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8 November, 17:33

3. Consider a 10-m-long smooth rectangular tube, with a = 50 mm and b=25 mm, that is maintained at a constant surface temperature. Liquid water enters the tube at 20C with a mass flow rate of 0.01 kg/s. Determine the tube surface temperature necessary to heat the water to the desired outlet temperature of 80C.

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  1. 8 November, 17:38
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    surface temperature is 86.31c

    Explanation:

    given that

    l = 10m

    width a = 0.05m

    height b = 0.025

    inlet flow temperature T1 = 20c

    mass flow rate, m = 0.01kg/s

    outlet flow temperature, T2 = 80c

    Average temperature

    Tave = T1 + T2/2

    = 20 + 80/2

    = 50C

    Hydraulic diameter of a rectanguar tube

    Dh = 4Ac/p

    = 4 (a x b) / 2 (a + b)

    4 (0.05 x 0.025) / 2 (0.05 + 0.025)

    = 0.0333m

    Average velocity

    Vavg = m / pAc

    Vavg = 0.01/988.1 x (0.05 x 0.025)

    =8.096 x 10∧-3m/s

    properties of water at 1 atm and average temperature

    Density, p = 988.1kg/m³

    specific heat, cp = 4181 j/kg-k

    kinematic viscosity, ∪ = 0. 547 x 10 ∧-3 kg/m-s

    pranditl number, pr = 3.55

    Reynolds number

    Re = pVavgDh / ∪

    = 988.1 x 8.096 x 10∧-3 x0.0333/0.54 x 10 ∧-3

    = 487.02.

    As the obtained reynolds number is less than the 2300, the flow is laminar.

    Hydraulic entry lenght

    lh = 0.05 x Re x Dh

    Lh = 0.05 x 487.02 x 0.0333

    = 0.811m

    therefore,

    thermal entry length = pr x lh

    = 3.55 x 0.811

    = 2.879m

    laminar flow now, a/b value is

    a/b = 0.05/0.0025

    a/b = 2

    note, the nusselt number for a/b = 2 is

    Nu = 3.39

    Heat transfer coefficient

    h = k x Nu/Dh

    h = 0.644 x3.39/0.333

    = 65.56 w/m²k

    The surface area = A

    As = 2 (a + b) L

    A = 2 (0.05 + 0.025) 10

    A = 1.5m²

    the tube temperature

    Te = Ts - (Ts-T1) esp [hAs/mcp]

    80 = Ts - (Ts - 20) esp[ 65.56 x 15/0.01 x 4181]

    80 = Ts - (Ts - 20) exp[-2.352]

    by solving this we get T = 86.31c

    hence the surface temperature is 86.31c
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