Verbrennungsmotoren

4 ^ei

* kann Spuren von Katzen enthalten nicht für Humorallergiker geeignet alle Angaben ohne Gewehr ^*

Verbrennungsmotoren Formelsammlung

ηe=

ηi

z }| {

η_V −∆η_BV −∆η_W−∆η_U−

| {z }

ηi,HD

∆η_LW−∆η_R

1. Grundlagen

g_i= m_i P

im_i ni=mi

=Mi M_i P

iψ_iM_iψ_i r_i= V_i P

iV_i ψ_i= n_i P

in_i M=X

i

ψ_iM_i= X i

g_i M_i

!₋₁

R_i=c_p,i−c_v,i & κ_i= c_p,i

c_v,i ⇒c_v,i= R_i κ_i−1

Rg=X i

giRi=







Otto: R_L+g_B(R_B−R_L) Diesel: R_L

mit









 R_i=^RM

Mi g_B=₁₊¹

λ LSt

ρ= 1 v=m

V

2. Kreisprozesse

p_i+1=ε^κp_i T_i+1=ε^κ−1T_i

η_th,GD= 1− 1 κ·q^∗

"

q^∗ ε^κ−1+ 1

!_κ

−1

#

q^∗= q_zu c_p·T₁ w_t,GD=−X

i

q_i,i+1=− R

κ−1[κ(T₃−T₂)−(T₄−T₁)]

η_th,GR= 1−T4−T1

T₃−T₂= 1− 1 ε^κ−1 w_t,GR=−X

i

q_i,i+1=− R

κ−1[(T₃−T₂)−(T₄−T₁)]

η_th,SEI= 1− (T₅−T₁) (T₃−T₂) +κ(T₄−T₃) w_t,SEI=− R

κ−1[(T3−T2) +κ(T4−T3)−(T5−T1)]

T_4,SEI=



 p₃′

p₁ ·T₁ ^1−κ

κ ·T₃′



 κ

3. Kenngr¨ oßen

P_(e,i,r)=p_m(e,i,r)i n V_H P_e=P_i−P_r=M ω ε= V_h+Vc(+V_KM) Vc(+V_KM)

η_(e,i)= P_(e,i) m

.

_BHu

= 1

b_(e,i)Hu ηm= P_e

Pi

= p_me pmi

=η_e ηi

= 1−p_mr pmi

= b_i be

p_m(e,i)=η_(e,i)λ_aH_G

p_mi=W_KA V_h =











Otto: ^{ηi VG HG}

Vh =η_iλ_aH_G=η_i ^{mF Z} ρth Vh Hu ρG

λ LSt+1 Diesel: ^{ηi VL}_Vh^HG¯ =ηiλaH¯_G=ηi mF Z

ρth Vh Hu ρL λ LSt pme=2π

i M

V_H =pmi−pmr

H_G= m_BH_u

V_G = m_BH_uρ_G m_G =











Otto: ^{mB Hu ρG}_mB+mL =₁₊^{Hu ρG}_{λ LSt}

Diesel: ^{mB Hu ρL} mL =^{Hu ρL}

λ LSt

b_(e,i)= m

.

_B P_(e,i)= 1

η_(e,i)H_u B_S=

m

.

_B v_{F zg} =m_B

s =b_eP_e (ρ)v

λ_a= m_G V_Hρ_th =

m

.

_G V

.

_Gρ_th λ_l= m_Z

V_hρ_th mit m_Z=m_ZL+m_ZB

λ= m

.

_L

m

.

_B·L_St = m_L m_B·L_St =

m

.

_L P

i

m

.

_B,i·L_St,i

!

λ_Z= m_ZL

m_ZB·L_St = m_ZL m_B·L_St

Sp¨ulung

̸= λ

C_xH_yO_z+λo_min

O₂+79 21N₂

→xCO₂+y

2H₂O+ (λ−1)o_minO₂ +λ o_min79

21N₂

o_min=

x+y 4+q−z

2

o_min,vol=l_min·r_{O2,f l} r_{O2,f l}= 0,21 o_min,grav=l_min·g_{O2,f l} g_{O2,f l}= 0,23142

Lst= M_O2 g_O

2,f l·M_B

x+y 4+q−z

2

= (2,664c+ 7,937h+ 0,998s−o) g_O

2,f l c=m_C

m_B =M_C

M_Bx h=M_H2 M_By= M_H

2M_By

4. Ladungswechsel 4-Takt

p_mi=p_mi,HD+p_mi,LW =p_mi,HD+|p_E−p_A| p_mi,HD=

m

.

_B·H_u i n V_H ·η_i,HD m

.

_G,AS,Zyl=

m

.

_G,ges iz

360^◦KW

∆α m_G,AS,Zyl=m_G,AS

z =

m

.

_G,ges i n z

α_(K,V)= A_S A_(K,V) A_S=

m

.

_G,AS,Zyl c_S·ρ_S

c_S= v u u u t

2κ κ−1R0T0



1− p_zm

p₀ ^κ−1

κ





ρ_S=ρ₀ p_zm

p₀ ¹

κ

dα dt

≈∆α

∆t = 2π n= 360^◦KW·n

∆t=∆s

v = 2l_rohr

pκR_LT =m_G,AS,Zyl m

.

_S

x_R= m_R m_R+m_Z m_R=x_spuel· p_ZVc

R_GT_Z

5. Ladungswechsel 2-Takt

Λ_S= V_Z

V_Zyl= m_Z

ρ_zmV_ES Λ_a= V_G V_Zyl = V_G

V_ES

6. Gemischbildung

x_AGR= m_AGR m_L+m_B+m_AGR

λ_vergaser= 1 L_St

A_L A_B

α_L α_Bε

s∆p_L

∆p_B s∆ρ_L

∆ρ_L λ_injektor=

m

.

_L L_St·A_B·α_B·t_E·i·n·z·p

2∆p ρ_B

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7. Aufladung

λ_{L,AT L}=λ_L,Saug T_{E,AT L} T_E,Saug

!_a ε

ε−1 a∈[0,2 ; 0,5]

p_{mi,AT L}=p_mi,Saug p_{E,AT L} p_E,Saug

T_{E,AT L} T_E,Saug

!1−a ε ε−1 +p_{E,AT L}−p_{A,AT L}

P_V =

.

m_Vw_V 1

η_mV =

.

m_Vw_sV 1 η_siVη_mV w_sV =c_pVT₁



 p2

p1 κV−1

κV −1





η_siV =T_2s−T₁ T₂−T₁ =

T2s T1 −1 T2 T1−1

= _p

2 p1

^κV

−1 κV −1 T2 T1−1 m

.

_V,red=_m

.

V pT_ein

p_ein

m

.

_V,korr=_m

.

red p_ref pT_ref

P_T=

.

m_Tw_T 1

η_mT =

.

m_Tw_sTη_siTη_mT

w_sT =c_pTT₅



1− p₆

p₅ ^κT−1

κT





m

.

_{T ,red}=

.

m_T

√T₅ p₅

m

.

_{T ,korr}=

.

m_red p_ref

pT_ref

η^∗_ges= m

.

_A m

.

_V

ηAT L z }| { η_siTη_siVη_mTη_mV

| {z } ηmAT L

= 1 +λL_St

λL_St η_siTη_siVη_{mAT L}

= w_sV w_sT =T₁

T₅ c_pV



 _p

p21 ^κV−1

κV −1





c_pT



1−_p 6 p5

^κT

−1 κT





8. Energiebilanz

∆

.

H_A=X

i

n

.

_i[h_i(T_A)−h_i(T₀)] =

.

n_AX

i

ψ_i[h_i(T_A)−h_i(T₀)]

=

.

m_AX

i g_i

M_i[h_i(T_A)−h_i(T₀)]

∆

.

H_A=

.

H_A(T_A)−

.

H_AH(T_Eintritt)

1 = P_e m

.

_BH_u

| {z } ηe

+ ∆

.

H_A m

.

_BH_u+∆

.

H_KW m

.

_BH_u +∆

.

H_LLK m

.

_BH_u +

Q

.

_R m

.

_BH_u

P_e=X i

P_an,i

η_an,i P_chem=

.

m_BH_u=X i

m

.

_B,iH_u,i

9. W¨ arme¨ ubertragung

φ_m=φ_Konv=α_mi[T_αi−T_{mW i}] φ_{m(T R)}=φ_{W Leit}= λ

d_wand h

T_Wli−T_Wrei (·f_{T R})

f_{T R}= DA

Di −1 ln_DA

Di

φ_{m(T R)}= λ

B_{(T R)}(T_αi−Ta) B= λ

α_mi+δ+ λ

α_a mit δ=D_a−D_i 2 B_{T R}= λ

α_mi + δ f_{T R}+ λ

α_a D_i Da

10. Kr¨ afte

F_Ko,osz=mosz(cos(α) +λ_Scos(2α))r ω²

→mosz=m_Ko+m_{P l,osz}

→m_{P l,osz}=l_1,(sp↔kw) l_2,(sp↔ko) m_{P l,rot}

→m_{P l,rot}=l_2,(sp↔ko) l m_{P l} F_{P l}= F_Ko,osz

cos(β)

F_T=F_{P l}sin(α+β)≈F_KO,res·

sin(α) +λ_S 2

·sin(2α)

F_R=F_{P l}cos(α+β)≈F_KO,res·

−λ_S

2 +cos(α) +λ_S 2

·cos(2α)

FKo,Anwechsel=F_G−F_Ko,osz(±F_Reib)= 0^!

11. Massenausgleich

F_(1,2),res=q F²

+(1,2),res+F_−(1,2),res² F(±)(1,2),res=q

F_(±)(1,2),x² +F_(±)(1,2),y² F_+(1,2)=F_−(1,2)=1

2F_01,(02)= 1

2m_oszr ω²(λ_S) M_(1,2),res=q

M_+(1,2),res² +M_−(1,2),res² M(±)(1,2),res=q

M_(±)(1,2),x² +M_(±)(1,2),y²

Fosz=F_Ko,osz=m a=moszr ω²(cos(α) +λ_Scos(2α)) Frot=mrota=mrotr ω²= m_KK,red+m_{P l,rot}

r ω² m_KK,red= r₁

r m_KW

M_1,osz= M_1,min

·2 M_1,rot=

M_1,max −

M_1,min

12. Kinematik

s_exakt(α) =r

"

(1−cos(α)) + 1 λ_S

1−q

1−λ²_Ssin²(α) #

sapprox(α) =r

(1−cos(α)) +λ_S

4 (1−cos(2α))

λS→0

≈ r(1−cos(α)) λ_S=r

l = s 2l

c_exakt(α) =r ω





sin(α) +λ_Ssin)(α) cos(α) q

1−λ²_Ssin²(α)







capprox(α) =r ω

sin(α) +λ_S 4 sin(2α)

λS→0

≈ r ωsin(α) cmax=r ω

q

1 +λ²_S ∢(r, l)→ ⊥ α(c_max) = arctan

l r

= arctan 1 λ_S

!

c_m= 2s n= 4r n

a_approx(α) =r ω²[cos(α) +λ_S cos(2α)]

λS→0

≈ r ω²cos(α)

13. Ventiltrieb

F_N=F_F,red+m_N,red a_N

|{z}

=

..

sN

+p_{A ¨O}π 4d²_Vi

_!

≥0 i= l_1,h↔v l_2,h↔n

⇒ω_{N W} = s

− F_F,red m_N,reds^′′_V

F_F,red=F_Fi= (sv0+s_Ni)i c_F = (sv0+s_V)i c_F m_N,red=

m1,v+m_2,f

2 +m_3,(t,k)

i²+ J_K

l²_2,(h↔n)+m5,st+m6,stoe

s^′= ds dα

⇒

..

_s

N=a=s^′′_Nω²_{N W} =s^′′_N(ω_{N W}S_n)² s_N=1

is_V

.

s_N= 1

i s

.

_V

..

s_N=1 i

..

s_V

s_N=1

is_V s^′_N= 1

is^′_V s^′′_N=1 is^′′_V s_(v,n),2=

s^′′_(v,n),2 s^′′_(v,n),1s_(v,n),1

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14. Brennverlaufsanalyse

dQ_B= κ

κ−1p_α1dV+ 1

κ−1V_α1dp+α_iA_W

T_α1−T_{mW i} dt

T_i,(α

1 )=p_α1V_α1 m_GR_G

A_W,α1=A_Kolben+A_Kopf+A_{W and}(+A_Kompr)

| {z }

=πD(s+(ss)) V_α1=V_c+π

4D²s_α1 v(α) =vc+ s(α)

s(180^◦KW)

·(v₁−v₂)

dt= ∆α

360^◦KW·n dm_B= dQ_B

Hu

15. ¨ Ahnlichkeit

c_m=const. p(α) =const. T_αi=const.

s₂

s₁ =x A₂

A₁ =x² V₂ V₁ =x³ σ_m∝c²_m t_¨u∝D=s

16. Spannungen

S= σ_B σ_mech+σ_therm σ_mech=p_Z·D

2δ σ_therm=E·α_L

1−ν

T_{mW i}−T_{mW a} 2

17. Allgemein

y_g−y₁

x_g−x₁= y₂−y₁

x₂−x₁ x_1,2=−p 2±

s p²

4 −q= −b±

√ b²–4ac 2a

∆X_%=X_neu−X_alt X_alt ·100%

i_antrieb=n_mot

n_rad = M_rad M_mot·η_antrieb

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