A small metal bead (radius 0.5 {mm} ), initially at 100^{\circ} {C}, when placed in a stream of fluid at 20^{\circ} {C}, attains a temperature of 28^{\circ} {C} in 35 seconds. The density and specific heat of the metal are 8500 {kg} {/} {m}^{3} and 400 {J} {/} {kg} \cdot {K}, respectively. If the bead is considered as lumped system, the convective heat transfer coefficient (in {W} {/} {m}^{ {B} \cdot {K}) " between the metal bead and the fluid stream is

GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 36 of 65

Marks: +1, -0

A small metal bead (radius

0.5 {mm}

), initially at

100^{∘} {C}

, when placed in a stream of fluid at

20^{∘} {C}

, attains a temperature of

28^{∘} {C}

in 35 seconds. The density and specific heat of the metal are

8500 {kg} {/} {m}^{3}

and

400 {J} {/} {kg} ⋅ {K}

, respectively. If the bead is considered as lumped system, the convective heat transfer coefficient (in

{W} {/} {m}^{ {B} ⋅ {K})

" between the metal bead and the fluid stream is

Solution:

r=0.5 {mm}

T_{0}=100^{∘} {C}

T_{∞}=20^{∘} {C}

T=28^{∘} {C} \;\text " after "\; {t}=4.35 {sec}

ρ=8500 {kg} {/}{m}^{3}

C_{p}=400 {J} {/} {kgk}

{h}=?

As it is lumped system

\;{T-T_{∞}}/{T_{0}-T_{∞}}=e^{-\;{h A}/{ρ V C_{p}} × t}

\;{28-20}/{100-20}=e^{-(\;{h × 4 π r^{2}}/{ρ × \;{4}/{3} π r^{3} × c_{p}} × t)}

\;{8}/{80}=e^{-\;{8500 × {0.5 × 10^{-3}}/{3} × 400}/{}}

h=299.95 {w}{/} {m}^{2} {k}

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