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4 years ago

5

Air in a spring-loaded piston/cylinder setup has a pressure that is linear with volume, P = A + BV. With an initial state of P =

150 kPa, V = 1 L and a final state of 800 kPa, V = 1.5 L, it is similar to the setup in Problem 3.48. Find the work done by the air. 3.54 Helium gas expands from 125 kPa, 350 K and 0.25 m3 to 100 kPa in a polytropic process with n = 1.667. How much work does it give out?

1 answer:

ohaa [14]4 years ago

7 0

Answer:

3.48) Work done by the air is 1.05 kJ

3.54) 4.12 kJ of work is given out

Explanation:

3.48) Work done (W) = P2V2 - P1V1

P1 is initial pressure of air = 150 kPa

V1 is initial volume of air = 1 L = 1/1000 = 0.001 m^3

P2 is final pressure of air = 800 kPa

V2 is final volume of air = 1.5 L = 1.5/1000 = 0.0015 m^3

W = (800×0.0015) - (150×0.001) = 1.2 - 0.15 = 1.05 kJ

3.54) For a polytropic process

W = (P1V1 - P2V2)/(n - 1)

P1 is initial pressure of helium = 125 kPa

V1 is initial volume of helium = 0.25 m^3

P2 is final pressure of helium = 100 kPa

n is polytropic exponent = 1.667

V2 is final volume of helium = V1(P1/P2)^(1/n) = 0.25(125/100)^(1/1.667) = 0.25(1.25)^0.5999 = 0.25(1.14) = 0.285 m^3

W = (125×0.25 - 100×0.285)/(1.667 - 1) = (31.25 - 28.5)/0.667 = 2.75/0.667 = 4.12 kJ

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6 0

3 years ago

Using a forked rod, a 0.5-kg smooth peg P is forced to move along the vertical slotted path r = (0.5 θ) m, whereθ is in radians.

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Answer:

N_c = 3.03 N

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Explanation:

Given:

- The attachment missing from the question is given:

- The given expressions for the radial and θ direction of motion:

r = 0.5*θ

θ = 0.5*t^2 ...... (correction for the question)

- Mass of peg m = 0.5 kg

Find:

a) Determine the magnitude of the force of the rod on the peg at the instant t = 2 s.

b) Determine the magnitude of the normal force of the slot on the peg.

Solution:

- Determine the expressions for radial kinematics:

dr/dt = 0.5*dθ/dt

d^2r/dt^2 = 0.5*d^2θ/dt^2

- Similarly the expressions for θ direction kinematics:

dθ/dt = t

d^2θ/dt^2 = 1

- Evaluate each at time t = 2 s.

θ = 0.5*t^2 = 0.5*2^2 = 2 rad -----> 114.59°

r = 1 m , dr / dt = 1 m/s , d^2 r / dt^2 = 0.5 m/s^2

- Evaluate the angle ψ between radial and horizontal direction:

tan Ψ = r / (dr/dθ) = 1 / 0.5

Ψ = 63.43°

- Develop a free body diagram (attached) and the compute the radial and θ acceleration:

a_r = d^2r / dt^2 - r * dθ/dt

a_r = 0.5 - 1*(2)^2 = -3.5 m/s^2

a_θ = r * (d^2θ/dt^2) + 2 * (dr/dt) * (dθ/dt)

a_θ = 1(1) + 2*(1)*(2) = 5 m/s^2

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Radial direction: N_c * cos(26.57) - W*cos(24.59) = m*a_r

θ direction: F - N_c * sin(26.57) + W*sin(24.59) = m*a_θ

Where, F is the force on the peg by rod and N_c is the normal force on peg by the slot. W is the weight of the peg. Using radial equation:

N_c * cos(26.57) - 4.905*cos(24.59) = 0.5*-3.5

N_c = 3.03 N

F - 3.03 * sin(26.57) + 4.905*sin(24.59) = 0.5*5

F = 1.81 N

4 0

3 years ago

Air at 400kPa, 970 K enters a turbine operating at steady state and exits at 100 kPa, 670 K. Heat transfer from the turbine occu

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Answer:

a

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b

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Explanation:

From the question we are told

The rate at which heat is transferred is $\frac{\r Q}{\r m } = - 30KJ/kg$

the negative sign because the heat is transferred from the turbine

The specific heat capacity of air is $c_p = 1.1KJ/kg \cdot K$

The inlet temperature is $T_1 = 970K$

The outlet temperature is $T_2 = 670K$

The pressure at the inlet of the turbine is $p_1 = 400 kPa$

The pressure at the exist of the turbine is $p_2 = 100kPa$

The temperature at outer surface is $T_s = 315K$

The individual gas constant of air R with a constant value $R = 0.287kJ/kg \cdot K$

The general equation for the turbine operating at steady state is \

$\r Q - \r W + \r m (h_1 - h_2) = 0$

h is the enthalpy of the turbine and it is mathematically represented as

$h = c_p T$

The above equation becomes

$\r Q - \r W + \r m c_p(T_1 - T_2) = 0$

$\frac{\r W}{\r m} = \frac{\r Q}{\r m} + c_p (T_1 -T_2)$

Where $\r Q$ is the heat transfer from the turbine

$\r W$ is the work output from the turbine

$\r m$ is the mass flow rate of air

$\frac{\r W}{\r m}$ is the rate of work developed

Substituting values

$\frac{\r W}{\r m} = (-30)+1.1(970-670)$

$\frac{\r W}{\r m}= 300kJ/kg$

The general balance equation for an entropy rate is represented mathematically as

$\frac{\r Q}{T_s} + \r m (s_1 -s_2) + \sigma = 0$

=> $\frac{\sigma}{\r m} = - \frac{\r Q}{\r m T_s} + (s_1 -s_2)$

generally $(s_1 -s_2) = \Delta s = c_p\ ln[\frac{T_2}{T_1} ] + R \ ln[\frac{v_2}{v_1} ]$

substituting for $(s_1 -s_2)$

$\frac{\sigma}{\r m} = \frac{-\r Q}{\r m} * \frac{1}{T_s} + c_p\ ln[\frac{T_2}{T_1} ] - R \ ln[\frac{p_2}{p_1} ]$

Where $\frac{\sigma}{\r m}$ is the rate of entropy produced within the turbine

substituting values

$\frac{\sigma}{\r m} = - (-30) * \frac{1}{315} + 1.1 * ln\frac{670}{970} - 0.287 * ln [\frac{100kPa}{400kPa} ]$

$\frac{\sigma}{\r m}= 0.0861kJ/kg \cdot K$

5 0

4 years ago

a cubical box 20-cm on a side is contructed from 1.2 cm thick concrete panels. A 100-W light bulb is sealed inside the box. What

Flura [38]

Answer:

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4 years ago