Green roofs to Mitigate the Urban Heat Island Adewunmi Fareo, Tim - PowerPoint PPT Presentation

Green roofs to Mitigate the Urban Heat Island Adewunmi Fareo, Tim Myers, Neville Fowkes, Graeme Hocking Hermane Mambili, Narenee Mewalal, Sicelo Goqo, Tresia Holtzhausen MISG2020 January 17, 2020 1 / 19

Introduction Urban Heat Island 2 / 19

Introduction 3 / 19

Mathematical Model The use of green roofs to reduce Urban heat Island AIM: How much energy is absorbed over one day? 4 / 19

Model Consider the heat equation ∂ t = k ∂ 2 T ρ c ∂ T ∂ x 2 subject to the following BC’s → T av , as T − x − → ∞ and − k ∂ T ∂ x = (1 − γ ) Q sun + H ( T − T a ) + ǫσ ( T 4 − T 4 a ) − ρ w L e ˙ m Heat flux = heat from sun + convective heat transfer from surface to air + radiative heat transfer to air + evaporative energy 5 / 19

Model Nobody likes T 4 so we linearise − k ∂ T (1 − γ ) Q sun + ( H + 4 ǫσ T 3 = a )( T − T a ) − ρ w L e ˙ m ∂ x (1 − γ ) Q sun − ( H + 4 ǫσ T 3 � � = a ) T a − ρ w L e ˙ m ( H + 4 ǫσ T 3 � � + a ) T 6 / 19

Model Non-dimensionalise T = T − T av x = x t = t ˆ ˆ ˆ ∆ T L τ We choose to work over a time-scale τ = 3600s ∂ 2 T ρ c ∂ T ∂ t = k L 2 ∂ x 2 τ � Hence L = k τ/ρ c ≈ 4cm 7 / 19

Model − k ∆ T ∂ T (1 − γ ) Q sun − ( H + 4 ǫσ T 3 � � = a )( T av − T a ) − ρ w L e ˙ m ∂ x L ( H + 4 ǫσ T 3 � � + a ) ∆ T . T Here we may choose ∆ T as the whole of the first square bracket or (1 − γ ) Q sun . Find ∆ T ≈ 10K. With whole of first bracket − ∂ T = 1 + β T ∂ x where � L ( H + 4 ǫσ T 3 � β = a ) k 8 / 19

Approximate Solution A fundamental solution using Green’s functions is given by: x 2 − � t 4 k ′ ( t − t ′ ) 1 q ( t ′ ) e dt ′ T ( x , t ) = √ √ t − t ′ ρ c π k 0 where q represents the heat flux. Then on the surface we have: � t q ( t ′ ) 1 t − t ′ dt ′ T (0 , t ) = T s ( t ) = √ √ ρ c π k 0 9 / 19

Approximate solution We get : � t 1 + β T s ( t ′ ) dt ′ , T s ( t ) = √ t − t ′ 0 An approximate solution is found by setting β = 0. T s = 2 √ t , putting this expression back in the equation above and integrating for τ s gives √ √ √ T s ( t ) = 2 t − 2 2 β 1 − t , 10 / 19

Approximate solution 11 / 19

Laplace Transform An exact solution may be found to the system using Laplace transforms � x � x � � �� T = 1 √ e β ( β ( t − x )) erfc 2 √ t − β t − erfc 2 √ t β From this we see the exact behaviour of the temperature with space and time The solution apears to be controlled only by β , but the temperature scale depends on values we wish to change (1 − γ ) Q sun − ( H + 4 ǫσ T 3 � � ∆ T = a )( T av − T a ) − ρ w L e ˙ m L / k ( H + 4 ǫσ T 3 � � β = a ) L / k β ≈ 0 . 03 12 / 19

Laplace Transform Can compare with surface solution by first setting β → 0 � x � � t T = 2 e − x 2 / (4 t ) π − x erfc + O ( β ) 2 √ t At x = 0 � t π + β t + O ( β 2 ) T = 2 Leading order ∝ √ t as with previous. Easy to carry on the series 13 / 19

Solution Temperature variation with air at 25C, average temperature 20C after 2 and 10 hours. Also shown is 10 hour curve with evaporation rate of 2mm/12 hours. 14 / 19

So, how can we use this? The energy absorbed/unit area is � ∞ E = ρ c ( T − T av ) dx 0 We may plot this over time, to see the increase during the day or ... Calculate E for different scenarios, changing albedo, adding evaporation etc This will then tell us how much we may change the energy storage in a city under different scenarios. 15 / 19

Solution Effect of changing albedo 16 / 19

Solution Fixed albedo γ = 0 . 15, with evaporation 17 / 19

Conclusion Can find exact solution for temperature in a semi-infinite concrete slab (could also do finite). Green roofs/cool roofs do not have a noticeable effect in the street (if roof above 10m). Actual temperature profile in a city way beyond our skills but ... Exact solution allows us to find differences in absorbed energy and shows effect of albedo and evaporation. Most important terms in (1 − γ ) Q sun − ( H + 4 ǫσ T 3 � � � a )( T av − T a ) − ρ w L e ˙ τ/ρ ck m Green roofs can reduce heat (soil layer will absorb much less heat). Evaporation also helps. High reflective/cool roofs may reflect more heat away. However, green roofs as well as reducing heat absorption also reduce CO2 and provide habitat for birds and insects. 18 / 19

The End 19 / 19