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Table of Contents Principle
3
Objective
3
Background
3
•
Newton’s law of cooling
Experimental Setup •
3 5
Description of the Combined Convection and Radiation Heat Transfer Equipment:
5
Useful Data
6
Procedure
7
1.
2 .
Free convection experiments experiments
Observations
8
Analysis of results
8
Comparison to theoretical correlations
9
Forced convection experiments
10
Observations
10
Analysis of results
10
Comparison to theoretical correlations
11
Discussion
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University of Puerto Rico Mayagüez Campus Department of Mechanical Engineering INME 4032 - LABORATORY II Spring 2004 Instructor: Guillermo Araya
Experiment 2: Natural And Forced Convection Experiment
Principle This experiment is designed to illustrate the Newton’s law of cooling by convection and to understand how the heat transfer coefficient is obtained experimentally. experimentally. Natural Natural and forced convection over a heated cylinder is analyzed and experimental results are compared with standard correlations.
Objective Determine the heat transfer coefficient for a flow around a cylinder under free and forced convection. Understand the correlation between Nu, Nu, Reynolds and Rayleigh numbers. Compare with standard correlation from textbooks on heat transfer. The effect of thermal radiation is also included.
Background Newton’s law of cooling For convective heat transfer, the rate equation is known as Newton’s law of cooling and is expressed as: q ′′ =h (T s −T ∞ )
Where Ts is the surface temperature, T ∞ the fluid temperature, h the convection heat transfer coefficient and
q ′′
the convective heat flux. The heat
transfer coefficient h is a function of the fluid flow, so, it is influenced by the surface geometry, the fluid motion in the boundary layer and the fluid properties as well.
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From the normalized momentum and energy equation in the boundary layer: *
U *
*
∂U ∂ x
+ V
*
*
∂ y
U
∂T ∂ x
*
=−
*
*
*
*
∂U
∂ P ∂ x
*
+
*
+ V
*
∂T ∂ y
*
=
2
*
∂ U
1
Momentum equation
Re L ∂ y * 2 ∂
1
2
T *
Energy equation
Re L Pr ∂ y * 2
Independently of the solution of these equations for a particular case, the functional form for U* and T* can be written as: U* = f(x*,y*,ReL, dp*/dx*) and T* = f(x*,y*,ReL, Pr, dp*/dx*) Heat transfer, due to the no-slip condition at the wall surface of the boundary layer, occurs by conduction;
qs
"
∂ = −∂ k f
T y
y
By combining with the Newton’s law of cooling, we obtain: k f h =−
* Since T * was defined as T
h
=
∂T ∂y
y =0
Ts −T∞
T − Ts T∞ − Ts
can be written in terms of the dimensionless temperature profile T *
h =−
k f (T ∞ −T s ) ∂T * L(T s −T ∞ ) ∂ y *
= y * =0
k f ∂T * L ∂ y *
y * =0
This expression suggests defining a dimensionless parameter;
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0
Nu =
hL
=
k f
∂T ∂y
*
* y* =0
From the dimensionless temperature profiles, we can imply a functional form for the Nusselt number, Nu = f(x*,ReL*,Pr,dp*/dx*) To calculate an average heat transfer coefficient, we have to integrate over x*, so the average Nusselt number becomes independent of x*. F o r a prescribed geometry,
Nu
L
dp
*
dx
*
is specified and
= f (Re L , Pr)
This means that the Nusselt number, for a prescribed geometry is a universal function of the Reynolds and Prandtl numbers. Doing a similar analysis for free convection, it can be shown that, Nu
f (Gr , Pr)
=
or
Nu
f (Ra , Pr)
=
Where Gr is the Grashof number and Ra is the Rayleigh number. The Rayleigh number is simply the product of Grashof and Prandtl numbers ( Ra = Gr Pr )
Then, for free convection the Nusselt number is a universal function of the Grashof and Prandtl numbers or Rayleigh and Prandtl numbers.
Experimental setup Description
of the
Combined Convection and
Radiation
Heat
Transfer Equipment: The combined convection and radiation heat transfer equipment allows investigate the heat transfer of a radiant cylinder located in flow of air (cross
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flow) and the effect of increasing the surface temperature. The unit allows investigation
of
both
natural
convection
with
radiation
and
forced
convection. The mounting arrangement is designed such that heat loss by conduction through the wall of the duct is minimized. A thermocouple (T 10) is attached to the surface of the cylinder. The surface of the cylinder is coated with a matt black finished, which gives an emissivity close to 1.0. The cylinder mounting allows the cylinder and thermocouple (T10) position to be turned 360° and locked in any position using a screw. An index mark on the end of the mounting allows the actual position of the surface to be determined. The cylinder can reach in excess 600°C when operated at maximum voltage and in still air. However the recommended maximum for the normal operation is 500°C .
Useful Data: Cylinder diameter D = 0.01 m Cylinder heated length L = 0.07 m Effective air velocity local to cylinder due to blockage effect Ue = (1.22) × (Ua ) Physical Properties of Air at Atmospheric Pressure T K 300 350 400 450 500 550 600
V 2
m /s 1.568E-5 1.568E-5 2.076E-5 2.076E-5 2.590E-5 2.590E-5 2.886E-5 2.886E-5 3.790E-5 3.790E-5 4.434E-5 4.434E-5 5.134E-5 5.134E-5
k W/mK 0.02624 0.03003 0.03365 0.03707 0.04038 0.04360 0.04659
Pr 0.708 0.697 0.689 0.683 0.68 0.68 0.68
Where:
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T is the absolute temperature, V is the Dynamic viscosity of air, k is the thermal conductivity and Pr is the Prandtl number.
Combined Convection and Radiation Heat Transfer Equipment
Schematic Diagram showing the Combined Convection and Radiation Heat Heat Transfer E ui ment
Procedure a ) Connect instruments to the heat transfer transfer unit unit b ) Measure the reading for the surface temperature of the cylinder, the
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temperature and velocity of the air flow and the power supplied by the heater. c ) Repeat steps 1 and 2 for different velocities the air flow and power
input.
Free convection experiments Observations Set 1 2 3 4
V Volts 4 8 12 16
I Amp
T9 °C
T10 °C
Analysis of results Set 1 2 3 4
Qinput W 4 8 12 16
hr W/m2K
hC1th W/m2K
hC2th W/m2K
The total heat input is: Qinput = V× I The heat transfer rate by radiation is: Qrad = ε
σ
A (Ts4 – Ta4) = hr A (Ts – Ta)
So,
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hr
=
4 ε σ s
(T
T s
− T a4 )
− T a
The heat transfer rate by convection is: Qconv = Qinput - Qrad From Newton’s law of cooling Qconv
hc A(T s
=
T a )
−
And hc =
Qconv A(T s − T a )
Comparison to theoretical correlations For an isothermal long horizontal cylinder, Morgan suggests a correlation of the form, Nu
D
=
hD k
= cRa
n D
(1)
c and n are coefficients that depend on the Rayleigh number Rayleigh number 10-10 – 10-2 10-2 – 102 102 – 104 104 – 107 107 – 1012
c
n
0.675 1.02 0.850 0.480 0.125
0.058 0.148 0.188 0.250 0.333
The Rayleigh number is calculated from,
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Ra
=
gβ(Ts
− Ta )D 3 Pr υ2
where β=
1 Tfilm
and Tfilm =
Ts + Ta 2
Churchill and Chu recommend a single correlation for a wide range of Rayleigh number, 2
Nu D
1/ 6 0.387 Ra = 0.60 + 9 / 16 8 / 27 [ ] 1 ( 0 . 559 / Pr) +
Ra ≤ 10
12
(2)
From correlation (1) and (2) we can determine h C1th and hC2th and compare with hc obtained from the experiment.
Forced convection Observations Set 1 2 3 4 5 6 7
V Volts 20 20 20 20 20 20 20
I Amp
Va m/s 0.5 1 2 3 4 5 6
T9 °C
T10 °C
Analysis of results Experiment2
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Qinput W
Set
hr W/m2K
hC W/m2K
Re -
Nu1 -
Nu2 -
hC1th -
hC2th -
1 2 3 4 5 6 7
The total heat input is: Qinput = V× I The heat transfer rate by radiation is: Qrad = ε So,
hr
σ
=
A (Ts4 – Ta4) = hr A (Ts – Ta)
4 ε σ s
(T
T s
− T a4 )
− T a
The heat transfer rate by convection is: Qconv = Qinput - Qrad From Newton’s law of cooling Qconv = hc A(T s −T a )
and hc =
Q conv A(Ts − Ta )
Comparison with theoretical correlations For an isothermal long horizontal cylinder, Hilper suggests, Nu
D
=
hD k
1/ 3 = C Re m D Pr
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where C and m are coefficient that depend on the Reynolds number: ReD
C 0.989 0.911 0.683 0.193 0.027
0.4-4 4-40 40-4000 4000-400000 40000-400000
m 0.330 0.385 0.466 0.618 0.805
All properties are evaluated at the film temperature Tfilm =
Ts + Ta 2
Churchill and Bernstein proposed the following correlation for Re Pr>0.2
Nu D
= 0.3 +
0.62 Re1 / 2 Pr 1 / 3
0.4 2 / 3 1 + Pr
1/ 4
Re D 5 / 8 1 + 282000
4/ 5
(4)
where all properties are evaluated at the film temperature. From correlation (3) and (4) we can determine h C1th and hC2th and compare with hc obtained from the experiment.
Discussion
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