𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛

𝑞𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛

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 ‘V’, and can be put equal to free stream velocity 𝑢∞ 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 𝛿 from the plate surface. Beyond this height𝛿, 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,

𝑢 = 0 𝑎𝑡 𝑦 = 0

𝑢 = 𝑢∞ 𝑎𝑡 𝑦 = 𝛿

The region of the flow above the plate bounded in 𝛿 in which the effects of the viscous shearing effects are felt is called the velocity boundary layer. The boundary layer thickness, 𝛿, is usually defined as the distance ‘y’ from the surface at which 𝑢 = 0.99 𝑢∞.

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,

11 | www.mindvis.in

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,

𝑃𝑟 =𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚

𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 ℎ𝑒𝑎𝑡=

𝜇𝐶𝑃

𝑘

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,

𝑅𝑒 =𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒𝑠

𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠=

𝜌𝑉𝐿𝐶

𝜇

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 Reynold’s 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 𝑢∞.

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, (𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑖𝑛) = (𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑜𝑢𝑡)

Mathematically the conservation of mass equation is given as,

𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦= 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 Newton’s 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.

(𝑚𝑎𝑠𝑠)(𝑎𝑐𝑐𝑒𝑙𝑒𝑎𝑟𝑖𝑜𝑛 𝑖𝑛 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛)= (𝑛𝑒𝑡 𝑓𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑎𝑡 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛)

The momentum conservation in x-direction is mathematically expressed as,

𝜌 (𝑢𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦) = 𝜇

𝜕2𝑢

𝜕𝑦2

−𝜕𝑃

𝜕𝑥, 𝑤ℎ𝑒𝑟𝑒 𝑃 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

Conservation of Energy equation

12 | www.mindvis.in

The energy equation for a steady two-dimensional flow of a fluid with constant properties and negligible shear stresses is expressed as,

𝜌𝐶𝑃 (𝑢𝜕𝑇

𝜕𝑥+ 𝑣

𝜕𝑇

𝜕𝑦) = 𝑘 (

𝜕2𝑇

𝜕𝑥2 +𝜕2𝑇

𝜕𝑦2)

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 𝑇∞. The thickness of the thermal boundary layer is designated as 𝛿𝑡. At the wall, the velocity of the fluid is zero, and the heat transfer into the fluid takes place via conduction,

𝑞 = −𝑘𝐴𝑑𝑇

𝑑𝑦|

𝑠𝑢𝑟𝑓𝑎𝑐𝑒

From Newton’s law of cooling,

𝑞𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 = ℎ𝐴𝑆(𝑇𝑆 − 𝑇∞)

Equating the above heat transfer rates, we get,

ℎ =

(−𝑘𝑑𝑇𝑑𝑦

|𝑠𝑢𝑟𝑓𝑎𝑐𝑒

)

𝑇𝑆 − 𝑇∞

The following boundary conditions apply,

𝑇 = 𝑇𝑆 𝑎𝑡 𝑦 = 0

𝑑𝑇

𝑑𝑦= 0 𝑎𝑡 𝑦 = 𝛿𝑡

𝑇 = 𝑇∞ 𝑎𝑡 𝑦 = 𝛿𝑡

After mathematical manipulation and substitution, we get the following relation for the temperature profile, which indicates a cubic curve,

𝑇 − 𝑇𝑆

𝑇𝑆 − 𝑇∞=

𝜃

𝜃∞=

3

2(

𝑦

𝛿𝑡) −

1

2(

𝑦

𝛿𝑡)

3

To find out the value of thermal boundary layer thickness, use the following formula,

𝛿𝑡

𝛿=

1

1.026𝑃𝑟−1/3

Where 𝛿 = 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,

𝑇𝑓 =𝑇𝑆 + 𝑇∞

2

13 | www.mindvis.in

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 (λ) and frequency (ν) are related as,

𝜆 =𝑐

𝜈

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 ν is considered to have an energy of,

𝑒 = ℎ𝜈 =ℎ𝑐

𝜆

Where h = 6.6256 X 10-34 J.sec is the Planck’s 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.

14 | www.mindvis.in

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,

𝐸𝑏(𝑇) = 𝜎𝑇4

Where σ = 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-Boltzmann’s law, blackbody emissive power is ONLY a function of absolute temperature of the surface, but as per Planck’s law, the blackbody emissive power is a function of wavelength and temperature, as follows,

𝐸𝑏𝜆(𝜆, 𝑇) =𝐶1

𝜆5 [exp (𝐶2𝜆𝑇

) − 1] 𝑊/𝑚2 . 𝜇𝑚

Where,

𝐶1 = 2𝜋ℎ𝑐𝑜2 = 3.742 𝑋 108𝑊. 𝜇𝑚4/𝑚2

𝐶2 =ℎ𝑐𝑜

𝑘= 1.439 𝑋 108𝜇𝑚. 𝐾

Where T = absolute temperature of the surface λ = wavelength of the radiation emitted k = Boltzmann’s 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,

𝐶1 𝑏𝑦𝐶1

𝑛2

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,

15 | www.mindvis.in

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 Wein’s displacement law as,

(𝜆𝑇)max 𝑝𝑜𝑤𝑒𝑟 = 2897.8 𝜇 𝑚. 𝐾

Surfaces at T≤ 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 Ebλ and λ 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 θ and azimuth angle φ, ass illustrated in the figure below

16 | www.mindvis.in

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 ‘ε’ and its value varies between 0 and 1. It is a measure of how closely a surface approximates a blackbody.

For a blackbody, ε = 1

Incident radiation on a surface is called IRRADIATION and is denoted by ‘G’

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 (α),

𝛼 =𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛

𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛, 0 ≤ 𝛼 ≤ 1

The fraction of incident radiation which is reflected is called reflectivity (ρ),

𝜌 =𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛

𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛, 0 ≤ 𝜌 ≤ 1

The fraction of incident radiation which is transmitted is called transmissivity (τ),

𝜏 =𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛

𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛, 0 ≤ 𝜏 ≤ 1

𝛼 + 𝜌 + 𝜏 = 1

For opaque surface, 𝜏 = 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,

17 | www.mindvis.in

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 𝐹𝑖→𝑗 or

just 𝐹𝑖𝑗 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 (β)

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

18 | www.mindvis.in

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,

𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙 = 𝑅𝑖 + 𝑅𝑤𝑎𝑙𝑙 + 𝑅𝑜 =1

ℎ𝑖𝐴𝑖+

[ln (𝐷𝑜𝐷𝑖

)]

2𝜋𝑘𝐿+

1

ℎ𝑜𝐴𝑜

Where Ai = inner surface area of the wall = 𝜋𝐷𝑖𝐿 Ao = outer surface area of the wall = 𝜋𝐷𝑜𝐿 L = length of the tube The heat transfer rate between two fluids is expressed as,

�̇� =∆𝑇

𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙= 𝑈𝐴∆𝑇

𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙 =1

𝑈𝐴

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

19 | www.mindvis.in

areas are almost equal, Ai =Ao, then the overall heat transfer coefficient is,

1

𝑈≈

1

ℎ𝑖+

1

ℎ𝑜

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

𝑈𝐴= 𝑅𝑡ℎ𝑒𝑟𝑚𝑎𝑙 =

1

ℎ𝑖𝐴𝑖+

𝑅𝑓,𝑖

𝐴𝑖+

[ln (𝐷𝑜𝐷𝑖

)]

2𝜋𝑘𝐿+

𝑅𝑓,𝑜

𝐴𝑜

+1

ℎ𝑜𝐴𝑜

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,

�̇� = 𝑈𝐴∆𝑇𝑙𝑛

Where, ΔTln = LMTD and it is expressed as,

∆𝑇𝑙𝑛 =∆𝑇1 − ∆𝑇2

[ln (∆𝑇1∆𝑇2

)]

The values of ∆𝑇1 and ∆𝑇2 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,

∆𝑇𝑚𝑒𝑎𝑛 =∆𝑇1 + ∆𝑇2

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 ‘heat transfer rate’ and the ‘outlet temperatures’ 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

20 | www.mindvis.in

this method, we define a dimensionless parameter, heat transfer effectiveness as,

휀 =𝑎𝑐𝑡𝑢𝑎𝑙 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑟𝑎𝑡𝑒

𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑟𝑎𝑡𝑒=

�̇�

𝑄𝑚𝑎𝑥̇

�̇� = 𝐶𝐶(𝑇𝑐,𝑜𝑢𝑡 − 𝑇𝑐,𝑖𝑛) = 𝐶ℎ(𝑇ℎ,𝑖𝑛 − 𝑇ℎ,𝑜𝑢𝑡)

𝐶𝐶 = 𝑚𝐶̇ 𝑐𝑝𝑐 𝑎𝑛𝑑 𝐶ℎ = 𝑚ℎ̇ 𝑐𝑝ℎ

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,

𝑄𝑚𝑎𝑥̇ = 𝐶𝑚𝑖𝑛(𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛)

Going by the above given relations, we can determine the actual heat transfer rate, if the heat exchanger effectiveness is given, as,

�̇� = 휀 𝑄𝑚𝑎𝑥̇

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,

휀𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑓𝑙𝑜𝑤 =

1 − exp [−𝑈𝐴𝑆

𝐶𝑚𝑖𝑛(1 + (

𝐶𝑚𝑖𝑛

𝐶𝑚𝑎𝑥))]

1 + (𝐶𝑚𝑖𝑛

𝐶𝑚𝑎𝑥)

The quantity −𝑈𝐴𝑆

𝐶𝑚𝑖𝑛 is a dimensionless parameter called the

number of transfer units (NTU),

𝑁𝑇𝑈 =−𝑈𝐴𝑆

𝐶𝑚𝑖𝑛

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),

𝑐 =𝐶𝑚𝑖𝑛

𝐶𝑚𝑎𝑥

Effectiveness is a function of the NTU and the

capacity ratio (c) as,

휀 = 𝑓 (𝑈𝐴𝑆

𝐶𝑚𝑖𝑛,

𝐶𝑚𝑖𝑛

𝐶𝑚𝑎𝑥) = 𝑓(𝑁𝑇𝑈, 𝑐)

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 𝐹13 is 0.17. The view factor 𝐹12 in this case will be

𝑟𝐷

(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 𝑄.4 and 𝑄.5 Heat is being transferred by convection 𝑓𝑖: 𝑜𝑚 water at 48𝑜𝐶 to a glass plate whose surface that is exposed to the water is at 40𝑜 C. The thermal conductivity of water is 0.6 𝑊/𝑚𝐾 and the thermal conductivity of glass is 1.2 𝑊/𝑚𝐾. The spatial gradient of temperature in the water at the water‐ glass interface is 𝑑𝑇𝑙𝑑𝑦 = 1 × 104𝐾/𝑚.

21 | www.mindvis.in

4. The value of the temperature gradient in the glass at

the water‐glass interface in 𝐾/𝑚 is (A) −2 × 104 (B) 0.0 × 104

(C) 0. 5 × 104 (𝐷) 2 × 104

5. The heat transfer coefficient ℎ in 𝑊/𝑚2𝐾 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 𝑈∞. At some distance 𝑥 from the leading edge the velocity boundary layer thickness is 𝛿𝑣 and the thermal boundary layer thickness is 𝛿𝒯 . If the Prandtl number is greater than 1, then (A) 𝛿𝑣 > 𝛿𝑇 (B) 𝛿𝑇 > 𝛿𝑣

(C) 𝛿𝑣 ≈ 𝛿𝑖 ∼ (𝑈∞𝑥)−1/2 (D) 𝛿𝑣 ≈ 𝛿𝑖 ∼ 𝑥−1/2

7. In a counter flow heat exchanger, for the hot fluid the heat capacity = 2𝑘𝐽/𝑘𝑔𝐾, mass flow rate =5𝑘𝑔/𝑠, inlet temperature = 150𝑜𝐶, outlet temperature = 100𝑜 C. For the cold fluid, heat capacity = 4𝑘𝐽/𝑘𝑔𝐾, mass flow rate = 10𝑘𝑔/𝑠, inlet temperature = 20𝑜 C. Neglecting heat transfer to the surroundings, the outlet temperature of the cold fluid in 0𝐶 is

(A) 7.5 (B) 32.5 (C) 45.5 (D) 70.0 8. A plate having 10 𝑐𝑚2 area each side is hanging in

the middle of a room of 100 𝑚2𝑡 ot al surface area. 𝑇 he plat et emp erature and emissivity are respectively 800 𝐾 and 0.6. The temperature and emissivity values for the surfaces of the room are 300 𝐾 and 0.3 respectively. Boltzmann’s constant 𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4 The total heat loss ffom the two surfaces of the plate is

(A) 13.66 𝑊 (B) 27.32 𝑊 (C) 27.87 𝑊 (D) 13.66 MW 9. In a condenser, water enters at 30𝑜𝐶 and flows at

the rate 1500 𝑘𝑔/ℎ𝑟. The condensing steam is at a temperature of 120𝑜𝐶 and cooling water leaves the condenser at 80𝑜 C. Specific heat of water is 4. 187 𝑘𝐽/𝑘𝑔𝐾. If the overall heat transfer coefficient is 2000 𝑊/𝑚2𝐾, then heat transfer area is

(A) 0.707 𝑚2 (B) 7.07 𝑚2 (C) 70.7 𝑚2 (D) 141.4 𝑚2 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 co‐efficient on the 𝑏 ead surface is 400 𝑊/𝑚2𝐾. 𝑇 hermo‐physical properties of thermocouple material are 𝑘 = 20𝑊/𝑚𝐾, 𝑐 =400𝐽/ kg 𝐾 and 𝜌 = 8500𝑘𝑔/𝑚3 If the thermocouple initially at 30𝑜𝐶 is placed in a hot stream of 300𝑜𝐶, then time taken by the bead to reach 298𝑜𝐶, is

(A) 2.35 𝑠 (B) 4.9 𝑠 (C) 14.7 𝑠 (D) 29.4 𝑠 11. A stainless steel tube (𝑘𝑠 = 19𝑊/𝑚𝐾) of 2 cm 𝐼𝐷

and 5 cm 𝑂𝐷 is insulated with 3 cm thick asbestos (𝑘𝑎 = 0.2𝑊/𝑚𝐾) . If the temperature difference between the innermost and outermost surfaces is 600𝑜𝐶, the heat transfer rate per unit length is

(A) 0.94 𝑊

𝑚 (B) 9.44 𝑊/𝑚

(C) 944.72 𝑊

𝑚 (D) 9447.21 𝑊/𝑚

12. One dimensional unsteady state heat transfer

equation for a sphere with heat generation at the rate of 𝑞’ can be written as

(A) 1

𝑟

𝜕

𝜕𝑟(𝑟

𝜃𝑇

𝜕𝑟) +

𝑞

𝑘=

1𝜕𝛤

𝛼𝜕𝑡

(B) 1𝜕

𝑟2𝜕𝑟(𝑟2 𝜃𝑇

𝜕𝑟) +

𝑞

𝑘=

1𝑓𝑓

𝛼𝜕𝑡

(C) 𝜕2𝑇

𝜕𝑟2 +𝑞

𝑘=

1𝑓𝑓

𝛼𝜕𝑡

(D) 𝜕2

𝜕𝑟2 (𝑟𝑇) +𝑞

𝑘=

1𝑓𝑓

𝛼𝜕𝑡

Common Data For 𝑄.13 and 𝑄.14 An uninsulated air conditioning duct of rectangular cross section 1 × 0.5𝑚 , carrying air at 20𝑜𝐶 with a velocity of 10 𝑚/𝑠, is exposed to an ambient of 30𝑜 C. Neglect the effect of duct construction material. For air in the range of 20 − 30𝑜𝐶, data are as follows; thermal conductivity = 0.025𝑊/𝑚𝐾 ; viscosity = 18𝜇𝑃 as, 𝑃 randtl number = 0.73; density = 1.2𝑘𝑔/𝑚3 𝑇 he laminar flow Nusselt number is 3.4 for constant wall temperature conditions and for turbulent flow, Nu = 0.023Re08𝑃𝑟033 13. The Reynolds number for the flow is (A) 444 (B) 890 (C) 4. 44 × 105 (𝐷) 5. 33 × 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 𝑓𝑖: 𝑜𝑚80 to 50𝑜𝐶 in an oil cooler

which uses air as the coolant. The air temperature rises ffom 30 to 40𝑜 C. The designer uses a LMTD value of 26𝑜 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 𝑚, while the cylinder has a diameter and length of 0.5 𝑚 each. The radiation configuration factor 𝐹11 is

(A) 0.375 (B) 0.625

22 | www.mindvis.in

(C) 0.75 (D) 1 17. A small copper ball of 5 mm diameter at 500 𝐾 is

dropped into an oil bath whose temperature is 300 K. The thermal conductivity of copper is 400 𝑊/𝑚𝐾, its density 9000 𝑘𝑔/𝑚3 and its specific heat 385 𝐽/𝑘𝑔𝐾. If the heat transfer coefficient is 250 𝑊/𝑚2𝐾 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 𝐾/𝑠,

(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 𝑚. The 𝑘 values are in 𝑊/𝑚𝐾. The overall thermal resistance in 𝐾/𝑊 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 𝑇1, 𝑇2 and 𝑇3(𝑇1 > 𝑇2 > 𝑇3) .

The conclusion is that the measurements are (A) correct because the maxima in 𝐸𝑏𝜆 show the correct trend (B) correct because Planck’s law is satisfied (C) wrong because the Stefan Boltzmann law is not satisfied (D) wrong because Wien’s displacement law is not satisfied 20. In a case of one dimensional heat conduction in a

medium with constant properties, 𝑇 is the

temperature at position 𝑥, at time 𝑡. Then 𝜃𝑇

𝜕𝑡 is

proportional to

(A) 𝑇

𝑥 (B)

𝜃𝑇

𝜕𝑥

(C) 𝜕2𝑇

𝜕𝑥𝜕𝑡 (D)

𝜕2𝑇

𝜕𝑥2

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 20𝑜𝐶 and relative humidity 5%. An extract of steam tables is given below.

Temp (∘𝐶)

−15 −10 −5 0.01 5 10 15 20

Saturation Pressure (𝑘𝑃𝑎)

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. 3𝑜𝐶

(B) −10. 3𝑜𝐶

(C) −14. 5𝑜𝐶

(D) 14. 5𝑜𝐶

23. 𝐴 100 𝑊 electric bulb was switched on in a 2.5 𝑚 ×3𝑚 × 3𝑚 size thermally insulated room having a temperature of 20𝑜 C. The room temperature at the end of 24 hours will be

(A) 321𝑜𝐶

(B) 341𝑜𝐶

(C) 450𝑜𝐶

(D) 470𝑜𝐶

24. In a composite slab, the temperature at the interface (𝑇𝑖𝑛𝑡𝑒𝑟) between two material is equalto the average ofthe temperature at the two ends. Assuming steady one‐dimensional heat conduction, which of the following statements is true about the respective thermal conductivities?

(A) 2𝑘1 = 𝑘2 (B) 𝑘1 = 𝑘2

(C) 2𝑘1 = 3𝑘2 (D) 𝑘1 = 2𝑘2

0.5 m

1 m

23 | www.mindvis.in

Common Data For 𝑄.25 and 𝑄.26 Consider steady one‐dimensional heat flow in a plate of 20 mm thickness with a uniform heat generation of 80 𝑀𝑊/𝑚3 The lefl and right faces are kept at constant temperatures of 160𝑜𝐶 and 120𝑜𝐶 respectively. The plate has a constant thermal conductivity of 200 𝑊/𝑚𝐾.

25. The location of maximum temperature within the plate 𝑓𝑟𝑜𝑚 its left face is

(A) 15 mm (B) 10 mm (C) 5 mm (D) 0 mm

26. The maximum temperature within the plate in 0𝐶 is (A) 160 (B) 165 (C) 200 (D) 250

27. The average heat transfer co‐efficient on a thin hot vertical plate suspended in still air can be determined 𝑓𝑖: 𝑜𝑚 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 25𝑜𝐶, the plat has a total surface area of 0.1 𝑚2 and a mass of 4 𝑘𝑔. The specific heat ofthe plate material is 2.5 𝑘𝐽/𝑘𝑔𝐾. The convective heat transfer co‐efficient in 𝑊/𝑚2𝐾, at the instant when the plate temperature is 225𝑜𝐶 and the change in plate temperature with time 𝑑𝑇𝑙𝑑𝑡 = −0.02𝐾/𝑠, is

(A) 200 (B) 20 (C) 15 (D) 10 28. In a counter flow heat exchanger, hot fluid enters at

60𝑜𝐶 and cold fluid leaves at 30𝑜 C. Mass flow rate of the fluid is 1 𝑘𝑔/𝑠 and that of the cold fluid is 2 𝑘𝑔/𝑠. Specific heat of the hot fluid is 10 𝑘𝐽/𝑘𝑔𝐾 and that of the cold fluid is 5 𝑘𝐽/𝑘𝑔𝐾. The Log Mean Temperature Difference (LMTD) for the heat exchanger in 0𝐶 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

𝑇 − 𝑇𝑤

𝑇∞ − 𝑇𝑤=

3

2(

𝑦

𝛿𝑡) −

1

2(

𝑦

𝛿𝑡)3,

where 𝑇𝑤 and 𝑇∞ are the temperature ofplate and 𝑓𝑖: 𝑒𝑒 stream respectively, and 𝑦 is the normal distance measured ffom the plate. The local Nusselt number based on the thermal boundary layer thickness 𝛿𝑡 is given by (A) 1.33 (B) 1.50 (C) 2.0 (D) 4.64 30. Steady two‐dimensional heat conduction takes

place in the body shown in the figure below. The normal temperature gradients over surfaces 𝑃 and 𝑄 can be considered to be uniform. The temperature gradient 𝜕𝑇𝑙𝜕𝑥 at surface 𝑄 is equal to 10 𝐾/𝑚. Surfaces 𝑃 and 𝑄 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

𝑊/𝑚𝐾. The values of 𝜕𝑇

𝜕𝑥 and

𝜃𝑇

𝜕𝑦 at surface 𝑃 are

𝑦

(A)

𝜕𝑇

𝜕𝑥= 20𝐾/𝑚,

𝜕𝑇

𝜕𝑦= 0𝐾/𝑚

(B) 𝜃𝑇

𝜕𝑥= 0𝐾/𝑚,

𝜃𝑇

𝜕𝑦= 10𝐾/𝑚

(C) 𝜕𝑇

𝜕𝑥= 10𝐾/𝑚,

𝜕𝑇

𝜕𝑦= 10𝐾/𝑚

(D) 𝜕𝑇

𝜕𝑥= 0𝐾/𝑚, 𝑓 = 20𝐾/𝑚

31. A hollow enclosure is formed between two infinitely long concentric cylinders of radii 1 𝑚 and 2 𝑚, respectively. Radiative heat exchange takes place between the inner surface of the larger cylinder (surface‐2) and the outer surface ofthe smaller cylinder (surface‐l). The radiating surfaces are difffise and the medium in the enclosure is non‐participating. 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 three‐dimensional object shown in the figure

below, five faces are insulated. The sixth face (𝑃𝑄𝑅𝑆), which is not insulated, interacts thermally with the ambient, with a convective heat transfer coefficient of 10 𝑊/𝑚2𝐾 The ambient temperature is 30𝑜 C. Heat is uniformly generated inside the object at the rate of 100 𝑊/𝑚3 Assuming the face 𝑃𝑄𝑅𝑆 to be at uniform temperature, its steady state temperature is

Surface ‐ 2

24 | www.mindvis.in

𝑆||

|1𝑚|𝐻

(A) 10𝑜𝐶 (B) 20𝑜𝐶

(C) 30𝑜𝐶 (D) 40𝑜𝐶

33. The logarithmic mean temperature difference

(LMTD) of a counter flow heat exchanger is 20𝑜 C. The cold fluid enters at 20𝑜𝐶 and the hot fluid enters at 100𝑜 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 40𝑜𝐶

(B) is 60𝑜𝐶

(C) is 80𝑜𝐶 (D) cannot be determined 34. For flow of fluid over a heated plate, the following

fluid properties are known Viscosity = 0.001 𝑃𝑎 − 𝑠;

Specific heat at constant pressure = 1𝑘𝐽/ kg.𝐾; Thermal conductivity = l𝑊/m‐𝐾 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 𝑄.35 and 𝑄.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 727𝑜𝐶 , the lower plate (plate 2) is a diffise and gray surface with an emissivity of 0.7 and is kept at 227𝑜𝐶. Assume that the surfaces are sufficiently large to form a two‐surface enclosure and steady‐state conditions to exits. Stefan‐Boltzmann constant is given as 5.67 × 10−8𝑊/𝑚2𝐾4

35. The irradiation (𝑖𝑛 𝑘𝑊/𝑚2) 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 (𝑖𝑛 𝑘𝑊/𝑚2) between plate 1 and plate 2 is

(A) 17.0 (B) 19.5 (C) 23.0 (D) 31.7 37. Consider steady‐state conduction across the

thickness in a plane composite wall (as shown in the figure) exposed to convection conditions on both sides.

ℎ𝑖 , 𝑇∞𝑖

Given: ℎ𝑖 = 20𝑊/𝑚2𝐾, ℎ𝑜 = 50𝑊/𝑚2𝐾;, 𝑇∞𝑗 =

20𝑜𝐶; 𝑇∞,0 = −2𝑜𝐶, 𝑘1 = 20𝑊/𝑚𝐾; 𝑘2 = 50𝑊/𝑚𝐾; 𝐿1 = 0.30𝑚 and 𝐿2 = 0.15𝑚. Assuming negligible contact resistance between the wall surfaces, the interface temperature, 𝑇 (in 0𝐶), of the two

walls will be (A) −0.50 (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 𝑘𝑔/𝑠 with 𝑐𝑝 = 4𝑘𝐽/ kg 𝐾, enters

the heat exchanger at 102𝑜𝐶 while the cold fluid has an inlet temperature of 15𝑜 C. The overall heat transfer coefficient for the heat exchanger is estimated to be 1 𝑘𝑊/𝑚2𝐾 and the corresponding heat transfer surface area is 5 𝑚2 Neglect heat transfer between the heat exchanger and the ambient. The heat exchanger is characterized by the following relations:

2휀 = − exp ( −2 NTU) The exit temperature (in 0𝐶) for the cold fluid is

(A) 45 (B) 55 (C) 65 (D) 75 39. A coolant fluid at 30𝑜𝐶 flows over a heated flat plate

maintained at constant temperature of 100𝑜 C. The boundary layer temperature distribution at a given location on the plate may be approximated as 𝑇 =30 + 70 exp (−𝑦) where 𝑦 (in m) is the distance normal to the plate and 𝑇 is in ∘ C. Ifthermal conductivity of the fluid is 1. 0𝑊/𝑚𝐾, the local convective heat transfer coefficient (𝑖𝑛 𝑊/𝑚2𝐾) 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 −1𝐾 − 1 One end ofthe fm is maintained at 130𝑜𝐶 and its remaining surface is exposed to ambient air at 30𝑜 C. If the convective heat transfer coefficient is 40 Wm −2𝐾 − 1, the heat loss (in 𝑊) from the fm is

(A) 0.08 (B) 5.0 (C) 7.0 (D) 7.8

1 2

L2 L1

ℎ𝑖, 𝑇∞,𝑖 ℎ𝑜, 𝑇∞,𝑜

25 | www.mindvis.in

41. The ratios of the laminar hydrodynamic boundary layer thickness to thermal boundary layer thickness of flows of two fluids 𝑃 and 𝑄 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 𝑃 are 1/8 and 35 respectively. The Prandtl and Nusselt numbers for 𝑄 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 𝑊/𝑚2 K. The thermal conductivity of steel is 20 𝑊/𝑚𝐾. 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 𝑊/𝑚2 K. It is proposed to reduce the heat loss ffom the pipe by adding insulation having a thermal conductivity of 0.05 𝑊/𝑚 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 60𝑜 C. The cooling water enters at 30𝑜𝐶 and leaves at 45𝑜 C. The logarithmic mean temperature difference (LMTD) of the condenser is

(A) 16. 2𝑜𝐶 (B) 21. 6𝑜𝐶 (C) 30𝑜𝐶 (D) 37. 5𝑜𝐶 45. Water (𝑐𝑝 = 4.18𝑘𝐽/𝑘𝑔𝐾) at 80𝑜𝐶 enters a

counter flow heat exchanger with a mass flow rate of 0.5 𝑘𝑔/𝑠. Air (𝑐𝑝 = 1𝑘𝐽/𝑘𝑔𝐾) enters at 30𝑜𝐶

with a mass flow rate of 2.09 𝑘𝑔/𝑠. If the effectiveness of the heat exchanger is 0.8, the LMTD (in𝐶) 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 𝐷1 and 𝐷2 are the diameters of the inner and outer tubes respectively, then the view factor 𝐹22 is give by

(A) (𝐷2

𝐷1) − 1 (B) zero

(C) (𝐷1

𝐷2) (𝐷) 1 − (

𝐷1

𝐷2)

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 (𝛼) ,

transmissivity (𝜏) and reflectivity (𝜌) are related by the equation:

(A) 𝛼 + 𝜌 = 𝜏

(B) 𝜌 + 𝛼 + 𝜏 = 0

(C) 𝛼 + 𝜌 = 1

(D) 𝛼 + 𝜌 = 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