thesis equations

7/31/2019 Thesis Equations

1/13

Fouriers Law

(

.1)

Where the constant of proportionality is the thermal conductivity of the material

Newtons Law of Cooling

(

.2)

Where:

is the convective heat transfer coefficient in

is the surface area through which transfer of heat by convection takes place;

is the surface temperature; andis the temperature of the fluid sufficiently far from the surface.

Radiant Heat Transfer

(

.3)

Where:

is the emissivity of solid surface;

is the Stephen Boltzmann constant

is the surface area completely enclosed by its surroundings;

and are the absolute temperatures of the solid surface and the surroundings

respectively.


2/13

The radiation heat exchange expression in the form of Newtons law of cooling

(

.4)

Where is the radiative heat transfer coefficient in

The evaporative heat transfer

Where is the evaporative heat transfer coefficient and relates to the convective heat transfer

coefficient by the Lewis number (ISO 9920 2009) i.e. is the surface

area through which evaporation takes place, is the saturated vapour pressure at the surface of

the body, and is the partial vapour pressure in the environment.

Heat balance equation

the heat balance equation at the skin surface of a human body

The metabolic rate of the body (M) provides the energy to enable the body to do mechanical

work (W) and the remainder (M-W) is released as heat to the environment through the skin

surface (Qsk) and as a result of the respiratory process(Qres), with any extra or deficit stored (S),

causing the body's temperature to increase or decrease.Where:

M is the rate of metabolic energy production,

W is the rate of mechanical work,

Qsk

is the total rate of heat loss from the skin,Qres is the total rate of heat loss from respiration,


3/13

S is the heat stored in the body in surplus or deficit,

C is the rate of convective heat loss from the skin,

R is the rate of radiative heat loss from the skin,

Esk is the rate of total evaporative heat loss from the skin,

Cres is the rate of convective heat loss from respiration,

Eres is the rate of evaporative heat loss from respiration,

Ssk is the rate of heat storage in skin compartment and

Scr is the rate of heat storage in core compartment


4/13

Operative temperature

Where, the Operative temperature can be defined as the average of the mean radiant and

ambient temperatures, weighted by their respective heat transfer coefficients.

)

Overall Heat Transfer Coefficeint

Dry Heat Transfer in (Skin-Clothing-Environment)

)

Evaporative Heat Transfer in (Skin-Clothing-Environment)

Evaporative heat loss from the skin depends upon the amount of moisture on the skin and

the difference between the water vapour pressure at the skin and in the ambient environment:

)

Where:

= skin wetness (dimensionless);

= saturated vapor pressure at skin (at ;

= vapor pressure in ambient air;

= evaporative heat transfer resistance of clothing;

= evaporative heat transfer resistance between the clothing and the

environment.


5/13

Resistance of Metal Wire

)

Metal Resistivity

where = electron charge ( coulombs);

= electron density (number per unit volume);

= mobility, relates to how electron moves through conductor by interacting with other

electron and molecular structure of conductor.

As the electron charge is always the same, so the resistivity of any metal depends upon the

product ofand . For most metals over a large range of temperatures, the product of and

decreases with increasing temperature, thus an increase in resistance establishes a positive

temperature coefficient.

Metal Resistivity and Temperature

Metal Resistance and Temperature

Temperature Coefficient of Resistivity

)


6/13

RTD Resistance and Temperature (-200 to 800 C)

RTD Resistance and Temperature (-200 to 800 C)

RTD Resistance and Temperature Equation Constants

The constants A and Bare defined like this (Honeywell):

are constants and are defined as (Honeywell):

)

Modified RT Relationship of TSF

)

)

)

)


7/13

Equation of the fitted line

Here M and B are the slope and the intercept of the fitted line..

Standard error in Resistance

is a measure of the amount of error in the prediction of resistance for an individual temperature

value. It can be calculated as:

Where, stands for the sum of the square of the residuals with respect to the fitted line.

Standard errors in the slope and Intercept:

Where, and represent individual temperature points and the means of all the

temperature points, respectively. The number of data points used in the regression process is

denoted by . ) and are in fact the standard deviation of slope and intercept

respectively.

Temperature Coefficient of Resistivity (),

Usually RTD sensing elements are specified with an alpha value between 0C and100C:


8/13

The Alpha value may also be calculated directly from the TR equation as:

Considering the testing range, the reference temperature, was preferred over for analysis

and comparison of samples. The value of for TSF samples made of the same kind of sensing

element will always be lower than their corresponding values; can be calculated by

following expressions:

r2-value

is known as the coefficient of determination, and is defined as:

Where, SSE stands for the sum of the square of the residuals with respect to the fitted line while

SST means the the sum of the square of the residuals with respect to the average resistance

value.

95% Slope and Intercept Confidence Deviation

were calculated by multiplying the t-value by their respective standard errors:

95% Resistance Confidence Deviation

is similar to the confidence deviation of the slope and the intercept and can also be calculated by

the product of the t-value and the standard error in resistance (


9/13

95% Temperature Confidence Deviation )

Calibration Equation

Manufacturing Uncertainty

The length of the sensing element in each sample was calculated using:

And compared with the target length of sensing element defined as:

Where and denotes the width and length of the sensing area of the TSF while stands

for the number of inlays.


10/13

1D Steady State Mathematical Modelling

)

The thermal resistance is the resistance of a material to the conduction or convection of thermal

energy and is definedas:

(For conduction) (

(For convection) )

Expression ) can also be expressed in terms of thermal resistance as:

(

Expression ( can be rearranged as:

(

In equation (, the values of parameters; were known. The rest of

the parameters may be derived as below:

(

(

(

(

(

(

Temperature of a Sensing Element

Applying Fouriers law across the TSF, would yield:

()


11/13

(

The minus sign indicates the decrease in temperature as heat flows towards the positive x-

direction. Now, suppose that the temperature of the underside of the TSF is indicated bywhen . Similarly represents the temperature of the upper side of the TSF when .

After integrating both sides of equation ( with the limits of

and the result is:

)

Assuming that is the temperature at a certain distance from the underside of the TSF. Afterintegrating equation ( again between the limits of and the result is:

(

Now comparing equations and ruling out the factor of , the temperature of the TSF at a

certain thickness ) can be expressed as:

(

Biot number

The Biot numberis a dimensionless parameter, usually used to classify the component as lumped

or not, and can be defined as:

()

Where and represents the convective heat transfer coefficient at the surface of the component

and its thermal conductivity. While is the characteristic length of the component, defined as the

ratio of volume and surface area of the component.


12/13

Modelled TR relationship of a Sensing Element

TR relationship of a sensing element in terms of dimensions of the TSF and wire can be modelled

as:

()

The term actually describes the nominal resistance at 20 C . Equation ()

can be expressed in a simplified form as:

)

The power required to raise the temperature of the sensing element depends upon the net

heat transfer through it and the self heating , and may be expressed as:

()

Where and are the mass and specific heat capacity of the sensing element and their product

is known as Thermal Capacitance. is the rate of change of the sensing element

temperature with respect to time. is the excitation current passing through the sensing element.

The net heat transfer through the sensing element can be expressed as:

()

Where

- Heat entered by conduction from layer one of TSF

- Heat escaped by conduction to layer two of TSF

- Heat escaped from edges to environment by convection

After rearranging, equation can be expressed as:

(

Equations are cross domain equations. In order to determine the and with respect to time,

the model would solve both equations simultaneously throughout the duration of the experiment,

as the output of one equation depends upon the input of the other equation.


13/13

Strain Testing

The extension of the TSF sample was calculated by considering the initial and final

lengths of the TSF (distance between the clamps):

)

Calibration equation

The calibration equation of a TSF sample can be created by first generating the regression

equation of Resistance(R) and Temperature (T) data acquired during rig testing (as explained in

chapter 5) in the following form:

and after rearranging the constants, converting the regression equation into a calibration equation

as:

)

where M and B are the slope and the intercept of the regression equation, whilst ) and

are the constants of the calibration equation. Each TSF sample would have a different

calibration constant, which should be calculated before using it in the application scenario.

Thermal Time Constant

TTC is directly related to the thermal mass (product of mass and specific heat) of a material and

inversely related to the surface area:

thesis equations

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