excerpt from, analyzing convective heat transfer

__________________________ 1Student, Department of Mechanical Engineering, Clemson, SC 29631

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The following excerpt is from a senior-level mechanical engineering lab report. The experiment and

report was a group assignment; however, I was responsible for the theoretical analysis that is presented

in Section C as well as Appendix A, B and C. I have removed text that was written by my teammates, but

have included the Abstract and Introduction to provide the reader with some background information

about the assignment.

Excerpt From,

“Analyzing Convective Heat Transfer” ME 444

November 2012

Heat transfer analysis has many applications in engineering design, especially where the time necessary to

change an objects temperature is needed. As an example of this analysis, the experiment examined a solid

sphere and cuboid exposed to free and forced convection. The convective heat transfer coefficient and time

required for heat transfer was determined from experimental data for both specimens in each mode of

convection. Theoretical values for these parameters were also determined, and then compared with

experimental results. Two analytical models were used to calculate the theoretical values: the lumped

parameter model and a transient 1-dimensional analysis. Overall, both analytical models appropriately

represented the system, but the results from the transient analysis were most accurate. Uncertainty analysis

and error propagation were conducted to determine the confidence in the reported values.

I. Introduction

HE transfer of heat between a solid object and a fluid is known as convection. Convection is classified

according to the nature of the fluid’s flow. Forced convection is used to describe circumstances when the flow

is caused by an external force, such as a fan or the earth’s atmosphere. Free convection is used to describe situations

when the fluid flow is caused by buoyant forces, which are created by the temperature induced density gradient in

the fluid. Both methods of convection can be analyzed to determine specific properties and parameters of the heat

transfer. One valuable parameter is the convection heat transfer coefficient, which is proportional to the rate of heat

transfer between the solid and the fluid. This coefficient is useful for determining the time required for convection to

raise or lower an object’s temperature and has many practical design applications.

Concepts of convection and the heat transfer coefficient were explored using a metal sphere and cuboid. Each

solid’s temperature was lowered near 0° C, and then exposed to free or forced convection until their temperature

reached steady state (room temperature ~ 23°C). The convection heat transfer coefficient for each solid in each

method of convection was determined from the experimental data. The heat transfer coefficient was also analytically

determined using an appropriate thermal model, and these values were compared. Similar thermal analysis can be

T

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applied to similarly shaped objects such as coolers, homes, electrical components, or vehicles. Analysis can

determine the time required for liquid in a cooler to become warm, a home to warm up at night, or time for a

computer or engine to overheat in given conditions.

[…text removed…]

C. Analysis Procedure

1. Lumped Parameter Model

The average convection heat transfer coefficient for each solid in each method of convection was analytically

determined to validate experimental data. The lumped parameter model was used for this analysis. This model

assumes that the temperature of the solid is uniform at any instant in time during the heat transfer process1.

Essentially, this model neglects any temperature gradient within the solid and allows the temperature measured by

the thermocouple to represent the temperature of the entire solid.

The following condition was used to verify that the lumped parameter model provided reasonable results:

(1)

where h is the heat transfer coefficient, Lc is the characteristic length, k is the fluid’s thermal conductivity, and Bi is

the Biot number. The Biot number represents the measure of temperature drop within a solid relative to the

temperature difference between the solid and its surroundings1. For the circumstances in this analysis, the Biot

number was approximately 0.012; therefore, the error created by the lumped parameter model was small and the

model was appropriate to use.

a.) Free Convection - Theoretical Heat Transfer Coefficient

The following assumptions were made before applying the lumped parameter model to each solid in free

convection:

1. The solids’ surface is isothermal

2. Thermal effects due to radiation are negligible

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3. The solid is made of stainless steel (AISI 304)

4. The surrounding air is quiescent

The average heat transfer coefficient of a solid in free convection is dependent upon several

dimensionless parameters. These parameters are shown in Table 2 (see Appendix A for each expression’s

conditions). The average convective heat transfer coefficient was then found using the following

relationship:

(2)

where h is the heat transfer coefficient, L is the characteristic length, k is the fluid’s thermal conductivity,

and Nu is the Nusselt number (a dimensionless parameter).

Table 2 Dimensionless Parameters for Free Convection

N

o

t

i

c

e

Notice that three different Nusselt numbers were needed to model the cuboid. Different expressions were

needed because the surfaces of the cube do not experience the same heat transfer because of their orientation

relative to buoyant forces (gravity). Therefore, each surface had an individual heat transfer coefficient, and

these values were combined using the following expression to determine the average heat transfer coefficient

for the entire solid:

( ) ( ) ( )

( ) (3)

Sphere Cuboid

Rayleigh

Number

RaD g Ti T D3

Nusselt

Number

NuD 20.589RaD

1/4

1 0.469/Pr 9/1 6

4 /9

RaL g Ti T L3

Nutop 0.27RaL1/4

Nubottom 0.15RaL1/3

Nuside 0.680.670RaL

1/ 4

1 0.492/Pr 9 /16

4 / 9

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b.) Forced Convection – Theoretical Heat Transfer Coefficient

The analysis for each solid in forced convection was similar to the free convection analysis; however, the

Rayleigh number was replaced with the flow’s Reynolds number. The equations used to calculate the

average heat transfer coefficient for the sphere and cuboid in forced convection is shown in Table 3 (see

Appendix A for each expression’s conditions).

Table 3 Dimensionless Parameters for Forced Convection

T

h

e front and back surfaces of the cuboid in forced convection were modeled as surfaces in free convection.

This was appropriate because these surfaces are oriented perpendicular to the fluid’s flow; therefore, the

fluid is stagnant at these surfaces and no forced convection takes place. The parameters in Table 3 show the

equations for the remaining four surfaces that were parallel to the fluid flow.

The average heat transfer coefficient for each solid was determined using the same relationships for

that were shown for free convection.

Also, the following assumptions were used during forced convection analysis:

1. The solid’s surface is isothermal

2. Thermal effects due to radiation are negligible

3. The solid is made of stainless steel (AISI 304)

c.) Time Required for Heat Transfer

The time required to heat each solid to room temperature was determined by using the following

equation:

(

) (4)

Sphere Cuboid

Reynolds

Number

ReVD

ReVD

Nusselt

Number

NuD 2 0.4ReD1/ 20.06ReD

2 / 3 Pr0.4

film

NuD 0.644Re1/2Pr1/3

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where ρ and Cp are material properties evaluated at room temperature, A is the solid’s surface area, and is

the solid’s volume. Details for the calculations are given in Appendix A.

2. Experimental Analysis

Analyzing the experimental data, heat transfer coefficients for each scenario were found based on initial and

final temperatures of the specimen, ambient air temperature, and time for the temperature change. Using these

values and manipulating Eq. 4, an overall heat transfer coefficient was found from the actual data. The temperature

curves using the calculated theoretical coefficients and the empirical coefficients were compared and can be seen in

Section III. Details for the calculations are given in Appendix B.

3. Transient 1-Dimensional Heat Transfer

Although it was shown that the lumped parameter model provides appropriate results, the sphere in free

convection was also analyzed with a transient internal temperature distribution. This additional analysis is a more

complicated alternative to the lumped parameter model, but is regarded as a more accurate analysis. The cuboid was

not analyzed because of its more complicated geometry.

Transient 1-dimensional analysis was used to determine the average convective heat transfer coefficient that

provided a cooling time most similar to the experimental cooling time. The coefficient was then used to validate

other analytical and experimental results. The cooling time for transient analysis is shown in Equation (5), where Fo

is the dimensionless Fourier number, ro is the sphere’s radius, and α is the metal’s thermal diffusivity.

(5)

The Fourier number is expressed by the following equation, where ζ and C1 are tabulated values that rely on the heat

transfer coefficient:

[

] (6)

Details for the calculations are given in Appendix C.

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Appendix A

Theoretical heat transfer analysis.

clear

clc

%% Measured Temperatures

Tinf = 296; % K (room temperature)

Ti = 273; % K (initial surface temperature)

Tf = 284.5; % K (film temperature = (Ti+Tinf)/2))

Tfinal = 291.5;

% Tfinal = Tinf-0.02*Tinf % K (Threshold for steady state temp.)

%% Test Specimen Dimensions

Ds = 0.0381; % m (sphere diameter)

Vol = 0.0254*0.02536*0.02536; % m^3 (Volume of cube)

At = 0.00064313; % m^2 (surface area of top & bottom surfaces of cube)

As = 0.00064414; % m^2 (surface area of side of cube)

Ac = (4*As)+At+At; % m^2 (surface area of entire cube)

Per = 0.1014; % m (perimeter of top & bottom surfaces of cube)

Ltb = At/Per; % m (charac. length of top & bottom surfaces of cube)

Ls = 0.0254; % m (charac. length of vertical side of cube)

Ls_perp = 0.02536; % m (charac. length of side in forced convec.)

%% Metal Properties at 300K

% Stainless Steel (AISI 304) (Table A.1 Incropera)

den_ss = 7900; % kg/m^3

Cp_ss = 477; % J/kgK

k_ss = 14.9; % W/mK

%% Air Properties

% Air at Room Temperature (Tinf = 296K)

mu_inf = 0.00001826; % Ns/m^2

nu_inf = 0.00001553; % m^2/s

k_inf = 0.0257; % W/mK

Pr_inf = 0.708;

% Air at Film Temperature (Tf = (Ti + Tinf)/2 = 284.5K)

mu_f = 0.00001769; % Ns/m^2

nu_f = 0.00001451; % m^2/s

k_f = 0.0251; % W/mk

alpha_f = 0.00002045; % m^2/s

Pr_f = 0.716;

%% Experimental Constants

g = 9.81; % m/s^2

V = 2.8; % m/s (air speed)

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%% Sphere in Free Convections

Ra_sfree = abs((g*(1/Tinf)*(Ti-Tinf)*(Ds^3))/(alpha_f*nu_f));

Nu_sfree = 2 + (0.589*(Ra_sfree^0.25))/(1 + (0.469/Pr_f)^(9/16))^(4/9);

h_sfree = Nu_sfree*(k_f/Ds)

%% Sphere in Forced Convection

Re_sforc = V*Ds/nu_inf;

Nu_sforc = 2 + (0.4*sqrt(Re_sforc) +

0.06*Re_sforc^(2/3))*(Pr_inf^0.4)*(mu_inf/mu_f);

h_sforc = Nu_sforc*(k_inf/Ds)

%% Cuboid in Free Convection

% Top Surface

Ra_top_fre = abs((g*(1/Tinf)*(Ti-Tinf)*(Ltb^3))/(alpha_f*nu_f));

Nu_top_fre = 0.27*(Ra_top_fre^.25);

h_top_fre = Nu_top_fre*(k_f/Ltb);

% Bottom Surface

Ra_bott_fre = Ra_top_fre;

Nu_bott_fre = 0.15*(Ra_bott_fre^(1/3));

h_bott_fre = Nu_bott_fre*(k_f/Ltb);

% Side Surface

Ra_Csfree = abs((g*(1/Tinf)*(Ti-Tinf)*(Ls^3))/(alpha_f*nu_f));

Nu_Csfree = 0.68 + (0.670*(Ra_Csfree^0.25))/(1 + (0.469/Pr_f)^(9/16))^(4/9);

h_Csfree = Nu_Csfree*(k_f/Ls);

% Total Area

h_cfree = ((4*h_Csfree*As)+(h_top_fre*At)+(h_bott_fre*At))/(4*As + At + At)

%% Cuboid in Forced Convection

% Surfaces parallel with flow direction

Re_CPforc = V*Ls_perp/nu_inf;

Nu_CPforc = 0.664*sqrt(Re_CPforc)*(Pr_inf^(1/3));

h_CPforc = Nu_CPforc*k_f/Ls_perp;

% Surfaces perpindicular to flow direction (modeled as free convection)

h_CPerforc = h_Csfree;

% Total Area

h_cforc = (2*As*h_CPerforc + (2*As+2*At)*h_CPforc)/(4*As + 2*At)

%% Time to reach steady state

% Free Sphere

t_sfree = ((den_ss*Cp_ss*Ds)/(h_sfree*6))*log((Ti-Tinf)/(Tfinal-Tinf))/60

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% Forced Sphere

t_sforc = ((den_ss*Cp_ss*Ds)/(h_sforc*6))*log((Ti-Tinf)/(Tfinal-Tinf))/60

% Free Cube

t_cfree = ((den_ss*Cp_ss*Vol)/(Ac*h_cfree))*log((Ti-Tinf)/(Tfinal-Tinf))/60

% Forced Cube

t_cforc = ((den_ss*Cp_ss*Vol)/((2*At+2*As)*h_CPforc))*log((Ti-Tinf)/(Tfinal-

Tinf))/60

Appendix B

Experimental heat transfer analysis.

%Free Sphere % Define constants

% Stainless Steel AISI 304 at 300K (from Table A.1) den = 7769.8; % kg/m^3 Cp = 477; % J/kgK

% Sample Diameter D = 0.0381; % meter

Ti = 273; % Specimen's initial temp [K] Tinf = 296; % Temp of Ambient Air [K]

%Theoretical hbar hbar = 7.585; % Define relationship between Temperature (T) and time (t) t = 0:0.5:5400; % time [0-90min every 0.5 second] T = Tinf+((Ti-Tinf)*exp((-6*hbar*t)/(den*Cp*D)))-273; % Specimen's Temp [K]

% Plot Sample's Temperature with respect to time figure(1) hold on plot(t,T,'--r')

%Experimental hbar %Define experimental parameters Texp = 291.4; %Final experimental temperature texp = 4000; %Total experimental time

%Solve for experimental hbar hbar_exp = den*Cp*D*log((Ti-Tinf)/(Texp-Tinf))/(6*texp);

% Define relationship between Temperature (T) and time (t) texp = 0:0.5:5400; % time [0-90min every 0.5 second] Texp = Tinf+((Ti-Tinf)*exp((-6*hbar_exp*t)/(den*Cp*D)))-273; % Specimen's

Temp [K]

plot(texp,Texp,'--b')

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%Plot Experimental Data plot(frstime,frstemp)

%Theoretical Dry Curve %Define dry parameters Tdry = 291.4; %Final experimental temperature tdry = 1200; %Total dry curve time Tidry = 289.5; %Start of dry curve

%Solve for experimental hbar hbar_dry = den*Cp*D*log((Tidry-Tinf)/(Tdry-Tinf))/(6*tdry);

% Define relationship between Temperature (T) and time (t) tdry = 0:0.5:5400; % time [0-90min every 0.5 second] Tdry = Tinf+((Ti-Tinf)*exp((-6*hbar_dry*t)/(den*Cp*D)))-273; % Specimen's

Temp [K]

plot(tdry,Tdry,'--g')

%Free Cube % Define constants


% Sample Dimensions Vol = 0.0254*0.02536*0.02536; % m^3 (Volume of cube) At = 0.00064313; % m^2 (surface area of top & bottom surfaces of cube) As = 0.00064414; % m^2 (surface area of side of cube) Ac = (4*As)+At+At; % m^2 (surface area of entire cube)


%Theoretical hbar hbar = 7.16; % Define relationship between Temperature (T) and time (t) t = 0:0.5:5400; % time [0-90min every 0.5 second] T = Tinf+((Ti-Tinf)*exp((-Ac*hbar*t)/(den*Cp*Vol)))-273; % Specimen's Temp

[K]



%Solve for experimental hbar

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hbar_exp = den*Cp*Vol*log((Ti-Tinf)/(Texp-Tinf))/(Ac*texp);

% Define relationship between Temperature (T) and time (t) texp = 0:0.5:5400; % time [0-90min every 0.5 second] Texp = Tinf+((Ti-Tinf)*exp((-Ac*hbar_exp*t)/(den*Cp*Vol)))-273; % Specimen's

Temp [K]


%Plot Experimental Data plot(frctime,frctemp)

%Theoretical Dry Curve %Define dry parameters Tdry = 293.4; %Final experimental temperature tdry = 2150; %Total dry curve time Tidry = 290; %Start of dry curve

%Solve for experimental hbar hbar_dry = den*Cp*Vol*log((Tidry-Tinf)/(Tdry-Tinf))/(Ac*tdry);

% Define relationship between Temperature (T) and time (t) tdry = 0:0.5:5400; % time [0-90min every 0.5 second] Tdry = Tinf+((Ti-Tinf)*exp((-Ac*hbar_dry*t)/(den*Cp*Vol)))-273; % Specimen's

Temp [K]

plot(tdry,Tdry,'--g')

%Forced Sphere % Define constants


% Sample Diameter D = 0.0381; % meter


%Theoretical hbar hbar = 34.6; % Define relationship between Temperature (T) and time (t) t = 0:0.5:3500; % time [0-90min every 0.5 second] T = Tinf+((Ti-Tinf)*exp((-6*hbar*t)/(den*Cp*D)))-273; % Specimen's Temp [K]


%Experimental hbar

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%Define experimental parameters Texp = 294.5; %Final experimental temperature texp = 2300; %Total experimental time

%Solve for experimental hbar hbar_exp = den*Cp*D*log((Ti-Tinf)/(Texp-Tinf))/(6*texp);

% Define relationship between Temperature (T) and time (t) texp = 0:0.5:3500; % time [0-90min every 0.5 second] Texp = Tinf+((Ti-Tinf)*exp((-6*hbar_exp*t)/(den*Cp*D)))-273; % Specimen's

Temp [K]


%Plot Experimental Data plot(fostime,fostemp)

%Forced Cube % Define constants


% Sample Dimensions Vol = 0.0254*0.02536*0.02536; % m^3 (Volume of cube) At = 0.00064313; % m^2 (surface area of top & bottom surfaces of cube) As = 0.00064414; % m^2 (surface area of side of cube) Ac = (4*As)+At+At; % m^2 (surface area of entire cube)


%Theoretical hbar hbar = 29.09; % Define relationship between Temperature (T) and time (t) t = 0:0.5:3500; % time [0-90min every 0.5 second] T = Tinf+((Ti-Tinf)*exp((-Ac*hbar*t)/(den*Cp*Vol)))-273; % Specimen's Temp

[K]



%Solve for experimental hbar hbar_exp = den*Cp*Vol*log((Ti-Tinf)/(Texp-Tinf))/(Ac*texp);

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% Define relationship between Temperature (T) and time (t) texp = 0:0.5:3500; % time [0-90min every 0.5 second] Texp = Tinf+((Ti-Tinf)*exp((-Ac*hbar_exp*t)/(den*Cp*Vol)))-273; % Specimen's

Temp [K]


%Plot Experimental Data plot(foctime,foctemp)

Appendix C

Theoretical analysis using a transient 1-dimensional model.

%% Sphere - Free Convection with transient heat transfer

clear

clc

%% Metal Properties at 300K

% Stainless Steel (AISI 304) (Table A.1 Incropera)

den_ss = 7900; % kg/m^3

alpha_ss = 0.00000394; % J/kgK

k_ss = 14.9; % W/mK

r = 0.0381/2; % radius (meter)

%% Measured Temperatures

Tinf = 296; % K (room temperature)

Ti = 273; % K (initial surface temperature)

Tf = 284.5; % K (film temperature = (Ti+Tinf)/2))

Tfinal = 291.5;

% Tfinal = Tinf-0.02*Tinf % K (Threshold for steady state temp.)

%% Coefficients for transient 1-Dimensional conduction (h = 9.4, Bi = 0.012)

Bi1 = 0.010;

h1 = Bi1*k_ss/r

z1 = 0.1730; % rad

C1 = 1.0030;

Bi2 = 0.011;

h2 = Bi2*k_ss/r

z2 = 0.1802; % rad

C2 = 1.0033;

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Bi3 = 0.012;

h3 = Bi3*k_ss/r

z3 = 0.1880; % rad

C3 = 1.0036;

Bi4 = 0.013;

h4 = Bi4*k_ss/r

z4 = 0.1945; % rad

C4 = 1.0039;

Bi5 = 0.014;

h5 = Bi5*k_ss/r

z5 = 0.2016; % rad

C5 = 1.0042;

%% Analysis

Fo1 = -(1/(z1^2))*log((1/C1)*((Tfinal-Tinf)/(Ti-Tinf)));

t1 = (r^2)*(Fo1/alpha_ss)/60 % min









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References

1 Incropera, Frank P., and David P. DeWitt. Fundamentals of Heat and Mass Transfer. New York: Wiley, 1990.

Print. 2

Wagner, J., Ph.D., P.E., and Freeman, T., ME 444 – Mechanical Engineering Lab III Student Manual, Fall 2012,

Clemson University, Clemson, South Carolina, Chap. 2.

excerpt from, analyzing convective heat transfer

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