Home Experiment 1 Report

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ME 331: Introduction to Heat Transfer

04/21/2017

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1. Introduction

Thermo-receptors in human fingers can indicate an object’s temperature relative to the skin temperature.

The experiment described in this report takes advantage of thermo-reception to estimate the

temperature of a long thin metallic cylindrical wire (pin fin) fashioned from a conventional wire clothes

hanger. One end of the wire was kept at a constant temperature of 100oC, and the distance at which the

wire’s temperature was close to the skin’s temperature was estimated using tactile thermo-reception.

Observations were consistent with fin heat conduction as applied to an infinitely long wire undergoing

natural convection, with radiation heat transfer as well. Given the uncertainties with the wire’s

composition and properties, we note that such a rough approximation without any sophisticated

equipment could potentially be used in other practical engineering situations, for a quick assessment

where temperatures are comparable to those of the human body.

Sensing of temperature differences relative to skin temperature is enabled by the presence of hot and

cold receptors under the skin surface. Human thermo-reception relies on at least three types of sensory

receptors: cold receptors, warmth receptors and pain receptors (Hall, 2016). Figure 1 shows the response

of the different types of receptors, as well as the subjective experiences of humans responding to

temperatures in a range from 5oC to 60oC.

Studies indicate there is a neutral zone from 31-36oC in which human subjects experience neither warmth

nor cold. Thermal sensations of warmth depend on the starting skin temperature. If the initial skin

temperature is 31oC, then a half-degree increase will cause a warming sensation, but if the initial skin

temperature is 36oC, a fifth of a degree decrease will result in a subjective sensation of cooling (Schmidt,

1986).

2. Theory

In this report, perceived temperatures are compared with estimates derived from heat transfer theory.

For a well-polished metal surface, the radiation term may be small as the emissivity approaches zero. The

temperature distribution in the wire can be derived from conservation of energy. For a wire of constant

cross-sectional area, and under the assumptions of one-dimensional, steady-state conduction with

radiation incorporated into an overall heat transfer coefficient, the governing equation is (see section

3.6.2 of Bergman & Incropera, 2011):

Figure 1: Discharge frequencies at different skin temperatures from a cold-pain fiber, a cold fiber, a warmth fiber, and a heat-pain fiber. Notice that the subjective experience is labeled as indifferent around 35oC (taken from Hall, 2016).

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𝑑2𝑇

𝑑𝑥2−

ℎ𝑃

𝑘𝐴𝑐(𝑇 − 𝑇∞) = 0 (1)

where ℎ is the overall heat transfer coefficient (convection and radiation), 𝐴𝑐 is the cross sectional area,

𝑃 is the perimeter, 𝑘 is the thermal conductivity and 𝑇∞ is the free-stream air temperature. Setting:

𝜃(𝑥) ≡ 𝑇(𝑥) − 𝑇∞ (2)

with a constant temperature Tb at the base (𝑥 = 0) for an “infinitely long” wire (pin fin), the temperature

distribution along the wire’s length coordinate x is:

𝜃

𝜃𝑏= 𝑒−𝑚𝑥 (3)

where 𝑚 = √ℎ𝑃

𝑘𝐴𝑐 .

The heat flux and heat rates are, respectively:

𝑞′′ = √ℎ𝑘𝑃

𝐴𝑐(𝑇𝑏 − 𝑇∞) (4)

𝑞 = √ℎ𝑘𝑃𝐴𝑐(𝑇𝑏 − 𝑇∞) (5)

An empirical correlation from Chapter 9 is used to estimate the natural convection heat transfer

coefficient. For Rayleigh numbers less than 1012 (see equation 9.34 in section 9.6.3 of Bergman &

Incropera, 2011):

𝑁𝑢̅̅ ̅̅ 𝐷 =ℎ̅𝐷

𝑘𝑎𝑖𝑟=

{

0.60 +0.387𝑅𝑎𝐷

16

[1+(0.559/𝑃𝑟)916]

827

}

2

(6)

where 𝑅𝑎𝐷 =𝑔𝛽(𝑇𝑠−𝑇∞)𝐷

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𝜈𝑎𝑖𝑟𝛼𝑎𝑖𝑟 , (7)

and 𝛼 =𝑘

𝜌𝑐 . (8)

Using this correlation, the natural convection heat transfer coefficient is estimated at h = 15 W/m2K.

3. Procedure

A straight long wire (shown in Figure 2) was fashioned from a wire (clothing hanger) of unknown metallic

composition. The wire was polished with sandpaper to remove rust or residual coating.

Figure 2: Long wire, L = 0.90 m, made from sanded metal wire hanger.

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The experimental setup is shown in Figure 3. The wire was bent at a distance of approximately 11 cm from

one end and inserted into a conventional kitchen pot filled with boiling water on an electric stove-top.

The wire was supported in a horizontal position by fabricated supports from paint-coated wire hangers.

Aluminum foil was attached to the wire near its base in the pot to avoid heating of the wire by vapor and

to shield the wire from the radiating element of the stove.

After the bent end of the wire was submerged, an estimated time of 30 minutes elapsed until a steady-

state temperature distribution in the wire was achieved, with a surrounding ambient temperature of T

= 19.4oC measured by a home thermostat. Using our finger-tips, we sensed the approximate location along

the wire’s length at which the temperature was approximately equal to our skin temperature, where we

had neither a cold nor a warm sensation. Finally, the distance from the bent end of the wire to the “neutral”

location was measured using the measuring tape shown in figure 2.

Figure 3: Diagram of experimental setup (top). Actual experimental setup (bottom).

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4. Results

The geometric properties of the wire as well as other relevant experimental parameters are listed in Table

1. Uncertainty ranges were estimated considering both operator and measurement device limitations,

including limited temperature sensitivity in the fingers.

Table 1: Results for Home Experiment 1.

Wire’s Length, 𝐿 0.90 +/- 0.005m

Wire’s Diameter 1.10 +/- 0.05 mm

Distance from submersed end to bend 0.11 +/- 0.005 m

Room Temperature, 𝑇∞ 19.4 +/- 0.3oC

Distance from base at which wire’s temperature was approximately equal to skin temperature, 𝑥𝑡𝑠

0.16 +/- 0.04 m

5. Analysis

To compare experimental observations with predictions from equations 1 - 8, additional information is

needed regarding the wire’s composition. The metal hanger was likely fabricated from an iron / carbon

steel alloy or aluminum alloy. Using properties for air and equations 6 - 9, an estimate of the natural

convection heat transfer coefficient was obtained. This estimate of the heat transfer coefficient was

somewhat different from the value of h = 22 W/m2K obtained in Home Experiment 2.

The temperature profiles in Figure 4 are for steel or aluminum composition, each material with two

different estimates (from Home Experiments 1 and 2) of the natural convection coefficient.

6. Discussion

Reasonable agreement was obtained between observation and calculated values for Home Experiment 1,

but only under the assumption that the wire was made of aluminum.

The most significant source of uncertainty is lack of knowledge on the wire’s composition. It would have

been straightforward to measure the wire’s mass and then determine whether it was an aluminum or

carbon-steel alloy based on its density. Unfortunately, this simple measurement was not done.

There was also uncertainty in our thermo-reception sensing. As mentioned in the introduction, subjective

sensations of warmth and cold depend on the initial skin temperature, which varies among individuals

and environmental conditions.

Moreover, in the present analysis we neglected radiation. Radiation may play a major part in heat transfer

especially for coated hangers for which emissivity can approach that of a blackbody. An initial estimate of

the average radiative heat flux for a black body could be 𝑞′′̅̅̅̅𝑟𝑎𝑑

= 𝜎(�̅�4 − 𝑇𝑠𝑢𝑟𝑟4 ) =

𝜎 ((373.15𝐾+292.55𝐾

2)4

− (292.55𝐾)4) = 281 𝑊/𝑚2 , where the bar superscript indicates mean

quantities. Whereas, the average convective heat flux could be estimated as 𝑞′′̅̅̅̅𝑐𝑜𝑛𝑣

= ℎ(�̅� − 𝑇∞) =

15𝑊

𝑚2𝐾∗ (332.85𝐾 − 292.55𝐾) = 603 𝑊/𝑚2 . This rough calculation would indicate that for a

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blackbody, radiation is not at all negligible in this case. A more rigorous treatment of radiation can be

found in Appendix 2.

Even with the limitations of this home experiment, first order approximations to heat transfer processes

can be deduced from simple human thermo-reception.

Figure 4: Predicted temperature profiles for an “infinitely long” pin fin made of steel or aluminum, with natural convection heat transfer coefficients obtained from empirical correlation (h1) and from an estimate based on experiment 2 (h2). Predicted

distance at which the pin fin equals skin temperature falls within the uncertainty bounds of observation only for aluminum.

7. Conclusion

Observations based on human thermo-reception were consistent with the temperature distribution

predicted along a very long metal pin fin, presumably made of an aluminum alloy. The experiments and

analysis presented here illustrate how our senses can be used to assess temperature distributions of

objects close to the human body temperature and how such assessments can be used to quickly check a

mathematical approach to a practical engineering situation.

References

Bergman, T., & Incropera, Frank P. (2011). Fundamentals of heat and mass transfer. (7th ed. /

Theodore L. Bergman [and others]. ed.). Hoboken, NJ: Wiley.

Hall, J. (2016). Guyton and Hall textbook of medical physiology (13th ed., Guyton Physiology).

Philadelphia, PA: Elsevier.

Schmidt, R., & Altner, Helmut. (1986). Fundamentals of sensory physiology (3rd, rev. and

expanded ed.). Berlin ; New York: Springer-Verlag.

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APPENDIX 1: MATLAB CODE

close all; clear; clc %% Lab Experiment 1 %% Define properties

R = (0.09*2.54e-2)/2; % radius of the wire in m L = 0.9; % Length in m Tinf = 19.4+273.15; % room temperature in K xtb = 0.16; % distance at which wire was at room temperature Tb = 34+273.15; % body (skin) temperature in K To = 100+273.15; % temp at base in K (boiling temp)

P = 2*pi*R; A = pi*(R^2);

%% to find h % use empirical correlations

g = 9.81; Tf = ((To+Tinf)/2+Tinf)/2; % film temp at point of Tbody % for air at 300K from table A.4 alpha = 22.5e-6; kair = 26.3e-3; Pr = 0.707; nu = 15.89e-6; beta = 3.43e-3; % volumetric thermal expansion coefficient (1/K) Ra = (g*beta*(Tf-Tinf)*(2*R)^3)/(nu*alpha) NuD = ( 0.60 + (0.387*Ra^(1/6))/(1+(0.559/Pr)^(9/16))^(8/27))^2

h1 = NuD*kair/(2*R)

%guessed h from experiment 2 h2 = 21.7484 % W/m^2-K

%% guessed quantities (big uncertainties with big effects) %% assume thermal conductivity steel k= 15; rho = 7700; c = 500; %stainless steel

m1 = sqrt(h1*P/(k*A)); % for eq. 3.84 m2 = sqrt(h2*P/(k*A)); xtb1 = (-1/m1)*log( (Tb-Tinf)/(To-Tinf) ); xtb2 = (-1/m2)*log( (Tb-Tinf)/(To-Tinf) );

disp(['steel, h1, calculated distance is ' num2str(xtb1*100) ' cm']) disp(['steel, h2, calculated distance is ' num2str(xtb2*100) ' cm'])

x = linspace(0,0.35,200); T = Tinf + (To-Tinf)*exp(-m1*x); plot(x,T-273.15,'Linewidth',1.6); hold on,

grid on T = Tinf + (To-Tinf)*exp(-m2*x); plot(x,T-273.15,'Linewidth',1.6);

%% % assume thermal conductivity of aluminum k = 200; rho = 2700; c = 900; % aluminum

%guessed h from experiment 2

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h2 = 13.7269 % W/m^2-K

m1 = sqrt(h1*P/(k*A)); % for eq. 3.84 m2 = sqrt(h2*P/(k*A));

xtb1 = (-1/m1)*log( (Tb-Tinf)/(To-Tinf) ); xtb2 = (-1/m2)*log( (Tb-Tinf)/(To-Tinf) );

disp(['Al, h1, calculated distance is ' num2str(xtb1*100) ' cm']) disp(['Al, h2, calculated distance is ' num2str(xtb2*100) ' cm'])

x = linspace(0,0.35,200); T = Tinf + (To-Tinf)*exp(-m1*x); plot(x,T-273.15,'Linewidth',1.6); T = Tinf + (To-Tinf)*exp(-m2*x); plot(x,T-273.15,'Linewidth',1.6); plot(xtb,34,'ko','Markerfacecolor','k'); legend('steel, h1','steel, h2','Al, h1','Al, h2','observed') set(gca,'Fontsize',14) xlabel('x (m)'); ylabel('Temp (^oC)') xlim([0 0.35]); ylim([10 100])

plot([xtb-0.04 xtb+0.04] ,[34 34],'k'); plot([xtb-0.04 xtb-0.04] ,[31

37],'k') plot([xtb+0.04 xtb+0.04] ,[31 37],'k') plot([xtb xtb] ,[31 37],'k'); plot([xtb-0.01 xtb+0.01] ,[31 31],'k'); plot([xtb-0.01 xtb+0.01] ,[37 37],'k');

plot([0 0.35],[34 34],'--','color',0.1*[1 1 1])

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APPENDIX 2: Including Radiation

Following section 3.6.1 of Incropera we can state the energy balance including radiation as follows:

𝑞𝑥 = 𝑞𝑥+𝑑𝑥 + 𝑑𝑞𝑐𝑜𝑛𝑣 + 𝑑𝑞𝑟𝑎𝑑

𝑑𝑞𝑐𝑜𝑛𝑣 = ℎ𝑑𝐴𝑠(𝑇 − 𝑇∞)

𝑑𝑞𝑟𝑎𝑑 = 𝜖𝜎(𝑇4 − 𝑇𝑠𝑢𝑟𝑟

4 )𝑑𝐴𝑠

−𝑘𝐴𝑐𝑑𝑇

𝑑𝑥= −𝑘𝐴𝑐

𝑑𝑇

𝑑𝑥− 𝑘

𝑑

𝑑𝑥(𝐴𝑐

𝑑𝑇

𝑑𝑥)𝑑𝑥 + ℎ𝑑𝐴𝑠(𝑇 − 𝑇∞) + 𝜖𝜎(𝑇

4 − 𝑇𝑠𝑢𝑟𝑟4 )𝑑𝐴𝑠

𝑑

𝑑𝑥(𝐴𝑐

𝑑𝑇

𝑑𝑥) −

ℎ

𝑘

𝑑𝐴𝑠𝑑𝑥

(𝑇 − 𝑇∞) −𝜖𝜎

𝑘

𝑑𝐴𝑠𝑑𝑥

(𝑇4 − 𝑇𝑠𝑢𝑟𝑟4 ) = 0

For a pin fin of circular cross section 𝐴𝑠 = 𝑃𝑥 and 𝑑𝐴𝑠

𝑑𝑥= 𝑃 and 𝐴𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐴𝑐𝑑2𝑇

𝑑𝑥2−ℎ

𝑘𝑃(𝑇 − 𝑇∞) −

𝜖𝜎

𝑘𝑃(𝑇4 − 𝑇𝑠𝑢𝑟𝑟

4 ) = 0

𝐴𝑐𝑑2𝑇

𝑑𝑥2=ℎ

𝑘

𝑃

𝐴𝑐(𝑇 − 𝑇∞) +

𝜖𝜎

𝑘

𝑃

𝐴𝑐(𝑇4 − 𝑇𝑠𝑢𝑟𝑟

4 )

And we have a mixed boundary value problem

𝑇|𝑥=0 = 𝑇𝑏

And heat conduction at the tip equals convection and radiation at the tip:

𝑑𝑇

𝑑𝑥|𝑥=𝐿

= −ℎ

𝑘(𝑇|𝑥=𝐿 − 𝑇∞) −

𝜖𝜎

𝑘(𝑇4|𝑥=𝐿 − 𝑇𝑠𝑢𝑟𝑟

4 )

This is a hard problem to solve even numerically as it is very nonlinear and stiff. Figure A.1 includes results

assuming properties of aluminum and an emissivity of 0.375. The integration was done using a fourth

order Runge Kutta method, and shooting for the boundary condition at the pin fin tip using the bi-section

method. The Matlab code is appended.

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Figure A.1: Temperature profiles with and without radiation assuming properties of aluminum and an emissivity of 0.375

clear; close all; clc

%% add radiation

R = (0.09*2.54e-2)/2; % radius of the wire in m L = 0.9; % Length in m Tinf = 19.4+273.15; % room temperature in K Tsurr = Tinf; % Temperature of the surroundings xtb = 0.16; % distance at which wire was at room temperature Tb = 34+273.15; % body (skin) temperature in K To = 100+273.15; % temp at base in K (boiling temp)

P = 2*pi*R; A = pi*(R^2);

%% assume thermal conductivity of aluminum k = 200; rho = 2700; c = 900; % aluminum

epsi = 0.375; % emmisivity [0.02,0.91]

g = 9.81; Tf = ((To+Tinf)/2+Tinf)/2; % film temp at point of Tbody

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% for air at 300K from table A.4 alpha = 22.5e-6; kair = 26.3e-3; Pr = 0.707; nu = 15.89e-6; beta = 3.43e-3; % volumetric thermal expansion coefficient (1/K) Ra = (g*beta*(Tf-Tinf)*(2*R)^3)/(nu*alpha) NuD = ( 0.60 + (0.387*Ra^(1/6))/(1+(0.559/Pr)^(9/16))^(8/27))^2

h1 = NuD*kair/(2*R)

m = sqrt(h1*P/(k*A)); % for eq. 3.84

sig = 5.67e-8; %stefan boltzmann constant

xspan = [0,L];

% opts = odeset('RelTol',1e-2,'AbsTol',1e-4);

% checa = -(h/k)*(Trad(end,1)-Tinf)-(epsi*sig/k)*(Trad(end,1)^4-Tsurr^4); a = -1.09*m*(To-Tinf); b = -m*(To-Tinf); tole = 1; checa = 10*tole; % yo = [To;a]; % [x,Trad] = ode45(@(x,y)

pinfinrad( x,y,h1,k,R,Tinf,epsi,Tsurr,sig),xspan,yo); while abs(checa)>tole c = (a+b)/2; yo = [To;a]; [x,Trada] = ode45(@(x,y)

pinfinrad( x,y,h1,k,R,Tinf,epsi,Tsurr,sig),xspan,yo); yo = [To;c]; [x,Tradc] = ode45(@(x,y)

pinfinrad( x,y,h1,k,R,Tinf,epsi,Tsurr,sig),xspan,yo);

if sign(Trada(end,1)-Tinf) == sign(Tradc(end,1)-Tinf) a = c; else b = c; end

% checa = Tradc(end,1)-Tinf; flujo = Tradc(end,2); che = -(h1/k)*(Tradc(end,1)-Tinf)-(epsi*sig/k)*(Tradc(end,1)^4-Tsurr^4); checa = flujo-che; c end % %% Trad = Tradc; % flujo = Trad(end,2); % checa = -(h/k)*(Trad(end,1)-Tinf)-(epsi*sig/k)*(Trad(end,1)^4-Tsurr^4);

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plot(x,Trad(:,1)-273.15,'Linewidth',1.6); hold on; grid on T = Tinf + (To-Tinf)*exp(-m*x); plot(x,T-273.15,'Linewidth',1.6); % plot(x,T-273.15,'*','Linewidth',1.6); legend('rad','no rad') set(gca,'Fontsize',14) xlabel('x (m)'); ylabel('Temp (^oC)') xlim([0 0.6])

% xlim([0 0.3]);

function [ dydx ] = pinfinrad( x,y,h,k,R,Tinf,epsi,Tsu,sig) %[ dydx ] = pinfinrad( x,y,h,k,R,Tinf,epsi,Tsu,sig)

dydx = zeros(2,1);

dydx(1) = y(2); dydx(2) = (2*h/(k*R))*(y(1)-Tinf)+(2*epsi*sig/(k*R))*(y(1).^4-Tsu^4);

end