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In-Class Example 2.6 Steady-state and transient heat transfer calculation approaches under varying conditions

Given data:

Data.hr In Class 2.6 Data.xlsx Sheet1 importData.XLSX :

Data.5min In Class 2.6 Data.xlsx Sheet2 importData.XLSX :

Data.hr 9.8 - 6.4 - 4.1 - 3.1 - 3.1 - 3.1 - 61 mat

Data.5min 12.917 - 12.633 - 12.35 - 12.067 - 11.783 - 11.5 - 61 mat

0.1 meter thick concrete wall with surface 1 at the temperature T2 shown below and T3 at 20°C constant. Calculate the temperature at the middle T2 and the heat flux density entering surface 3.

Given constants:

k.cnc 1.2 W m K * / * :

ρ.cnc 2200 kg m 3^/ * :

c.cnc 800 J kg K * / * :

l.cnc 0.1 m * :

T.3 20 °C * :

Defining variables for plot:

T.1.SS Data.hr °C * :

Δt.SS 60 min * :

i.SS 1 Data.hr rows range :

T.1.TS Data.5min °C * :

Δt.TS 5 min * :

i.TS 1 Data.5min rows range :

..."SS" = steady state ..."TS" = transient state

i.SS T.1.SS 273.15 K * - augment i.TS T.1.TS 273.15 K * - augment 2 1 sys

(1) Steady-State (SS) Heat Transfer

We can assume steady-state heat transfer in each hour, for the entire simulation period. Under this approach, we can use hourly average surface temperatures as our boundary condition.

r.1_3 l.cnc k.cnc / : m 2^K * W / 0.0833

r.1_2 r.1_3 2 / : m 2^K * W / 0.0417

r.2_3 r.1_2 :

q.SS T.1.SS T.3 - (r.1_3 / :

... the heat flow entering surface 3

Recall...

T.2.SS T.1.SS q.SS r.1_2 * - : 278.25 K * 279.95 K * 281.1 K * 31 mat

T.2.SS.alternatively T.1.SS r.1_2 / T.3 r.2_3 / + 1 r.1_2 / 1 r.2_3 / + / : 278.25 K * 279.95 K * 281.1 K * 31 mat

i.SS T.2.SS 273.15 K * - augment i.SS q.SS augment 2 1 sys

... the negative values of the heat flow into surface 3 means the actual direction of the heat flow is from node 3 to node 2.

(2) Transient State (TS) Heat Transfer

Under this approach, we will use the 5-minute timestep surface temperatures as our boundary conditions. We will divide the wall into 3 control volumes. We don't know the initial temperature of T2 because not enough information prior to timestep 1 is given. We will need to make an assumption. The temperature of T2 will react to the boundary conditions timestep after timestep, and the effect of the initial guess will diminish over time.

T.2.TS 1 el 0 °C * :

... first T2 assumption

q.2_3.TS 1 el T.2.TS 1 el T.3 - r.2_3 / : W m 2^/ 480 - 11 mat

Setting up layer geometries:

ly 1 el l.cnc 3 / :

C.ly 1 el ρ.cnc c.cnc * ly 1 el * :

ly 2 el ly 1 el :

C.ly 2 el C.ly 1 el :

ly 3 el ly 1 el :

ly 0.0333 m * 0.0333 m * 0.0333 m * 31 mat

C.ly 3 el C.ly 1 el :

C.ly J m 2^K * / 58666.6667 58666.6667 58666.6667 31 mat

... capital C is the heat capacity per unit wall area

Recall ...

Δt.TS min 5

TandQ i.t 1 Data.5min rows 1 - (range T.2.TS i.t 1 + el T.1.TS i.t el T.2.TS i.t el - r.1_2 / (T.3 T.2.TS i.t el - r.2_3 / (Δt.TS * + C.ly 2 el / T.2.TS i.t el + : q.2_3.TS i.t 1 + el T.2.TS i.t 1 + el T.3 - r.2_3 / : 21 line for T.2.TS K / q.2_3.TS W m 2^/ / 21 mat 21 line :

TandQ 1 el #