Question

Water at 15°C is to be heated to 65°C by passing it over a bundle of 7-m-long, 1-cm-diameter resistance heater rods maintained at 90°C. Water approaches the heater rod bundle in normal direction at a mean velocity of 0.8 m/s. The rods are arranged in-line with longitudinal and transverse pitches of SL = 4 cm and ST = 3 cm. Determine the number of tube rows NL in the flow direction needed to achieve the indicated temperature rise.

Answer 1

NL = 207

Step-by-step explanation:

Solution

Now,

The mean temperature is measured as:

Tm = (T₁ - T₀)/2

= (15 + 65)/2

= 40°C

So,

we find all the thermo-physical properties of water from the table, that is properties of saturated water at T =40°C

Thermo conductivity, k = 0.631 W/m . K

Specific heat Cp = 4179 J/kg . K

Density р = 992. 1 kg/m³

The dynamic viscosity, μ = 0.653 * 10 ^⁻3 kg/m *s

Prandtl number, Pr = 4.32

At T = 15°C

рi = 992.1 kg/m³

At T = 90°C

Prandtl number, Prs = 1.96

Thus,

The maximum flow of velocity is known from the equation stated as:

Vmax = ST/ST - D *V

Here,

ST is refereed to as the transverse pitch for inline arrangements of the rods

so,

Vmax = 3/3-1 * 0.8

= 1.2 m/s

Now

The Reynolds number is determined from the equation given below

ReD =ρVmax D/μ

= 922.1 * 1.2 *(1 *10^⁻²)/ 0.653 * 10^⁻³

= 18231.55

From the table, The Nusselt number correlations fro cross flow over the tube banks for inline arrangement over the range of ReD is shown as

1000 - 2 * 10⁵

Now, the Nusselt number is determined by

NuD = 0.27ReD ^0.63 Pr^ 0.36 (Pr/Prs)^0.25

= 0.27 * (18231.55)^0.63 (4.32)^0.36 * (4.32/1.96)^0.25

=269.32

Then,

The convective transfer of heat water coefficient is determined from the equation shown by Diametral Nusselt Number

NuD =hD/k

So,

we re-write and solve for h

h = NuD * k/D

=269.32 * 0.631/(1 * 10 ^⁻2)

=16993.9 W/m² .K

Now,

The heat transfer surface area for a tube in a row is NT = 1

As = NT NLπDL

= 1*NL* π * (1 * 10^⁻2) * 4

= 0.1257NL

The logarithmic mean temperature of water is represented as

ΔTlm = Te - Ti/ln (Ts - Ti/Ts- Te)

= 65- 15/ln (90 -15/ 90 -65) = 45.51°C

Thus,

The rate of the heat transfer is determined from the equation shown below,

Q =hAsΔTlm

=16993.9 *0.1257 * NL* 45.51.......equation (1)

The mas flow rate of water is determined by the equation below

m =ρiAcV

= ρi * (STL) * V

= 999.1 8 ( 3* 10^⁻2 * 4) * 0.8

= 95.91 kg/s

The rate of heat transfer of water is determined by the equation below

Q = mcp (Te- Ti)

= 95.91 * 4179 * (65-15)

=20041146.72 W..........(Equation 2)

Now,

The number of tube rows in the direction flow is determined by measuring both equations 1 and 2 as

97219.61 NL = 20041146.72

NL =206.14

NL = 207

Therefore, the number of tube rows NL in the flow direction needed to achieve the indicated temperature rise is NL = 207

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