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aliina [53]
3 years ago
12

A spherical, stainless steel (k 16 W m1 K-1) tank has a wall thickness of 0.2 cm and an inside diameter of 10 cm. The inside sur

face of the tank wall is held constant at 25 oC and the outside surface heat transfer coefficient is 6 W m2 K. Calculate the rate of heat loss from the tank when the outside air temperature is 15°C.
Engineering
1 answer:
SOVA2 [1]3 years ago
8 0

Answer:

the rate of heat loss is 2.037152 W

Explanation:

Given data

stainless steel K = 16 W m^{-1}K^{-1}

diameter (d1) = 10 cm

so radius (r1)  = 10 /2 = 5 cm = 5 × 10^{-2}

radius (r2)  = 0.2 + 5 = 5.2 cm = 5.2 × 10^{-2}

temperature = 25°C

surface heat transfer coefficient = 6 6 W m^{-2}K^{-1}

outside air temperature = 15°C

To find out

the rate of heat loss

Solution

we know current is pass in series from temperature = 25°C to  15°C

first pass through through resistance R1  i.e.

R1  = ( r2 -  r1 ) / 4\pi  × r1 × r2 × K

R1  = ( 5.2 - 5 ) 10^{-2}  / 4\pi  × 5 × 5.2 × 16 × 10^{-4}

R1  = 3.825 ×  10^{-3}

same like we calculate for resistance R2 we know   i.e.

R2 = 1 / ( h × area )

here area = 4 \pi r2²

area = 4 \pi (5.2 × 10^{-2})²  =  0.033979

so R2 = 1 / ( h × area ) = 1 / ( 6 × 0.033979  )

R2 = 4.90499

now we calculate the heat flex rate by the initial and final temp and R1 and R2

i.e.

heat loss = T1 -T2 / R1 + R2

heat loss = 25 -15 / 3.825 ×  10^{-3} + 4.90499

heat loss =  2.037152 W

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Answer:

T = 167 ° C

Explanation:

To solve the question we have the following known variables

Type of surface = plane wall ,

Thermal conductivity k = 25.0 W/m·K,  

Thickness L = 0.1 m,

Heat generation rate q' = 0.300 MW/m³,

Heat transfer coefficient hc = 400 W/m² ·K,

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We are to determine the maximum temperature in the wall

Assumptions for the calculation are as follows

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  • Steady state system
  • One dimensional conduction across the wall

Therefore by the one dimensional conduction equation we have

k\frac{d^{2}T }{dx^{2} } +q'_{G} = \rho c\frac{dT}{dt}

During steady state

\frac{dT}{dt} = 0 which gives k\frac{d^{2}T }{dx^{2} } +q'_{G} = 0

From which we have \frac{d^{2}T }{dx^{2} }  = -\frac{q'_{G}}{k}

Considering the boundary condition at x =0 where there is no heat loss

 \frac{dT}{dt} = 0 also at the other end of the plane wall we have

-k\frac{dT }{dx } = hc (T - T∞) at point x = L

Integrating the equation we have

\frac{dT }{dx }  = \frac{q'_{G}}{k} x+ C_{1} from which C₁ is evaluated from the first boundary condition thus

0 = \frac{q'_{G}}{k} (0)+ C_{1}  from which C₁ = 0

From the second integration we have

T  = -\frac{q'_{G}}{2k} x^{2} + C_{2}

From which we can solve for C₂ by substituting the T and the first derivative into the second boundary condition s follows

-k\frac{q'_{G}L}{k} = h_{c}( -\frac{q'_{G}L^{2} }{k}  + C_{2}-T∞) → C₂ = q'_{G}L(\frac{1}{h_{c} }+ \frac{L}{2k} } )+T∞

T(x) = \frac{q'_{G}}{2k} x^{2} + q'_{G}L(\frac{1}{h_{c} }+ \frac{L}{2k} } )+T∞ and T(x) = T∞ + \frac{q'_{G}}{2k} (L^{2}+(\frac{2kL}{h_{c} }} )-x^{2} )

∴ Tmax → when x = 0 = T∞ + \frac{q'_{G}}{2k} (L^{2}+(\frac{2kL}{h_{c} }} ))

Substituting the values we get

T = 167 ° C

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Answer:

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(b) The net work output, W_{net}, is 308.07 kJ/kg

(c) The thermal efficiency is 58.8%

(d) The Mean Effective Pressure, MEP, is 393.209 kPa

Explanation:

(a) The assumptions made are;

c_p = 1.005 kJ/(kg·K), c_v = 0.718 kJ/(kg·K), R = 0.287 kJ/(kg·K),

Process 1 to 2 is isentropic compression, therefore;

T_{2}= T_{1}\left (\dfrac{v_{1}}{v_{2}}  \right )^{k-1} = 300.15\times 9.2^{0.4} = 729.21 \, K

From;

\dfrac{p_{1}\times v_{1}}{T_{1}} = \dfrac{p_{2}\times v_{2}}{T_{2} }

We have;

p_{2} = \dfrac{p_{1}\times v_{1}\times T_{2}}{T_{1} \times v_{2}} = \dfrac{98\times 9.2\times 729.21}{300.15 } = 2190.43 \, kPa

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\dfrac{p_3}{T_3} =\dfrac{p_2}{T_2}

p₃ = 2 × p₂ = 2 × 2190.43 = 4380.86 kPa

T_3 = \dfrac{p_3 \times T_2}{p_2} =\dfrac{4380.86  \times 729.21}{2190.43} = 1458.42 \, K

Process 3 to 4 is isentropic expansion, therefore;

T_{3}= T_{4}\left (\dfrac{v_{4}}{v_{3}}  \right )^{k-1}

1458.42= T_{4} \times \left (9.2 \right )^{0.4}

T_4 = \dfrac{1458.42}{(9.2)^{0.4}}  = 600.3 \, K

q_{out} = m \times c_v \times (T_4 - T_1) = 0.718  \times (600.3 - 300.15) = 215.5077 \, kJ/kg

The amount of heat transferred to the air, q_{out} = 215.5077 kJ/kg

(b) The net work output, W_{net}, is found as follows;

W_{net} = q_{in} - q_{out}

q_{in} = m \times c_v \times (T_3 - T_2) = 0.718  \times (1458.42 - 729.21) = 523.574 \, kJ/kg

\therefore W_{net} = 523.574 - 215.5077 = 308.07 \, kJ/kg

(c) The thermal efficiency is given by the relation;

\eta_{th} = \dfrac{W_{net}}{q_{in}} \times 100=  \dfrac{308.07}{523.574} \times 100= 58.8\%

(d) From the general gas equation, we have;

V_{1} = \dfrac{m\times R\times T_{1}}{p_{1}} = \dfrac{1\times 0.287\times 300.15}{98} =0.897\, m^{3}/kg

The Mean Effective Pressure, MEP, is given as follows;

MEP =\dfrac{W_{net}}{V_1 - V_2} = \dfrac{W_{net}}{V_1 \times (1- 1/r)}= \dfrac{308.07}{0.897\times (1- 1/9.2)} = 393.209 \, kPa

The Mean Effective Pressure, MEP = 393.209 kPa.

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