HEAT TRANSFER SPREADSHEET CALCULATIONS CONDUCTION

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Copy write, Heat Transfer Spreadsheet Calculations by John R Andrew, 12 June 2011

Heat travels three ways: * Conduction - by spreading through solids. * Convection is heat transfer by the movement of heated gasses and liquids. * Radiation - is heat in the form of radiation and travels through space at the speed of light.Heat always travels from an area of higher temperature to an area of lower temperature. Heat transfer (Q) is the flow rate of heat and is measured in Watts or Btu's per hour.

Spread Sheet Method: new Excel version1. Type in values for the Input Data. 2. Excel will make the Calculations.

Excel's GOAL SEEKExcel's, "Goal Seek" adjusts one Input value to cause a Calculated formula cell to equal a given value. When using Excel's Goal Seek, unprotect the spread sheet by selecting: Drop down menu: Home > Format > Unprotect Sheet > OK When Excel's Goal Seek is not needed, restore protection with: Drop down menu: Home > Format > Protect Sheet > OK

When using Excel's Goal Seek, unprotect the spread sheet by selecting: Drop down menu: Tools > Protection > Unprotect Sheet > OK When Excel's Goal Seek is not needed, restore protection with: Drop down menu: Tools > Protection > Protect Sheet > OK

UNITS Symbol Q Q w K h Cp L L UnitsBtu/hr Btu/hr Btu Btu/hr-ft-deg F Btu/hr-ft^2-F lbm/ft^2 Btu/lbm-deg F lbm / sec-ft lbm / sec-ft ft in

Multiply by 0.2931 3.93E-04 778.2 1.730 5.5956 16.0185 4187 3600 1.488 0.3048 25.4

Units ObtainedW hp ft-lbf W / m-C W / m^2-C kg / m^3 J/kg-C lbm / hr-ft kg / sec-m m mm

L in F lbf w ft-lbf ft^2 / sec Heat transfer is measured in feet and meter units. Use the above units table from left to right:Input Data

0.0254 4.4482 1.3558 0.0929

m N N-m m^2 / s

Units = Multiply by = Units Obtained =

42 0.0254 Calculations 1.067

Use the above units table from right to left:Input Data

Units Obtained = Divide by = Units = Temperature is the intensity of heat: T= T= T=

22 5.5956 Calculations 3.932Input Data Input Data

52 deg C + 273.2 325.2Input Data

deg C deg K

T= 50 T = deg F + 460 T= 510Input Data

deg F deg R

T= 62 T = 5*(deg F - 32)/9 T= 16.67

deg F deg C

T= 60 deg C T = (9*deg C/5) + 32 T= 140 deg F

Conduction Example

Heat source on t1 side, Q = High temperature, t1 = Thermal conductivity, K = Dimension in inches, L1 = Dimension in inches, L2 = Thickness in inches, X =

Input Data 12 100 111 4.00 4.00 0.375

Watts deg C W / m-K in in in

Area, Answer: Heat transfer, Low temperature, Answer:

Calculation A = (L1*L2)*0.0254^2 A= 0.0103 m^2 Q = K*A*(t1 - t2) / (X*0.0254) t2 = t1 - (Q*X) / (K*A) t2 = 99.90 deg C

CONDUCTIVITIES & DENSITIES MATERIAL Copper, pure Brass, 70 Cu, 30 Zn Silver, 99.9% pure Duralumin, 3-5%Cu, trace Mg Carbon Steel, 1.0%C Bronze, 75 Cu, 25 Sn Stainless Steel,18 Cr, 8 Ni Concrete, stone, 1-2-4 mix Glass, window Brick, common building Wood, fir Wood, white pine Glass Wool, 1.5 lb/ft^3 Properties at 68 deg F K lbm/ft^3 Btu/hr-ft-F 559 223 532 64 657 235 174 95 487 25 541 15 488 9.4 0.79 0.45 0.40 0.063 0.065 0.022 K W/m*C 386 111 407 164 43 26 16.3 1.37 0.78 0.69 0.109 0.112 0.038

Heat Transmission Through Air Films and SolidsConduction through wall and air films on each side of the wall. Find the heat transfer rate Q and the inner and outer wall surface temperatures. OVERALL HEAT TRANSFER COEFFICIENT U

The heat flow rate, Q, is the same through each layer in the diagram above.

Thermal Resistances of LayersHeat transfer per unit area, Q / A = (t1 - t2) / (Xa / Ka) Heat intensity = Thermal resistance for layer a, Ra = Thermal resistance for layer o, Ro = t1 -t2 Xa / Ka 1/ ho Equation 1

Overall heat transfer rate, Q / A = (t1 -t4) / (Ra + Rb + Rc) Overall temperature difference, T= Overall heat transfer coefficient, U = Heat transfer per unit area, Q / A = t1 - t4 1 / (A * Rn) U*(T)

U.S. Units Wall length, L = Wall height, H = Wall thickness, X = Inside convective coefficient, ho = to = t5 = Thickness, Xa = Xb = Xc = Thermal conductivity, Ka = Thermal conductivity, Kb = Thermal conductivity, Kc = Outside convective coefficient, h5 =

Wall area, A = A= ft^2 Thermal Resistances Ro = 1 / ho Ro = 0.500 Ra = Xa / Ka Ra = 1.2821 Rb = Xb / Kb Rb = 0.8333 Rc = Xc / Kc Rc = 0.2232 R5 = 1 / h5 R5 = 0.172 Overall heat transfer coefficient 1 / U = 1/ho + Xa / Ka + Xb / Kb + Xc / Kc + 1/h5 1 / U = Ro+R1+Ra+Rb+Rc+R5

Input Data 300.00 48.00 7.00 2 70.0 36.0 1.00 4.00 0.75 0.065 0.400 0.280 5.80 Calculation L*H / 144 100.00

in in in Btu/hr-ft^2-F deg F deg F in in in Btu/hr-ft-F Btu/hr-ft-F Btu/hr-ft-F Btu/hr-ft^2-F

1/U= 3.0110 Answer: U = 0.332 T = to - t5 Answer: T = 34 deg F Heat transfer per unit area, Q / A = U*(T) Answer: Q / A = 11.29 Heat transfer, Q = U*A*(T) Answer: Q = 1129 Btu/hr Answer: Q = 331 Watts Surface temperature is found from Equation-1 above: Heat transfer per unit area, Q / A = (to - t1) / (1 / ho) t1 = to - (Q/A)*(1 / ho) Answer: t1 = 64.4 deg F Internal temperature follows: Heat transfer per unit area, Q / A = (t1 - t2) / (Xa / Ka) t2 = t1 - (Q/A)*(Xa / Ka) Answer: t2 = 49.9 deg F

Thermal Resistances of Layers - continuedWall length, L = Wall height, H = Wall thickness, X = Inside temperature, t1 = Outside temperature, t4 = Wall material conductivity, K = Inside convective coefficient, ha = Outside convective coefficient, hc = Wall area, A = A= Ra = Ra = Rb = Rb = Rc = Rc = Overall thermal resistance, R = R = Overall temperature difference, T = T = Input Data 12.00 12.00 0.5 70 20 0.263 2.00 10.00 Calculations L*H / 144 1.00 1 / ha 0.50 X/K 1.90 1 / hc 0.10 Ra + Rb + Rc 2.50 t1 - t4 50.0 in in ft deg F deg F Btu/hr-ft-F Btu/hr-ft^2-F Btu/hr-ft^2-F

ft^2

deg F

Heat transfer per unit area, Q / A = T / R Q/A= 20.00 Btu / hr-ft^2 Heat transfer, Q = A*T / R Q= 20.00 Btu/hr Q= 5.86 Watts Internal temperatures are found from Equation-1 above: Heat transfer per unit area, Q / A = (t1 - t2) / (Ra)

t2 = Answer: t2 =

t1 -(Q/A)*(Ra) 60.00

deg F

Heat transfer per unit area, Q / A = (t3 - t4) / (Rc) t3 = t4 + (Q/A)*(Rc) Answer: t3 = 22.00

deg F

2D HEAT TRANSFER EXCEL'S SOLVER > see MATH TOOLS tab below.The steady state energy balance on the interior nodal point N is: 0 = Q1-N + Q2-N + Q3-N + Q4-N L = Thickness into page. Finite difference equations for each conductive flux: Q1-N = K*L*(Y)*(T1 - TN) / (X) Q2-N = K*L*(X)*(T2 - TN) / (Y) Q3-N = K*L*(Y)*(T3 - TN) / (X) Q4-N = K*L*(X)*(T4 - TN) / (Y) If X equals Y: 0 = T1 +T2 + T3 + T4 -4*TN

Node 1, 0 = 400 + 500 + T2 + T4 - 4*T1 Node 2, 0 = T1 + 500 + 200 + T3 - 4*T2

Node 3, 0 = T4 + T2 + 200 + 300 - 4*T3 Node 4, 0 = 400 + T1 + T3 + 300 - 4*T4

Node 1, -900 = T2 + T4 - 4*T1 Node 2, -700 = T1 + T3 - 4*T2 Node 3, -500 = T4 + T2 - 4*T3 Node 4, -700 = T1 + T3 - 4*T4

Node 1 Node 2 Node 3 Node 4

Equations -900.0 -700.0 -500.0 -700.0

Constants -900 -700 -500 -700

Solution T1 = T2 = T3 = T4 =

Column E 400 350 300 350

deg C deg C deg C deg C

CLICK THE "MATH TOOLS" TAB BELOW FOR INFORMATION ABOUT "SOLVER"

Node 1 Node 2 Node 3 Node 4

Equations 0.0 0.0 0.0 0.0

Constants -900 -700 -500 -700

Solution T1 = T2 = T3 = T4 =

Column E deg C deg C deg C deg C

Node 1, Node 2, Node 3, Node 4, Node 5, Node 6, Node 7, Node 8, Node 9,

0= 0= 0= 0= 0= 0= 0= 0= 0=

400 + 100 + T2 + T4 - 4*T1 T1 + 100 + T3 + T5 - 4*T2 T2 + 100 + 100 + T6 - 4*T3 400 + T1 + T5 + T7 - 4*T4 T4 + T2 + T6 + T8 - 4*T5 T5 + T3 + 100 + T9 - 4*T6 400 + T4 + T8 + 200 - 4*T7 T7 + T5 + T9 + 200 - 4*T8 T8 + T6 + 100 + 200 - 4*T9

Node 1, Node 2, Node 3, Node 4, Node Node 6, Node 7, Node 8,

-500 = -100 = -200 = -400 = 5, 0 = -100 = -600 = -200 =

T2 + T4 - 4*T1 T1 + T3 + T5 - 4*T2 T2 + T6 - 4*T3 T1 + T5 + T7 - 4*T4 T4 + T2 + T6 + T8 - 4*T5 T5 + T3 + T9 - 4*T6 T4 + T8 - 4*T7 T7 + T5 + T9 - 4*T8

Node 9, -300 = T8 + T6 - 4*T9 Equations -500.0 -100.0 -200.0 -400.0 0.0 -100.0 -600.0 -200.0 -300.0 Constants -500 -100 -200 -400 0 -100 -600 -200 -300 Solution T1 = T2 = T3 = T4 = T5 = T6 = T7 = T8 = T9 = Column E 235.7 166.1 128.6 276.8 200.0 148.2 271.4 208.9 164.3

Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9

deg C deg C deg C deg C deg C deg C deg C deg C deg C

BOUNDARY CONDITIONS

0 = (T1 + T2)/ 2 + (h*X / k)*T - (h*X / k + 1)*Tn Input Data Ambient temperature, T = 90.0 Convective heat transfer coefficient, hc = 9.1 Conductivity, k = 2.0 Grid spacing, X = Y = 1.5

0 = T1 + T4 + (T2 + T3)/ 2 + (h*X / k)*T - (h*X / k + 3)*Tn

0 = (T1 + T2)/ 2 + T3 - 2*Tn

0 = T1 / a*(a + 1) + T2 / (b + 1) + T3 / (a + 1) + T4 / b*(b + 1) - (1/a + 1/b)*Tn

Linear Thermal Expansion

Length, L = Material Coefficient, = Temperature Change, t =

Input Data 120 0.000012 100

Units in in/in deg F

or mm mm/mm deg C

Coefficients of Linear Expansion in the range 0 to 100C Aluminum = 0.0000238 Bronze = 0.0000175 Copper = 0.0000165 Mild Steel = 0.000012 Porcelain = 0.000004

Length change, L = Answer: L = This is the end of this spread sheet.

Calculations L * * t 0.144

in or mm

Spread Sheet Method: Excel-97 2003 - old version1. Type in values for the Input Data. 2. Excel will make the Calculations.

Excel's GOAL SEEKExcel's, "Goal Seek" adjusts one Input value to cause a Calculated formula cell to equal a given value. When using Excel's Goal Seek, unprotect the spread sheet by selecting: Drop down menu: Tools > Protection > Unprotect Sheet > OK When Excel's Goal Seek is not needed, restore protection with: Drop down menu: Tools > Protection > Protect Sheet > OK

Metric Units

Wall length, L = Wall height, H = Wall thickness, X = Inside convective coefficient, ho = to = t5 = Xa = Xb = Xc = Ka = Kb = Kc = Outside convective coefficient, h5 =

Input Data 300.00 in 48.00 in 6.00 in 9.38 Btu/hr-ft^2-F 22.2 deg F -1.0 deg F 1.00 in 0.66 in 0.50 in 0.10 Btu/hr-ft-F 0.10 Btu/hr-ft-F 0.52 Btu/hr-ft-F 34.10 Btu/hr-ft^2-F Calculation Wall area, A = L*H*0.0929 / 144 A= 9.29 m^2 Thermal Resistances: Ro = 1 / ho Ro = 0.107 Ra = Xa / Ka Ra = 0.2540 Rb = Xb / Kb Rb = 0.1676 Rc = Xc / Kc Rc = 0.0244 R5 = 1 / h5 R5 = 0.029 Overall heat transfer coefficient 1 / U = 1/ho + Xa / Ka + Xb / Kb + Xc / Kc + 1/h5 1 / U = Ro+R1+Ra+Rb+Rc+R5

1/U= Answer: U = T = Answer: T = Heat transfer per unit area, Q / A = Answer: Q / A = Heat transfer, Q = Answer: Q =

0.5820 1.718 to - t5 23.2 U*(T) 39.86 U*A*(T) 370

deg C

Watts

Surface temperature is found from Equation-1 above: Heat transfer per unit area, Q / A = (to - t1) / (1 / ho) t1 = to - (Q/A)*(1 / ho) Answer: t1 = 18.0 deg F Internal temperature follows: Heat transfer per unit area, Q / A = (t1 - t2) / (Xa / Ka) t2 = t1 - (Q/A)*(Xa / Ka) Answer: t2 = 7.8 deg F

Node 1, 0 = 300 + 400 + T2 + T4 - 4*T1 Node 2, 0 = T1 + 400 + 100 + T3 - 4*T2

Node 3, 0 = T4 + T2 + 100 + 200 - 4*T3 Node 4, 0 = 300 + T1 + T3 + 200 - 4*T4

Node 1, -700 = T2 + T4 - 4*T1 Node 2, -500 = T1 + T3 - 4*T2 Node 3, -300 = T4 + T2 - 4*T3 Node 4, -500 = T1 + T3 - 4*T4

Node 1 Node 2 Node 3 Node 4

Equations 0.0 0.0 0.0 0.0

Constants -700 -500 -300 -500

Solution Column E T1 = T2 = T3 = T4 =

deg C deg C deg C deg C

HEAT TRANSFER SPREADSHEET CALCULATIONS CONVECTIONConvection is heat transfer by the movement of heated gasses and liquids.

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Copy write, Heat Transfer Spreadsheet Calculations by John R Andrew, 12 June 2011

Measuring Air Film Coefficient Heat source, Q = Surface area, A = Inside air temp. thermocouple, t1 = Inside surface temp. thermocouple, t2 =

Input Data 100 0.4 65.3 20.0

Watts sq m C C

Calculations Heat convection, air layer, Q = h * A * (t1 - t2) Watts h = Q / (A*(t1 - t2)) S.I. Answer: h = 5.52 W/m^2*C h = (W/m^2*K)/5.596 U.S. Answer: h = 0.986 Btu/hr-ft^2*F Boundary layer thickness = Air flow velocity = X V mm m/s

Convective Heat Transfer Coefficient Convective heat transfer coefficient, h = k*C*(Gr*Pr)^n / L Description Vertical Plate or Vertical Cylinder Horizontal Plate hot surface facing up Horizontal Plate hot surface facing down Length L Gr C < 10^4 1.36 10^4 Protection > Protect Sheet > OK

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Copy write, Heat Transfer Spreadsheet Calculations by John R Andrew, 12 June 2011

GOAL SEEK - Trial and Error by Excel Spread Sheet T4 = 38.7 deg. C in the calculation below when input heat flow Q = 5.0 Watts. The objective is to find the input heat Q that will result in the temperature T4 = 25 deg. C. Excel spread sheets will make a trial and error iteration automatically with the tool called, "Goal Seek". 1. Select the calculated answer at red cell, T4 = 38.7 below. 2. Select: Tools > Goal Seek > Pick "To value:" > 25 > By changing: > Pick green cell, Q = 16.0 > Okay.

EXAMPLE - LOCKEDQ= A= T1 = h1 = h2 = k= L= R1 = R1 = R2 = R2 = R3 = R3 = Q/A= T4 = Answer: T4 = Input Data 5.0 Watts 0.25 sq m 45 deg. C 9.00 W/sq m C 5.00 W/sq m C 164 W/mC 0.25 m Calculations 1 / h1 sq m C/W 0.111 sq m C/W L/k m*C / W 0.0015 m*C / W 1 / h2 sq m C/W 0.200 sq m C/W (T1 -T4) / (R1 + R2 + R3) T1 - ((Q / A)*(R1 + R2 + R3)) 38.7 deg. C

PROBLEM - UNLOCKEDPractice Goal Seek below: Input Data Q= 16.0 A= 0.25 T1 = 45 Watts sq m deg. C

h1 = h2 = k= L= R1 = R1 = R2 = R2 = R3 = R3 = Q/A= T4 = Answer: T4 =

9.00 W/sq m C 5.00 W/sq m C 164 W/mC 0.25 m Calculations 1 / h1 sq m C/W 0.111 sq m C/W L/k m*C / W 0.0015 m*C / W 1 / h2 sq m C/W 0.200 sq m C/W (T1 -T4) / (R1 + R2 + R3) T1 - ((Q / A)*(R1 + R2 + R3)) 25.0 deg. C

EXCEL'S SOLVER > see 2D CONDUCTIONTo install Solver, click the Microsoft Office Button, click Excel Options, and click Add-Ins. In the Manage box at the bottom of the window, select Excel Add-ins, and click Go. Check the Solver Add-in box in the Add-Ins dialog box, and click OK. After Solver is installed, you can run Solver by clicking Solver in the Analysis group on the Data tab.

Step-1 Clik the "Excel Button" top left > Click "Excel Options".

This is the end of this spread sheet.