thesis equations
TRANSCRIPT
-
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.
-
7/31/2019 Thesis Equations
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,
-
7/31/2019 Thesis Equations
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
-
7/31/2019 Thesis Equations
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.
-
7/31/2019 Thesis Equations
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
)
-
7/31/2019 Thesis Equations
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
)
)
)
)
-
7/31/2019 Thesis Equations
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:
-
7/31/2019 Thesis Equations
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 (
-
7/31/2019 Thesis Equations
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.
-
7/31/2019 Thesis Equations
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:
()
-
7/31/2019 Thesis Equations
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.
-
7/31/2019 Thesis Equations
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.
-
7/31/2019 Thesis Equations
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: