# Conduction

 Question 1
In a furnace, the inner and outer sides of the brick wall ($k_1 = 2.5 W/mK$) are maintained at $1100^{\circ}C$ and $700^{\circ}C$ respectively as shown in figure.

The brick wall is covered by an insulating material of thermal conductivity $k_2$. The thickness of the insulation is $1/4^{th}$ of the thickness of the brick wall. The outer surface of the insulation is at $200^{\circ}C$. The heat flux through the composite walls is 2500 $W/m^2$.

The value of $k_2$ is ________ W/mK (round off to 2 decimal places).
 A 0.25 B 0.5 C 0.75 D 1
GATE ME 2020 SET-2   Heat Transfer
Question 1 Explanation:
Given, $L_{2}=\frac{L_{1}}{4}$
Assuming steady state, one-dimensional conduction heat transfer through composite slab,
Thermal circuit:
$\begin{array}{l} \Rightarrow \quad q=\frac{1100-700}{\frac{L_{1}}{k_{1} A}}=\frac{700-200}{\frac{L_{2}}{k_{2} A}} \\ \Rightarrow \quad \frac{400}{\frac{L_{2}}{2.5}}=\frac{500}{\frac{L_{2}}{k_{2}}}\\ \Rightarrow \quad k_{2}=0.5 \mathrm{W}-\mathrm{mK} \end{array}$
 Question 2
One-dimensional steady state heat conduction takes place through a solid whose cross-sectional area varies linearly in the direction of heat transfer. Assume there is no heat generation in the solid and the thermal conductivity of the material is constant and independent of temperature. The temperature distribution in the solid is
 A Linear B Quadratic C Logarithmic D Exponential
GATE ME 2019 SET-2   Heat Transfer
Question 2 Explanation:
$\begin{array}{c} \text { Area }(\mathrm{A}) \propto \mathrm{x} \\ \mathrm{A}=\mathrm{c} \mathrm{x} \end{array}$
According to Fourier's law of heat conduction
$\mathrm{Q}=-\mathrm{k} \mathrm{A} \frac{\mathrm{d} \mathrm{T}}{\mathrm{d} \mathrm{x}}$
$\begin{array}{l} \mathrm{Q}=-\mathrm{k} \cdot \mathrm{c} \mathrm{x} \frac{\mathrm{d} \mathrm{T}}{\mathrm{d} \mathrm{x}} \\ \mathrm{Q} \frac{\mathrm{d} \mathrm{x}}{\mathrm{x}}=-\mathrm{k} \mathrm{c} \mathrm{d} \mathrm{T} \\ \int \mathrm{d} \mathrm{T}=-\frac{\mathrm{Q}}{\mathrm{k} \mathrm{c}} \int \frac{\mathrm{d} \mathrm{x}}{\mathrm{x}} \\ \mathrm{T}=-\frac{\mathrm{Q}}{\mathrm{k}_{\mathrm{r}}} \ln \mathrm{x}+\mathrm{c}_{1} \end{array}$
Temperature distribution is logarithmic.
(or)
In hollow cylinder, area is linearly proportional to radius.
$A = 2\pi rL$
Temperature profile is logarithmic in case of hollow cylinder with no heat generation.
 Question 3
A slender rod of length L, diameter d ($L \gt \gt d$) and thermal conductivity $k_1$ is joined with another rod of identical dimensions, but of thermal conductivity $k_2$, to form a composite cylindrical rod of length 2L. The heat transfer in radial direction and contact resistance are negligible. The effective thermal conductivity of the composite rod is
 A $k_1+k_2$ B $\sqrt{k_1k_2}$ C $\frac{k_1k_2}{k_1+k_2}$ D $\frac{2k_1k_2}{k_1+k_2}$
GATE ME 2019 SET-1   Heat Transfer
Question 3 Explanation:

Area is constant in the direction of heat flow. (similar like slab)
Thermal circuit :

$\begin{array}{l} \text { Heat transfer rate, }(Q)=\frac{T_{1}-T_{3}}{\frac{L}{k_{1} A}+\frac{L}{k_{2} A}}=\frac{T_{1}-T_{3}}{\frac{2 L}{k_{e q} A}} \\ \frac{L}{k_{1} A}+\frac{L}{k_{2} A}=\frac{2 L}{k_{e q} A} \\ \frac{2}{k_{e q}}=\frac{1}{k_{1}}+\frac{1}{k_{2}} \\ k_{e q}=\frac{2 k_{1} k_{2}}{k_{1}+k_{2}} \end{array}$
 Question 4
A 0.2 m thick infinite black plate having a thermal conductivity of 3.96 W/m-K is exposed to two infinite black surfaces at 300 K and 400 K as shown in the figure. At steady state, the surface temperature of the plate facing the cold side is 350 K. The value of Stefan- Boltzmann constant, $\sigma$ , is $5.67 \times 10^{-8} \: W/m^{2}\: K^{4}$ Assuming 1-D heat conduction, the magnitude of heat flux through the plate (in W/$m^{2}$) is ________ (correct to two decimal places).
 A 258.78 B 963.25 C 391.61 D 147.93
GATE ME 2018 SET-2   Heat Transfer
Question 4 Explanation:
Under steady state condition, all rate of heat transfer i.e. from surface at 400 K to black plate (via radiation), inside black plate (via conduction) and from black plate to surface at 300 K (via radiation) are equal.
So, heat flux through wall= flux from wall to surface at 300 K
$=\frac{\sigma\left(350^{4}-300^{4}\right)}{\frac{1}{1}+\frac{1}{1}-1}=391.612 \text { watt/m }^{2}$
 Question 5
Heat is generated uniformly in a long solid cylindrical rod (diameter = 10mm) at the rate of $4\,\times\,10^{7}$ W/$m^{3}$. The thermal conductivity of the rod material is 25W/m.K. Under steady state conditions, the temperature difference between the centre and the surface of the rod is _________ $^{\circ}C$.
 A 7 B 8 C 9 D 10
GATE ME 2017 SET-1   Heat Transfer
Question 5 Explanation:

\begin{aligned} T_{0}&= \text{ Temp. of cylinder at the axis}\\ T_{S} &=\text { Surface Temp. of Rod (cylindel) } \\ T_{O}-T_{S} &=\frac{\dot{q}}{4 k} R^{2} \end{aligned}
$\dot{q}$ uniform heat generation rate per unit volumne
$\left(w / m^{3}\right)$
$T_{0}-T_{S}=\frac{4 \times 10^{7}}{4 \times 25} \times\left(\frac{5}{1000}\right)^{2}=10^{\circ} \mathrm{C}$
 Question 6
Steady one-dimensional heat conduction takes place across the faces 1 and 3 of a composite slab consisting of slabs A and B in perfect contact as shown in the figure, where $k_{A},k_{B}$ denote the respective thermal conductivities. Using the data as given in the figure, the interface temperature T2 (in $^{\circ}C$) is __________

 A 67.5 C B 68.5 C C 70.6 C D 54.6 C
GATE ME 2016 SET-3   Heat Transfer
Question 6 Explanation:

\begin{aligned} Q_{A} &=Q_{B} \\ \frac{k_{A}\left[T_{1}-T_{2}\right] A}{L_{1}} &=\frac{k_{B}\left[T_{2}-T_{3}\right] A}{L_{2}} \\ \frac{20 \times\left[130-T_{2}\right]}{0.1} &=\frac{100 \times\left[T_{2}-30\right]}{0.3} \\ 3\left[130-T_{2}\right] &=5 T_{2}-150 \\ 390-3 T_{2} &=5 T_{2}-150 \\ 540 &=8 T_{2} \\ T_{2} &=67.5^{\circ} \mathrm{C} \end{aligned}
 Question 7
A hollow cylinder has length L, inner radius $r_{1}$, outer radius $r_{2}$, and thermal conductivity k. The thermal resistance of the cylinder for radial conduction is
 A $\frac{\ln( r_{2}/r_{1} )}{2\pi kL}$ B $\frac{\ln( r_{1}/r_{2} )}{2\pi kL}$ C $\frac{2\pi kL}{\ln( r_{2}/r_{1})}$ D $\frac{2\pi kL}{\ln( r_{1}/r_{2})}$
GATE ME 2016 SET-2   Heat Transfer
Question 7 Explanation:
$Q=\frac{\Delta T}{\text { Thermal resistance }}$

Thermal resistance $=\int_{t_{1}}^{t_{2}} \frac{L}{k A}$
\begin{aligned} \quad&=\int_{t_{1}}^{r_{2}} \frac{d r}{k \times 2 \pi r l} \\ \quad&=\frac{1}{2 \pi / k} \int_{r_{1}}^{r_{2}} \frac{d r}{r} \\ \quad&=\frac{1}{2 \pi l k}[\ln r]_{r_{1}}^{r_{2}} \\ \quad&=\frac{1}{2 \pi l k} \ln \frac{r_{2}}{r_{1}} \\ \end{aligned}
 Question 8
A plastic sleeve of outer radius $r_{0}=1mm$ covers a wire (radius r = 0.5 mm) carrying electric current. Thermal conductivity of the plastic is 0.15 W/m-K. The heat transfer coefficient on the outer surface of the sleeve exposed to air is 25 W/$m^{2}$-K. Due to the addition of the plastic cover, the heat transfer from the wire to the ambient will
 A increase B remain the same C decrease D be zero
GATE ME 2016 SET-1   Heat Transfer
Question 8 Explanation:
\begin{aligned} \text { Given data: } r_{o} &=1 \mathrm{mm} \\ \text { Inner radius: } r_{i} &=0.5 \mathrm{mm} \\ k &=0.15 \mathrm{W} / \mathrm{mK} \\ h_{o} &=25 \mathrm{W} / \mathrm{m}^{2} \mathrm{K}\\ \text{Critical radius,}\\ r_{c} &=\frac{k}{h_{0}}=\frac{0.15}{25} \\ &=6 \times 10^{-3} \mathrm{m}=6 \mathrm{mm} \end{aligned}

As $r_{o} \lt r_{c^{\prime}}$ due to addition of plastic cover heat transfer will increase till $r_{o}=r_{c}$ beyond that heat transfer will decrease.
 Question 9
A brick wall $(k= 0.9\frac{W}{m.K})$ of thickness 0.18m separate the warm air in a room from the cold ambient air. On a particular winter day, the outside air temperature is -5$^{\circ}$C and the room needs to be maintained at 27$^{\circ}$C. The heat transfer coefficient associated with outside air is $20\frac{W}{m^{2} K}$ . Neglecting the convective resistance of the air inside the room, the heat loss, in$\frac{W}{m^{2}}$, is
 A 88 B 110 C 128 D 160
GATE ME 2015 SET-3   Heat Transfer
Question 9 Explanation:
Given data:
\begin{aligned} k&=0.9 \mathrm{W} / \mathrm{mK} \\ \delta&=0.18 \mathrm{m} \end{aligned}

$T_{0}=-5^{\circ} \mathrm{C} ; \quad T_{1}=27^{\circ} \mathrm{C} ; \quad h_{0}=20 \mathrm{W} / \mathrm{m}^{2} \mathrm{K}$
\begin{aligned} \text{Heat loss: }Q&=\frac{T_{1}-T_{0}}{\frac{\delta}{\mathrm{k} A}+\frac{1}{h_{0} A}} \\ \frac{Q}{A} &=\frac{T_{1}-T_{0}}{\frac{\delta}{k}+\frac{1}{h_{0}}} \\ q&=\frac{27-(-5)}{0.18+\frac{1}{20}} \quad \because q=\frac{Q}{A} \\ &=\frac{32}{0.2+0.05} = \frac{32}{0.25}\\ &=128 \mathrm{W} / \mathrm{m}^{2} \end{aligned}
 Question 10
A cylindrical uranium fuel rod of radius 5 mm in a nuclear reactor is generating heat at the rate of $4\times 10^{7} Wm^{3}$ . The rod is cooled by a liquid (convective heat transfer coefficient 1000 W/$m^{2}$-K) at $25^{\circ}C$. At steady state, the surface temperature (in K) of the rod is
 A 308 B 398 C 418 D 448
GATE ME 2015 SET-2   Heat Transfer
Question 10 Explanation:
Given data:
\begin{aligned} r&=5 \mathrm{mm}=0.005 \mathrm{m}\\ \therefore \quad &=2 r=2 \times 0.005=0.010 \mathrm{m} \\ q_{G} &=4 \times 10^{7} \mathrm{W} / \mathrm{m}^{3} \\ h_{0} &=1000 \mathrm{W} / \mathrm{m}^{2} \mathrm{K} \\ T_{0} &=25^{\circ} \mathrm{C} \end{aligned}
\begin{aligned} q_{G} \times \text { volume of } \operatorname{rod} &=h_{0} A\left(T_{s}-T_{0}\right) \\ q_{G} \times \frac{\pi}{4} d^{2} l^{\prime} &=h_{0} \times \pi d l\left(T_{s}-T_{0}\right) \\ \frac{d q_{G}}{4} &=h_{0}\left(T_{s}-T_{0}\right) \\ \frac{0.010 \times 4 \times 10^{7}}{4} &=1000\left(T_{s}-25\right) \\ \text{or }\quad T_{s}-25 &=100\\ \text{or }\quad T_{s}&=100+25=125^{\circ} \mathrm{C}\\ &=(125+273) \mathrm{K}=398 \mathrm{K} \end{aligned}