Keywords

1 Introduction

Due to global warming, the average indoor and outdoor temperature has increased. Mechanical air-conditioning systems are used for comfort in India. Energy consumption in conventional domestic housing in India is high, and the equipment is highly energy intensive. Planted roofs are now an established technology for improvement of thermal efficiency or heat reduction process of buildings.

A detailed literature review about the thermal performance of planted roofs, planted roof materials and construction, feasibility of planted roofs in arid regions, the effect of planted roof in energy efficiency of the buildings, etc., has been done. Paper demonstrates the formulation of experimental data-based model and the influence of the individual pie terms based on its indices to optimize the performance of planted roof so as to control the heat flow through planted roof. The mathematical model investigates the effectiveness of green roof in terms of energy efficiency on small building depending on the heat flux. The formulated mathematical equation of the green roof highlights the most effective pie terms so that accordingly the care of that pie term will be taken to obtain best performance of planted roof [1].

2 Working Principal of Planted Roof

Green roof consists of different layers such as

Planted Roof Layer is as follows:

  1. (1)

    Plants (chosen depending on unique qualities).

  2. (2)

    Mixture of soil, minerals and nutrients.

  3. (3)

    Retention medium to hold the water.

  4. (4)

    Drainage layer that sometimes has a built in reservoir.

  5. (5)

    Water proofing membrane with a root barrier.

  6. (6)

    Original roof structural material (Concrete slab)

Figure 1 shows the different layers of the planted roof under the extensive roofing with thin medium layer for plantation of different plants and vegetation [2,3,4].

Fig. 1
figure 1

Different layers of green roofing

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3 Model Formulation

Model formulation of any activities related to field or experimental investigation plays an important role in the analysis. The recorded observations were studied by following the analysis steps of formulation as given below.

  • Recognize various inputs and corresponding outputs related to the system.

  • As it is difficult to handle many inputs in the equation, dimensionless analysis is used to minimize the number of inputs under the group of suitable pie.

  • Once the pie terms are selected, it is necessary to fix the range.

  • The data which is not within the range is rejected.

  • Final stage is to correlate the data by formulating the mathematical model.

The independent variable or parameters involved in the experimental or field data are identified as the Theories of Experimentation. The inputs and output involved in the green roof is identify to effects of these variables.

4 Variables of Planted Roof

The input variables of the green roof and corresponding output are given in Table 1.

Table 1 Planted roof–variable
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5 Dimensionless Pie Terms

5.1 Formation of Pie Terms

All the input terms involved in the experimental set-up are grouped to the individual pie term so that the effect of each pie term, i.e. the group of variable, can be studied effectively. The group of these variables is given in Table 2.

Table 2 Grouped independent pie terms
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5.2 Formulation of Experimental Data-Based Model

All the variables described in Table 1 are grouped as given in Table 2. Further model formulation is done to identify the curve fitting constant and the indices of each pie term. The total nine pies are P1, P2, P3, P4, P5, P6, P7, P8 and P9 with dependent pie term πD. The approximate mathematical model is presented as follow [5],

$$\begin{aligned} \left( {z_{1} } \right) & = k*[\left( {P_{1} } \right)^{{a_{1} }} *\left( {P_{2} } \right)^{{b_{1} }} *\left( {P_{3} } \right)^{{c_{1} }} *\left( {P_{4} } \right)^{{d_{1} }} *\left( {P_{5} } \right)^{{e_{1} }} *\left( {P_{6} } \right)^{{f_{1} }} \\ & \quad *\left( {P_{7} } \right)^{{g_{1} }} *\left( {P_{8} } \right)^{{h_{1} }} *\left( {P_{9} } \right)^{{i_{1} }} ] \\ \end{aligned}$$
(1)

Equation 1 shows the exponential form of model formed between the pie terms. In the above equation, dependent variable z1 represents heat flux through planted roof. Independent variable π1 represents solar radiation data, π2 represents wind velocity data, π3 represents relative humidity, π4 represents ambient temperature, π5 represents roof plant layer data, π6 represents roof soil layer data, π7 represents roof retention layer data, π8 represents roof drainage layer data and π9 represents specification concrete layer data. To determine the values of k, a1, b1, c1, d1, e1, f1, g1, h1 and i1 to arrive at the regression hyper plane, the above equations are presented as follows. Equation 2 is formed by taking the log on both sides of Eq. 1.

$$\begin{aligned} {\text{Log}}\,z_{1} & = \log \, k + \, a_{1} \log \, P_{1} + \, b_{1} \log \, P_{2} + \, c_{1} \log \, P_{3} + \, d_{1} \log \, P_{4} \\ & \quad + \, e_{1} \log \, P_{5} + \, f_{1} \log \, P_{6} + \, g_{1} \log \, P_{7} + \, h_{1} \log \, P_{8} + \, i_{1} \log \, P_{9} \\ \end{aligned}$$
(2)

Let, Z1 = log z1, K = log k1, A = log P1, B = log P2, C = log P3, D = log P4, E = log P5, F = log P6,, G = log P7, H = log P8, and I = log P9.

Substituting the above terms in Eq. 2 to simplify the equation to find out the unknown.

$$Z_{1} = K + a_{1} A + b_{1} B + c_{1} C + d_{1} + e_{1} E + f_{1} F + g_{1} G + h_{1} H + i_{1} I$$
(3)

As in the experimental set-up, the total nine pie terms with curve fitting constant K can be evaluated by formulating the total ten equations. This is done by taking the summation of each pie term as shown in Eq. 4.

$$\begin{aligned} \sum Z_{1} & = nK + a_{1} \sum A + b_{1} \sum B + c_{1} \sum C + d_{1} \sum D + e_{1} \sum E \\ & \quad + f_{1} \sum F + g_{1} \sum G + h_{1} \sum H + i_{1} \sum I \\ \end{aligned}$$
(4)

In the above equations, n represents the number of reading and A, B, C, D, E, F, G, H and I represent the independent pi terms P1, P2, P3, P4, P5, P6, P7, P8 and P9 while Z represents dependent pi term. All the equations are represented in the matrix form as shown below.

$$\left[ Z \right] = \, \left[ W \right]*\left[ X \right]$$

All the equations are arranged in the matrix form as presented above. The matrix W is a 10 × 10 matrix with the multipliers of k, a1, b1, c1, d1, e1, f1, g1, h1 and i1.

Matrix

$$\begin{array}{*{20}c} {Z_{1} } & x & {\left[ {\begin{array}{*{20}c} n \\ A \\ B \\ C \\ D \\ E \\ F \\ G \\ H \\ I \\ \end{array} } \right]} & = & {\left[ {\begin{array}{*{20}c} n & A & B & C & D & E & F & G & H & I \\ A & {A^{2} } & {BA} & {CA} & {DA} & {EA} & {FA} & {GA} & {HA} & {IA} \\ B & {AB} & {B^{2} } & {CB} & {DB} & {EB} & {FB} & {GB} & {HB} & {IB} \\ C & {AC} & {BC} & {C^{2} } & {DC} & {EC} & {FC} & {GC} & {HC} & {IC} \\ D & {AD} & {BD} & {CD} & {D^{2} } & {ED} & {FD} & {GD} & {HD} & {ID} \\ E & {AE} & {BE} & {CE} & {DE} & {E^{2} } & {FE} & {GE} & {HE} & {IE} \\ F & {AF} & {BF} & {CF} & {DF} & {EF} & {F^{2} } & {GF} & {HF} & {IF} \\ G & {AG} & {BG} & {CG} & {DG} & {EG} & {FG} & {G^{2} } & {HG} & {IG} \\ H & {AH} & {BH} & {CH} & {DH} & {EH} & {FH} & {GH} & {H^{2} } & {IH} \\ I & {AI} & {BI} & {CI} & {DI} & {EI} & {FI} & {GI} & {HI} & {I^{2} } \\ \end{array} } \right]} & x & {\left[ {\begin{array}{*{20}c} {k_{{}} } \\ {a_{1} } \\ {b_{1} } \\ {c_{1} } \\ {d_{1} } \\ {e_{1} } \\ {f_{1} } \\ {g_{1} } \\ {h_{1} } \\ {i_{1} } \\ \end{array} } \right]} \\ \end{array}$$

The unknown of the above matrix is evaluated with the help of MATLAB software and substituted in the exponential mathematical model to form the final mathematical model of green roof.

5.3 Proposed Form of Model for Dependent Variables of Planted Roof

The proposed form of models for dependent variables of heat flux of planted roof is as under.

$$\begin{aligned} Q = \left( {Z_{1} } \right) & = 3.25*\left[ {\left( {P_{1} } \right)^{2.5} \left( {P_{2} } \right)^{ - 0.4} \left( {P_{3} } \right)^{0.33} \left( {P_{4} } \right)^{1.7} \left( {P_{5} } \right)^{1.32} \left( {P_{6} } \right)^{ - 2.5 }} \right. \\ & \quad \left. { \left( {P_{7} } \right)^{ - 0.5} \left( {P_{8} } \right)^{0.3} \left( {P_{9} } \right)^{ - 3.7} } \right] \\ \end{aligned}$$

In the above equations, (Z1) is relating to response variable for heat flux of planted roof. As the soil depth increases, it reduces the temperature in the room, at the same time thicker roof would increase the dead weight upon the concrete roof. Around 20–30 cm thickness of soil gives the best performance. Heat transfer is controlled by leaf cover area. With the increase in leaf area index, heat transfer reduces. The experimental investigation shows that the room air temperature of green roofs is always lower than normal roof throughout the experimentation. The average difference can be predicted by varying the experimental data, i.e. pie terms of independent variables. Thus, the model can be utilized as a tool to find the optimum parameter and shows that green roof is one through which energy consumption impacts the residential building.

6 Results and Conclusions

The indices of the mathematical models of green roof show the effect of the individual pie term so that the index with high value or low value will indicate the dominating nature over the output heat flux of the green roof.

Curve fitting constant k and the indices of each pie term show the influence of the causes on the effects. The curve fitting constant k is the collectively combined effect of the entire extra variable which is not included in the experimentation. If the value of k is 1, it means the model is perfect. If it is too low, the causes are overestimated; if it is too high, the causes are underestimated. This would decide when to repeat the investigation again or to refine the approach in subsequent attempts. The magnitude of exponents of the causes indicates the degree of influences of those causes on the specific response. The results indicate that the planted roof can greatly affect the room temperature profile. Planted roofs are potentially good for climates in terms of energy and cost.

The value of the curve fitting constant in this model for (Z1) is 3.25. This collectively represents the combined effect of all extraneous variables such as soil properties, concrete materials and plant types. Further, as it is positive, this indicates that these causes have an increasing influence on heat flux through planted roof.

figure a

Flow chart of performance of planted roof

  1. 1.

    The absolute index of π1 is the highest, viz. 2.5. Thus, the term related to the specification of the solar radiation involved the most influencing π term in this model. The value of this index is positive indicating that the heat flux through planted roof (Z1) is directly proportional to term related to the specification of heat generation by respiration and evaporation rate. Heat flux increases with the increase in heat generation by respiration and evaporation rate.

  2. 2.

    The absolute index of π3 is the lowest, viz. 0.3. Thus, the term related is the effect influencing π term in this model. The value of this index is positive indicating that the heat flux through planted roof (Z1) is directly proportional to the term related to temperature gradient across drainage layer [π8]. The heat flux through planted roof increases as [π8] increases on the effect of predetermining parameters such as temperature, evaporation rate, area of flow of water and rate of flow of water. Suggestions regarding selection of appropriate area of flow of water will reduce the heat flux.

  3. 3.

    The sequence of influence of another independent π term present in this model is π4, π5, π3, π8, π2, π6 and π9 having absolute indices as 1.7, 1.32, 0.33, 0.3, −0.4, −2.5 and −3.7 in the order, respectively.