Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab

Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab James Robinson December 6, 2007 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 1/42

Outline 1 Introduction Objectives Justification 2 Theory Fourier s Equation 3 Model Application Validation 4 Results Verification Sensitivity Analysis Optimization Data Analysis Normal Distribution using k Normal Distribution using Cp 5 Conclusion Further Objectives James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 2/42

Aim of Study Objectives Model 1-D heat transport using Fourier s Equation via Finite Difference. Assess impacts of discretization. Optimize node numbers Time needed to reach liveable temperature Use Monte Carlo Simulation: determine probability of uncertainty Develop a probability density function(pdf): classify the uncertainty of the thermal conductivity constant. James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 3/42

Reason for study Radiant heating is important for cold climates such as Canada (Good et al., 2005) Benefits include: an even temperature profile, optimal comfort level for occupants, energy savings Thermal conductivity has a narrow range of variation for most pure substances however most substances are far from pure. (McQuiston, et al., 2005) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 4/42

System Figure: Schematic of flooring system James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 5/42

Fourier s Heat Transport Model The Fourier Equation θ t = α 2 θ (1) α = K C p ρ (2) (SCBR, 1983) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 6/42

Fourier s Heat Transport Model Parameters Defined Constants of Fourier s Equation θ Temperature ( C) α Thermal diffusivity k ( m2 C p ρ k concrete Thermal Conductivity ( W C p Specific heat ( kj C kg ρ Density, ( kg ) m 3 ) C s ) C m James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 7/42

Assumptions Assumptions 1 1-D, all conduction in the upward (z) direction 2 Homogeneous limestone concrete 3 Perfectly sealed and insulated room 4 No other flooring overlay 5 Uniform, constant temperature from hydronic heating elements below the slab James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 8/42

Simplified Equation Fourier Equation θ t = θ α 2 (3) z 2 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 9/42

Boundary Conditions/Initial Conditions BCs and IC θ(0, t) = 70 C θ(h, t) = dθ dz z=h = 0 θ(z, 0) = 15 C James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 10/42

Finite Difference Method Steady-State Second Order Difference: with a vector-matrix form of: Error O( z 2 ) 0 = α( θ i+1 2θ i + θ i 1 z 2 ) (4) Aθ + f = 0 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 11/42

Finite Difference Method Transient Forward Difference: θ t+1 θ t t System of equations in Vector-Matrix form: (A I t )θt = f t 1 θ t 1 t (5) Error O( z 2, t) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 12/42

Monte Carlo Simulation Steps of MCS Build a validated model Generate a population of n random variables Choose a spatial point at a given time Run the model n times for each value Analyze n state variables at a given space and time for probability of occurrence James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 13/42

Monte Carlo Simulation From this process, a coefficient of variance (c.o.v.= σ P ) can be generated based on (σ) and P by: σ 2 P = P(1 P) n (1 c.o.v.( P) = P) n P The percent error can also be computed from using: 1 %error = 200 P n P (6) (7) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 14/42

Validation Steady-State Model 70 68 66 64 Temperature ( C) 62 60 58 56 54 52 50 0 50 100 150 200 250 300 350 400 450 500 node Figure: Steady state condition James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 15/42

Validation Analytical Solution Analytical Equation z T (z, t) = (T o T s )erf ( ) + T s (8) 4αt (Carslaw and Jaeger, 1986) where T(z,t) T s T o unknown temp at node n and time t T(1,t)=70 C temp at z ; equals initial condition =15 C James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 16/42

Validation Analytical Solution Error Function where: erf = 2 π z u argument of erf; in this case u = z 4αt z position in the slab foundation α diffusivity constant t time 0 e u2 du (9) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 17/42

Validation Applied Analytical Solution Example z T (z, t) = (T o T s )erf ( ) + T s 4( k ) t C p ρ z T (z, t) = (15 70)erf ( ) + 70 1.14 4( )t 0.84 1920 (10) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 18/42

Verification Analytical Graph 70 60 Temperature ( C) 50 40 30 20 10 0 50 100 150 200 250 300 350 400 450 500 node Figure: Analytical solution to model James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 19/42

Verification Numerical Graph 70 60 Temperature ( C) 50 40 30 20 10 0 50 100 150 200 250 300 350 400 450 500 Node Figure: Numerical solution to the model James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 20/42

Sensitivity Analysis Effect of Discretization 70 70 60 60 Temperature ( C) 50 40 30 Temperature ( C) 50 40 30 20 20 10 0 20 40 60 80 100 120 node 10 0 20 40 60 80 100 120 node Figure: t=0.2 Figure: t=10 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 21/42

Verification Error Analysis Equations %error = S N S A S N 100 LS = (S A S N ) 2 SSD t = Σ n i=1 LS(n) where S A S N LS SSD Analytical solution Numerical solution Least square method Sum of squared difference James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 22/42

Verification Error Analysis 60 50 40 ge % error Averag 30 20 10 0 0 100 200 300 400 500 600 700 800 900 1000 Time (sec) Figure: % error between analytical and numerical solution James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 23/42

Optimization Least Squares Method Figure: Multiple node numbers at the same time step. LS t = Σ n i=1(p(i) Q(2i)) 2 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 24/42

Optimization 70000 60000 50000 ared difference Sum of Squ 40000 30000 20000 10000 0 Number of Nodes James Robinson Fourier s Law Figure: Applied tooptimization the Heat Equation for Conduction of nodeofnumbers Heat Through a Concrete Slab 25/42

Probabilistic Analysis Histogram and Cumulative Distribution Functions 16 120% 14 100% 12 10 80% quency Freq 8 6 60% 40% Frequency Cumulative % 4 2 20% 0 37.2 37.4 37.6 37.8 38 38.2 38.4 38.6 38.88 39 39.2 39.4 39.6 More Temperature ( C) 0% Figure: Histogram of population of temperature values based on normally distributed thermal conductivity James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 26/42

Probabilistic Analysis Plot Thermal Conductivity vs. Temperature James Robinson Figure: Fourier s LawNormally Applied to thedistributed Heat Equation for Conduction Conductivity of Heat Through Population a Concrete Slab 27/42 48 46 44 Temperat ture (Deg C) 42 40 38 36 34 32 30 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Conducitivty Constant (k)

Probabilistic Analysis Eureka! 48 46 44 Temperat ture (Deg C) 42 40 38 36 34 32 30 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Conducitivty Constant (k) James Robinson Figure: Fourier s LawNormally Applied to thedistributed Heat Equation for Conduction Conductivity of Heat Through Population a Concrete Slab 28/42

Probabilistic Analysis Hypothesis Test χ 2 FA1NEW - 2.182 Table A.4-80% confidence 2.182.lt. 5.9886 Possible Type II error - 95% confidence to accept null hypothesis - RVs are normally distributed (Ang and Tang, 2007) James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 29/42

Probabilistic Analysis Thermal Conductivity: Normal Distribution f 0.9 0.8 0.7 0.6 N(38.362, 0.5067) fx(x) 0.5 0.4 0.3 0.2 0.1 0 37 37.5 38 38.5 39 39.5 Temperature ( C) Figure: PDF of normal distribution James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 30/42

Probabilistic Analysis Thermal Conductivity: Normal Distribution GWPE Probability Game Example S = x µ σ P(θ < X ) = φ(s) P(38 C < θ < 39 39 38.36 38.36 C) = φ( ) φ(38 0.5067 0.5067 ) = φ(1.259) φ( 0.7144) = 0.8962 (1 0.7611) = 65.73% James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 31/42

Probabilistic Analysis Thermal Conductivity: Normal Distribution Figure: Probability of temperature between 38 and 39 C is James65.73% Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 32/42

Probabilistic Analysis Thermal Conductivity: Normal Distribution Time at θ = 25 C 35 120% 30 100% quency Freq 25 20 15 10 80% 60% 40% Frequency Cumulative % 5 20% 0 22 23 24 25 26 27 More Time (sec) 0% Figure: After 24 sec 62% chance of reaching 25 C James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 33/42

Probabilistic Analysis Thermal Conductivity: Normal Distribution Time at θ = 25 C Figure: Time elapsed to reach 25 C James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 34/42

Probabilistic Analysis Specific Heat: Normal distribution 0.6 0.5 N(37.43, 0.7207) 0.4 X(x) fx 0.3 0.2 0.1 0 35 35.5 36 36.5 37 37.5 38 38.5 39 39.5 Temperature ( C) Figure: PDF based on specific heat as a random variable James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 35/42

Probabilistic Analysis More GWP Games Figure: 67.29% probability temperature will be body temperature James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 36/42

Probabilistic Analysis More GWPE Games 0.6 n $ 0.3 0 35 35.5 36 36.5 37 37.5 38 38.5 39 39.5 Temperature ("C) Figure: Minimum temperature value with 90% occurrence is 38.90 C; represented by the compliment James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 37/42

Summary Variable temp t(sec) µ σ skewness Temp 60 38.36 0.5076 9.37E-02 Temp 25 24.96 0.09114 t 25 24.06 1.048 Cp 60 37.45 0.7207 2.35E-03 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 38/42

Conclusion Conclusions Results of this study demonstrated: 1 Model tested and validated for the steady-state and transient case. 2 The optimal number of spatial nodes used to discretize the 10cm concrete slab is around 120. 3 Uncertainty in conductivity is narrow over range used. 4 Probability of upper surface reaching comfortable temperature after 60 sec is zero for range of RVs. James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 39/42

Further Objectives Further Objectives Fix program at 1st node Graph the variation in the last node over all time for each RV Alter conductivity (material used) to slow the heating/cooling James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 40/42

References Ang, A. H-S. and W.H. Tang, (2007), Probability Concepts in Engineering, Chapter 5:Computer-Based Numerical and Simulation Methods in Probability Carslaw, H.S., Jaeger, J.C. (1986) Conduction of Heat in Solids. Clarendon Press, Oxford Good, J., V.I. Ugursal, A. Fung (2005) Simulation strategy and sensitivity analysis of an in-floor radiant heating model. Ninth International IBPSA Conference, Montreal, Canada. pp:341-348 Kreyszig, Erwin.(1993) Advanced Engineering Mathematics(7th ed.). John Wiley and Sons, Inc., U.S.A. pp:552,645-646 McQuiston, F.C., Parker, J.D., Spitler, J.D. (2005) Heating, Ventilation and Air Conditioning. John Wiley and Sons, Inc pp:126-127 Swedish Council for Building Research (SCBR). (1983) Calculation Methods to Predict Energy Savings in Residential Buildings. International Energy Agency, Annex III. Stockholm, Sweden pp. 19,89,98 Thomas, Linden C.(1999) Heat Transfer Professional Edition (2nd ed.). Capstone Publishing, Tulsa, OK. pp:26 James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 41/42

End Questions? Typset in L A TEX James Robinson Fourier s Law Applied to the Heat Equation for Conduction of Heat Through a Concrete Slab 42/42