3 heat addition eqs

Ampliació de Motors a Reacció – Curs 2012-2013

1 Oriol Lizandra i Dalmases

HEAT ADDITION (STEADY STATE)

Consider a duct to which heat is added (numbering 3 the inlet section and 4 the outlet

section).

Conservation of momentum:

�� = − ��

Conservation of energy:

�� ℎ�� = � �� + ��

Conservation of mass flow:

�� = ��

For simplicity, we’ll take flow area constant: � = ��

In addition

� = � �

�� = �� + � �� = �� − ��

Momentum:

�� = − �� − ��

� �� = − �� − ��

�� = −� �� − 1��

Energy:

�� = �� − �� + ��

��

��

��

��

!�� !�



�� − � �� = −�� + �� = �� 1�� − 1� + ��

�� "�� − 1# �� = �� 1�� − 1� �� + ��

�� $ 1� − 1 − 1�� − 1% = ��

�� & �� − �� − 1�� − 1�' = ��

�� & �� − 1�� − 1' = ��

→ )*)+ = ,-./.01234 − ,34 − , )5)+

� �� = − �� − 1 = − 1�� − 1 ��

→ / )/)+ = − ,-./.�2 − ,�3434 − , )5)+

1� �� = 12�� = 12 � �� = 12 � �� 2�� − ��

1� �� = 12 1�� 2� �� − ��

1� �� = 12 1�� 2� �� − ��

1� �� = − 12 1�� &2 1�� − 1�� − 1 �� + �� 1��

�� − 1�� − 1 ��'

1� �� = − 12 1�� &2 �� − 1�� − 1 �� + �� − 1� �� − 1�� − 1 ��'

→ ,3 )3)+ = − ,4-./.01* , + 23434 − , )5)+

The momentum equation multiplied per � and added to energy equation yields

�� ℎ�� + �� = �� → �� ℎ�� + � �� = �� ℎ + ��2 � = �� ℎ�� = ��



�� = ��

So adding heat always raises total temperature, and

�� = �� + �� → *89*8. = , + 5-./.01*8.

By definition, total pressure is

�� = � $1 + � − 12 ��% ::;<

1�� = 1� �� + �� − 1 �� − 1��

1 + � − 12 ��

Using the momentum equation

1�� = − 1� �� + ��

1 + � − 12 ��

1�� = − 1 � � �� + ��

1 + � − 12 ��

1�� = 1 � 1��

�� − 1�� − 1 �� − �� − 1�2�� 1 + � − 12 ��

1 + �� − 1 ��

1�� = = 2�� − 1 − ��

1 + � − 12 ��1 + �� − 1 > �� − 1�2��

1�� = =2 + �� − 1�� − 1 − ��

"1 + � − 12 ��# �� − 1� > �� − 1��2��

1�� = − 12�� − 1��

"1 + � − 12 ��#��

1�� = − �� − 1��2��

��

→ ,18)18)+ = − 2344-./.01*8

)5)+



Combining this expression with

,3 )3)+ = − ,4-./.01* , + 23434 − , )5)+ = − , + 23434 − , ", + 2 − ,4 34#4-./.01*8)5)+

We get

23�34 − ,��, + 234� ", + 2 − ,4 34#

)3)+ = ,18)18)+

This in turn can be written as

1�� = = −2��1 + �� + ��

"1 + � − 12 ��#> ��

which can easily be integrated between states 2 and 4, giving

18918. = , + 23.4, + 2394 ?, + 2 − ,4 394, + 2 − ,4 3.4

@22;,

By definition of total pressure it’s easy to find

191. = , + 23.4, + 2394

In a similar way

1� �� = −2 �� − 1��1 + �� → 1� �� = 2 � �� − 2�1 + ��

→ *9*. = �393.�4 �, + 23.4, + 2394�4

From these results one can find the remaining relations, the velocity ratio and the total

temperature ratio

/9/. = -.-9 = *9*. �191.�;, = �393.�4 , + 23.4, + 2394

*89*8. = �393.�4 �, + 23.4, + 2394�4 , + 2 − ,4 394, + 2 − ,4 3.4



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0 0.5 1 1.5 2 2.5 3 3.5

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Heat addition

Tt/Tt*

Pt/pt*

3 heat addition eqs

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