Formulario di Fisica Tecnica, voto conseguito 28

Appunti di termodinamica e trasmissione del calore

Q = C̅ ⋅ ΔTC̅ = c̅ ⋅ mδQ = m ⋅ c̅ ⋅ dTdU = dQ - δLdm = q̇ - l̇dm/dt = q̇ - l̇

Ciclo

Q̇ = L̇Q̇im - Q̇ad = L̇P V = R Tv̅ = 1/δdU = m c dT - Pδ V = m c dT - m P dV1 dU = c v dTCp = Cv + R

Politropiche

P V^m = Costc = Cv + R/μ (1- m)

Sistema chiuso e areato

dU + δK + δL g= δQ - δL

Fermo e areato

dU = δQ - δL + ̇ ard ( ue + ke + ge) - ̇ au ( uu + ku + gu)dm = ̇ me - ̇ mudm/dt = ̇ mdot e - ̇ mdot u

«dm/dt =∑f pf wf Aj =∑f ̇ meu J -∑f ̇ mux JdU/dt = q̇ { L̇ [∑J ̇ me J (uJ+keJ+geJ)]!-∑J [ ̇ mu J (u J+ke J+ge J)]} dϵ/dt = ̇ me e V ( eu - P Vmu)dυ/dt = q̇ - L̇ vc + [∑ ( ̇ me J (u J+pe VJ+ke J+1f J)]-ΣJ [ ̇ muJ ( uJ + pJ vJ + kJ + gJ) ] hj = u + pVdu = cv dT dla = pM oT

Caso struet­tord

» dm/dt = O, Allora= {mfin (h0 − he) + ( Km−Ke) + ( gu−ge) ] = Q̇ − l̇ vcδL = PDVL = Lvc+ P(xsuu−tes) Lvc= diersostenδLvc= −VdpvcδL= δLvc+ d(Pv)x = mvapore / msemez = PV/RT

Q = C̅ ΔTC̅ = c̅ mdL = dQ - δLδQ = m c dTdmdm/dt = Q̇ - L̇ciclo: Q = LQ̇int = Q̇ass + Q̇cscP V = R TV = 1/δdU = m c dT - P dVdu = c v dTCp = c v + R

Politropiche

P V^m = costc = cv + R/μ-m

Sistema chiuso e aperto

dU + dK + dG = δQ - δL

Fermo e aperto

dUm = δQ - δL + Σṁe (u e + ke + ge) - Σṁu (u u + ku + gu)dm / dt = Σṁe - Σṁudm / dt = Σf ρf Vf Af = Σf ṁe,f - Σf ṁu,f

1° principio per sls teni aperti

dm / dt = ii = L̇u - L̇e + Vcdu / dt = i̇ - Lu/P ru mudu / dt = i i̇ - LvLp = μ + Pvdu = cv dTdU = _edL = -

Caso stazionario

dm / dt = ∫ ṁ (h u-h e) + (Rm-K2) + (gu-ge) = i Q̇ - Lv cδL = PdVLv c lavoro tecnicoX = V mvolume / m totalez = Pv / Rt

Conduzione

dQ = -λ ^dT⁄_dn da_n ∅̇ = δQ ⁄ δt = -λ ^dT ⁄_dn flujo isotropo_1∫ f(l)T= ∫(ρ, C_p, 1) dQ̇^∇T = ∇̇q̇ ⇔ (q̇ ∝ ∇̇T)q̇ = -λ ^dT ⁄_dn Ȧ_{(T f(l))} conv. dif T finesee (A regime δ̂ ėt controf) δq̇ = ρ V = 1 ⁄ α ^dT ⁄_dt qV=0 α= λ ⁄ C_p = A retone. ^dT ⁄_dt >0 ∇̇T≠0 (eqiscawoe)

Condiciona al confianza

Dirichlet: Ta(dt)l= Tb(tb)Neumȧn: Ia ^dT ⁄_dn⁄e = Ib^dT⁄_dn⁄b-l^dT⁄_dn=h(T=Trad)](q̇= l^{T₁ -T₂}⁄_S)

Politropiche

PVⁿ = costd= _PV2-PV1/^1-dTV^n-1 = costc_x = c_V + _Ri/^1-dQ-L = ΔUC = c ⋅ mC = δQ/δTT_eq = _{C1T1 + C2T2}/^{C₁ + C₂}δQ₁ = m c_p (T_o - T_f)δQ₂ = m c_v (T_f - T_i)δQ = -mc (T_B - T_A)θ = m -2H = U + VdV = ^c_pdL = mc v⋅dT dL = dV⋅dLdH = _(δH)/^δt δTδ(δH/p) = (δp/p)₁(H/V)_T = c_p = _(δH) /^δT

Adiabacita reversibile

δq = 0c = yPV = costΔU = -pdV = widespread

Efficienza di carnot

C.LQ₁/T₁ = PV%dn% - Δ%ΔS = Q = -ΔU⋅dSΔS = L = _[dΦQ -Δs = C.MaQ/TQ ⋅ Q₁/Q₂qL₁ ^|L1| = Δs_MM = L_|L||Q| = L_1−M|Q|_MΔs_U = −_{Qc |Qf|} = ΔT•|ΔS|

Entropie

Δs = m R_m ln_{VB + VA}Reviso CâmaraΔs = m c_v ln_{[TB] TA}iso Câmaram = L_|L| ln_VBΔs = T_VB^γtf ISO BARRÊΔS = mcP lnl^c = dell_cágua

Irraggiamento

G G_A G_T G_n ≠ G_no r_o = ^G_no⁄_G σ = ^G_A⁄_G t = ^G_E⁄_G n = ^G_n⁄_Gσ + α + t = 1G_λ = ^G_Δλ⁄_Δλ = lim_Δλ→∞ ^G_Δλ⁄_Δλ dG = G_λ dλp_λ = G_nλ⁄G_λ α_λ = ^G_Aλ⁄_Gλ σ_λ = ^G_Eλ⁄_Gλ n_λ + t_λ + ε_λ = 1

r₀ = ^{∫₀¹∞ G_noλ dλ}⁄_{∫0¹∞ Gλ dλ} n = ^{∫₀¹∞ n_λ G_λ}⁄_{∫0¹∞ Gλ dλ}n_λ = f(N_λ, T) a_λ = f(M, T, A) r₀ = f[N, T, G_λ] n = [N, T, (φ, G_λ)^E_λ⁄_Eλ^m = f(λ, T) = ^{E_λm(λ, T)}⁄_ΦE_λ^m = ν₀ T⁴ σ = 5.67 .10^-8λ = ^B⁄_T B ν 2898 μm KE_λ0 = ^C₁⁄_{λ⁵(e^C2⁄λT -1 )} del corpo nero

Ε_λ = ^E_λ⁄_Eλ^m Ε = ^{E(T, N)}⁄_Em(T) a_λ = ^E_λ⁄_Eλm a_λ = Ε_λ- in generale Ε ≠ a perché le funzioni peso sono^Ε = ^{∫₀¹∞ Ε_λ(λ, T, N)(E_λm(λ, T)) dλ}⁄_{∫0¹∞ Eλ^m(λ, T) dλ}a = ^{∫₀¹∞ a_λ(U, T, N) G_λ}dλ⁄_{∫0¹∞ Gλ dλ}J = E + G_a

q̇_i = ^v_o/_{¹/ε1 + ¹/ε2} (T₁⁴ - T₂⁴)x gruppo selez tra le kosteq̇_i dimessaq̇_i = ^v_o/₁ (T₁⁴ - T₂⁴)q̇₁ = v_o ( ¹/_ε1 + ¹/_ε2J - 1) + ∑_i=1^M ( ¹/_εiJ + ¹/_ε2J - 1)q̇ = q̇_inn + q̇_oud (x metto are a di esempio)q̇_i = q̇_inn + ^λ/_S (T₁ - T₂)dQ_i = E_i dA_i - a_i G_i dA_iq̇_i = E_i A_i - a_i G_i A_ij°_i - J_o cosφ

Legge di Lambert

del cosenoΩ = ^S/_R²F₂₁ A₂ = F₁₂ A₁F₂₁ = ^{A₁ cosφ₁ cosφ₂}/_{π r12²}

Sistema risolutivo

J_c = E_i + (1 - a_i) ∑_J J_J F_iJq̇_i = E_c - a_i ∑_J J_iJ F_iJq̇ = ^v_o/ ₁ (T₁⁴ - T₂⁴)¹/_ε1 + ^A₁/_A2 ( ¹/_ε2 - 1)

Legge di Bouver

P(x) = P_o e^-αx

Condizione

dQ = 0 → 1 ∫_S dT dA𛉩 = -1 ∫_S dT dmdT = costT(x = 1) = T₁T(x = S) = T₂λ dT/dx |₁ = λ dT/dx |₂

Cilindro cavo

qλ = A λ dT/dr = -λ 2π rl dT/drqλ = (T₁-T₂)/(ln(R₂/R₁)) 2π λα = 1/cρs∆²T = 0 → T = c₁ x + c₂λ dT/dx |_b = -b^b dT/dx |_aT_c=c₁x+c₂x=0     -λ     dT/dx|₀ =    -h₁   (T-T₄)    T₍₀₎x=s     -λ    dT/dx|_s = ± h₂(T-T₂)    T_(s)dT/dlx = T₂-T₁/sq̇ = λ T₁-T₂/sT₁= c_ax + c_2aT₂= c_bx + c_2b

1. -λ_ac_1a= θh₁ (-c_1as_a + tc_2a -T₁)
2. -λ_bc_2b= h₁ (s_bc_1b + c_2b -T₂)
3. 1 T₁(x=0)=c_2a= c_2b= T₂(x=d)
4. -λ_ac_a= λ_bc_1b

q̇= λ ΔT/spessore      q̇= θ(λ) ΔT/ forma1/R_j=s_j/λ_j     1/R_l = 1/q̇Q̇ = ΔT/R = A ΔT/R_i T_i>T₂Q̇ = T₁-T₂/      1/h₂πD₃L + ln(D₃/D₂)/2πL λ_b+ ln(D₂/D₁)/2πL λ_a + 1/h₁πD₁L⁽³⁾r_λ=b2 = λ_b/h₂

Generazione interna di calore piane

∆T + ^qV⁄_λ = 0T = -^qV⁄_2λ x² + c₁x + c₂T₁ = c₂T₂ = -^qV⁄_2λ s² + c₁s + c₂ΏTi⁄ΏTtot = ^{R i cosnumeri}⁄_Rtot

Guscio sferico

q = ^{T₁ - T₂}⁄_{(¹⁄r₁ - ¹⁄r₂) 4πK}

Gener. di calore interna

^d²⁄_dr² + ¹⁄_r ^dT⁄_dr + ^qV⁄_λ = 0T = -^qV⁄_4λ r² + ^qVr²⁄_4λ + c₃¹⁄_r ^d⁄_dr ( r ^dT⁄_dr ) + ^qV⁄_λ = 0

Irraggiamento

G = G_n + G_c + Eq̇ = E = AGλ_max = 2898J = G_R + ER_λ + a_λ + E_λ = 1corpo neroa_λ = 1E_M =σ₀(T₁⁴ - T₂⁴)

1. ¹/_ε1 - 1

n caloriE_M - J_jq̇_ij =J̇ = J₀ + J₁cavità 2 superfici grigieσ₀((T₁⁴ - T₂⁴))q̇₁₂ =doppio tubo con schermo radiattivo(σ(T₁⁴ - T₂⁴))2 superfici con neri semitrasparentiϕ_i = ______J_i = J_R + 1/(A₁F_iR)J_R - J_R = 1/(A₁F_1R)P(x) = P₀ e^-KXe = 1/A₁ = γ₀ 4τ_m³ + 1/Sq_i = σ_R (T₁ - T₂)ϕ̇ = ΔT/R’ΔT/(T₁ + T₂)

1 atm = 101 325 Pa1 bar ≅ 1 atmdU = δQ - δLδQ = mc dTdU/dt = Ė - ḢP V = R TR = R₀/m (R = c_p - c_v)Te = 1/θ(dU) = c_V dTP V^M = costK = c_p/c_vc = c_v + R/(1 - m)[ṁ] = kg/sṁ = ρ A w = Au/Tdm/dτ = ∑_J ṁ_eJ - ∑_J ṁ_uJ

Per sistemi aperti

dU/dτ = Ė - Ŀ + ∑_J ṁ_eJ (μ_J + k_J + g_fJ) - ∑_J ṁ_uJ (μ_J + k_J + g_fJ)Ŀ = ∑_J ṁ_uJ P_J γ_J - ∑_J ṁ_eJ P_J γ_J + Ŀ_vcdU/dτ = Ė - Ŀ_vc + ∑_J ṁ_eJ (h_J + k_J + g_fJ) - ∑_J ṁ_uJ (h_MJ + k_J + g_fJ)

Regime stazionario

δL = p dVδL_vc = -V dpΔK + ΔH + ΔG = Q - Ŀ_vc

Lastra piana

Forzato moto laminare

Pn > 0,6Re < 3∙10⁵Nu ̄ = 0,664 Re^1/2 Pr^1/3T ̄ = T_∞ + 0.958[ Tp - T_∞ ]

Moto turbolento

Pn > 0,6Re > 5∙10⁵M ̄ = 0,0366 Re^0,8 Pn^0,33T ̄ = T_∞ - 0,1 Pn + 40Pn + 32

Convezione naturale moto laminare

0,6 < Pn < 100Pr < Ra < 10⁸Nu_c = 0,59 √ Returbulento0,6 < Pn < 10010⁸ > Ra > 10¹³Nu_w = 0,13 R^1/3N_u(Re,Pn)corrente facile inter-superficeN_u, 0,82√Re ( h/d )^-1/4per h/d > 2 laminareN_u = 0,05 √( h/d )^-1/42 ∙10⁴ < Gr < 2 ∙10⁵2 ∙10⁵ < Gr < 1,3 ∙10⁷δ - laminareδ = 5x ( Re_x )^-1/2Nu = 0,332 Re^1/2 Pn^1/3con 0,6 ≤ Pn ≤ 50per replicare α, L/d > 10Lunghezza, L = 0,05 Re_LD1D ≤ L ≤ 60Ddottus-boelta0,6 ≤ Ra ≤ 60Nu = 0,023 R^0,4 Pr^0,3PnM = 0,4 T_S < T_BM = 0,3 T_S < T_CQ = ∫_A^B ρ V dTT_p > T_∞∂²TQ = ∫ ^{λ ∂2Tt}τ = μ du dy∫∫ du ∂t = ∂P dxdy>dz + μ d²W dy² dxdy>dz_τP = P_o - ρβg2P + 1² s t t g g = costPr = L_√ LRe << G>>

Aletta di raffreddamento

\(\dot{q}_x - \dot{q}_{x+dx} = d\dot{a}\)

\(d\dot{a} = \dot{q}_x dx\)

\(d\dot{a} = \rho dx h (T - T_\infty)\)

\(\lambda \frac{d^2 T}{dx^2} dx = \rho dx h (T - T_\infty)\)

\(\Theta = T_x - T_\infty\)

\(\frac{d\Theta}{dx} = \frac{dT}{dx} - \frac{dT_\infty}{dx}\)

\(\frac{d^2 \Theta}{dx^2} = \frac{d^2 T}{dx^2}\)

\(\frac{d^2 \Theta}{dx^2} = \frac{\rho h}{\lambda A} \Theta \)

\(\Theta(x) = \Theta_b e^{-mx}\)

Efficacia

\(\varepsilon = \frac{\dot{Q}_{delta}}{\dot{Q}_{bare}}\)

\(\dot{Q}_{bare} = hA\Theta_b\)

con \(\Theta_b = T_b - T_\infty\)

\(\dot{Q}_{delta} = \lambda A m \Theta_b\)

\(\left(\frac{m\lambda}{h}\right) \approx \frac{\sqrt{\rho A}}{A} > 0\)

Considero aletta finita

con estremo adiabatico

\(\rho x=L\)

\(e^{-2mL}\)

\(\Theta(L) = 0\)

\(\frac{d\Theta}{dx}|_L = 0\)

1 condizione2

\(c_1 + c_2 = \Theta_b\)

\(\Theta = \Theta_b \frac{e^{-mL}}{1+e^{-2mL}}\)

moltiplica sopra e sotto per \(e^{-mL}\)