HANDLY & FREQUENTLY USED FORMULAS FOR THERMAL ENGINEERS

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1 HANDLY & FREQUENTLY USED FORMULAS FOR THERMAL ENGINEERS GEOMETRY & MATH GEOMETRI & MATEMATIK Cylindrical (Tube) Volume V = p / 4 d 2 L [m 3 ] Cylindrical (Tube) Surface A = p d L [m 2 ] Rectangular Triangle A = 90 : Geometrical Vector Sum a 2 = b 2 + c 2 a = (b 2 + c 2 ) [m] Diameter d = (4 A / p) [m] cos B = c / a ; sin B = b / a ; tan B = b / c

2 STRENGTH & STATICS STYRKELÆRE & STATIK Force F = m a [N] Stress s = F / A [N/m 2 ] Stress s = e E [N/m 2 ] Hook s Law Strain e = DL / L [-] A = Cross section area [m 2 ] E = Elasticity Modulus [N/m 2 ] m = Mass [kg] a = Acceleration / Gravity Acceleration [m/s 2 ] L = Deformation in Length [m] ; L= Length [m] Bending Stress in beams s = M T / W [N/m 2 ] M T = Torque [Nm] W = Section Modulus [m 3 ] profile depending Simple Supported Beam Uniform spread load Max. torque M T, MAX = P L / 8 [N m] at middle of the beam Max. Reflection U = 5 M L 2 / (48 E I) [m] at middle Cantilever Beam Uniform spread load Max. torque M T, MAX = P L / 2 [N m] at the fixed support in the wall Max. Reflection U = M T, MAX L 2 / (4 E I) [m] at free end of the beam P = Total uniform load of beam [N] I = Moment of Inertia [m 4 ]

3 HEAT & TEMPERATURES VARME & TEMPERATUR Absolute Temperature (Kelvin) T = t + 273,15 [K] t = Temperature [ C] Heat / Heat Content Q = m C p (t 2 t 1 ) [W] [J] m = Mass Flow [kg/s] / Mass [kg] C p = Specific Heat [J/(kgK] t 1 and t 2 = Temperatures Inlet and Outlet [K] [ C] Linearly Heat Expansion of Materials DL = L a L Dt [m] Volumetric Heat Expansion of Materials DV = V b V Dt [m 3 ] L = Length [m] ; V = Volume [m 3 ] ; α L = Length Expansion Coefficient [1/ C] β V = Volume Expansion Coefficient [1/ C] t = Temperature Change [ C] For Ideal Gasses : p v = R T = p 0 v 0 (1+ t / 273,15) Specific Volume v = 1 / r [m 3 /kg] p = Pressure (bar abs.) ; ρ = Density [kg/ m 3 ] T = Absolute Temperature [K] p 0 v 0 : Pressure and Specific volume at 0 C R = Gas Coefficient [J/(kg K)] : Air = 287,1 J/(kg K) Steam = 461,5 J/(kg K) 1 kmol equals a volume of 22,4138 m³ m = n M [kg] V n = n V mol [m n 3 ] at 0 C and 1,01325 bar r = m / V n [kg/m n 3 ] M = Mol mass [kg/mol] ; ρ = Density [kg/m n 3 ] V n = Normal Volume [m n 3 ] ; n = Number of mol V mol = Molar Volume [m n 3 /mol] ; m = mass [kg]

4 HEAT TRANSFER VARMEOVERFØRING BY CONVECTION VED KONVEKTION Heat Transfer by Convection Q = k F DJ [W] F = Heat Surface Total wall area [m 2 ] Heat Transmission Coefficient k = 1 / (1/a 1 + 1/a 2 + e/l + f 1 + f 2 ) [W/(m 2 K)] α 1 = Heat Transfer Coefficient Fluid 1 [W/(m 2 K)] α 2 = Heat Transfer Coefficient Fluid 2 [W/(m 2 K)] λ = Heat Conductivity Wall Material [W/( mk)] e = Wall Thickness [m] f 1 = Fouling Coefficient for the wall of fluid 1 [m 2 K/W)] f 2 = Fouling Coefficient for the wall of fluid 2 [m 2 K/W)] BY RADIATION VED STRÅLING Radiation Heat between two surfaces 1 and 2 F = C 12 F 1 ((T 1 /100) 4 (T 2 /100) 4 ) [W] Radiation Coefficient C 12 = 1 / (1/C 1 + 1/C 2-1/C S ) [W/(m 2 K)] C = e C S [W/(m 2 K)] ε = Emission ratio [-] C S = Radiation Coefficient for the absolute black surface [-] T = Absolute temperature [K] Individual Thermal Solutions Logarithmic Middle Temperature Difference DJ = (Dt 1 -Dt 2 ) / ln (Dt 1 /Dt 2 ) ; all values in [K] [ C] t 1 = Difference in Temperatures of Fluid1 and Fluid 2 at 1 t 2 = Difference in Temperatures of Fluid1 and Fluid 2 at 2 1 and 2 being the physical positions of the inlets and outlets of heat exchanger in current or counter flow types Nusselt s Number Nu = a L F / l [-] a = Nu l / L F α = Heat Transfer Coefficient [W/(m 2 K)] L F = Flow Length [m] e.g. diameter or plate length λ = Heat Conductivity Fluid [W/(m K)] General expression for forced circulation Nu = K 1 Re K2 Pr K3 General expression for natural circulation Nu = K 5 Gr K4 Pr K3 Prandtl s Number Pr = r C p n / l [-] Grashoff s Number Gr = g r DV Dt L F 3 / n [-] g= Gravity acceleration [m/s²] K 1, K2, K3, K4 and K5 are different constants and equations based on tests and depending on the type of heat transfer.

5 MECHANICS OF FLUIDS STRØMNING & VÆSKEFYSIK Total pressure p T = p S + p D [N/m 2 ] Dynamic pressure p D = ½ c 2 r [N/m 2 ] Pressure Height p H = g r H [N/m 2 ] p S = Static pressure [N/m 2 ] g = Gravity acceleration [m/s 2 ] H = Height / Altitude [m] Bernoulli s Law about constancy in pressure. All in [N/m 2 ] p S,1 + p D,1 + p H,1 = p S,2 + p D,2 + p H,2 p S,1 + ½ c 12 r + g r H 1 = p S,2 + ½ c 22 r + g r H 2 For Ideal Gasses: Dynamic Viscosity h 0 (273 + t) / 273 [N s/m 2 ] t = Temperature [ C] Dynamic Viscosity h = n r [Pa s] [kg/(m s)] Reynold s Number Re = c L F / n [-] ν = Kinematic Viscosity [m 2 /s] ; ρ = Density [kg/m 3 ] c = Velocity [m/s]; L F = Flow Length [m] Pressure Drop in tube Dp TB = l p D L T / d = l ½ r c 2 L T / d [N/m 2 ] λ = Friction Coefficient [-]; L T = Tube Length [m] d = Internal Tube Diameter [m]; ρ = Density [kg/m 3 ] c = Velocity [m/s] Pump Capacity P = h T q V S Dp [W] Total Efficiency h T = (h PUMP h MOTOR ) Efficiency h = P PERFORMED / P ABSORB q V = Volume flow [m 3 /s] Σ p = Sum of all pressure drops in the circuit [Pa]

6 ELECTRICITY ELECTRICITET Power / Capacity of a 1-Phase System : P = U PH I PH [W] Power / Capacity of a 3-Phase System : P = 3 U N I N cos j [W] Power, Voltage and Current in Conventional Resistances Ohm s Law U = R I [V] U N = Net Voltage [V] ; I N = Net Current [A] U PH = Phase Voltage [V] ; I PH = Phase Current [A] cos ϕ = Phase Angel [-] cos ϕ = 1 for Heating elements and other simple resist. cos ϕ < 1 for Electrical Motors (inductive resistance). Power expressed by the resistance P = U I = U 2 / R = I 2 R U = Voltage [V] ; I = Current [A] R = Resistance [Ω ] [Ohm] [W]

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