Dynamic ThermalSystems: Heat Transfer and Calculations ...jcardell/Courses/EGR326/... · Dynamic ThermalSystems: Heat Transfer and Calculations ... to its initial temperature •When

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Dynamic Thermal Systems: Heat Transfer and Calculations

EGR 100/290 à EGR 326March 1, 2017

Next HW Modeling Question

• Thermal system• Model the thermal dynamics of a

small solar building that has§ Solar thermal energy input§ Heater heat energy input§ Heat flow out

• What is the internal temperature of the cabin?

Modeling Questions

• What are the dynamic variables?• What are the types of equations we

need?• What are the parameters we need

values for?• What type of system behavior do we

expect?• How do we control this system?

Dynamics in Thermal Systems? • Heat Transfer

• Conduction• Convection • Radiation

• Heat storage• Dynamic variable and equation

• Temperature Difference • The dynamic variable

• Energy balance, Heat balance equation• The interconnecting equation

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Heat Transfer: Conduction• Conduction

§ The ability of a solid to conduct heat• Rate of heat transfer by conduction

across boundaries of a unit volume

§ Where A is the cross-sectional area, L is the distance for heat transfer (~ Δx), and k is the thermal conductivity

§ k ~ 10-2 for insulators; ~ 102 for metals

( )21 TTLkAQhk -÷øö

çèæ=

1D Steady-State Heat Conduction

( )21 TTLkAQhk -÷øö

çèæ=

Heat Transfer: Convection• The process of heat transfer

between a surface of a solid and a fluid exposed to that surface§ Associated with the transfer of mass

• Conduction from the solid to the fluid; then the density of the heated fluid changes so it flows away, bringing in fresh fluid to be heated/cooled, etc.

Heat Transfer: Convection

• Qh/A = h(Ts – T∞) § h is the convection coefficient, representing

the entire heat transfer mechanism§ Ts is the temperature at surface§ T∞ is the ambient temperature of the free fluid

at a distance, and is constant

Qh/A

TsT∞solid

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FOR US:Lumped Parameter Model

– Thermal Resistance– Thermal Capacitance

Thermal Resistance

• Thermal resistance to heat flow rate, Qh between two points of differing temperatures, T1 and T2§ Rh = (T1 – T2) / Qh à Q = ΔT / R

• Conduction§ Qh = (kA / L) (T1 – T2)§ R ≡ L / kA

• Convection§ Qh = hA(T1 – T2)§ R ≡ 1 / hA

Thermal Capacitance

• The ability of matter to store or hold heat is the thermal capacity of the material§ Behaves like a thermal capacitance

• Qh = CpM dT/dt§ Cp = specific heat (on a per mass basis)§ M = mass

Thermal Capacitance• Thermal capacitance Ch is defined as:

§ Ch = MCp• Cp is specific heat capacity

– Equal to the ratio of the heat added to (or removed from) an object to the resulting temperature change, units of J/kg∙K

§ A thermal capacitance stores heat proportional to its initial temperature

• When stored energy is released, temp changes as a function of time, such that§ dT/dt = Qh / CpM, or § Qh = Ch dT/dt

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Heat Transfer

Distance

25

0

20

15

10

5

30

Tem

pera

ture

Inside OutsideWal

lC

ondu

ctio

nConvection Convection

Lumped Parameter Model

Time [seconds]

Time [seconds]

Tem

pera

ture

[K]

Ener

gy [J

]

E = MCDT = 12600 J

12600 J0.8 W

2 W

5.6 W

Message in the Bottle Message in the Bottle

Temperature Difference DT [K]

Heat

Los

s Q

[W

]Qout = DT / Rth

Rth= 5 K/W

Q = DT / 5

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Thermal System: Dynamic Variables• Temperature represents the dynamic

variable• Heat transfer, relates to flow rate• Note that thermal inductance does

not exist• Passive (non-controlled) thermal

systems do not have resonance or overshoot§ Though they can be described by 2nd– and

higher order equations

Dynamic Interconnecting Equation

• Qrate stored within sysem = + Qrate flow in

– Qrate flow out

+ Qrate generated in

+ Qrate work done upon system

• Ch dT/dt = Qhin (t) – Qhout(t)+ Qhgen + dW/dt

§ Recall thermal capacitance: Ch = MCp

Question: Watermelon Warming

• How long will a watermelon remain cool when sitting outside at a picnic?§ Mass = 4 kg§ Ti = 5 ˚C; Tf = 30 ˚C

§ We will find how long it takes the watermelon to reach 63% of the temperature rise, to 20.75 ˚C§ Cp = 4200 J/kg ˚K§ h = 15W / m2 ˚K

Watermelon Warming

• The mass of the melon acts as a thermal capacitor§ Qh = CpM dTi / dt

• Next, equate the heat transfer released by the thermal capacitance to the convection heat transfer§ Qh = CpMTi

’ = hA(Ti – T∞)§ τTi

’ = Ti – T∞• τ = CpM / hA

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Simple Building Models

WarmQin Qout Cooling

Qout

Steady-State Transient

Heat Loss Model

Qout

Tin

Rth

Tout

Qout = (Tin - Tout) / Rth

Large Rth = Small Q LossSmall Rth = Large Q Loss

Energy & Heat in an Electric Circuit

P = V·I = (IR)·I = I2R

I

V

Qin = V·IR

Simple Building

QoutWarm

PowerSupply

+ -Qin

I

DT / Rth

DT = Tin - Tout

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V·I(heater)

DT / Rth(heat flow out)

Tin

Simple Thermal Dynamic Building

(Tout)

Homework

• Model the dynamics of a thermal system that consists of a small building.

• The building receives energy from the sun as well as from a small electric heater.

• There is heat loss through the walls to the outside.

• The house itself can store a certain amount of energy, acting as a thermal capacitor.

To Hand In• Using the heat energy balance equation develop a

dynamic model of the small house system. § Put your model into Simulink using integrator/delay, gain,

summation, etc. blocks, and run it with a Matlab script§ Label/comment your model.§ Include plots of the behavior of the system – from Matlab (Not

Simulink ‘scope’ output)

• Comment upon the behavior of your system.§ Including but not at all limited to: § What order of system is this (1st, 2nd, 3rd …)?§ What is the time constant for this system, and what does the time

constant represent?§ Brief discussion on why the natural response of a thermal system

will not oscillate.