Gelöste Aufgaben/JUMP/Battery - Versionsgeschichte

Mechaniker: /* Parameters */

2021-03-22T07:57:25Z

Parameters

← Nächstältere Version		Version vom 22. März 2021, 07:57 Uhr
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	::<math>T_{B,max} = 60^\circ C</math>.		::<math>T_{B,max} = 60^\circ C</math>.


	{\| class="wikitable" style="border-collapse: collapse;"		{\| class="wikitable" style="border-collapse: collapse;"

Mechaniker: /* Parameters */

2021-03-22T06:44:19Z

Parameters

← Nächstältere Version		Version vom 22. März 2021, 06:44 Uhr
Zeile 280:		Zeile 280:

	::<math>T_{B,max} = 60^\circ C</math>.		::<math>T_{B,max} = 60^\circ C</math>.


			{\| class="wikitable" style="border-collapse: collapse;"
			!style="border-right: solid 0px;"\|name
			!symbol
			!value
			!unit
			\|-
			\|battery mass
			\|''m<sub>B</sub>''
			\|340
			\|g
			\|-
			\|specific heat capacity (averaged)
			\|''c<sub>pB</sub>''
			\|1015
			\|J/(kg*K)
			\|-
			\|resistor 1
			\|''R<sub>1</sub>''
			\|0.05
			\|Ω
			\|-
			\|resistor 2
			\|''R<sub>2</sub>''
			\|0.025
			\|Ω
			\|-
			\|capacity
			\|''Q<sub>B0</sub>''
			\|5000
			\|Ah
			\|-
			\|battery discharge rating
			\|''1C''
			\|25
			\|A
			\|-
			\|battery reference potential voltage
			\|''U<sub>0</sub>''
			\|8
			\|V
			\|-
			\|ambient temperature
			\|''T<sub>0</sub>''
			\|20
			\|°C
			\|-
			\|time it takes to let the battery cool down by 50%
			\|''T<sub>50</sub>''
			\|180
			\|s
			\|-
			\|correction value for current ''I<sub>B</sub>''
			\|''I<sub>α</sub>''
			\|100
			\|A
			\|- T[beta] 100
			\|correction value for current ''T<sub>B</sub>''
			\|''T<sub>β</sub>''
			\|100
			\|°C
			\|}

	<hr/>		<hr/>

Mechaniker: /* Summary */

2021-03-17T08:07:43Z

Summary

← Nächstältere Version		Version vom 17. März 2021, 08:07 Uhr
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	::<math>\begin{array}{ll}		::<math>\begin{array}{ll}
	\dot{Q}_H &= R_1\cdot I_B^2 + R_{R2} \cdot I_2^2 \text{ where } I_{R2} = \frac{\displaystyle U_2}{\displaystyle R_2}\\		\dot{Q}_H &= R_1\cdot I_B^2 + R_{2} \cdot I_2^2 \text{ where } I_{2} = \frac{\displaystyle U_2}{\displaystyle R_2}\\
	U_B &= R_1 \cdot I_B + U_{2} + E(S_D),\\		U_B &= R_1 \cdot I_B + U_{2} + E(S_D),\\
	\dot{Q}_D &= H_B \cdot (T_B(t) - T_0).		\dot{Q}_D &= H_B \cdot (T_B(t) - T_0).

Mechaniker: /* Model */

2021-03-17T08:03:05Z

Model

← Nächstältere Version		Version vom 17. März 2021, 08:03 Uhr
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	::<math>\begin{array}{lcl}		::<math>\begin{array}{lcl}
	e(S_D) &:= &\frac{\displaystyle E(Q_B)}{\displaystyle U_0}\\		e(S_D) &:= &\frac{\displaystyle E(Q_B)}{\displaystyle U_0}\\
	&=&\displaystyle \sum_0^4 C_i \cdot S_D^i		&=&\displaystyle \sum_0^5 C_i \cdot S_D^i
	\end{array}</math>		\end{array}</math>

Mechaniker: /* Parameters */

2021-03-10T10:11:36Z

Parameters

← Nächstältere Version		Version vom 10. März 2021, 10:11 Uhr
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	The coefficients ''C<sub>i</sub>'' for ''e(S<sub>D</sub>)'' have been computed from conditions for a 5th-order polynomial by defining function-values and gradients at reference points:		The coefficients ''C<sub>i</sub>'' for ''e(S<sub>D</sub>)'' have been computed from conditions for a 5th-order polynomial by defining function-values and gradients at reference points:

	::<math>C_0=\frac{11}{10},C_1=-1,C_2=\frac{1177}{240},C_3=-\frac{28489}{2160},C_4=\frac{7891}{432},C_5=-\frac{4355}{432}</math>[[Datei:JUMP-battery-characteristic-implemented.png\|mini\|Implemented Characteristic]]The plot right shows the resultant characteristic of ''e(S<sub>D</sub>)'':		::<math>C_0=\frac{11}{10},C_1=-1,C_2=\frac{1177}{240},C_3=-\frac{28489}{2160},C_4=\frac{7891}{432},C_5=-\frac{4355}{432}</math>

			[[Datei:JUMP-battery-characteristic-implemented.png\|mini\|Implemented Characteristic]]
			The plot right shows the resultant characteristic of ''e(S<sub>D</sub>)'':

	For the remaining parameters, no measurements were available. We adopted those to obtain a performance similar to what has been described in [1], the resistors were chosen as to “deliver” reasonable amounts of heat when the battery is discharged with rating of C=25.		For the remaining parameters, no measurements were available. We adopted those to obtain a performance similar to what has been described in [1], the resistors were chosen as to “deliver” reasonable amounts of heat when the battery is discharged with rating of C=25.
Zeile 272:		Zeile 275:
	With the values given above, we find		With the values given above, we find

	''h<sub>B</sub>=2.6E-3 J/(K''s)		::''h<sub>B</sub>=2.6E-3 J/(K''s)

	Let the tolerable, maximum battery temperature be		Let the tolerable, maximum battery temperature be

Mechaniker: /* Parameters */

2021-03-10T10:10:43Z

Parameters

Mechaniker: /* Summary */

2021-03-10T10:10:04Z

Summary

← Nächstältere Version		Version vom 10. März 2021, 10:10 Uhr
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	Thus, the battery-coordinates are		Thus, the battery-coordinates are

	<math>\underline{q}_B = \left(		::<math>\underline{q}_B = \left(
	\begin{array}{c}		\begin{array}{c}
	Q_B(t)\\		Q_B(t)\\
Zeile 197:		Zeile 197:
	with the associated equations of motion		with the associated equations of motion

	<math>\begin{array}{ll}		::<math>\begin{array}{ll}
	\dot{Q}_B &= - \alpha(I_B)\cdot \beta(T)\cdot I_B \text{ and } S_D &= \frac{\displaystyle Q_B(t) - Q_{B0}}{\displaystyle Q_{B0}},\\\dot{U}_2 &= \frac{\displaystyle I_C}{\displaystyle C},\\		\dot{Q}_B &= - \alpha(I_B)\cdot \beta(T)\cdot I_B \text{ and } S_D &= \frac{\displaystyle Q_B(t) - Q_{B0}}{\displaystyle Q_{B0}},\\\dot{U}_2 &= \frac{\displaystyle I_C}{\displaystyle C},\\
	\dot{T}_B &= \frac{\displaystyle 1}{\displaystyle m_B \; c_B} \cdot \left( \dot{Q}_H - \dot{Q}_D\right)\\		\dot{T}_B &= \frac{\displaystyle 1}{\displaystyle m_B \; c_B} \cdot \left( \dot{Q}_H - \dot{Q}_D\right)\\
Zeile 204:		Zeile 204:
	and additional algebraic equation		and additional algebraic equation

	<math>\begin{array}{ll}		::<math>\begin{array}{ll}
	\dot{Q}_H &= R_1\cdot I_B^2 + R_{R2} \cdot I_2^2 \text{ where } I_{R2} = \frac{\displaystyle U_2}{\displaystyle R_2}\\		\dot{Q}_H &= R_1\cdot I_B^2 + R_{R2} \cdot I_2^2 \text{ where } I_{R2} = \frac{\displaystyle U_2}{\displaystyle R_2}\\
	U_B &= R_1 \cdot I_B + U_{2} + E(S_D),\\		U_B &= R_1 \cdot I_B + U_{2} + E(S_D),\\

Mechaniker: /* Scope */

2021-03-10T10:08:58Z

Scope

← Nächstältere Version		Version vom 10. März 2021, 10:08 Uhr
Zeile 18:		Zeile 18:
	The C-rating refers to the rate of energy the battery can safely discharge continuously, represented in terms of current as a multiple of its overall capacity. The maximum continuous current for this battery is		The C-rating refers to the rate of energy the battery can safely discharge continuously, represented in terms of current as a multiple of its overall capacity. The maximum continuous current for this battery is

	<math>\begin{array}{ll}		::<math>\begin{array}{ll}
	I_{B,max} & = 5000 \; mAh \cdot 25 \cdot 1/h\\		I_{B,max} & = 5000 \; mAh \cdot 25 \cdot 1/h\\
	& = 125 A		& = 125 A
	\end{array}</math>		\end{array}</math>

	Also we need to model the battery heat to avoid excessive temperatures ''T<sub>B</sub>'' of the battery.		Also we need to model the battery heat to avoid excessive temperatures ''T<sub>B</sub>'' of the battery.

	==Structure==		==Structure==

Mechaniker am 10. März 2021 um 10:08 Uhr

2021-03-10T10:08:41Z

← Nächstältere Version		Version vom 10. März 2021, 10:08 Uhr
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	'''Power Delivery'''		'''Power Delivery'''
	[[Datei:JUMP-Battery.png\|mini\|Battery Equivalent Circuit Diagram\|alternativtext=\|206x206px]]We employ a standard battery model which accounts for		[[Datei:JUMP-Battery.png\|mini\|Battery Equivalent Circuit Diagram\|alternativtext=\|206x206px]]We employ a standard battery model which accounts for
	~~{\| class="wikitable"~~
	\|
	* ''E'': the equilibrium potential		* ''E'': the equilibrium potential
	* ''R<sub>1</sub>, R<sub>2</sub>'': two internal resistances and		* ''R<sub>1</sub>, R<sub>2</sub>'': two internal resistances and
	* ''C'': an effective capacitance.		* ''C'': an effective capacitance.
	\|
	\|}
	In a steady-state-condition, the battery voltage ''U<sub>B</sub>'' is thus a function of the battery current ''I<sub>B</sub>''.		In a steady-state-condition, the battery voltage ''U<sub>B</sub>'' is thus a function of the battery current ''I<sub>B</sub>''.

Zeile 152:		Zeile 150:


			{{MyCodeBlock
			\|title=Temperature
			\|text=
			The thermodynamic balance-equation of heat yields

			::<math>\dot{Q}_T = \dot{Q}_H - \dot{Q}_D</math>.

	~~The thermodynamic balance~~-~~equation of heat yields~~		{{MyTip\|title=Nomenclature "Q"\|text=Note that Q is used here as the heat of the battery – not as the battery-capacity as above!}}

	~~<math>\dot{Q}_T = \dot{Q}_H - \dot{Q}_D</math>.~~
	~~{\| class="wikitable"~~
	~~\|''Note that Q is used here as the heat of the battery – not as the battery-capacity as above!''~~
	\|}
	The battery heat is a function of		The battery heat is a function of

Zeile 167:		Zeile 167:
	so that		so that

	<math>\dot{Q}_T = m_B\cdot c_B\cdot \frac{\displaystyle d T}{\displaystyle dt}</math>.		::<math>\dot{Q}_T = m_B\cdot c_B\cdot \frac{\displaystyle d T}{\displaystyle dt}</math>.

	Heat generation inside the battery be		Heat generation inside the battery be

	<math>\dot{Q}_H = R_1\cdot I_B^2+R_2\cdot I_{R2}^2</math>.		::<math>\dot{Q}_H = R_1\cdot I_B^2+R_2\cdot I_{R2}^2</math>.

	and heat dissipation		and heat dissipation

	<math>\dot{Q}_D = h_B\cdot (T_B-T_0)</math>		::<math>\dot{Q}_D = h_B\cdot (T_B-T_0)</math>

	with ambient air temperature ''T<sub>0</sub>'' and the coefficient of heat transfer ''h<sub>B</sub>''.~~{{MyCodeBlock~~		with ambient air temperature ''T<sub>0</sub>'' and the coefficient of heat transfer ''h<sub>B</sub>''.
	~~\|title=Temperature~~
	~~\|text=Text~~
	\|code=		\|code=
	<syntaxhighlight lang="lisp" line start=1>		<syntaxhighlight lang="lisp" line start=1>
	~~1+1=2~~		/* none given */
	</syntaxhighlight>		</syntaxhighlight>
	}}		}}


	== Summary ==		== Summary ==

Mechaniker am 10. März 2021 um 10:04 Uhr

2021-03-10T10:04:00Z

← Nächstältere Version		Version vom 10. März 2021, 10:04 Uhr
Zeile 48:		Zeile 48:
	In a steady-state-condition, the battery voltage ''U<sub>B</sub>'' is thus a function of the battery current ''I<sub>B</sub>''.		In a steady-state-condition, the battery voltage ''U<sub>B</sub>'' is thus a function of the battery current ''I<sub>B</sub>''.

	'''Temperature'''		'''Temperature'''[[Datei:JUMP-battery-02.png\|mini\|Battery Heat Rates.\|alternativtext=\|123x123px\|links]]

	Our simple lumped-body model accounts for heat generation inside the battery and heat dissipation over its surface.		Our simple lumped-body model accounts for heat generation inside the battery and heat dissipation over its surface.
Zeile 177:		Zeile 177:
	<math>\dot{Q}_D = h_B\cdot (T_B-T_0)</math>		<math>\dot{Q}_D = h_B\cdot (T_B-T_0)</math>

	with ambient air temperature ''T<sub>0</sub>'' and the coefficient of heat transfer ''h<sub>B</sub>''.~~[[Datei:JUMP-battery-02.png\|mini\|Battery Heat Rates.\|alternativtext=\|123x123px]]~~		with ambient air temperature ''T<sub>0</sub>'' and the coefficient of heat transfer ''h<sub>B</sub>''.{{MyCodeBlock

	{{MyCodeBlock
	\|title=Temperature		\|title=Temperature
	\|text=Text		\|text=Text