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Oduvanchick [21]
3 years ago
8

procedure for determining the thermal conductivity of a solid involves embedding thermocouple in a thick slab of the material an

d measuring the response to a prescribed change in temperature at one surface. Consider an arrangement for which the thermocouple is embedded 10 mm from a surface that is suddenly brought to a temperature of 100degreeC by exposure to boiling water. If the initial temperature of the slab was 30degreeC and the thermocouple measures a temperature of 65degreeC, 2 minutes after the surface is brought to 100degreeC, what is the thermal conductivity. The density of the material is 2200 kg/m3 and the specific heat is 700 J/M- Find: What is the thermal conductivity of the material
Physics
1 answer:
Viefleur [7K]3 years ago
7 0

Answer:

The thermal conductivity  k  = 1.4094   W/ m\cdot K

Explanation:

From the question we are told that

  The  depth of the thermocouple from the surface is  x =  10 mm  = 0.01 m

   The  temperature is  T_f  =  100 ^o C

   The  initial temperature is   T_i  =  30 ^o C

    The  temperature of the thermocouple after t =  2 minutes( 2 * 60 =  120 \ seconds)  is   T_t  =  65 ^o C

    The  density of the material  is  \rho =  2200 kg/m^3

     The specific heat of the solid c_s  =  700 J/kg \cdot K

Generally  the  equation for  semi -infinite medium  is mathematically as  

    \frac{T_s - T }{T_i - T} =  erf [\frac{x}{2 \sqrt{\alpha  * t} } ]

     \frac{65 - 100 }{30 - 100} =  erf [\frac{x}{2 \sqrt{\alpha  * t} } ]

        0.5 =  erf [\frac{0.01}{2 \sqrt{\alpha  * 120} } ]

Here \alpha is a constant with unit m^2 /s

  \frac{0.01}{ 2 (\sqrt{\alpha  *  120 } )}   this is from the Gaussian function table  

    0.0 1 =  0.954 * (\sqrt{\alpha * 120  } )

=>   \sqrt{\alpha  * 120  } =  \frac{0.01 }{0.954 }

=>   \alpha =  9.1525 *10^{-7} \  m^2 /s

Generally the thermal  conductivity is mathematically represented as

     k  =  \alpha  *  \rho * c_s

      k  = 9.1525 *10^{-7}   *  2200 * 700

     k  = 1.4094   W/ m\cdot K

   

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Answer:

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Explanation:

First we find the components of F₁:

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For y-component:

F₁y = F₁ Sin θ₁

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Now, for the resultant force on ball to be zero, the sum of x-components of the forces and the sum of the y-component of the forces must also be equal to zero:

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<u>F₃ = 122.88 N</u>

and the direction is given as:

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<u>θ₃ = 20.63°</u>

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