A Nichrome Element Settling

solved TemperatureDependenceOfResistance

Q. A heating element using nichrome $α_{h} = α_{c}$ , $R_{0,h} = R_{0,c}$ connected to a 230 V $V_{h} = 230$ V supply draws an initial current $V_{c} = V_{h}$ , $T_{c} = T_{0}$ of 3.2 A $I_{c} = 3.2$ A which settles after a few seconds to a steady value of 2.8 A $I_{h} = 2.8$ A. The room temperature is 27.0 °C $T_{0} = 27.0$ °C, and the temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹ $α_{c} = 1.70 \times 10^{−4}$ °C⁻¹. What is the steady temperature of the heating element, in °C?

Inputs

supply

V_{h}

230 V

initial

I_{c}

3.2 A

steady

I_{h}

2.8 A

room

T_{0}

27 °C

coefficient

α_{c}

0.00017 °C⁻¹

Outputs

temperature

T_{h}

How it's solved

Solved with TemperatureDependenceOfResistance

Given

$V_{h} = 230$ V
$I_{c} = 3.2$ A
$I_{h} = 2.8$ A
$T_{0} = 27.0$ °C
$α_{c} = 1.70 \times 10^{−4}$ °C⁻¹
$α_{h} = α_{c}$ from “A heating element using nichrome”
$R_{0,h} = R_{0,c}$ from “A heating element using nichrome”
$V_{c} = V_{h}$ from “draws an initial current”
$T_{c} = T_{0}$ from “draws an initial current”

Find $T_{h}$ .

Working

$T_{0} = 27.0$ °C $= 300.15$ K
$α_{c} = 1.70 \times 10^{−4}$ °C⁻¹ $= 1.7 \times 10^{−4}$ K⁻¹
ohms law(hot) $V_{h} = I_{h} \cdot R_{h}$
$230 = 2.8 \cdot R_{h}$
$R_{h} = 82.1428571$ Ω
draws an initial current (voltage of cold) $V_{c} = V_{h}$
$V_{c} = 230$
$V_{c} = 230$ V
ohms law(cold) $V_{c} = I_{c} \cdot R_{c}$
$230 = 3.2 \cdot R_{c}$
$R_{c} = 71.875$ Ω
draws an initial current (temperature of cold) $T_{c} = T_{0}$
$T_{c} = 300.15$
$T_{c} = 300.15$ K
resistance grows with temperature(cold) $R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))$
$71.875 = R_{0,c} \cdot (1 + 1.7 \times 10^{−4} \cdot (300.15 - 300.15))$
$R_{0,c} = 71.875$ Ω
A heating element using nichrome (reference resistance of hot) $R_{0,h} = R_{0,c}$
$R_{0,h} = 71.875$
$R_{0,h} = 71.875$ Ω
A heating element using nichrome (temperature coefficient of hot) $α_{h} = α_{c}$
$α_{h} = 1.7 \times 10^{−4}$
$α_{h} = 1.7 \times 10^{−4}$ K⁻¹
resistance grows with temperature(hot) $R_{h} = R_{0,h} \cdot (1 + α_{h} \cdot (T_{h} - T_{0}))$
$82.1428571 = 71.875 \cdot (1 + 1.7 \times 10^{−4} \cdot (T_{h} - 300.15))$
$T_{h} = 867.3$ °C
$T_{h} = 1140.48613$ K $= 867.336134$ °C
$T_{h} \approx 867$ °Cto 3 significant figures

Conclusion

At first the element is at room temperature and passes more current; as it heats up its resistance grows until the current settles, and that growth tells how hot it is.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_nichrome_element_settling do4  kind :numerical5  describes "A heating element using nichrome connected to a %{supply} supply draws an initial current of " \6            "%{initial} which settles after a few seconds to a steady value of %{steady}. The room temperature is " \7            "%{room}, and the temperature coefficient of resistance of nichrome averaged over the temperature range " \8            "involved is %{coefficient}. %{temperature}, in °C?"9  given supply: "230 V"10  given initial: "3.2 A"11  given steady: "2.8 A"12  given room: "27.0 °C"13  given coefficient: "1.70e-4 °C-1"14  asks temperature: "What is the steady temperature of the heating element"15end

solution.rb

1require_relative "../questions/a_nichrome_element_settling"2require_relative "../science/current_electricity/temperature_dependence_of_resistance"34solve :a_nichrome_element_settling, using: TemperatureDependenceOfResistance do5  objects :cold, :hot67  take supply: hot.voltage8  take initial: cold.current9  take steady: hot.current10  take room: T_011  take coefficient: cold.temperature_coefficient12  deduce_from "A heating element using nichrome", hot.temperature_coefficient => cold.temperature_coefficient,13                                                 hot.reference_resistance => cold.reference_resistance1415  deduce_from "draws an initial current", cold.voltage => hot.voltage, cold.temperature => T_01617  temperature = hot.temperature18  answer temperature:, in: "°C", figures: 31920  conclusion "At first the element is at room temperature and passes more current; as it heats up its " \21             "resistance grows until the current settles, and that growth tells how hot it is."22end

playground.rb

1require_relative "../solutions/a_nichrome_element_settling"23play_with :a_nichrome_element_settling do4  title "A Nichrome Element Settling"5end

The laws it drew on

quantity	written	units	within	from
voltage	$V$	V	—	OhmsLaw
current	$I$	A	—	OhmsLaw
resistance	$R$	Ω	0 … ∞	OhmsLaw
length of the conductor	$l$	m	—	OhmsLaw
cross section	$A$	m²	—	OhmsLaw
resistivity	$ρ$	Ω m	—	OhmsLaw
temperature coefficient	$α$	K⁻¹	—	TemperatureDependenceOfResistance
reference resistance	$R_{0}$	Ω	—	TemperatureDependenceOfResistance
reference temperature	$T_{0}$	K	—	TemperatureDependenceOfResistance
temperature	$T$	K	0 … ∞	TemperatureDependenceOfResistance
voltage of ground	$V_{g}$	V	—
current of ground	$I_{g}$	A	—
resistance of ground	$R_{g}$	Ω	0 … ∞
voltage of cold	$V_{c}$	V	—
current of cold	$I_{c}$	A	—
resistance of cold	$R_{c}$	Ω	0 … ∞
voltage of hot	$V_{h}$	V	—
current of hot	$I_{h}$	A	—
resistance of hot	$R_{h}$	Ω	0 … ∞
resistivity of ground	$ρ_{g}$	Ω m	—
length of the conductor of ground	$l_{g}$	m	—
cross section of ground	$A_{g}$	m²	—
resistivity of cold	$ρ_{c}$	Ω m	—
length of the conductor of cold	$l_{c}$	m	—
cross section of cold	$A_{c}$	m²	—
resistivity of hot	$ρ_{h}$	Ω m	—
length of the conductor of hot	$l_{h}$	m	—
cross section of hot	$A_{h}$	m²	—
reference resistance of ground	$R_{0,g}$	Ω	—
temperature coefficient of ground	$α_{g}$	K⁻¹	—
temperature of ground	$T_{g}$	K	0 … ∞
reference resistance of cold	$R_{0,c}$	Ω	—
temperature coefficient of cold	$α_{c}$	K⁻¹	—
temperature of cold	$T_{c}$	K	0 … ∞
reference resistance of hot	$R_{0,h}$	Ω	—
temperature coefficient of hot	$α_{h}$	K⁻¹	—
temperature of hot	$T_{h}$	K	0 … ∞

V_{c} = I_{c} \cdot R_{c}

ohms law

R_{c} = \frac{ρ_{c} \cdot l_{c}}{A_{c}}

resistance from resistivity

R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))

resistance grows with temperature

V_{g} = I_{g} \cdot R_{g}

ohms law(ground)

V_{c} = I_{c} \cdot R_{c}

ohms law(cold)

V_{h} = I_{h} \cdot R_{h}

ohms law(hot)

R_{g} = \frac{ρ_{g} \cdot l_{g}}{A_{g}}

resistance from resistivity(ground)

R_{c} = \frac{ρ_{c} \cdot l_{c}}{A_{c}}

resistance from resistivity(cold)

R_{h} = \frac{ρ_{h} \cdot l_{h}}{A_{h}}

resistance from resistivity(hot)

R_{g} = R_{0,g} \cdot (1 + α_{g} \cdot (T_{g} - T_{0}))

resistance grows with temperature(ground)

R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))

resistance grows with temperature(cold)

R_{h} = R_{0,h} \cdot (1 + α_{h} \cdot (T_{h} - T_{0}))

resistance grows with temperature(hot)

α_{h} = α_{c}

A heating element using nichrome (temperature coefficient of hot)

R_{0,h} = R_{0,c}

A heating element using nichrome (reference resistance of hot)

V_{c} = V_{h}

draws an initial current (voltage of cold)

T_{c} = T_{0}

draws an initial current (temperature of cold)