## Conduction

### GATE-CH-1994-4-k-ht-1mark

1994-4-k-ht

Critical thickness of insulation for

• I. Sphere

• II. Cylinder

### GATE-CH-1989-4-i-a-ht-1mark

1989-4-i-a-ht

The thermal conductivity is minimum for:

• Silver

• Chrome-nickel steel

• Aluminium

• Carbon steel

### GATE-CH-1994-1-n-ht-1mark

1994-1-n-ht

A metal wire of 0.01 m dia and thermal conductivity 200 W/m.K is exposed to a fluid stream with a convective heat transfer coefficient 1000 W/m$$^2$$.K. The Biot number is

• 5.6

• 0.0125

• 3.5

• 0.0035

### GATE-CH-1997-1-11-ht-1mark

1997-1-11-ht

According to kinetic theory, the thermal conductivity of a monoatomic gas is proportional to

• $$T$$

• $$T^{0.5}$$

• $$T^{1.5}$$

• $$T^2$$

### GATE-CH-1997-2-9-ht-2mark

1997-2-9-ht

At steady state, the temperature variation in a plane wall, made of two different solids I and II is shown below:

Then, the thermal conductivity of material I

• is smaller than that of II

• is greater than that of II

• is equal to that of II

• can be greater than or smaller than that of II

[Index]

### GATE-CH-1998-1-6-ht-1mark

1998-1-6-ht

The variation of thermal conductivity of a metal with temperature is often correlated using an expression of the form $k = k_o + aT$ Where $$k$$ is the thermal conductivity, and $$T$$ is the temperature in K. The unit of ‘$$a$$’ in the SI system will be

• W/(m.K)

• W/m

• W/(m.K$$^2$$)

• None; ‘$$a$$’ is just a number

### GATE-CH-1999-2-10-ht-2mark

1999-2-10-ht

Walls of a cubical oven are of thickness $$L$$, and they are made of material of thermal conductivity $$k$$. The temperature inside the oven is 100$$^\circ$$C and the inside heat transfer coefficient is $$3k/L$$. If the wall temperature on the outside is held at 25$$^\circ$$C, what is the inside wall temperature in degrees C?

• 35.5

• 43.75

• 81.25

• 48.25

### GATE-CH-2000-2-12-ht-2mark

2000-2-12-ht

A composite flat wall of a furnace is made of two materials $$A$$ and $$B$$. The thermal conductivity of $$A$$ is twice of that of material $$B$$, while the thickness of layer $$A$$ is half of that of $$B$$. If the temperature at the two sides of the wall are 400 and 1200 K, then the temperature drop (in K) across the layer of material $$A$$ is

• 125

• 133

• 150

• 160

### GATE-CH-2001-2-11-ht-2mark

2001-2-11-ht

The heat flux (from outside to inside) across an insulating wall with thermal conductivity $$k$$ = 0.04 W/m.K and thickness 0.16 m is 10 W/m2. The temperature of the inside wall is -5oC . The outside wall temperature is

• 25oC

• 30oC

• 35oC

• 40oC

### GATE-CH-2002-2-18-ht-2mark

2002-2-18-ht

A composite wall consists of two plates $$A$$ and $$B$$ placed in series normal to the flow of heat. The thermal conductivities are $$k_A$$ and $$k_B$$ and the the specific heat capacities are $$C_{PA}$$ and $$C_{PB}$$, for plates $$A$$ and $$B$$ respectively. Plate $$B$$ has twice the thickness of plate $$A$$. At steady state, the temperature difference across plate $$A$$ is greater than that across plate $$B$$ when

• $$C_{PA} > C_{PB}$$

• $$C_{PA} < C_{PB}$$

• $$k_A < 0.5k_B$$

• $$k_A > 2k_B$$

[Index]

### GATE-CH-2003-13-ht-1mark

2003-13-ht

Three solid objects of the same material and of equal mass:  a sphere, a cylinder (length = diameter) and a cube are at 500oC initially. These are dropped in a quenching bath containing a large volume of cooling oil each attaining the bath temperature eventually. The time required for 90% change of temperature is smallest for

• cube

• cylinder

• sphere

• equal for all the three

### GATE-CH-2003-15-ht-1mark

2003-15-ht

The units of resistance to heat transfer are

• J.m$$^{-2}$$.K$$^{-1}$$

• J.m$$^{-1}$$.K$$^{-1}$$

• W.m$$^{-2}$$.K$$^{-1}$$

• W$$^{-1}$$.m$$^2$$.K

### GATE-CH-2003-57-ht-2mark

2003-57-ht

The inner wall of a furnace is at a temperature of 700oC . The composite wall is made of two substances, 10 and 20 cm thick with thermal conductivities of 0.05 and 0.1 W/(m.oC ) respectively. The ambient air is at 30oC and the heat transfer coefficient between the outer surface of wall and air is 20 W/(m2.oC ). The rate of heat loss from the outer surface in W/m2 is

• 165.4

• 167.5

• 172.8

• 175

### GATE-CH-2003-61-ht-2mark-MANY

2003-61-ht

For a given ambient air temperature with increase in thickness of insulation of a hot cylindrical pipe, the rate of heat loss from the surface would

• decrease

• increase

• first decrease and then increase

• first increase and then decrease

### GATE-CH-2004-58-ht-2mark

2004-58-ht

The left face of a one dimensional slab of thickness 0.2 m is maintained at 80oC and the right face is exposed to air at 30oC. The thermal conductivity of the slab is 1.2 W/(m.K) and the heat transfer coefficient from the right face is 10 W/(m2.K). At steady state, the temperature of the right face in oC is

• 77.2

• 71.2

• 63.8

• 48.7

[Index]

### GATE-CH-2005-58-ht-2mark

2005-58-ht

A circular tube of outer diameter 5 cm and inner diameter 4 cm is used to convey hot fluid. The inner surface of the wall of the tube is at a temperature of 80oC, while the outer surface of the wall of the tube is at 25oC. What is the rate of heat transport across the tube wall per meter length of the tube of steady state, if the thermal conductivity of the tube wall is 10 W/(m.K) ?

• 13823 W/m

• 15487 W/m

• 17279 W/m

• 27646 W/m

### GATE-CH-2006-10-ht-1mark

2006-10-ht

A stagnant liquid film of 0.4 mm thickness is held between two parallel plates. The top plate is maintained at 40oC and the bottom plate is maintained at 30oC. If the thermal conductivity of the liquid is 0.14 W/(m.K), then the steady state heat flux (in W/m2) assuming one dimensional heat transfer is

• 3.5

• 350

• 3500

• 7000

### GATE-CH-2007-42-ht-2mark

2007-42-ht

The composite wall of an oven consists of three materials $$A, B$$ and $$C$$. Under steady state operating conditions, the outer surface temperature $$T_{S,O}$$ is 20oC, the inner surface temperature $$T_{S,I}$$ is 600oC and the oven air temperature is $$T_\infty = 800$$oC. For the following data

thermal conductivities $$k_A = 20$$ W/(m.K) and $$k_C = 50$$ W/(m.K),
thickness $$L_A = 0.3$$ m, $$L_B = 0.15$$ m and $$L_C = 0.15$$ m
inner-wall heat transfer coefficient $$h = 25$$ W/(m2.K),

the thermal conductivity $$k_B$$ in W/(m.K) of the material $$B$$, is calculated as

• 35

• 1.53

• 0.66

• 0.03

### GATE-CH-2008-15-ht-1mark

2008-15-ht

Transient three-dimensional heat conduction is governed by ONE of the following differential equations ($$\alpha$$ - thermal diffusivity, $$k$$ - thermal conductivity and $$\psi$$ - volumetric rate of heat generation).

• $${\displaystyle \frac {1}{\alpha } \frac {\partial T}{\partial t} = \nabla T + \psi k }$$

• $${\displaystyle \frac {1}{\alpha } \frac {\partial T}{\partial t} = \nabla T + \frac {\psi }{k} }$$

• $${\displaystyle \frac {1}{\alpha } \frac {\partial T}{\partial t} = \nabla ^2 T + \psi k }$$

• $${\displaystyle \frac {1}{\alpha } \frac {\partial T}{\partial t} = \nabla ^2 T + \frac {\psi }{k} }$$

### GATE-CH-2008-43-ht-2mark

2008-43-ht

Two plates of equal thickness $$(t)$$ and cross-sectional area, are joined together to form a composite as shown in the figure. If the thermal conductivities of the plates are $$k$$ and $$2k$$ then, the effective thermal conductivity of the composite is

• $$3k/2$$

• $$4k/3$$

• $$3k/4$$

• $$2k/3$$

[Index]

### GATE-CH-2008-46-ht-2mark

2008-46-ht

The temperature profile for heat transfer from one fluid to another separated by a solid wall is

• (a)

• (b)

• (c)

• (d)

### GATE-CH-2009-9-ht-1mark-MANY

2009-9-ht

During the transient convective cooling of a solid object, Biot number $$\rightarrow 0$$ indicates

• uniform temperature throughout the object

• negligible convection at the surface of the object

• significant thermal resistance within the object

• significant temperature gradient within the object

### GATE-CH-2012-14-ht-1mark

2012-14-ht

For heat transfer across a solid-fluid interface, which one of the following statement is NOT true when the Biot number is very small compared to 1?

• Conduction resistance in the solid is very small compared to convection resistance in the fluid

• Temperature profile within the solid is nearly uniform

• Temperature drop in the fluid is significant

• Temperature drop in the solid is significant

### GATE-CH-2012-15-ht-1mark

2012-15-ht

A solid sphere with an initial temperature $$T_i$$ is immersed in a large thermal reservoir of temperature $$T_o$$. The sphere reaches a steady temperature after a certain time $$t_1$$. If the radius of the sphere is doubled, the time required to reach steady-state will be

• $$t_1/4$$

• $$t_1/2$$

• $$2t_1$$

• $$4t_1$$

### GATE-CH-2012-26-ht-2mark

2012-26-ht

The one-dimensional unsteady state heat conduction equation for a hollow cylinder with a constant heat source $$q$$ is $\frac {\partial T}{\partial t} = \frac {1}{r}\frac {\partial }{\partial r}\left (r\frac {\partial T}{\partial r}\right ) + q$ If $$A$$ and $$B$$ are arbitrary constants, then the steady state solution to the above equation is

• $$\displaystyle T(r) = -\frac {qr^2}{2} + \frac {A}{r} + B$$

• $$\displaystyle T(r) = -\frac {qr^2}{4} + A\ln r + B$$

• $$\displaystyle T(r) = A\ln r+ B$$

• $$\displaystyle T(r) = \frac {qr^2}{4} + A\ln r + B$$

[Index]

### GATE-CH-2017-12-ht-1mark

2017-12-ht

The one-dimensional unsteady heat conduction equation is $\rho C_P \frac {\partial T}{\partial t} = \frac {1}{r^n}\frac {\partial }{\partial r}\left (r^n k\frac {\partial T}{\partial r}\right )$ where $$T$$ - temperature, $$t$$ - time, $$r$$ - radial position, $$k$$ - thermal conductivity, $$\rho$$ - density, and $$C_P$$ - specific heat.

For the cylindrical coordinate system, the value of $$n$$ in the above equation is

• 0

• 1

• 2

• 3

### GATE-CH-ME-2008-50-ht-2mark

ME-2008-50-ht

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/m2.K. The ambient temperature is 30oC. Heat is uniformly generated inside the object at the rate of 100 W/m3. Assume the face PQRS to be at uniform temperature, its steady state temperature is

• 10oC

• 20oC

• 30oC

• 40oC

### GATE-CH-ME-2011-40-ht-2mark

ME-2011-40-ht

A spherical steel ball of 12 mm diameter is initially at 1000 K. It is slowly cooled in a surrounding of 300 K. The heat transfer coefficient between the steel ball and surrounding is 5 W/m2.K. The thermal conductivity of steel is 20 W/m.K. The temperature difference between the centre and the surface of the steel ball is

• large because conduction resistance is far higher than the convective resistance

• large because conduction resistance is far less than the convective resistance

• small because conduction resistance is far higher than the convective resistance

• small because conduction resistance is far less than the convective resistance

### GATE-CH-2016-9-ht-1mark

2016-9-ht

A composite wall is made of four different materials of construction in the fashion shown below. The resistance (in K/W) of each of the sections of the wall is indicated in the diagram.

The overall resistance (in K/W, rounded off to the first decimal place) of the composite wall, in the direction of heat flow, is ____________

### GATE-CH-2009-53-54-ht-4mark

2009-53-54-ht

A slab of thickness $$L$$ with one side $$(x = 0)$$ insulated and the other side $$(x = L)$$ maintained at a constant temperature $$T_0$$ is shown below.

A uniformly distributed internal heat source produces heat in the slab at the rate of $$S$$ W/m3. Assume the heat conduction to be steady and 1-D along the $$x$$-direction.

(i) The maximum temperature in the slab occurs at $$x$$ equal to

{#1}

(ii) The heat flux at $$x = L$$ is

{#2}

[Index]

### GATE-CH-PI-2008-76-77-ht-4mark

PI-2008-76-77-ht

A wall is heated uniformly at a volumetric heat generation rate of 1 kW/m3. The temperature distribution across the 1 m thick wall at a certain instant of time is given by: $T(x)=a+bx+cx^2$ where $$a=900$$oC, $$b=-300$$oC/m, and $$c=-50$$oC/m2. The wall has an area of 10 m2 (as shown in the figure) and a thermal conductivity of 40 W/m.K.

(i) The rate of heat transfer (in kW) into the wall (at $$x=0$$) is

{#1}

(ii) The rate of change of energy storage (in kW) in the wall is

{#2}

### GATE-CH-1999-2-8-ht-2mark

1999-2-8-ht

Rate of heat transfer through a pipe wall is given by $Q = \frac {2\pi k(T_i-T_o)}{\ln (r_i/r_o)}$ For cylinders of very thin wall, $$Q$$ can be approximated by

• $$Q = \dfrac {2\pi k\big [(T_i+T_o)/2\big ]}{\ln (r_i/r_o)}$$

• $$Q = \dfrac {2\pi r_i k (T_i-T_o)}{(r_o-r_i)}$$

• $$Q = \dfrac {2\pi k (T_i-T_o)}{(r_o-r_i)}$$

• $$Q = \dfrac {2\pi k (T_i-T_o)}{\big [(r_i+r_o)/2\big ]}$$

### GATE-CH-2008-44-ht-2mark

2008-44-ht

A metallic ball ($$\rho = 2700$$ kg/m3 and $$C_P = 0.9$$ kJ/kg.oC) of diameter 7.5 cm is allowed to cool in air at 25oC. When the temperature of the ball is 125oC, it is found to cool at the rate of 4oC per minute. If thermal gradients inside the ball are neglected, the heat transfer coefficient (in W/m2.oC) is

• 2.034

• 20.34

• 81.36

• 203.4

### GATE-CH-2009-35-ht-2mark

2009-35-ht

For the composite wall shown below (Case 1), the steady state interface temperature is 180oC. If the thickness of layer P is doubled (Case 2), then the rate of heat transfer (assuming 1-D conduction) is reduced by

• 20%

• 40%

• 50%

• 70%

### GATE-CH-2010-36-ht-2mark

2010-36-ht

The figure below shows steady state temperature profiles for one dimensional heat transfer within a solid slab for the following cases:

• [P:] uniform heat generation with left surface perfectly insulted
• [Q:] uniform heat generation with right surface perfectly insulted
• [R:] uniform heat consumption with left surface perfectly insulted
• [S:] uniform heat consumption with right surface perfectly insulted

Match the profiles with appropriate cases.

• P-I, Q-III, R-II, S-IV

• P-II, Q-III, R-I, S-IV

• P-I, Q-IV, R-II, S-III

• P-II, Q-IV, R-I, S-III

[Index]

### GATE-CH-2012-34-ht-2mark

2012-34-ht

Heat is generated at a steady rate of 100 W due to resistance heating in a long wire (length = 5 m, diameter = 2 mm). This wire is wrapped with an insulation of thickness 1 mm that has a thermal conductivity of 0.1 W/m.K. The insulated wire is exposed to air at 30oC. The convective heat transfer between the wire and surrounding air is characterized by a heat transfer coefficient of 10 W/m2.K. The temperature (in oC) at the interface between the wire and the insulation is

• 211.2

• 242.1

• 311.2

• 484.2

### GATE-CH-2015-39-ht-2mark

2015-39-ht

A heated solid copper sphere (of surface area $$A$$ and volume $$V$$) is immersed in a large body of cold fluid. Assume the resistance to heat transfer inside the sphere to be negligible and heat transfer coefficient ($$h$$), density ($$\rho$$), heat capacity ($$C$$), and thermal conductivity ($$k$$) to be constant. Then, at time $$t$$, the temperature difference between the sphere and the fluid is proportional to:

• $$\displaystyle \exp \left (-\frac {hA}{\rho VC}t \right )$$

• $$\displaystyle \exp \left (-\frac {\rho VC}{hA}t \right )$$

• $$\displaystyle \exp \left (-\frac {4\pi k}{\rho CA}t \right )$$

• $$\displaystyle \exp \left (-\frac {\rho CA}{4\pi k}t \right )$$

### GATE-CH-2015-41-ht-2mark

2015-41-ht

Consider a solid block of unit thickness for which the thermal conductivity decreases with an increase in temperature. The opposite faces of the block are maintained at constant but different temperatures: $$T(x=0) > T(x=1)$$. Heat transfer is by steady state conduction in $$x$$-direction only. There is no source or sink of heat inside the block. In the figure below, identify the correct temperature profile in the block.

• I

• II

• III

• IV

### GATE-CH-ME-2010-41-ht-2mark

ME-2010-41-ht

A fin has 5 mm diameter and 100 mm length. The thermal conductivity of fin material is 400 W/m.K. One end of the fin is maintained at 130oC and its remaining surface is exposed to ambient air at 30oC. If the convective heat transfer coefficient is 40 W/m2.K, the heat loss (in W) from the fin is

• 0.08

• 5.0

• 7.0

• 7.8

### GATE-CH-1988-14-iii-ht-6mark

1988-14-iii-ht

A thermopane window consists of two sheets of glass each 6 mm thick, separated by a layer of stagnant air also 6 mm thick. Find the percentage reduction in heat loss from this pane as compared to that of a single sheet of glass 6 mm thickness. The temperature drop between inside and outside remains same at 15$$^\circ$$C. Thermal conductivity of glass is 30 times that of air.

[Index]

### GATE-CH-1994-18-ht-5mark

1994-18-ht

An asbestos pad, square in cross-section, measures 0.05 m on a side and increases linearly to 0.1 m on the side at the other end (see the figure). The length of the pad is 0.15 m.

If the small end is held at 600 K and the larger end at 300 K. What will be the heat flow rate (in W) if the other four sides are insulated. Assume one directional heat flow. Thermal conductivity of asbestos is 0.173 W/m.K.

### GATE-CH-1996-20-ht-5mark

1996-20-ht

A thermocouple junction may be approximated as a sphere of diameter 2 mm with thermal conductivity 30 W/(m.$$^\circ$$C), density 8600 kg/m$$^3$$ and specific heat 0.4 kJ/(kg.$$^\circ$$C). The heat transfer coefficient between the gas stream and the junction is 280 W/(m$$^2$$.$$^\circ$$C). How long (in s) will it take for the thermocouple to record 98% of the applied temperature difference?

### GATE-CH-1997-16-ht-5mark

1997-16-ht

It is proposed to reduce the heat loss from a rectangular furnace wall by doubling its wall thickness as shown in figure. The temperature of the hot surface of the wall is 723 K and it loses heat from the other side exposed to air at 308 K. In case I, the temperature of the wall surface exposed to air is 453 K.

Estimate the % reduction in heat loss due to the doubling of wall thickness. Neglect the radiation losses and assume 1-D conduction. Also assume the thermal conductivity ($$k$$) of furnace wall and the convective heat transfer coefficient ($$h$$) to be constant.

### GATE-CH-1998-16-ht-5mark

1998-16-ht

The wall of a cold storage unit comprises a brick layer (thickness $$\delta _B$$ = 0.1 m, thermal conductivity $$k_B$$ = 1.4 W/m.K) and an inner layer of polyurethane foam (thickness $$\delta _P$$ = 0.05 m, thermal conductivity $$k_P$$ = 0.015 W/m.K). Assume one dimensional heat transfer by conduction through the composite wall, and that the inner surface of the polyurethane layer is at a temperature $$T_c$$ and the outer surface of the brick layer is at temperature $$T_h$$.
Derive an expression for the heat flux through the wall.
Calculate the rate of heat gain (in W) when $$T_c = -10^\circ$$C and $$T_h = 40^\circ$$C.
The surface area for heat transfer is 260 m$$^2$$.

### GATE-CH-2000-20-ht-2mark

2000-20-ht

The outside surface temperature of a pipe (radius = 0.1m) is 400 K. The pipe is losing heat to atmosphere, which is at 300 K. The film heat transfer coefficient is 10 W/(m$$^2$$.K). To reduce the rate of heat loss, the pipe is insulated by a 50 mm thick layer of asbestos ($$k$$ = 0.5 W/(m.K)). Calculate the percentage reduction in the rate of heat loss.

[Index]

### GATE-CH-2014-47-ht-2mark

2014-47-ht

The bottom face of a horizontal slab of thickness 6 mm is maintained at 300oC. The top face is exposed to a flowing gas at 30oC. The thermal conductivity of the slab is 1.5 W/(m.K) and the convective heat transfer coefficient is 30 W/(m2.K). At steady state, the temperature (in oC) of the top face is ____________

### GATE-CH-2005-60-ht-2mark-DIFFICULT

2005-60-ht

A semi-infinite slab occupying the region $$x=0$$ and $$x=\infty$$ is at an initial temperature $$T_0$$. At time $$t=0$$, the surface of the slab at $$x=0$$ is brought into contact with a heat bath at a temperature $$T_H$$. The temperature $$T(x,t)$$ of the slab rises according to the equation $\frac {T_H-T(x,t)}{T_H-T_0} = \frac {2}{\sqrt {\pi }} \int _0^{x/\sqrt {4\alpha t}} e^{-\eta ^2}d\eta$ where $$x$$ is position and $$t$$ is time. The heat flux at the surface $$x=0$$ is proportional to

• $$t^{-1/2}$$

• $$t^{1/2}$$

• $$t$$

• $$t^{3/2}$$

### GATE-CH-1987-14-i-ht-2mark

1987-14-i-ht

A metallic slab of thickness $$2R$$, initially at a uniform temperature $$T_i$$ throughout, is immersed in

1. large pool of fluid, at a temperature $$T_f$$, flowing at a very large velocity.
2. a large pool of fluid, at a temperature $$T_f$$, flowing at a small velocity.
It is required to obtain the temperature distribution in the slab, in each of the above cases. Write down the governing equation that needs to be solved, together with all relevant initial and or boundary conditions necessary for the solution, in each of the above cases. Consider unidirectional conduction only and neglect radiation effects.

### GATE-CH-1993-21-b-ht-5mark

1993-21-b-ht

A pipe of 20 mm inner diameter and 30 mm outer diameter is insulated with 35 mm thick insulation. The thermal conductivity of insulating material is 0.15 W/(m.K) and the convective heat transfer coefficient of outside air is 3 W/(m$$^2$$.K). The temperature of bare pipe is 200$$^\circ$$C and the ambient air temperature is 30$$^\circ$$C. The heat transfer resistance of the pipe metal can be neglected.

1. Comment with reasoning about the heat transfer rates with and without insulation.
2. If the same insulating material is used, what is the minimum thickness above which there is a reduction in heat loss as compared to the bare pipe?
3. For optimum design, what conductivity of insulating material do you suggest for the conditions given in the problem?

### GATE-CH-1999-12-ht-5mark

1999-12-ht

Obtain expressions for steady-state temperature profile and heat transfer rate for a hollow spherical container. The inner surface (at $$r = r_i$$) is maintained at $$T = T_i$$ and the outer surface (at $$r = r_o$$) is maintained at $$T = T_o$$.

[Index]